Subgroups — #34088624

Order 19648.

252 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23754254	C₂	1	2	yes	maximal, complemented
2	#23754241	C₂	1	3	yes	maximal, complemented
3	#23754232	C₂	1	6	yes	maximal, complemented
4	#23754228	C₂	1	3	yes	maximal, complemented
5	#127×C₃₀₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×D₃₀₇	C₂²	1	1	yes	characteristic
7	C₂×C₂²⋊D₃₀₇	C₂²	1	6	yes	—
8	C₂²×Dic₃₀₇	C₂²	1	3	yes	complemented
9	Dic₃₀₇⋊C₄	C₂²	1	6	yes	complemented
10	C₂²⋊Dic₃₀₇	C₂²	1	3	yes	—
11	C₂²⋊Dic₃₀₇	C₂²	1	3	yes	complemented
12	(C₂×D₃₀₇)⋊C₄	C₂²	1	6	yes	—
13	C₂×D₄×C₃₀₇	C₂²	1	3	yes	—
14	C₂×D₄×C₃₀₇	C₂²	1	3	yes	complemented
15	C₄²×C₃₀₇	C₂²	1	1	yes	characteristic, complemented
16	C₂²×D₃₀₇	—	4	3	no	—
17	C₂²×D₃₀₇	C₂³	1	2	yes	—
18	C₂²⋊D₃₀₇	—	4	6	no	—
19	C₂×Dic₃₀₇	—	4	3	no	—
20	C₂×Dic₃₀₇	C₂³	1	6	yes	—
21	C₂³×C₃₀₇	C₂³	1	1	yes	characteristic
22	C₂³×C₃₀₇	C₂³	1	3	yes	—
23	D₄×C₃₀₇	—	4	3	no	—
24	D₄×C₃₀₇	—	4	3	no	—
25	C₂×C₄×C₃₀₇	C₂³	1	3	yes	—
26	C₂²×C₃₀₇	—	2	3	no	—
27	C₂²×C₃₀₇	—	2	6	no	—
28	C₂²×C₃₀₇	—	2	3	no	—
29	C₂²×C₃₀₇	C₂⁴	1	1	yes	characteristic
30	C₂×D₃₀₇	—	2	6	no	—
31	C₂×D₃₀₇	—	4	4	no	—
32	C₄×C₃₀₇	—	2	3	no	—
33	Dic₃₀₇	—	2	6	no	—
34	C₂×C₃₀₇	—	4	1	no	—
35	C₂×C₃₀₇	—	4	3	no	—
36	C₂×C₃₀₇	H(5,𝔽₂)	1	3	yes	—
37	D₃₀₇	—	4	2	no	—
38	C₃₀₇	#560	1	1	yes	characteristic, complemented, Sylow
39	#560	—	307	1	no	maximal, Sylow
40	#127	D₃₀₇	1	1	yes	characteristic, complemented
41	#126	—	307	2	no	—
42	#123	—	307	3	no	—
43	#121	—	307	6	no	—
44	#120	—	307	3	no	—
45	C₂⁴	—	307	1	no	—
46	C₂×D₄	—	307	6	no	—
47	C₂×D₄	C₂×D₃₀₇	1	3	yes	—
48	C₂×D₄	C₂×D₃₀₇	1	3	yes	complemented
49	C₂²×C₄	—	307	3	no	—
50	C₄⋊C₄	—	307	6	no	—
51	C₂²⋊C₄	—	307	3	no	—
52	C₂²⋊C₄	—	307	3	no	—
53	C₂²⋊C₄	—	307	6	no	—
54	C₄²	C₂×D₃₀₇	1	1	yes	characteristic, complemented
55	C₂³	—	307	2	no	—
56	C₂³	—	1228	3	no	—
57	C₂³	C₂²×D₃₀₇	1	1	yes	characteristic
58	C₂³	C₂²×D₃₀₇	1	3	yes	—
59	D₄	—	4	3	no	—
60	D₄	—	4	3	no	—
61	D₄	—	1228	6	no	—
62	C₂×C₄	—	307	6	no	—
63	C₂×C₄	—	1228	3	no	—
64	C₂×C₄	C₂²×D₃₀₇	1	3	yes	—
65	C₂²	—	2	3	no	—
66	C₂²	—	2	6	no	—
67	C₂²	—	2	3	no	—
68	C₂²	—	614	6	no	—
69	C₂²	—	1228	4	no	—
70	C₂²	C₂³×D₃₀₇	1	1	yes	characteristic
71	C₄	—	2	3	no	—
72	C₄	—	614	6	no	—
73	C₂	—	4	1	no	—
74	C₂	—	4	3	no	—
75	C₂	—	1228	2	no	—
76	C₂	#23754305	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃₀₇
commutator, socle, cosocle	C₂²×C₃₀₇	C₂⁴
Fitting	#127×C₃₀₇	C₂
hypercenter	#127	D₃₀₇
hypocenter	C₃₀₇	#560
LCS	G ▷ C₂²×C₃₀₇ ▷ C₃₀₇
UCS	C₁ ◁ C₂² ◁ #127

Complements

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

N	H ≅ G/N	type
1. #23754254	C₂	semidirect
2. #23754241	C₂	semidirect
3. #23754232	C₂	semidirect
4. #23754228	C₂	semidirect
5. #127×C₃₀₇	C₂	semidirect
8. C₂²×Dic₃₀₇	C₂²	semidirect
9. Dic₃₀₇⋊C₄	C₂²	semidirect
11. C₂²⋊Dic₃₀₇	C₂²	semidirect
14. C₂×D₄×C₃₀₇	C₂²	semidirect
15. C₄²×C₃₀₇	C₂²	semidirect
38. C₃₀₇	#560	semidirect
40. #127	D₃₀₇	semidirect
48. C₂×D₄	C₂×D₃₀₇	semidirect
54. C₄²	C₂×D₃₀₇	semidirect

Maximal subgroups

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

subgroup	index
#23754254	2
#23754241	2
#23754232	2
#23754228	2
#127×C₃₀₇	2
#560	307