Subgroups — C₂³×C₂²⋊(C₁₁⋊C₈) #15617200

Order 2816.

7646 conjugacy classes of subgroups.

The subgroup lattice has 176 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₁₁⋊C₈	C₂	1	2	yes	maximal, complemented
2	C₂²×C₂²⋊(C₁₁⋊C₈)	C₂	1	28	yes	maximal, direct product factor
3	C₂⁵×C₄×C₁₁	C₂	1	1	yes	maximal, characteristic
4	C₂⁶×C₁₁	C₄	1	1	yes	characteristic
5	C₂⁴×C₄×C₁₁	—	2	16	no	—
6	C₂⁴×C₄×C₁₁	C₄	1	1	yes	characteristic
7	C₂⁴×C₄×C₁₁	C₄	1	14	yes	—
8	C₂⁴×C₄×C₁₁	C₂²	1	14	yes	—
9	C₂⁴×C₄×C₁₁	C₂²	1	1	yes	characteristic
10	C₂³×C₁₁⋊C₈	—	2	16	no	—
11	C₂³×C₁₁⋊C₈	C₂²	1	28	yes	complemented
12	C₂×C₂²⋊(C₁₁⋊C₈)	C₂²	1	112	yes	direct product factor
13	C₂⁵×C₁₁	—	2	8	no	—
14	C₂⁵×C₁₁	—	2	8	no	—
15	C₂⁵×C₁₁	C₈	1	16	yes	complemented
16	C₂⁵×C₁₁	C₂×C₄	1	14	yes	—
17	C₂⁵×C₁₁	C₂×C₄	1	1	yes	characteristic
18	C₂³×C₄×C₁₁	—	2	8	no	—
19	C₂³×C₄×C₁₁	—	2	112	no	—
20	C₂³×C₄×C₁₁	—	2	112	no	—
21	C₂³×C₄×C₁₁	C₂×C₄	1	14	yes	—
22	C₂³×C₄×C₁₁	C₂×C₄	1	84	yes	—
23	C₂³×C₄×C₁₁	C₂×C₄	1	7	yes	—
24	C₂³×C₄×C₁₁	D₄	1	16	yes	—
25	C₂³×C₄×C₁₁	C₂³	1	28	yes	—
26	C₂³×C₄×C₁₁	C₂³	1	7	yes	—
27	C₂²×C₁₁⋊C₈	—	2	112	no	—
28	C₂²×C₁₁⋊C₈	C₂³	1	56	yes	complemented
29	C₂²⋊(C₁₁⋊C₈)	C₂³	1	64	yes	direct product factor
30	C₂³×C₂²⋊C₈	—	11	1	no	maximal, Sylow
31	C₂⁴×C₁₁	—	2	128	no	—
32	C₂⁴×C₁₁	—	2	56	no	—
33	C₂⁴×C₁₁	—	2	56	no	—
34	C₂⁴×C₁₁	C₂²⋊C₄	1	8	yes	—
35	C₂⁴×C₁₁	C₂×C₈	1	8	yes	—
36	C₂⁴×C₁₁	C₂×C₈	1	112	yes	complemented
37	C₂⁴×C₁₁	M₁₆	1	8	yes	—
38	C₂⁴×C₁₁	C₂²×C₄	1	28	yes	—
39	C₂⁴×C₁₁	C₂²×C₄	1	7	yes	—
40	C₂²×C₄×C₁₁	—	2	56	no	—
41	C₂²×C₄×C₁₁	—	2	336	no	—
42	C₂²×C₄×C₁₁	—	2	112	no	—
43	C₂²×C₄×C₁₁	C₂²⋊C₄	1	56	yes	—
44	C₂²×C₄×C₁₁	C₂²×C₄	1	28	yes	—
45	C₂²×C₄×C₁₁	C₂²×C₄	1	56	yes	—
46	C₂²×C₄×C₁₁	C₂²×C₄	1	21	yes	—
47	C₂²×C₄×C₁₁	C₂×D₄	1	56	yes	—
48	C₂²×C₄×C₁₁	C₂⁴	1	8	yes	—
49	C₂²×C₄×C₁₁	C₂⁴	1	7	yes	—
50	C₂×C₁₁⋊C₈	—	2	112	no	—
51	C₂×C₁₁⋊C₈	C₂⁴	1	16	yes	complemented
52	C₂⁵×C₄	D₁₁	1	1	yes	characteristic
53	C₂⁴×C₈	—	11	2	no	—
54	C₂²×C₂²⋊C₈	—	11	28	no	—
55	C₂³×C₁₁	—	2	56	no	—
56	C₂³×C₁₁	—	2	56	no	—
57	C₂³×C₁₁	—	2	448	no	—
58	C₂³×C₁₁	C₂²⋊C₈	1	64	yes	complemented
59	C₂³×C₁₁	C₂×C₂²⋊C₄	1	28	yes	—
60	C₂³×C₁₁	C₂²×C₈	1	28	yes	—
61	C₂³×C₁₁	C₂²×C₈	1	112	yes	complemented
62	C₂³×C₁₁	C₂×M₁₆	1	28	yes	—
63	C₂³×C₁₁	C₂³×C₄	1	8	yes	—
64	C₂³×C₁₁	C₂³×C₄	1	7	yes	—
65	C₂×C₄×C₁₁	—	2	56	no	—
66	C₂×C₄×C₁₁	—	2	112	no	—
67	C₂×C₄×C₁₁	—	2	16	no	—
68	C₂×C₄×C₁₁	C₂×C₂²⋊C₄	1	84	yes	—
69	C₂×C₄×C₁₁	C₂³×C₄	1	8	yes	—
70	C₂×C₄×C₁₁	C₂³×C₄	1	7	yes	—
71	C₂×C₄×C₁₁	C₂²×D₄	1	28	yes	—
72	C₂×C₄×C₁₁	C₂⁵	1	1	yes	characteristic
73	C₁₁⋊C₈	—	2	16	no	—
74	C₂⁶	Dic₁₁	1	1	yes	characteristic
75	C₂⁴×C₄	—	2	16	no	—
76	C₂⁴×C₄	Dic₁₁	1	1	yes	characteristic
77	C₂⁴×C₄	Dic₁₁	1	14	yes	—
78	C₂⁴×C₄	C₂×D₁₁	1	14	yes	—
79	C₂⁴×C₄	C₂×D₁₁	1	1	yes	characteristic
80	C₂³×C₈	—	11	28	no	—
81	C₂³×C₈	—	22	16	no	—
82	C₂×C₂²⋊C₈	—	11	112	no	—
83	C₂²×C₁₁	—	2	224	no	—
84	C₂²×C₁₁	—	2	8	no	—
85	C₂²×C₁₁	—	2	8	no	—
86	C₂²×C₁₁	C₂×C₂²⋊C₈	1	112	yes	complemented
87	C₂²×C₁₁	C₂²×C₂²⋊C₄	1	14	yes	—
88	C₂²×C₁₁	C₂³×C₈	1	14	yes	—
89	C₂²×C₁₁	C₂³×C₈	1	16	yes	complemented
90	C₂²×C₁₁	C₂²×M₁₆	1	14	yes	—
91	C₂²×C₁₁	C₂⁴×C₄	1	1	yes	characteristic
92	C₄×C₁₁	—	2	8	no	—
93	C₄×C₁₁	C₂²×C₂²⋊C₄	1	14	yes	—
94	C₄×C₁₁	C₂³×D₄	1	2	yes	—
95	C₂⁵	—	2	8	no	—
96	C₂⁵	—	2	8	no	—
97	C₂⁵	C₁₁⋊C₈	1	16	yes	complemented
98	C₂⁵	C₂×Dic₁₁	1	14	yes	—
99	C₂⁵	C₂×Dic₁₁	1	1	yes	characteristic
100	C₂³×C₄	—	2	8	no	—
101	C₂³×C₄	—	2	112	no	—
102	C₂³×C₄	—	2	112	no	—
103	C₂³×C₄	C₂×Dic₁₁	1	14	yes	—
104	C₂³×C₄	C₂×Dic₁₁	1	84	yes	—
105	C₂³×C₄	C₂×Dic₁₁	1	7	yes	—
106	C₂³×C₄	C₂²⋊D₁₁	1	16	yes	—
107	C₂³×C₄	C₂²×D₁₁	1	28	yes	—
108	C₂³×C₄	C₂²×D₁₁	1	7	yes	—
109	C₂²×C₈	—	11	56	no	—
110	C₂²×C₈	—	22	112	no	—
111	C₂²⋊C₈	—	11	64	no	—
112	C₂×C₁₁	—	2	16	no	—
113	C₂×C₁₁	C₂²×C₂²⋊C₈	1	28	yes	complemented
114	C₂×C₁₁	C₂³×C₂²⋊C₄	1	1	yes	characteristic
115	C₂×C₁₁	C₂⁴×C₈	1	1	yes	characteristic
116	C₂×C₁₁	C₂³×M₁₆	1	1	yes	characteristic
117	C₂⁴	—	2	128	no	—
118	C₂⁴	—	2	56	no	—
119	C₂⁴	—	2	56	no	—
120	C₂⁴	C₂×C₁₁⋊C₈	1	8	yes	—
121	C₂⁴	C₂×C₁₁⋊C₈	1	112	yes	complemented
122	C₂⁴	#4318	1	8	yes	—
123	C₂⁴	C₂²⋊Dic₁₁	1	8	yes	—
124	C₂⁴	C₂²×Dic₁₁	1	28	yes	—
125	C₂⁴	C₂²×Dic₁₁	1	7	yes	—
126	C₂²×C₄	—	2	56	no	—
127	C₂²×C₄	—	2	336	no	—
128	C₂²×C₄	—	2	112	no	—
129	C₂²×C₄	C₂²⋊Dic₁₁	1	56	yes	—
130	C₂²×C₄	C₂²×Dic₁₁	1	28	yes	—
131	C₂²×C₄	C₂²×Dic₁₁	1	56	yes	—
132	C₂²×C₄	C₂²×Dic₁₁	1	21	yes	—
133	C₂²×C₄	C₂×C₂²⋊D₁₁	1	56	yes	—
134	C₂²×C₄	C₂³×D₁₁	1	8	yes	—
135	C₂²×C₄	C₂³×D₁₁	1	7	yes	—
136	C₂×C₈	—	11	16	no	—
137	C₂×C₈	—	22	112	no	—
138	C₁₁	C₂³×C₂²⋊C₈	1	1	yes	characteristic, complemented, Sylow
139	C₂³	—	2	56	no	—
140	C₂³	—	2	448	no	—
141	C₂³	—	2	56	no	—
142	C₂³	C₂²⋊(C₁₁⋊C₈)	1	64	yes	direct product factor
143	C₂³	C₂²×C₁₁⋊C₈	1	28	yes	—
144	C₂³	C₂²×C₁₁⋊C₈	1	112	yes	complemented
145	C₂³	C₂×#4318	1	28	yes	—
146	C₂³	C₂×C₂²⋊Dic₁₁	1	28	yes	—
147	C₂³	C₂³×Dic₁₁	1	8	yes	—
148	C₂³	C₂³×Dic₁₁	1	7	yes	—
149	C₂×C₄	—	2	56	no	—
150	C₂×C₄	—	2	112	no	—
151	C₂×C₄	—	2	16	no	—
152	C₂×C₄	C₂×C₂²⋊Dic₁₁	1	84	yes	—
153	C₂×C₄	C₂³×Dic₁₁	1	8	yes	—
154	C₂×C₄	C₂³×Dic₁₁	1	7	yes	—
155	C₂×C₄	C₂²×C₂²⋊D₁₁	1	28	yes	—
156	C₂×C₄	C₂⁴×D₁₁	1	1	yes	characteristic
157	C₈	—	22	16	no	—
158	C₂²	—	2	224	no	—
159	C₂²	—	2	8	no	—
160	C₂²	—	2	8	no	—
161	C₂²	C₂×C₂²⋊(C₁₁⋊C₈)	1	112	yes	direct product factor
162	C₂²	C₂³×C₁₁⋊C₈	1	14	yes	—
163	C₂²	C₂³×C₁₁⋊C₈	1	16	yes	complemented
164	C₂²	C₂²×#4318	1	14	yes	—
165	C₂²	C₂²×C₂²⋊Dic₁₁	1	14	yes	—
166	C₂²	C₂⁴×Dic₁₁	1	1	yes	characteristic
167	C₄	—	2	8	no	—
168	C₄	C₂²×C₂²⋊Dic₁₁	1	14	yes	—
169	C₄	C₂³×C₂²⋊D₁₁	1	2	yes	—
170	C₂	—	2	16	no	—
171	C₂	C₂²×C₂²⋊(C₁₁⋊C₈)	1	28	yes	direct product factor
172	C₂	C₂³×C₂²⋊Dic₁₁	1	1	yes	characteristic
173	C₂	C₂⁴×C₁₁⋊C₈	1	1	yes	characteristic
174	C₂	C₂³×#4318	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴×C₄	C₂×D₁₁
commutator	C₂×C₁₁	C₂⁴×C₈
Frattini	C₂×C₄	C₂⁴×D₁₁
Fitting	C₂⁵×C₄×C₁₁	C₂
socle	C₂⁵×C₁₁	C₂×C₄
prime radical	C₂⁶×C₁₁	C₄
cosocle	C₂×C₄×C₁₁	C₂⁵
hypercenter	C₂⁵×C₄	D₁₁
hypocenter	C₁₁	C₂³×C₂²⋊C₈
LCS	G ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂⁴×C₄ ◁ C₂⁵×C₄

Complements

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

N	H ≅ G/N	type
1. C₂⁴×C₁₁⋊C₈	C₂	semidirect
2. C₂²×C₂²⋊(C₁₁⋊C₈)	C₂	direct (N × H)
11. C₂³×C₁₁⋊C₈	C₂²	semidirect
12. C₂×C₂²⋊(C₁₁⋊C₈)	C₂²	direct (N × H)
15. C₂⁵×C₁₁	C₈	semidirect
28. C₂²×C₁₁⋊C₈	C₂³	semidirect
29. C₂²⋊(C₁₁⋊C₈)	C₂³	direct (N × H)
36. C₂⁴×C₁₁	C₂×C₈	semidirect
51. C₂×C₁₁⋊C₈	C₂⁴	semidirect
58. C₂³×C₁₁	C₂²⋊C₈	semidirect
61. C₂³×C₁₁	C₂²×C₈	semidirect
86. C₂²×C₁₁	C₂×C₂²⋊C₈	semidirect
89. C₂²×C₁₁	C₂³×C₈	semidirect
97. C₂⁵	C₁₁⋊C₈	semidirect
113. C₂×C₁₁	C₂²×C₂²⋊C₈	semidirect
121. C₂⁴	C₂×C₁₁⋊C₈	semidirect
138. C₁₁	C₂³×C₂²⋊C₈	semidirect
142. C₂³	C₂²⋊(C₁₁⋊C₈)	direct (N × H)
144. C₂³	C₂²×C₁₁⋊C₈	semidirect
161. C₂²	C₂×C₂²⋊(C₁₁⋊C₈)	direct (N × H)
163. C₂²	C₂³×C₁₁⋊C₈	semidirect
171. C₂	C₂²×C₂²⋊(C₁₁⋊C₈)	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×C₁₁⋊C₈	2
C₂²×C₂²⋊(C₁₁⋊C₈)	2
C₂⁵×C₄×C₁₁	2
C₂³×C₂²⋊C₈	11

Subgroups — C23×C22⋊(C11⋊C8) #15617200

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₂²⋊(C₁₁⋊C₈) #15617200