Subgroups — #14908353

Order 2816.

642 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10622517	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#10622514	C₂	1	1	yes	maximal, characteristic, complemented
3	#13078337	C₂	1	2	yes	maximal, complemented
4	#13070747	C₂	1	4	yes	maximal, complemented
5	#13070742	C₂	1	2	yes	maximal, complemented
6	#13070737	C₂	1	2	yes	maximal, complemented
7	#13070363	C₂	1	2	yes	maximal, complemented
8	C₂×C₁₁×#368	C₂	1	1	yes	maximal, characteristic, complemented
9	#10623279	—	2	2	no	—
10	C₂×#66957	C₂²	1	2	yes	—
11	C₂×#66954	C₂²	1	1	yes	characteristic
12	C₂²×C₁₁×C₁₆	C₂²	1	1	yes	characteristic
13	C₂×C₁₁×#108	C₂²	1	2	yes	—
14	#10622745	—	2	2	no	—
15	C₂×#66876	C₂²	1	1	yes	characteristic
16	C₂×Dic₈₈	C₂²	1	1	yes	characteristic, complemented
17	#10622517	—	2	1	no	—
18	#10622517	—	2	1	no	—
19	#10622517	—	2	1	no	—
20	#10622517	—	2	1	no	—
21	C₂×#66864	C₂²	1	2	yes	complemented
22	C₂×D₁₇₆	C₂²	1	1	yes	characteristic, complemented
23	#10622514	—	2	1	no	—
24	#10622514	—	2	1	no	—
25	#10622514	C₂²	1	2	yes	—
26	#10622514	C₂²	1	2	yes	—
27	C₂×#66862	—	2	1	no	—
28	C₂×C₁₆×D₁₁	—	2	1	no	—
29	#10622511	—	2	4	no	—
30	#10622510	C₂²	1	4	yes	—
31	#10622509	C₂²	1	4	yes	—
32	C₁₁×#368	C₂²	1	2	yes	complemented
33	C₁₁×#368	C₂²	1	2	yes	complemented
34	#10622134	C₂²	1	4	yes	complemented
35	#10622133	C₂²	1	4	yes	complemented
36	C₂×#4339	—	2	2	no	—
37	C₂×#4339	D₄	1	1	yes	characteristic
38	C₂×C₁₁×M₁₆	—	2	2	no	—
39	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
40	C₂×#4318	D₄	1	1	yes	characteristic
41	#66962	—	2	4	no	—
42	#66962	—	4	2	no	—
43	#66961	—	2	4	no	—
44	#66961	—	4	2	no	—
45	C₂×Dic₄₄	C₂³	1	2	yes	—
46	#66957	—	2	2	no	—
47	#66957	—	2	2	no	—
48	#66957	—	2	2	no	—
49	#66957	—	2	2	no	—
50	#66957	—	4	4	no	—
51	C₂×D₈₈	C₂³	1	2	yes	—
52	C₂×#4314	—	2	2	no	—
53	#66954	—	2	1	no	—
54	#66954	—	2	1	no	—
55	#66954	D₄	1	2	yes	—
56	#66954	C₂³	1	2	yes	—
57	C₂×#4313	—	2	1	no	—
58	C₂×C₈×D₁₁	—	2	1	no	—
59	C₂×C₁₁×C₁₆	D₄	1	1	yes	characteristic
60	C₂×C₁₁×C₁₆	D₄	1	1	yes	characteristic
61	C₂×C₁₁×C₁₆	D₄	1	1	yes	characteristic
62	C₂×C₁₁×C₁₆	D₄	1	1	yes	characteristic
63	C₂×C₁₁×C₁₆	C₂³	1	1	yes	characteristic
64	C₂×C₁₁×C₁₆	C₂³	1	1	yes	characteristic
65	C₁₁×#108	—	2	2	no	—
66	C₁₁×#108	C₂³	1	4	yes	—
67	#66888	—	2	4	no	—
68	#66887	—	2	4	no	—
69	#66876	—	2	1	no	—
70	#66876	C₂³	1	2	yes	—
71	C₂×C₁₁⋊C₁₆	—	2	1	no	—
72	Dic₈₈	—	2	1	no	—
73	Dic₈₈	—	2	1	no	—
74	#66864	—	2	2	no	—
75	#66864	—	2	2	no	—
76	D₁₇₆	—	2	1	no	—
77	D₁₇₆	—	2	1	no	—
78	#66862	—	2	2	no	—
79	#66862	—	2	2	no	—
80	C₁₆×D₁₁	—	2	2	no	—
81	C₁₆×D₁₁	—	2	2	no	—
82	#33617	—	11	1	no	maximal, Sylow
83	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
84	C₂×C₂²⋊D₁₁	—	2	1	no	—
85	C₂×C₂²⋊D₁₁	—	4	2	no	—
86	#4339	—	2	1	no	—
87	#4339	—	2	1	no	—
88	#4339	—	4	2	no	—
89	#4339	—	4	2	no	—
90	#4339	—	4	2	no	—
91	#4339	—	4	2	no	—
92	#4339	D₈	1	2	yes	—
93	#4339	C₂×D₄	1	2	yes	—
94	C₂×D₄₄	—	2	2	no	—
95	C₂×D₄₄	D₈	1	1	yes	characteristic
96	C₂×C₄×D₁₁	—	2	1	no	—
97	C₂×C₄×D₁₁	—	4	2	no	—
98	C₂×Dic₂₂	—	2	2	no	—
99	C₂×Dic₂₂	D₈	1	1	yes	characteristic
100	C₁₁×M₁₆	—	2	4	no	—
101	C₁₁×M₁₆	—	4	2	no	—
102	C₂×C₈×C₁₁	—	4	2	no	—
103	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
104	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₁₁	#41	1	1	yes	characteristic
107	C₂×C₈×C₁₁	#41	1	1	yes	characteristic
108	C₂×C₈×C₁₁	C₂⁴	1	1	yes	characteristic
109	#4318	—	2	1	no	—
110	#4318	C₂×D₄	1	2	yes	—
111	C₂×C₁₁⋊C₈	—	2	1	no	—
112	Dic₄₄	—	2	2	no	—
113	Dic₄₄	—	2	2	no	—
114	Dic₄₄	—	4	4	no	—
115	D₈₈	—	2	2	no	—
116	D₈₈	—	2	2	no	—
117	D₈₈	—	4	4	no	—
118	#4314	—	4	4	no	—
119	#4314	—	4	4	no	—
120	#4314	—	4	2	no	—
121	#4314	—	4	2	no	—
122	#4313	—	2	2	no	—
123	#4313	—	2	2	no	—
124	C₈×D₁₁	—	2	2	no	—
125	C₈×D₁₁	—	2	2	no	—
126	C₁₁×C₁₆	C₂×D₄	1	1	yes	characteristic
127	C₁₁×C₁₆	C₂×D₄	1	1	yes	characteristic
128	C₁₁×C₁₆	C₂×D₄	1	1	yes	characteristic
129	C₁₁×C₁₆	C₂×D₄	1	1	yes	characteristic
130	C₁₁⋊C₁₆	—	2	2	no	—
131	C₂×#508	—	11	1	no	—
132	C₂×#504	—	11	1	no	—
133	#3067	—	11	2	no	—
134	#2223	—	11	4	no	—
135	#2222	—	11	2	no	—
136	#2221	—	11	2	no	—
137	C₂×#368	D₁₁	1	1	yes	characteristic, complemented
138	#2148	—	11	2	no	—
139	C₂³×C₁₁	#135	1	1	yes	characteristic
140	C₂²×D₁₁	—	2	1	no	—
141	C₂²×D₁₁	—	4	2	no	—
142	C₂×C₄×C₁₁	#121	1	1	yes	characteristic
143	C₂×C₄×C₁₁	#121	1	1	yes	characteristic
144	C₂×C₄×C₁₁	C₂×D₈	1	1	yes	characteristic
145	C₂×C₄×C₁₁	C₂×D₈	1	1	yes	characteristic
146	C₂×C₄×C₁₁	#135	1	1	yes	characteristic
147	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
148	C₂²⋊D₁₁	—	2	2	no	—
149	C₂²⋊D₁₁	—	2	2	no	—
150	C₂²⋊D₁₁	—	8	2	no	—
151	C₂²⋊D₁₁	—	8	2	no	—
152	C₂×Dic₁₁	—	2	1	no	—
153	C₂×Dic₁₁	—	4	2	no	—
154	D₄₄	—	2	1	no	—
155	D₄₄	—	4	2	no	—
156	D₄₄	—	4	2	no	—
157	D₄₄	C₂×D₈	1	2	yes	—
158	C₄×D₁₁	—	2	2	no	—
159	C₄×D₁₁	—	2	2	no	—
160	C₄×D₁₁	—	8	2	no	—
161	C₄×D₁₁	—	8	2	no	—
162	Dic₂₂	—	2	1	no	—
163	Dic₂₂	—	4	2	no	—
164	Dic₂₂	—	4	2	no	—
165	Dic₂₂	C₂×D₈	1	2	yes	—
166	C₈×C₁₁	—	4	4	no	—
167	C₈×C₁₁	#121	1	1	yes	characteristic
168	C₈×C₁₁	#121	1	1	yes	characteristic
169	C₈×C₁₁	C₂²×D₄	1	1	yes	characteristic
170	C₈×C₁₁	C₂×#41	1	1	yes	characteristic
171	C₁₁⋊C₈	—	2	2	no	—
172	#575	—	22	2	no	—
173	C₂×#135	—	11	2	no	—
174	C₂×#131	—	11	1	no	—
175	#508	—	22	1	no	—
176	#508	—	22	1	no	—
177	#508	—	22	1	no	—
178	#508	—	22	1	no	—
179	C₂×Q₃₂	—	11	1	no	—
180	C₂×SD₃₂	—	11	2	no	—
181	C₂×D₁₆	—	11	1	no	—
182	#504	—	11	2	no	—
183	#504	—	11	2	no	—
184	#504	—	22	1	no	—
185	#504	—	22	1	no	—
186	C₂×M₃₂	—	11	1	no	—
187	C₂×M₃₂	—	22	1	no	—
188	C₂²×C₁₆	—	22	1	no	—
189	C₂²×C₁₆	C₂×D₁₁	1	1	yes	characteristic
190	#473	—	11	4	no	—
191	#472	—	11	4	no	—
192	#471	—	22	4	no	—
193	C₂×#108	C₂×D₁₁	1	2	yes	—
194	#413	—	22	2	no	—
195	#368	C₂×D₁₁	1	2	yes	complemented
196	#368	C₂×D₁₁	1	2	yes	complemented
197	#362	—	11	4	no	—
198	#361	—	11	4	no	—
199	C₂²×C₁₁	—	4	1	no	—
200	C₂²×C₁₁	#466	1	1	yes	characteristic
201	C₂²×C₁₁	#466	1	1	yes	characteristic
202	C₂²×C₁₁	C₂×#135	1	1	yes	characteristic
203	C₂×D₁₁	—	2	2	no	—
204	C₂×D₁₁	—	4	2	no	—
205	C₂×D₁₁	—	8	2	no	—
206	C₂×D₁₁	—	16	2	no	—
207	C₄×C₁₁	#466	1	1	yes	characteristic
208	C₄×C₁₁	#466	1	1	yes	characteristic
209	C₄×C₁₁	C₂×#121	1	1	yes	characteristic
210	C₄×C₁₁	C₂²×D₈	1	1	yes	characteristic
211	Dic₁₁	—	2	2	no	—
212	Dic₁₁	—	8	2	no	—
213	C₂×#41	—	11	1	no	—
214	C₂×#41	—	22	2	no	—
215	#137	—	22	4	no	—
216	#137	—	44	2	no	—
217	AGL(1,ℤ/8)	—	22	4	no	—
218	AGL(1,ℤ/8)	—	44	2	no	—
219	#135	—	22	2	no	—
220	#135	—	22	2	no	—
221	#135	—	22	2	no	—
222	#135	—	22	2	no	—
223	#135	—	44	4	no	—
224	C₂×Q₁₆	—	11	2	no	—
225	C₂×SD₁₆	—	22	2	no	—
226	C₂×D₈	—	11	2	no	—
227	#131	—	11	2	no	—
228	#131	—	11	2	no	—
229	#131	—	22	1	no	—
230	#131	—	22	1	no	—
231	C₂×M₁₆	—	2	2	no	—
232	C₂×M₁₆	—	11	1	no	—
233	C₂×M₁₆	—	22	1	no	—
234	C₂²×C₈	—	22	1	no	—
235	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
236	Q₃₂	—	22	1	no	—
237	Q₃₂	—	22	1	no	—
238	SD₃₂	—	22	2	no	—
239	SD₃₂	—	22	2	no	—
240	D₁₆	—	22	1	no	—
241	D₁₆	—	22	1	no	—
242	M₃₂	—	11	2	no	—
243	M₃₂	—	22	1	no	—
244	M₃₂	—	22	2	no	—
245	M₃₂	—	22	2	no	—
246	C₂×C₁₆	—	22	1	no	—
247	C₂×C₁₆	—	22	2	no	—
248	C₂×C₁₆	—	22	2	no	—
249	C₂×C₁₆	D₄₄	1	1	yes	characteristic
250	C₂×C₁₆	D₄₄	1	1	yes	characteristic
251	C₂×C₁₆	D₄₄	1	1	yes	characteristic
252	C₂×C₁₆	D₄₄	1	1	yes	characteristic
253	C₂×C₁₆	C₂²×D₁₁	1	1	yes	characteristic
254	C₂×C₁₆	C₂²×D₁₁	1	1	yes	characteristic
255	#108	—	2	2	no	—
256	#108	C₂²×D₁₁	1	4	yes	—
257	#101	—	22	4	no	—
258	#100	—	22	4	no	—
259	C₂×C₁₁	—	2	1	no	—
260	C₂×C₁₁	—	2	1	no	—
261	C₂×C₁₁	—	2	1	no	—
262	C₂×C₁₁	C₂×#466	1	1	yes	characteristic
263	D₁₁	—	4	2	no	—
264	D₁₁	—	16	2	no	—
265	#41	—	11	2	no	—
266	#41	—	11	2	no	—
267	#41	—	22	1	no	—
268	#41	—	22	1	no	—
269	#41	—	44	2	no	—
270	#41	—	44	2	no	—
271	#41	—	44	2	no	—
272	#41	—	44	2	no	—
273	C₂×Q₈	—	11	1	no	—
274	C₂×Q₈	—	22	2	no	—
275	C₂×D₄	—	11	1	no	—
276	C₂×D₄	—	22	1	no	—
277	C₂×D₄	—	22	2	no	—
278	C₂×D₄	—	44	2	no	—
279	C₂²×C₄	—	22	1	no	—
280	C₂²×C₄	—	44	2	no	—
281	C₂²×C₄	D₄×D₁₁	1	1	yes	characteristic
282	Q₁₆	—	22	2	no	—
283	Q₁₆	—	22	2	no	—
284	Q₁₆	—	44	4	no	—
285	SD₁₆	—	44	4	no	—
286	SD₁₆	—	44	4	no	—
287	SD₁₆	—	44	2	no	—
288	SD₁₆	—	44	2	no	—
289	D₈	—	22	2	no	—
290	D₈	—	22	2	no	—
291	D₈	—	44	4	no	—
292	M₁₆	—	2	4	no	—
293	M₁₆	—	4	2	no	—
294	M₁₆	—	11	2	no	—
295	M₁₆	—	22	1	no	—
296	M₁₆	—	22	2	no	—
297	M₁₆	—	22	2	no	—
298	C₂×C₈	—	4	2	no	—
299	C₂×C₈	—	22	1	no	—
300	C₂×C₈	—	22	2	no	—
301	C₂×C₈	—	22	2	no	—
302	C₂×C₈	C₂×D₄₄	1	1	yes	characteristic
303	C₂×C₈	C₂×D₄₄	1	1	yes	characteristic
304	C₂×C₈	D₄×D₁₁	1	1	yes	characteristic
305	C₂×C₈	#4343	1	1	yes	characteristic
306	C₂×C₈	#4343	1	1	yes	characteristic
307	C₂×C₈	C₂³×D₁₁	1	1	yes	characteristic
308	C₁₆	—	22	2	no	—
309	C₁₆	C₂×D₄₄	1	1	yes	characteristic
310	C₁₆	C₂×D₄₄	1	1	yes	characteristic
311	C₁₆	C₂×D₄₄	1	1	yes	characteristic
312	C₁₆	C₂×D₄₄	1	1	yes	characteristic
313	C₁₁	#33617	1	1	yes	characteristic, complemented, Sylow
314	C₂³	—	22	1	no	—
315	C₂³	—	44	2	no	—
316	C₂³	#66972	1	1	yes	characteristic
317	Q₈	—	11	2	no	—
318	Q₈	—	22	1	no	—
319	Q₈	—	44	2	no	—
320	Q₈	—	44	2	no	—
321	D₄	—	11	2	no	—
322	D₄	—	22	2	no	—
323	D₄	—	22	2	no	—
324	D₄	—	22	1	no	—
325	D₄	—	44	2	no	—
326	D₄	—	44	2	no	—
327	D₄	—	88	2	no	—
328	D₄	—	88	2	no	—
329	C₂×C₄	—	22	1	no	—
330	C₂×C₄	—	22	2	no	—
331	C₂×C₄	—	22	2	no	—
332	C₂×C₄	—	44	2	no	—
333	C₂×C₄	—	88	2	no	—
334	C₂×C₄	—	88	2	no	—
335	C₂×C₄	#66948	1	1	yes	characteristic
336	C₂×C₄	#66948	1	1	yes	characteristic
337	C₂×C₄	D₈×D₁₁	1	1	yes	characteristic
338	C₂×C₄	D₈×D₁₁	1	1	yes	characteristic
339	C₂×C₄	#66972	1	1	yes	characteristic
340	C₂×C₄	C₂×D₄×D₁₁	1	1	yes	characteristic
341	C₈	—	4	4	no	—
342	C₈	—	22	2	no	—
343	C₈	#66948	1	1	yes	characteristic
344	C₈	#66948	1	1	yes	characteristic
345	C₈	C₂²×D₄₄	1	1	yes	characteristic
346	C₈	C₂×#4343	1	1	yes	characteristic
347	C₂²	—	4	1	no	—
348	C₂²	—	22	2	no	—
349	C₂²	—	44	2	no	—
350	C₂²	—	88	2	no	—
351	C₂²	—	176	2	no	—
352	C₂²	#10622496	1	1	yes	characteristic
353	C₂²	#10622496	1	1	yes	characteristic
354	C₂²	C₂×#66972	1	1	yes	characteristic
355	C₄	—	22	2	no	—
356	C₄	—	88	2	no	—
357	C₄	#10622496	1	1	yes	characteristic
358	C₄	#10622496	1	1	yes	characteristic
359	C₄	C₂×#66948	1	1	yes	characteristic
360	C₄	C₂×D₈×D₁₁	1	1	yes	characteristic
361	C₂	—	2	1	no	—
362	C₂	—	2	1	no	—
363	C₂	—	2	1	no	—
364	C₂	—	44	2	no	—
365	C₂	—	176	2	no	—
366	C₂	C₂×#10622496	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#10622496
commutator, cosocle	C₂×C₈×C₁₁	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₁
Fitting	C₂×C₁₁×#368	C₂
socle	C₂×C₁₁	C₂×#466
hypercenter	C₂×#368	D₁₁
hypocenter	C₁₁	#33617
LCS	G ▷ C₂×C₈×C₁₁ ▷ C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#368

Complements

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

N	H ≅ G/N	type
1. C₂×#10622517	C₂	semidirect
2. C₂×#10622514	C₂	semidirect
3. #13078337	C₂	semidirect
4. #13070747	C₂	semidirect
5. #13070742	C₂	semidirect
6. #13070737	C₂	semidirect
7. #13070363	C₂	semidirect
8. C₂×C₁₁×#368	C₂	semidirect
16. C₂×Dic₈₈	C₂²	semidirect
21. C₂×#66864	C₂²	semidirect
22. C₂×D₁₇₆	C₂²	semidirect
32. C₁₁×#368	C₂²	semidirect
33. C₁₁×#368	C₂²	semidirect
34. #10622134	C₂²	semidirect
35. #10622133	C₂²	semidirect
137. C₂×#368	D₁₁	semidirect
195. #368	C₂×D₁₁	semidirect
196. #368	C₂×D₁₁	semidirect
313. C₁₁	#33617	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10622517	2
C₂×#10622514	2
#13078337	2
#13070747	2
#13070742	2
#13070737	2
#13070363	2
C₂×C₁₁×#368	2
#33617	11