Subgroups — #14911186

Order 2816.

494 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13081019	C₂	1	2	yes	maximal, complemented
2	#13070895	C₂	1	1	yes	maximal, characteristic, complemented
3	#13070808	C₂	1	2	yes	maximal, complemented
4	#13070802	C₂	1	2	yes	maximal, complemented
5	#13070774	C₂	1	2	yes	maximal, complemented
6	#13070702	C₂	1	1	yes	maximal, characteristic, complemented
7	#13070702	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×D₁₇₆	C₂	1	1	yes	maximal, characteristic, complemented
9	#13070400	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#10622142	C₂	1	1	yes	maximal, characteristic, complemented
11	D₄×C₁₁×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
12	#10623096	—	2	2	no	—
13	C₂²×C₁₁×C₁₆	C₂²	1	1	yes	characteristic
14	C₂²×C₁₁×C₁₆	C₂²	1	1	yes	characteristic
15	C₈×D₄×C₁₁	C₂²	1	1	yes	characteristic
16	#10622728	C₂²	1	2	yes	—
17	#10622728	C₂²	1	2	yes	—
18	#10622719	C₂²	1	2	yes	—
19	C₂×#66882	C₂²	1	2	yes	—
20	C₂×D₁₇₆	C₂²	1	1	yes	characteristic
21	#10622460	—	2	2	no	—
22	#10622350	—	2	2	no	—
23	#10622348	—	2	2	no	—
24	C₄×D₈₈	C₂²	1	2	yes	—
25	#10622296	C₂²	1	2	yes	—
26	C₁₁×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
27	C₁₁×C₂²⋊C₁₆	C₂²	1	1	yes	characteristic
28	C₁₁×C₂²⋊C₁₆	C₂²	1	1	yes	characteristic
29	C₄×C₁₁×C₁₆	C₂²	1	1	yes	characteristic
30	#10622147	C₂²	1	2	yes	—
31	#10622147	C₂²	1	2	yes	—
32	#10622147	C₂²	1	2	yes	—
33	#10622143	C₂²	1	2	yes	—
34	#10622143	C₂²	1	2	yes	—
35	#10622142	C₂²	1	1	yes	characteristic
36	#10622142	C₂²	1	1	yes	characteristic
37	#10622142	C₂²	1	1	yes	characteristic
38	#10622142	C₂²	1	2	yes	—
39	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
40	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
41	C₄×D₄×C₁₁	D₄	1	1	yes	characteristic
42	#66992	—	4	2	no	—
43	#66983	—	2	2	no	—
44	#66983	—	2	2	no	—
45	#66979	—	2	2	no	—
46	C₂×C₄⋊Dic₁₁	—	2	2	no	—
47	C₂×D₈₈	C₂³	1	2	yes	—
48	#66951	—	4	2	no	—
49	#66936	—	4	2	no	—
50	(C₂²⋊D₁₁)⋊C₄	—	4	2	no	—
51	C₄×D₄₄	—	2	2	no	—
52	#66923	—	2	2	no	—
53	C₂×C₁₁×C₁₆	—	2	2	no	—
54	C₂×C₁₁×C₁₆	—	2	2	no	—
55	C₂×C₁₁×C₁₆	C₂³	1	1	yes	characteristic
56	C₂×C₁₁×C₁₆	C₂³	1	1	yes	characteristic
57	C₂×C₁₁×C₁₆	C₂³	1	1	yes	characteristic
58	C₂×C₁₁×C₁₆	C₂³	1	1	yes	characteristic
59	C₁₁×C₄⋊C₈	D₄	1	1	yes	characteristic
60	C₁₁×C₂²⋊C₈	D₄	1	1	yes	characteristic
61	C₁₁×C₂²⋊C₈	D₄	1	1	yes	characteristic
62	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
63	#66885	—	2	2	no	—
64	#66885	—	2	2	no	—
65	#66885	—	2	2	no	—
66	#66882	—	2	2	no	—
67	#66882	C₂³	1	2	yes	—
68	#66882	C₂³	1	2	yes	—
69	#66882	C₂³	1	2	yes	—
70	#66881	—	2	2	no	—
71	#66881	—	2	2	no	—
72	D₁₇₆	—	2	2	no	—
73	#33906	—	11	1	no	maximal, Sylow
74	C₂×D₄×C₁₁	D₈	1	1	yes	characteristic
75	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
77	C₂×C₂²⋊D₁₁	—	4	2	no	—
78	C₂×C₂²⋊D₁₁	—	4	2	no	—
79	C₂²×Dic₁₁	—	4	2	no	—
80	C₂×D₄₄	—	2	2	no	—
81	C₂×C₄×D₁₁	—	4	2	no	—
82	C₂×C₈×C₁₁	—	2	1	no	—
83	C₂×C₈×C₁₁	—	2	1	no	—
84	C₂×C₈×C₁₁	—	2	1	no	—
85	C₂×C₈×C₁₁	—	2	1	no	—
86	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
87	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₁₁	C₂⁴	1	1	yes	characteristic
90	C₁₁×C₄⋊C₄	D₈	1	1	yes	characteristic
91	C₁₁×C₂²⋊C₄	D₈	1	1	yes	characteristic
92	C₁₁×C₂²⋊C₄	D₈	1	1	yes	characteristic
93	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
94	C₂²⋊Dic₁₁	—	4	2	no	—
95	(C₂×D₁₁)⋊C₄	—	4	2	no	—
96	(C₂×D₁₁)⋊C₄	—	4	2	no	—
97	(C₂×D₁₁)⋊C₄	—	4	2	no	—
98	C₄⋊Dic₁₁	—	2	2	no	—
99	C₄⋊Dic₁₁	—	2	2	no	—
100	C₄⋊Dic₁₁	—	2	2	no	—
101	C₄⋊Dic₁₁	—	4	2	no	—
102	Dic₁₁⋊C₄	—	4	2	no	—
103	Dic₁₁⋊C₄	—	4	2	no	—
104	C₄×Dic₁₁	—	4	2	no	—
105	D₈₈	—	2	2	no	—
106	D₈₈	—	2	2	no	—
107	C₁₁×C₁₆	—	2	1	no	—
108	C₁₁×C₁₆	—	2	1	no	—
109	C₁₁×C₁₆	—	2	1	no	—
110	C₁₁×C₁₆	#41	1	2	yes	—
111	#3339	—	11	2	no	—
112	#2253	—	11	1	no	—
113	#2234	—	11	2	no	—
114	#2233	—	11	2	no	—
115	#2228	—	11	2	no	—
116	#2215	—	11	1	no	—
117	#2215	—	11	1	no	—
118	C₄×D₁₆	—	11	1	no	—
119	D₄×C₁₆	D₁₁	1	1	yes	characteristic, complemented
120	#2158	—	11	1	no	—
121	C₂×#366	—	11	1	no	—
122	C₂³×C₁₁	C₂×D₈	1	1	yes	characteristic
123	C₂³×C₁₁	C₂×D₈	1	1	yes	characteristic
124	C₂²×D₁₁	—	4	2	no	—
125	D₄×C₁₁	—	2	2	no	—
126	C₂×C₄×C₁₁	—	2	1	no	—
127	C₂×C₄×C₁₁	—	2	1	no	—
128	C₂×C₄×C₁₁	—	2	1	no	—
129	C₂×C₄×C₁₁	—	2	1	no	—
130	C₂×C₄×C₁₁	C₂×D₈	1	1	yes	characteristic
131	C₂×C₄×C₁₁	C₂×D₈	1	1	yes	characteristic
132	C₂×C₄×C₁₁	C₂×D₈	1	1	yes	characteristic
133	C₂×C₄×C₁₁	C₂×D₈	1	1	yes	characteristic
134	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
135	C₂²⋊D₁₁	—	8	4	no	—
136	C₂²⋊D₁₁	—	16	2	no	—
137	C₂×Dic₁₁	—	4	2	no	—
138	C₂×Dic₁₁	—	4	2	no	—
139	C₂×Dic₁₁	—	4	2	no	—
140	C₂×Dic₁₁	—	16	2	no	—
141	D₄₄	—	4	2	no	—
142	D₄₄	—	4	2	no	—
143	C₄×D₁₁	—	16	2	no	—
144	C₈×C₁₁	—	2	1	no	—
145	C₈×C₁₁	—	2	1	no	—
146	C₈×C₁₁	—	2	1	no	—
147	C₈×C₁₁	C₂×#41	1	1	yes	characteristic
148	C₈×C₁₁	2₋¹⁺⁴	1	1	yes	characteristic
149	#553	—	22	2	no	—
150	C₂×D₁₆	—	11	1	no	—
151	C₂²×C₁₆	C₂×D₁₁	1	1	yes	characteristic
152	C₂²×C₁₆	C₂×D₁₁	1	1	yes	characteristic
153	#499	—	11	2	no	—
154	#482	—	22	2	no	—
155	#481	—	22	2	no	—
156	#478	—	22	2	no	—
157	#466	—	11	2	no	—
158	#466	—	11	2	no	—
159	C₄×D₈	—	11	2	no	—
160	C₈×D₄	C₂×D₁₁	1	1	yes	characteristic
161	#427	—	11	2	no	—
162	C₂×#107	—	11	2	no	—
163	#367	—	11	2	no	—
164	#367	—	11	2	no	—
165	#366	—	11	1	no	—
166	#366	—	11	1	no	—
167	#366	—	11	2	no	—
168	#366	—	11	1	no	—
169	C₄⋊C₁₆	C₂×D₁₁	1	1	yes	characteristic
170	#357	—	11	2	no	—
171	#357	—	11	2	no	—
172	#357	—	11	2	no	—
173	C₂²⋊C₁₆	C₂×D₁₁	1	1	yes	characteristic
174	C₂²⋊C₁₆	C₂×D₁₁	1	1	yes	characteristic
175	C₄×C₁₆	C₂×D₁₁	1	1	yes	characteristic
176	C₂²×C₁₁	—	2	1	no	—
177	C₂²×C₁₁	—	2	1	no	—
178	C₂²×C₁₁	—	2	1	no	—
179	C₂²×C₁₁	—	2	1	no	—
180	C₂²×C₁₁	—	2	1	no	—
181	C₂²×C₁₁	#508	1	2	yes	—
182	C₂²×C₁₁	C₂²×D₈	1	1	yes	characteristic
183	C₂×D₁₁	—	8	2	no	—
184	C₂×D₁₁	—	8	2	no	—
185	C₂×D₁₁	—	8	2	no	—
186	C₄×C₁₁	—	2	1	no	—
187	C₄×C₁₁	—	2	1	no	—
188	C₄×C₁₁	—	2	1	no	—
189	C₄×C₁₁	—	2	1	no	—
190	C₄×C₁₁	#547	1	1	yes	characteristic
191	C₄×C₁₁	#578	1	1	yes	characteristic
192	Dic₁₁	—	8	2	no	—
193	Dic₁₁	—	8	2	no	—
194	Dic₁₁	—	8	2	no	—
195	C₂×D₈	—	11	2	no	—
196	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
197	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
198	#126	—	44	2	no	—
199	#125	—	22	2	no	—
200	#123	—	44	2	no	—
201	#121	—	22	2	no	—
202	#121	—	22	2	no	—
203	#121	—	44	2	no	—
204	C₄×D₄	—	22	2	no	—
205	C₄×D₄	—	44	2	no	—
206	C₄×D₄	D₄₄	1	1	yes	characteristic
207	#117	—	22	2	no	—
208	C₂×C₄⋊C₄	—	22	2	no	—
209	D₁₆	—	22	2	no	—
210	C₂×C₁₆	—	2	2	no	—
211	C₂×C₁₆	—	2	2	no	—
212	C₂×C₁₆	C₂²×D₁₁	1	1	yes	characteristic
213	C₂×C₁₆	C₂²×D₁₁	1	1	yes	characteristic
214	C₂×C₁₆	C₂²×D₁₁	1	1	yes	characteristic
215	C₂×C₁₆	C₂²×D₁₁	1	1	yes	characteristic
216	#107	—	11	2	no	—
217	#107	—	11	2	no	—
218	#107	—	11	2	no	—
219	#107	—	22	2	no	—
220	#106	—	22	2	no	—
221	#106	—	22	2	no	—
222	C₄⋊C₈	D₄₄	1	1	yes	characteristic
223	#102	—	22	2	no	—
224	#102	—	22	2	no	—
225	#102	—	22	2	no	—
226	C₂²⋊C₈	D₄₄	1	1	yes	characteristic
227	C₂²⋊C₈	D₄₄	1	1	yes	characteristic
228	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
229	C₂×C₁₁	—	2	2	no	—
230	C₂×C₁₁	—	4	1	no	—
231	C₂×C₁₁	#3334	1	1	yes	characteristic
232	C₂×C₁₁	C₂×#508	1	1	yes	characteristic
233	C₂×C₁₁	#3415	1	1	yes	characteristic
234	D₁₁	—	16	2	no	—
235	C₂×D₄	—	22	2	no	—
236	C₂×D₄	—	44	2	no	—
237	C₂×D₄	—	44	2	no	—
238	C₂×D₄	D₈₈	1	1	yes	characteristic
239	C₂²×C₄	—	44	2	no	—
240	C₂²×C₄	—	44	2	no	—
241	C₂²×C₄	C₂×D₄₄	1	1	yes	characteristic
242	C₂²×C₄	C₂×D₄₄	1	1	yes	characteristic
243	D₈	—	22	2	no	—
244	D₈	—	22	2	no	—
245	C₂×C₈	—	2	1	no	—
246	C₂×C₈	—	2	1	no	—
247	C₂×C₈	—	2	1	no	—
248	C₂×C₈	—	2	1	no	—
249	C₂×C₈	C₂×D₄₄	1	1	yes	characteristic
250	C₂×C₈	C₂×D₄₄	1	1	yes	characteristic
251	C₂×C₈	C₂×D₄₄	1	1	yes	characteristic
252	C₂×C₈	C₂³×D₁₁	1	1	yes	characteristic
253	C₄⋊C₄	—	22	2	no	—
254	C₄⋊C₄	—	22	2	no	—
255	C₄⋊C₄	—	22	2	no	—
256	C₄⋊C₄	—	44	2	no	—
257	C₄⋊C₄	—	44	2	no	—
258	C₄⋊C₄	—	44	2	no	—
259	C₄⋊C₄	D₈₈	1	1	yes	characteristic
260	C₂²⋊C₄	—	44	2	no	—
261	C₂²⋊C₄	—	44	2	no	—
262	C₂²⋊C₄	—	44	2	no	—
263	C₂²⋊C₄	—	44	2	no	—
264	C₂²⋊C₄	D₈₈	1	1	yes	characteristic
265	C₂²⋊C₄	D₈₈	1	1	yes	characteristic
266	C₄²	—	44	2	no	—
267	C₄²	C₂×D₄₄	1	1	yes	characteristic
268	C₁₆	—	2	1	no	—
269	C₁₆	—	2	1	no	—
270	C₁₆	—	2	1	no	—
271	C₁₆	#4341	1	2	yes	—
272	C₁₁	#33906	1	1	yes	characteristic, complemented, Sylow
273	C₂³	—	44	2	no	—
274	C₂³	C₂×D₈₈	1	1	yes	characteristic
275	C₂³	C₂×D₈₈	1	1	yes	characteristic
276	D₄	—	2	2	no	—
277	D₄	—	44	2	no	—
278	D₄	—	44	2	no	—
279	D₄	—	88	4	no	—
280	D₄	—	176	2	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	44	2	no	—
286	C₂×C₄	—	44	2	no	—
287	C₂×C₄	—	44	2	no	—
288	C₂×C₄	—	176	2	no	—
289	C₂×C₄	—	176	2	no	—
290	C₂×C₄	C₂×D₈₈	1	1	yes	characteristic
291	C₂×C₄	C₂×D₈₈	1	1	yes	characteristic
292	C₂×C₄	C₂×D₈₈	1	1	yes	characteristic
293	C₂×C₄	C₂×D₈₈	1	1	yes	characteristic
294	C₂×C₄	C₂²×D₄₄	1	1	yes	characteristic
295	C₈	—	2	1	no	—
296	C₈	—	2	1	no	—
297	C₈	—	2	1	no	—
298	C₈	C₂×#4341	1	1	yes	characteristic
299	C₈	#67043	1	1	yes	characteristic
300	C₂²	—	2	1	no	—
301	C₂²	—	2	1	no	—
302	C₂²	—	2	1	no	—
303	C₂²	—	2	1	no	—
304	C₂²	—	2	1	no	—
305	C₂²	—	88	2	no	—
306	C₂²	—	88	2	no	—
307	C₂²	—	88	2	no	—
308	C₂²	#10622517	1	2	yes	—
309	C₂²	C₂²×D₈₈	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	2	1	no	—
313	C₄	—	2	1	no	—
314	C₄	—	88	2	no	—
315	C₄	—	88	2	no	—
316	C₄	—	88	2	no	—
317	C₄	#10623086	1	1	yes	characteristic
318	C₄	#10623284	1	1	yes	characteristic
319	C₂	—	2	2	no	—
320	C₂	—	4	1	no	—
321	C₂	—	176	2	no	—
322	C₂	#13080964	1	1	yes	characteristic
323	C₂	C₂×#10622517	1	1	yes	characteristic
324	C₂	#13081757	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₈₈
commutator, cosocle	C₂×C₈×C₁₁	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₁
Fitting	D₄×C₁₁×C₁₆	C₂
socle	C₂²×C₁₁	C₂²×D₈
hypercenter	D₄×C₁₆	D₁₁
hypocenter	C₁₁	#33906
LCS	G ▷ C₂×C₈×C₁₁ ▷ C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ D₄×C₁₆

Complements

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

N	H ≅ G/N	type
1. #13081019	C₂	semidirect
2. #13070895	C₂	semidirect
3. #13070808	C₂	semidirect
4. #13070802	C₂	semidirect
5. #13070774	C₂	semidirect
6. #13070702	C₂	semidirect
7. #13070702	C₂	semidirect
8. C₄×D₁₇₆	C₂	semidirect
9. #13070400	C₂	semidirect
10. C₂×#10622142	C₂	semidirect
11. D₄×C₁₁×C₁₆	C₂	semidirect
119. D₄×C₁₆	D₁₁	semidirect
272. C₁₁	#33906	semidirect

Maximal subgroups

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

subgroup	index
#13081019	2
#13070895	2
#13070808	2
#13070802	2
#13070774	2
#13070702	2
C₄×D₁₇₆	2
#13070400	2
C₂×#10622142	2
D₄×C₁₁×C₁₆	2
#33906	11