Subgroups — C₂×#66888 #10622749

Order 704.

146 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₂₂	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₁×M₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂×#4318	C₂	1	1	yes	maximal, characteristic
4	#66888	C₂	1	4	yes	maximal, direct product factor
5	C₂²×C₄×C₁₁	C₂²	1	1	yes	characteristic
6	C₂²×Dic₁₁	C₄	1	2	yes	—
7	C₂×Dic₂₂	—	2	1	no	—
8	C₂×Dic₂₂	—	2	1	no	—
9	C₂×Dic₂₂	—	2	1	no	—
10	C₂×Dic₂₂	—	2	1	no	—
11	C₂×Dic₂₂	C₄	1	2	yes	—
12	C₂×Dic₂₂	C₂²	1	2	yes	—
13	C₁₁×M₁₆	—	2	1	no	—
14	C₁₁×M₁₆	C₂²	1	2	yes	—
15	C₂×C₈×C₁₁	—	2	1	no	—
16	#4318	—	2	1	no	—
17	#4318	C₂²	1	2	yes	—
18	C₂×C₁₁⋊C₈	—	2	1	no	—
19	C₂³×C₁₁	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×C₁₁	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×C₁₁	D₄	1	1	yes	characteristic
22	C₂×C₄×C₁₁	D₄	1	1	yes	characteristic
23	C₂×C₄×C₁₁	D₄	1	1	yes	characteristic
24	C₂×C₄×C₁₁	D₄	1	1	yes	characteristic
25	C₂×C₄×C₁₁	C₂³	1	1	yes	characteristic
26	C₂×Dic₁₁	—	2	2	no	—
27	C₂×Dic₁₁	—	2	2	no	—
28	C₂×Dic₁₁	C₂×C₄	1	4	yes	—
29	Dic₂₂	—	2	2	no	—
30	Dic₂₂	—	2	2	no	—
31	Dic₂₂	—	2	2	no	—
32	Dic₂₂	—	2	2	no	—
33	C₈×C₁₁	—	2	2	no	—
34	C₁₁⋊C₈	—	2	2	no	—
35	C₂×#101	—	11	1	no	maximal, Sylow
36	C₂²×C₁₁	—	2	2	no	—
37	C₂²×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
38	C₂²×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
39	C₂²×C₁₁	C₂²×C₄	1	1	yes	characteristic
40	C₄×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
41	C₄×C₁₁	C₂²⋊C₄	1	1	yes	characteristic
42	C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
43	C₄×C₁₁	C₂×D₄	1	1	yes	characteristic
44	Dic₁₁	—	2	4	no	—
45	C₂²×Q₈	—	11	1	no	—
46	C₂×M₁₆	—	11	1	no	—
47	C₂×M₁₆	D₁₁	1	1	yes	characteristic
48	#101	—	11	4	no	—
49	C₂×C₁₁	—	2	1	no	—
50	C₂×C₁₁	—	2	1	no	—
51	C₂×C₁₁	#101	1	2	yes	complemented
52	C₂×C₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
53	C₂×Q₈	—	11	2	no	—
54	C₂×Q₈	—	11	2	no	—
55	C₂×Q₈	—	22	1	no	—
56	C₂×Q₈	—	22	1	no	—
57	C₂×Q₈	—	22	1	no	—
58	C₂×Q₈	—	22	1	no	—
59	C₂²×C₄	—	11	2	no	—
60	C₂²×C₄	C₂×D₁₁	1	1	yes	characteristic
61	M₁₆	—	2	1	no	—
62	M₁₆	—	11	2	no	—
63	M₁₆	—	22	1	no	—
64	M₁₆	C₂×D₁₁	1	2	yes	—
65	C₂×C₈	—	2	1	no	—
66	C₂×C₈	—	22	1	no	—
67	C₁₁	C₂×#101	1	1	yes	characteristic, complemented, Sylow
68	C₂³	C₄×D₁₁	1	1	yes	characteristic
69	Q₈	—	22	2	no	—
70	Q₈	—	22	2	no	—
71	Q₈	—	22	2	no	—
72	Q₈	—	22	2	no	—
73	C₂×C₄	—	11	4	no	—
74	C₂×C₄	—	22	2	no	—
75	C₂×C₄	—	22	2	no	—
76	C₂×C₄	C₄×D₁₁	1	1	yes	characteristic
77	C₂×C₄	D₄₄	1	1	yes	characteristic
78	C₂×C₄	D₄₄	1	1	yes	characteristic
79	C₂×C₄	C₂²⋊D₁₁	1	1	yes	characteristic
80	C₂×C₄	C₂²⋊D₁₁	1	1	yes	characteristic
81	C₂×C₄	C₂²×D₁₁	1	1	yes	characteristic
82	C₈	—	2	2	no	—
83	C₈	—	22	2	no	—
84	C₂²	—	2	2	no	—
85	C₂²	(C₂×D₁₁)⋊C₄	1	1	yes	characteristic
86	C₂²	(C₂×D₁₁)⋊C₄	1	1	yes	characteristic
87	C₂²	C₂×C₄×D₁₁	1	1	yes	characteristic
88	C₄	—	22	4	no	—
89	C₄	(C₂×D₁₁)⋊C₄	1	1	yes	characteristic
90	C₄	(C₂×D₁₁)⋊C₄	1	1	yes	characteristic
91	C₄	C₂×D₄₄	1	1	yes	characteristic
92	C₄	C₂×C₂²⋊D₁₁	1	1	yes	characteristic
93	C₂	—	2	1	no	—
94	C₂	—	2	1	no	—
95	C₂	#66888	1	2	yes	direct product factor
96	C₂	C₂×(C₂×D₁₁)⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	(C₂×D₁₁)⋊C₄
commutator	C₂²×C₁₁	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₁₁
Fitting	C₂×C₁₁×M₁₆	C₂
socle	C₂²×C₁₁	C₂²⋊C₄
prime radical	C₂³×C₁₁	C₂×C₄
cosocle	C₂×C₄×C₁₁	C₂³
hypercenter	C₂×M₁₆	D₁₁
hypocenter	C₁₁	C₂×#101
LCS	G ▷ C₂²×C₁₁ ▷ C₂×C₁₁ ▷ C₁₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
4. #66888	C₂	direct (N × H)
51. C₂×C₁₁	#101	semidirect
67. C₁₁	C₂×#101	semidirect
95. C₂	#66888	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×Dic₂₂	2
C₂×C₁₁×M₁₆	2
C₂×#4318	2
#66888	2
C₂×#101	11

Subgroups — C2×#66888 #10622749

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#66888 #10622749