Subgroups — #30637022

Order 16256.

196 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22549929	C₂	1	2	yes	maximal
2	#22549598	C₂	1	4	yes	maximal, complemented
3	C₂×C₄×C₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₄⋊(C₁₂₇⋊C₈)	C₄	1	4	yes	—
5	C₂²⋊(C₁₂₇⋊C₈)	C₄	1	4	yes	—
6	C₄×C₁₂₇⋊C₈	C₂²	1	4	yes	—
7	C₂²×C₈×C₁₂₇	C₄	1	2	yes	—
8	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
9	C₄×C₈×C₁₂₇	C₄	1	2	yes	—
10	C₄×C₈×C₁₂₇	C₂²	1	2	yes	—
11	C₂×C₁₂₇⋊C₈	C₂×C₄	1	8	yes	—
12	C₂²×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
13	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
14	C₂×C₈×C₁₂₇	—	2	4	no	—
15	C₂×C₈×C₁₂₇	C₂×C₄	1	4	yes	—
16	C₄²×C₁₂₇	C₂×C₄	1	2	yes	—
17	C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
18	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
19	C₁₂₇⋊C₈	—	2	8	no	—
20	C₂³×C₁₂₇	C₄²	1	1	yes	characteristic
21	C₂×C₄×C₁₂₇	—	2	4	no	—
22	C₂×C₄×C₁₂₇	C₄²	1	2	yes	—
23	C₂×C₄×C₁₂₇	C₄²	1	1	yes	characteristic
24	C₂×C₄×C₁₂₇	M₁₆	1	4	yes	—
25	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
26	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
27	C₈×C₁₂₇	—	2	4	no	—
28	C₂²×C₁₂₇	—	2	2	no	—
29	C₂²×C₁₂₇	#97	1	2	yes	—
30	C₂²×C₁₂₇	C₂×C₄²	1	1	yes	characteristic
31	C₄×C₁₂₇	—	2	2	no	—
32	C₄×C₁₂₇	#97	1	2	yes	—
33	C₄×C₁₂₇	C₂×M₁₆	1	2	yes	—
34	C₄×C₁₂₇	#131	1	4	yes	—
35	C₂×C₁₂₇	—	2	2	no	—
36	C₂×C₁₂₇	C₂×#97	1	1	yes	characteristic
37	C₂×C₁₂₇	#405	1	2	yes	—
38	#1452	—	127	1	no	maximal, Sylow
39	C₁₂₇	#1452	1	1	yes	characteristic, complemented, Sylow
40	#431	—	127	2	no	—
41	C₂×C₄×C₈	D₁₂₇	1	1	yes	characteristic
42	#322	—	127	4	no	—
43	C₂²×C₈	Dic₁₂₇	1	2	yes	—
44	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
45	C₄⋊C₈	—	127	4	no	—
46	C₂²⋊C₈	—	127	4	no	—
47	C₄×C₈	—	127	4	no	—
48	C₄×C₈	Dic₁₂₇	1	2	yes	—
49	C₄×C₈	C₂×D₁₂₇	1	2	yes	—
50	C₂²×C₄	C₄×D₁₂₇	1	2	yes	—
51	C₂²×C₄	C₂×Dic₁₂₇	1	1	yes	characteristic
52	C₂×C₈	—	2	4	no	—
53	C₂×C₈	—	127	8	no	—
54	C₂×C₈	C₂×Dic₁₂₇	1	4	yes	—
55	C₄²	C₄×D₁₂₇	1	2	yes	—
56	C₄²	C₂×Dic₁₂₇	1	1	yes	characteristic
57	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
58	C₂³	C₄×Dic₁₂₇	1	1	yes	characteristic
59	C₂×C₄	—	2	4	no	—
60	C₂×C₄	C₄×Dic₁₂₇	1	2	yes	—
61	C₂×C₄	C₄×Dic₁₂₇	1	1	yes	characteristic
62	C₂×C₄	#14523232	1	4	yes	—
63	C₂×C₄	C₂×C₄×D₁₂₇	1	2	yes	—
64	C₂×C₄	C₂²×Dic₁₂₇	1	1	yes	characteristic
65	C₈	—	2	4	no	—
66	C₈	—	254	8	no	—
67	C₂²	—	2	2	no	—
68	C₂²	#16861120	1	2	yes	—
69	C₂²	C₂×C₄×Dic₁₂₇	1	1	yes	characteristic
70	C₄	—	2	2	no	—
71	C₄	#16861120	1	2	yes	—
72	C₄	C₂×#14523232	1	2	yes	—
73	C₄	#16861215	1	4	yes	—
74	C₂	—	2	2	no	—
75	C₂	C₂×#16861120	1	1	yes	characteristic
76	C₂	#22549831	1	2	yes	—

	subgroup	G/H
center, Frattini	C₄²	C₂²×D₁₂₇
commutator, socle	C₂²×C₁₂₇	C₂×C₄²
Fitting	C₂×C₄×C₈×C₁₂₇	C₂
prime radical	C₂³×C₁₂₇	C₄²
cosocle	C₄²×C₁₂₇	C₂³
hypercenter	C₂×C₄×C₈	D₁₂₇
hypocenter	C₁₂₇	#1452
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #22549598	C₂	semidirect
39. C₁₂₇	#1452	semidirect

Maximal subgroups

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

subgroup	index
#22549929	2
#22549598	2
C₂×C₄×C₈×C₁₂₇	2
#1452	127