Subgroups — #30647117

Order 16256.

400 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22549774	C₂	1	4	yes	maximal
2	#22549760	C₂	1	8	yes	maximal
3	#22549742	C₂	1	2	yes	maximal
4	C₄³×C₁₂₇	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊Dic₁₂₇	C₂²	1	4	yes	—
6	C₂×Dic₁₂₇⋊C₄	C₂²	1	8	yes	—
7	C₁₂₇⋊#95	C₂²	1	8	yes	—
8	C₁₂₇⋊#95	C₂²	1	8	yes	—
9	C₂×C₄²×C₁₂₇	C₂²	1	4	yes	—
10	C₂×C₄²×C₁₂₇	C₂²	1	2	yes	—
11	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
12	C₂²×Dic₁₂₇	C₂³	1	8	yes	—
13	C₄⋊Dic₁₂₇	—	2	8	no	—
14	Dic₁₂₇⋊C₄	—	2	16	no	—
15	C₂²×C₄×C₁₂₇	C₂³	1	2	yes	—
16	C₂²×C₄×C₁₂₇	C₂³	1	4	yes	—
17	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
18	C₄²×C₁₂₇	—	2	8	no	—
19	C₄²×C₁₂₇	—	2	4	no	—
20	C₄²×C₁₂₇	Q₈	1	4	yes	—
21	C₂×Dic₁₂₇	—	2	16	no	—
22	C₂×Dic₁₂₇	—	2	8	no	—
23	C₂³×C₁₂₇	C₂⁴	1	1	yes	characteristic
24	C₂×C₄×C₁₂₇	—	2	4	no	—
25	C₂×C₄×C₁₂₇	—	2	4	no	—
26	C₂×C₄×C₁₂₇	—	2	4	no	—
27	C₂×C₄×C₁₂₇	C₂×Q₈	1	4	yes	—
28	C₂×C₄×C₁₂₇	C₂×Q₈	1	2	yes	—
29	C₂×C₄×C₁₂₇	#41	1	8	yes	—
30	C₂×C₄×C₁₂₇	#41	1	4	yes	—
31	C₂²×C₁₂₇	C₂²×Q₈	1	1	yes	characteristic
32	C₂²×C₁₂₇	C₂×#41	1	2	yes	—
33	C₂²×C₁₂₇	C₂×#41	1	4	yes	—
34	C₄×C₁₂₇	—	2	8	no	—
35	C₄×C₁₂₇	—	2	4	no	—
36	C₄×C₁₂₇	#125	1	4	yes	—
37	Dic₁₂₇	—	4	8	no	—
38	C₂×C₁₂₇	C₂×#125	1	1	yes	characteristic
39	C₂×C₁₂₇	#529	1	4	yes	—
40	C₂×C₁₂₇	#533	1	2	yes	—
41	#2864	—	127	1	no	maximal, Sylow
42	C₁₂₇	#2864	1	1	yes	characteristic, complemented, Sylow
43	#389	—	127	4	no	—
44	#387	—	127	8	no	—
45	#382	—	127	2	no	—
46	C₄³	D₁₂₇	1	1	yes	characteristic
47	C₂×C₄⋊C₄	—	127	4	no	—
48	C₂×C₄⋊C₄	—	127	8	no	—
49	C₂×C₄²	C₂×D₁₂₇	1	4	yes	—
50	C₂×C₄²	C₂×D₁₂₇	1	2	yes	—
51	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
52	#95	—	127	8	no	—
53	#95	—	127	8	no	—
54	C₂²×C₄	—	127	8	no	—
55	C₂²×C₄	C₂²×D₁₂₇	1	2	yes	—
56	C₂²×C₄	C₂²×D₁₂₇	1	4	yes	—
57	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
58	C₄⋊C₄	—	254	16	no	—
59	C₄⋊C₄	—	254	8	no	—
60	C₄²	—	2	8	no	—
61	C₄²	—	2	4	no	—
62	C₄²	Dic₂₅₄	1	4	yes	—
63	C₂³	C₂³×D₁₂₇	1	1	yes	characteristic
64	C₂×C₄	—	2	4	no	—
65	C₂×C₄	—	2	4	no	—
66	C₂×C₄	—	2	4	no	—
67	C₂×C₄	—	254	8	no	—
68	C₂×C₄	—	254	16	no	—
69	C₂×C₄	C₂×Dic₂₅₄	1	4	yes	—
70	C₂×C₄	C₂×Dic₂₅₄	1	2	yes	—
71	C₂×C₄	#14523248	1	8	yes	—
72	C₂×C₄	#14523248	1	4	yes	—
73	C₂²	C₂²×Dic₂₅₄	1	1	yes	characteristic
74	C₂²	C₂×#14523248	1	2	yes	—
75	C₂²	C₂×#14523248	1	4	yes	—
76	C₄	—	2	8	no	—
77	C₄	—	2	4	no	—
78	C₄	—	508	8	no	—
79	C₄	#16861201	1	4	yes	—
80	C₂	C₂×#16861201	1	1	yes	characteristic
81	C₂	#22550410	1	4	yes	—
82	C₂	#22550433	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₂₇
commutator, socle, prime radical, cosocle	C₂³×C₁₂₇	C₂⁴
Fitting	C₄³×C₁₂₇	C₂
hypercenter	C₄³	D₁₂₇
hypocenter	C₁₂₇	#2864
LCS	G ▷ C₂³×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂³ ◁ C₄³

Complements

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

N	H ≅ G/N	type
42. C₁₂₇	#2864	semidirect

Maximal subgroups

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

subgroup	index
#22549774	2
#22549760	2
#22549742	2
C₄³×C₁₂₇	2
#2864	127