Subgroups — C₂×C₂₃×#3284 #19208670

Order 5888.

2858 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₂₃×#135	C₂	1	2	yes	maximal, complemented
2	C₂×C₂₃×#550	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₂₃×#547	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#3284	C₂	1	16	yes	maximal, direct product factor
5	C₂×C₂₃×#492	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₂₃×#465	C₂	1	2	yes	maximal, complemented
7	C₂×C₂₃×#462	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₂₃×#459	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₂₃×#452	C₂	1	2	yes	maximal, complemented
10	C₂×C₂₃×#449	C₂	1	2	yes	maximal, complemented
11	C₂×C₄×SD₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₈×D₄×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×#41×C₂₃	C₂²	1	2	yes	—
14	C₂²×#41×C₂₃	C₂²	1	2	yes	—
15	C₂×C₂₃×#135	—	2	16	no	—
16	C₂×C₂₃×#135	C₂²	1	16	yes	complemented
17	C₂²×Q₁₆×C₂₃	C₂²	1	2	yes	—
18	C₂²×SD₁₆×C₂₃	C₂²	1	2	yes	—
19	C₂²×SD₁₆×C₂₃	C₂²	1	2	yes	—
20	C₂²×SD₁₆×C₂₃	C₂²	1	1	yes	characteristic
21	C₂²×D₈×C₂₃	C₂²	1	2	yes	—
22	C₂³×C₈×C₂₃	C₂²	1	2	yes	—
23	C₂₃×#550	—	2	4	no	—
24	C₂₃×#550	C₂²	1	8	yes	complemented
25	C₂₃×#547	—	2	4	no	—
26	C₂₃×#547	C₂²	1	8	yes	complemented
27	C₂×C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
28	C₂×C₂₃×#127	—	2	1	no	—
29	C₂×C₂₃×#127	C₂²	1	1	yes	characteristic
30	C₂×C₂₃×#123	—	2	2	no	—
31	C₂×C₂₃×C₂²⋊Q₈	C₂²	1	2	yes	—
32	C₂×C₂₃×#121	—	2	1	no	—
33	C₂×C₂₃×#121	—	2	2	no	—
34	C₂×C₂₃×#121	C₂²	1	2	yes	—
35	C₂×C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
36	C₂×C₄×D₄×C₂₃	—	2	1	no	—
37	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
38	C₂×C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
39	C₂²×C₄⋊C₄×C₂₃	—	2	1	no	—
40	C₂₃×#492	C₂²	1	8	yes	complemented
41	C₂₃×#465	C₂²	1	16	yes	complemented
42	C₂₃×#462	C₂²	1	8	yes	complemented
43	C₂₃×#459	C₂²	1	8	yes	complemented
44	C₂₃×#452	C₂²	1	16	yes	complemented
45	C₂₃×#449	C₂²	1	16	yes	complemented
46	C₄×SD₁₆×C₂₃	C₂²	1	8	yes	complemented
47	C₈×D₄×C₂₃	C₂²	1	8	yes	complemented
48	C₂×C₂₃×#106	C₂²	1	1	yes	characteristic
49	C₂×C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
50	C₂×C₂₃×#103	C₂²	1	1	yes	characteristic
51	C₂×C₂₃×#103	C₂²	1	2	yes	—
52	C₂×C₂₃×#102	C₂²	1	1	yes	characteristic
53	C₂×C₂₃×#102	C₂²	1	2	yes	—
54	C₂×C₂₃×C₂²⋊C₈	C₂²	1	2	yes	—
55	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
56	C₂×#41×C₂₃	—	2	4	no	—
57	C₂×#41×C₂₃	—	2	4	no	—
58	C₂×#41×C₂₃	—	2	8	no	—
59	C₂×#41×C₂₃	—	2	8	no	—
60	C₂×#41×C₂₃	—	4	4	no	—
61	C₂×#41×C₂₃	—	4	4	no	—
62	C₂×#41×C₂₃	C₂³	1	8	yes	—
63	C₂×#41×C₂₃	C₂³	1	8	yes	—
64	C₂²×Q₈×C₂₃	C₂³	1	2	yes	—
65	C₂²×Q₈×C₂₃	C₂³	1	1	yes	characteristic
66	C₂²×D₄×C₂₃	—	2	2	no	—
67	C₂²×D₄×C₂₃	—	2	2	no	—
68	C₂²×D₄×C₂₃	—	2	2	no	—
69	C₂²×D₄×C₂₃	—	2	1	no	—
70	C₂²×D₄×C₂₃	—	2	1	no	—
71	C₂²×D₄×C₂₃	D₄	1	1	yes	characteristic
72	C₂²×D₄×C₂₃	C₂³	1	2	yes	—
73	C₂²×D₄×C₂₃	C₂³	1	1	yes	characteristic
74	C₂³×C₄×C₂₃	—	2	2	no	—
75	C₂³×C₄×C₂₃	—	2	2	no	—
76	C₂³×C₄×C₂₃	—	2	1	no	—
77	C₂³×C₄×C₂₃	C₂³	1	2	yes	—
78	C₂₃×#135	—	2	64	no	—
79	C₂×Q₁₆×C₂₃	—	2	8	no	—
80	C₂×Q₁₆×C₂₃	C₂³	1	8	yes	—
81	C₂×SD₁₆×C₂₃	—	2	8	no	—
82	C₂×SD₁₆×C₂₃	—	4	4	no	—
83	C₂×SD₁₆×C₂₃	D₄	1	8	yes	complemented
84	C₂×SD₁₆×C₂₃	C₂³	1	8	yes	—
85	C₂×SD₁₆×C₂₃	C₂³	1	8	yes	—
86	C₂×SD₁₆×C₂₃	C₂³	1	4	yes	—
87	C₂×D₈×C₂₃	—	2	8	no	—
88	C₂×D₈×C₂₃	C₂³	1	8	yes	—
89	C₂²×C₈×C₂₃	—	2	8	no	—
90	C₂²×C₈×C₂₃	C₂³	1	2	yes	—
91	C₂²×C₈×C₂₃	C₂³	1	8	yes	—
92	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
93	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
94	C₂₃×C₄⋊Q₈	—	2	2	no	—
95	C₂₃×C₄⋊Q₈	C₂³	1	4	yes	—
96	C₂₃×#127	—	2	8	no	—
97	C₂₃×#127	—	2	2	no	—
98	C₂₃×#127	C₂³	1	4	yes	—
99	C₂₃×#123	—	2	16	no	—
100	C₂₃×C₂²⋊Q₈	—	2	4	no	—
101	C₂₃×C₂²⋊Q₈	C₂³	1	8	yes	—
102	C₂₃×#121	—	2	8	no	—
103	C₂₃×#121	—	2	4	no	—
104	C₂₃×#121	—	2	16	no	—
105	C₂₃×#121	C₂³	1	8	yes	—
106	C₄×Q₈×C₂₃	—	2	2	no	—
107	C₄×Q₈×C₂₃	C₂³	1	4	yes	—
108	C₄×D₄×C₂₃	—	2	8	no	—
109	C₄×D₄×C₂₃	—	2	2	no	—
110	C₄×D₄×C₂₃	D₄	1	4	yes	—
111	C₄×D₄×C₂₃	C₂³	1	4	yes	—
112	C₄×D₄×C₂₃	C₂³	1	4	yes	—
113	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
114	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
115	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
116	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
117	C₂×C₄⋊C₄×C₂₃	—	2	8	no	—
118	C₂×C₄⋊C₄×C₂₃	D₄	1	1	yes	characteristic
119	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
120	C₂×C₄⋊C₄×C₂₃	C₂³	1	1	yes	characteristic
121	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
122	C₂×C₂²⋊C₄×C₂₃	—	2	2	no	—
123	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
124	C₂×C₂²⋊C₄×C₂₃	D₄	1	2	yes	—
125	C₂×C₄²×C₂₃	—	2	1	no	—
126	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
127	C₂₃×#106	C₂³	1	4	yes	—
128	C₂₃×C₄⋊C₈	C₂³	1	4	yes	—
129	C₂₃×#103	C₂³	1	4	yes	—
130	C₂₃×#103	C₂³	1	8	yes	—
131	C₂₃×#102	C₂³	1	4	yes	—
132	C₂₃×#102	C₂³	1	8	yes	—
133	C₂₃×C₂²⋊C₈	C₂³	1	8	yes	—
134	C₄×C₈×C₂₃	C₂³	1	4	yes	—
135	C₂⁴×C₂₃	—	2	1	no	—
136	C₂⁴×C₂₃	—	2	2	no	—
137	C₂⁴×C₂₃	C₂×D₄	1	2	yes	—
138	#41×C₂₃	—	2	16	no	—
139	#41×C₂₃	—	2	16	no	—
140	#41×C₂₃	—	4	8	no	—
141	#41×C₂₃	—	4	8	no	—
142	#41×C₂₃	—	4	16	no	—
143	#41×C₂₃	—	4	16	no	—
144	C₂×Q₈×C₂₃	—	2	1	no	—
145	C₂×Q₈×C₂₃	—	2	2	no	—
146	C₂×Q₈×C₂₃	—	2	2	no	—
147	C₂×Q₈×C₂₃	—	2	4	no	—
148	C₂×Q₈×C₂₃	—	2	4	no	—
149	C₂×Q₈×C₂₃	C₂×D₄	1	4	yes	—
150	C₂×Q₈×C₂₃	C₂⁴	1	4	yes	—
151	C₂×Q₈×C₂₃	C₂⁴	1	2	yes	—
152	C₂×D₄×C₂₃	—	2	8	no	—
153	C₂×D₄×C₂₃	—	2	8	no	—
154	C₂×D₄×C₂₃	—	2	8	no	—
155	C₂×D₄×C₂₃	—	2	4	no	—
156	C₂×D₄×C₂₃	—	2	1	no	—
157	C₂×D₄×C₂₃	—	2	4	no	—
158	C₂×D₄×C₂₃	—	2	4	no	—
159	C₂×D₄×C₂₃	—	2	2	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	—	4	8	no	—
164	C₂×D₄×C₂₃	—	4	8	no	—
165	C₂×D₄×C₂₃	—	4	4	no	—
166	C₂×D₄×C₂₃	—	8	2	no	—
167	C₂×D₄×C₂₃	—	8	4	no	—
168	C₂×D₄×C₂₃	C₂×D₄	1	4	yes	—
169	C₂×D₄×C₂₃	C₂×D₄	1	4	yes	—
170	C₂×D₄×C₂₃	C₂⁴	1	4	yes	—
171	C₂×D₄×C₂₃	C₂⁴	1	2	yes	—
172	C₂²×C₄×C₂₃	—	2	1	no	—
173	C₂²×C₄×C₂₃	—	2	4	no	—
174	C₂²×C₄×C₂₃	—	2	4	no	—
175	C₂²×C₄×C₂₃	—	2	1	no	—
176	C₂²×C₄×C₂₃	—	2	1	no	—
177	C₂²×C₄×C₂₃	—	2	2	no	—
178	C₂²×C₄×C₂₃	—	2	4	no	—
179	C₂²×C₄×C₂₃	—	2	8	no	—
180	C₂²×C₄×C₂₃	—	2	8	no	—
181	C₂²×C₄×C₂₃	—	4	8	no	—
182	C₂²×C₄×C₂₃	—	4	8	no	—
183	C₂²×C₄×C₂₃	—	4	4	no	—
184	C₂²×C₄×C₂₃	C₂×D₄	1	4	yes	—
185	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
186	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
187	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
188	C₂²×C₄×C₂₃	C₂⁴	1	4	yes	—
189	C₂²×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
190	Q₁₆×C₂₃	—	2	16	no	—
191	SD₁₆×C₂₃	—	2	16	no	—
192	SD₁₆×C₂₃	—	4	8	no	—
193	SD₁₆×C₂₃	C₂×D₄	1	16	yes	complemented
194	D₈×C₂₃	—	2	16	no	—
195	C₂×C₈×C₂₃	—	2	4	no	—
196	C₂×C₈×C₂₃	—	2	2	no	—
197	C₂×C₈×C₂₃	—	2	16	no	—
198	C₂×C₈×C₂₃	C₂×D₄	1	4	yes	—
199	C₂×C₈×C₂₃	C₂⁴	1	4	yes	—
200	C₂×C₈×C₂₃	C₂⁴	1	2	yes	—
201	C₂×C₈×C₂₃	C₂⁴	1	2	yes	—
202	C₄⋊C₄×C₂₃	—	2	4	no	—
203	C₄⋊C₄×C₂₃	—	2	8	no	—
204	C₄⋊C₄×C₂₃	—	2	4	no	—
205	C₄⋊C₄×C₂₃	—	2	4	no	—
206	C₄⋊C₄×C₂₃	—	2	1	no	—
207	C₄⋊C₄×C₂₃	—	2	1	no	—
208	C₄⋊C₄×C₂₃	C₂×D₄	1	4	yes	—
209	C₄⋊C₄×C₂₃	C₂⁴	1	2	yes	—
210	C₄⋊C₄×C₂₃	C₂⁴	1	2	yes	—
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	C₂×D₄	1	8	yes	—
215	C₄²×C₂₃	—	2	4	no	—
216	C₄²×C₂₃	C₂×D₄	1	2	yes	—
217	C₄²×C₂₃	C₂⁴	1	2	yes	—
218	C₂×#3284	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
219	C₂³×C₂₃	—	2	4	no	—
220	C₂³×C₂₃	—	2	4	no	—
221	C₂³×C₂₃	—	2	4	no	—
222	C₂³×C₂₃	—	2	4	no	—
223	C₂³×C₂₃	—	2	2	no	—
224	C₂³×C₂₃	—	2	4	no	—
225	C₂³×C₂₃	—	4	2	no	—
226	C₂³×C₂₃	—	4	2	no	—
227	C₂³×C₂₃	—	4	4	no	—
228	C₂³×C₂₃	—	4	4	no	—
229	C₂³×C₂₃	—	4	4	no	—
230	C₂³×C₂₃	C₂²×D₄	1	4	yes	—
231	C₂³×C₂₃	C₂²×D₄	1	1	yes	characteristic
232	Q₈×C₂₃	—	2	4	no	—
233	Q₈×C₂₃	—	2	2	no	—
234	Q₈×C₂₃	—	2	4	no	—
235	Q₈×C₂₃	—	2	4	no	—
236	Q₈×C₂₃	—	4	4	no	—
237	Q₈×C₂₃	C₂²×D₄	1	4	yes	—
238	D₄×C₂₃	—	2	4	no	—
239	D₄×C₂₃	—	2	8	no	—
240	D₄×C₂₃	—	2	4	no	—
241	D₄×C₂₃	—	2	2	no	—
242	D₄×C₂₃	—	2	4	no	—
243	D₄×C₂₃	—	4	16	no	—
244	D₄×C₂₃	—	4	16	no	—
245	D₄×C₂₃	—	4	4	no	—
246	D₄×C₂₃	—	4	8	no	—
247	D₄×C₂₃	—	8	4	no	—
248	D₄×C₂₃	—	8	8	no	—
249	D₄×C₂₃	C₂²×D₄	1	4	yes	—
250	C₂×C₄×C₂₃	—	2	4	no	—
251	C₂×C₄×C₂₃	—	2	4	no	—
252	C₂×C₄×C₂₃	—	2	2	no	—
253	C₂×C₄×C₂₃	—	2	2	no	—
254	C₂×C₄×C₂₃	—	2	8	no	—
255	C₂×C₄×C₂₃	—	2	4	no	—
256	C₂×C₄×C₂₃	—	2	4	no	—
257	C₂×C₄×C₂₃	—	2	2	no	—
258	C₂×C₄×C₂₃	—	2	2	no	—
259	C₂×C₄×C₂₃	—	2	4	no	—
260	C₂×C₄×C₂₃	—	4	16	no	—
261	C₂×C₄×C₂₃	—	4	16	no	—
262	C₂×C₄×C₂₃	—	4	4	no	—
263	C₂×C₄×C₂₃	—	4	2	no	—
264	C₂×C₄×C₂₃	—	4	2	no	—
265	C₂×C₄×C₂₃	—	4	8	no	—
266	C₂×C₄×C₂₃	#135	1	4	yes	complemented
267	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
268	C₂×C₄×C₂₃	C₂²×D₄	1	4	yes	—
269	C₂×C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
270	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
271	C₂×C₄×C₂₃	C₂²×D₄	1	2	yes	—
272	C₂×C₄×C₂₃	H(5,𝔽₂)	1	2	yes	—
273	C₂×C₄×C₂₃	C₂⁵	1	1	yes	characteristic
274	C₈×C₂₃	—	2	4	no	—
275	C₈×C₂₃	—	2	2	no	—
276	C₈×C₂₃	C₂²×D₄	1	4	yes	—
277	C₂²×#135	C₂×C₂₃	1	2	yes	complemented
278	C₂×#550	C₂×C₂₃	1	1	yes	characteristic, complemented
279	C₂×#547	C₂×C₂₃	1	1	yes	characteristic, complemented
280	#3284	C₂×C₂₃	1	16	yes	direct product factor
281	C₂×#492	C₂×C₂₃	1	1	yes	characteristic, complemented
282	C₂×#465	C₂×C₂₃	1	2	yes	complemented
283	C₂×#462	C₂×C₂₃	1	1	yes	characteristic, complemented
284	C₂×#459	C₂×C₂₃	1	1	yes	characteristic, complemented
285	C₂×#452	C₂×C₂₃	1	2	yes	complemented
286	C₂×#449	C₂×C₂₃	1	2	yes	complemented
287	C₂×C₄×SD₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
288	C₂×C₈×D₄	C₂×C₂₃	1	1	yes	characteristic, complemented
289	C₂²×C₂₃	—	2	4	no	—
290	C₂²×C₂₃	—	2	4	no	—
291	C₂²×C₂₃	—	2	4	no	—
292	C₂²×C₂₃	—	2	4	no	—
293	C₂²×C₂₃	—	4	8	no	—
294	C₂²×C₂₃	—	4	8	no	—
295	C₂²×C₂₃	—	4	2	no	—
296	C₂²×C₂₃	—	4	2	no	—
297	C₂²×C₂₃	—	4	4	no	—
298	C₂²×C₂₃	—	4	4	no	—
299	C₂²×C₂₃	—	4	4	no	—
300	C₂²×C₂₃	—	8	8	no	—
301	C₂²×C₂₃	D₄²	1	2	yes	—
302	C₂²×C₂₃	C₂×#135	1	2	yes	—
303	C₂²×C₂₃	#577	1	2	yes	—
304	C₂²×C₂₃	C₂³×D₄	1	1	yes	characteristic
305	C₄×C₂₃	—	2	4	no	—
306	C₄×C₂₃	—	2	4	no	—
307	C₄×C₂₃	—	2	2	no	—
308	C₄×C₂₃	—	4	4	no	—
309	C₄×C₂₃	—	4	2	no	—
310	C₄×C₂₃	—	4	2	no	—
311	C₄×C₂₃	D₄²	1	2	yes	—
312	C₄×C₂₃	C₂×#135	1	4	yes	complemented
313	C₄×C₂₃	C₂³×D₄	1	1	yes	characteristic
314	C₄×C₂₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
315	C₂²×#41	C₂²×C₂₃	1	2	yes	—
316	C₂²×#41	C₂²×C₂₃	1	2	yes	—
317	C₂×#135	—	2	16	no	—
318	C₂×#135	C₂²×C₂₃	1	16	yes	complemented
319	C₂²×Q₁₆	C₂²×C₂₃	1	2	yes	—
320	C₂²×SD₁₆	C₂²×C₂₃	1	2	yes	—
321	C₂²×SD₁₆	C₂²×C₂₃	1	2	yes	—
322	C₂²×SD₁₆	C₂²×C₂₃	1	1	yes	characteristic
323	C₂²×D₈	C₂²×C₂₃	1	2	yes	—
324	C₂³×C₈	C₂²×C₂₃	1	2	yes	—
325	#550	—	2	4	no	—
326	#550	C₂²×C₂₃	1	8	yes	complemented
327	#547	—	2	4	no	—
328	#547	C₂²×C₂₃	1	8	yes	complemented
329	C₂×C₄⋊Q₈	C₂²×C₂₃	1	1	yes	characteristic
330	C₂×#127	—	2	1	no	—
331	C₂×#127	C₂²×C₂₃	1	1	yes	characteristic
332	C₂×#123	—	2	2	no	—
333	C₂×C₂²⋊Q₈	C₂²×C₂₃	1	2	yes	—
334	C₂×#121	—	2	1	no	—
335	C₂×#121	—	2	2	no	—
336	C₂×#121	C₂²×C₂₃	1	2	yes	—
337	C₂×C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
338	C₂×C₄×D₄	—	2	1	no	—
339	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
340	C₂×C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
341	C₂²×C₄⋊C₄	—	2	1	no	—
342	#492	C₂²×C₂₃	1	8	yes	complemented
343	#465	C₂²×C₂₃	1	16	yes	complemented
344	#462	C₂²×C₂₃	1	8	yes	complemented
345	#459	C₂²×C₂₃	1	8	yes	complemented
346	#452	C₂²×C₂₃	1	16	yes	complemented
347	#449	C₂²×C₂₃	1	16	yes	complemented
348	C₄×SD₁₆	C₂²×C₂₃	1	8	yes	complemented
349	C₈×D₄	C₂²×C₂₃	1	8	yes	complemented
350	C₂×#106	C₂²×C₂₃	1	1	yes	characteristic
351	C₂×C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
352	C₂×#103	C₂²×C₂₃	1	1	yes	characteristic
353	C₂×#103	C₂²×C₂₃	1	2	yes	—
354	C₂×#102	C₂²×C₂₃	1	1	yes	characteristic
355	C₂×#102	C₂²×C₂₃	1	2	yes	—
356	C₂×C₂²⋊C₈	C₂²×C₂₃	1	2	yes	—
357	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
358	C₂×C₂₃	—	4	4	no	—
359	C₂×C₂₃	—	4	4	no	—
360	C₂×C₂₃	—	8	4	no	—
361	C₂×C₂₃	#3284	1	4	yes	direct product factor
362	C₂×C₂₃	C₂×D₄²	1	1	yes	characteristic
363	C₂×C₂₃	C₂²×#135	1	1	yes	characteristic
364	C₂×C₂₃	C₂×#577	1	1	yes	characteristic
365	C₂×#41	—	2	4	no	—
366	C₂×#41	—	2	4	no	—
367	C₂×#41	—	2	8	no	—
368	C₂×#41	—	2	8	no	—
369	C₂×#41	—	4	4	no	—
370	C₂×#41	—	4	4	no	—
371	C₂×#41	C₂³×C₂₃	1	8	yes	—
372	C₂×#41	C₂³×C₂₃	1	8	yes	—
373	C₂²×Q₈	C₂³×C₂₃	1	2	yes	—
374	C₂²×Q₈	C₂³×C₂₃	1	1	yes	characteristic
375	C₂²×D₄	—	2	2	no	—
376	C₂²×D₄	—	2	2	no	—
377	C₂²×D₄	—	2	2	no	—
378	C₂²×D₄	—	2	1	no	—
379	C₂²×D₄	—	2	1	no	—
380	C₂²×D₄	D₄×C₂₃	1	1	yes	characteristic
381	C₂²×D₄	C₂³×C₂₃	1	2	yes	—
382	C₂²×D₄	C₂³×C₂₃	1	1	yes	characteristic
383	C₂³×C₄	—	2	1	no	—
384	C₂³×C₄	—	2	2	no	—
385	C₂³×C₄	—	2	2	no	—
386	C₂³×C₄	C₂³×C₂₃	1	2	yes	—
387	#135	—	2	64	no	—
388	C₂×Q₁₆	—	2	8	no	—
389	C₂×Q₁₆	C₂³×C₂₃	1	8	yes	—
390	C₂×SD₁₆	—	2	8	no	—
391	C₂×SD₁₆	—	4	4	no	—
392	C₂×SD₁₆	D₄×C₂₃	1	8	yes	complemented
393	C₂×SD₁₆	C₂³×C₂₃	1	8	yes	—
394	C₂×SD₁₆	C₂³×C₂₃	1	8	yes	—
395	C₂×SD₁₆	C₂³×C₂₃	1	4	yes	—
396	C₂×D₈	—	2	8	no	—
397	C₂×D₈	C₂³×C₂₃	1	8	yes	—
398	C₂²×C₈	—	2	8	no	—
399	C₂²×C₈	C₂³×C₂₃	1	2	yes	—
400	C₂²×C₈	C₂³×C₂₃	1	8	yes	—
401	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
402	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
403	C₄⋊Q₈	—	2	2	no	—
404	C₄⋊Q₈	C₂³×C₂₃	1	4	yes	—
405	#127	—	2	8	no	—
406	#127	—	2	2	no	—
407	#127	C₂³×C₂₃	1	4	yes	—
408	#123	—	2	16	no	—
409	C₂²⋊Q₈	—	2	4	no	—
410	C₂²⋊Q₈	C₂³×C₂₃	1	8	yes	—
411	#121	—	2	8	no	—
412	#121	—	2	4	no	—
413	#121	—	2	16	no	—
414	#121	C₂³×C₂₃	1	8	yes	—
415	C₄×Q₈	—	2	2	no	—
416	C₄×Q₈	C₂³×C₂₃	1	4	yes	—
417	C₄×D₄	—	2	8	no	—
418	C₄×D₄	—	2	2	no	—
419	C₄×D₄	D₄×C₂₃	1	4	yes	—
420	C₄×D₄	C₂³×C₂₃	1	4	yes	—
421	C₄×D₄	C₂³×C₂₃	1	4	yes	—
422	C₂×C₄⋊C₄	—	2	1	no	—
423	C₂×C₄⋊C₄	—	2	2	no	—
424	C₂×C₄⋊C₄	—	2	1	no	—
425	C₂×C₄⋊C₄	—	2	1	no	—
426	C₂×C₄⋊C₄	—	2	8	no	—
427	C₂×C₄⋊C₄	D₄×C₂₃	1	1	yes	characteristic
428	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
429	C₂×C₄⋊C₄	C₂³×C₂₃	1	1	yes	characteristic
430	C₂×C₂²⋊C₄	—	2	2	no	—
431	C₂×C₂²⋊C₄	—	2	2	no	—
432	C₂×C₂²⋊C₄	—	2	1	no	—
433	C₂×C₂²⋊C₄	D₄×C₂₃	1	2	yes	—
434	C₂×C₄²	—	2	1	no	—
435	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
436	#106	C₂³×C₂₃	1	4	yes	—
437	C₄⋊C₈	C₂³×C₂₃	1	4	yes	—
438	#103	C₂³×C₂₃	1	4	yes	—
439	#103	C₂³×C₂₃	1	8	yes	—
440	#102	C₂³×C₂₃	1	4	yes	—
441	#102	C₂³×C₂₃	1	8	yes	—
442	C₂²⋊C₈	C₂³×C₂₃	1	8	yes	—
443	C₄×C₈	C₂³×C₂₃	1	4	yes	—
444	C₂₃	C₂×#3284	1	1	yes	characteristic, direct product factor, Sylow
445	C₂⁴	—	2	1	no	—
446	C₂⁴	—	2	2	no	—
447	C₂⁴	C₂×D₄×C₂₃	1	2	yes	—
448	#41	—	2	16	no	—
449	#41	—	2	16	no	—
450	#41	—	4	8	no	—
451	#41	—	4	8	no	—
452	#41	—	4	16	no	—
453	#41	—	4	16	no	—
454	C₂×Q₈	—	2	1	no	—
455	C₂×Q₈	—	2	2	no	—
456	C₂×Q₈	—	2	2	no	—
457	C₂×Q₈	—	2	4	no	—
458	C₂×Q₈	—	2	4	no	—
459	C₂×Q₈	C₂×D₄×C₂₃	1	4	yes	—
460	C₂×Q₈	C₂⁴×C₂₃	1	4	yes	—
461	C₂×Q₈	C₂⁴×C₂₃	1	2	yes	—
462	C₂×D₄	—	2	8	no	—
463	C₂×D₄	—	2	8	no	—
464	C₂×D₄	—	2	8	no	—
465	C₂×D₄	—	2	4	no	—
466	C₂×D₄	—	2	1	no	—
467	C₂×D₄	—	2	4	no	—
468	C₂×D₄	—	2	4	no	—
469	C₂×D₄	—	2	2	no	—
470	C₂×D₄	—	2	4	no	—
471	C₂×D₄	—	2	2	no	—
472	C₂×D₄	—	2	4	no	—
473	C₂×D₄	—	4	8	no	—
474	C₂×D₄	—	4	8	no	—
475	C₂×D₄	—	4	4	no	—
476	C₂×D₄	—	8	2	no	—
477	C₂×D₄	—	8	4	no	—
478	C₂×D₄	C₂×D₄×C₂₃	1	4	yes	—
479	C₂×D₄	C₂×D₄×C₂₃	1	4	yes	—
480	C₂×D₄	C₂⁴×C₂₃	1	4	yes	—
481	C₂×D₄	C₂⁴×C₂₃	1	2	yes	—
482	C₂²×C₄	—	2	1	no	—
483	C₂²×C₄	—	2	4	no	—
484	C₂²×C₄	—	2	4	no	—
485	C₂²×C₄	—	2	1	no	—
486	C₂²×C₄	—	2	1	no	—
487	C₂²×C₄	—	2	2	no	—
488	C₂²×C₄	—	2	4	no	—
489	C₂²×C₄	—	2	8	no	—
490	C₂²×C₄	—	2	8	no	—
491	C₂²×C₄	—	4	8	no	—
492	C₂²×C₄	—	4	8	no	—
493	C₂²×C₄	—	4	4	no	—
494	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
495	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
496	C₂²×C₄	C₂×D₄×C₂₃	1	4	yes	—
497	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
498	C₂²×C₄	C₂⁴×C₂₃	1	4	yes	—
499	C₂²×C₄	C₂⁴×C₂₃	1	1	yes	characteristic
500	Q₁₆	—	2	16	no	—
501	SD₁₆	—	2	16	no	—
502	SD₁₆	—	4	8	no	—
503	SD₁₆	C₂×D₄×C₂₃	1	16	yes	complemented
504	D₈	—	2	16	no	—
505	C₂×C₈	—	2	4	no	—
506	C₂×C₈	—	2	2	no	—
507	C₂×C₈	—	2	16	no	—
508	C₂×C₈	C₂×D₄×C₂₃	1	4	yes	—
509	C₂×C₈	C₂⁴×C₂₃	1	2	yes	—
510	C₂×C₈	C₂⁴×C₂₃	1	4	yes	—
511	C₂×C₈	C₂⁴×C₂₃	1	2	yes	—
512	C₄⋊C₄	—	2	4	no	—
513	C₄⋊C₄	—	2	8	no	—
514	C₄⋊C₄	—	2	4	no	—
515	C₄⋊C₄	—	2	4	no	—
516	C₄⋊C₄	—	2	1	no	—
517	C₄⋊C₄	—	2	1	no	—
518	C₄⋊C₄	C₂×D₄×C₂₃	1	4	yes	—
519	C₄⋊C₄	C₂⁴×C₂₃	1	2	yes	—
520	C₄⋊C₄	C₂⁴×C₂₃	1	2	yes	—
521	C₂²⋊C₄	—	2	8	no	—
522	C₂²⋊C₄	—	2	8	no	—
523	C₂²⋊C₄	—	2	4	no	—
524	C₂²⋊C₄	C₂×D₄×C₂₃	1	8	yes	—
525	C₄²	—	2	4	no	—
526	C₄²	C₂×D₄×C₂₃	1	2	yes	—
527	C₄²	C₂⁴×C₂₃	1	2	yes	—
528	C₂³	—	2	4	no	—
529	C₂³	—	2	4	no	—
530	C₂³	—	2	2	no	—
531	C₂³	—	2	4	no	—
532	C₂³	—	2	4	no	—
533	C₂³	—	2	4	no	—
534	C₂³	—	4	2	no	—
535	C₂³	—	4	2	no	—
536	C₂³	—	4	4	no	—
537	C₂³	—	4	4	no	—
538	C₂³	—	4	4	no	—
539	C₂³	C₂²×D₄×C₂₃	1	4	yes	—
540	C₂³	C₂²×D₄×C₂₃	1	1	yes	characteristic
541	Q₈	—	2	2	no	—
542	Q₈	—	2	4	no	—
543	Q₈	—	2	4	no	—
544	Q₈	—	2	4	no	—
545	Q₈	—	4	4	no	—
546	Q₈	C₂²×D₄×C₂₃	1	4	yes	—
547	D₄	—	2	8	no	—
548	D₄	—	2	4	no	—
549	D₄	—	2	2	no	—
550	D₄	—	2	4	no	—
551	D₄	—	2	4	no	—
552	D₄	—	4	16	no	—
553	D₄	—	4	16	no	—
554	D₄	—	4	8	no	—
555	D₄	—	4	4	no	—
556	D₄	—	8	4	no	—
557	D₄	—	8	8	no	—
558	D₄	C₂²×D₄×C₂₃	1	4	yes	—
559	C₂×C₄	—	2	2	no	—
560	C₂×C₄	—	2	2	no	—
561	C₂×C₄	—	2	4	no	—
562	C₂×C₄	—	2	4	no	—
563	C₂×C₄	—	2	4	no	—
564	C₂×C₄	—	2	4	no	—
565	C₂×C₄	—	2	8	no	—
566	C₂×C₄	—	2	2	no	—
567	C₂×C₄	—	2	2	no	—
568	C₂×C₄	—	2	4	no	—
569	C₂×C₄	—	4	16	no	—
570	C₂×C₄	—	4	16	no	—
571	C₂×C₄	—	4	2	no	—
572	C₂×C₄	—	4	2	no	—
573	C₂×C₄	—	4	4	no	—
574	C₂×C₄	—	4	8	no	—
575	C₂×C₄	C₂₃×#135	1	4	yes	complemented
576	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
577	C₂×C₄	C₂²×D₄×C₂₃	1	4	yes	—
578	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
579	C₂×C₄	C₂²×D₄×C₂₃	1	2	yes	—
580	C₂×C₄	C₂²×D₄×C₂₃	1	1	yes	characteristic
581	C₂×C₄	C₂₃×H(5,𝔽₂)	1	2	yes	—
582	C₂×C₄	C₂⁵×C₂₃	1	1	yes	characteristic
583	C₈	—	2	4	no	—
584	C₈	—	2	2	no	—
585	C₈	C₂²×D₄×C₂₃	1	4	yes	—
586	C₂²	—	2	4	no	—
587	C₂²	—	2	4	no	—
588	C₂²	—	2	4	no	—
589	C₂²	—	2	4	no	—
590	C₂²	—	4	8	no	—
591	C₂²	—	4	8	no	—
592	C₂²	—	4	2	no	—
593	C₂²	—	4	2	no	—
594	C₂²	—	4	4	no	—
595	C₂²	—	4	4	no	—
596	C₂²	—	4	4	no	—
597	C₂²	—	8	8	no	—
598	C₂²	D₄²×C₂₃	1	2	yes	—
599	C₂²	C₂×C₂₃×#135	1	2	yes	—
600	C₂²	C₂₃×#577	1	2	yes	—
601	C₂²	C₂³×D₄×C₂₃	1	1	yes	characteristic
602	C₄	—	2	4	no	—
603	C₄	—	2	4	no	—
604	C₄	—	2	2	no	—
605	C₄	—	4	4	no	—
606	C₄	—	4	2	no	—
607	C₄	—	4	2	no	—
608	C₄	D₄²×C₂₃	1	2	yes	—
609	C₄	C₂×C₂₃×#135	1	4	yes	complemented
610	C₄	C₂³×D₄×C₂₃	1	1	yes	characteristic
611	C₄	C₂×C₂₃×H(5,𝔽₂)	1	1	yes	characteristic
612	C₂	—	4	4	no	—
613	C₂	—	4	4	no	—
614	C₂	—	8	4	no	—
615	C₂	C₂₃×#3284	1	4	yes	direct product factor
616	C₂	C₂×D₄²×C₂₃	1	1	yes	characteristic
617	C₂	C₂²×C₂₃×#135	1	1	yes	characteristic
618	C₂	C₂×C₂₃×#577	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₂²×C₂₃×#135	C₂	semidirect
2. C₂×C₂₃×#550	C₂	semidirect
3. C₂×C₂₃×#547	C₂	semidirect
4. C₂₃×#3284	C₂	direct (N × H)
5. C₂×C₂₃×#492	C₂	semidirect
6. C₂×C₂₃×#465	C₂	semidirect
7. C₂×C₂₃×#462	C₂	semidirect
8. C₂×C₂₃×#459	C₂	semidirect
9. C₂×C₂₃×#452	C₂	semidirect
10. C₂×C₂₃×#449	C₂	semidirect
11. C₂×C₄×SD₁₆×C₂₃	C₂	semidirect
12. C₂×C₈×D₄×C₂₃	C₂	semidirect
16. C₂×C₂₃×#135	C₂²	semidirect
24. C₂₃×#550	C₂²	semidirect
26. C₂₃×#547	C₂²	semidirect
40. C₂₃×#492	C₂²	semidirect
41. C₂₃×#465	C₂²	semidirect
42. C₂₃×#462	C₂²	semidirect
43. C₂₃×#459	C₂²	semidirect
44. C₂₃×#452	C₂²	semidirect
45. C₂₃×#449	C₂²	semidirect
46. C₄×SD₁₆×C₂₃	C₂²	semidirect
47. C₈×D₄×C₂₃	C₂²	semidirect
83. C₂×SD₁₆×C₂₃	D₄	semidirect
193. SD₁₆×C₂₃	C₂×D₄	semidirect
218. C₂×#3284	C₂₃	direct (N × H)
266. C₂×C₄×C₂₃	#135	semidirect
277. C₂²×#135	C₂×C₂₃	semidirect
278. C₂×#550	C₂×C₂₃	semidirect
279. C₂×#547	C₂×C₂₃	semidirect
280. #3284	C₂×C₂₃	direct (N × H)
281. C₂×#492	C₂×C₂₃	semidirect
282. C₂×#465	C₂×C₂₃	semidirect
283. C₂×#462	C₂×C₂₃	semidirect
284. C₂×#459	C₂×C₂₃	semidirect
285. C₂×#452	C₂×C₂₃	semidirect
286. C₂×#449	C₂×C₂₃	semidirect
287. C₂×C₄×SD₁₆	C₂×C₂₃	semidirect
288. C₂×C₈×D₄	C₂×C₂₃	semidirect
312. C₄×C₂₃	C₂×#135	semidirect
318. C₂×#135	C₂²×C₂₃	semidirect
326. #550	C₂²×C₂₃	semidirect
328. #547	C₂²×C₂₃	semidirect
342. #492	C₂²×C₂₃	semidirect
343. #465	C₂²×C₂₃	semidirect
344. #462	C₂²×C₂₃	semidirect
345. #459	C₂²×C₂₃	semidirect
346. #452	C₂²×C₂₃	semidirect
347. #449	C₂²×C₂₃	semidirect
348. C₄×SD₁₆	C₂²×C₂₃	semidirect
349. C₈×D₄	C₂²×C₂₃	semidirect
361. C₂×C₂₃	#3284	direct (N × H)
392. C₂×SD₁₆	D₄×C₂₃	semidirect
444. C₂₃	C₂×#3284	direct (N × H)
503. SD₁₆	C₂×D₄×C₂₃	semidirect
575. C₂×C₄	C₂₃×#135	semidirect
609. C₄	C₂×C₂₃×#135	semidirect
615. C₂	C₂₃×#3284	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₂₃×#135	2
C₂×C₂₃×#550	2
C₂×C₂₃×#547	2
C₂₃×#3284	2
C₂×C₂₃×#492	2
C₂×C₂₃×#465	2
C₂×C₂₃×#462	2
C₂×C₂₃×#459	2
C₂×C₂₃×#452	2
C₂×C₂₃×#449	2
C₂×C₄×SD₁₆×C₂₃	2
C₂×C₈×D₄×C₂₃	2
C₂×#3284	23

Subgroups — C2×C23×#3284 #19208670

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₂₃×#3284 #19208670