Subgroups — #36326395

Order 21376.

196 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19128775	C₂	1	4	yes	maximal, complemented
2	#24923646	C₂	1	1	yes	maximal, characteristic
3	C₄×C₁₆₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#494×C₁₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₁₃₃₆	—	2	1	no	—
6	C₂×D₁₃₃₆	C₂²	1	2	yes	—
7	#19128829	—	2	1	no	—
8	#19128775	—	2	8	no	—
9	C₂×C₁₆₇⋊C₁₆	C₂²	1	2	yes	—
10	C₂×Q₁₆×C₁₆₇	—	2	1	no	—
11	C₂×Q₁₆×C₁₆₇	C₂²	1	2	yes	—
12	C₄⋊Q₈×C₁₆₇	—	2	1	no	—
13	C₄×C₈×C₁₆₇	C₂²	1	1	yes	characteristic
14	C₂×D₆₆₈	—	2	2	no	—
15	C₂×D₆₆₈	—	4	2	no	—
16	D₁₃₃₆	—	2	4	no	—
17	D₁₃₃₆	—	4	2	no	—
18	C₁₆₇⋊C₁₆	D₄	1	4	yes	complemented
19	C₂×Q₈×C₁₆₇	—	2	2	no	—
20	Q₁₆×C₁₆₇	—	2	4	no	—
21	Q₁₆×C₁₆₇	—	4	2	no	—
22	C₂×C₈×C₁₆₇	D₄	1	1	yes	characteristic
23	C₂×C₈×C₁₆₇	C₂³	1	1	yes	characteristic
24	C₄⋊C₄×C₁₆₇	—	4	2	no	—
25	C₄²×C₁₆₇	D₄	1	1	yes	characteristic
26	C₂²×D₁₆₇	—	4	2	no	—
27	D₆₆₈	—	4	2	no	—
28	D₆₆₈	—	4	2	no	—
29	D₆₆₈	—	8	4	no	—
30	Q₈×C₁₆₇	—	4	2	no	—
31	Q₈×C₁₆₇	—	4	2	no	—
32	C₂×C₄×C₁₆₇	—	4	2	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	C₂×D₈	1	1	yes	characteristic
38	C₂×D₁₆₇	—	8	2	no	—
39	C₂×D₁₆₇	—	8	4	no	—
40	C₄×C₁₆₇	—	8	2	no	—
41	C₄×C₁₆₇	SD₃₂	1	4	yes	complemented
42	C₄×C₁₆₇	#127	1	1	yes	characteristic
43	C₄×C₁₆₇	C₂×D₈	1	1	yes	characteristic
44	C₂×C₁₆₇	#493	1	1	yes	characteristic
45	C₂×C₁₆₇	C₂×SD₃₂	1	2	yes	—
46	D₁₆₇	—	16	2	no	—
47	C₁₆₇	#2248	1	1	yes	characteristic, complemented, Sylow
48	#2248	—	167	1	no	maximal, Sylow
49	C₂×SD₃₂	—	167	4	no	—
50	#494	D₁₆₇	1	1	yes	characteristic, complemented
51	#493	—	167	1	no	—
52	C₄×C₁₆	—	167	1	no	—
53	C₂×Q₁₆	—	2	1	no	—
54	C₂×Q₁₆	C₂×D₁₆₇	1	2	yes	—
55	C₂×D₈	—	334	1	no	—
56	C₂×D₈	—	167	2	no	—
57	C₄⋊Q₈	—	2	1	no	—
58	#127	—	334	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	—	2	2	no	—
63	C₂×D₄	—	334	2	no	—
64	C₂×D₄	—	668	2	no	—
65	Q₁₆	—	2	4	no	—
66	Q₁₆	—	4	2	no	—
67	D₈	—	334	4	no	—
68	D₈	—	668	2	no	—
69	C₂×C₈	C₂²⋊D₁₆₇	1	1	yes	characteristic
70	C₂×C₈	C₂²×D₁₆₇	1	1	yes	characteristic
71	C₄⋊C₄	—	4	2	no	—
72	C₄²	C₂²⋊D₁₆₇	1	1	yes	characteristic
73	C₁₆	—	167	4	no	—
74	C₂³	—	668	2	no	—
75	Q₈	—	4	2	no	—
76	Q₈	—	4	2	no	—
77	D₄	—	1336	4	no	—
78	D₄	—	668	2	no	—
79	D₄	—	668	2	no	—
80	C₂×C₄	—	4	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	—	1336	4	no	—
86	C₂²	—	1336	2	no	—
87	C₂²	C₂×#14581935	1	1	yes	characteristic
88	C₄	—	8	2	no	—
89	C₄	#19128775	1	4	yes	complemented
90	C₄	#19128828	1	1	yes	characteristic
91	C₄	C₂×#14581935	1	1	yes	characteristic
92	C₂	—	2672	2	no	—
93	C₂	#24923645	1	1	yes	characteristic
94	C₂	C₂×#19128775	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#14581935
commutator, cosocle	C₂×C₈×C₁₆₇	C₂³
Frattini	C₂×C₈	C₂²×D₁₆₇
Fitting	#494×C₁₆₇	C₂
socle	C₂²×C₁₆₇	C₂×D₈
prime radical	#24923646	C₂
hypercenter	#494	D₁₆₇
hypocenter	C₁₆₇	#2248
LCS	G ▷ C₂×C₈×C₁₆₇ ▷ C₄×C₁₆₇ ▷ C₂×C₁₆₇ ▷ C₁₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #494

Complements

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

N	H ≅ G/N	type
1. C₂×#19128775	C₂	semidirect
3. C₄×C₁₆₇⋊C₁₆	C₂	semidirect
4. #494×C₁₆₇	C₂	semidirect
18. C₁₆₇⋊C₁₆	D₄	semidirect
41. C₄×C₁₆₇	SD₃₂	semidirect
47. C₁₆₇	#2248	semidirect
50. #494	D₁₆₇	semidirect
89. C₄	#19128775	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19128775	2
#24923646	2
C₄×C₁₆₇⋊C₁₆	2
#494×C₁₆₇	2
#2248	167