Subgroups — C₂×C₂₃×#3215 #19208410

Order 5888.

2242 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂₃×#533	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₂₃×#124	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#3215	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₂₃×#497	C₂	1	2	yes	maximal, complemented
5	C₂×C₂₃×#489	C₂	1	2	yes	maximal, complemented
6	C₂×C₂₃×#464	C₂	1	4	yes	maximal, complemented
7	C₂×C₂₃×#451	C₂	1	2	yes	maximal, complemented
8	C₂×C₂₃×#450	C₂	1	2	yes	maximal
9	C₂×C₂₃×#432	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂³×Q₈×C₂₃	C₂²	1	1	yes	characteristic
11	C₂³×D₄×C₂₃	C₂²	1	1	yes	characteristic
12	C₂²×Q₁₆×C₂₃	C₂²	1	4	yes	complemented
13	C₂²×SD₁₆×C₂₃	C₂²	1	4	yes	—
14	C₂₃×#533	—	2	4	no	—
15	C₂₃×#533	C₂²	1	8	yes	complemented
16	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
17	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
18	C₂×C₂₃×#125	—	2	1	no	—
19	C₂×C₂₃×#124	—	2	1	no	—
20	C₂×C₂₃×#124	—	2	1	no	—
21	C₂×C₂₃×#124	—	2	4	no	—
22	C₂×C₂₃×#124	C₂²	1	8	yes	—
23	C₂×C₂₃×#124	C₂²	1	2	yes	—
24	C₂×C₂₃×#124	C₂²	1	2	yes	—
25	C₂×C₂₃×C₂²⋊Q₈	—	2	1	no	—
26	C₂×C₂₃×C₂²⋊Q₈	C₂²	1	2	yes	—
27	C₂×C₄×Q₈×C₂₃	—	2	1	no	—
28	C₂×C₄×Q₈×C₂₃	C₂²	1	2	yes	complemented
29	C₂×C₂₃×#117	—	2	1	no	—
30	C₂²×C₂²⋊C₄×C₂₃	—	2	1	no	—
31	C₂²×C₂²⋊C₄×C₂₃	—	2	1	no	—
32	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
33	C₂₃×#497	C₂²	1	16	yes	complemented
34	C₂₃×#489	C₂²	1	16	yes	complemented
35	C₂₃×#464	C₂²	1	32	yes	complemented
36	C₂₃×#451	C₂²	1	16	yes	complemented
37	C₂₃×#450	C₂²	1	16	yes	—
38	C₂₃×#432	C₂²	1	8	yes	complemented
39	C₂×C₂₃×C₄⋊C₈	C₂²	1	2	yes	complemented
40	C₂×C₂₃×#103	C₂²	1	4	yes	complemented
41	C₂×C₂₃×#102	C₂²	1	4	yes	—
42	C₂×C₂₃×C₂²⋊C₈	C₂²	1	2	yes	—
43	C₂×C₂₃×#97	C₂²	1	2	yes	complemented
44	C₂⁵×C₂₃	—	2	1	no	—
45	C₂²×Q₈×C₂₃	—	2	2	no	—
46	C₂²×Q₈×C₂₃	—	2	1	no	—
47	C₂²×Q₈×C₂₃	—	2	1	no	—
48	C₂²×Q₈×C₂₃	—	4	4	no	—
49	C₂²×Q₈×C₂₃	C₂³	1	2	yes	—
50	C₂²×Q₈×C₂₃	C₂³	1	2	yes	—
51	C₂²×Q₈×C₂₃	C₂³	1	4	yes	—
52	C₂²×D₄×C₂₃	—	2	2	no	—
53	C₂²×D₄×C₂₃	—	2	1	no	—
54	C₂²×D₄×C₂₃	—	4	4	no	—
55	C₂²×D₄×C₂₃	C₂³	1	2	yes	—
56	C₂²×D₄×C₂₃	C₂³	1	4	yes	—
57	C₂³×C₄×C₂₃	—	2	1	no	—
58	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
59	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
60	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
61	C₂×Q₁₆×C₂₃	—	4	8	no	—
62	C₂×Q₁₆×C₂₃	C₂³	1	16	yes	complemented
63	C₂×SD₁₆×C₂₃	—	4	8	no	—
64	C₂×SD₁₆×C₂₃	C₂³	1	16	yes	—
65	C₂²×C₈×C₂₃	C₂³	1	4	yes	—
66	C₂₃×C₄⋊Q₈	—	2	2	no	—
67	C₂₃×C₄⋊Q₈	—	2	2	no	—
68	C₂₃×C₄⋊Q₈	C₂³	1	4	yes	complemented
69	C₂₃×C₄⋊Q₈	C₂³	1	4	yes	complemented
70	C₂₃×#125	—	2	8	no	—
71	C₂₃×#124	—	2	8	no	—
72	C₂₃×#124	—	2	4	no	—
73	C₂₃×#124	—	2	8	no	—
74	C₂₃×#124	—	2	4	no	—
75	C₂₃×#124	C₂³	1	8	yes	—
76	C₂₃×#124	C₂³	1	8	yes	—
77	C₂₃×C₂²⋊Q₈	—	2	8	no	—
78	C₂₃×C₂²⋊Q₈	—	2	4	no	—
79	C₂₃×C₂²⋊Q₈	C₂³	1	8	yes	—
80	C₄×Q₈×C₂₃	—	2	8	no	—
81	C₄×Q₈×C₂₃	—	2	4	no	—
82	C₄×Q₈×C₂₃	C₂³	1	8	yes	complemented
83	C₂₃×#117	—	2	8	no	—
84	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
85	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
86	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
87	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
88	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
89	C₂×C₄⋊C₄×C₂₃	C₂³	1	2	yes	—
90	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
91	C₂×C₂²⋊C₄×C₂₃	—	2	8	no	—
92	C₂×C₂²⋊C₄×C₂₃	—	2	8	no	—
93	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
94	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
95	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
96	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
97	C₂×C₄²×C₂₃	—	2	2	no	—
98	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
99	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
100	C₂×C₄²×C₂₃	D₄	1	4	yes	—
101	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
102	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
103	C₂×C₄²×C₂₃	C₂³	1	4	yes	—
104	C₂₃×C₄⋊C₈	C₂³	1	8	yes	complemented
105	C₂₃×#103	C₂³	1	16	yes	complemented
106	C₂₃×#102	C₂³	1	16	yes	—
107	C₂₃×C₂²⋊C₈	C₂³	1	8	yes	—
108	C₂₃×#97	C₂³	1	8	yes	complemented
109	C₂⁴×C₂₃	—	2	4	no	—
110	C₂⁴×C₂₃	—	2	2	no	—
111	C₂⁴×C₂₃	—	4	4	no	—
112	C₂⁴×C₂₃	—	4	4	no	—
113	C₂⁴×C₂₃	—	4	4	no	—
114	C₂⁴×C₂₃	C₂×D₄	1	1	yes	characteristic
115	C₂×Q₈×C₂₃	—	2	2	no	—
116	C₂×Q₈×C₂₃	—	2	4	no	—
117	C₂×Q₈×C₂₃	—	2	4	no	—
118	C₂×Q₈×C₂₃	—	2	2	no	—
119	C₂×Q₈×C₂₃	—	2	8	no	—
120	C₂×Q₈×C₂₃	—	2	8	no	—
121	C₂×Q₈×C₂₃	—	4	8	no	—
122	C₂×Q₈×C₂₃	—	4	4	no	—
123	C₂×Q₈×C₂₃	—	4	4	no	—
124	C₂×Q₈×C₂₃	—	4	8	no	—
125	C₂×Q₈×C₂₃	C₂⁴	1	4	yes	—
126	C₂×Q₈×C₂₃	C₂⁴	1	4	yes	—
127	C₂×D₄×C₂₃	—	2	4	no	—
128	C₂×D₄×C₂₃	—	2	2	no	—
129	C₂×D₄×C₂₃	—	2	8	no	—
130	C₂×D₄×C₂₃	—	4	8	no	—
131	C₂×D₄×C₂₃	—	4	4	no	—
132	C₂×D₄×C₂₃	—	4	8	no	—
133	C₂×D₄×C₂₃	C₂⁴	1	4	yes	—
134	C₂²×C₄×C₂₃	—	2	4	no	—
135	C₂²×C₄×C₂₃	—	2	4	no	—
136	C₂²×C₄×C₂₃	—	2	4	no	—
137	C₂²×C₄×C₂₃	—	2	2	no	—
138	C₂²×C₄×C₂₃	—	2	2	no	—
139	C₂²×C₄×C₂₃	—	2	2	no	—
140	C₂²×C₄×C₂₃	—	4	2	no	—
141	C₂²×C₄×C₂₃	—	4	4	no	—
142	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
143	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
144	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
145	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
146	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
147	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
148	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
149	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
150	C₂²×C₄×C₂₃	C₂⁴	1	2	yes	—
151	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
152	Q₁₆×C₂₃	—	4	16	no	—
153	SD₁₆×C₂₃	—	4	16	no	—
154	C₂×C₈×C₂₃	—	2	8	no	—
155	C₂×C₈×C₂₃	C₂⁴	1	8	yes	—
156	C₄⋊C₄×C₂₃	—	2	4	no	—
157	C₄⋊C₄×C₂₃	—	2	4	no	—
158	C₄⋊C₄×C₂₃	—	2	4	no	—
159	C₄⋊C₄×C₂₃	—	2	8	no	—
160	C₄⋊C₄×C₂₃	—	2	8	no	—
161	C₄⋊C₄×C₂₃	—	2	2	no	—
162	C₄⋊C₄×C₂₃	C₂⁴	1	4	yes	—
163	C₂²⋊C₄×C₂₃	—	2	8	no	—
164	C₂²⋊C₄×C₂₃	—	2	8	no	—
165	C₂²⋊C₄×C₂₃	—	2	8	no	—
166	C₂²⋊C₄×C₂₃	—	2	8	no	—
167	C₂²⋊C₄×C₂₃	—	2	8	no	—
168	C₄²×C₂₃	—	2	8	no	—
169	C₄²×C₂₃	C₂×D₄	1	4	yes	—
170	C₄²×C₂₃	C₂×D₄	1	2	yes	—
171	C₄²×C₂₃	C₂×D₄	1	4	yes	—
172	C₄²×C₂₃	C₂×D₄	1	2	yes	—
173	C₄²×C₂₃	C₂⁴	1	2	yes	—
174	C₄²×C₂₃	C₂⁴	1	2	yes	—
175	C₂×#3215	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
176	C₂³×C₂₃	—	2	2	no	—
177	C₂³×C₂₃	—	2	2	no	—
178	C₂³×C₂₃	—	2	4	no	—
179	C₂³×C₂₃	—	4	8	no	—
180	C₂³×C₂₃	—	4	8	no	—
181	C₂³×C₂₃	—	4	4	no	—
182	C₂³×C₂₃	—	4	8	no	—
183	C₂³×C₂₃	—	4	4	no	—
184	C₂³×C₂₃	—	4	4	no	—
185	C₂³×C₂₃	—	8	16	no	—
186	C₂³×C₂₃	#137	1	4	yes	—
187	C₂³×C₂₃	C₂²×D₄	1	2	yes	—
188	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
189	Q₈×C₂₃	—	2	8	no	—
190	Q₈×C₂₃	—	2	8	no	—
191	Q₈×C₂₃	—	4	8	no	—
192	Q₈×C₂₃	—	4	8	no	—
193	Q₈×C₂₃	—	4	4	no	—
194	Q₈×C₂₃	—	4	4	no	—
195	D₄×C₂₃	—	2	8	no	—
196	D₄×C₂₃	—	4	8	no	—
197	D₄×C₂₃	—	4	4	no	—
198	C₂×C₄×C₂₃	—	2	2	no	—
199	C₂×C₄×C₂₃	—	2	8	no	—
200	C₂×C₄×C₂₃	—	2	2	no	—
201	C₂×C₄×C₂₃	—	2	2	no	—
202	C₂×C₄×C₂₃	—	2	2	no	—
203	C₂×C₄×C₂₃	—	2	2	no	—
204	C₂×C₄×C₂₃	—	2	4	no	—
205	C₂×C₄×C₂₃	—	2	4	no	—
206	C₂×C₄×C₂₃	—	2	4	no	—
207	C₂×C₄×C₂₃	—	2	4	no	—
208	C₂×C₄×C₂₃	—	2	4	no	—
209	C₂×C₄×C₂₃	—	4	4	no	—
210	C₂×C₄×C₂₃	—	4	8	no	—
211	C₂×C₄×C₂₃	—	4	4	no	—
212	C₂×C₄×C₂₃	—	4	4	no	—
213	C₂×C₄×C₂₃	—	4	4	no	—
214	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
215	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
216	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
217	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
218	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
219	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
220	C₂×C₄×C₂₃	H(5,𝔽₂)	1	4	yes	—
221	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
222	C₈×C₂₃	—	2	8	no	—
223	C₂×#533	C₂×C₂₃	1	1	yes	characteristic, complemented
224	C₂²×#124	C₂×C₂₃	1	1	yes	characteristic
225	#3215	C₂×C₂₃	1	16	yes	direct product factor
226	C₂×#497	C₂×C₂₃	1	2	yes	complemented
227	C₂×#489	C₂×C₂₃	1	2	yes	complemented
228	C₂×#464	C₂×C₂₃	1	4	yes	complemented
229	C₂×#451	C₂×C₂₃	1	2	yes	complemented
230	C₂×#450	C₂×C₂₃	1	2	yes	—
231	C₂×#432	C₂×C₂₃	1	1	yes	characteristic, complemented
232	C₂²×C₂₃	—	2	2	no	—
233	C₂²×C₂₃	—	2	8	no	—
234	C₂²×C₂₃	—	2	2	no	—
235	C₂²×C₂₃	—	4	4	no	—
236	C₂²×C₂₃	—	4	4	no	—
237	C₂²×C₂₃	—	4	4	no	—
238	C₂²×C₂₃	—	8	16	no	—
239	C₂²×C₂₃	—	8	8	no	—
240	C₂²×C₂₃	#535	1	2	yes	—
241	C₂²×C₂₃	C₂×#137	1	4	yes	—
242	C₂²×C₂₃	C₂×#137	1	2	yes	—
243	C₂²×C₂₃	#575	1	2	yes	—
244	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
245	C₄×C₂₃	—	2	2	no	—
246	C₄×C₂₃	—	2	2	no	—
247	C₄×C₂₃	—	2	2	no	—
248	C₄×C₂₃	—	2	2	no	—
249	C₄×C₂₃	—	2	2	no	—
250	C₄×C₂₃	—	4	4	no	—
251	C₄×C₂₃	—	4	4	no	—
252	C₄×C₂₃	—	4	4	no	—
253	C₄×C₂₃	#535	1	2	yes	—
254	C₄×C₂₃	C₂×H(5,𝔽₂)	1	2	yes	—
255	C₂³×Q₈	C₂²×C₂₃	1	1	yes	characteristic
256	C₂³×D₄	C₂²×C₂₃	1	1	yes	characteristic
257	C₂²×Q₁₆	C₂²×C₂₃	1	4	yes	complemented
258	C₂²×SD₁₆	C₂²×C₂₃	1	4	yes	—
259	#533	—	2	4	no	—
260	#533	C₂²×C₂₃	1	8	yes	complemented
261	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
262	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic, complemented
263	C₂×#125	—	2	1	no	—
264	C₂×#124	—	2	1	no	—
265	C₂×#124	—	2	1	no	—
266	C₂×#124	—	2	4	no	—
267	C₂×#124	C₂²×C₂₃	1	2	yes	—
268	C₂×#124	C₂²×C₂₃	1	8	yes	—
269	C₂×#124	C₂²×C₂₃	1	2	yes	—
270	C₂×C₂²⋊Q₈	—	2	1	no	—
271	C₂×C₂²⋊Q₈	C₂²×C₂₃	1	2	yes	—
272	C₂×C₄×Q₈	—	2	1	no	—
273	C₂×C₄×Q₈	C₂²×C₂₃	1	2	yes	complemented
274	C₂×#117	—	2	1	no	—
275	C₂²×C₂²⋊C₄	—	2	1	no	—
276	C₂²×C₂²⋊C₄	—	2	1	no	—
277	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
278	#497	C₂²×C₂₃	1	16	yes	complemented
279	#489	C₂²×C₂₃	1	16	yes	complemented
280	#464	C₂²×C₂₃	1	32	yes	complemented
281	#451	C₂²×C₂₃	1	16	yes	complemented
282	#450	C₂²×C₂₃	1	16	yes	—
283	#432	C₂²×C₂₃	1	8	yes	complemented
284	C₂×C₄⋊C₈	C₂²×C₂₃	1	2	yes	complemented
285	C₂×#103	C₂²×C₂₃	1	4	yes	complemented
286	C₂×#102	C₂²×C₂₃	1	4	yes	—
287	C₂×C₂²⋊C₈	C₂²×C₂₃	1	2	yes	—
288	C₂×#97	C₂²×C₂₃	1	2	yes	complemented
289	C₂×C₂₃	—	2	4	no	—
290	C₂×C₂₃	—	8	4	no	—
291	C₂×C₂₃	#3215	1	4	yes	direct product factor
292	C₂×C₂₃	C₂×#535	1	1	yes	characteristic
293	C₂×C₂₃	C₂²×#137	1	1	yes	characteristic
294	C₂×C₂₃	C₂×#575	1	1	yes	characteristic
295	C₂⁵	—	2	1	no	—
296	C₂²×Q₈	—	2	1	no	—
297	C₂²×Q₈	—	2	1	no	—
298	C₂²×Q₈	—	2	2	no	—
299	C₂²×Q₈	—	4	4	no	—
300	C₂²×Q₈	C₂³×C₂₃	1	2	yes	—
301	C₂²×Q₈	C₂³×C₂₃	1	2	yes	—
302	C₂²×Q₈	C₂³×C₂₃	1	4	yes	—
303	C₂²×D₄	—	2	1	no	—
304	C₂²×D₄	—	2	2	no	—
305	C₂²×D₄	—	4	4	no	—
306	C₂²×D₄	C₂³×C₂₃	1	2	yes	—
307	C₂²×D₄	C₂³×C₂₃	1	4	yes	—
308	C₂³×C₄	—	2	1	no	—
309	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
310	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
311	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
312	C₂×Q₁₆	—	4	8	no	—
313	C₂×Q₁₆	C₂³×C₂₃	1	16	yes	complemented
314	C₂×SD₁₆	—	4	8	no	—
315	C₂×SD₁₆	C₂³×C₂₃	1	16	yes	—
316	C₂²×C₈	C₂³×C₂₃	1	4	yes	—
317	C₄⋊Q₈	—	2	2	no	—
318	C₄⋊Q₈	—	2	2	no	—
319	C₄⋊Q₈	C₂³×C₂₃	1	4	yes	complemented
320	C₄⋊Q₈	C₂³×C₂₃	1	4	yes	complemented
321	#125	—	2	8	no	—
322	#124	—	2	8	no	—
323	#124	—	2	4	no	—
324	#124	—	2	8	no	—
325	#124	—	2	4	no	—
326	#124	C₂³×C₂₃	1	8	yes	—
327	#124	C₂³×C₂₃	1	8	yes	—
328	C₂²⋊Q₈	—	2	8	no	—
329	C₂²⋊Q₈	—	2	4	no	—
330	C₂²⋊Q₈	C₂³×C₂₃	1	8	yes	—
331	C₄×Q₈	—	2	8	no	—
332	C₄×Q₈	—	2	4	no	—
333	C₄×Q₈	C₂³×C₂₃	1	8	yes	complemented
334	#117	—	2	8	no	—
335	C₂×C₄⋊C₄	—	2	1	no	—
336	C₂×C₄⋊C₄	—	2	1	no	—
337	C₂×C₄⋊C₄	—	2	1	no	—
338	C₂×C₄⋊C₄	—	2	2	no	—
339	C₂×C₄⋊C₄	—	2	2	no	—
340	C₂×C₄⋊C₄	C₂³×C₂₃	1	2	yes	—
341	C₂×C₂²⋊C₄	—	2	2	no	—
342	C₂×C₂²⋊C₄	—	2	2	no	—
343	C₂×C₂²⋊C₄	—	2	2	no	—
344	C₂×C₂²⋊C₄	—	2	2	no	—
345	C₂×C₂²⋊C₄	—	2	2	no	—
346	C₂×C₂²⋊C₄	—	2	8	no	—
347	C₂×C₂²⋊C₄	—	2	8	no	—
348	C₂×C₄²	—	2	2	no	—
349	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
350	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
351	C₂×C₄²	D₄×C₂₃	1	4	yes	—
352	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
353	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
354	C₂×C₄²	C₂³×C₂₃	1	4	yes	—
355	C₄⋊C₈	C₂³×C₂₃	1	8	yes	complemented
356	#103	C₂³×C₂₃	1	16	yes	complemented
357	#102	C₂³×C₂₃	1	16	yes	—
358	C₂²⋊C₈	C₂³×C₂₃	1	8	yes	—
359	#97	C₂³×C₂₃	1	8	yes	complemented
360	C₂₃	C₂×#3215	1	1	yes	characteristic, direct product factor, Sylow
361	C₂⁴	—	2	4	no	—
362	C₂⁴	—	2	2	no	—
363	C₂⁴	—	4	4	no	—
364	C₂⁴	—	4	4	no	—
365	C₂⁴	—	4	4	no	—
366	C₂⁴	C₂×D₄×C₂₃	1	1	yes	characteristic
367	C₂×Q₈	—	2	2	no	—
368	C₂×Q₈	—	2	4	no	—
369	C₂×Q₈	—	2	4	no	—
370	C₂×Q₈	—	2	2	no	—
371	C₂×Q₈	—	2	8	no	—
372	C₂×Q₈	—	2	8	no	—
373	C₂×Q₈	—	4	8	no	—
374	C₂×Q₈	—	4	4	no	—
375	C₂×Q₈	—	4	8	no	—
376	C₂×Q₈	—	4	4	no	—
377	C₂×Q₈	C₂⁴×C₂₃	1	4	yes	—
378	C₂×Q₈	C₂⁴×C₂₃	1	4	yes	—
379	C₂×D₄	—	2	4	no	—
380	C₂×D₄	—	2	2	no	—
381	C₂×D₄	—	2	8	no	—
382	C₂×D₄	—	4	8	no	—
383	C₂×D₄	—	4	4	no	—
384	C₂×D₄	—	4	8	no	—
385	C₂×D₄	C₂⁴×C₂₃	1	4	yes	—
386	C₂²×C₄	—	2	4	no	—
387	C₂²×C₄	—	2	4	no	—
388	C₂²×C₄	—	2	2	no	—
389	C₂²×C₄	—	2	2	no	—
390	C₂²×C₄	—	2	4	no	—
391	C₂²×C₄	—	2	2	no	—
392	C₂²×C₄	—	4	2	no	—
393	C₂²×C₄	—	4	4	no	—
394	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
395	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
396	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
397	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
398	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
399	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
400	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
401	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
402	C₂²×C₄	C₂⁴×C₂₃	1	2	yes	—
403	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
404	Q₁₆	—	4	16	no	—
405	SD₁₆	—	4	16	no	—
406	C₂×C₈	—	2	8	no	—
407	C₂×C₈	C₂⁴×C₂₃	1	8	yes	—
408	C₄⋊C₄	—	2	4	no	—
409	C₄⋊C₄	—	2	4	no	—
410	C₄⋊C₄	—	2	4	no	—
411	C₄⋊C₄	—	2	8	no	—
412	C₄⋊C₄	—	2	8	no	—
413	C₄⋊C₄	—	2	2	no	—
414	C₄⋊C₄	C₂⁴×C₂₃	1	4	yes	—
415	C₂²⋊C₄	—	2	8	no	—
416	C₂²⋊C₄	—	2	8	no	—
417	C₂²⋊C₄	—	2	8	no	—
418	C₂²⋊C₄	—	2	8	no	—
419	C₂²⋊C₄	—	2	8	no	—
420	C₄²	—	2	8	no	—
421	C₄²	C₂×D₄×C₂₃	1	4	yes	—
422	C₄²	C₂×D₄×C₂₃	1	2	yes	—
423	C₄²	C₂×D₄×C₂₃	1	4	yes	—
424	C₄²	C₂×D₄×C₂₃	1	2	yes	—
425	C₄²	C₂⁴×C₂₃	1	2	yes	—
426	C₄²	C₂⁴×C₂₃	1	2	yes	—
427	C₂³	—	2	2	no	—
428	C₂³	—	2	2	no	—
429	C₂³	—	2	4	no	—
430	C₂³	—	4	8	no	—
431	C₂³	—	4	8	no	—
432	C₂³	—	4	8	no	—
433	C₂³	—	4	4	no	—
434	C₂³	—	4	4	no	—
435	C₂³	—	4	4	no	—
436	C₂³	—	8	16	no	—
437	C₂³	C₂₃×#137	1	4	yes	—
438	C₂³	C₂²×D₄×C₂₃	1	2	yes	—
439	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
440	Q₈	—	2	8	no	—
441	Q₈	—	2	8	no	—
442	Q₈	—	4	8	no	—
443	Q₈	—	4	8	no	—
444	Q₈	—	4	4	no	—
445	Q₈	—	4	4	no	—
446	D₄	—	2	8	no	—
447	D₄	—	4	8	no	—
448	D₄	—	4	4	no	—
449	C₂×C₄	—	2	8	no	—
450	C₂×C₄	—	2	2	no	—
451	C₂×C₄	—	2	2	no	—
452	C₂×C₄	—	2	2	no	—
453	C₂×C₄	—	2	2	no	—
454	C₂×C₄	—	2	2	no	—
455	C₂×C₄	—	2	4	no	—
456	C₂×C₄	—	2	4	no	—
457	C₂×C₄	—	2	4	no	—
458	C₂×C₄	—	2	4	no	—
459	C₂×C₄	—	2	4	no	—
460	C₂×C₄	—	4	4	no	—
461	C₂×C₄	—	4	8	no	—
462	C₂×C₄	—	4	4	no	—
463	C₂×C₄	—	4	4	no	—
464	C₂×C₄	—	4	4	no	—
465	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
466	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
467	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
468	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
469	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
470	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
471	C₂×C₄	C₂₃×H(5,𝔽₂)	1	4	yes	—
472	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
473	C₈	—	2	8	no	—
474	C₂²	—	2	2	no	—
475	C₂²	—	2	8	no	—
476	C₂²	—	2	2	no	—
477	C₂²	—	4	4	no	—
478	C₂²	—	4	4	no	—
479	C₂²	—	4	4	no	—
480	C₂²	—	8	16	no	—
481	C₂²	—	8	8	no	—
482	C₂²	C₂₃×#535	1	2	yes	—
483	C₂²	C₂×C₂₃×#137	1	4	yes	—
484	C₂²	C₂×C₂₃×#137	1	2	yes	—
485	C₂²	C₂₃×#575	1	2	yes	—
486	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
487	C₄	—	2	2	no	—
488	C₄	—	2	2	no	—
489	C₄	—	2	2	no	—
490	C₄	—	2	2	no	—
491	C₄	—	2	2	no	—
492	C₄	—	4	4	no	—
493	C₄	—	4	4	no	—
494	C₄	—	4	4	no	—
495	C₄	C₂₃×#535	1	2	yes	—
496	C₄	C₂×C₂₃×H(5,𝔽₂)	1	2	yes	—
497	C₂	—	2	4	no	—
498	C₂	—	8	4	no	—
499	C₂	C₂₃×#3215	1	4	yes	direct product factor
500	C₂	C₂×C₂₃×#535	1	1	yes	characteristic
501	C₂	C₂²×C₂₃×#137	1	1	yes	characteristic
502	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	C₂³×D₄×C₂₃	C₂²
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×#533	C₂	semidirect
3. C₂₃×#3215	C₂	direct (N × H)
4. C₂×C₂₃×#497	C₂	semidirect
5. C₂×C₂₃×#489	C₂	semidirect
6. C₂×C₂₃×#464	C₂	semidirect
7. C₂×C₂₃×#451	C₂	semidirect
9. C₂×C₂₃×#432	C₂	semidirect
12. C₂²×Q₁₆×C₂₃	C₂²	semidirect
15. C₂₃×#533	C₂²	semidirect
16. C₂×C₂₃×C₄⋊Q₈	C₂²	semidirect
17. C₂×C₂₃×C₄⋊Q₈	C₂²	semidirect
28. C₂×C₄×Q₈×C₂₃	C₂²	semidirect
33. C₂₃×#497	C₂²	semidirect
34. C₂₃×#489	C₂²	semidirect
35. C₂₃×#464	C₂²	semidirect
36. C₂₃×#451	C₂²	semidirect
38. C₂₃×#432	C₂²	semidirect
39. C₂×C₂₃×C₄⋊C₈	C₂²	semidirect
40. C₂×C₂₃×#103	C₂²	semidirect
43. C₂×C₂₃×#97	C₂²	semidirect
62. C₂×Q₁₆×C₂₃	C₂³	semidirect
68. C₂₃×C₄⋊Q₈	C₂³	semidirect
69. C₂₃×C₄⋊Q₈	C₂³	semidirect
82. C₄×Q₈×C₂₃	C₂³	semidirect
104. C₂₃×C₄⋊C₈	C₂³	semidirect
105. C₂₃×#103	C₂³	semidirect
108. C₂₃×#97	C₂³	semidirect
175. C₂×#3215	C₂₃	direct (N × H)
223. C₂×#533	C₂×C₂₃	semidirect
225. #3215	C₂×C₂₃	direct (N × H)
226. C₂×#497	C₂×C₂₃	semidirect
227. C₂×#489	C₂×C₂₃	semidirect
228. C₂×#464	C₂×C₂₃	semidirect
229. C₂×#451	C₂×C₂₃	semidirect
231. C₂×#432	C₂×C₂₃	semidirect
257. C₂²×Q₁₆	C₂²×C₂₃	semidirect
260. #533	C₂²×C₂₃	semidirect
261. C₂×C₄⋊Q₈	C₂²×C₂₃	semidirect
262. C₂×C₄⋊Q₈	C₂²×C₂₃	semidirect
273. C₂×C₄×Q₈	C₂²×C₂₃	semidirect
278. #497	C₂²×C₂₃	semidirect
279. #489	C₂²×C₂₃	semidirect
280. #464	C₂²×C₂₃	semidirect
281. #451	C₂²×C₂₃	semidirect
283. #432	C₂²×C₂₃	semidirect
284. C₂×C₄⋊C₈	C₂²×C₂₃	semidirect
285. C₂×#103	C₂²×C₂₃	semidirect
288. C₂×#97	C₂²×C₂₃	semidirect
291. C₂×C₂₃	#3215	direct (N × H)
313. C₂×Q₁₆	C₂³×C₂₃	semidirect
319. C₄⋊Q₈	C₂³×C₂₃	semidirect
320. C₄⋊Q₈	C₂³×C₂₃	semidirect
333. C₄×Q₈	C₂³×C₂₃	semidirect
355. C₄⋊C₈	C₂³×C₂₃	semidirect
356. #103	C₂³×C₂₃	semidirect
359. #97	C₂³×C₂₃	semidirect
360. C₂₃	C₂×#3215	direct (N × H)
499. C₂	C₂₃×#3215	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×#533	2
C₂²×C₂₃×#124	2
C₂₃×#3215	2
C₂×C₂₃×#497	2
C₂×C₂₃×#489	2
C₂×C₂₃×#464	2
C₂×C₂₃×#451	2
C₂×C₂₃×#450	2
C₂×C₂₃×#432	2
C₂×#3215	23

Subgroups — C2×C23×#3215 #19208410

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#3215 #19208410