Subgroups — #14772672

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	Q₈×#4318	C₂	1	2	yes	maximal
2	#13069392	C₂	1	4	yes	maximal
3	#13069392	C₂	1	2	yes	maximal
4	#13068791	C₂	1	2	yes	maximal
5	#13068788	C₂	1	1	yes	maximal, characteristic
6	#13068522	C₂	1	2	yes	maximal
7	C₄×#66883	C₂	1	1	yes	maximal, characteristic
8	C₂×C₁₁×#446	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Q₈×C₁₁	C₂²	1	1	yes	characteristic
10	C₁₁×#446	C₄	1	4	yes	—
11	C₁₁×#446	C₄	1	4	yes	—
12	C₂×C₁₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₁₁×C₄⋊C₈	C₂²	1	2	yes	—
14	C₂×C₁₁×#97	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₁₁	C₂²	1	1	yes	characteristic
16	C₂×#66883	C₂²	1	1	yes	characteristic
17	C₂×#66883	C₂²	1	2	yes	—
18	C₂×#66883	C₂²	1	2	yes	—
19	C₂×#66883	C₂²	1	4	yes	—
20	C₂×#66883	C₂²	1	1	yes	characteristic
21	#10622637	—	2	2	no	—
22	#10622637	—	2	4	no	—
23	Q₈×C₁₁⋊C₈	—	2	2	no	—
24	#10622537	C₂²	1	2	yes	—
25	#10622537	C₂²	1	4	yes	—
26	C₄×#4318	C₂²	1	2	yes	—
27	C₄×#4318	C₂²	1	4	yes	—
28	#10622190	C₂²	1	2	yes	—
29	#10622190	C₂²	1	4	yes	—
30	C₂²×Q₈×C₁₁	Q₈	1	1	yes	characteristic
31	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
32	C₂²×C₈×C₁₁	C₂³	1	2	yes	—
33	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
34	C₄×Q₈×C₁₁	C₂×C₄	1	2	yes	—
35	C₄×Q₈×C₁₁	C₂×C₄	1	2	yes	—
36	C₄×Q₈×C₁₁	D₄	1	2	yes	—
37	C₄×Q₈×C₁₁	D₄	1	2	yes	—
38	C₂×C₁₁×C₄⋊C₄	Q₈	1	1	yes	characteristic
39	C₂×C₁₁×C₄⋊C₄	Q₈	1	2	yes	—
40	C₂×C₄²×C₁₁	C₂³	1	1	yes	characteristic
41	C₂×C₄²×C₁₁	C₂³	1	2	yes	—
42	C₂×#4318	C₂³	1	2	yes	—
43	C₂×#4318	C₂³	1	4	yes	—
44	C₂×#4318	C₂³	1	2	yes	—
45	C₁₁×C₄⋊C₈	C₂×C₄	1	2	yes	—
46	C₁₁×C₄⋊C₈	C₂×C₄	1	2	yes	—
47	C₁₁×C₄⋊C₈	C₂×C₄	1	8	yes	—
48	C₁₁×#97	C₂×C₄	1	8	yes	—
49	C₄×C₈×C₁₁	C₂×C₄	1	2	yes	—
50	C₄×C₈×C₁₁	C₂×C₄	1	2	yes	—
51	#66883	—	2	2	no	—
52	#66883	—	2	4	no	—
53	#66883	—	4	2	no	—
54	#66883	—	4	4	no	—
55	#66883	Q₈	1	4	yes	complemented
56	C₄⋊(C₁₁⋊C₈)	—	2	2	no	—
57	C₄⋊(C₁₁⋊C₈)	—	2	4	no	—
58	C₄⋊(C₁₁⋊C₈)	—	2	4	no	—
59	C₄⋊(C₁₁⋊C₈)	—	2	2	no	—
60	#66867	—	2	4	no	—
61	#66867	—	2	2	no	—
62	C₄×C₁₁⋊C₈	—	2	2	no	—
63	C₄×C₁₁⋊C₈	—	2	4	no	—
64	#22733	—	11	1	no	maximal, Sylow
65	C₂×Q₈×C₁₁	—	4	2	no	—
66	C₂×Q₈×C₁₁	C₄⋊C₄	1	2	yes	—
67	C₂×Q₈×C₁₁	C₄⋊C₄	1	2	yes	—
68	C₂²×C₄×C₁₁	C₂×Q₈	1	1	yes	characteristic
69	C₂²×C₄×C₁₁	C₂×Q₈	1	1	yes	characteristic
70	C₂²×C₄×C₁₁	C₂×Q₈	1	2	yes	—
71	C₂²×C₄×C₁₁	C₂×Q₈	1	2	yes	—
72	C₂²×C₄×C₁₁	C₂⁴	1	1	yes	characteristic
73	C₂×C₈×C₁₁	—	2	1	no	—
74	C₂×C₈×C₁₁	—	2	2	no	—
75	C₂×C₈×C₁₁	—	2	1	no	—
76	C₂×C₈×C₁₁	—	2	2	no	—
77	C₂×C₈×C₁₁	C₂²×C₄	1	2	yes	—
78	C₂×C₈×C₁₁	C₂²×C₄	1	4	yes	—
79	C₂×C₈×C₁₁	C₂²×C₄	1	2	yes	—
80	C₂×C₈×C₁₁	C₂×Q₈	1	2	yes	—
81	C₂×C₈×C₁₁	#41	1	2	yes	—
82	C₁₁×C₄⋊C₄	C₄⋊C₄	1	4	yes	—
83	C₁₁×C₄⋊C₄	C₄⋊C₄	1	4	yes	—
84	C₁₁×C₄⋊C₄	C₄⋊C₄	1	2	yes	—
85	C₁₁×C₄⋊C₄	C₄⋊C₄	1	2	yes	—
86	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
87	C₄²×C₁₁	C₂²×C₄	1	1	yes	characteristic
88	C₄²×C₁₁	C₂²×C₄	1	4	yes	—
89	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
90	C₄²×C₁₁	C₂×D₄	1	4	yes	—
91	C₄²×C₁₁	C₂×D₄	1	1	yes	characteristic
92	#4318	—	2	2	no	—
93	#4318	—	2	4	no	—
94	#4318	—	2	4	no	—
95	#4318	—	2	2	no	—
96	#4318	—	2	2	no	—
97	#4318	C₂×Q₈	1	4	yes	—
98	C₂×C₁₁⋊C₈	—	2	2	no	—
99	C₂×C₁₁⋊C₈	—	2	2	no	—
100	C₂×C₁₁⋊C₈	—	2	4	no	—
101	Q₈×M₁₆	—	11	2	no	—
102	C₂×#446	D₁₁	1	1	yes	characteristic
103	#1950	—	11	4	no	—
104	#1950	—	11	2	no	—
105	#1849	—	11	2	no	—
106	#1848	—	11	1	no	—
107	#1778	—	11	2	no	—
108	C₄×#108	—	11	1	no	—
109	C₂³×C₁₁	C₂²×Q₈	1	1	yes	characteristic
110	Q₈×C₁₁	—	2	4	no	—
111	Q₈×C₁₁	—	4	2	no	—
112	C₂×C₄×C₁₁	—	2	2	no	—
113	C₂×C₄×C₁₁	—	4	1	no	—
114	C₂×C₄×C₁₁	—	4	2	no	—
115	C₂×C₄×C₁₁	—	4	2	no	—
116	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	4	yes	—
119	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	4	yes	—
121	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
123	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
124	C₂×C₄×C₁₁	C₂²×Q₈	1	1	yes	characteristic
125	C₂×C₄×C₁₁	C₂×#41	1	1	yes	characteristic
126	C₂×C₄×C₁₁	H(5,𝔽₂)	1	1	yes	characteristic
127	C₂×C₄×C₁₁	2₋¹⁺⁴	1	1	yes	characteristic
128	C₈×C₁₁	—	2	2	no	—
129	C₈×C₁₁	—	2	4	no	—
130	C₈×C₁₁	C₄×Q₈	1	4	yes	—
131	C₁₁⋊C₈	—	2	4	no	—
132	C₁₁⋊C₈	—	4	2	no	—
133	C₁₁⋊C₈	—	4	4	no	—
134	C₂×C₄×Q₈	C₂×D₁₁	1	1	yes	characteristic
135	#446	—	22	2	no	—
136	#446	—	22	4	no	—
137	#446	Dic₁₁	1	4	yes	—
138	#446	Dic₁₁	1	4	yes	—
139	C₈×Q₈	—	22	2	no	—
140	C₂×#108	—	11	1	no	—
141	C₂×#108	—	11	2	no	—
142	C₂×#108	—	11	1	no	—
143	C₂×#108	—	11	2	no	—
144	C₂×#108	—	11	4	no	—
145	#423	—	11	2	no	—
146	#423	—	11	4	no	—
147	C₂×C₄⋊C₈	C₂×D₁₁	1	1	yes	characteristic
148	C₂×C₄⋊C₈	C₂×D₁₁	1	2	yes	—
149	C₄×M₁₆	—	11	2	no	—
150	C₄×M₁₆	—	11	4	no	—
151	C₂×#97	C₂×D₁₁	1	2	yes	—
152	C₂×C₄×C₈	C₂×D₁₁	1	1	yes	characteristic
153	#341	—	11	2	no	—
154	#341	—	11	4	no	—
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	4	no	—
162	C₄×C₁₁	—	2	1	no	—
163	C₄×C₁₁	—	2	1	no	—
164	C₄×C₁₁	—	2	4	no	—
165	C₄×C₁₁	#430	1	2	yes	—
166	C₄×C₁₁	C₂×C₄×Q₈	1	1	yes	characteristic
167	C₄×C₁₁	#520	1	1	yes	characteristic
168	C₄×C₁₁	D₄×Q₈	1	1	yes	characteristic
169	C₄×C₁₁	#550	1	1	yes	characteristic
170	C₂²×Q₈	Dic₂₂	1	1	yes	characteristic
171	C₂×M₁₆	—	11	2	no	—
172	C₂×M₁₆	—	11	2	no	—
173	C₂×M₁₆	—	11	4	no	—
174	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
175	C₂²×C₈	C₂²×D₁₁	1	2	yes	—
176	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
177	C₄×Q₈	D₄₄	1	2	yes	—
178	C₄×Q₈	D₄₄	1	2	yes	—
179	C₄×Q₈	C₂×Dic₁₁	1	2	yes	—
180	C₄×Q₈	C₂×Dic₁₁	1	2	yes	—
181	C₂×C₄⋊C₄	Dic₂₂	1	1	yes	characteristic
182	C₂×C₄⋊C₄	Dic₂₂	1	2	yes	—
183	C₂×C₄²	C₂²×D₁₁	1	1	yes	characteristic
184	C₂×C₄²	C₂²×D₁₁	1	2	yes	—
185	#108	—	11	4	no	—
186	#108	—	22	4	no	—
187	#108	—	22	2	no	—
188	#108	—	44	2	no	—
189	#108	—	44	4	no	—
190	C₄⋊C₈	—	22	2	no	—
191	C₄⋊C₈	—	22	4	no	—
192	C₄⋊C₈	—	22	4	no	—
193	C₄⋊C₈	—	22	2	no	—
194	C₄⋊C₈	C₂×Dic₁₁	1	2	yes	—
195	C₄⋊C₈	C₂×Dic₁₁	1	2	yes	—
196	C₄⋊C₈	C₂×Dic₁₁	1	8	yes	—
197	#97	—	22	4	no	—
198	#97	—	22	2	no	—
199	#97	C₂×Dic₁₁	1	8	yes	—
200	C₄×C₈	—	22	2	no	—
201	C₄×C₈	—	22	4	no	—
202	C₄×C₈	C₂×Dic₁₁	1	2	yes	—
203	C₄×C₈	C₂×Dic₁₁	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	—	4	2	no	—
210	C₂×Q₈	C₄⋊Dic₁₁	1	2	yes	—
211	C₂×Q₈	C₄⋊Dic₁₁	1	2	yes	—
212	C₂²×C₄	C₂×Dic₂₂	1	1	yes	characteristic
213	C₂²×C₄	C₂×Dic₂₂	1	1	yes	characteristic
214	C₂²×C₄	C₂×Dic₂₂	1	2	yes	—
215	C₂²×C₄	C₂×Dic₂₂	1	2	yes	—
216	C₂²×C₄	C₂³×D₁₁	1	1	yes	characteristic
217	M₁₆	—	11	4	no	—
218	M₁₆	—	22	2	no	—
219	M₁₆	—	22	4	no	—
220	M₁₆	—	22	2	no	—
221	M₁₆	—	22	4	no	—
222	M₁₆	—	22	2	no	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	2	1	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	22	2	no	—
228	C₂×C₈	—	22	2	no	—
229	C₂×C₈	—	22	4	no	—
230	C₂×C₈	Q₈×D₁₁	1	2	yes	—
231	C₂×C₈	#4343	1	2	yes	—
232	C₂×C₈	C₂²×Dic₁₁	1	2	yes	—
233	C₂×C₈	C₂²×Dic₁₁	1	4	yes	—
234	C₂×C₈	C₂²×Dic₁₁	1	2	yes	—
235	C₄⋊C₄	C₄⋊Dic₁₁	1	4	yes	—
236	C₄⋊C₄	C₄⋊Dic₁₁	1	2	yes	—
237	C₄⋊C₄	C₄⋊Dic₁₁	1	4	yes	—
238	C₄⋊C₄	C₄⋊Dic₁₁	1	2	yes	—
239	C₄²	C₂×D₄₄	1	1	yes	characteristic
240	C₄²	C₂×D₄₄	1	4	yes	—
241	C₄²	C₂×D₄₄	1	1	yes	characteristic
242	C₄²	C₂²×Dic₁₁	1	1	yes	characteristic
243	C₄²	C₂²×Dic₁₁	1	1	yes	characteristic
244	C₄²	C₂²×Dic₁₁	1	4	yes	—
245	C₁₁	#22733	1	1	yes	characteristic, complemented, Sylow
246	C₂³	C₂²×Dic₂₂	1	1	yes	characteristic
247	Q₈	—	2	4	no	—
248	Q₈	—	4	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	4	1	no	—
251	C₂×C₄	—	4	2	no	—
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	C₂×C₄⋊Dic₁₁	1	1	yes	characteristic
254	C₂×C₄	C₂×C₄⋊Dic₁₁	1	4	yes	—
255	C₂×C₄	C₂×C₄⋊Dic₁₁	1	1	yes	characteristic
256	C₂×C₄	C₂×C₄⋊Dic₁₁	1	1	yes	characteristic
257	C₂×C₄	C₂×C₄⋊Dic₁₁	1	4	yes	—
258	C₂×C₄	C₂×C₄⋊Dic₁₁	1	1	yes	characteristic
259	C₂×C₄	C₂²×D₄₄	1	1	yes	characteristic
260	C₂×C₄	C₂×Q₈×D₁₁	1	1	yes	characteristic
261	C₂×C₄	C₂×#4343	1	1	yes	characteristic
262	C₂×C₄	#67042	1	1	yes	characteristic
263	C₂×C₄	#67043	1	1	yes	characteristic
264	C₂×C₄	C₂³×Dic₁₁	1	1	yes	characteristic
265	C₈	—	2	2	no	—
266	C₈	—	2	4	no	—
267	C₈	—	22	4	no	—
268	C₈	—	44	2	no	—
269	C₈	—	44	4	no	—
270	C₈	Q₈×Dic₁₁	1	4	yes	—
271	C₂²	—	2	1	no	—
272	C₂²	—	2	1	no	—
273	C₂²	Q₈×Dic₂₂	1	1	yes	characteristic
274	C₂²	#10623101	1	1	yes	characteristic
275	C₂²	C₂²×C₄⋊Dic₁₁	1	1	yes	characteristic
276	C₄	—	2	1	no	—
277	C₄	—	2	4	no	—
278	C₄	—	2	1	no	—
279	C₄	—	2	4	no	—
280	C₄	—	2	1	no	—
281	C₄	#10622738	1	2	yes	—
282	C₄	Q₈×D₄₄	1	1	yes	characteristic
283	C₄	#10623108	1	1	yes	characteristic
284	C₄	C₂×Q₈×Dic₁₁	1	1	yes	characteristic
285	C₄	#10623358	1	1	yes	characteristic
286	C₂	—	2	1	no	—
287	C₂	—	2	1	no	—
288	C₂	Q₈×C₄⋊Dic₁₁	1	1	yes	characteristic
289	C₂	C₂×#10622738	1	1	yes	characteristic
290	C₂	#13077227	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₄₄
commutator	C₂×C₄×C₁₁	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₁
Fitting	C₂×C₁₁×#446	C₂
socle	C₂²×C₁₁	C₂²×C₄⋊C₄
prime radical	C₂³×C₁₁	C₂²×Q₈
cosocle	C₂²×C₄×C₁₁	C₂⁴
hypercenter	C₂×#446	D₁₁
hypocenter	C₁₁	#22733
LCS	G ▷ C₂×C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×Q₈ ◁ C₂×#446

Complements

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

N	H ≅ G/N	type
55. #66883	Q₈	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
Q₈×#4318	2
#13069392	2
#13068791	2
#13068788	2
#13068522	2
C₄×#66883	2
C₂×C₁₁×#446	2
#22733	11