Subgroups — #30636617

Order 16256.

190 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂²⋊(C₁₂₇⋊C₈)	C₂	1	2	yes	maximal
2	#407×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₁₂₇⋊C₈	—	2	2	no	—
4	C₂²⋊(C₁₂₇⋊C₈)	—	2	2	no	—
5	C₂²⋊(C₁₂₇⋊C₈)	C₄	1	4	yes	—
6	C₂³×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
7	C₂×M₁₆×C₁₂₇	C₄	1	2	yes	—
8	C₂²⋊C₈×C₁₂₇	—	2	2	no	—
9	C₂×C₁₂₇⋊C₈	—	2	4	no	—
10	C₂×C₁₂₇⋊C₈	—	4	4	no	—
11	C₂⁴×C₁₂₇	Q₈	1	1	yes	characteristic
12	C₂²×C₄×C₁₂₇	—	4	2	no	—
13	C₂²×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
14	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₄×C₁₂₇	D₄	1	2	yes	—
16	C₂²×C₄×C₁₂₇	D₄	1	1	yes	characteristic
17	M₁₆×C₁₂₇	—	4	2	no	—
18	C₂×C₈×C₁₂₇	—	2	2	no	—
19	C₁₂₇⋊C₈	—	4	4	no	—
20	C₂³×C₁₂₇	—	2	4	no	—
21	C₂³×C₁₂₇	—	4	1	no	—
22	C₂³×C₁₂₇	C₄⋊C₄	1	2	yes	—
23	C₂³×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
24	C₂×C₄×C₁₂₇	—	2	2	no	—
25	C₂×C₄×C₁₂₇	—	2	4	no	—
26	C₂×C₄×C₁₂₇	—	4	2	no	—
27	C₂×C₄×C₁₂₇	—	4	1	no	—
28	C₂×C₄×C₁₂₇	C₄²	1	1	yes	characteristic
29	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	2	yes	—
30	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
31	C₈×C₁₂₇	—	4	2	no	—
32	C₂²×C₁₂₇	—	2	2	no	—
33	C₂²×C₁₂₇	—	2	2	no	—
34	C₂²×C₁₂₇	—	2	2	no	—
35	C₂²×C₁₂₇	—	2	1	no	—
36	C₂²×C₁₂₇	—	4	4	no	—
37	C₂²×C₁₂₇	#95	1	1	yes	characteristic
38	C₂²×C₁₂₇	#108	1	4	yes	—
39	C₄×C₁₂₇	—	2	1	no	—
40	C₄×C₁₂₇	—	2	2	no	—
41	C₄×C₁₂₇	#99	1	2	yes	—
42	C₂×C₁₂₇	—	2	4	no	—
43	C₂×C₁₂₇	—	4	1	no	—
44	C₂×C₁₂₇	#341	1	2	yes	—
45	C₂×C₁₂₇	#342	1	1	yes	characteristic
46	#1298	—	127	1	no	maximal, Sylow
47	C₁₂₇	#1298	1	1	yes	characteristic, complemented, Sylow
48	#407	D₁₂₇	1	1	yes	characteristic
49	C₂×C₂²⋊C₈	—	127	2	no	—
50	C₂³×C₄	C₂×D₁₂₇	1	1	yes	characteristic
51	C₂×M₁₆	Dic₁₂₇	1	2	yes	—
52	C₂²×C₈	—	254	2	no	—
53	C₂²⋊C₈	—	2	2	no	—
54	C₂²⋊C₈	—	127	4	no	—
55	C₂²⋊C₈	—	254	2	no	—
56	C₂⁴	Dic₂₅₄	1	1	yes	characteristic
57	C₂²×C₄	—	4	2	no	—
58	C₂²×C₄	C₄×D₁₂₇	1	2	yes	—
59	C₂²×C₄	C₂×Dic₁₂₇	1	1	yes	characteristic
60	C₂²×C₄	C₂²⋊D₁₂₇	1	2	yes	—
61	C₂²×C₄	D₅₀₈	1	1	yes	characteristic
62	M₁₆	—	4	2	no	—
63	C₂×C₈	—	2	2	no	—
64	C₂×C₈	—	508	4	no	—
65	C₂×C₈	—	254	4	no	—
66	C₂³	—	2	4	no	—
67	C₂³	—	4	1	no	—
68	C₂³	Dic₁₂₇⋊C₄	1	2	yes	—
69	C₂³	C₄⋊Dic₁₂₇	1	1	yes	characteristic
70	C₂×C₄	—	2	4	no	—
71	C₂×C₄	—	2	2	no	—
72	C₂×C₄	—	4	2	no	—
73	C₂×C₄	—	4	1	no	—
74	C₂×C₄	C₄×Dic₁₂₇	1	1	yes	characteristic
75	C₂×C₄	(C₂×D₁₂₇)⋊C₄	1	2	yes	—
76	C₂×C₄	C₂²⋊Dic₁₂₇	1	1	yes	characteristic
77	C₈	—	4	2	no	—
78	C₈	—	508	4	no	—
79	C₂²	—	2	2	no	—
80	C₂²	—	2	2	no	—
81	C₂²	—	2	2	no	—
82	C₂²	—	2	1	no	—
83	C₂²	—	4	4	no	—
84	C₂²	C₁₂₇⋊#95	1	1	yes	characteristic
85	C₂²	#16861144	1	4	yes	—
86	C₄	—	2	2	no	—
87	C₄	—	2	1	no	—
88	C₄	#16861124	1	2	yes	—
89	C₂	—	2	4	no	—
90	C₂	—	4	1	no	—
91	C₂	#22549640	1	2	yes	—
92	C₂	#22549643	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊Dic₁₂₇
commutator	C₂×C₄×C₁₂₇	C₄²
Frattini	C₂³×C₄	C₂×D₁₂₇
Fitting	#407×C₁₂₇	C₂
socle	C₂²×C₁₂₇	#95
prime radical	C₂⁴×C₁₂₇	Q₈
cosocle	C₂³×C₄×C₁₂₇	C₂²
hypercenter	#407	D₁₂₇
hypocenter	C₁₂₇	#1298
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #407

Complements

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

N	H ≅ G/N	type
47. C₁₂₇	#1298	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂²⋊(C₁₂₇⋊C₈)	2
#407×C₁₂₇	2
#1298	127