Subgroups — #37554097

Order 23488.

202 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×(C₂×D₃₆₇)⋊C₄	C₂	1	3	yes	maximal
2	C₃₆₇⋊#95	C₂	1	3	yes	maximal, complemented
3	C₂²×Q₈×C₃₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂³×D₃₆₇	C₂²	1	1	yes	characteristic
5	C₂²×Dic₃₆₇	C₂²	1	3	yes	—
6	(C₂×D₃₆₇)⋊C₄	—	2	6	no	—
7	C₂×Q₈×C₃₆₇	—	2	6	no	—
8	C₂²×C₄×C₃₆₇	C₂²	1	3	yes	—
9	C₂²×D₃₆₇	—	2	3	no	—
10	C₂²×D₃₆₇	—	2	3	no	—
11	C₂²×D₃₆₇	Q₈	1	2	yes	complemented
12	C₂×Dic₃₆₇	—	2	3	no	—
13	C₂×Dic₃₆₇	—	2	6	no	—
14	C₂³×C₃₆₇	C₂³	1	1	yes	characteristic
15	Q₈×C₃₆₇	—	2	8	no	—
16	C₂×C₄×C₃₆₇	—	2	6	no	—
17	C₂×C₄×C₃₆₇	D₄	1	6	yes	—
18	C₂²×C₃₆₇	C₂×D₄	1	3	yes	—
19	C₂²×C₃₆₇	C₂×Q₈	1	1	yes	characteristic
20	C₂²×C₃₆₇	#41	1	3	yes	—
21	C₂×D₃₆₇	—	2	6	no	—
22	C₂×D₃₆₇	—	4	1	no	—
23	C₂×D₃₆₇	—	4	3	no	—
24	C₄×C₃₆₇	—	2	6	no	—
25	Dic₃₆₇	—	4	3	no	—
26	C₂×C₃₆₇	#120	1	1	yes	characteristic
27	C₂×C₃₆₇	C₂²⋊Q₈	1	3	yes	—
28	C₂×C₃₆₇	#124	1	3	yes	—
29	D₃₆₇	—	4	2	no	—
30	C₃₆₇	#393	1	1	yes	characteristic, complemented, Sylow
31	#393	—	367	1	no	maximal, Sylow
32	C₂²×Q₈	D₃₆₇	1	1	yes	characteristic, complemented
33	C₂×C₂²⋊C₄	—	367	3	no	—
34	#95	—	367	3	no	—
35	C₂⁴	—	367	1	no	—
36	C₂×Q₈	—	2	6	no	—
37	C₂²×C₄	—	367	3	no	—
38	C₂²×C₄	C₂×D₃₆₇	1	3	yes	—
39	C₂²⋊C₄	—	734	6	no	—
40	C₂³	—	367	2	no	—
41	C₂³	—	734	3	no	—
42	C₂³	—	734	3	no	—
43	C₂³	C₂²×D₃₆₇	1	1	yes	characteristic
44	Q₈	—	2	8	no	—
45	C₂×C₄	—	2	6	no	—
46	C₂×C₄	—	734	3	no	—
47	C₂×C₄	—	734	6	no	—
48	C₂×C₄	C₂²⋊D₃₆₇	1	6	yes	—
49	C₂²	—	1468	1	no	—
50	C₂²	—	1468	3	no	—
51	C₂²	—	734	6	no	—
52	C₂²	C₂×C₂²⋊D₃₆₇	1	3	yes	—
53	C₂²	Q₈×D₃₆₇	1	1	yes	characteristic
54	C₂²	#19154578	1	3	yes	—
55	C₄	—	2	6	no	—
56	C₄	—	1468	3	no	—
57	C₂	—	1468	2	no	—
58	C₂	#26044230	1	1	yes	characteristic
59	C₂	#26044237	1	3	yes	—
60	C₂	#26044250	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₃₆₇
commutator, socle, cosocle	C₂³×C₃₆₇	C₂³
Fitting	C₂²×Q₈×C₃₆₇	C₂
prime radical	C₂³×D₃₆₇	C₂²
hypercenter	C₂²×Q₈	D₃₆₇
hypocenter	C₃₆₇	#393
LCS	G ▷ C₂³×C₃₆₇ ▷ C₃₆₇
UCS	C₁ ◁ C₂³ ◁ C₂²×Q₈

Complements

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

N	H ≅ G/N	type
2. C₃₆₇⋊#95	C₂	semidirect
3. C₂²×Q₈×C₃₆₇	C₂	semidirect
11. C₂²×D₃₆₇	Q₈	semidirect
30. C₃₆₇	#393	semidirect
32. C₂²×Q₈	D₃₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×(C₂×D₃₆₇)⋊C₄	2
C₃₆₇⋊#95	2
C₂²×Q₈×C₃₆₇	2
#393	367