Subgroups — #451×C₁₆₇ #24922852

Order 10688.

148 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×Q₁₆×C₁₆₇	C₂	1	2	yes	maximal, complemented
3	C₂²⋊Q₈×C₁₆₇	C₂	1	1	yes	maximal, characteristic
4	#103×C₁₆₇	C₂	1	2	yes	maximal, complemented
5	C₂²⋊C₈×C₁₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×Q₈×C₁₆₇	—	2	4	no	—
7	C₂×Q₈×C₁₆₇	—	2	1	no	—
8	C₂×Q₈×C₁₆₇	C₂²	1	1	yes	characteristic
9	C₂×Q₈×C₁₆₇	C₂²	1	2	yes	—
10	C₂²×C₄×C₁₆₇	—	2	1	no	—
11	C₂²×C₄×C₁₆₇	C₂²	1	1	yes	characteristic
12	Q₁₆×C₁₆₇	—	2	4	no	—
13	C₂×C₈×C₁₆₇	C₂²	1	2	yes	—
14	C₄⋊C₄×C₁₆₇	—	2	1	no	—
15	C₄⋊C₄×C₁₆₇	C₂²	1	1	yes	characteristic
16	C₂²⋊C₄×C₁₆₇	—	2	1	no	—
17	C₂³×C₁₆₇	D₄	1	1	yes	characteristic
18	Q₈×C₁₆₇	—	2	4	no	—
19	Q₈×C₁₆₇	—	2	2	no	—
20	Q₈×C₁₆₇	—	2	2	no	—
21	Q₈×C₁₆₇	D₄	1	4	yes	—
22	C₂×C₄×C₁₆₇	—	2	4	no	—
23	C₂×C₄×C₁₆₇	—	2	2	no	—
24	C₂×C₄×C₁₆₇	—	2	2	no	—
25	C₂×C₄×C₁₆₇	—	2	1	no	—
26	C₂×C₄×C₁₆₇	—	2	1	no	—
27	C₂×C₄×C₁₆₇	D₄	1	1	yes	characteristic
28	C₂×C₄×C₁₆₇	C₂³	1	1	yes	characteristic
29	C₈×C₁₆₇	—	2	2	no	—
30	C₂²×C₁₆₇	—	2	1	no	—
31	C₂²×C₁₆₇	—	2	1	no	—
32	C₂²×C₁₆₇	Q₁₆	1	2	yes	complemented
33	C₂²×C₁₆₇	C₂×D₄	1	1	yes	characteristic
34	C₄×C₁₆₇	—	2	4	no	—
35	C₄×C₁₆₇	—	2	1	no	—
36	C₄×C₁₆₇	—	4	1	no	—
37	C₄×C₁₆₇	—	4	1	no	—
38	C₄×C₁₆₇	C₂×D₄	1	2	yes	—
39	C₂×C₁₆₇	—	2	2	no	—
40	C₂×C₁₆₇	#120	1	1	yes	characteristic
41	C₂×C₁₆₇	C₂×Q₁₆	1	1	yes	characteristic
42	C₂×C₁₆₇	#137	1	1	yes	characteristic
43	C₁₆₇	#451	1	1	yes	characteristic, direct product factor, Sylow
44	#451	C₁₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
45	C₂²×Q₈	C₂×C₁₆₇	1	1	yes	characteristic
46	C₂×Q₁₆	C₂×C₁₆₇	1	2	yes	complemented
47	C₂²⋊Q₈	C₂×C₁₆₇	1	1	yes	characteristic
48	#103	C₂×C₁₆₇	1	2	yes	complemented
49	C₂²⋊C₈	C₂×C₁₆₇	1	1	yes	characteristic
50	C₂×Q₈	—	2	4	no	—
51	C₂×Q₈	—	2	1	no	—
52	C₂×Q₈	C₂²×C₁₆₇	1	1	yes	characteristic
53	C₂×Q₈	C₂²×C₁₆₇	1	2	yes	—
54	C₂²×C₄	—	2	1	no	—
55	C₂²×C₄	C₂²×C₁₆₇	1	1	yes	characteristic
56	Q₁₆	—	2	4	no	—
57	C₂×C₈	C₂²×C₁₆₇	1	2	yes	—
58	C₄⋊C₄	—	2	1	no	—
59	C₄⋊C₄	C₂²×C₁₆₇	1	1	yes	characteristic
60	C₂²⋊C₄	—	2	1	no	—
61	C₂³	D₄×C₁₆₇	1	1	yes	characteristic
62	Q₈	—	2	4	no	—
63	Q₈	—	2	2	no	—
64	Q₈	—	2	2	no	—
65	Q₈	D₄×C₁₆₇	1	4	yes	—
66	C₂×C₄	—	2	4	no	—
67	C₂×C₄	—	2	2	no	—
68	C₂×C₄	—	2	2	no	—
69	C₂×C₄	—	2	1	no	—
70	C₂×C₄	—	2	1	no	—
71	C₂×C₄	D₄×C₁₆₇	1	1	yes	characteristic
72	C₂×C₄	C₂³×C₁₆₇	1	1	yes	characteristic
73	C₈	—	2	2	no	—
74	C₂²	—	2	1	no	—
75	C₂²	—	2	1	no	—
76	C₂²	Q₁₆×C₁₆₇	1	2	yes	complemented
77	C₂²	C₂×D₄×C₁₆₇	1	1	yes	characteristic
78	C₄	—	2	4	no	—
79	C₄	—	2	1	no	—
80	C₄	—	4	1	no	—
81	C₄	—	4	1	no	—
82	C₄	C₂×D₄×C₁₆₇	1	2	yes	—
83	C₂	—	2	2	no	—
84	C₂	#120×C₁₆₇	1	1	yes	characteristic
85	C₂	C₂×Q₁₆×C₁₆₇	1	1	yes	characteristic
86	C₂	#137×C₁₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₆₇	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂³×C₁₆₇
prime radical	C₂³×C₁₆₇	D₄
LCS	G ▷ C₂×C₄ ▷ 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
2. C₂×Q₁₆×C₁₆₇	C₂	semidirect
4. #103×C₁₆₇	C₂	semidirect
32. C₂²×C₁₆₇	Q₁₆	semidirect
43. C₁₆₇	#451	direct (N × H)
44. #451	C₁₆₇	direct (N × H)
46. C₂×Q₁₆	C₂×C₁₆₇	semidirect
48. #103	C₂×C₁₆₇	semidirect
76. C₂²	Q₁₆×C₁₆₇	semidirect

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×Q₁₆×C₁₆₇	2
C₂²⋊Q₈×C₁₆₇	2
#103×C₁₆₇	2
C₂²⋊C₈×C₁₆₇	2
#451	167

Subgroups — #451×C167 #24922852

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #451×C₁₆₇ #24922852