Subgroups — C₂×C₂₃×#3211 #19208402

Order 5888.

2294 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#529	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₂₃×#124	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂₃×#3211	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₂₃×#495	C₂	1	2	yes	maximal, complemented
5	C₂×C₂₃×#488	C₂	1	2	yes	maximal, complemented
6	C₂×C₂₃×#464	C₂	1	2	yes	maximal, complemented
7	C₂×C₂₃×#463	C₂	1	2	yes	maximal, complemented
8	C₂×C₂₃×#451	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₂₃×#448	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₂₃×#450	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₂₃×#447	C₂	1	1	yes	maximal, characteristic
12	C₂×C₂₃×#431	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂³×Q₈×C₂₃	C₂²	1	1	yes	characteristic
14	C₂³×D₄×C₂₃	C₂²	1	1	yes	characteristic
15	C₂²×Q₁₆×C₂₃	C₂²	1	2	yes	complemented
16	C₂²×SD₁₆×C₂₃	C₂²	1	2	yes	—
17	C₂²×SD₁₆×C₂₃	C₂²	1	2	yes	complemented
18	C₂²×D₈×C₂₃	C₂²	1	2	yes	—
19	C₂₃×#529	—	2	4	no	—
20	C₂₃×#529	C₂²	1	8	yes	complemented
21	C₂×C₂₃×#126	—	2	1	no	—
22	C₂×C₂₃×#125	C₂²	1	1	yes	characteristic, complemented
23	C₂×C₂₃×#124	—	2	1	no	—
24	C₂×C₂₃×#124	—	2	1	no	—
25	C₂×C₂₃×#124	—	2	4	no	—
26	C₂×C₂₃×#124	C₂²	1	2	yes	—
27	C₂×C₂₃×#124	C₂²	1	2	yes	—
28	C₂×C₂₃×#124	C₂²	1	8	yes	complemented
29	C₂×C₂₃×#124	C₂²	1	1	yes	characteristic, complemented
30	C₂×C₂₃×#123	—	2	1	no	—
31	C₂×C₂₃×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
32	C₂×C₂₃×#121	C₂²	1	1	yes	characteristic
33	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic, complemented
34	C₂×C₄×D₄×C₂₃	—	2	1	no	—
35	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic, complemented
36	C₂×C₂₃×#117	—	2	1	no	—
37	C₂²×C₂²⋊C₄×C₂₃	—	2	1	no	—
38	C₂²×C₂²⋊C₄×C₂₃	—	2	1	no	—
39	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
40	C₂₃×#495	C₂²	1	16	yes	complemented
41	C₂₃×#488	C₂²	1	16	yes	complemented
42	C₂₃×#464	C₂²	1	16	yes	complemented
43	C₂₃×#463	C₂²	1	16	yes	complemented
44	C₂₃×#451	C₂²	1	8	yes	complemented
45	C₂₃×#450	C₂²	1	8	yes	complemented
46	C₂₃×#448	C₂²	1	8	yes	complemented
47	C₂₃×#447	C₂²	1	8	yes	—
48	C₂₃×#431	C₂²	1	8	yes	complemented
49	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	complemented
50	C₂×C₂₃×#103	C₂²	1	2	yes	complemented
51	C₂×C₂₃×#103	C₂²	1	2	yes	complemented
52	C₂×C₂₃×#102	C₂²	1	2	yes	—
53	C₂×C₂₃×#102	C₂²	1	2	yes	—
54	C₂×C₂₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
55	C₂×C₂₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
56	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	complemented
57	C₂⁵×C₂₃	—	2	1	no	—
58	C₂²×Q₈×C₂₃	—	2	2	no	—
59	C₂²×Q₈×C₂₃	—	2	1	no	—
60	C₂²×Q₈×C₂₃	—	4	4	no	—
61	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
62	C₂²×Q₈×C₂₃	C₂³	1	2	yes	—
63	C₂²×Q₈×C₂₃	C₂³	1	4	yes	—
64	C₂²×D₄×C₂₃	—	2	1	no	—
65	C₂²×D₄×C₂₃	—	2	2	no	—
66	C₂²×D₄×C₂₃	—	2	1	no	—
67	C₂²×D₄×C₂₃	—	4	4	no	—
68	C₂²×D₄×C₂₃	C₂³	1	1	yes	characteristic
69	C₂²×D₄×C₂₃	C₂³	1	2	yes	—
70	C₂²×D₄×C₂₃	C₂³	1	4	yes	—
71	C₂³×C₄×C₂₃	—	2	1	no	—
72	C₂³×C₄×C₂₃	—	2	1	no	—
73	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
74	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
75	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
76	C₂×Q₁₆×C₂₃	—	4	4	no	—
77	C₂×Q₁₆×C₂₃	C₂³	1	8	yes	complemented
78	C₂×SD₁₆×C₂₃	—	4	4	no	—
79	C₂×SD₁₆×C₂₃	—	4	4	no	—
80	C₂×SD₁₆×C₂₃	C₂³	1	8	yes	—
81	C₂×SD₁₆×C₂₃	C₂³	1	8	yes	complemented
82	C₂×D₈×C₂₃	—	4	4	no	—
83	C₂×D₈×C₂₃	C₂³	1	8	yes	—
84	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
85	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
86	C₂₃×#126	—	2	8	no	—
87	C₂₃×#125	—	2	2	no	—
88	C₂₃×#125	C₂³	1	4	yes	complemented
89	C₂₃×#124	—	2	4	no	—
90	C₂₃×#124	—	2	4	no	—
91	C₂₃×#124	—	2	8	no	—
92	C₂₃×#124	—	2	8	no	—
93	C₂₃×#124	—	2	2	no	—
94	C₂₃×#124	C₂³	1	8	yes	—
95	C₂₃×#124	C₂³	1	8	yes	—
96	C₂₃×#124	C₂³	1	4	yes	complemented
97	C₂₃×#123	—	2	8	no	—
98	C₂₃×C₂²⋊Q₈	—	2	2	no	—
99	C₂₃×C₂²⋊Q₈	C₂³	1	4	yes	—
100	C₂₃×#121	—	2	2	no	—
101	C₂₃×#121	C₂³	1	4	yes	—
102	C₄×Q₈×C₂₃	—	2	2	no	—
103	C₄×Q₈×C₂₃	C₂³	1	4	yes	complemented
104	C₄×D₄×C₂₃	—	2	8	no	—
105	C₄×D₄×C₂₃	—	2	2	no	—
106	C₄×D₄×C₂₃	C₂³	1	4	yes	complemented
107	C₂₃×#117	—	2	8	no	—
108	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
109	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
110	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
111	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
112	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
113	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
114	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
115	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
116	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
117	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
118	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
119	C₂×C₂²⋊C₄×C₂₃	—	2	8	no	—
120	C₂×C₂²⋊C₄×C₂₃	—	2	8	no	—
121	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
122	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
123	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
124	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
125	C₂×C₄²×C₂₃	—	2	1	no	—
126	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
127	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
128	C₂×C₄²×C₂₃	D₄	1	4	yes	—
129	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
130	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
131	C₂×C₄²×C₂₃	C₂³	1	4	yes	—
132	C₂₃×C₄⋊C₈	C₂³	1	8	yes	complemented
133	C₂₃×#103	C₂³	1	8	yes	complemented
134	C₂₃×#103	C₂³	1	8	yes	complemented
135	C₂₃×#102	C₂³	1	8	yes	—
136	C₂₃×#102	C₂³	1	8	yes	—
137	C₂₃×C₂²⋊C₈	C₂³	1	4	yes	—
138	C₂₃×C₂²⋊C₈	C₂³	1	4	yes	—
139	C₄×C₈×C₂₃	C₂³	1	8	yes	complemented
140	C₂⁴×C₂₃	—	2	4	no	—
141	C₂⁴×C₂₃	—	2	1	no	—
142	C₂⁴×C₂₃	—	2	2	no	—
143	C₂⁴×C₂₃	—	4	4	no	—
144	C₂⁴×C₂₃	—	4	4	no	—
145	C₂⁴×C₂₃	—	4	4	no	—
146	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
147	C₂×Q₈×C₂₃	—	2	1	no	—
148	C₂×Q₈×C₂₃	—	2	4	no	—
149	C₂×Q₈×C₂₃	—	2	2	no	—
150	C₂×Q₈×C₂₃	—	2	4	no	—
151	C₂×Q₈×C₂₃	—	2	4	no	—
152	C₂×Q₈×C₂₃	—	2	4	no	—
153	C₂×Q₈×C₂₃	—	4	8	no	—
154	C₂×Q₈×C₂₃	—	4	4	no	—
155	C₂×Q₈×C₂₃	—	4	8	no	—
156	C₂×Q₈×C₂₃	C₂⁴	1	4	yes	—
157	C₂×Q₈×C₂₃	C₂⁴	1	2	yes	—
158	C₂×D₄×C₂₃	—	2	4	no	—
159	C₂×D₄×C₂₃	—	2	1	no	—
160	C₂×D₄×C₂₃	—	2	4	no	—
161	C₂×D₄×C₂₃	—	2	2	no	—
162	C₂×D₄×C₂₃	—	2	4	no	—
163	C₂×D₄×C₂₃	—	2	4	no	—
164	C₂×D₄×C₂₃	—	2	4	no	—
165	C₂×D₄×C₂₃	—	4	8	no	—
166	C₂×D₄×C₂₃	—	4	4	no	—
167	C₂×D₄×C₂₃	—	4	8	no	—
168	C₂×D₄×C₂₃	—	4	4	no	—
169	C₂×D₄×C₂₃	C₂⁴	1	4	yes	—
170	C₂×D₄×C₂₃	C₂⁴	1	2	yes	—
171	C₂²×C₄×C₂₃	—	2	4	no	—
172	C₂²×C₄×C₂₃	—	2	4	no	—
173	C₂²×C₄×C₂₃	—	2	4	no	—
174	C₂²×C₄×C₂₃	—	2	1	no	—
175	C₂²×C₄×C₂₃	—	2	1	no	—
176	C₂²×C₄×C₂₃	—	2	2	no	—
177	C₂²×C₄×C₂₃	—	2	1	no	—
178	C₂²×C₄×C₂₃	—	2	4	no	—
179	C₂²×C₄×C₂₃	—	4	2	no	—
180	C₂²×C₄×C₂₃	—	4	4	no	—
181	C₂²×C₄×C₂₃	—	4	4	no	—
182	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
183	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
184	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
185	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
186	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
187	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
188	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
189	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
190	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
191	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
192	Q₁₆×C₂₃	—	4	8	no	—
193	SD₁₆×C₂₃	—	4	8	no	—
194	SD₁₆×C₂₃	—	4	8	no	—
195	D₈×C₂₃	—	4	8	no	—
196	C₂×C₈×C₂₃	—	2	4	no	—
197	C₂×C₈×C₂₃	—	2	4	no	—
198	C₂×C₈×C₂₃	C₂⁴	1	4	yes	—
199	C₂×C₈×C₂₃	C₂⁴	1	4	yes	—
200	C₄⋊C₄×C₂₃	—	2	4	no	—
201	C₄⋊C₄×C₂₃	—	2	4	no	—
202	C₄⋊C₄×C₂₃	—	2	4	no	—
203	C₄⋊C₄×C₂₃	—	2	4	no	—
204	C₄⋊C₄×C₂₃	—	2	1	no	—
205	C₄⋊C₄×C₂₃	—	2	1	no	—
206	C₄⋊C₄×C₂₃	C₂⁴	1	2	yes	—
207	C₄⋊C₄×C₂₃	C₂⁴	1	2	yes	—
208	C₂²⋊C₄×C₂₃	—	2	4	no	—
209	C₂²⋊C₄×C₂₃	—	2	8	no	—
210	C₂²⋊C₄×C₂₃	—	2	8	no	—
211	C₂²⋊C₄×C₂₃	—	2	8	no	—
212	C₂²⋊C₄×C₂₃	—	2	8	no	—
213	C₂²⋊C₄×C₂₃	—	2	4	no	—
214	C₂²⋊C₄×C₂₃	—	2	4	no	—
215	C₂²⋊C₄×C₂₃	—	2	4	no	—
216	C₂²⋊C₄×C₂₃	—	2	4	no	—
217	C₄²×C₂₃	—	2	4	no	—
218	C₄²×C₂₃	C₂×D₄	1	4	yes	—
219	C₄²×C₂₃	C₂×D₄	1	2	yes	—
220	C₄²×C₂₃	C₂×D₄	1	2	yes	—
221	C₄²×C₂₃	C₂×D₄	1	4	yes	—
222	C₄²×C₂₃	C₂⁴	1	2	yes	—
223	C₄²×C₂₃	C₂⁴	1	2	yes	—
224	C₂×#3211	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
225	C₂³×C₂₃	—	2	2	no	—
226	C₂³×C₂₃	—	2	2	no	—
227	C₂³×C₂₃	—	2	2	no	—
228	C₂³×C₂₃	—	2	4	no	—
229	C₂³×C₂₃	—	4	8	no	—
230	C₂³×C₂₃	—	4	8	no	—
231	C₂³×C₂₃	—	4	4	no	—
232	C₂³×C₂₃	—	4	2	no	—
233	C₂³×C₂₃	—	4	2	no	—
234	C₂³×C₂₃	—	4	8	no	—
235	C₂³×C₂₃	—	4	4	no	—
236	C₂³×C₂₃	—	4	2	no	—
237	C₂³×C₂₃	—	4	4	no	—
238	C₂³×C₂₃	—	8	16	no	—
239	C₂³×C₂₃	#135	1	4	yes	—
240	C₂³×C₂₃	C₂²×D₄	1	2	yes	—
241	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
242	Q₈×C₂₃	—	2	4	no	—
243	Q₈×C₂₃	—	2	4	no	—
244	Q₈×C₂₃	—	2	4	no	—
245	Q₈×C₂₃	—	4	8	no	—
246	Q₈×C₂₃	—	4	2	no	—
247	Q₈×C₂₃	—	4	4	no	—
248	D₄×C₂₃	—	2	4	no	—
249	D₄×C₂₃	—	2	4	no	—
250	D₄×C₂₃	—	2	4	no	—
251	D₄×C₂₃	—	4	8	no	—
252	D₄×C₂₃	—	4	8	no	—
253	D₄×C₂₃	—	4	2	no	—
254	D₄×C₂₃	—	4	4	no	—
255	C₂×C₄×C₂₃	—	2	2	no	—
256	C₂×C₄×C₂₃	—	2	8	no	—
257	C₂×C₄×C₂₃	—	2	2	no	—
258	C₂×C₄×C₂₃	—	2	2	no	—
259	C₂×C₄×C₂₃	—	2	2	no	—
260	C₂×C₄×C₂₃	—	2	2	no	—
261	C₂×C₄×C₂₃	—	2	4	no	—
262	C₂×C₄×C₂₃	—	2	4	no	—
263	C₂×C₄×C₂₃	—	2	2	no	—
264	C₂×C₄×C₂₃	—	2	2	no	—
265	C₂×C₄×C₂₃	—	2	4	no	—
266	C₂×C₄×C₂₃	—	2	2	no	—
267	C₂×C₄×C₂₃	—	4	4	no	—
268	C₂×C₄×C₂₃	—	4	8	no	—
269	C₂×C₄×C₂₃	—	4	8	no	—
270	C₂×C₄×C₂₃	—	4	4	no	—
271	C₂×C₄×C₂₃	—	4	2	no	—
272	C₂×C₄×C₂₃	—	4	2	no	—
273	C₂×C₄×C₂₃	—	4	2	no	—
274	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
275	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
276	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
277	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
278	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
279	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
280	C₂×C₄×C₂₃	H(5,𝔽₂)	1	4	yes	—
281	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
282	C₈×C₂₃	—	2	4	no	—
283	C₈×C₂₃	—	2	4	no	—
284	C₂×#529	C₂×C₂₃	1	1	yes	characteristic, complemented
285	C₂²×#124	C₂×C₂₃	1	1	yes	characteristic, complemented
286	#3211	C₂×C₂₃	1	16	yes	direct product factor
287	C₂×#495	C₂×C₂₃	1	2	yes	complemented
288	C₂×#488	C₂×C₂₃	1	2	yes	complemented
289	C₂×#464	C₂×C₂₃	1	2	yes	complemented
290	C₂×#463	C₂×C₂₃	1	2	yes	complemented
291	C₂×#451	C₂×C₂₃	1	1	yes	characteristic, complemented
292	C₂×#448	C₂×C₂₃	1	1	yes	characteristic, complemented
293	C₂×#450	C₂×C₂₃	1	1	yes	characteristic, complemented
294	C₂×#447	C₂×C₂₃	1	1	yes	characteristic
295	C₂×#431	C₂×C₂₃	1	1	yes	characteristic, complemented
296	C₂²×C₂₃	—	2	2	no	—
297	C₂²×C₂₃	—	2	8	no	—
298	C₂²×C₂₃	—	2	2	no	—
299	C₂²×C₂₃	—	4	4	no	—
300	C₂²×C₂₃	—	4	2	no	—
301	C₂²×C₂₃	—	4	2	no	—
302	C₂²×C₂₃	—	4	4	no	—
303	C₂²×C₂₃	—	4	2	no	—
304	C₂²×C₂₃	—	4	4	no	—
305	C₂²×C₂₃	—	8	16	no	—
306	C₂²×C₂₃	—	8	4	no	—
307	C₂²×C₂₃	—	8	8	no	—
308	C₂²×C₂₃	#535	1	2	yes	—
309	C₂²×C₂₃	C₂×#135	1	4	yes	—
310	C₂²×C₂₃	C₂×#135	1	2	yes	—
311	C₂²×C₂₃	#575	1	2	yes	—
312	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
313	C₄×C₂₃	—	2	2	no	—
314	C₄×C₂₃	—	2	2	no	—
315	C₄×C₂₃	—	2	2	no	—
316	C₄×C₂₃	—	2	2	no	—
317	C₄×C₂₃	—	2	2	no	—
318	C₄×C₂₃	—	4	4	no	—
319	C₄×C₂₃	—	4	2	no	—
320	C₄×C₂₃	—	4	2	no	—
321	C₄×C₂₃	—	4	2	no	—
322	C₄×C₂₃	#535	1	2	yes	—
323	C₄×C₂₃	C₂×H(5,𝔽₂)	1	2	yes	—
324	C₂³×Q₈	C₂²×C₂₃	1	1	yes	characteristic
325	C₂³×D₄	C₂²×C₂₃	1	1	yes	characteristic
326	C₂²×Q₁₆	C₂²×C₂₃	1	2	yes	complemented
327	C₂²×SD₁₆	C₂²×C₂₃	1	2	yes	—
328	C₂²×SD₁₆	C₂²×C₂₃	1	2	yes	complemented
329	C₂²×D₈	C₂²×C₂₃	1	2	yes	—
330	#529	—	2	4	no	—
331	#529	C₂²×C₂₃	1	8	yes	complemented
332	C₂×#126	—	2	1	no	—
333	C₂×#125	C₂²×C₂₃	1	1	yes	characteristic, complemented
334	C₂×#124	—	2	1	no	—
335	C₂×#124	—	2	1	no	—
336	C₂×#124	—	2	4	no	—
337	C₂×#124	C₂²×C₂₃	1	2	yes	—
338	C₂×#124	C₂²×C₂₃	1	2	yes	—
339	C₂×#124	C₂²×C₂₃	1	8	yes	complemented
340	C₂×#124	C₂²×C₂₃	1	1	yes	characteristic, complemented
341	C₂×#123	—	2	1	no	—
342	C₂×C₂²⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
343	C₂×#121	C₂²×C₂₃	1	1	yes	characteristic
344	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
345	C₂×C₄×D₄	—	2	1	no	—
346	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic, complemented
347	C₂×#117	—	2	1	no	—
348	C₂²×C₂²⋊C₄	—	2	1	no	—
349	C₂²×C₂²⋊C₄	—	2	1	no	—
350	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
351	#495	C₂²×C₂₃	1	16	yes	complemented
352	#488	C₂²×C₂₃	1	16	yes	complemented
353	#464	C₂²×C₂₃	1	16	yes	complemented
354	#463	C₂²×C₂₃	1	16	yes	complemented
355	#451	C₂²×C₂₃	1	8	yes	complemented
356	#450	C₂²×C₂₃	1	8	yes	complemented
357	#448	C₂²×C₂₃	1	8	yes	complemented
358	#447	C₂²×C₂₃	1	8	yes	—
359	#431	C₂²×C₂₃	1	8	yes	complemented
360	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	complemented
361	C₂×#103	C₂²×C₂₃	1	2	yes	complemented
362	C₂×#103	C₂²×C₂₃	1	2	yes	complemented
363	C₂×#102	C₂²×C₂₃	1	2	yes	—
364	C₂×#102	C₂²×C₂₃	1	2	yes	—
365	C₂×C₂²⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
366	C₂×C₂²⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
367	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	complemented
368	C₂×C₂₃	—	2	4	no	—
369	C₂×C₂₃	—	8	2	no	—
370	C₂×C₂₃	—	8	4	no	—
371	C₂×C₂₃	#3211	1	4	yes	direct product factor
372	C₂×C₂₃	C₂×#535	1	1	yes	characteristic
373	C₂×C₂₃	C₂²×#135	1	1	yes	characteristic
374	C₂×C₂₃	C₂×#575	1	1	yes	characteristic
375	C₂⁵	—	2	1	no	—
376	C₂²×Q₈	—	2	1	no	—
377	C₂²×Q₈	—	2	2	no	—
378	C₂²×Q₈	—	4	4	no	—
379	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
380	C₂²×Q₈	C₂³×C₂₃	1	2	yes	—
381	C₂²×Q₈	C₂³×C₂₃	1	4	yes	—
382	C₂²×D₄	—	2	1	no	—
383	C₂²×D₄	—	2	1	no	—
384	C₂²×D₄	—	2	2	no	—
385	C₂²×D₄	—	4	4	no	—
386	C₂²×D₄	C₂³×C₂₃	1	1	yes	characteristic
387	C₂²×D₄	C₂³×C₂₃	1	2	yes	—
388	C₂²×D₄	C₂³×C₂₃	1	4	yes	—
389	C₂³×C₄	—	2	1	no	—
390	C₂³×C₄	—	2	1	no	—
391	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
392	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
393	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
394	C₂×Q₁₆	—	4	4	no	—
395	C₂×Q₁₆	C₂³×C₂₃	1	8	yes	complemented
396	C₂×SD₁₆	—	4	4	no	—
397	C₂×SD₁₆	—	4	4	no	—
398	C₂×SD₁₆	C₂³×C₂₃	1	8	yes	—
399	C₂×SD₁₆	C₂³×C₂₃	1	8	yes	complemented
400	C₂×D₈	—	4	4	no	—
401	C₂×D₈	C₂³×C₂₃	1	8	yes	—
402	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
403	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
404	#126	—	2	8	no	—
405	#125	—	2	2	no	—
406	#125	C₂³×C₂₃	1	4	yes	complemented
407	#124	—	2	4	no	—
408	#124	—	2	4	no	—
409	#124	—	2	8	no	—
410	#124	—	2	8	no	—
411	#124	—	2	2	no	—
412	#124	C₂³×C₂₃	1	8	yes	—
413	#124	C₂³×C₂₃	1	8	yes	—
414	#124	C₂³×C₂₃	1	4	yes	complemented
415	#123	—	2	8	no	—
416	C₂²⋊Q₈	—	2	2	no	—
417	C₂²⋊Q₈	C₂³×C₂₃	1	4	yes	—
418	#121	—	2	2	no	—
419	#121	C₂³×C₂₃	1	4	yes	—
420	C₄×Q₈	—	2	2	no	—
421	C₄×Q₈	C₂³×C₂₃	1	4	yes	complemented
422	C₄×D₄	—	2	8	no	—
423	C₄×D₄	—	2	2	no	—
424	C₄×D₄	C₂³×C₂₃	1	4	yes	complemented
425	#117	—	2	8	no	—
426	C₂×C₄⋊C₄	—	2	1	no	—
427	C₂×C₄⋊C₄	—	2	1	no	—
428	C₂×C₄⋊C₄	—	2	1	no	—
429	C₂×C₄⋊C₄	—	2	1	no	—
430	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
431	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
432	C₂×C₂²⋊C₄	—	2	1	no	—
433	C₂×C₂²⋊C₄	—	2	2	no	—
434	C₂×C₂²⋊C₄	—	2	2	no	—
435	C₂×C₂²⋊C₄	—	2	2	no	—
436	C₂×C₂²⋊C₄	—	2	2	no	—
437	C₂×C₂²⋊C₄	—	2	8	no	—
438	C₂×C₂²⋊C₄	—	2	8	no	—
439	C₂×C₂²⋊C₄	—	2	1	no	—
440	C₂×C₂²⋊C₄	—	2	1	no	—
441	C₂×C₂²⋊C₄	—	2	1	no	—
442	C₂×C₂²⋊C₄	—	2	1	no	—
443	C₂×C₄²	—	2	1	no	—
444	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
445	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
446	C₂×C₄²	D₄×C₂₃	1	4	yes	—
447	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
448	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
449	C₂×C₄²	C₂³×C₂₃	1	4	yes	—
450	C₄⋊C₈	C₂³×C₂₃	1	8	yes	complemented
451	#103	C₂³×C₂₃	1	8	yes	complemented
452	#103	C₂³×C₂₃	1	8	yes	complemented
453	#102	C₂³×C₂₃	1	8	yes	—
454	#102	C₂³×C₂₃	1	8	yes	—
455	C₂²⋊C₈	C₂³×C₂₃	1	4	yes	—
456	C₂²⋊C₈	C₂³×C₂₃	1	4	yes	—
457	C₄×C₈	C₂³×C₂₃	1	8	yes	complemented
458	C₂₃	C₂×#3211	1	1	yes	characteristic, direct product factor, Sylow
459	C₂⁴	—	2	1	no	—
460	C₂⁴	—	2	4	no	—
461	C₂⁴	—	2	2	no	—
462	C₂⁴	—	4	4	no	—
463	C₂⁴	—	4	4	no	—
464	C₂⁴	—	4	4	no	—
465	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
466	C₂×Q₈	—	2	1	no	—
467	C₂×Q₈	—	2	4	no	—
468	C₂×Q₈	—	2	2	no	—
469	C₂×Q₈	—	2	4	no	—
470	C₂×Q₈	—	2	4	no	—
471	C₂×Q₈	—	2	4	no	—
472	C₂×Q₈	—	4	8	no	—
473	C₂×Q₈	—	4	4	no	—
474	C₂×Q₈	—	4	8	no	—
475	C₂×Q₈	C₂⁴×C₂₃	1	4	yes	—
476	C₂×Q₈	C₂⁴×C₂₃	1	2	yes	—
477	C₂×D₄	—	2	4	no	—
478	C₂×D₄	—	2	1	no	—
479	C₂×D₄	—	2	4	no	—
480	C₂×D₄	—	2	2	no	—
481	C₂×D₄	—	2	4	no	—
482	C₂×D₄	—	2	4	no	—
483	C₂×D₄	—	2	4	no	—
484	C₂×D₄	—	4	8	no	—
485	C₂×D₄	—	4	4	no	—
486	C₂×D₄	—	4	8	no	—
487	C₂×D₄	—	4	4	no	—
488	C₂×D₄	C₂⁴×C₂₃	1	4	yes	—
489	C₂×D₄	C₂⁴×C₂₃	1	2	yes	—
490	C₂²×C₄	—	2	4	no	—
491	C₂²×C₄	—	2	4	no	—
492	C₂²×C₄	—	2	1	no	—
493	C₂²×C₄	—	2	1	no	—
494	C₂²×C₄	—	2	4	no	—
495	C₂²×C₄	—	2	2	no	—
496	C₂²×C₄	—	2	1	no	—
497	C₂²×C₄	—	2	4	no	—
498	C₂²×C₄	—	4	2	no	—
499	C₂²×C₄	—	4	4	no	—
500	C₂²×C₄	—	4	4	no	—
501	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
502	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
503	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
504	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
505	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
506	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
507	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
508	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
509	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
510	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
511	Q₁₆	—	4	8	no	—
512	SD₁₆	—	4	8	no	—
513	SD₁₆	—	4	8	no	—
514	D₈	—	4	8	no	—
515	C₂×C₈	—	2	4	no	—
516	C₂×C₈	—	2	4	no	—
517	C₂×C₈	C₂⁴×C₂₃	1	4	yes	—
518	C₂×C₈	C₂⁴×C₂₃	1	4	yes	—
519	C₄⋊C₄	—	2	4	no	—
520	C₄⋊C₄	—	2	4	no	—
521	C₄⋊C₄	—	2	4	no	—
522	C₄⋊C₄	—	2	4	no	—
523	C₄⋊C₄	—	2	1	no	—
524	C₄⋊C₄	—	2	1	no	—
525	C₄⋊C₄	C₂⁴×C₂₃	1	2	yes	—
526	C₄⋊C₄	C₂⁴×C₂₃	1	2	yes	—
527	C₂²⋊C₄	—	2	4	no	—
528	C₂²⋊C₄	—	2	8	no	—
529	C₂²⋊C₄	—	2	8	no	—
530	C₂²⋊C₄	—	2	8	no	—
531	C₂²⋊C₄	—	2	8	no	—
532	C₂²⋊C₄	—	2	4	no	—
533	C₂²⋊C₄	—	2	4	no	—
534	C₂²⋊C₄	—	2	4	no	—
535	C₂²⋊C₄	—	2	4	no	—
536	C₄²	—	2	4	no	—
537	C₄²	C₂×D₄×C₂₃	1	4	yes	—
538	C₄²	C₂×D₄×C₂₃	1	2	yes	—
539	C₄²	C₂×D₄×C₂₃	1	2	yes	—
540	C₄²	C₂×D₄×C₂₃	1	4	yes	—
541	C₄²	C₂⁴×C₂₃	1	2	yes	—
542	C₄²	C₂⁴×C₂₃	1	2	yes	—
543	C₂³	—	2	2	no	—
544	C₂³	—	2	2	no	—
545	C₂³	—	2	2	no	—
546	C₂³	—	2	4	no	—
547	C₂³	—	4	8	no	—
548	C₂³	—	4	8	no	—
549	C₂³	—	4	2	no	—
550	C₂³	—	4	8	no	—
551	C₂³	—	4	4	no	—
552	C₂³	—	4	2	no	—
553	C₂³	—	4	4	no	—
554	C₂³	—	4	4	no	—
555	C₂³	—	4	2	no	—
556	C₂³	—	8	16	no	—
557	C₂³	C₂₃×#135	1	4	yes	—
558	C₂³	C₂²×D₄×C₂₃	1	2	yes	—
559	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
560	Q₈	—	2	4	no	—
561	Q₈	—	2	4	no	—
562	Q₈	—	2	4	no	—
563	Q₈	—	4	8	no	—
564	Q₈	—	4	4	no	—
565	Q₈	—	4	2	no	—
566	D₄	—	2	4	no	—
567	D₄	—	2	4	no	—
568	D₄	—	2	4	no	—
569	D₄	—	4	8	no	—
570	D₄	—	4	8	no	—
571	D₄	—	4	4	no	—
572	D₄	—	4	2	no	—
573	C₂×C₄	—	2	8	no	—
574	C₂×C₄	—	2	2	no	—
575	C₂×C₄	—	2	2	no	—
576	C₂×C₄	—	2	2	no	—
577	C₂×C₄	—	2	2	no	—
578	C₂×C₄	—	2	2	no	—
579	C₂×C₄	—	2	4	no	—
580	C₂×C₄	—	2	4	no	—
581	C₂×C₄	—	2	2	no	—
582	C₂×C₄	—	2	2	no	—
583	C₂×C₄	—	2	4	no	—
584	C₂×C₄	—	2	2	no	—
585	C₂×C₄	—	4	4	no	—
586	C₂×C₄	—	4	8	no	—
587	C₂×C₄	—	4	8	no	—
588	C₂×C₄	—	4	2	no	—
589	C₂×C₄	—	4	2	no	—
590	C₂×C₄	—	4	4	no	—
591	C₂×C₄	—	4	2	no	—
592	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
593	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
594	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
595	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
596	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
597	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
598	C₂×C₄	C₂₃×H(5,𝔽₂)	1	4	yes	—
599	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
600	C₈	—	2	4	no	—
601	C₈	—	2	4	no	—
602	C₂²	—	2	2	no	—
603	C₂²	—	2	8	no	—
604	C₂²	—	2	2	no	—
605	C₂²	—	4	4	no	—
606	C₂²	—	4	2	no	—
607	C₂²	—	4	2	no	—
608	C₂²	—	4	4	no	—
609	C₂²	—	4	2	no	—
610	C₂²	—	4	4	no	—
611	C₂²	—	8	16	no	—
612	C₂²	—	8	8	no	—
613	C₂²	—	8	4	no	—
614	C₂²	C₂₃×#535	1	2	yes	—
615	C₂²	C₂×C₂₃×#135	1	4	yes	—
616	C₂²	C₂×C₂₃×#135	1	2	yes	—
617	C₂²	C₂₃×#575	1	2	yes	—
618	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
619	C₄	—	2	2	no	—
620	C₄	—	2	2	no	—
621	C₄	—	2	2	no	—
622	C₄	—	2	2	no	—
623	C₄	—	2	2	no	—
624	C₄	—	4	4	no	—
625	C₄	—	4	2	no	—
626	C₄	—	4	2	no	—
627	C₄	—	4	2	no	—
628	C₄	C₂₃×#535	1	2	yes	—
629	C₄	C₂×C₂₃×H(5,𝔽₂)	1	2	yes	—
630	C₂	—	2	4	no	—
631	C₂	—	8	4	no	—
632	C₂	—	8	2	no	—
633	C₂	C₂₃×#3211	1	4	yes	direct product factor
634	C₂	C₂×C₂₃×#535	1	1	yes	characteristic
635	C₂	C₂²×C₂₃×#135	1	1	yes	characteristic
636	C₂	C₂×C₂₃×#575	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₂₃
prime radical	C₂×C₂₃×#447	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂²×C₄²×C₂₃ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂×C₂₃×#529	C₂	semidirect
2. C₂²×C₂₃×#124	C₂	semidirect
3. C₂₃×#3211	C₂	direct (N × H)
4. C₂×C₂₃×#495	C₂	semidirect
5. C₂×C₂₃×#488	C₂	semidirect
6. C₂×C₂₃×#464	C₂	semidirect
7. C₂×C₂₃×#463	C₂	semidirect
8. C₂×C₂₃×#451	C₂	semidirect
9. C₂×C₂₃×#448	C₂	semidirect
10. C₂×C₂₃×#450	C₂	semidirect
12. C₂×C₂₃×#431	C₂	semidirect
15. C₂²×Q₁₆×C₂₃	C₂²	semidirect
17. C₂²×SD₁₆×C₂₃	C₂²	semidirect
20. C₂₃×#529	C₂²	semidirect
22. C₂×C₂₃×#125	C₂²	semidirect
28. C₂×C₂₃×#124	C₂²	semidirect
29. C₂×C₂₃×#124	C₂²	semidirect
33. C₂×C₄×Q₈×C₂₃	C₂²	semidirect
35. C₂×C₄×D₄×C₂₃	C₂²	semidirect
40. C₂₃×#495	C₂²	semidirect
41. C₂₃×#488	C₂²	semidirect
42. C₂₃×#464	C₂²	semidirect
43. C₂₃×#463	C₂²	semidirect
44. C₂₃×#451	C₂²	semidirect
45. C₂₃×#450	C₂²	semidirect
46. C₂₃×#448	C₂²	semidirect
48. C₂₃×#431	C₂²	semidirect
49. C₂×C₂₃×C₄⋊C₈	C₂²	semidirect
50. C₂×C₂₃×#103	C₂²	semidirect
51. C₂×C₂₃×#103	C₂²	semidirect
56. C₂×C₄×C₈×C₂₃	C₂²	semidirect
77. C₂×Q₁₆×C₂₃	C₂³	semidirect
81. C₂×SD₁₆×C₂₃	C₂³	semidirect
88. C₂₃×#125	C₂³	semidirect
96. C₂₃×#124	C₂³	semidirect
103. C₄×Q₈×C₂₃	C₂³	semidirect
106. C₄×D₄×C₂₃	C₂³	semidirect
132. C₂₃×C₄⋊C₈	C₂³	semidirect
133. C₂₃×#103	C₂³	semidirect
134. C₂₃×#103	C₂³	semidirect
139. C₄×C₈×C₂₃	C₂³	semidirect
224. C₂×#3211	C₂₃	direct (N × H)
284. C₂×#529	C₂×C₂₃	semidirect
285. C₂²×#124	C₂×C₂₃	semidirect
286. #3211	C₂×C₂₃	direct (N × H)
287. C₂×#495	C₂×C₂₃	semidirect
288. C₂×#488	C₂×C₂₃	semidirect
289. C₂×#464	C₂×C₂₃	semidirect
290. C₂×#463	C₂×C₂₃	semidirect
291. C₂×#451	C₂×C₂₃	semidirect
292. C₂×#448	C₂×C₂₃	semidirect
293. C₂×#450	C₂×C₂₃	semidirect
295. C₂×#431	C₂×C₂₃	semidirect
326. C₂²×Q₁₆	C₂²×C₂₃	semidirect
328. C₂²×SD₁₆	C₂²×C₂₃	semidirect
331. #529	C₂²×C₂₃	semidirect
333. C₂×#125	C₂²×C₂₃	semidirect
339. C₂×#124	C₂²×C₂₃	semidirect
340. C₂×#124	C₂²×C₂₃	semidirect
344. C₂×C₄×Q₈	C₂²×C₂₃	semidirect
346. C₂×C₄×D₄	C₂²×C₂₃	semidirect
351. #495	C₂²×C₂₃	semidirect
352. #488	C₂²×C₂₃	semidirect
353. #464	C₂²×C₂₃	semidirect
354. #463	C₂²×C₂₃	semidirect
355. #451	C₂²×C₂₃	semidirect
356. #450	C₂²×C₂₃	semidirect
357. #448	C₂²×C₂₃	semidirect
359. #431	C₂²×C₂₃	semidirect
360. C₂×C₄⋊C₈	C₂²×C₂₃	semidirect
361. C₂×#103	C₂²×C₂₃	semidirect
362. C₂×#103	C₂²×C₂₃	semidirect
367. C₂×C₄×C₈	C₂²×C₂₃	semidirect
371. C₂×C₂₃	#3211	direct (N × H)
395. C₂×Q₁₆	C₂³×C₂₃	semidirect
399. C₂×SD₁₆	C₂³×C₂₃	semidirect
406. #125	C₂³×C₂₃	semidirect
414. #124	C₂³×C₂₃	semidirect
421. C₄×Q₈	C₂³×C₂₃	semidirect
424. C₄×D₄	C₂³×C₂₃	semidirect
450. C₄⋊C₈	C₂³×C₂₃	semidirect
451. #103	C₂³×C₂₃	semidirect
452. #103	C₂³×C₂₃	semidirect
457. C₄×C₈	C₂³×C₂₃	semidirect
458. C₂₃	C₂×#3211	direct (N × H)
633. C₂	C₂₃×#3211	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₂₃×#529	2
C₂²×C₂₃×#124	2
C₂₃×#3211	2
C₂×C₂₃×#495	2
C₂×C₂₃×#488	2
C₂×C₂₃×#464	2
C₂×C₂₃×#463	2
C₂×C₂₃×#451	2
C₂×C₂₃×#448	2
C₂×C₂₃×#450	2
C₂×C₂₃×#447	2
C₂×C₂₃×#431	2
C₂×#3211	23

Subgroups — C2×C23×#3211 #19208402

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#3211 #19208402