Subgroups — #14772676

Order 2816.

876 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×C₁₁⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₄×Q₈×Dic₁₁	C₂	1	1	yes	maximal, characteristic
3	#13069334	C₂	1	3	yes	maximal
4	#13069332	C₂	1	3	yes	maximal
5	#13068732	C₂	1	3	yes	maximal
6	C₄×Dic₁₁⋊C₈	C₂	1	3	yes	maximal
7	C₂×C₈×Q₈×C₁₁	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄×Q₈×C₁₁	C₂²	1	1	yes	characteristic
9	C₂×Q₈×Dic₁₁	C₄	1	1	yes	characteristic
10	C₂×Q₈×Dic₁₁	C₄	1	1	yes	characteristic
11	C₈×Q₈×C₁₁	—	2	4	no	—
12	C₂×C₁₁×C₄⋊C₈	C₂²	1	3	yes	—
13	C₂×C₄×C₈×C₁₁	C₂²	1	3	yes	—
14	C₂×Dic₁₁⋊C₈	C₂²	1	3	yes	—
15	C₂×Dic₁₁⋊C₈	C₂²	1	3	yes	—
16	Q₈×C₁₁⋊C₈	—	2	4	no	—
17	C₄⋊C₄×Dic₁₁	C₄	1	3	yes	—
18	C₄⋊C₄×Dic₁₁	C₄	1	3	yes	—
19	C₄×C₄⋊Dic₁₁	C₂²	1	3	yes	—
20	C₄²×Dic₁₁	C₂²	1	3	yes	—
21	C₂×C₄⋊(C₁₁⋊C₈)	C₂²	1	3	yes	—
22	C₂×C₄×C₁₁⋊C₈	C₂²	1	3	yes	—
23	#10622188	C₂²	1	6	yes	—
24	#10622188	C₂²	1	1	yes	characteristic
25	#10622188	C₂²	1	3	yes	—
26	C₂²×Q₈×C₁₁	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₁	C₂³	1	3	yes	—
28	C₂²×C₈×C₁₁	C₂³	1	1	yes	characteristic
29	C₄×Q₈×C₁₁	D₄	1	4	yes	—
30	C₄×Q₈×C₁₁	Q₈	1	4	yes	—
31	C₂×C₁₁×C₄⋊C₄	C₂×C₄	1	3	yes	—
32	C₂×C₄²×C₁₁	C₂³	1	3	yes	—
33	Q₈×Dic₁₁	—	2	4	no	—
34	Q₈×Dic₁₁	C₈	1	8	yes	complemented
35	C₂×C₄⋊Dic₁₁	C₂×C₄	1	3	yes	—
36	C₂×C₄⋊Dic₁₁	C₂×C₄	1	3	yes	—
37	C₂×C₄×Dic₁₁	C₂×C₄	1	3	yes	—
38	C₂×C₄×Dic₁₁	C₂×C₄	1	3	yes	—
39	C₂×C₄×Dic₁₁	C₂×C₄	1	3	yes	—
40	C₂×C₄×Dic₁₁	C₂×C₄	1	3	yes	—
41	C₂×C₄×Dic₁₁	C₂³	1	1	yes	characteristic
42	C₂×C₄×Dic₁₁	C₂³	1	3	yes	—
43	C₂²×C₁₁⋊C₈	C₂³	1	3	yes	—
44	C₂²×C₁₁⋊C₈	C₂³	1	1	yes	characteristic
45	C₁₁×C₄⋊C₈	—	2	6	no	—
46	C₄×C₈×C₁₁	—	2	6	no	—
47	C₁₁⋊#95	C₂×C₄	1	6	yes	—
48	Dic₁₁⋊C₈	—	2	6	no	—
49	Dic₁₁⋊C₈	—	2	6	no	—
50	C₄⋊(C₁₁⋊C₈)	—	2	6	no	—
51	C₄×C₁₁⋊C₈	—	2	6	no	—
52	#22734	—	11	1	no	maximal, Sylow
53	C₂×Q₈×C₁₁	C₄⋊C₄	1	4	yes	—
54	C₂×Q₈×C₁₁	C₂×C₈	1	4	yes	—
55	C₂×Q₈×C₁₁	M₁₆	1	4	yes	—
56	C₂²×C₄×C₁₁	C₂²×C₄	1	3	yes	—
57	C₂²×C₄×C₁₁	C₂²×C₄	1	3	yes	—
58	C₂²×C₄×C₁₁	C₂⁴	1	1	yes	characteristic
59	C₂²×Dic₁₁	C₂²×C₄	1	3	yes	—
60	C₂²×Dic₁₁	C₂²×C₄	1	3	yes	—
61	C₂²×Dic₁₁	C₂²×C₄	1	1	yes	characteristic
62	C₂²×Dic₁₁	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₈×C₁₁	—	2	1	no	—
64	C₂×C₈×C₁₁	—	2	3	no	—
65	C₂×C₈×C₁₁	—	2	3	no	—
66	C₂×C₈×C₁₁	—	2	1	no	—
67	C₂×C₈×C₁₁	—	2	3	no	—
68	C₂×C₈×C₁₁	#41	1	2	yes	—
69	C₁₁×C₄⋊C₄	C₄⋊C₄	1	12	yes	—
70	C₄²×C₁₁	C₂×D₄	1	6	yes	—
71	C₄²×C₁₁	C₂×Q₈	1	6	yes	—
72	C₄⋊Dic₁₁	—	2	6	no	—
73	C₄⋊Dic₁₁	C₂×C₈	1	12	yes	—
74	C₄×Dic₁₁	—	2	6	no	—
75	C₄×Dic₁₁	—	2	6	no	—
76	C₄×Dic₁₁	—	2	3	no	—
77	C₄×Dic₁₁	—	2	3	no	—
78	C₄×Dic₁₁	—	2	1	no	—
79	C₄×Dic₁₁	—	2	1	no	—
80	C₄×Dic₁₁	—	4	3	no	—
81	C₄×Dic₁₁	C₂×C₈	1	12	yes	—
82	C₂×C₁₁⋊C₈	—	2	1	no	—
83	C₂×C₁₁⋊C₈	—	2	3	no	—
84	C₂×C₁₁⋊C₈	—	2	3	no	—
85	C₂×C₁₁⋊C₈	—	2	1	no	—
86	C₂×C₁₁⋊C₈	—	2	3	no	—
87	C₂×C₁₁⋊C₈	#41	1	2	yes	—
88	C₂×C₈×Q₈	—	11	1	no	—
89	C₂×C₈×Q₈	D₁₁	1	1	yes	characteristic
90	C₄²×Q₈	—	11	1	no	—
91	#1939	—	11	3	no	—
92	#1939	—	11	3	no	—
93	#1831	—	11	3	no	—
94	C₄×C₄⋊C₈	—	11	3	no	—
95	C₂³×C₁₁	C₂³×C₄	1	1	yes	characteristic
96	Q₈×C₁₁	C₄⋊C₈	1	16	yes	complemented
97	C₂×C₄×C₁₁	—	2	3	no	—
98	C₂×C₄×C₁₁	—	2	3	no	—
99	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	6	yes	—
100	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	6	yes	—
101	C₂×C₄×C₁₁	C₂²×C₈	1	6	yes	—
102	C₂×C₄×C₁₁	C₂×M₁₆	1	6	yes	—
103	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
104	C₂×C₄×C₁₁	C₂²×Q₈	1	1	yes	characteristic
105	C₂×C₄×C₁₁	C₂×#41	1	1	yes	characteristic
106	C₂×C₄×C₁₁	C₂×#41	1	1	yes	characteristic
107	C₂×C₄×C₁₁	2₋¹⁺⁴	1	1	yes	characteristic
108	C₂×C₄×C₁₁	2₋¹⁺⁴	1	1	yes	characteristic
109	C₂×Dic₁₁	—	2	3	no	—
110	C₂×Dic₁₁	—	2	1	no	—
111	C₂×Dic₁₁	—	2	1	no	—
112	C₂×Dic₁₁	—	2	1	no	—
113	C₂×Dic₁₁	—	2	3	no	—
114	C₂×Dic₁₁	—	2	1	no	—
115	C₂×Dic₁₁	—	2	3	no	—
116	C₂×Dic₁₁	—	2	3	no	—
117	C₂×Dic₁₁	—	2	3	no	—
118	C₂×Dic₁₁	C₂²×C₈	1	2	yes	—
119	C₂×Dic₁₁	C₂²×C₈	1	6	yes	—
120	C₂×Dic₁₁	#131	1	2	yes	—
121	C₈×C₁₁	—	2	2	no	—
122	C₈×C₁₁	—	4	3	no	—
123	C₁₁⋊C₈	—	2	2	no	—
124	C₁₁⋊C₈	—	4	3	no	—
125	C₂×C₄×Q₈	—	11	1	no	—
126	C₂×C₄×Q₈	—	11	1	no	—
127	C₂×C₄×Q₈	C₂×D₁₁	1	1	yes	characteristic
128	C₈×Q₈	—	2	4	no	—
129	C₈×Q₈	—	22	4	no	—
130	C₂×C₄⋊C₈	—	11	3	no	—
131	C₂×C₄⋊C₈	—	11	3	no	—
132	C₂×C₄⋊C₈	—	11	3	no	—
133	C₂×C₄⋊C₈	C₂×D₁₁	1	3	yes	—
134	C₂×C₄×C₈	—	11	3	no	—
135	C₂×C₄×C₈	C₂×D₁₁	1	3	yes	—
136	C₄×C₄⋊C₄	—	11	3	no	—
137	C₄×C₄⋊C₄	—	11	3	no	—
138	C₄×C₄⋊C₄	—	11	3	no	—
139	C₄³	—	11	3	no	—
140	#336	—	11	3	no	—
141	#336	—	11	6	no	—
142	#336	—	11	1	no	—
143	C₂²×C₁₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
144	C₂²×C₁₁	C₄×#41	1	1	yes	characteristic
145	C₂²×C₁₁	#519	1	1	yes	characteristic
146	C₂²×C₁₁	C₂³×C₈	1	1	yes	characteristic
147	C₂²×C₁₁	C₂²×M₁₆	1	1	yes	characteristic
148	C₂²×C₁₁	C₂×#131	1	1	yes	characteristic
149	C₂²×C₁₁	#568	1	1	yes	characteristic
150	C₄×C₁₁	—	2	6	no	—
151	C₄×C₁₁	C₂×C₄⋊C₈	1	12	yes	—
152	C₄×C₁₁	#548	1	1	yes	characteristic
153	C₄×C₁₁	#548	1	1	yes	characteristic
154	C₄×C₁₁	#557	1	1	yes	characteristic
155	C₄×C₁₁	#557	1	1	yes	characteristic
156	Dic₁₁	—	2	6	no	—
157	Dic₁₁	—	2	2	no	—
158	Dic₁₁	—	2	2	no	—
159	Dic₁₁	—	4	3	no	—
160	C₂²×Q₈	C₄×D₁₁	1	1	yes	characteristic
161	C₂²×C₈	—	11	1	no	—
162	C₂²×C₈	—	11	3	no	—
163	C₂²×C₈	C₂²×D₁₁	1	3	yes	—
164	C₂²×C₈	C₂²×D₁₁	1	1	yes	characteristic
165	C₄×Q₈	—	11	8	no	—
166	C₄×Q₈	—	22	4	no	—
167	C₄×Q₈	Dic₂₂	1	4	yes	—
168	C₄×Q₈	C₂²⋊D₁₁	1	4	yes	—
169	C₂×C₄⋊C₄	—	11	3	no	—
170	C₂×C₄⋊C₄	—	11	3	no	—
171	C₂×C₄⋊C₄	C₄×D₁₁	1	3	yes	—
172	C₂×C₄²	—	11	3	no	—
173	C₂×C₄²	—	11	3	no	—
174	C₂×C₄²	—	11	3	no	—
175	C₂×C₄²	—	11	1	no	—
176	C₂×C₄²	—	11	3	no	—
177	C₂×C₄²	—	11	3	no	—
178	C₂×C₄²	C₂²×D₁₁	1	3	yes	—
179	C₄⋊C₈	—	2	6	no	—
180	C₄⋊C₈	—	22	6	no	—
181	C₄⋊C₈	—	22	6	no	—
182	C₄⋊C₈	—	22	6	no	—
183	C₄×C₈	—	2	6	no	—
184	C₄×C₈	—	22	6	no	—
185	#95	—	11	6	no	—
186	C₂×C₁₁	#2351	1	1	yes	characteristic
187	C₂×C₁₁	C₂²×C₄⋊C₈	1	1	yes	characteristic
188	C₂×C₁₁	#2906	1	1	yes	characteristic
189	C₂×C₁₁	C₈×#41	1	1	yes	characteristic
190	C₂×C₁₁	#2971	1	1	yes	characteristic
191	C₂×C₁₁	#2982	1	1	yes	characteristic
192	C₂×C₁₁	#2984	1	1	yes	characteristic
193	C₂×Q₈	C₈×D₁₁	1	4	yes	—
194	C₂×Q₈	#4313	1	4	yes	—
195	C₂×Q₈	Dic₁₁⋊C₄	1	4	yes	—
196	C₂²×C₄	—	11	3	no	—
197	C₂²×C₄	—	11	1	no	—
198	C₂²×C₄	—	11	1	no	—
199	C₂²×C₄	—	11	3	no	—
200	C₂²×C₄	C₂×C₄×D₁₁	1	3	yes	—
201	C₂²×C₄	C₂×C₄×D₁₁	1	3	yes	—
202	C₂²×C₄	C₂³×D₁₁	1	1	yes	characteristic
203	C₂×C₈	—	2	1	no	—
204	C₂×C₈	—	2	1	no	—
205	C₂×C₈	—	2	3	no	—
206	C₂×C₈	—	2	3	no	—
207	C₂×C₈	—	2	3	no	—
208	C₂×C₈	—	11	2	no	—
209	C₂×C₈	—	22	1	no	—
210	C₂×C₈	—	22	1	no	—
211	C₂×C₈	—	22	3	no	—
212	C₂×C₈	—	22	3	no	—
213	C₂×C₈	—	22	3	no	—
214	C₂×C₈	#4343	1	2	yes	—
215	C₄⋊C₄	—	11	12	no	—
216	C₄⋊C₄	—	22	6	no	—
217	C₄⋊C₄	Dic₁₁⋊C₄	1	12	yes	—
218	C₄²	—	11	12	no	—
219	C₄²	—	22	6	no	—
220	C₄²	—	22	6	no	—
221	C₄²	—	22	3	no	—
222	C₄²	—	22	1	no	—
223	C₄²	—	22	3	no	—
224	C₄²	—	22	1	no	—
225	C₄²	—	44	3	no	—
226	C₄²	C₂×Dic₂₂	1	6	yes	—
227	C₄²	C₂×C₂²⋊D₁₁	1	6	yes	—
228	C₁₁	#22734	1	1	yes	characteristic, complemented, Sylow
229	C₂³	C₂²×C₄×D₁₁	1	1	yes	characteristic
230	Q₈	Dic₁₁⋊C₈	1	16	yes	complemented
231	C₂×C₄	—	2	3	no	—
232	C₂×C₄	—	2	3	no	—
233	C₂×C₄	—	11	2	no	—
234	C₂×C₄	—	11	6	no	—
235	C₂×C₄	—	11	2	no	—
236	C₂×C₄	—	22	1	no	—
237	C₂×C₄	—	22	3	no	—
238	C₂×C₄	—	22	1	no	—
239	C₂×C₄	—	22	1	no	—
240	C₂×C₄	—	22	1	no	—
241	C₂×C₄	—	22	3	no	—
242	C₂×C₄	—	22	3	no	—
243	C₂×C₄	—	22	3	no	—
244	C₂×C₄	—	22	3	no	—
245	C₂×C₄	C₂×C₈×D₁₁	1	6	yes	—
246	C₂×C₄	C₂×#4313	1	6	yes	—
247	C₂×C₄	C₂×Dic₁₁⋊C₄	1	6	yes	—
248	C₂×C₄	C₂×Dic₁₁⋊C₄	1	6	yes	—
249	C₂×C₄	C₂²×Dic₂₂	1	1	yes	characteristic
250	C₂×C₄	C₂×#4343	1	1	yes	characteristic
251	C₂×C₄	#67040	1	1	yes	characteristic
252	C₂×C₄	#41×D₁₁	1	1	yes	characteristic
253	C₂×C₄	#67043	1	1	yes	characteristic
254	C₂×C₄	C₂²×C₂²⋊D₁₁	1	1	yes	characteristic
255	C₈	—	2	2	no	—
256	C₈	—	4	3	no	—
257	C₈	—	22	2	no	—
258	C₈	—	44	3	no	—
259	C₂²	#10623103	1	1	yes	characteristic
260	C₂²	C₄×#4343	1	1	yes	characteristic
261	C₂²	C₂²×C₈×D₁₁	1	1	yes	characteristic
262	C₂²	C₂²×#4313	1	1	yes	characteristic
263	C₂²	D₁₁×#131	1	1	yes	characteristic
264	C₂²	#10623281	1	1	yes	characteristic
265	C₂²	C₂²×Dic₁₁⋊C₄	1	1	yes	characteristic
266	C₄	—	2	6	no	—
267	C₄	—	22	6	no	—
268	C₄	—	22	2	no	—
269	C₄	—	22	2	no	—
270	C₄	—	44	3	no	—
271	C₄	C₂×Dic₁₁⋊C₈	1	12	yes	—
272	C₄	#10623100	1	1	yes	characteristic
273	C₄	#10623101	1	1	yes	characteristic
274	C₄	#10623348	1	1	yes	characteristic
275	C₄	#10623362	1	1	yes	characteristic
276	C₂	#13071534	1	1	yes	characteristic
277	C₂	C₂²×Dic₁₁⋊C₈	1	1	yes	characteristic
278	C₂	#13077182	1	1	yes	characteristic
279	C₂	C₈×#4343	1	1	yes	characteristic
280	C₂	#13077635	1	1	yes	characteristic
281	C₂	#13077695	1	1	yes	characteristic
282	C₂	#13077704	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₁₁
commutator	C₂²×C₁₁	C₂³×C₈
Fitting	C₂×C₈×Q₈×C₁₁	C₂
socle, prime radical	C₂³×C₁₁	C₂³×C₄
cosocle	C₂²×C₄×C₁₁	C₂⁴
hypercenter	C₂×C₈×Q₈	D₁₁
hypocenter	C₁₁	#22734
LCS	G ▷ C₂²×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
34. Q₈×Dic₁₁	C₈	semidirect
96. Q₈×C₁₁	C₄⋊C₈	semidirect
228. C₁₁	#22734	semidirect
230. Q₈	Dic₁₁⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₈×C₁₁⋊C₈	2
C₄×Q₈×Dic₁₁	2
#13069334	2
#13069332	2
#13068732	2
C₄×Dic₁₁⋊C₈	2
C₂×C₈×Q₈×C₁₁	2
#22734	11