Subgroups — C₁₁×#22538 #14599731

Order 2816.

970 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₁×#557	C₂	1	1	yes	maximal, characteristic
2	Q₈×C₁₁×M₁₆	C₂	1	2	yes	maximal
3	C₁₁×#1966	C₂	1	6	yes	maximal
4	C₁₁×#1801	C₂	1	3	yes	maximal
5	C₄×C₁₁×#101	C₂	1	3	yes	maximal
6	C₁₁×#557	—	2	2	no	—
7	C₁₁×#557	—	2	6	no	—
8	C₂×C₁₁×C₄⋊Q₈	C₂²	1	3	yes	—
9	C₂×C₁₁×#125	C₄	1	6	yes	—
10	C₂×C₄×Q₈×C₁₁	C₄	1	2	yes	—
11	C₂×C₄×Q₈×C₁₁	C₂²	1	3	yes	—
12	C₂×C₄×Q₈×C₁₁	C₂²	1	1	yes	characteristic
13	C₁₁×#446	—	2	6	no	—
14	C₈×Q₈×C₁₁	—	2	2	no	—
15	C₁₁×#423	C₂²	1	6	yes	—
16	C₂×C₁₁×#101	C₂²	1	6	yes	—
17	C₂×C₁₁×#101	C₂²	1	6	yes	—
18	C₄×C₁₁×M₁₆	C₂²	1	6	yes	—
19	C₁₁×#343	C₂²	1	1	yes	characteristic
20	C₁₁×#343	C₂²	1	3	yes	—
21	C₂²×Q₈×C₁₁	C₂×C₄	1	1	yes	characteristic
22	C₂²×Q₈×C₁₁	C₂³	1	3	yes	—
23	C₂×C₁₁×M₁₆	C₂³	1	6	yes	—
24	C₂×C₁₁×M₁₆	C₂³	1	2	yes	—
25	C₁₁×C₄⋊Q₈	—	2	6	no	—
26	C₁₁×C₄⋊Q₈	—	2	6	no	—
27	C₁₁×#125	—	2	12	no	—
28	C₁₁×#125	—	2	12	no	—
29	C₄×Q₈×C₁₁	—	2	6	no	—
30	C₄×Q₈×C₁₁	—	2	2	no	—
31	C₄×Q₈×C₁₁	—	2	6	no	—
32	C₄×Q₈×C₁₁	—	2	6	no	—
33	C₄×Q₈×C₁₁	D₄	1	2	yes	—
34	C₄×Q₈×C₁₁	D₄	1	6	yes	—
35	C₂×C₁₁×C₄⋊C₄	C₂×C₄	1	6	yes	—
36	C₂×C₁₁×C₄⋊C₄	C₂×C₄	1	6	yes	—
37	C₂×C₁₁×C₄⋊C₄	C₂×C₄	1	6	yes	—
38	C₂×C₁₁×C₄⋊C₄	C₂×C₄	1	3	yes	—
39	C₂×C₁₁×C₄⋊C₄	C₂³	1	1	yes	characteristic
40	C₂×C₄²×C₁₁	C₂×C₄	1	6	yes	—
41	C₂×C₄²×C₁₁	C₂³	1	3	yes	—
42	C₁₁×C₄⋊C₈	—	2	6	no	—
43	C₁₁×C₄⋊C₈	—	2	6	no	—
44	C₁₁×#101	—	2	12	no	—
45	C₁₁×#101	—	4	6	no	—
46	C₁₁×#97	—	2	6	no	—
47	C₄×C₈×C₁₁	—	2	6	no	—
48	#22538	C₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
49	C₂×Q₈×C₁₁	—	2	4	no	—
50	C₂×Q₈×C₁₁	—	2	6	no	—
51	C₂×Q₈×C₁₁	—	2	6	no	—
52	C₂×Q₈×C₁₁	—	2	3	no	—
53	C₂×Q₈×C₁₁	—	2	3	no	—
54	C₂×Q₈×C₁₁	C₂²⋊C₄	1	3	yes	—
55	C₂×Q₈×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₁	C₂²×C₄	1	6	yes	—
57	C₂²×C₄×C₁₁	C₂²×C₄	1	3	yes	—
58	C₂²×C₄×C₁₁	C₂²×C₄	1	2	yes	—
59	C₂²×C₄×C₁₁	C₂²×C₄	1	3	yes	—
60	C₂²×C₄×C₁₁	C₂⁴	1	1	yes	characteristic
61	C₁₁×M₁₆	—	2	6	no	—
62	C₁₁×M₁₆	—	2	6	no	—
63	C₁₁×M₁₆	—	2	2	no	—
64	C₁₁×M₁₆	#41	1	4	yes	—
65	C₂×C₈×C₁₁	—	2	2	no	—
66	C₂×C₈×C₁₁	—	2	6	no	—
67	C₁₁×C₄⋊C₄	—	2	12	no	—
68	C₁₁×C₄⋊C₄	—	2	6	no	—
69	C₁₁×C₄⋊C₄	—	2	6	no	—
70	C₁₁×C₄⋊C₄	—	2	12	no	—
71	C₁₁×C₄⋊C₄	—	2	2	no	—
72	C₁₁×C₄⋊C₄	C₂²⋊C₄	1	6	yes	—
73	C₁₁×C₄⋊C₄	C₂²⋊C₄	1	6	yes	—
74	C₄²×C₁₁	—	2	6	no	—
75	C₄²×C₁₁	—	2	6	no	—
76	C₄²×C₁₁	C₂×D₄	1	6	yes	—
77	C₄²×C₁₁	C₂×D₄	1	6	yes	—
78	C₂×#557	C₂×C₁₁	1	1	yes	characteristic
79	Q₈×M₁₆	C₂×C₁₁	1	2	yes	—
80	#1966	C₂×C₁₁	1	6	yes	—
81	#1801	C₂×C₁₁	1	3	yes	—
82	C₄×#101	C₂×C₁₁	1	3	yes	—
83	C₂³×C₁₁	C₂³×C₄	1	1	yes	characteristic
84	Q₈×C₁₁	—	2	6	no	—
85	Q₈×C₁₁	—	4	6	no	—
86	Q₈×C₁₁	—	4	6	no	—
87	Q₈×C₁₁	#101	1	4	yes	complemented
88	C₂×C₄×C₁₁	—	2	6	no	—
89	C₂×C₄×C₁₁	—	2	2	no	—
90	C₂×C₄×C₁₁	—	2	6	no	—
91	C₂×C₄×C₁₁	—	2	6	no	—
92	C₂×C₄×C₁₁	—	2	6	no	—
93	C₂×C₄×C₁₁	—	4	3	no	—
94	C₂×C₄×C₁₁	—	4	2	no	—
95	C₂×C₄×C₁₁	C₂×C₂²⋊C₄	1	3	yes	—
96	C₂×C₄×C₁₁	C₂×C₂²⋊C₄	1	6	yes	—
97	C₂×C₄×C₁₁	C₂×C₂²⋊C₄	1	3	yes	—
98	C₂×C₄×C₁₁	C₂²×D₄	1	2	yes	—
99	C₂×C₄×C₁₁	C₂×#41	1	2	yes	—
100	C₂×C₄×C₁₁	H(5,𝔽₂)	1	1	yes	characteristic
101	C₂×C₄×C₁₁	2₋¹⁺⁴	1	1	yes	characteristic
102	C₈×C₁₁	—	2	4	no	—
103	C₈×C₁₁	—	4	6	no	—
104	#557	—	2	2	no	—
105	#557	—	2	6	no	—
106	C₂×C₄⋊Q₈	C₂²×C₁₁	1	3	yes	—
107	C₂×#125	C₄×C₁₁	1	6	yes	—
108	C₂×C₄×Q₈	C₄×C₁₁	1	2	yes	—
109	C₂×C₄×Q₈	C₂²×C₁₁	1	3	yes	—
110	C₂×C₄×Q₈	C₂²×C₁₁	1	1	yes	characteristic
111	#446	—	2	6	no	—
112	C₈×Q₈	—	2	2	no	—
113	#423	C₂²×C₁₁	1	6	yes	—
114	C₂×#101	C₂²×C₁₁	1	6	yes	—
115	C₂×#101	C₂²×C₁₁	1	6	yes	—
116	C₄×M₁₆	C₂²×C₁₁	1	6	yes	—
117	#343	C₂²×C₁₁	1	3	yes	—
118	#343	C₂²×C₁₁	1	1	yes	characteristic
119	C₂²×C₁₁	—	2	1	no	—
120	C₂²×C₁₁	—	2	1	no	—
121	C₂²×C₁₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
122	C₂²×C₁₁	C₄×#41	1	1	yes	characteristic
123	C₂²×C₁₁	#520	1	1	yes	characteristic
124	C₄×C₁₁	—	2	3	no	—
125	C₄×C₁₁	—	2	6	no	—
126	C₄×C₁₁	—	4	6	no	—
127	C₄×C₁₁	—	4	2	no	—
128	C₄×C₁₁	C₂×#101	1	6	yes	—
129	C₄×C₁₁	#548	1	2	yes	—
130	C₄×C₁₁	#550	1	2	yes	—
131	C₂²×Q₈	C₂×C₄×C₁₁	1	1	yes	characteristic
132	C₂²×Q₈	C₂³×C₁₁	1	3	yes	—
133	C₂×M₁₆	C₂³×C₁₁	1	6	yes	—
134	C₂×M₁₆	C₂³×C₁₁	1	2	yes	—
135	C₄⋊Q₈	—	2	6	no	—
136	C₄⋊Q₈	—	2	6	no	—
137	#125	—	2	12	no	—
138	#125	—	2	12	no	—
139	C₄×Q₈	—	2	6	no	—
140	C₄×Q₈	—	2	2	no	—
141	C₄×Q₈	—	2	6	no	—
142	C₄×Q₈	—	2	6	no	—
143	C₄×Q₈	D₄×C₁₁	1	2	yes	—
144	C₄×Q₈	D₄×C₁₁	1	6	yes	—
145	C₂×C₄⋊C₄	C₂×C₄×C₁₁	1	6	yes	—
146	C₂×C₄⋊C₄	C₂×C₄×C₁₁	1	6	yes	—
147	C₂×C₄⋊C₄	C₂×C₄×C₁₁	1	6	yes	—
148	C₂×C₄⋊C₄	C₂×C₄×C₁₁	1	3	yes	—
149	C₂×C₄⋊C₄	C₂³×C₁₁	1	1	yes	characteristic
150	C₂×C₄²	C₂×C₄×C₁₁	1	6	yes	—
151	C₂×C₄²	C₂³×C₁₁	1	3	yes	—
152	C₄⋊C₈	—	2	6	no	—
153	C₄⋊C₈	—	2	6	no	—
154	#101	—	2	12	no	—
155	#101	—	4	6	no	—
156	#97	—	2	6	no	—
157	C₄×C₈	—	2	6	no	—
158	C₂×C₁₁	—	2	1	no	—
159	C₂×C₁₁	—	2	1	no	—
160	C₂×C₁₁	#2341	1	1	yes	characteristic
161	C₂×C₁₁	C₂²×#101	1	1	yes	characteristic
162	C₂×C₁₁	#2889	1	1	yes	characteristic
163	C₂×Q₈	—	2	4	no	—
164	C₂×Q₈	—	2	6	no	—
165	C₂×Q₈	—	2	6	no	—
166	C₂×Q₈	—	2	3	no	—
167	C₂×Q₈	—	2	3	no	—
168	C₂×Q₈	C₁₁×C₂²⋊C₄	1	3	yes	—
169	C₂×Q₈	C₁₁×C₂²⋊C₄	1	1	yes	characteristic
170	C₂²×C₄	C₂²×C₄×C₁₁	1	6	yes	—
171	C₂²×C₄	C₂²×C₄×C₁₁	1	3	yes	—
172	C₂²×C₄	C₂²×C₄×C₁₁	1	2	yes	—
173	C₂²×C₄	C₂²×C₄×C₁₁	1	3	yes	—
174	C₂²×C₄	C₂⁴×C₁₁	1	1	yes	characteristic
175	M₁₆	—	2	2	no	—
176	M₁₆	—	2	6	no	—
177	M₁₆	—	2	6	no	—
178	M₁₆	C₁₁×#41	1	4	yes	—
179	C₂×C₈	—	2	2	no	—
180	C₂×C₈	—	2	6	no	—
181	C₄⋊C₄	—	2	12	no	—
182	C₄⋊C₄	—	2	6	no	—
183	C₄⋊C₄	—	2	6	no	—
184	C₄⋊C₄	—	2	12	no	—
185	C₄⋊C₄	—	2	2	no	—
186	C₄⋊C₄	C₁₁×C₂²⋊C₄	1	6	yes	—
187	C₄⋊C₄	C₁₁×C₂²⋊C₄	1	6	yes	—
188	C₄²	—	2	6	no	—
189	C₄²	—	2	6	no	—
190	C₄²	C₂×D₄×C₁₁	1	6	yes	—
191	C₄²	C₂×D₄×C₁₁	1	6	yes	—
192	C₁₁	#22538	1	1	yes	characteristic, direct product factor, Sylow
193	C₂³	C₂³×C₄×C₁₁	1	1	yes	characteristic
194	Q₈	—	2	6	no	—
195	Q₈	—	4	6	no	—
196	Q₈	—	4	6	no	—
197	Q₈	C₁₁×#101	1	4	yes	complemented
198	C₂×C₄	—	2	6	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	6	no	—
201	C₂×C₄	—	2	6	no	—
202	C₂×C₄	—	2	6	no	—
203	C₂×C₄	—	4	3	no	—
204	C₂×C₄	—	4	2	no	—
205	C₂×C₄	C₂×C₁₁×C₂²⋊C₄	1	3	yes	—
206	C₂×C₄	C₂×C₁₁×C₂²⋊C₄	1	6	yes	—
207	C₂×C₄	C₂×C₁₁×C₂²⋊C₄	1	3	yes	—
208	C₂×C₄	C₂²×D₄×C₁₁	1	2	yes	—
209	C₂×C₄	C₂×C₁₁×#41	1	2	yes	—
210	C₂×C₄	C₁₁×H(5,𝔽₂)	1	1	yes	characteristic
211	C₂×C₄	C₁₁×2₋¹⁺⁴	1	1	yes	characteristic
212	C₈	—	2	4	no	—
213	C₈	—	4	6	no	—
214	C₂²	—	2	1	no	—
215	C₂²	—	2	1	no	—
216	C₂²	C₂²×C₁₁×C₂²⋊C₄	1	1	yes	characteristic
217	C₂²	C₄×C₁₁×#41	1	1	yes	characteristic
218	C₂²	C₁₁×#520	1	1	yes	characteristic
219	C₄	—	2	3	no	—
220	C₄	—	2	6	no	—
221	C₄	—	4	6	no	—
222	C₄	—	4	2	no	—
223	C₄	C₂×C₁₁×#101	1	6	yes	—
224	C₄	C₁₁×#548	1	2	yes	—
225	C₄	C₁₁×#550	1	2	yes	—
226	C₂	—	2	1	no	—
227	C₂	—	2	1	no	—
228	C₂	C₁₁×#2341	1	1	yes	characteristic
229	C₂	C₂²×C₁₁×#101	1	1	yes	characteristic
230	C₂	C₁₁×#2889	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₁	C₂²×C₂²⋊C₄
commutator	C₂³	C₂³×C₄×C₁₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₁
prime radical	C₂³×C₁₁	C₂³×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₁ ◁ C₂×C₄×Q₈×C₁₁ ◁ G

Complements

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

N	H ≅ G/N	type
48. #22538	C₁₁	direct (N × H)
87. Q₈×C₁₁	#101	semidirect
192. C₁₁	#22538	direct (N × H)
197. Q₈	C₁₁×#101	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₁₁×#557	2
Q₈×C₁₁×M₁₆	2
C₁₁×#1966	2
C₁₁×#1801	2
C₄×C₁₁×#101	2
#22538	11

Subgroups — C11×#22538 #14599731

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₁×#22538 #14599731