Subgroups — #14772679

Order 2816.

876 conjugacy classes of subgroups.

The subgroup lattice has 234 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	2	yes	maximal
2	#13069335	C₂	1	6	yes	maximal
3	#13068733	C₂	1	3	yes	maximal
4	C₄×C₄⋊(C₁₁⋊C₈)	C₂	1	3	yes	maximal
5	C₄²×Q₈×C₁₁	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Q₈×C₁₁	C₄	1	1	yes	characteristic
7	C₂×C₄×Q₈×C₁₁	C₄	1	1	yes	characteristic
8	C₂×C₄×Q₈×C₁₁	C₂²	1	1	yes	characteristic
9	C₄×C₁₁×C₄⋊C₄	C₄	1	3	yes	—
10	C₄×C₁₁×C₄⋊C₄	C₄	1	3	yes	—
11	C₄×C₁₁×C₄⋊C₄	C₂²	1	3	yes	—
12	C₄³×C₁₁	C₂²	1	3	yes	—
13	Q₈×C₁₁⋊C₈	—	2	8	no	—
14	C₂×C₄⋊(C₁₁⋊C₈)	C₂²	1	6	yes	—
15	C₂×C₄⋊(C₁₁⋊C₈)	C₂²	1	6	yes	—
16	C₂×C₄×C₁₁⋊C₈	C₂²	1	6	yes	—
17	#10622162	C₂²	1	6	yes	—
18	#10622162	C₂²	1	1	yes	characteristic
19	#10622162	C₂²	1	3	yes	—
20	C₂²×Q₈×C₁₁	C₂×C₄	1	1	yes	characteristic
21	C₄×Q₈×C₁₁	—	2	4	no	—
22	C₄×Q₈×C₁₁	C₈	1	8	yes	complemented
23	C₄×Q₈×C₁₁	D₄	1	4	yes	—
24	C₄×Q₈×C₁₁	Q₈	1	4	yes	—
25	C₂×C₁₁×C₄⋊C₄	C₂×C₄	1	3	yes	—
26	C₂×C₁₁×C₄⋊C₄	C₂×C₄	1	3	yes	—
27	C₂×C₁₁×C₄⋊C₄	C₂×C₄	1	3	yes	—
28	C₂×C₄²×C₁₁	C₂×C₄	1	3	yes	—
29	C₂×C₄²×C₁₁	C₂×C₄	1	3	yes	—
30	C₂×C₄²×C₁₁	C₂×C₄	1	3	yes	—
31	C₂×C₄²×C₁₁	C₂×C₄	1	3	yes	—
32	C₂×C₄²×C₁₁	C₂³	1	3	yes	—
33	C₂×C₄²×C₁₁	C₂³	1	1	yes	characteristic
34	C₂×C₄²×C₁₁	C₂³	1	3	yes	—
35	C₂²×C₁₁⋊C₈	C₂³	1	6	yes	—
36	C₂²×C₁₁⋊C₈	C₂³	1	2	yes	—
37	C₁₁×#95	C₂×C₄	1	6	yes	—
38	C₄⋊(C₁₁⋊C₈)	—	2	12	no	—
39	C₄⋊(C₁₁⋊C₈)	—	2	12	no	—
40	C₄×C₁₁⋊C₈	—	2	12	no	—
41	#22734	—	11	1	no	maximal, Sylow
42	C₂×Q₈×C₁₁	C₄⋊C₄	1	4	yes	—
43	C₂×Q₈×C₁₁	C₂×C₈	1	4	yes	—
44	C₂×Q₈×C₁₁	M₁₆	1	4	yes	—
45	C₂²×C₄×C₁₁	C₂²×C₄	1	3	yes	—
46	C₂²×C₄×C₁₁	C₂²×C₄	1	3	yes	—
47	C₂²×C₄×C₁₁	C₂²×C₄	1	3	yes	—
48	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₁	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₁₁	C₂²×C₄	1	3	yes	—
51	C₂²×C₄×C₁₁	C₂⁴	1	1	yes	characteristic
52	C₁₁×C₄⋊C₄	—	2	6	no	—
53	C₁₁×C₄⋊C₄	C₄⋊C₄	1	12	yes	—
54	C₁₁×C₄⋊C₄	C₂×C₈	1	12	yes	—
55	C₄²×C₁₁	—	2	6	no	—
56	C₄²×C₁₁	—	2	6	no	—
57	C₄²×C₁₁	—	2	6	no	—
58	C₄²×C₁₁	—	2	2	no	—
59	C₄²×C₁₁	—	4	3	no	—
60	C₄²×C₁₁	C₂×C₈	1	12	yes	—
61	C₄²×C₁₁	C₂×D₄	1	6	yes	—
62	C₄²×C₁₁	C₂×Q₈	1	6	yes	—
63	C₂×C₁₁⋊C₈	—	2	2	no	—
64	C₂×C₁₁⋊C₈	—	2	6	no	—
65	C₂×C₁₁⋊C₈	—	2	6	no	—
66	C₂×C₁₁⋊C₈	—	2	2	no	—
67	C₂×C₁₁⋊C₈	—	2	6	no	—
68	C₂×C₁₁⋊C₈	#41	1	4	yes	—
69	C₂×C₈×Q₈	—	11	2	no	—
70	C₄²×Q₈	D₁₁	1	1	yes	characteristic
71	#1939	—	11	6	no	—
72	#1831	—	11	3	no	—
73	C₄×C₄⋊C₈	—	11	3	no	—
74	C₂³×C₁₁	C₂³×C₄	1	1	yes	characteristic
75	Q₈×C₁₁	C₄⋊C₈	1	16	yes	complemented
76	C₂×C₄×C₁₁	—	2	3	no	—
77	C₂×C₄×C₁₁	—	2	1	no	—
78	C₂×C₄×C₁₁	—	2	1	no	—
79	C₂×C₄×C₁₁	—	2	1	no	—
80	C₂×C₄×C₁₁	—	2	3	no	—
81	C₂×C₄×C₁₁	—	2	3	no	—
82	C₂×C₄×C₁₁	—	2	1	no	—
83	C₂×C₄×C₁₁	—	2	3	no	—
84	C₂×C₄×C₁₁	—	2	3	no	—
85	C₂×C₄×C₁₁	—	2	3	no	—
86	C₂×C₄×C₁₁	—	2	3	no	—
87	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	6	yes	—
88	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	6	yes	—
89	C₂×C₄×C₁₁	C₂²×C₈	1	2	yes	—
90	C₂×C₄×C₁₁	C₂²×C₈	1	6	yes	—
91	C₂×C₄×C₁₁	C₂²×C₈	1	6	yes	—
92	C₂×C₄×C₁₁	C₂×M₁₆	1	6	yes	—
93	C₂×C₄×C₁₁	#131	1	2	yes	—
94	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
95	C₂×C₄×C₁₁	C₂²×Q₈	1	1	yes	characteristic
96	C₂×C₄×C₁₁	C₂×#41	1	2	yes	—
97	C₂×C₄×C₁₁	2₋¹⁺⁴	1	2	yes	—
98	C₁₁⋊C₈	—	2	4	no	—
99	C₁₁⋊C₈	—	4	6	no	—
100	C₂×C₄×Q₈	Dic₁₁	1	1	yes	characteristic
101	C₂×C₄×Q₈	Dic₁₁	1	1	yes	characteristic
102	C₂×C₄×Q₈	C₂×D₁₁	1	1	yes	characteristic
103	C₈×Q₈	—	22	8	no	—
104	C₂×C₄⋊C₈	—	11	6	no	—
105	C₂×C₄⋊C₈	—	11	6	no	—
106	C₂×C₄×C₈	—	11	6	no	—
107	C₄×C₄⋊C₄	Dic₁₁	1	3	yes	—
108	C₄×C₄⋊C₄	Dic₁₁	1	3	yes	—
109	C₄×C₄⋊C₄	C₂×D₁₁	1	3	yes	—
110	C₄³	C₂×D₁₁	1	3	yes	—
111	#336	—	11	3	no	—
112	#336	—	11	6	no	—
113	#336	—	11	1	no	—
114	C₂²×C₁₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
115	C₂²×C₁₁	C₄×#41	1	1	yes	characteristic
116	C₂²×C₁₁	#519	1	1	yes	characteristic
117	C₂²×C₁₁	C₂³×C₈	1	1	yes	characteristic
118	C₂²×C₁₁	C₂²×M₁₆	1	1	yes	characteristic
119	C₂²×C₁₁	C₂×#131	1	1	yes	characteristic
120	C₂²×C₁₁	#568	1	1	yes	characteristic
121	C₄×C₁₁	—	2	6	no	—
122	C₄×C₁₁	—	2	6	no	—
123	C₄×C₁₁	—	2	2	no	—
124	C₄×C₁₁	—	2	2	no	—
125	C₄×C₁₁	—	4	3	no	—
126	C₄×C₁₁	C₂×C₄⋊C₈	1	12	yes	—
127	C₄×C₁₁	#548	1	2	yes	—
128	C₄×C₁₁	#557	1	2	yes	—
129	C₂²×Q₈	C₂×Dic₁₁	1	1	yes	characteristic
130	C₂²×C₈	—	11	2	no	—
131	C₂²×C₈	—	11	6	no	—
132	C₄×Q₈	—	2	4	no	—
133	C₄×Q₈	C₁₁⋊C₈	1	8	yes	complemented
134	C₄×Q₈	Dic₂₂	1	4	yes	—
135	C₄×Q₈	D₄₄	1	4	yes	—
136	C₂×C₄⋊C₄	C₂×Dic₁₁	1	3	yes	—
137	C₂×C₄⋊C₄	C₂×Dic₁₁	1	3	yes	—
138	C₂×C₄⋊C₄	C₂×Dic₁₁	1	3	yes	—
139	C₂×C₄²	C₂×Dic₁₁	1	3	yes	—
140	C₂×C₄²	C₂×Dic₁₁	1	3	yes	—
141	C₂×C₄²	C₂×Dic₁₁	1	3	yes	—
142	C₂×C₄²	C₂×Dic₁₁	1	3	yes	—
143	C₂×C₄²	C₂²×D₁₁	1	1	yes	characteristic
144	C₂×C₄²	C₂²×D₁₁	1	3	yes	—
145	C₂×C₄²	C₂²×D₁₁	1	3	yes	—
146	C₄⋊C₈	—	22	12	no	—
147	C₄⋊C₈	—	22	12	no	—
148	C₄×C₈	—	22	12	no	—
149	#95	C₂×Dic₁₁	1	6	yes	—
150	C₂×C₁₁	#2351	1	1	yes	characteristic
151	C₂×C₁₁	C₂²×C₄⋊C₈	1	1	yes	characteristic
152	C₂×C₁₁	#2906	1	1	yes	characteristic
153	C₂×C₁₁	C₈×#41	1	1	yes	characteristic
154	C₂×C₁₁	#2971	1	1	yes	characteristic
155	C₂×C₁₁	#2982	1	1	yes	characteristic
156	C₂×C₁₁	#2984	1	1	yes	characteristic
157	C₂×Q₈	C₂×C₁₁⋊C₈	1	4	yes	—
158	C₂×Q₈	#4318	1	4	yes	—
159	C₂×Q₈	C₄⋊Dic₁₁	1	4	yes	—
160	C₂²×C₄	C₂²×Dic₁₁	1	3	yes	—
161	C₂²×C₄	C₂²×Dic₁₁	1	3	yes	—
162	C₂²×C₄	C₂²×Dic₁₁	1	1	yes	characteristic
163	C₂²×C₄	C₂²×Dic₁₁	1	3	yes	—
164	C₂²×C₄	C₂²×Dic₁₁	1	1	yes	characteristic
165	C₂²×C₄	C₂²×Dic₁₁	1	3	yes	—
166	C₂²×C₄	C₂³×D₁₁	1	1	yes	characteristic
167	C₂×C₈	—	11	4	no	—
168	C₂×C₈	—	22	2	no	—
169	C₂×C₈	—	22	2	no	—
170	C₂×C₈	—	22	6	no	—
171	C₂×C₈	—	22	6	no	—
172	C₂×C₈	—	22	6	no	—
173	C₄⋊C₄	—	2	6	no	—
174	C₄⋊C₄	C₂×C₁₁⋊C₈	1	12	yes	—
175	C₄⋊C₄	C₄⋊Dic₁₁	1	12	yes	—
176	C₄²	—	2	6	no	—
177	C₄²	—	2	6	no	—
178	C₄²	—	2	2	no	—
179	C₄²	—	2	6	no	—
180	C₄²	—	4	3	no	—
181	C₄²	C₂×C₁₁⋊C₈	1	12	yes	—
182	C₄²	C₂×Dic₂₂	1	6	yes	—
183	C₄²	C₂×D₄₄	1	6	yes	—
184	C₁₁	#22734	1	1	yes	characteristic, complemented, Sylow
185	C₂³	C₂³×Dic₁₁	1	1	yes	characteristic
186	Q₈	C₄⋊(C₁₁⋊C₈)	1	16	yes	complemented
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	3	no	—
191	C₂×C₄	—	2	3	no	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	2	3	no	—
194	C₂×C₄	—	2	3	no	—
195	C₂×C₄	—	2	3	no	—
196	C₂×C₄	—	2	3	no	—
197	C₂×C₄	—	2	3	no	—
198	C₂×C₄	C₂²×C₁₁⋊C₈	1	2	yes	—
199	C₂×C₄	C₂²×C₁₁⋊C₈	1	6	yes	—
200	C₂×C₄	C₂²×C₁₁⋊C₈	1	6	yes	—
201	C₂×C₄	C₂×#4318	1	6	yes	—
202	C₂×C₄	C₂×C₄⋊Dic₁₁	1	6	yes	—
203	C₂×C₄	C₂×C₄⋊Dic₁₁	1	6	yes	—
204	C₂×C₄	#67001	1	2	yes	—
205	C₂×C₄	C₂²×Dic₂₂	1	1	yes	characteristic
206	C₂×C₄	C₂²×D₄₄	1	1	yes	characteristic
207	C₂×C₄	#67040	1	2	yes	—
208	C₂×C₄	#41×D₁₁	1	2	yes	—
209	C₈	—	22	4	no	—
210	C₈	—	44	6	no	—
211	C₂²	C₂³×C₁₁⋊C₈	1	1	yes	characteristic
212	C₂²	C₂²×#4318	1	1	yes	characteristic
213	C₂²	C₂²×C₄⋊Dic₁₁	1	1	yes	characteristic
214	C₂²	#10623343	1	1	yes	characteristic
215	C₂²	C₂×#67001	1	1	yes	characteristic
216	C₂²	#10623350	1	1	yes	characteristic
217	C₂²	#41×Dic₁₁	1	1	yes	characteristic
218	C₄	—	2	6	no	—
219	C₄	—	2	6	no	—
220	C₄	—	2	2	no	—
221	C₄	—	2	2	no	—
222	C₄	—	4	3	no	—
223	C₄	C₂×C₄⋊(C₁₁⋊C₈)	1	12	yes	—
224	C₄	#10623100	1	2	yes	—
225	C₄	#10623107	1	2	yes	—
226	C₂	#13071537	1	1	yes	characteristic
227	C₂	C₂²×C₄⋊(C₁₁⋊C₈)	1	1	yes	characteristic
228	C₂	#13077184	1	1	yes	characteristic
229	C₂	#41×C₁₁⋊C₈	1	1	yes	characteristic
230	C₂	#13077636	1	1	yes	characteristic
231	C₂	#13077697	1	1	yes	characteristic
232	C₂	#13077705	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₁₁
commutator	C₂²×C₁₁	C₂³×C₈
Fitting	C₄²×Q₈×C₁₁	C₂
socle, prime radical	C₂³×C₁₁	C₂³×C₄
cosocle	C₂²×C₄×C₁₁	C₂⁴
hypercenter	C₄²×Q₈	D₁₁
hypocenter	C₁₁	#22734
LCS	G ▷ C₂²×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄²×Q₈

Complements

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

N	H ≅ G/N	type
22. C₄×Q₈×C₁₁	C₈	semidirect
75. Q₈×C₁₁	C₄⋊C₈	semidirect
133. C₄×Q₈	C₁₁⋊C₈	semidirect
184. C₁₁	#22734	semidirect
186. Q₈	C₄⋊(C₁₁⋊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
#13069335	2
#13068733	2
C₄×C₄⋊(C₁₁⋊C₈)	2
C₄²×Q₈×C₁₁	2
#22734	11