Subgroups — #381×C₃₆₇ #37553699

Order 23488.

218 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	C₂×C₂²⋊C₄×C₃₆₇	C₂	1	2	yes	maximal
3	#95×C₃₆₇	C₂	1	4	yes	maximal, complemented
4	C₂⁴×C₃₆₇	C₂²	1	1	yes	characteristic
5	C₂²×C₄×C₃₆₇	—	2	4	no	—
6	C₂²×C₄×C₃₆₇	C₄	1	4	yes	complemented
7	C₂²×C₄×C₃₆₇	C₂²	1	4	yes	—
8	C₂²×C₄×C₃₆₇	C₂²	1	2	yes	—
9	C₂²⋊C₄×C₃₆₇	—	2	4	no	—
10	C₂³×C₃₆₇	—	2	4	no	—
11	C₂³×C₃₆₇	C₂×C₄	1	2	yes	—
12	C₂³×C₃₆₇	D₄	1	4	yes	—
13	C₂³×C₃₆₇	C₂³	1	1	yes	characteristic
14	C₂×C₄×C₃₆₇	—	2	8	no	—
15	C₂×C₄×C₃₆₇	—	2	4	no	—
16	C₂×C₄×C₃₆₇	—	2	4	no	—
17	C₂×C₄×C₃₆₇	—	2	8	no	—
18	C₂×C₄×C₃₆₇	C₂×C₄	1	4	yes	—
19	C₂²×C₃₆₇	—	2	4	no	—
20	C₂²×C₃₆₇	—	2	8	no	—
21	C₂²×C₃₆₇	C₂²⋊C₄	1	4	yes	—
22	C₂²×C₃₆₇	C₂²×C₄	1	1	yes	characteristic
23	C₂²×C₃₆₇	C₂×D₄	1	2	yes	—
24	C₂²×C₃₆₇	#41	1	4	yes	—
25	C₄×C₃₆₇	—	2	4	no	—
26	C₄×C₃₆₇	—	4	4	no	—
27	C₂×C₃₆₇	—	2	4	no	—
28	C₂×C₃₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
29	C₂×C₃₆₇	#117	1	2	yes	—
30	C₂×C₃₆₇	#123	1	4	yes	—
31	C₃₆₇	#381	1	1	yes	characteristic, direct product factor, Sylow
32	#381	C₃₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
33	C₂³×C₄	C₂×C₃₆₇	1	1	yes	characteristic
34	C₂×C₂²⋊C₄	C₂×C₃₆₇	1	2	yes	—
35	#95	C₂×C₃₆₇	1	4	yes	complemented
36	C₂⁴	C₂²×C₃₆₇	1	1	yes	characteristic
37	C₂²×C₄	—	2	4	no	—
38	C₂²×C₄	C₄×C₃₆₇	1	4	yes	complemented
39	C₂²×C₄	C₂²×C₃₆₇	1	4	yes	—
40	C₂²×C₄	C₂²×C₃₆₇	1	2	yes	—
41	C₂²⋊C₄	—	2	4	no	—
42	C₂³	—	2	4	no	—
43	C₂³	C₂×C₄×C₃₆₇	1	2	yes	—
44	C₂³	D₄×C₃₆₇	1	4	yes	—
45	C₂³	C₂³×C₃₆₇	1	1	yes	characteristic
46	C₂×C₄	—	2	8	no	—
47	C₂×C₄	—	2	4	no	—
48	C₂×C₄	—	2	4	no	—
49	C₂×C₄	—	2	8	no	—
50	C₂×C₄	C₂×C₄×C₃₆₇	1	4	yes	—
51	C₂²	—	2	4	no	—
52	C₂²	—	2	8	no	—
53	C₂²	C₂²⋊C₄×C₃₆₇	1	4	yes	—
54	C₂²	C₂²×C₄×C₃₆₇	1	1	yes	characteristic
55	C₂²	C₂×D₄×C₃₆₇	1	2	yes	—
56	C₂²	#41×C₃₆₇	1	4	yes	—
57	C₄	—	2	4	no	—
58	C₄	—	4	4	no	—
59	C₂	—	2	4	no	—
60	C₂	C₂×C₂²⋊C₄×C₃₆₇	1	1	yes	characteristic
61	C₂	#117×C₃₆₇	1	2	yes	—
62	C₂	#123×C₃₆₇	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
3. #95×C₃₆₇	C₂	semidirect
6. C₂²×C₄×C₃₆₇	C₄	semidirect
31. C₃₆₇	#381	direct (N × H)
32. #381	C₃₆₇	direct (N × H)
35. #95	C₂×C₃₆₇	semidirect
38. C₂²×C₄	C₄×C₃₆₇	semidirect

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
C₂×C₂²⋊C₄×C₃₆₇	2
#95×C₃₆₇	2
#381	367

Subgroups — #381×C367 #37553699

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #381×C₃₆₇ #37553699