Subgroups — #14772670

Order 2816.

638 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	M₁₆×Dic₂₂	C₂	1	2	yes	maximal
2	C₂×#10622292	C₂	1	1	yes	maximal, characteristic
3	#13069391	C₂	1	2	yes	maximal
4	#13069390	C₂	1	4	yes	maximal
5	#13068789	C₂	1	2	yes	maximal
6	#13068788	C₂	1	1	yes	maximal, characteristic
7	#13068524	C₂	1	2	yes	maximal
8	C₄×C₁₁×#108	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Dic₂₂	C₂²	1	1	yes	characteristic
10	C₂×C₁₁×#108	C₂²	1	1	yes	characteristic
11	C₂×C₁₁×#108	C₂²	1	1	yes	characteristic
12	C₄×C₁₁×M₁₆	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
14	C₂×#66886	C₂²	1	4	yes	—
15	#10622731	C₂²	1	4	yes	—
16	M₁₆×Dic₁₁	C₂²	1	4	yes	—
17	C₂×#66883	C₂²	1	2	yes	—
18	C₂×#66883	C₂²	1	2	yes	—
19	C₂×#66879	C₂²	1	2	yes	—
20	C₂×Dic₁₁⋊C₈	C₂²	1	2	yes	—
21	#10622537	C₂²	1	2	yes	—
22	C₂×C₄⋊(C₁₁⋊C₈)	C₂²	1	1	yes	characteristic
23	#10622314	—	2	4	no	—
24	#10622292	—	2	2	no	—
25	#10622292	C₄	1	4	yes	—
26	#10622292	C₄	1	4	yes	—
27	C₈×Dic₂₂	—	2	2	no	—
28	C₁₁×#341	C₂²	1	2	yes	—
29	#10622196	C₂²	1	4	yes	—
30	C₂²×Dic₂₂	Q₈	1	1	yes	characteristic
31	C₂×C₁₁×M₁₆	C₂³	1	2	yes	—
32	C₂×C₁₁×M₁₆	C₂³	1	2	yes	—
33	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
34	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
35	C₂×C₄²×C₁₁	C₂³	1	1	yes	characteristic
36	C₂×C₄⋊Dic₁₁	Q₈	1	1	yes	characteristic
37	C₂×Dic₁₁⋊C₄	Q₈	1	2	yes	—
38	C₂×C₄×Dic₁₁	C₂³	1	2	yes	—
39	C₂×#4318	C₂³	1	4	yes	—
40	C₂²×C₁₁⋊C₈	C₂³	1	2	yes	—
41	C₄×Dic₂₂	C₂×C₄	1	2	yes	—
42	C₄×Dic₂₂	C₂×C₄	1	2	yes	—
43	C₄×Dic₂₂	D₄	1	2	yes	—
44	C₄×Dic₂₂	D₄	1	2	yes	—
45	C₁₁×#108	—	2	2	no	—
46	C₁₁×#108	Q₈	1	4	yes	complemented
47	C₁₁×#97	—	2	2	no	—
48	C₄×C₈×C₁₁	—	2	2	no	—
49	C₄×C₈×C₁₁	C₂×C₄	1	2	yes	—
50	C₄×C₈×C₁₁	C₂×C₄	1	2	yes	—
51	#66886	—	4	4	no	—
52	#66883	—	2	4	no	—
53	#66883	—	4	2	no	—
54	#66879	—	2	4	no	—
55	#66879	C₂×C₄	1	8	yes	—
56	Dic₁₁⋊C₈	—	2	4	no	—
57	Dic₁₁⋊C₈	—	2	4	no	—
58	Dic₁₁⋊C₈	C₂×C₄	1	8	yes	—
59	C₈×Dic₁₁	—	2	4	no	—
60	C₄⋊(C₁₁⋊C₈)	—	2	2	no	—
61	C₄⋊(C₁₁⋊C₈)	—	2	2	no	—
62	C₄⋊(C₁₁⋊C₈)	C₂×C₄	1	2	yes	—
63	C₄⋊(C₁₁⋊C₈)	C₂×C₄	1	2	yes	—
64	#22733	—	11	1	no	maximal, Sylow
65	C₂²×C₄×C₁₁	C₂×Q₈	1	1	yes	characteristic
66	C₂²×C₄×C₁₁	C₂×Q₈	1	1	yes	characteristic
67	C₂²×C₄×C₁₁	C₂⁴	1	1	yes	characteristic
68	C₂²×Dic₁₁	C₂×Q₈	1	2	yes	—
69	C₂²×Dic₁₁	C₂×Q₈	1	2	yes	—
70	C₂×Dic₂₂	—	4	2	no	—
71	C₂×Dic₂₂	C₄⋊C₄	1	2	yes	—
72	C₂×Dic₂₂	C₄⋊C₄	1	2	yes	—
73	C₁₁×M₁₆	—	2	2	no	—
74	C₁₁×M₁₆	—	2	2	no	—
75	C₁₁×M₁₆	—	2	2	no	—
76	C₁₁×M₁₆	C₂×Q₈	1	4	yes	—
77	C₂×C₈×C₁₁	—	2	2	no	—
78	C₂×C₈×C₁₁	—	2	2	no	—
79	C₂×C₈×C₁₁	—	2	1	no	—
80	C₂×C₈×C₁₁	—	2	1	no	—
81	C₂×C₈×C₁₁	C₂²×C₄	1	2	yes	—
82	C₂×C₈×C₁₁	C₂²×C₄	1	2	yes	—
83	C₂×C₈×C₁₁	C₂×Q₈	1	2	yes	—
84	C₂×C₈×C₁₁	#41	1	2	yes	—
85	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
86	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
87	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
88	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
89	C₄⋊Dic₁₁	C₄⋊C₄	1	2	yes	—
90	C₄⋊Dic₁₁	C₄⋊C₄	1	2	yes	—
91	Dic₁₁⋊C₄	C₄⋊C₄	1	4	yes	—
92	Dic₁₁⋊C₄	C₄⋊C₄	1	4	yes	—
93	C₄×Dic₁₁	C₂²×C₄	1	4	yes	—
94	C₄×Dic₁₁	C₂×D₄	1	4	yes	—
95	#4318	—	2	4	no	—
96	#4318	—	2	4	no	—
97	C₂×C₁₁⋊C₈	—	2	4	no	—
98	C₂×C₁₁⋊C₈	—	2	2	no	—
99	C₂×C₁₁⋊C₈	—	2	2	no	—
100	C₂×C₁₁⋊C₈	C₂²×C₄	1	4	yes	—
101	Q₈×M₁₆	—	11	2	no	—
102	C₂×#446	—	11	1	no	—
103	#1950	—	11	2	no	—
104	#1950	—	11	4	no	—
105	#1849	—	11	2	no	—
106	#1848	—	11	1	no	—
107	#1778	—	11	2	no	—
108	C₄×#108	D₁₁	1	1	yes	characteristic
109	C₂³×C₁₁	C₂²×Q₈	1	1	yes	characteristic
110	C₂×C₄×C₁₁	—	2	2	no	—
111	C₂×C₄×C₁₁	—	4	1	no	—
112	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
117	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
118	C₂×C₄×C₁₁	C₂²×Q₈	1	1	yes	characteristic
119	C₂×C₄×C₁₁	C₂×#41	1	1	yes	characteristic
120	C₂×C₄×C₁₁	H(5,𝔽₂)	1	1	yes	characteristic
121	C₂×C₄×C₁₁	2₋¹⁺⁴	1	1	yes	characteristic
122	C₂×Dic₁₁	—	4	2	no	—
123	C₂×Dic₁₁	—	4	2	no	—
124	C₂×Dic₁₁	C₂×C₄⋊C₄	1	4	yes	—
125	C₂×Dic₁₁	C₂×C₄⋊C₄	1	4	yes	—
126	Dic₂₂	—	2	4	no	—
127	Dic₂₂	—	4	2	no	—
128	C₈×C₁₁	—	2	2	no	—
129	C₈×C₁₁	—	2	4	no	—
130	C₈×C₁₁	—	4	2	no	—
131	C₈×C₁₁	C₄×Q₈	1	4	yes	—
132	C₁₁⋊C₈	—	2	4	no	—
133	C₁₁⋊C₈	—	4	4	no	—
134	C₂×C₄×Q₈	—	11	1	no	—
135	#446	—	11	4	no	—
136	#446	—	11	4	no	—
137	#446	—	22	2	no	—
138	#446	—	22	4	no	—
139	C₈×Q₈	—	22	2	no	—
140	C₂×#108	—	11	2	no	—
141	C₂×#108	—	11	4	no	—
142	C₂×#108	—	11	2	no	—
143	C₂×#108	C₂×D₁₁	1	1	yes	characteristic
144	C₂×#108	C₂×D₁₁	1	1	yes	characteristic
145	#423	—	11	4	no	—
146	#423	—	11	2	no	—
147	C₂×C₄⋊C₈	—	11	2	no	—
148	C₂×C₄⋊C₈	—	11	1	no	—
149	C₄×M₁₆	—	11	4	no	—
150	C₄×M₁₆	C₂×D₁₁	1	2	yes	—
151	C₂×#97	—	11	2	no	—
152	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
153	#341	—	11	4	no	—
154	#341	C₂×D₁₁	1	2	yes	—
155	C₂²×C₁₁	—	2	1	no	—
156	C₂²×C₁₁	—	2	1	no	—
157	C₂²×C₁₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
158	C₂²×C₁₁	#557	1	1	yes	characteristic
159	C₂²×C₁₁	Q₈²	1	1	yes	characteristic
160	C₄×C₁₁	—	2	1	no	—
161	C₄×C₁₁	—	2	1	no	—
162	C₄×C₁₁	—	2	1	no	—
163	C₄×C₁₁	#430	1	2	yes	—
164	C₄×C₁₁	C₂×C₄×Q₈	1	1	yes	characteristic
165	C₄×C₁₁	#520	1	1	yes	characteristic
166	C₄×C₁₁	D₄×Q₈	1	1	yes	characteristic
167	C₄×C₁₁	#550	1	1	yes	characteristic
168	Dic₁₁	—	2	4	no	—
169	Dic₁₁	—	2	4	no	—
170	C₂²×Q₈	—	11	1	no	—
171	C₂×M₁₆	—	11	4	no	—
172	C₂×M₁₆	C₂²×D₁₁	1	2	yes	—
173	C₂×M₁₆	C₂²×D₁₁	1	2	yes	—
174	C₂²×C₈	—	11	2	no	—
175	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
176	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
177	C₄×Q₈	—	11	2	no	—
178	C₄×Q₈	—	11	2	no	—
179	C₄×Q₈	—	11	2	no	—
180	C₄×Q₈	—	11	2	no	—
181	C₂×C₄⋊C₄	—	11	2	no	—
182	C₂×C₄⋊C₄	—	11	1	no	—
183	C₂×C₄²	—	11	2	no	—
184	C₂×C₄²	C₂²×D₁₁	1	1	yes	characteristic
185	#108	—	2	2	no	—
186	#108	—	22	4	no	—
187	#108	—	44	2	no	—
188	#108	—	44	4	no	—
189	#108	Dic₂₂	1	4	yes	complemented
190	C₄⋊C₈	—	11	8	no	—
191	C₄⋊C₈	—	11	2	no	—
192	C₄⋊C₈	—	11	2	no	—
193	C₄⋊C₈	—	22	4	no	—
194	C₄⋊C₈	—	22	4	no	—
195	C₄⋊C₈	—	22	2	no	—
196	C₄⋊C₈	—	22	2	no	—
197	#97	—	2	2	no	—
198	#97	—	11	8	no	—
199	#97	—	22	4	no	—
200	C₄×C₈	—	2	2	no	—
201	C₄×C₈	—	22	4	no	—
202	C₄×C₈	C₄×D₁₁	1	2	yes	—
203	C₄×C₈	C₄×D₁₁	1	2	yes	—
204	C₂×C₁₁	—	2	1	no	—
205	C₂×C₁₁	—	2	1	no	—
206	C₂×C₁₁	Q₈×C₄⋊C₄	1	1	yes	characteristic
207	C₂×C₁₁	C₂×#430	1	1	yes	characteristic
208	C₂×C₁₁	#2916	1	1	yes	characteristic
209	C₂×Q₈	—	11	2	no	—
210	C₂×Q₈	—	11	2	no	—
211	C₂×Q₈	—	44	2	no	—
212	C₂²×C₄	—	11	2	no	—
213	C₂²×C₄	—	11	2	no	—
214	C₂²×C₄	Q₈×D₁₁	1	1	yes	characteristic
215	C₂²×C₄	Q₈×D₁₁	1	1	yes	characteristic
216	C₂²×C₄	C₂³×D₁₁	1	1	yes	characteristic
217	M₁₆	—	2	2	no	—
218	M₁₆	—	2	2	no	—
219	M₁₆	—	2	2	no	—
220	M₁₆	—	22	4	no	—
221	M₁₆	—	22	4	no	—
222	M₁₆	C₂×Dic₂₂	1	4	yes	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	1	no	—
225	C₂×C₈	—	2	1	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	11	4	no	—
228	C₂×C₈	—	22	2	no	—
229	C₂×C₈	—	22	2	no	—
230	C₂×C₈	—	22	4	no	—
231	C₂×C₈	C₂×Dic₂₂	1	2	yes	—
232	C₂×C₈	C₂×C₄×D₁₁	1	2	yes	—
233	C₂×C₈	C₂×C₄×D₁₁	1	2	yes	—
234	C₂×C₈	#4339	1	2	yes	—
235	C₄⋊C₄	—	11	4	no	—
236	C₄⋊C₄	—	11	4	no	—
237	C₄⋊C₄	—	11	2	no	—
238	C₄⋊C₄	—	11	2	no	—
239	C₄²	—	11	4	no	—
240	C₄²	—	11	4	no	—
241	C₄²	C₂×C₄×D₁₁	1	1	yes	characteristic
242	C₄²	C₂×C₄×D₁₁	1	1	yes	characteristic
243	C₄²	D₄×D₁₁	1	1	yes	characteristic
244	C₄²	D₄×D₁₁	1	1	yes	characteristic
245	C₁₁	#22733	1	1	yes	characteristic, complemented, Sylow
246	C₂³	C₂×Q₈×D₁₁	1	1	yes	characteristic
247	Q₈	—	22	4	no	—
248	Q₈	—	44	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	4	1	no	—
251	C₂×C₄	—	11	4	no	—
252	C₂×C₄	—	11	4	no	—
253	C₂×C₄	—	44	2	no	—
254	C₂×C₄	—	44	2	no	—
255	C₂×C₄	C₄⋊C₄×D₁₁	1	1	yes	characteristic
256	C₂×C₄	C₄⋊C₄×D₁₁	1	1	yes	characteristic
257	C₂×C₄	C₄⋊C₄×D₁₁	1	1	yes	characteristic
258	C₂×C₄	C₄⋊C₄×D₁₁	1	1	yes	characteristic
259	C₂×C₄	C₂²×Dic₂₂	1	1	yes	characteristic
260	C₂×C₄	C₂²×C₄×D₁₁	1	1	yes	characteristic
261	C₂×C₄	C₂×#4339	1	1	yes	characteristic
262	C₂×C₄	C₂×D₄×D₁₁	1	1	yes	characteristic
263	C₂×C₄	#67042	1	1	yes	characteristic
264	C₂×C₄	#67043	1	1	yes	characteristic
265	C₈	—	2	2	no	—
266	C₈	—	2	4	no	—
267	C₈	—	4	2	no	—
268	C₈	—	22	4	no	—
269	C₈	—	44	4	no	—
270	C₈	C₄×Dic₂₂	1	4	yes	—
271	C₂²	—	2	1	no	—
272	C₂²	—	2	1	no	—
273	C₂²	C₂×C₄⋊C₄×D₁₁	1	1	yes	characteristic
274	C₂²	Q₈×Dic₂₂	1	1	yes	characteristic
275	C₂²	#10623098	1	1	yes	characteristic
276	C₄	—	2	1	no	—
277	C₄	—	2	1	no	—
278	C₄	—	2	1	no	—
279	C₄	—	22	4	no	—
280	C₄	—	22	4	no	—
281	C₄	#10622506	1	2	yes	—
282	C₄	C₂×C₄×Dic₂₂	1	1	yes	characteristic
283	C₄	#10623054	1	1	yes	characteristic
284	C₄	D₄×Dic₂₂	1	1	yes	characteristic
285	C₄	#10623084	1	1	yes	characteristic
286	C₂	—	2	1	no	—
287	C₂	—	2	1	no	—
288	C₂	C₄⋊C₄×Dic₂₂	1	1	yes	characteristic
289	C₂	C₂×#10622506	1	1	yes	characteristic
290	C₂	#13077224	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₁₁
commutator	C₂×C₄×C₁₁	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₁
Fitting	C₄×C₁₁×#108	C₂
socle	C₂²×C₁₁	C₂²×C₄⋊C₄
prime radical	C₂³×C₁₁	C₂²×Q₈
cosocle	C₂²×C₄×C₁₁	C₂⁴
hypercenter	C₄×#108	D₁₁
hypocenter	C₁₁	#22733
LCS	G ▷ C₂×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#108

Complements

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

N	H ≅ G/N	type
46. C₁₁×#108	Q₈	semidirect
189. #108	Dic₂₂	semidirect
245. C₁₁	#22733	semidirect

Maximal subgroups

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

subgroup	index
M₁₆×Dic₂₂	2
C₂×#10622292	2
#13069391	2
#13069390	2
#13068789	2
#13068788	2
#13068524	2
C₄×C₁₁×#108	2
#22733	11