Subgroups — C₂×#117×C₃₆₇ #37553832

Order 23488.

330 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₄×C₃₆₇	C₂	1	1	yes	maximal, characteristic
2	#117×C₃₆₇	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₄⋊C₄×C₃₆₇	C₂	1	2	yes	maximal, complemented
4	C₂×C₂²⋊C₄×C₃₆₇	C₂	1	2	yes	maximal
5	C₂×C₄²×C₃₆₇	C₂	1	2	yes	maximal, complemented
6	C₂⁴×C₃₆₇	C₂²	1	1	yes	characteristic
7	C₂²×C₄×C₃₆₇	C₄	1	8	yes	complemented
8	C₂²×C₄×C₃₆₇	C₂²	1	4	yes	—
9	C₂²×C₄×C₃₆₇	C₂²	1	1	yes	characteristic
10	C₂²×C₄×C₃₆₇	C₂²	1	4	yes	—
11	C₂²×C₄×C₃₆₇	C₂²	1	1	yes	characteristic
12	C₄⋊C₄×C₃₆₇	C₂²	1	8	yes	complemented
13	C₂²⋊C₄×C₃₆₇	C₂²	1	8	yes	—
14	C₄²×C₃₆₇	C₂²	1	8	yes	complemented
15	C₂³×C₃₆₇	—	2	4	no	—
16	C₂³×C₃₆₇	C₂×C₄	1	4	yes	—
17	C₂³×C₃₆₇	C₂³	1	2	yes	—
18	C₂³×C₃₆₇	C₂³	1	1	yes	characteristic
19	C₂×C₄×C₃₆₇	—	2	8	no	—
20	C₂×C₄×C₃₆₇	C₂×C₄	1	4	yes	—
21	C₂×C₄×C₃₆₇	C₂×C₄	1	4	yes	—
22	C₂×C₄×C₃₆₇	C₂×C₄	1	16	yes	complemented
23	C₂×C₄×C₃₆₇	C₂³	1	8	yes	—
24	C₂×C₄×C₃₆₇	C₂³	1	2	yes	—
25	C₂×C₄×C₃₆₇	C₂³	1	2	yes	—
26	C₂²×C₃₆₇	—	2	8	no	—
27	C₂²×C₃₆₇	—	2	4	no	—
28	C₂²×C₃₆₇	C₂²×C₄	1	4	yes	—
29	C₂²×C₃₆₇	C₂²×C₄	1	2	yes	—
30	C₂²×C₃₆₇	#41	1	4	yes	—
31	C₂²×C₃₆₇	C₂⁴	1	1	yes	characteristic
32	C₄×C₃₆₇	—	2	8	no	—
33	C₄×C₃₆₇	C₂²×C₄	1	4	yes	—
34	C₄×C₃₆₇	C₂²×C₄	1	4	yes	—
35	C₂×C₃₆₇	—	2	4	no	—
36	C₂×C₃₆₇	#117	1	4	yes	direct product factor
37	C₂×C₃₆₇	C₂³×C₄	1	1	yes	characteristic
38	C₂×C₃₆₇	C₂×#41	1	2	yes	—
39	C₃₆₇	C₂×#117	1	1	yes	characteristic, direct product factor, Sylow
40	C₂×#117	C₃₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
41	C₂³×C₄	C₂×C₃₆₇	1	1	yes	characteristic
42	#117	C₂×C₃₆₇	1	8	yes	direct product factor
43	C₂×C₄⋊C₄	C₂×C₃₆₇	1	2	yes	complemented
44	C₂×C₂²⋊C₄	C₂×C₃₆₇	1	2	yes	—
45	C₂×C₄²	C₂×C₃₆₇	1	2	yes	complemented
46	C₂⁴	C₂²×C₃₆₇	1	1	yes	characteristic
47	C₂²×C₄	C₄×C₃₆₇	1	8	yes	complemented
48	C₂²×C₄	C₂²×C₃₆₇	1	4	yes	—
49	C₂²×C₄	C₂²×C₃₆₇	1	1	yes	characteristic
50	C₂²×C₄	C₂²×C₃₆₇	1	1	yes	characteristic
51	C₂²×C₄	C₂²×C₃₆₇	1	4	yes	—
52	C₄⋊C₄	C₂²×C₃₆₇	1	8	yes	complemented
53	C₂²⋊C₄	C₂²×C₃₆₇	1	8	yes	—
54	C₄²	C₂²×C₃₆₇	1	8	yes	complemented
55	C₂³	—	2	4	no	—
56	C₂³	C₂×C₄×C₃₆₇	1	4	yes	—
57	C₂³	C₂³×C₃₆₇	1	2	yes	—
58	C₂³	C₂³×C₃₆₇	1	1	yes	characteristic
59	C₂×C₄	—	2	8	no	—
60	C₂×C₄	C₂×C₄×C₃₆₇	1	4	yes	—
61	C₂×C₄	C₂×C₄×C₃₆₇	1	4	yes	—
62	C₂×C₄	C₂×C₄×C₃₆₇	1	16	yes	complemented
63	C₂×C₄	C₂³×C₃₆₇	1	8	yes	—
64	C₂×C₄	C₂³×C₃₆₇	1	2	yes	—
65	C₂×C₄	C₂³×C₃₆₇	1	2	yes	—
66	C₂²	—	2	4	no	—
67	C₂²	—	2	8	no	—
68	C₂²	C₂²×C₄×C₃₆₇	1	4	yes	—
69	C₂²	C₂²×C₄×C₃₆₇	1	2	yes	—
70	C₂²	#41×C₃₆₇	1	4	yes	—
71	C₂²	C₂⁴×C₃₆₇	1	1	yes	characteristic
72	C₄	—	2	8	no	—
73	C₄	C₂²×C₄×C₃₆₇	1	4	yes	—
74	C₄	C₂²×C₄×C₃₆₇	1	4	yes	—
75	C₂	—	2	4	no	—
76	C₂	#117×C₃₆₇	1	4	yes	direct product factor
77	C₂	C₂³×C₄×C₃₆₇	1	1	yes	characteristic
78	C₂	C₂×#41×C₃₆₇	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₃₆₇	C₂²
commutator	C₂	C₂³×C₄×C₃₆₇
Frattini, cosocle	C₂²	C₂⁴×C₃₆₇
socle	C₂³×C₃₆₇	C₂³
prime radical	C₂⁴×C₃₆₇	C₂²
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₃₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. #117×C₃₆₇	C₂	direct (N × H)
3. C₂×C₄⋊C₄×C₃₆₇	C₂	semidirect
5. C₂×C₄²×C₃₆₇	C₂	semidirect
7. C₂²×C₄×C₃₆₇	C₄	semidirect
12. C₄⋊C₄×C₃₆₇	C₂²	semidirect
14. C₄²×C₃₆₇	C₂²	semidirect
22. C₂×C₄×C₃₆₇	C₂×C₄	semidirect
36. C₂×C₃₆₇	#117	direct (N × H)
39. C₃₆₇	C₂×#117	direct (N × H)
40. C₂×#117	C₃₆₇	direct (N × H)
42. #117	C₂×C₃₆₇	direct (N × H)
43. C₂×C₄⋊C₄	C₂×C₃₆₇	semidirect
45. C₂×C₄²	C₂×C₃₆₇	semidirect
47. C₂²×C₄	C₄×C₃₆₇	semidirect
52. C₄⋊C₄	C₂²×C₃₆₇	semidirect
54. C₄²	C₂²×C₃₆₇	semidirect
62. C₂×C₄	C₂×C₄×C₃₆₇	semidirect
76. C₂	#117×C₃₆₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₄×C₃₆₇	2
#117×C₃₆₇	2
C₂×C₄⋊C₄×C₃₆₇	2
C₂×C₂²⋊C₄×C₃₆₇	2
C₂×C₄²×C₃₆₇	2
C₂×#117	367

Subgroups — C2×#117×C367 #37553832

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#117×C₃₆₇ #37553832