Subgroups — #36326393

Order 21376.

196 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19128776	C₂	1	4	yes	maximal, complemented
2	#24923649	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₁₆₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#493×C₁₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₆₆₈	—	2	1	no	—
6	C₂×Dic₆₆₈	C₂²	1	2	yes	—
7	#19128832	—	2	1	no	—
8	#19128776	—	2	8	no	—
9	C₂×C₁₆₇⋊C₁₆	C₂²	1	2	yes	—
10	C₂×D₈×C₁₆₇	—	2	1	no	—
11	C₂×D₈×C₁₆₇	C₂²	1	2	yes	—
12	#127×C₁₆₇	—	2	1	no	—
13	C₄×C₈×C₁₆₇	C₂²	1	1	yes	characteristic
14	C₂×Dic₃₃₄	—	2	2	no	—
15	Dic₆₆₈	—	2	4	no	—
16	Dic₆₆₈	—	4	2	no	—
17	C₄⋊Dic₁₆₇	—	4	2	no	—
18	C₁₆₇⋊C₁₆	D₄	1	4	yes	complemented
19	C₂×D₄×C₁₆₇	—	2	2	no	—
20	C₂×D₄×C₁₆₇	—	4	2	no	—
21	D₈×C₁₆₇	—	2	4	no	—
22	D₈×C₁₆₇	—	4	2	no	—
23	C₂×C₈×C₁₆₇	D₄	1	1	yes	characteristic
24	C₂×C₈×C₁₆₇	C₂³	1	1	yes	characteristic
25	C₄²×C₁₆₇	D₄	1	1	yes	characteristic
26	Dic₃₃₄	—	4	2	no	—
27	Dic₃₃₄	—	4	2	no	—
28	C₂×Dic₁₆₇	—	4	2	no	—
29	C₂³×C₁₆₇	—	4	2	no	—
30	D₄×C₁₆₇	—	4	2	no	—
31	D₄×C₁₆₇	—	4	2	no	—
32	D₄×C₁₆₇	—	8	4	no	—
33	C₂×C₄×C₁₆₇	D₈	1	2	yes	—
34	C₂×C₄×C₁₆₇	C₂×D₄	1	1	yes	characteristic
35	C₈×C₁₆₇	D₈	1	2	yes	—
36	C₈×C₁₆₇	C₂×D₄	1	2	yes	—
37	C₂²×C₁₆₇	—	8	2	no	—
38	C₂²×C₁₆₇	—	8	4	no	—
39	C₂²×C₁₆₇	C₂×D₈	1	1	yes	characteristic
40	C₄×C₁₆₇	SD₃₂	1	4	yes	complemented
41	C₄×C₁₆₇	#127	1	1	yes	characteristic
42	C₄×C₁₆₇	C₂×D₈	1	1	yes	characteristic
43	Dic₁₆₇	—	8	2	no	—
44	C₂×C₁₆₇	—	16	2	no	—
45	C₂×C₁₆₇	#493	1	1	yes	characteristic
46	C₂×C₁₆₇	C₂×SD₃₂	1	2	yes	—
47	C₁₆₇	#2248	1	1	yes	characteristic, complemented, Sylow
48	#2248	—	167	1	no	maximal, Sylow
49	C₂×SD₃₂	—	167	4	no	—
50	#494	—	167	1	no	—
51	#493	D₁₆₇	1	1	yes	characteristic
52	C₄×C₁₆	—	167	1	no	—
53	C₂×Q₁₆	—	334	1	no	—
54	C₂×Q₁₆	—	167	2	no	—
55	C₂×D₈	—	2	1	no	—
56	C₂×D₈	C₂×D₁₆₇	1	2	yes	—
57	C₄⋊Q₈	—	334	1	no	—
58	#127	—	2	1	no	—
59	SD₃₂	—	334	8	no	—
60	C₂×C₁₆	—	167	2	no	—
61	C₄×C₈	C₂×D₁₆₇	1	1	yes	characteristic
62	C₂×Q₈	—	334	2	no	—
63	C₂×D₄	—	2	2	no	—
64	C₂×D₄	—	4	2	no	—
65	Q₁₆	—	334	4	no	—
66	Q₁₆	—	668	2	no	—
67	D₈	—	2	4	no	—
68	D₈	—	4	2	no	—
69	C₂×C₈	C₂²⋊D₁₆₇	1	1	yes	characteristic
70	C₂×C₈	C₂²×D₁₆₇	1	1	yes	characteristic
71	C₄⋊C₄	—	668	2	no	—
72	C₄²	C₂²⋊D₁₆₇	1	1	yes	characteristic
73	C₁₆	—	167	4	no	—
74	C₂³	—	4	2	no	—
75	Q₈	—	668	2	no	—
76	Q₈	—	668	2	no	—
77	D₄	—	4	2	no	—
78	D₄	—	4	2	no	—
79	D₄	—	8	4	no	—
80	C₂×C₄	—	668	2	no	—
81	C₂×C₄	#14581935	1	2	yes	—
82	C₂×C₄	C₂×C₂²⋊D₁₆₇	1	1	yes	characteristic
83	C₈	#14581935	1	2	yes	—
84	C₈	D₄×D₁₆₇	1	2	yes	—
85	C₂²	—	8	4	no	—
86	C₂²	—	8	2	no	—
87	C₂²	C₂×#14581935	1	1	yes	characteristic
88	C₄	—	1336	2	no	—
89	C₄	#19128776	1	4	yes	complemented
90	C₄	#19128828	1	1	yes	characteristic
91	C₄	C₂×#14581935	1	1	yes	characteristic
92	C₂	—	16	2	no	—
93	C₂	#24923645	1	1	yes	characteristic
94	C₂	C₂×#19128776	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#14581935
commutator, cosocle	C₂×C₈×C₁₆₇	C₂³
Frattini	C₂×C₈	C₂²×D₁₆₇
Fitting, prime radical	#493×C₁₆₇	C₂
socle	C₂²×C₁₆₇	C₂×D₈
hypercenter	#493	D₁₆₇
hypocenter	C₁₆₇	#2248
LCS	G ▷ C₂×C₈×C₁₆₇ ▷ C₄×C₁₆₇ ▷ C₂×C₁₆₇ ▷ C₁₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #493

Complements

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

N	H ≅ G/N	type
1. C₂×#19128776	C₂	semidirect
2. #24923649	C₂	semidirect
3. C₄×C₁₆₇⋊C₁₆	C₂	semidirect
18. C₁₆₇⋊C₁₆	D₄	semidirect
40. C₄×C₁₆₇	SD₃₂	semidirect
47. C₁₆₇	#2248	semidirect
89. C₄	#19128776	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19128776	2
#24923649	2
C₄×C₁₆₇⋊C₁₆	2
#493×C₁₆₇	2
#2248	167