Subgroups — C₂²×#19128772 #36337234

Order 21376.

400 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19128772	C₂	1	12	yes	maximal, direct product factor
2	C₂²×C₁₆₇⋊C₁₆	C₂	1	2	yes	maximal, complemented
3	C₂³×C₈×C₁₆₇	C₂	1	1	yes	maximal, characteristic
4	#19128772	C₂²	1	16	yes	direct product factor
5	C₂×C₁₆₇⋊C₁₆	C₂²	1	12	yes	complemented
6	C₂³×C₄×C₁₆₇	C₄	1	1	yes	characteristic
7	C₂²×C₈×C₁₆₇	C₄	1	1	yes	characteristic
8	C₂²×C₈×C₁₆₇	C₄	1	6	yes	—
9	C₂²×C₈×C₁₆₇	C₂²	1	6	yes	—
10	C₂²×C₈×C₁₆₇	C₂²	1	1	yes	characteristic
11	C₁₆₇⋊C₁₆	C₂³	1	8	yes	complemented
12	C₂⁴×C₁₆₇	C₈	1	1	yes	characteristic
13	C₂²×C₄×C₁₆₇	C₈	1	6	yes	—
14	C₂²×C₄×C₁₆₇	C₈	1	1	yes	characteristic
15	C₂²×C₄×C₁₆₇	C₂×C₄	1	6	yes	—
16	C₂²×C₄×C₁₆₇	C₂×C₄	1	1	yes	characteristic
17	C₂×C₈×C₁₆₇	C₂×C₄	1	12	yes	—
18	C₂×C₈×C₁₆₇	C₂×C₄	1	6	yes	—
19	C₂×C₈×C₁₆₇	C₂×C₄	1	3	yes	—
20	C₂×C₈×C₁₆₇	C₂³	1	3	yes	—
21	C₂×C₈×C₁₆₇	C₂³	1	4	yes	—
22	C₂³×C₁₆₇	—	2	4	no	—
23	C₂³×C₁₆₇	C₂×C₈	1	6	yes	—
24	C₂³×C₁₆₇	C₂×C₈	1	1	yes	characteristic
25	C₂×C₄×C₁₆₇	C₂×C₈	1	12	yes	—
26	C₂×C₄×C₁₆₇	C₂×C₈	1	6	yes	—
27	C₂×C₄×C₁₆₇	C₂×C₈	1	3	yes	—
28	C₂×C₄×C₁₆₇	C₂²×C₄	1	4	yes	—
29	C₂×C₄×C₁₆₇	C₂²×C₄	1	3	yes	—
30	C₈×C₁₆₇	C₂²×C₄	1	3	yes	—
31	C₈×C₁₆₇	C₂²×C₄	1	4	yes	—
32	C₈×C₁₆₇	C₂⁴	1	1	yes	characteristic
33	C₂²×C₁₆₇	—	2	12	no	—
34	C₂²×C₁₆₇	M₃₂	1	4	yes	complemented
35	C₂²×C₁₆₇	C₂²×C₈	1	4	yes	—
36	C₂²×C₁₆₇	C₂²×C₈	1	3	yes	—
37	C₄×C₁₆₇	C₂²×C₈	1	3	yes	—
38	C₄×C₁₆₇	C₂²×C₈	1	4	yes	—
39	C₄×C₁₆₇	C₂³×C₄	1	1	yes	characteristic
40	C₂×C₁₆₇	—	2	4	no	—
41	C₂×C₁₆₇	C₂×M₃₂	1	6	yes	complemented
42	C₂×C₁₆₇	C₂³×C₈	1	1	yes	characteristic
43	C₁₆₇	C₂²×M₃₂	1	1	yes	characteristic, complemented, Sylow
44	C₂²×M₃₂	—	167	1	no	maximal, Sylow
45	C₂³×C₈	D₁₆₇	1	1	yes	characteristic
46	C₂×M₃₂	—	167	12	no	—
47	C₂²×C₁₆	—	167	2	no	—
48	C₂³×C₄	Dic₁₆₇	1	1	yes	characteristic
49	C₂²×C₈	Dic₁₆₇	1	1	yes	characteristic
50	C₂²×C₈	Dic₁₆₇	1	6	yes	—
51	C₂²×C₈	C₂×D₁₆₇	1	6	yes	—
52	C₂²×C₈	C₂×D₁₆₇	1	1	yes	characteristic
53	M₃₂	—	167	16	no	—
54	C₂×C₁₆	—	167	12	no	—
55	C₂⁴	C₁₆₇⋊C₈	1	1	yes	characteristic
56	C₂²×C₄	C₁₆₇⋊C₈	1	6	yes	—
57	C₂²×C₄	C₁₆₇⋊C₈	1	1	yes	characteristic
58	C₂²×C₄	C₂×Dic₁₆₇	1	6	yes	—
59	C₂²×C₄	C₂×Dic₁₆₇	1	1	yes	characteristic
60	C₂×C₈	C₂×Dic₁₆₇	1	6	yes	—
61	C₂×C₈	C₂×Dic₁₆₇	1	12	yes	—
62	C₂×C₈	C₂×Dic₁₆₇	1	3	yes	—
63	C₂×C₈	C₂²×D₁₆₇	1	3	yes	—
64	C₂×C₈	C₂²×D₁₆₇	1	4	yes	—
65	C₁₆	—	167	8	no	—
66	C₂³	—	2	4	no	—
67	C₂³	C₂×C₁₆₇⋊C₈	1	6	yes	—
68	C₂³	C₂×C₁₆₇⋊C₈	1	1	yes	characteristic
69	C₂×C₄	C₂×C₁₆₇⋊C₈	1	12	yes	—
70	C₂×C₄	C₂×C₁₆₇⋊C₈	1	6	yes	—
71	C₂×C₄	C₂×C₁₆₇⋊C₈	1	3	yes	—
72	C₂×C₄	C₂²×Dic₁₆₇	1	4	yes	—
73	C₂×C₄	C₂²×Dic₁₆₇	1	3	yes	—
74	C₈	C₂²×Dic₁₆₇	1	4	yes	—
75	C₈	C₂²×Dic₁₆₇	1	3	yes	—
76	C₈	C₂³×D₁₆₇	1	1	yes	characteristic
77	C₂²	—	2	12	no	—
78	C₂²	#19128772	1	4	yes	direct product factor
79	C₂²	C₂²×C₁₆₇⋊C₈	1	4	yes	—
80	C₂²	C₂²×C₁₆₇⋊C₈	1	3	yes	—
81	C₄	C₂²×C₁₆₇⋊C₈	1	3	yes	—
82	C₄	C₂²×C₁₆₇⋊C₈	1	4	yes	—
83	C₄	C₂³×Dic₁₆₇	1	1	yes	characteristic
84	C₂	—	2	4	no	—
85	C₂	C₂×#19128772	1	6	yes	direct product factor
86	C₂	C₂³×C₁₆₇⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂×D₁₆₇
commutator	C₂×C₁₆₇	C₂³×C₈
Frattini	C₈	C₂³×D₁₆₇
Fitting	C₂³×C₈×C₁₆₇	C₂
socle	C₂³×C₁₆₇	C₂×C₈
prime radical	C₂⁴×C₁₆₇	C₈
cosocle	C₈×C₁₆₇	C₂⁴
hypercenter	C₂³×C₈	D₁₆₇
hypocenter	C₁₆₇	C₂²×M₃₂
LCS	G ▷ C₂×C₁₆₇ ▷ C₁₆₇
UCS	C₁ ◁ C₂²×C₈ ◁ C₂³×C₈

Complements

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

N	H ≅ G/N	type
1. C₂×#19128772	C₂	direct (N × H)
2. C₂²×C₁₆₇⋊C₁₆	C₂	semidirect
4. #19128772	C₂²	direct (N × H)
5. C₂×C₁₆₇⋊C₁₆	C₂²	semidirect
11. C₁₆₇⋊C₁₆	C₂³	semidirect
34. C₂²×C₁₆₇	M₃₂	semidirect
41. C₂×C₁₆₇	C₂×M₃₂	semidirect
43. C₁₆₇	C₂²×M₃₂	semidirect
78. C₂²	#19128772	direct (N × H)
85. C₂	C₂×#19128772	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#19128772	2
C₂²×C₁₆₇⋊C₁₆	2
C₂³×C₈×C₁₆₇	2
C₂²×M₃₂	167

Subgroups — C22×#19128772 #36337234

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#19128772 #36337234