Subgroups — C₂×#101×C₁₆₇ #24922813

Order 10688.

146 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Q₈×C₁₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×M₁₆×C₁₆₇	C₂	1	2	yes	maximal
3	#101×C₁₆₇	C₂	1	4	yes	maximal, direct product factor
4	C₂×Q₈×C₁₆₇	—	2	2	no	—
5	C₂×Q₈×C₁₆₇	—	2	2	no	—
6	C₂×Q₈×C₁₆₇	C₄	1	2	yes	—
7	C₂×Q₈×C₁₆₇	C₂²	1	2	yes	—
8	C₂²×C₄×C₁₆₇	C₄	1	2	yes	—
9	C₂²×C₄×C₁₆₇	C₂²	1	1	yes	characteristic
10	M₁₆×C₁₆₇	—	2	2	no	—
11	M₁₆×C₁₆₇	C₂²	1	4	yes	—
12	C₂×C₈×C₁₆₇	—	2	2	no	—
13	C₂³×C₁₆₇	C₂×C₄	1	1	yes	characteristic
14	Q₈×C₁₆₇	—	2	4	no	—
15	Q₈×C₁₆₇	—	2	4	no	—
16	C₂×C₄×C₁₆₇	—	2	2	no	—
17	C₂×C₄×C₁₆₇	—	2	2	no	—
18	C₂×C₄×C₁₆₇	C₂×C₄	1	4	yes	—
19	C₂×C₄×C₁₆₇	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×C₁₆₇	D₄	1	2	yes	—
21	C₂×C₄×C₁₆₇	D₄	1	2	yes	—
22	C₂×C₄×C₁₆₇	C₂³	1	1	yes	characteristic
23	C₈×C₁₆₇	—	2	4	no	—
24	C₂²×C₁₆₇	—	2	2	no	—
25	C₂²×C₁₆₇	C₂²⋊C₄	1	1	yes	characteristic
26	C₂²×C₁₆₇	C₂²⋊C₄	1	1	yes	characteristic
27	C₂²×C₁₆₇	C₂²×C₄	1	1	yes	characteristic
28	C₄×C₁₆₇	—	2	4	no	—
29	C₄×C₁₆₇	C₂²⋊C₄	1	2	yes	—
30	C₄×C₁₆₇	C₂×D₄	1	2	yes	—
31	C₂×C₁₆₇	—	2	1	no	—
32	C₂×C₁₆₇	—	2	1	no	—
33	C₂×C₁₆₇	#101	1	2	yes	direct product factor
34	C₂×C₁₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
35	C₁₆₇	C₂×#101	1	1	yes	characteristic, direct product factor, Sylow
36	C₂×#101	C₁₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
37	C₂²×Q₈	C₂×C₁₆₇	1	1	yes	characteristic
38	C₂×M₁₆	C₂×C₁₆₇	1	2	yes	—
39	#101	C₂×C₁₆₇	1	4	yes	direct product factor
40	C₂×Q₈	—	2	2	no	—
41	C₂×Q₈	—	2	2	no	—
42	C₂×Q₈	C₄×C₁₆₇	1	2	yes	—
43	C₂×Q₈	C₂²×C₁₆₇	1	2	yes	—
44	C₂²×C₄	C₄×C₁₆₇	1	2	yes	—
45	C₂²×C₄	C₂²×C₁₆₇	1	1	yes	characteristic
46	M₁₆	—	2	2	no	—
47	M₁₆	C₂²×C₁₆₇	1	4	yes	—
48	C₂×C₈	—	2	2	no	—
49	C₂³	C₂×C₄×C₁₆₇	1	1	yes	characteristic
50	Q₈	—	2	4	no	—
51	Q₈	—	2	4	no	—
52	C₂×C₄	—	2	2	no	—
53	C₂×C₄	—	2	2	no	—
54	C₂×C₄	C₂×C₄×C₁₆₇	1	4	yes	—
55	C₂×C₄	C₂×C₄×C₁₆₇	1	1	yes	characteristic
56	C₂×C₄	D₄×C₁₆₇	1	2	yes	—
57	C₂×C₄	D₄×C₁₆₇	1	2	yes	—
58	C₂×C₄	C₂³×C₁₆₇	1	1	yes	characteristic
59	C₈	—	2	4	no	—
60	C₂²	—	2	2	no	—
61	C₂²	C₂²⋊C₄×C₁₆₇	1	1	yes	characteristic
62	C₂²	C₂²⋊C₄×C₁₆₇	1	1	yes	characteristic
63	C₂²	C₂²×C₄×C₁₆₇	1	1	yes	characteristic
64	C₄	—	2	4	no	—
65	C₄	C₂²⋊C₄×C₁₆₇	1	2	yes	—
66	C₄	C₂×D₄×C₁₆₇	1	2	yes	—
67	C₂	—	2	1	no	—
68	C₂	—	2	1	no	—
69	C₂	#101×C₁₆₇	1	2	yes	direct product factor
70	C₂	C₂×C₂²⋊C₄×C₁₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₆₇	C₂²⋊C₄
commutator	C₂²	C₂²×C₄×C₁₆₇
Frattini, cosocle	C₂×C₄	C₂³×C₁₆₇
prime radical	C₂³×C₁₆₇	C₂×C₄
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₆₇ ◁ C₂²×C₄×C₁₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. #101×C₁₆₇	C₂	direct (N × H)
33. C₂×C₁₆₇	#101	direct (N × H)
35. C₁₆₇	C₂×#101	direct (N × H)
36. C₂×#101	C₁₆₇	direct (N × H)
39. #101	C₂×C₁₆₇	direct (N × H)
69. C₂	#101×C₁₆₇	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×Q₈×C₁₆₇	2
C₂×M₁₆×C₁₆₇	2
#101×C₁₆₇	2
C₂×#101	167

Subgroups — C2×#101×C167 #24922813

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#101×C₁₆₇ #24922813