Subgroups — #15683016

Order 2944.

548 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10625079	C₂	1	1	yes	maximal, characteristic, complemented
2	#13092928	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#10625101	C₂	1	1	yes	maximal, characteristic, complemented
4	#13092670	C₂	1	2	yes	maximal, complemented
5	#13092663	C₂	1	2	yes	maximal, complemented
6	#13092653	C₂	1	2	yes	maximal, complemented
7	#13092646	C₂	1	2	yes	maximal, complemented
8	#13092459	C₂	1	1	yes	maximal, characteristic, complemented
9	#13092423	C₂	1	1	yes	maximal, characteristic, complemented
10	#13092419	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂²×M₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
13	C₂²×C₂²⋊D₂₃	—	2	1	no	—
14	C₂×#67288	C₂²	1	1	yes	characteristic
15	C₂²×D₉₂	C₂²	1	1	yes	characteristic
16	C₂×M₁₆×C₂₃	—	2	4	no	—
17	C₂×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
18	C₂×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
19	C₂×M₁₆×C₂₃	C₂²	1	2	yes	complemented
20	C₂²×C₈×C₂₃	C₂²	1	2	yes	complemented
21	#10625124	—	2	1	no	—
22	#10625101	—	2	1	no	—
23	#10625101	—	2	1	no	—
24	#10625101	C₂²	1	2	yes	—
25	#10625101	C₂²	1	2	yes	—
26	#10625101	C₂²	1	1	yes	characteristic, complemented
27	#10625101	C₂²	1	1	yes	characteristic, complemented
28	#10625100	—	2	1	no	—
29	C₄×C₂²⋊D₂₃	—	2	1	no	—
30	C₄×C₂²⋊D₂₃	—	2	1	no	—
31	C₂×(C₂×D₂₃)⋊C₄	—	2	1	no	—
32	#10625097	C₂²	1	1	yes	characteristic
33	C₂×C₄⋊Dic₂₃	C₂²	1	1	yes	characteristic
34	#10625095	C₂²	1	1	yes	characteristic, complemented
35	#10625095	C₂²	1	1	yes	characteristic, complemented
36	#10625079	—	2	4	no	—
37	C₂×D₁₈₄	C₂²	1	2	yes	complemented
38	C₂×#67263	C₂²	1	2	yes	complemented
39	#10625003	C₂²	1	2	yes	complemented
40	#10625003	C₂²	1	2	yes	complemented
41	#10625003	C₂²	1	2	yes	complemented
42	#10625000	C₂²	1	2	yes	complemented
43	#10624999	C₂²	1	2	yes	complemented
44	#10624998	C₂²	1	2	yes	complemented
45	C₂⁴×C₂₃	D₄	1	1	yes	characteristic
46	C₂³×D₂₃	—	2	1	no	—
47	C₂²×C₄×C₂₃	—	2	2	no	—
48	C₂²×C₄×C₂₃	—	2	2	no	—
49	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
50	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
51	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
52	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
54	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
55	C₂×C₂²⋊D₂₃	—	2	1	no	—
56	C₂×C₂²⋊D₂₃	—	2	1	no	—
57	C₂×C₂²⋊D₂₃	—	2	1	no	—
58	C₂×C₂²⋊D₂₃	—	2	2	no	—
59	C₂×C₂²⋊D₂₃	—	2	2	no	—
60	C₂×C₂²⋊D₂₃	—	4	4	no	—
61	C₂²×Dic₂₃	—	2	1	no	—
62	#67288	—	2	2	no	—
63	#67288	—	2	2	no	—
64	C₂×D₉₂	—	2	2	no	—
65	C₂×D₉₂	—	2	2	no	—
66	C₂×D₉₂	—	2	1	no	—
67	C₂×D₉₂	C₂³	1	2	yes	—
68	C₂×D₉₂	C₂³	1	1	yes	characteristic
69	C₂×C₄×D₂₃	—	2	1	no	—
70	C₂×Dic₄₆	C₂³	1	1	yes	characteristic
71	M₁₆×C₂₃	—	2	2	no	—
72	M₁₆×C₂₃	—	2	4	no	—
73	M₁₆×C₂₃	D₄	1	4	yes	complemented
74	C₂×C₈×C₂₃	—	2	4	no	—
75	C₂×C₈×C₂₃	C₂³	1	2	yes	—
76	C₂×C₈×C₂₃	C₂³	1	2	yes	—
77	C₂²⋊Dic₂₃	—	2	1	no	—
78	C₂²⋊Dic₂₃	—	2	1	no	—
79	C₂²⋊Dic₂₃	—	2	1	no	—
80	(C₂×D₂₃)⋊C₄	—	2	2	no	—
81	(C₂×D₂₃)⋊C₄	—	2	2	no	—
82	(C₂×D₂₃)⋊C₄	—	2	1	no	—
83	(C₂×D₂₃)⋊C₄	—	2	1	no	—
84	C₄⋊Dic₂₃	—	2	1	no	—
85	C₄⋊Dic₂₃	C₂³	1	1	yes	characteristic
86	C₄⋊Dic₂₃	C₂³	1	1	yes	characteristic
87	C₄⋊Dic₂₃	C₂³	1	2	yes	—
88	Dic₂₃⋊C₄	—	2	1	no	—
89	Dic₂₃⋊C₄	—	2	1	no	—
90	C₄×Dic₂₃	—	2	1	no	—
91	D₁₈₄	—	2	4	no	—
92	#67263	—	2	4	no	—
93	C₂³×C₂₃	—	2	2	no	—
94	C₂³×C₂₃	—	2	2	no	—
95	C₂³×C₂₃	—	4	1	no	—
96	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
97	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
98	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
99	C₂²×D₂₃	—	2	1	no	—
100	C₂²×D₂₃	—	2	2	no	—
101	C₂²×D₂₃	—	4	2	no	—
102	C₂²×D₂₃	—	4	2	no	—
103	C₂²×D₂₃	—	4	2	no	—
104	C₂×C₄×C₂₃	—	2	2	no	—
105	C₂×C₄×C₂₃	—	2	2	no	—
106	C₂×C₄×C₂₃	—	2	2	no	—
107	C₂×C₄×C₂₃	—	2	1	no	—
108	C₂×C₄×C₂₃	—	2	1	no	—
109	C₂×C₄×C₂₃	—	2	1	no	—
110	C₂×C₄×C₂₃	—	2	1	no	—
111	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
112	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
113	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
114	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
115	C₂×C₄×C₂₃	#41	1	2	yes	—
116	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
117	C₂²⋊D₂₃	—	4	2	no	—
118	C₂²⋊D₂₃	—	4	4	no	—
119	C₂²⋊D₂₃	—	4	2	no	—
120	C₂²⋊D₂₃	—	4	2	no	—
121	C₂²⋊D₂₃	—	4	4	no	—
122	C₂×Dic₂₃	—	2	1	no	—
123	C₂×Dic₂₃	—	2	1	no	—
124	C₂×Dic₂₃	—	2	1	no	—
125	C₂×Dic₂₃	—	2	2	no	—
126	C₂×Dic₂₃	—	4	2	no	—
127	D₉₂	—	2	2	no	—
128	D₉₂	—	2	1	no	—
129	D₉₂	—	2	2	no	—
130	D₉₂	—	2	1	no	—
131	D₉₂	—	4	2	no	—
132	C₄×D₂₃	—	4	2	no	—
133	Dic₄₆	—	2	1	no	—
134	Dic₄₆	—	2	1	no	—
135	C₈×C₂₃	—	2	2	no	—
136	C₈×C₂₃	C₂×D₄	1	4	yes	complemented
137	#3055	—	23	1	no	maximal, Sylow
138	C₂²×C₂₃	—	2	1	no	—
139	C₂²×C₂₃	—	2	1	no	—
140	C₂²×C₂₃	—	2	1	no	—
141	C₂²×C₂₃	—	2	1	no	—
142	C₂²×C₂₃	—	2	1	no	—
143	C₂²×C₂₃	—	2	1	no	—
144	C₂²×C₂₃	—	2	1	no	—
145	C₂²×C₂₃	—	4	4	no	—
146	C₂²×C₂₃	#121	1	2	yes	—
147	C₂²×C₂₃	AGL(1,ℤ/8)	1	2	yes	complemented
148	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
149	C₂×D₂₃	—	4	1	no	—
150	C₂×D₂₃	—	4	2	no	—
151	C₂×D₂₃	—	4	2	no	—
152	C₂×D₂₃	—	4	1	no	—
153	C₂×D₂₃	—	4	1	no	—
154	C₂×D₂₃	—	4	2	no	—
155	C₂×D₂₃	—	8	4	no	—
156	C₄×C₂₃	—	2	1	no	—
157	C₄×C₂₃	—	2	1	no	—
158	C₄×C₂₃	#121	1	2	yes	—
159	C₄×C₂₃	C₂²×D₄	1	1	yes	characteristic
160	C₄×C₂₃	C₂×#41	1	1	yes	characteristic
161	Dic₂₃	—	4	1	no	—
162	Dic₂₃	—	4	2	no	—
163	Dic₂₃	—	4	1	no	—
164	Dic₂₃	—	4	1	no	—
165	C₂×AGL(1,ℤ/8)	—	23	1	no	—
166	C₂²×M₁₆	D₂₃	1	1	yes	characteristic, complemented
167	#525	—	23	1	no	—
168	C₂×#121	—	23	1	no	—
169	#469	—	23	2	no	—
170	#468	—	23	2	no	—
171	#466	—	23	2	no	—
172	#465	—	23	2	no	—
173	#428	—	23	1	no	—
174	#418	—	23	1	no	—
175	#417	—	23	1	no	—
176	C₂×C₂₃	—	2	2	no	—
177	C₂×C₂₃	—	2	2	no	—
178	C₂×C₂₃	—	4	1	no	—
179	C₂×C₂₃	C₂×#121	1	1	yes	characteristic
180	C₂×C₂₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
181	C₂×C₂₃	#575	1	1	yes	characteristic
182	D₂₃	—	8	1	no	—
183	D₂₃	—	8	2	no	—
184	C₂×#41	—	23	1	no	—
185	C₂²×D₄	—	23	1	no	—
186	C₂²×D₄	—	46	1	no	—
187	C₂³×C₄	C₂×D₂₃	1	1	yes	characteristic
188	AGL(1,ℤ/8)	—	46	4	no	—
189	C₂×SD₁₆	—	23	2	no	—
190	C₂×D₈	—	23	2	no	—
191	C₂×M₁₆	—	2	4	no	—
192	C₂×M₁₆	C₂×D₂₃	1	1	yes	characteristic
193	C₂×M₁₆	C₂×D₂₃	1	1	yes	characteristic
194	C₂×M₁₆	C₂×D₂₃	1	2	yes	complemented
195	C₂²×C₈	C₂×D₂₃	1	2	yes	complemented
196	#123	—	46	1	no	—
197	C₂²⋊Q₈	—	23	1	no	—
198	C₂²⋊Q₈	—	23	1	no	—
199	#121	—	23	2	no	—
200	#121	—	23	2	no	—
201	#121	—	23	1	no	—
202	#121	—	23	1	no	—
203	#121	—	46	1	no	—
204	#121	—	46	1	no	—
205	#120	—	46	1	no	—
206	C₄×D₄	—	46	1	no	—
207	C₄×D₄	—	46	1	no	—
208	#117	—	23	1	no	—
209	C₂×C₄⋊C₄	—	23	1	no	—
210	C₂×C₂²⋊C₄	—	46	1	no	—
211	#107	—	23	2	no	—
212	#106	—	23	2	no	—
213	#103	—	23	2	no	—
214	#102	—	23	2	no	—
215	#102	—	23	2	no	—
216	#102	—	23	2	no	—
217	C₂₃	#3055	1	1	yes	characteristic, complemented, Sylow
218	C₂⁴	—	46	1	no	—
219	C₂⁴	D₉₂	1	1	yes	characteristic
220	#41	—	46	2	no	—
221	#41	—	46	2	no	—
222	C₂×Q₈	—	23	1	no	—
223	C₂×D₄	—	23	1	no	—
224	C₂×D₄	—	23	2	no	—
225	C₂×D₄	—	46	1	no	—
226	C₂×D₄	—	46	1	no	—
227	C₂×D₄	—	46	1	no	—
228	C₂×D₄	—	46	2	no	—
229	C₂×D₄	—	46	2	no	—
230	C₂×D₄	—	46	1	no	—
231	C₂×D₄	—	46	2	no	—
232	C₂×D₄	—	46	2	no	—
233	C₂×D₄	—	92	4	no	—
234	C₂²×C₄	—	2	2	no	—
235	C₂²×C₄	—	2	2	no	—
236	C₂²×C₄	—	46	1	no	—
237	C₂²×C₄	—	46	1	no	—
238	C₂²×C₄	D₉₂	1	1	yes	characteristic
239	C₂²×C₄	D₉₂	1	1	yes	characteristic
240	C₂²×C₄	D₉₂	1	1	yes	characteristic
241	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
242	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
243	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
244	SD₁₆	—	46	4	no	—
245	D₈	—	46	4	no	—
246	M₁₆	—	2	2	no	—
247	M₁₆	—	2	4	no	—
248	M₁₆	C₂²⋊D₂₃	1	4	yes	complemented
249	C₂×C₈	—	2	4	no	—
250	C₂×C₈	C₂²×D₂₃	1	2	yes	—
251	C₂×C₈	C₂²×D₂₃	1	2	yes	—
252	C₄⋊C₄	—	23	1	no	—
253	C₄⋊C₄	—	23	1	no	—
254	C₄⋊C₄	—	23	2	no	—
255	C₄⋊C₄	—	46	1	no	—
256	C₄⋊C₄	—	46	1	no	—
257	C₄⋊C₄	—	46	1	no	—
258	C₂²⋊C₄	—	46	1	no	—
259	C₂²⋊C₄	—	46	1	no	—
260	C₂²⋊C₄	—	46	1	no	—
261	C₂²⋊C₄	—	46	2	no	—
262	C₂²⋊C₄	—	46	2	no	—
263	C₂²⋊C₄	—	46	1	no	—
264	C₂²⋊C₄	—	46	1	no	—
265	C₄²	—	46	1	no	—
266	C₂³	—	2	2	no	—
267	C₂³	—	2	2	no	—
268	C₂³	—	4	1	no	—
269	C₂³	—	46	1	no	—
270	C₂³	—	46	2	no	—
271	C₂³	—	92	2	no	—
272	C₂³	—	92	2	no	—
273	C₂³	—	92	2	no	—
274	C₂³	C₂×D₉₂	1	1	yes	characteristic
275	C₂³	C₂×D₉₂	1	1	yes	characteristic
276	C₂³	C₂×D₉₂	1	1	yes	characteristic
277	Q₈	—	46	1	no	—
278	Q₈	—	46	1	no	—
279	D₄	—	46	2	no	—
280	D₄	—	46	1	no	—
281	D₄	—	46	1	no	—
282	D₄	—	46	2	no	—
283	D₄	—	92	4	no	—
284	D₄	—	92	2	no	—
285	D₄	—	92	4	no	—
286	D₄	—	92	2	no	—
287	D₄	—	92	2	no	—
288	D₄	—	92	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	46	1	no	—
297	C₂×C₄	—	46	1	no	—
298	C₂×C₄	—	46	2	no	—
299	C₂×C₄	—	46	1	no	—
300	C₂×C₄	—	92	2	no	—
301	C₂×C₄	—	92	2	no	—
302	C₂×C₄	C₂×D₉₂	1	1	yes	characteristic
303	C₂×C₄	C₂×D₉₂	1	1	yes	characteristic
304	C₂×C₄	C₂×D₉₂	1	1	yes	characteristic
305	C₂×C₄	#67288	1	2	yes	—
306	C₂×C₄	C₂×C₂²⋊D₂₃	1	2	yes	—
307	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
308	C₈	—	2	2	no	—
309	C₈	C₂×C₂²⋊D₂₃	1	4	yes	complemented
310	C₂²	—	2	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	2	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	4	4	no	—
318	C₂²	—	92	1	no	—
319	C₂²	—	92	2	no	—
320	C₂²	—	92	2	no	—
321	C₂²	—	92	1	no	—
322	C₂²	—	92	1	no	—
323	C₂²	—	92	2	no	—
324	C₂²	—	184	4	no	—
325	C₂²	#10625079	1	2	yes	complemented
326	C₂²	#10625101	1	2	yes	—
327	C₂²	C₂²×D₉₂	1	1	yes	characteristic
328	C₄	—	2	1	no	—
329	C₄	—	2	1	no	—
330	C₄	—	92	1	no	—
331	C₄	—	92	2	no	—
332	C₄	—	92	1	no	—
333	C₄	—	92	1	no	—
334	C₄	#10625101	1	2	yes	—
335	C₄	C₂×#67288	1	1	yes	characteristic
336	C₄	C₂²×C₂²⋊D₂₃	1	1	yes	characteristic
337	C₂	—	2	2	no	—
338	C₂	—	2	2	no	—
339	C₂	—	4	1	no	—
340	C₂	—	184	2	no	—
341	C₂	—	184	1	no	—
342	C₂	C₂×#10625101	1	1	yes	characteristic
343	C₂	C₂×#10625079	1	1	yes	characteristic
344	C₂	#13093217	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₉₂
commutator, cosocle	C₂×C₄×C₂₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₃
Fitting	C₂²×M₁₆×C₂₃	C₂
socle	C₂²×C₂₃	C₂²×D₄
hypercenter	C₂²×M₁₆	D₂₃
hypocenter	C₂₃	#3055
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×#10625079	C₂	semidirect
2. #13092928	C₂	semidirect
3. C₂×#10625101	C₂	semidirect
4. #13092670	C₂	semidirect
5. #13092663	C₂	semidirect
6. #13092653	C₂	semidirect
7. #13092646	C₂	semidirect
8. #13092459	C₂	semidirect
9. #13092423	C₂	semidirect
10. #13092419	C₂	semidirect
11. C₂²×M₁₆×C₂₃	C₂	semidirect
19. C₂×M₁₆×C₂₃	C₂²	semidirect
20. C₂²×C₈×C₂₃	C₂²	semidirect
26. #10625101	C₂²	semidirect
27. #10625101	C₂²	semidirect
34. #10625095	C₂²	semidirect
35. #10625095	C₂²	semidirect
37. C₂×D₁₈₄	C₂²	semidirect
38. C₂×#67263	C₂²	semidirect
39. #10625003	C₂²	semidirect
40. #10625003	C₂²	semidirect
41. #10625003	C₂²	semidirect
42. #10625000	C₂²	semidirect
43. #10624999	C₂²	semidirect
44. #10624998	C₂²	semidirect
73. M₁₆×C₂₃	D₄	semidirect
136. C₈×C₂₃	C₂×D₄	semidirect
147. C₂²×C₂₃	AGL(1,ℤ/8)	semidirect
166. C₂²×M₁₆	D₂₃	semidirect
194. C₂×M₁₆	C₂×D₂₃	semidirect
195. C₂²×C₈	C₂×D₂₃	semidirect
217. C₂₃	#3055	semidirect
248. M₁₆	C₂²⋊D₂₃	semidirect
309. C₈	C₂×C₂²⋊D₂₃	semidirect
325. C₂²	#10625079	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10625079	2
#13092928	2
C₂×#10625101	2
#13092670	2
#13092663	2
#13092653	2
#13092646	2
#13092459	2
#13092423	2
#13092419	2
C₂²×M₁₆×C₂₃	2
#3055	23