Subgroups — #14908516

Order 2816.

524 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	#13079130	C₂	1	1	yes	maximal, characteristic, complemented
2	#13070792	C₂	1	4	yes	maximal, complemented
3	#13070787	C₂	1	4	yes	maximal, complemented
4	#13070390	C₂	1	2	yes	maximal, complemented
5	C₂×#10622101	C₂	1	1	yes	maximal, characteristic, complemented
6	#13070352	C₂	1	2	yes	maximal
7	C₂×C₁₁×#443	C₂	1	1	yes	maximal, characteristic
8	C₂×C₁₁×#135	C₂²	1	1	yes	characteristic
9	C₂×C₁₁×#131	—	2	1	no	—
10	#10623001	—	2	1	no	—
11	C₁₁×#443	—	2	1	no	—
12	C₁₁×#443	—	2	1	no	—
13	C₁₁×#443	C₂²	1	2	yes	—
14	C₁₁×#443	C₂²	1	2	yes	—
15	C₂×C₁₁×#108	C₂²	1	1	yes	characteristic
16	C₂×C₁₁×#104	—	2	1	no	—
17	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
18	#10622719	—	2	1	no	—
19	#10622719	C₂²	1	2	yes	—
20	C₂×#66876	C₂²	1	2	yes	—
21	#10622296	—	2	1	no	—
22	#10622296	C₂²	1	2	yes	complemented
23	#10622295	—	2	1	no	—
24	#10622293	C₂²	1	2	yes	complemented
25	#10622204	C₂²	1	8	yes	—
26	#10622132	C₂²	1	8	yes	complemented
27	#10622101	C₂²	1	2	yes	complemented
28	#10622101	C₂²	1	2	yes	complemented
29	C₂×C₁₁×#41	—	2	1	no	—
30	C₁₁×#135	—	2	1	no	—
31	C₁₁×#135	—	2	1	no	—
32	C₁₁×#135	Q₈	1	2	yes	complemented
33	C₁₁×#135	C₂³	1	2	yes	—
34	C₂×C₁₁×Q₁₆	Q₈	1	1	yes	characteristic, complemented
35	C₂×C₁₁×SD₁₆	—	2	1	no	—
36	C₂×C₁₁×D₈	Q₈	1	1	yes	characteristic, complemented
37	C₁₁×#131	—	2	2	no	—
38	C₁₁×#131	—	2	2	no	—
39	C₁₁×#131	—	4	1	no	—
40	C₁₁×#131	—	4	1	no	—
41	C₂×C₁₁×M₁₆	—	2	1	no	—
42	C₂×C₁₁×M₁₆	—	4	1	no	—
43	C₂²×C₈×C₁₁	—	4	1	no	—
44	C₂²×C₈×C₁₁	D₄	1	1	yes	characteristic
45	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₁₁	D₄	1	1	yes	characteristic
47	#66979	—	2	2	no	—
48	#66977	—	4	2	no	—
49	#66923	—	2	2	no	—
50	#66922	—	2	2	no	—
51	C₄×Dic₂₂	—	4	2	no	—
52	C₁₁×#108	—	2	1	no	—
53	C₁₁×#108	C₂³	1	2	yes	—
54	C₁₁×#104	—	2	2	no	—
55	C₁₁×#104	—	2	2	no	—
56	C₄×C₈×C₁₁	D₄	1	1	yes	characteristic
57	C₄×C₈×C₁₁	D₄	1	1	yes	characteristic
58	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
59	C₄×C₈×C₁₁	C₂³	1	1	yes	characteristic
60	#66882	—	2	2	no	—
61	#66882	—	2	2	no	—
62	#66881	—	2	2	no	—
63	#66881	C₂³	1	4	yes	—
64	#66876	—	2	2	no	—
65	#66876	C₂³	1	4	yes	—
66	C₂×C₁₁⋊C₁₆	—	2	2	no	—
67	#33637	—	11	1	no	maximal, Sylow
68	C₁₁×#41	—	2	2	no	—
69	C₁₁×#41	—	2	2	no	—
70	C₁₁×#41	—	4	1	no	—
71	C₁₁×#41	—	4	1	no	—
72	C₂×Q₈×C₁₁	—	2	1	no	—
73	C₂×D₄×C₁₁	—	2	1	no	—
74	C₂×D₄×C₁₁	—	4	1	no	—
75	C₂²×C₄×C₁₁	—	2	1	no	—
76	C₂²×C₄×C₁₁	—	4	1	no	—
77	C₂²×C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
78	C₂×Dic₂₂	—	4	2	no	—
79	C₁₁×Q₁₆	—	2	1	no	—
80	C₁₁×Q₁₆	C₂×Q₈	1	2	yes	—
81	C₁₁×SD₁₆	—	2	2	no	—
82	C₁₁×SD₁₆	—	2	2	no	—
83	C₁₁×D₈	—	2	1	no	—
84	C₁₁×D₈	C₂×Q₈	1	2	yes	—
85	C₁₁×M₁₆	—	2	2	no	—
86	C₁₁×M₁₆	—	4	2	no	—
87	C₁₁×M₁₆	—	4	2	no	—
88	C₁₁×M₁₆	—	4	1	no	—
89	C₂×C₈×C₁₁	—	4	2	no	—
90	C₂×C₈×C₁₁	—	4	2	no	—
91	C₂×C₈×C₁₁	—	4	1	no	—
92	C₂×C₈×C₁₁	SD₁₆	1	1	yes	characteristic
93	C₂×C₈×C₁₁	SD₁₆	1	1	yes	characteristic
94	C₂×C₈×C₁₁	SD₁₆	1	1	yes	characteristic
95	C₂×C₈×C₁₁	SD₁₆	1	1	yes	characteristic
96	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈×C₁₁	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈×C₁₁	C₂×Q₈	1	1	yes	characteristic
100	C₂×C₈×C₁₁	C₂×Q₈	1	1	yes	characteristic
101	C₂×C₈×C₁₁	#41	1	1	yes	characteristic
102	C₂×C₈×C₁₁	#41	1	1	yes	characteristic
103	C₂×C₈×C₁₁	C₂⁴	1	1	yes	characteristic
104	C₄²×C₁₁	—	2	1	no	—
105	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
106	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
107	C₂²⋊Dic₁₁	—	4	2	no	—
108	C₄⋊Dic₁₁	—	2	4	no	—
109	C₄⋊Dic₁₁	—	4	2	no	—
110	Dic₁₁⋊C₄	—	4	4	no	—
111	C₄×Dic₁₁	—	4	2	no	—
112	C₁₁⋊C₁₆	—	2	4	no	—
113	#3160	—	11	1	no	—
114	C₂×#443	D₁₁	1	1	yes	characteristic
115	#2231	—	11	4	no	—
116	#2230	—	11	4	no	—
117	#2155	—	11	2	no	—
118	C₂×#364	—	11	1	no	—
119	#2145	—	11	2	no	—
120	C₂³×C₁₁	—	4	1	no	—
121	C₂³×C₁₁	AGL(1,ℤ/8)	1	1	yes	characteristic
122	Q₈×C₁₁	—	2	2	no	—
123	Q₈×C₁₁	—	4	1	no	—
124	D₄×C₁₁	—	2	2	no	—
125	D₄×C₁₁	—	4	2	no	—
126	D₄×C₁₁	—	4	2	no	—
127	D₄×C₁₁	—	4	1	no	—
128	C₂×C₄×C₁₁	—	2	1	no	—
129	C₂×C₄×C₁₁	—	2	1	no	—
130	C₂×C₄×C₁₁	—	4	2	no	—
131	C₂×C₄×C₁₁	—	4	1	no	—
132	C₂×C₄×C₁₁	—	4	2	no	—
133	C₂×C₄×C₁₁	—	4	2	no	—
134	C₂×C₄×C₁₁	C₂²⋊Q₈	1	1	yes	characteristic
135	C₂×C₄×C₁₁	C₂²⋊Q₈	1	1	yes	characteristic
136	C₂×C₄×C₁₁	C₂×SD₁₆	1	1	yes	characteristic
137	C₂×C₄×C₁₁	C₂×SD₁₆	1	1	yes	characteristic
138	C₂×C₄×C₁₁	AGL(1,ℤ/8)	1	1	yes	characteristic
139	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
140	C₂×Dic₁₁	—	4	4	no	—
141	Dic₂₂	—	8	4	no	—
142	C₈×C₁₁	—	4	2	no	—
143	C₈×C₁₁	C₂²⋊Q₈	1	1	yes	characteristic
144	C₈×C₁₁	C₂²⋊Q₈	1	1	yes	characteristic
145	C₈×C₁₁	C₂×SD₁₆	1	1	yes	characteristic
146	C₈×C₁₁	C₂×SD₁₆	1	1	yes	characteristic
147	C₈×C₁₁	C₂×SD₁₆	1	1	yes	characteristic
148	C₈×C₁₁	C₂×SD₁₆	1	1	yes	characteristic
149	C₈×C₁₁	C₂²×Q₈	1	1	yes	characteristic
150	C₈×C₁₁	C₂×#41	1	1	yes	characteristic
151	C₂×#135	C₂×D₁₁	1	1	yes	characteristic
152	C₂×#131	—	2	1	no	—
153	#533	—	22	1	no	—
154	C₂×M₃₂	—	11	2	no	—
155	#500	—	22	1	no	—
156	#499	—	11	2	no	—
157	#499	—	22	1	no	—
158	#498	—	11	2	no	—
159	#443	—	2	1	no	—
160	#443	—	2	1	no	—
161	#443	C₂×D₁₁	1	2	yes	—
162	#443	C₂×D₁₁	1	2	yes	—
163	C₂×#108	C₂×D₁₁	1	1	yes	characteristic
164	#427	—	11	2	no	—
165	#427	—	22	1	no	—
166	C₂×#104	—	2	1	no	—
167	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
168	#365	—	11	8	no	—
169	#364	—	11	2	no	—
170	#364	—	11	2	no	—
171	#360	—	11	8	no	—
172	C₂²×C₁₁	—	4	1	no	—
173	C₂²×C₁₁	—	4	2	no	—
174	C₂²×C₁₁	—	8	2	no	—
175	C₂²×C₁₁	#476	1	1	yes	characteristic
176	C₂²×C₁₁	#476	1	1	yes	characteristic
177	C₂²×C₁₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
178	C₄×C₁₁	—	2	2	no	—
179	C₄×C₁₁	—	4	1	no	—
180	C₄×C₁₁	—	4	2	no	—
181	C₄×C₁₁	#476	1	1	yes	characteristic
182	C₄×C₁₁	#476	1	1	yes	characteristic
183	C₄×C₁₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
184	C₄×C₁₁	C₂²×SD₁₆	1	1	yes	characteristic
185	Dic₁₁	—	8	4	no	—
186	C₂×#41	—	2	1	no	—
187	#135	—	2	1	no	—
188	#135	—	2	1	no	—
189	#135	Dic₂₂	1	2	yes	complemented
190	#135	C₂²×D₁₁	1	2	yes	—
191	C₂×Q₁₆	Dic₂₂	1	1	yes	characteristic, complemented
192	C₂×SD₁₆	—	2	1	no	—
193	C₂×D₈	Dic₂₂	1	1	yes	characteristic, complemented
194	#131	—	2	2	no	—
195	#131	—	2	2	no	—
196	#131	—	4	1	no	—
197	#131	—	4	1	no	—
198	C₂×M₁₆	—	2	1	no	—
199	C₂×M₁₆	—	4	1	no	—
200	C₂²×C₈	—	4	1	no	—
201	C₂²×C₈	C₂²⋊D₁₁	1	1	yes	characteristic
202	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
203	C₄⋊Q₈	—	22	2	no	—
204	#125	—	22	2	no	—
205	C₂²⋊Q₈	—	44	2	no	—
206	C₄×Q₈	—	44	2	no	—
207	#117	—	22	2	no	—
208	C₂×C₄²	C₂²⋊D₁₁	1	1	yes	characteristic
209	M₃₂	—	11	4	no	—
210	M₃₂	—	22	2	no	—
211	C₂×C₁₆	—	22	2	no	—
212	#108	—	2	1	no	—
213	#108	C₂²×D₁₁	1	2	yes	—
214	#107	—	22	2	no	—
215	#107	—	22	2	no	—
216	#106	—	11	4	no	—
217	#106	—	22	2	no	—
218	#104	—	2	2	no	—
219	#104	—	2	2	no	—
220	C₄×C₈	C₂²⋊D₁₁	1	1	yes	characteristic
221	C₄×C₈	C₂²⋊D₁₁	1	1	yes	characteristic
222	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
223	C₄×C₈	C₂²×D₁₁	1	1	yes	characteristic
224	C₂×C₁₁	—	2	1	no	—
225	C₂×C₁₁	—	2	1	no	—
226	C₂×C₁₁	—	2	1	no	—
227	C₂×C₁₁	—	8	2	no	—
228	C₂×C₁₁	C₂×#476	1	1	yes	characteristic
229	#41	—	2	2	no	—
230	#41	—	2	2	no	—
231	#41	—	4	1	no	—
232	#41	—	4	1	no	—
233	C₂×Q₈	—	2	1	no	—
234	C₂×Q₈	—	44	2	no	—
235	C₂×D₄	—	2	1	no	—
236	C₂×D₄	—	4	1	no	—
237	C₂²×C₄	—	2	1	no	—
238	C₂²×C₄	—	4	1	no	—
239	C₂²×C₄	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
240	Q₁₆	—	2	1	no	—
241	Q₁₆	C₂×Dic₂₂	1	2	yes	—
242	SD₁₆	—	2	2	no	—
243	SD₁₆	—	2	2	no	—
244	D₈	—	2	1	no	—
245	D₈	C₂×Dic₂₂	1	2	yes	—
246	M₁₆	—	2	2	no	—
247	M₁₆	—	4	1	no	—
248	M₁₆	—	4	2	no	—
249	M₁₆	—	4	2	no	—
250	C₂×C₈	—	4	1	no	—
251	C₂×C₈	—	4	2	no	—
252	C₂×C₈	—	4	2	no	—
253	C₂×C₈	#4324	1	1	yes	characteristic
254	C₂×C₈	#4324	1	1	yes	characteristic
255	C₂×C₈	#4324	1	1	yes	characteristic
256	C₂×C₈	#4324	1	1	yes	characteristic
257	C₂×C₈	C₂×Dic₂₂	1	1	yes	characteristic
258	C₂×C₈	C₂×Dic₂₂	1	1	yes	characteristic
259	C₂×C₈	#4339	1	1	yes	characteristic
260	C₂×C₈	#4339	1	1	yes	characteristic
261	C₂×C₈	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
262	C₂×C₈	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
263	C₂×C₈	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
264	C₂×C₈	C₂³×D₁₁	1	1	yes	characteristic
265	C₄⋊C₄	—	22	4	no	—
266	C₄⋊C₄	—	44	4	no	—
267	C₄⋊C₄	—	44	2	no	—
268	C₂²⋊C₄	—	44	2	no	—
269	C₄²	—	2	1	no	—
270	C₄²	—	44	2	no	—
271	C₄²	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
272	C₄²	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
273	C₁₆	—	22	4	no	—
274	C₁₁	#33637	1	1	yes	characteristic, complemented, Sylow
275	C₂³	—	4	1	no	—
276	C₂³	#67002	1	1	yes	characteristic
277	Q₈	—	2	2	no	—
278	Q₈	—	4	1	no	—
279	Q₈	—	88	4	no	—
280	D₄	—	2	2	no	—
281	D₄	—	4	2	no	—
282	D₄	—	4	2	no	—
283	D₄	—	4	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	4	1	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	44	4	no	—
291	C₂×C₄	#66977	1	1	yes	characteristic
292	C₂×C₄	#66977	1	1	yes	characteristic
293	C₂×C₄	C₂×#4324	1	1	yes	characteristic
294	C₂×C₄	C₂×#4324	1	1	yes	characteristic
295	C₂×C₄	#67002	1	1	yes	characteristic
296	C₂×C₄	C₂²×C₂²⋊D₁₁	1	1	yes	characteristic
297	C₈	—	4	2	no	—
298	C₈	#66977	1	1	yes	characteristic
299	C₈	#66977	1	1	yes	characteristic
300	C₈	C₂×#4324	1	1	yes	characteristic
301	C₈	C₂×#4324	1	1	yes	characteristic
302	C₈	C₂×#4324	1	1	yes	characteristic
303	C₈	C₂×#4324	1	1	yes	characteristic
304	C₈	C₂²×Dic₂₂	1	1	yes	characteristic
305	C₈	C₂×#4339	1	1	yes	characteristic
306	C₂²	—	4	1	no	—
307	C₂²	—	4	2	no	—
308	C₂²	—	8	2	no	—
309	C₂²	#10622621	1	1	yes	characteristic
310	C₂²	#10622621	1	1	yes	characteristic
311	C₂²	C₂×#67002	1	1	yes	characteristic
312	C₄	—	2	2	no	—
313	C₄	—	4	1	no	—
314	C₄	—	4	2	no	—
315	C₄	—	88	4	no	—
316	C₄	#10622621	1	1	yes	characteristic
317	C₄	#10622621	1	1	yes	characteristic
318	C₄	C₂×#66977	1	1	yes	characteristic
319	C₄	C₂²×#4324	1	1	yes	characteristic
320	C₂	—	2	1	no	—
321	C₂	—	2	1	no	—
322	C₂	—	2	1	no	—
323	C₂	—	8	2	no	—
324	C₂	C₂×#10622621	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#10622621
commutator, cosocle	C₂×C₈×C₁₁	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₁
Fitting	C₂×C₁₁×#443	C₂
socle	C₂×C₁₁	C₂×#476
prime radical	C₂×C₁₁×#135	C₂²
hypercenter	C₂×#443	D₁₁
hypocenter	C₁₁	#33637
LCS	G ▷ C₂×C₈×C₁₁ ▷ C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#443

Complements

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

N	H ≅ G/N	type
1. #13079130	C₂	semidirect
2. #13070792	C₂	semidirect
3. #13070787	C₂	semidirect
4. #13070390	C₂	semidirect
5. C₂×#10622101	C₂	semidirect
22. #10622296	C₂²	semidirect
24. #10622293	C₂²	semidirect
26. #10622132	C₂²	semidirect
27. #10622101	C₂²	semidirect
28. #10622101	C₂²	semidirect
32. C₁₁×#135	Q₈	semidirect
34. C₂×C₁₁×Q₁₆	Q₈	semidirect
36. C₂×C₁₁×D₈	Q₈	semidirect
189. #135	Dic₂₂	semidirect
191. C₂×Q₁₆	Dic₂₂	semidirect
193. C₂×D₈	Dic₂₂	semidirect
274. C₁₁	#33637	semidirect

Maximal subgroups

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

subgroup	index
#13079130	2
#13070792	2
#13070787	2
#13070390	2
C₂×#10622101	2
#13070352	2
C₂×C₁₁×#443	2
#33637	11