Subgroups — #30638860

Order 16256.

216 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₁₂₇⋊C₈	C₂	1	1	yes	maximal, characteristic
2	#22549742	C₂	1	1	yes	maximal, characteristic
3	#22549638	C₂	1	4	yes	maximal
4	#382×C₁₂₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₁₂₇⋊C₈	C₂²	1	2	yes	—
6	C₂×C₄⋊Dic₁₂₇	C₂²	1	2	yes	—
7	C₄×C₁₂₇⋊C₈	C₄	1	4	yes	complemented
8	C₁₂₇⋊#95	—	2	2	no	—
9	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
10	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
11	#95×C₁₂₇	—	2	2	no	—
12	C₂²×Dic₁₂₇	—	2	2	no	—
13	C₂×C₁₂₇⋊C₈	—	2	2	no	—
14	C₂×C₁₂₇⋊C₈	—	2	2	no	—
15	C₂×C₁₂₇⋊C₈	C₂×C₄	1	4	yes	—
16	C₄⋊Dic₁₂₇	—	2	4	no	—
17	C₂²×C₄×C₁₂₇	—	2	2	no	—
18	C₂²×C₄×C₁₂₇	D₄	1	2	yes	—
19	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
20	C₄⋊C₄×C₁₂₇	—	2	4	no	—
21	C₄²×C₁₂₇	C₂×C₄	1	2	yes	—
22	C₄²×C₁₂₇	Q₈	1	2	yes	—
23	C₂×Dic₁₂₇	—	4	2	no	—
24	C₂×Dic₁₂₇	—	4	4	no	—
25	C₁₂₇⋊C₈	—	2	4	no	—
26	C₂³×C₁₂₇	C₂×D₄	1	1	yes	characteristic
27	C₂×C₄×C₁₂₇	—	2	4	no	—
28	C₂×C₄×C₁₂₇	—	4	2	no	—
29	C₂×C₄×C₁₂₇	—	4	4	no	—
30	C₂×C₄×C₁₂₇	C₄⋊C₄	1	4	yes	—
31	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
32	C₂×C₄×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
33	C₂×C₄×C₁₂₇	#41	1	2	yes	—
34	C₂×C₄×C₁₂₇	#41	1	2	yes	—
35	C₂²×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
36	C₂²×C₁₂₇	#124	1	1	yes	characteristic
37	C₂²×C₁₂₇	#124	1	1	yes	characteristic
38	C₂²×C₁₂₇	#135	1	4	yes	—
39	C₄×C₁₂₇	—	2	4	no	—
40	C₄×C₁₂₇	—	8	2	no	—
41	C₄×C₁₂₇	#117	1	2	yes	—
42	C₄×C₁₂₇	#125	1	2	yes	—
43	Dic₁₂₇	—	8	2	no	—
44	C₂×C₁₂₇	#382	1	1	yes	characteristic
45	C₂×C₁₂₇	#427	1	2	yes	—
46	C₂×C₁₂₇	#488	1	2	yes	—
47	C₂×C₁₂₇	#488	1	2	yes	—
48	#1835	—	127	1	no	maximal, Sylow
49	C₁₂₇	#1835	1	1	yes	characteristic, complemented, Sylow
50	C₂×C₄×C₈	—	127	1	no	—
51	#382	—	127	1	no	—
52	#382	D₁₂₇	1	1	yes	characteristic
53	#340	—	127	4	no	—
54	C₂²×C₈	—	127	2	no	—
55	C₂×C₄⋊C₄	—	127	2	no	—
56	C₂×C₄⋊C₄	C₂×D₁₂₇	1	2	yes	—
57	C₂×C₄²	C₂×D₁₂₇	1	1	yes	characteristic
58	C₄×C₈	—	127	4	no	—
59	#95	—	2	2	no	—
60	#95	—	254	2	no	—
61	C₂²×C₄	—	2	2	no	—
62	C₂²×C₄	—	254	2	no	—
63	C₂²×C₄	C₂²⋊D₁₂₇	1	2	yes	—
64	C₂²×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
65	C₂×C₈	—	127	4	no	—
66	C₂×C₈	—	254	2	no	—
67	C₂×C₈	—	254	2	no	—
68	C₄⋊C₄	—	2	4	no	—
69	C₄⋊C₄	—	254	4	no	—
70	C₄²	C₄×D₁₂₇	1	2	yes	—
71	C₄²	Dic₂₅₄	1	2	yes	—
72	C₂³	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
73	C₂×C₄	—	2	4	no	—
74	C₂×C₄	—	4	2	no	—
75	C₂×C₄	—	4	4	no	—
76	C₂×C₄	—	508	2	no	—
77	C₂×C₄	—	508	4	no	—
78	C₂×C₄	Dic₁₂₇⋊C₄	1	4	yes	—
79	C₂×C₄	C₂×C₄×D₁₂₇	1	1	yes	characteristic
80	C₂×C₄	C₂×Dic₂₅₄	1	1	yes	characteristic
81	C₂×C₄	#14523247	1	2	yes	—
82	C₂×C₄	#14523249	1	2	yes	—
83	C₈	—	254	4	no	—
84	C₂²	C₂×Dic₁₂₇⋊C₄	1	1	yes	characteristic
85	C₂²	#16861196	1	1	yes	characteristic
86	C₂²	#16861198	1	1	yes	characteristic
87	C₂²	#16861229	1	4	yes	—
88	C₄	—	2	4	no	—
89	C₄	—	8	2	no	—
90	C₄	—	1016	2	no	—
91	C₄	#16861167	1	2	yes	—
92	C₄	#16861200	1	2	yes	—
93	C₂	#22549741	1	1	yes	characteristic
94	C₂	#22549912	1	2	yes	—
95	C₂	#22550230	1	2	yes	—
96	C₂	#22550233	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₂₇
commutator	C₂×C₄×C₁₂₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₂₇
Fitting	#382×C₁₂₇	C₂
socle, prime radical	C₂³×C₁₂₇	C₂×D₄
cosocle	C₂²×C₄×C₁₂₇	C₂³
hypercenter	#382	D₁₂₇
hypocenter	C₁₂₇	#1835
LCS	G ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #382

Complements

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

N	H ≅ G/N	type
7. C₄×C₁₂₇⋊C₈	C₄	semidirect
49. C₁₂₇	#1835	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₁₂₇⋊C₈	2
#22549742	2
#22549638	2
#382×C₁₂₇	2
#1835	127