Subgroups — #416×C₃₆₇ #37553734

Order 23488.

158 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#41×C₃₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₈×C₃₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#117×C₃₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#103×C₃₆₇	C₂	1	2	yes	maximal, complemented
5	#102×C₃₆₇	C₂	1	2	yes	maximal, complemented
6	#41×C₃₆₇	—	2	2	no	—
7	#41×C₃₆₇	C₄	1	4	yes	complemented
8	C₂×Q₈×C₃₆₇	C₂²	1	1	yes	characteristic
9	C₂×D₄×C₃₆₇	—	2	1	no	—
10	C₂×D₄×C₃₆₇	C₂²	1	1	yes	characteristic
11	C₂²×C₄×C₃₆₇	—	2	1	no	—
12	C₂²×C₄×C₃₆₇	C₂²	1	1	yes	characteristic
13	C₂×C₈×C₃₆₇	—	2	2	no	—
14	C₂×C₈×C₃₆₇	C₂²	1	2	yes	—
15	C₄⋊C₄×C₃₆₇	C₂²	1	2	yes	—
16	C₂²⋊C₄×C₃₆₇	—	2	1	no	—
17	C₄²×C₃₆₇	—	2	1	no	—
18	C₂³×C₃₆₇	—	2	1	no	—
19	C₂³×C₃₆₇	D₄	1	1	yes	characteristic
20	Q₈×C₃₆₇	—	2	1	no	—
21	Q₈×C₃₆₇	C₂×C₄	1	2	yes	—
22	D₄×C₃₆₇	—	2	4	no	—
23	D₄×C₃₆₇	—	2	1	no	—
24	D₄×C₃₆₇	C₂×C₄	1	2	yes	—
25	C₂×C₄×C₃₆₇	—	2	1	no	—
26	C₂×C₄×C₃₆₇	—	2	4	no	—
27	C₂×C₄×C₃₆₇	—	2	2	no	—
28	C₂×C₄×C₃₆₇	C₂×C₄	1	2	yes	—
29	C₂×C₄×C₃₆₇	D₄	1	2	yes	—
30	C₂×C₄×C₃₆₇	D₄	1	1	yes	characteristic
31	C₂×C₄×C₃₆₇	C₂³	1	1	yes	characteristic
32	C₈×C₃₆₇	—	2	2	no	—
33	C₂²×C₃₆₇	—	2	2	no	—
34	C₂²×C₃₆₇	—	2	2	no	—
35	C₂²×C₃₆₇	—	4	2	no	—
36	C₂²×C₃₆₇	C₂²⋊C₄	1	2	yes	—
37	C₂²×C₃₆₇	C₂×D₄	1	1	yes	characteristic
38	C₄×C₃₆₇	—	2	2	no	—
39	C₄×C₃₆₇	—	4	2	no	—
40	C₄×C₃₆₇	C₂²⋊C₄	1	2	yes	—
41	C₄×C₃₆₇	C₂²×C₄	1	1	yes	characteristic
42	C₄×C₃₆₇	C₂×D₄	1	1	yes	characteristic
43	C₂×C₃₆₇	—	2	2	no	—
44	C₂×C₃₆₇	—	4	2	no	—
45	C₂×C₃₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₃₆₇	#135	1	2	yes	—
47	C₃₆₇	#416	1	1	yes	characteristic, direct product factor, Sylow
48	#416	C₃₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
49	C₂×#41	C₂×C₃₆₇	1	1	yes	characteristic
50	C₂²×C₈	C₂×C₃₆₇	1	1	yes	characteristic, complemented
51	#117	C₂×C₃₆₇	1	1	yes	characteristic, complemented
52	#103	C₂×C₃₆₇	1	2	yes	complemented
53	#102	C₂×C₃₆₇	1	2	yes	complemented
54	#41	—	2	2	no	—
55	#41	C₄×C₃₆₇	1	4	yes	complemented
56	C₂×Q₈	C₂²×C₃₆₇	1	1	yes	characteristic
57	C₂×D₄	—	2	1	no	—
58	C₂×D₄	C₂²×C₃₆₇	1	1	yes	characteristic
59	C₂²×C₄	—	2	1	no	—
60	C₂²×C₄	C₂²×C₃₆₇	1	1	yes	characteristic
61	C₂×C₈	—	2	2	no	—
62	C₂×C₈	C₂²×C₃₆₇	1	2	yes	—
63	C₄⋊C₄	C₂²×C₃₆₇	1	2	yes	—
64	C₂²⋊C₄	—	2	1	no	—
65	C₄²	—	2	1	no	—
66	C₂³	—	2	1	no	—
67	C₂³	D₄×C₃₆₇	1	1	yes	characteristic
68	Q₈	—	2	1	no	—
69	Q₈	C₂×C₄×C₃₆₇	1	2	yes	—
70	D₄	—	2	4	no	—
71	D₄	—	2	1	no	—
72	D₄	C₂×C₄×C₃₆₇	1	2	yes	—
73	C₂×C₄	—	2	1	no	—
74	C₂×C₄	—	2	4	no	—
75	C₂×C₄	—	2	2	no	—
76	C₂×C₄	C₂×C₄×C₃₆₇	1	2	yes	—
77	C₂×C₄	D₄×C₃₆₇	1	1	yes	characteristic
78	C₂×C₄	D₄×C₃₆₇	1	2	yes	—
79	C₂×C₄	C₂³×C₃₆₇	1	1	yes	characteristic
80	C₈	—	2	2	no	—
81	C₂²	—	2	2	no	—
82	C₂²	—	2	2	no	—
83	C₂²	—	4	2	no	—
84	C₂²	C₂²⋊C₄×C₃₆₇	1	2	yes	—
85	C₂²	C₂×D₄×C₃₆₇	1	1	yes	characteristic
86	C₄	—	2	2	no	—
87	C₄	—	4	2	no	—
88	C₄	C₂²⋊C₄×C₃₆₇	1	2	yes	—
89	C₄	C₂²×C₄×C₃₆₇	1	1	yes	characteristic
90	C₄	C₂×D₄×C₃₆₇	1	1	yes	characteristic
91	C₂	—	2	2	no	—
92	C₂	—	4	2	no	—
93	C₂	C₂×C₂²⋊C₄×C₃₆₇	1	1	yes	characteristic
94	C₂	#135×C₃₆₇	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₃₆₇	D₄
commutator	C₄	C₂²×C₄×C₃₆₇
Frattini, cosocle	C₂×C₄	C₂³×C₃₆₇
socle	C₂²×C₃₆₇	C₂×D₄
prime radical	C₂×#41×C₃₆₇	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₆₇ ◁ C₂²×C₄×C₃₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₈×C₃₆₇	C₂	semidirect
3. #117×C₃₆₇	C₂	semidirect
4. #103×C₃₆₇	C₂	semidirect
5. #102×C₃₆₇	C₂	semidirect
7. #41×C₃₆₇	C₄	semidirect
47. C₃₆₇	#416	direct (N × H)
48. #416	C₃₆₇	direct (N × H)
50. C₂²×C₈	C₂×C₃₆₇	semidirect
51. #117	C₂×C₃₆₇	semidirect
52. #103	C₂×C₃₆₇	semidirect
53. #102	C₂×C₃₆₇	semidirect
55. #41	C₄×C₃₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#41×C₃₆₇	2
C₂²×C₈×C₃₆₇	2
#117×C₃₆₇	2
#103×C₃₆₇	2
#102×C₃₆₇	2
#416	367

Subgroups — #416×C367 #37553734

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #416×C₃₆₇ #37553734