Subgroups — #36323807

Order 21376.

198 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24923672	C₂	1	2	yes	maximal, complemented
2	#24923649	C₂	1	1	yes	maximal, characteristic, complemented
3	#24923615	C₂	1	2	yes	maximal
4	#24922990	C₂	1	1	yes	maximal, characteristic, complemented
5	#492×C₁₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₆₆₈	—	2	2	no	—
7	#19128832	C₂²	1	2	yes	—
8	#19128765	—	2	4	no	—
9	#19128753	—	2	4	no	—
10	C₄×C₁₆₇⋊C₈	C₂²	1	2	yes	—
11	C₂×SD₁₆×C₁₆₇	—	2	2	no	—
12	C₄⋊Q₈×C₁₆₇	C₂²	1	1	yes	characteristic
13	#127×C₁₆₇	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₁₆₇	C₂²	1	1	yes	characteristic
15	C₂×Dic₃₃₄	—	2	2	no	—
16	Dic₆₆₈	—	4	4	no	—
17	C₂×C₁₆₇⋊C₈	D₄	1	4	yes	—
18	C₄⋊Dic₁₆₇	—	2	2	no	—
19	C₄⋊Dic₁₆₇	—	2	2	no	—
20	C₂×Q₈×C₁₆₇	—	2	1	no	—
21	C₂×D₄×C₁₆₇	—	2	1	no	—
22	C₂×D₄×C₁₆₇	—	2	2	no	—
23	SD₁₆×C₁₆₇	—	4	4	no	—
24	C₂×C₈×C₁₆₇	D₄	1	2	yes	—
25	C₄⋊C₄×C₁₆₇	—	2	2	no	—
26	C₄²×C₁₆₇	C₂³	1	1	yes	characteristic
27	Dic₃₃₄	—	4	4	no	—
28	C₂×Dic₁₆₇	—	4	2	no	—
29	C₁₆₇⋊C₈	—	2	4	no	—
30	C₂³×C₁₆₇	—	4	1	no	—
31	Q₈×C₁₆₇	—	4	2	no	—
32	D₄×C₁₆₇	—	4	2	no	—
33	D₄×C₁₆₇	—	4	4	no	—
34	C₂×C₄×C₁₆₇	—	4	1	no	—
35	C₂×C₄×C₁₆₇	C₂×D₄	1	2	yes	—
36	C₂×C₄×C₁₆₇	C₂×D₄	1	1	yes	characteristic
37	C₈×C₁₆₇	D₈	1	4	yes	—
38	C₂²×C₁₆₇	—	8	1	no	—
39	C₂²×C₁₆₇	—	8	2	no	—
40	C₂²×C₁₆₇	#127	1	1	yes	characteristic
41	C₄×C₁₆₇	—	8	1	no	—
42	C₄×C₁₆₇	C₂×D₈	1	2	yes	—
43	C₄×C₁₆₇	#137	1	4	yes	—
44	Dic₁₆₇	—	8	2	no	—
45	C₂×C₁₆₇	—	16	1	no	—
46	C₂×C₁₆₇	#493	1	1	yes	characteristic
47	C₂×C₁₆₇	#497	1	2	yes	—
48	C₁₆₇	#1722	1	1	yes	characteristic, complemented, Sylow
49	#1722	—	167	1	no	maximal, Sylow
50	#500	—	167	2	no	—
51	#494	—	167	1	no	—
52	#492	D₁₆₇	1	1	yes	characteristic
53	#486	—	167	2	no	—
54	#322	—	167	1	no	—
55	C₂×Q₁₆	—	334	2	no	—
56	C₂×SD₁₆	—	2	2	no	—
57	C₄⋊Q₈	—	167	2	no	—
58	C₄⋊Q₈	C₂×D₁₆₇	1	1	yes	characteristic
59	#127	C₂×D₁₆₇	1	1	yes	characteristic
60	#107	—	334	4	no	—
61	#102	—	334	4	no	—
62	C₄×C₈	—	167	2	no	—
63	C₄×C₈	C₂×D₁₆₇	1	1	yes	characteristic
64	C₂×Q₈	—	2	1	no	—
65	C₂×Q₈	—	334	2	no	—
66	C₂×D₄	—	2	2	no	—
67	C₂×D₄	—	2	1	no	—
68	Q₁₆	—	668	4	no	—
69	SD₁₆	—	4	4	no	—
70	C₂×C₈	—	167	4	no	—
71	C₂×C₈	C₂²⋊D₁₆₇	1	2	yes	—
72	C₄⋊C₄	—	2	2	no	—
73	C₄⋊C₄	—	334	2	no	—
74	C₄⋊C₄	—	334	2	no	—
75	C₄²	C₂²×D₁₆₇	1	1	yes	characteristic
76	C₂³	—	4	1	no	—
77	Q₈	—	4	2	no	—
78	Q₈	—	668	4	no	—
79	D₄	—	4	2	no	—
80	D₄	—	4	4	no	—
81	C₂×C₄	—	4	1	no	—
82	C₂×C₄	—	668	2	no	—
83	C₂×C₄	D₄×D₁₆₇	1	2	yes	—
84	C₂×C₄	C₂×C₂²⋊D₁₆₇	1	1	yes	characteristic
85	C₈	—	334	4	no	—
86	C₈	#14581935	1	4	yes	—
87	C₂²	—	8	2	no	—
88	C₂²	—	8	1	no	—
89	C₂²	#19128828	1	1	yes	characteristic
90	C₄	—	8	1	no	—
91	C₄	—	1336	2	no	—
92	C₄	C₂×#14581935	1	2	yes	—
93	C₄	#19128861	1	4	yes	—
94	C₂	—	16	1	no	—
95	C₂	#24923645	1	1	yes	characteristic
96	C₂	#24923660	1	2	yes	—

	subgroup	G/H
center	C₂²	#19128828
commutator, cosocle	C₄²×C₁₆₇	C₂³
Frattini	C₄²	C₂²×D₁₆₇
Fitting	#492×C₁₆₇	C₂
socle	C₂²×C₁₆₇	#127
prime radical	#127×C₁₆₇	C₂²
hypercenter	#492	D₁₆₇
hypocenter	C₁₆₇	#1722
LCS	G ▷ C₄²×C₁₆₇ ▷ C₂²×C₁₆₇ ▷ C₁₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #492

Complements

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

N	H ≅ G/N	type
1. #24923672	C₂	semidirect
2. #24923649	C₂	semidirect
4. #24922990	C₂	semidirect
48. C₁₆₇	#1722	semidirect

Maximal subgroups

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

subgroup	index
#24923672	2
#24923649	2
#24923615	2
#24922990	2
#492×C₁₆₇	2
#1722	167