Subgroups — #323×C₁₂₇ #22549331

Order 8128.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂²⋊C₄×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₂²⋊C₈×C₁₂₇	C₂	1	2	yes	maximal, complemented
3	C₂⁴×C₁₂₇	C₄	1	1	yes	characteristic
4	C₂²×C₄×C₁₂₇	C₄	1	1	yes	characteristic
5	C₂²×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
6	C₂×C₈×C₁₂₇	—	2	2	no	—
7	C₂²⋊C₄×C₁₂₇	—	2	2	no	—
8	C₂³×C₁₂₇	—	2	1	no	—
9	C₂³×C₁₂₇	—	2	1	no	—
10	C₂³×C₁₂₇	—	4	1	no	—
11	C₂³×C₁₂₇	—	4	1	no	—
12	C₂³×C₁₂₇	C₈	1	2	yes	complemented
13	C₂³×C₁₂₇	C₂×C₄	1	1	yes	characteristic
14	C₂×C₄×C₁₂₇	—	2	1	no	—
15	C₂×C₄×C₁₂₇	—	4	1	no	—
16	C₂×C₄×C₁₂₇	—	4	1	no	—
17	C₂×C₄×C₁₂₇	D₄	1	2	yes	—
18	C₈×C₁₂₇	—	4	2	no	—
19	C₂²×C₁₂₇	—	2	1	no	—
20	C₂²×C₁₂₇	—	2	1	no	—
21	C₂²×C₁₂₇	—	2	2	no	—
22	C₂²×C₁₂₇	—	4	2	no	—
23	C₂²×C₁₂₇	—	4	2	no	—
24	C₂²×C₁₂₇	—	4	1	no	—
25	C₂²×C₁₂₇	—	4	1	no	—
26	C₂²×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
27	C₂²×C₁₂₇	C₂×C₈	1	1	yes	characteristic
28	C₂²×C₁₂₇	M₁₆	1	1	yes	characteristic
29	C₄×C₁₂₇	—	2	2	no	—
30	C₄×C₁₂₇	—	4	1	no	—
31	C₂×C₁₂₇	—	2	1	no	—
32	C₂×C₁₂₇	—	2	1	no	—
33	C₂×C₁₂₇	—	4	2	no	—
34	C₂×C₁₂₇	C₂²⋊C₈	1	1	yes	characteristic
35	C₂×C₁₂₇	#99	1	1	yes	characteristic
36	C₂×C₁₂₇	#100	1	1	yes	characteristic
37	C₁₂₇	#323	1	1	yes	characteristic, direct product factor, Sylow
38	#323	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
39	C₂×C₂²⋊C₄	C₂×C₁₂₇	1	1	yes	characteristic
40	C₂²⋊C₈	C₂×C₁₂₇	1	2	yes	complemented
41	C₂⁴	C₄×C₁₂₇	1	1	yes	characteristic
42	C₂²×C₄	C₄×C₁₂₇	1	1	yes	characteristic
43	C₂²×C₄	C₂²×C₁₂₇	1	1	yes	characteristic
44	C₂×C₈	—	2	2	no	—
45	C₂²⋊C₄	—	2	2	no	—
46	C₂³	—	2	1	no	—
47	C₂³	—	2	1	no	—
48	C₂³	—	4	1	no	—
49	C₂³	—	4	1	no	—
50	C₂³	C₈×C₁₂₇	1	2	yes	complemented
51	C₂³	C₂×C₄×C₁₂₇	1	1	yes	characteristic
52	C₂×C₄	—	2	1	no	—
53	C₂×C₄	—	4	1	no	—
54	C₂×C₄	—	4	1	no	—
55	C₂×C₄	D₄×C₁₂₇	1	2	yes	—
56	C₈	—	4	2	no	—
57	C₂²	—	2	1	no	—
58	C₂²	—	2	1	no	—
59	C₂²	—	2	2	no	—
60	C₂²	—	4	2	no	—
61	C₂²	—	4	2	no	—
62	C₂²	—	4	1	no	—
63	C₂²	—	4	1	no	—
64	C₂²	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
65	C₂²	C₂×C₈×C₁₂₇	1	1	yes	characteristic
66	C₂²	M₁₆×C₁₂₇	1	1	yes	characteristic
67	C₄	—	2	2	no	—
68	C₄	—	4	1	no	—
69	C₂	—	2	1	no	—
70	C₂	—	2	1	no	—
71	C₂	—	4	2	no	—
72	C₂	C₂²⋊C₈×C₁₂₇	1	1	yes	characteristic
73	C₂	#99×C₁₂₇	1	1	yes	characteristic
74	C₂	#100×C₁₂₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₂₇	C₂²⋊C₄
commutator	C₂²	C₂×C₈×C₁₂₇
Frattini, cosocle	C₂²×C₄	C₂²×C₁₂₇
prime radical	C₂⁴×C₁₂₇	C₄
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₂₇ ◁ C₂²×C₄×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²⋊C₈×C₁₂₇	C₂	semidirect
12. C₂³×C₁₂₇	C₈	semidirect
37. C₁₂₇	#323	direct (N × H)
38. #323	C₁₂₇	direct (N × H)
40. C₂²⋊C₈	C₂×C₁₂₇	semidirect
50. C₂³	C₈×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂²⋊C₄×C₁₂₇	2
C₂²⋊C₈×C₁₂₇	2
#323	127

Subgroups — #323×C127 #22549331

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #323×C₁₂₇ #22549331