Subgroups — C₂²×#10622095 #14722739

Order 2816.

942 conjugacy classes of subgroups.

The subgroup lattice has 178 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊(C₁₁⋊C₈)	C₂	1	2	yes	maximal
2	C₂×#10622095	C₂	1	12	yes	maximal, direct product factor
3	C₂²×C₄×C₈×C₁₁	C₂	1	1	yes	maximal, characteristic
4	C₂³×C₈×C₁₁	C₄	1	1	yes	characteristic
5	C₂³×C₈×C₁₁	C₄	1	1	yes	characteristic
6	C₂²×C₄²×C₁₁	C₂²	1	1	yes	characteristic
7	C₂³×C₁₁⋊C₈	—	2	2	no	—
8	C₂×C₄×C₈×C₁₁	C₄	1	6	yes	—
9	C₂×C₄×C₈×C₁₁	C₂²	1	6	yes	—
10	C₂×C₄⋊(C₁₁⋊C₈)	—	2	6	no	—
11	C₂×C₄⋊(C₁₁⋊C₈)	C₂²	1	12	yes	—
12	#10622095	C₂²	1	16	yes	direct product factor
13	C₂³×C₄×C₁₁	C₂×C₄	1	1	yes	characteristic
14	C₂³×C₄×C₁₁	D₄	1	1	yes	characteristic
15	C₂³×C₄×C₁₁	Q₈	1	1	yes	characteristic
16	C₂²×C₈×C₁₁	—	2	4	no	—
17	C₂²×C₈×C₁₁	C₈	1	8	yes	complemented
18	C₂²×C₈×C₁₁	C₂×C₄	1	6	yes	—
19	C₂²×C₈×C₁₁	C₂×C₄	1	6	yes	—
20	C₂×C₄²×C₁₁	C₂×C₄	1	3	yes	—
21	C₂×C₄²×C₁₁	D₄	1	3	yes	—
22	C₂×C₄²×C₁₁	Q₈	1	3	yes	—
23	C₂×C₄²×C₁₁	C₂³	1	3	yes	—
24	C₂²×C₁₁⋊C₈	—	2	12	no	—
25	C₂²×C₁₁⋊C₈	—	4	8	no	—
26	C₄×C₈×C₁₁	C₂×C₄	1	12	yes	—
27	C₄×C₈×C₁₁	C₂³	1	4	yes	—
28	C₄⋊(C₁₁⋊C₈)	—	2	12	no	—
29	C₄⋊(C₁₁⋊C₈)	C₂³	1	8	yes	—
30	C₂²×#334	—	11	1	no	maximal, Sylow
31	C₂⁴×C₁₁	C₄⋊C₄	1	1	yes	characteristic
32	C₂²×C₄×C₁₁	—	2	4	no	—
33	C₂²×C₄×C₁₁	C₄⋊C₄	1	3	yes	—
34	C₂²×C₄×C₁₁	C₄⋊C₄	1	3	yes	—
35	C₂²×C₄×C₁₁	C₄⋊C₄	1	3	yes	—
36	C₂²×C₄×C₁₁	C₂×C₈	1	4	yes	—
37	C₂²×C₄×C₁₁	M₁₆	1	4	yes	—
38	C₂²×C₄×C₁₁	SD₁₆	1	8	yes	—
39	C₂²×C₄×C₁₁	C₂²×C₄	1	3	yes	—
40	C₂²×C₄×C₁₁	C₂×D₄	1	3	yes	—
41	C₂²×C₄×C₁₁	C₂×Q₈	1	3	yes	—
42	C₂×C₈×C₁₁	—	2	12	no	—
43	C₂×C₈×C₁₁	C₂×C₈	1	24	yes	complemented
44	C₂×C₈×C₁₁	C₂²×C₄	1	4	yes	—
45	C₂×C₈×C₁₁	C₂²×C₄	1	4	yes	—
46	C₄²×C₁₁	C₄⋊C₄	1	6	yes	—
47	C₄²×C₁₁	C₂²×C₄	1	3	yes	—
48	C₄²×C₁₁	C₂×D₄	1	3	yes	—
49	C₄²×C₁₁	C₂×Q₈	1	3	yes	—
50	C₄²×C₁₁	C₂⁴	1	1	yes	characteristic
51	C₂×C₁₁⋊C₈	—	2	8	no	—
52	C₂×C₁₁⋊C₈	—	4	24	no	—
53	C₂²×C₄⋊C₈	—	11	2	no	—
54	C₂²×C₄×C₈	D₁₁	1	1	yes	characteristic
55	C₂×#334	—	11	12	no	—
56	C₂³×C₁₁	C₄⋊C₈	1	4	yes	—
57	C₂³×C₁₁	#106	1	4	yes	—
58	C₂³×C₁₁	#108	1	4	yes	—
59	C₂³×C₁₁	C₂×C₄⋊C₄	1	3	yes	—
60	C₂×C₄×C₁₁	—	2	6	no	—
61	C₂×C₄×C₁₁	—	2	6	no	—
62	C₂×C₄×C₁₁	C₄⋊C₈	1	12	yes	—
63	C₂×C₄×C₁₁	#106	1	12	yes	—
64	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	3	yes	—
65	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	3	yes	—
66	C₂×C₄×C₁₁	C₂×C₄⋊C₄	1	3	yes	—
67	C₂×C₄×C₁₁	C₂²×C₈	1	6	yes	—
68	C₂×C₄×C₁₁	C₂×M₁₆	1	6	yes	—
69	C₂×C₄×C₁₁	C₂×SD₁₆	1	12	yes	—
70	C₂×C₄×C₁₁	C₂³×C₄	1	1	yes	characteristic
71	C₂×C₄×C₁₁	C₂²×D₄	1	1	yes	characteristic
72	C₂×C₄×C₁₁	C₂²×Q₈	1	1	yes	characteristic
73	C₈×C₁₁	—	2	4	no	—
74	C₈×C₁₁	C₂²×C₈	1	8	yes	complemented
75	C₁₁⋊C₈	—	4	8	no	—
76	C₂³×C₈	—	22	2	no	—
77	C₂³×C₈	Dic₁₁	1	1	yes	characteristic
78	C₂³×C₈	Dic₁₁	1	1	yes	characteristic
79	C₂²×C₄²	C₂×D₁₁	1	1	yes	characteristic
80	C₂×C₄⋊C₈	—	11	12	no	—
81	C₂×C₄⋊C₈	—	22	6	no	—
82	C₂×C₄×C₈	Dic₁₁	1	6	yes	—
83	C₂×C₄×C₈	C₂×D₁₁	1	6	yes	—
84	#334	—	11	16	no	—
85	C₂²×C₁₁	#334	1	16	yes	complemented
86	C₂²×C₁₁	C₂×C₄⋊C₈	1	6	yes	—
87	C₂²×C₁₁	C₂×#106	1	6	yes	—
88	C₂²×C₁₁	C₂×#108	1	6	yes	—
89	C₂²×C₁₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
90	C₄×C₁₁	—	2	3	no	—
91	C₄×C₁₁	—	2	1	no	—
92	C₄×C₁₁	C₂×C₄⋊C₈	1	6	yes	—
93	C₄×C₁₁	C₂×#106	1	6	yes	—
94	C₄×C₁₁	C₂³×C₈	1	1	yes	characteristic
95	C₄×C₁₁	C₂²×M₁₆	1	1	yes	characteristic
96	C₄×C₁₁	C₂²×SD₁₆	1	2	yes	—
97	C₂³×C₄	Dic₂₂	1	1	yes	characteristic
98	C₂³×C₄	D₄₄	1	1	yes	characteristic
99	C₂³×C₄	C₂×Dic₁₁	1	1	yes	characteristic
100	C₂²×C₈	—	2	4	no	—
101	C₂²×C₈	—	22	12	no	—
102	C₂²×C₈	—	44	8	no	—
103	C₂²×C₈	C₁₁⋊C₈	1	8	yes	complemented
104	C₂²×C₈	C₂×Dic₁₁	1	6	yes	—
105	C₂²×C₈	C₂×Dic₁₁	1	6	yes	—
106	C₂×C₄²	Dic₂₂	1	3	yes	—
107	C₂×C₄²	D₄₄	1	3	yes	—
108	C₂×C₄²	C₂×Dic₁₁	1	3	yes	—
109	C₂×C₄²	C₂²×D₁₁	1	3	yes	—
110	C₄⋊C₈	—	11	8	no	—
111	C₄⋊C₈	—	22	12	no	—
112	C₄×C₈	C₂×Dic₁₁	1	12	yes	—
113	C₄×C₈	C₂²×D₁₁	1	4	yes	—
114	C₂×C₁₁	C₂×#334	1	12	yes	complemented
115	C₂×C₁₁	C₂²×C₄⋊C₈	1	1	yes	characteristic
116	C₂×C₁₁	C₂²×#106	1	1	yes	characteristic
117	C₂×C₁₁	C₂²×#108	1	1	yes	characteristic
118	C₂⁴	C₄⋊Dic₁₁	1	1	yes	characteristic
119	C₂²×C₄	—	2	4	no	—
120	C₂²×C₄	#4314	1	8	yes	—
121	C₂²×C₄	C₂×C₁₁⋊C₈	1	4	yes	—
122	C₂²×C₄	#4318	1	4	yes	—
123	C₂²×C₄	C₄⋊Dic₁₁	1	3	yes	—
124	C₂²×C₄	C₄⋊Dic₁₁	1	3	yes	—
125	C₂²×C₄	C₄⋊Dic₁₁	1	3	yes	—
126	C₂²×C₄	C₂×Dic₂₂	1	3	yes	—
127	C₂²×C₄	C₂×D₄₄	1	3	yes	—
128	C₂²×C₄	C₂²×Dic₁₁	1	3	yes	—
129	C₂×C₈	—	2	12	no	—
130	C₂×C₈	—	22	8	no	—
131	C₂×C₈	—	44	24	no	—
132	C₂×C₈	C₂×C₁₁⋊C₈	1	24	yes	complemented
133	C₂×C₈	C₂²×Dic₁₁	1	4	yes	—
134	C₂×C₈	C₂²×Dic₁₁	1	4	yes	—
135	C₄²	C₄⋊Dic₁₁	1	6	yes	—
136	C₄²	C₂×Dic₂₂	1	3	yes	—
137	C₄²	C₂×D₄₄	1	3	yes	—
138	C₄²	C₂²×Dic₁₁	1	3	yes	—
139	C₄²	C₂³×D₁₁	1	1	yes	characteristic
140	C₁₁	C₂²×#334	1	1	yes	characteristic, complemented, Sylow
141	C₂³	C₄⋊(C₁₁⋊C₈)	1	4	yes	—
142	C₂³	#66881	1	4	yes	—
143	C₂³	#66883	1	4	yes	—
144	C₂³	C₂×C₄⋊Dic₁₁	1	3	yes	—
145	C₂×C₄	—	2	6	no	—
146	C₂×C₄	—	2	6	no	—
147	C₂×C₄	C₄⋊(C₁₁⋊C₈)	1	12	yes	—
148	C₂×C₄	#66881	1	12	yes	—
149	C₂×C₄	C₂×#4314	1	12	yes	—
150	C₂×C₄	C₂²×C₁₁⋊C₈	1	6	yes	—
151	C₂×C₄	C₂×#4318	1	6	yes	—
152	C₂×C₄	C₂×C₄⋊Dic₁₁	1	3	yes	—
153	C₂×C₄	C₂×C₄⋊Dic₁₁	1	3	yes	—
154	C₂×C₄	C₂×C₄⋊Dic₁₁	1	3	yes	—
155	C₂×C₄	C₂²×Dic₂₂	1	1	yes	characteristic
156	C₂×C₄	C₂²×D₄₄	1	1	yes	characteristic
157	C₂×C₄	C₂³×Dic₁₁	1	1	yes	characteristic
158	C₈	—	2	4	no	—
159	C₈	—	44	8	no	—
160	C₈	C₂²×C₁₁⋊C₈	1	8	yes	complemented
161	C₂²	#10622095	1	16	yes	direct product factor
162	C₂²	C₂×C₄⋊(C₁₁⋊C₈)	1	6	yes	—
163	C₂²	C₂×#66881	1	6	yes	—
164	C₂²	C₂×#66883	1	6	yes	—
165	C₂²	C₂²×C₄⋊Dic₁₁	1	1	yes	characteristic
166	C₄	—	2	3	no	—
167	C₄	—	2	1	no	—
168	C₄	C₂×C₄⋊(C₁₁⋊C₈)	1	6	yes	—
169	C₄	C₂×#66881	1	6	yes	—
170	C₄	C₂²×#4314	1	2	yes	—
171	C₄	C₂³×C₁₁⋊C₈	1	1	yes	characteristic
172	C₄	C₂²×#4318	1	1	yes	characteristic
173	C₂	C₂×#10622095	1	12	yes	direct product factor
174	C₂	C₂²×C₄⋊(C₁₁⋊C₈)	1	1	yes	characteristic
175	C₂	C₂²×#66881	1	1	yes	characteristic
176	C₂	C₂²×#66883	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄	D₄₄
commutator	C₄×C₁₁	C₂³×C₈
Frattini	C₄²	C₂³×D₁₁
Fitting	C₂²×C₄×C₈×C₁₁	C₂
socle, prime radical	C₂⁴×C₁₁	C₄⋊C₄
cosocle	C₄²×C₁₁	C₂⁴
hypercenter	C₂²×C₄×C₈	D₁₁
hypocenter	C₁₁	C₂²×#334
LCS	G ▷ C₄×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂³×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×#10622095	C₂	direct (N × H)
12. #10622095	C₂²	direct (N × H)
17. C₂²×C₈×C₁₁	C₈	semidirect
43. C₂×C₈×C₁₁	C₂×C₈	semidirect
74. C₈×C₁₁	C₂²×C₈	semidirect
85. C₂²×C₁₁	#334	semidirect
103. C₂²×C₈	C₁₁⋊C₈	semidirect
114. C₂×C₁₁	C₂×#334	semidirect
132. C₂×C₈	C₂×C₁₁⋊C₈	semidirect
140. C₁₁	C₂²×#334	semidirect
160. C₈	C₂²×C₁₁⋊C₈	semidirect
161. C₂²	#10622095	direct (N × H)
173. C₂	C₂×#10622095	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄⋊(C₁₁⋊C₈)	2
C₂×#10622095	2
C₂²×C₄×C₈×C₁₁	2
C₂²×#334	11

Subgroups — C22×#10622095 #14722739

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#10622095 #14722739