Subgroups — #30636725

Order 16256.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22550134	C₂	1	1	yes	maximal, characteristic
2	C₁₂₇⋊#338	C₂	1	1	yes	maximal, characteristic
3	#349×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₅₀₈	C₄	1	1	yes	characteristic
5	C₂×#14523232	—	2	1	no	—
6	#16861188	—	2	1	no	—
7	#16861141	C₄	1	1	yes	characteristic
8	#16861134	—	2	1	no	—
9	C₂×M₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
10	M₃₂×C₁₂₇	—	2	1	no	—
11	C₂×Dic₂₅₄	—	2	1	no	—
12	Dic₅₀₈	—	4	1	no	—
13	#14523232	—	4	1	no	—
14	#14523232	—	4	1	no	—
15	C₂×C₁₂₇⋊C₈	—	2	2	no	—
16	C₄⋊Dic₁₂₇	—	2	1	no	—
17	Dic₁₂₇⋊C₄	—	4	1	no	—
18	C₂²⋊Dic₁₂₇	—	4	1	no	—
19	C₂²×C₄×C₁₂₇	D₄	1	1	yes	characteristic
20	M₁₆×C₁₂₇	—	4	1	no	—
21	C₂×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
22	C₂×C₈×C₁₂₇	D₄	1	1	yes	characteristic
23	C₁₆×C₁₂₇	—	4	1	no	—
24	Dic₂₅₄	—	4	1	no	—
25	Dic₂₅₄	—	4	1	no	—
26	C₂×Dic₁₂₇	—	4	1	no	—
27	C₂×Dic₁₂₇	—	4	1	no	—
28	C₁₂₇⋊C₈	—	4	2	no	—
29	C₂³×C₁₂₇	SD₁₆	1	1	yes	characteristic
30	C₂×C₄×C₁₂₇	—	2	1	no	—
31	C₂×C₄×C₁₂₇	—	2	1	no	—
32	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₁₂₇	D₈	1	1	yes	characteristic
34	C₈×C₁₂₇	—	2	1	no	—
35	C₈×C₁₂₇	—	2	1	no	—
36	C₂²×C₁₂₇	—	2	1	no	—
37	C₂²×C₁₂₇	—	4	1	no	—
38	C₂²×C₁₂₇	#102	1	1	yes	characteristic
39	C₄×C₁₂₇	—	2	1	no	—
40	C₄×C₁₂₇	#99	1	1	yes	characteristic
41	C₄×C₁₂₇	#104	1	1	yes	characteristic
42	Dic₁₂₇	—	8	1	no	—
43	Dic₁₂₇	—	8	1	no	—
44	C₂×C₁₂₇	—	2	1	no	—
45	C₂×C₁₂₇	—	4	1	no	—
46	C₂×C₁₂₇	#327	1	1	yes	characteristic
47	#1342	—	127	1	no	maximal, Sylow
48	C₁₂₇	#1342	1	1	yes	characteristic, complemented, Sylow
49	#470	—	127	1	no	—
50	#349	D₁₂₇	1	1	yes	characteristic
51	#338	—	127	1	no	—
52	C₂×Q₁₆	—	127	1	no	—
53	C₂×M₁₆	—	254	1	no	—
54	C₂×M₁₆	C₂×D₁₂₇	1	1	yes	characteristic
55	C₂²⋊Q₈	—	254	1	no	—
56	M₃₂	—	2	1	no	—
57	#106	—	127	1	no	—
58	#103	—	254	1	no	—
59	C₂×Q₈	—	254	1	no	—
60	C₂²×C₄	D₅₀₈	1	1	yes	characteristic
61	Q₁₆	—	508	1	no	—
62	M₁₆	—	4	1	no	—
63	M₁₆	—	508	1	no	—
64	M₁₆	—	508	1	no	—
65	C₂×C₈	—	254	2	no	—
66	C₂×C₈	C₄×D₁₂₇	1	1	yes	characteristic
67	C₂×C₈	C₂²⋊D₁₂₇	1	1	yes	characteristic
68	C₄⋊C₄	—	508	1	no	—
69	C₄⋊C₄	—	254	1	no	—
70	C₂²⋊C₄	—	508	1	no	—
71	C₁₆	—	4	1	no	—
72	C₂³	#14523237	1	1	yes	characteristic
73	Q₈	—	508	1	no	—
74	Q₈	—	508	1	no	—
75	C₂×C₄	—	2	1	no	—
76	C₂×C₄	—	2	1	no	—
77	C₂×C₄	—	508	1	no	—
78	C₂×C₄	—	508	1	no	—
79	C₂×C₄	(C₂×D₁₂₇)⋊C₄	1	1	yes	characteristic
80	C₂×C₄	D₁₀₁₆	1	1	yes	characteristic
81	C₈	—	2	1	no	—
82	C₈	—	2	1	no	—
83	C₈	—	508	2	no	—
84	C₂²	—	2	1	no	—
85	C₂²	—	4	1	no	—
86	C₂²	#16861131	1	1	yes	characteristic
87	C₄	—	2	1	no	—
88	C₄	—	1016	1	no	—
89	C₄	—	1016	1	no	—
90	C₄	#16861123	1	1	yes	characteristic
91	C₄	#16861135	1	1	yes	characteristic
92	C₂	—	2	1	no	—
93	C₂	—	4	1	no	—
94	C₂	#22549611	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#22549611
commutator	C₂×C₈×C₁₂₇	C₂×C₄
Frattini	C₂×M₁₆	C₂×D₁₂₇
Fitting	#349×C₁₂₇	C₂
socle	C₂×C₁₂₇	#327
prime radical	C₂³×C₁₂₇	SD₁₆
cosocle	C₂×M₁₆×C₁₂₇	C₂²
hypercenter	#349	D₁₂₇
hypocenter	C₁₂₇	#1342
LCS	G ▷ C₂×C₈×C₁₂₇ ▷ C₂×C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×M₁₆ ◁ #349

Complements

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

N	H ≅ G/N	type
48. C₁₂₇	#1342	semidirect

Maximal subgroups

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

subgroup	index
#22550134	2
C₁₂₇⋊#338	2
#349×C₁₂₇	2
#1342	127