Subgroups — #540×C₁₆₇ #24922941

Order 10688.

240 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#127×C₁₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	#125×C₁₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#123×C₁₆₇	C₂	1	4	yes	maximal, complemented
4	#121×C₁₆₇	C₂	1	4	yes	maximal, complemented
5	#121×C₁₆₇	C₂	1	2	yes	maximal, complemented
6	C₄×D₄×C₁₆₇	C₂	1	2	yes	maximal, complemented
7	#117×C₁₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×D₄×C₁₆₇	C₂²	1	2	yes	—
9	C₂×D₄×C₁₆₇	C₂²	1	4	yes	—
10	C₂×D₄×C₁₆₇	C₂²	1	4	yes	—
11	C₂²×C₄×C₁₆₇	C₂²	1	1	yes	characteristic
12	C₂²×C₄×C₁₆₇	C₂²	1	4	yes	—
13	C₄⋊C₄×C₁₆₇	C₂²	1	2	yes	—
14	C₄⋊C₄×C₁₆₇	C₂²	1	2	yes	—
15	C₄⋊C₄×C₁₆₇	C₂²	1	4	yes	—
16	C₂²⋊C₄×C₁₆₇	C₂²	1	2	yes	—
17	C₂²⋊C₄×C₁₆₇	C₂²	1	4	yes	—
18	C₂²⋊C₄×C₁₆₇	C₂²	1	4	yes	—
19	C₄²×C₁₆₇	C₂²	1	1	yes	characteristic
20	C₄²×C₁₆₇	C₂²	1	1	yes	characteristic
21	C₂³×C₁₆₇	C₂³	1	1	yes	characteristic
22	C₂³×C₁₆₇	C₂³	1	4	yes	—
23	D₄×C₁₆₇	—	2	4	no	—
24	D₄×C₁₆₇	—	4	4	no	—
25	D₄×C₁₆₇	—	4	4	no	—
26	C₂×C₄×C₁₆₇	—	2	2	no	—
27	C₂×C₄×C₁₆₇	—	4	4	no	—
28	C₂×C₄×C₁₆₇	C₂³	1	2	yes	—
29	C₂×C₄×C₁₆₇	C₂³	1	2	yes	—
30	C₂×C₄×C₁₆₇	C₂³	1	1	yes	characteristic
31	C₂×C₄×C₁₆₇	C₂³	1	4	yes	—
32	C₂×C₄×C₁₆₇	C₂³	1	1	yes	characteristic
33	C₂²×C₁₆₇	—	2	1	no	—
34	C₂²×C₁₆₇	—	2	4	no	—
35	C₂²×C₁₆₇	—	2	4	no	—
36	C₂²×C₁₆₇	—	2	2	no	—
37	C₂²×C₁₆₇	—	2	4	no	—
38	C₂²×C₁₆₇	C₂⁴	1	1	yes	characteristic
39	C₄×C₁₆₇	—	2	1	no	—
40	C₄×C₁₆₇	—	2	2	no	—
41	C₄×C₁₆₇	—	2	4	no	—
42	C₄×C₁₆₇	—	2	2	no	—
43	C₄×C₁₆₇	#41	1	2	yes	—
44	C₂×C₁₆₇	—	4	1	no	—
45	C₂×C₁₆₇	—	4	4	no	—
46	C₂×C₁₆₇	C₂×#41	1	1	yes	characteristic
47	C₂×C₁₆₇	H(5,𝔽₂)	1	2	yes	—
48	C₁₆₇	#540	1	1	yes	characteristic, direct product factor, Sylow
49	#540	C₁₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
50	#127	C₂×C₁₆₇	1	1	yes	characteristic, complemented
51	#125	C₂×C₁₆₇	1	1	yes	characteristic, complemented
52	#123	C₂×C₁₆₇	1	4	yes	complemented
53	#121	C₂×C₁₆₇	1	4	yes	complemented
54	#121	C₂×C₁₆₇	1	2	yes	complemented
55	C₄×D₄	C₂×C₁₆₇	1	2	yes	complemented
56	#117	C₂×C₁₆₇	1	1	yes	characteristic, complemented
57	C₂×D₄	C₂²×C₁₆₇	1	2	yes	—
58	C₂×D₄	C₂²×C₁₆₇	1	4	yes	—
59	C₂×D₄	C₂²×C₁₆₇	1	4	yes	—
60	C₂²×C₄	C₂²×C₁₆₇	1	1	yes	characteristic
61	C₂²×C₄	C₂²×C₁₆₇	1	4	yes	—
62	C₄⋊C₄	C₂²×C₁₆₇	1	2	yes	—
63	C₄⋊C₄	C₂²×C₁₆₇	1	2	yes	—
64	C₄⋊C₄	C₂²×C₁₆₇	1	4	yes	—
65	C₂²⋊C₄	C₂²×C₁₆₇	1	2	yes	—
66	C₂²⋊C₄	C₂²×C₁₆₇	1	4	yes	—
67	C₂²⋊C₄	C₂²×C₁₆₇	1	4	yes	—
68	C₄²	C₂²×C₁₆₇	1	1	yes	characteristic
69	C₄²	C₂²×C₁₆₇	1	1	yes	characteristic
70	C₂³	C₂³×C₁₆₇	1	1	yes	characteristic
71	C₂³	C₂³×C₁₆₇	1	4	yes	—
72	D₄	—	2	4	no	—
73	D₄	—	4	4	no	—
74	D₄	—	4	4	no	—
75	C₂×C₄	—	2	2	no	—
76	C₂×C₄	—	4	4	no	—
77	C₂×C₄	C₂³×C₁₆₇	1	2	yes	—
78	C₂×C₄	C₂³×C₁₆₇	1	2	yes	—
79	C₂×C₄	C₂³×C₁₆₇	1	1	yes	characteristic
80	C₂×C₄	C₂³×C₁₆₇	1	4	yes	—
81	C₂×C₄	C₂³×C₁₆₇	1	1	yes	characteristic
82	C₂²	—	2	1	no	—
83	C₂²	—	2	4	no	—
84	C₂²	—	2	4	no	—
85	C₂²	—	2	2	no	—
86	C₂²	—	2	4	no	—
87	C₂²	C₂⁴×C₁₆₇	1	1	yes	characteristic
88	C₄	—	2	1	no	—
89	C₄	—	2	2	no	—
90	C₄	—	2	4	no	—
91	C₄	—	2	2	no	—
92	C₄	#41×C₁₆₇	1	2	yes	—
93	C₂	—	4	1	no	—
94	C₂	—	4	4	no	—
95	C₂	C₂×#41×C₁₆₇	1	1	yes	characteristic
96	C₂	H(5,𝔽₂)×C₁₆₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₆₇	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁴×C₁₆₇
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. #127×C₁₆₇	C₂	semidirect
2. #125×C₁₆₇	C₂	semidirect
3. #123×C₁₆₇	C₂	semidirect
4. #121×C₁₆₇	C₂	semidirect
5. #121×C₁₆₇	C₂	semidirect
6. C₄×D₄×C₁₆₇	C₂	semidirect
7. #117×C₁₆₇	C₂	semidirect
48. C₁₆₇	#540	direct (N × H)
49. #540	C₁₆₇	direct (N × H)
50. #127	C₂×C₁₆₇	semidirect
51. #125	C₂×C₁₆₇	semidirect
52. #123	C₂×C₁₆₇	semidirect
53. #121	C₂×C₁₆₇	semidirect
54. #121	C₂×C₁₆₇	semidirect
55. C₄×D₄	C₂×C₁₆₇	semidirect
56. #117	C₂×C₁₆₇	semidirect

Maximal subgroups

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

subgroup	index
#127×C₁₆₇	2
#125×C₁₆₇	2
#123×C₁₆₇	2
#121×C₁₆₇	2
C₄×D₄×C₁₆₇	2
#117×C₁₆₇	2
#540	167

Subgroups — #540×C167 #24922941

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #540×C₁₆₇ #24922941