Subgroups — #30636888

Order 16256.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22550266	C₂	1	1	yes	maximal, characteristic
2	#22549671	C₂	1	1	yes	maximal, characteristic, complemented
3	#355×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#14523235	—	2	1	no	—
5	C₂×#14523233	—	2	1	no	—
6	#16861206	C₄	1	1	yes	characteristic, complemented
7	#16861124	—	2	1	no	—
8	#16861120	C₄	1	1	yes	characteristic, complemented
9	#124×C₁₂₇	C₂²	1	1	yes	characteristic
10	#101×C₁₂₇	—	2	1	no	—
11	C₂×D₅₀₈	—	2	1	no	—
12	C₂×D₅₀₈	—	4	1	no	—
13	#14523235	—	4	2	no	—
14	#14523233	—	4	2	no	—
15	C₂×C₁₂₇⋊C₈	—	2	1	no	—
16	C₂²⋊Dic₁₂₇	—	4	1	no	—
17	C₂×Q₈×C₁₂₇	D₄	1	1	yes	characteristic
18	C₂×D₄×C₁₂₇	D₄	1	1	yes	characteristic
19	M₁₆×C₁₂₇	—	4	1	no	—
20	C₂²⋊C₄×C₁₂₇	—	4	1	no	—
21	C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
22	C₂²×D₁₂₇	—	4	1	no	—
23	D₅₀₈	—	4	1	no	—
24	D₅₀₈	—	4	1	no	—
25	D₅₀₈	—	8	2	no	—
26	C₂×Dic₁₂₇	—	8	1	no	—
27	C₁₂₇⋊C₈	—	2	2	no	—
28	C₂³×C₁₂₇	—	2	1	no	—
29	Q₈×C₁₂₇	—	4	1	no	—
30	D₄×C₁₂₇	—	4	1	no	—
31	C₂×C₄×C₁₂₇	—	2	1	no	—
32	C₂×C₄×C₁₂₇	—	2	1	no	—
33	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
34	C₈×C₁₂₇	—	8	1	no	—
35	C₂²×C₁₂₇	—	4	1	no	—
36	C₂²×C₁₂₇	—	8	1	no	—
37	C₂²×C₁₂₇	#99	1	1	yes	characteristic
38	C₂×D₁₂₇	—	8	2	no	—
39	C₂×D₁₂₇	—	8	1	no	—
40	C₄×C₁₂₇	—	2	1	no	—
41	C₄×C₁₂₇	—	4	1	no	—
42	C₄×C₁₂₇	—	4	1	no	—
43	Dic₁₂₇	—	16	1	no	—
44	C₂×C₁₂₇	—	2	1	no	—
45	C₂×C₁₂₇	—	8	1	no	—
46	C₂×C₁₂₇	#353	1	1	yes	characteristic
47	D₁₂₇	—	16	1	no	—
48	#1406	—	127	1	no	maximal, Sylow
49	C₁₂₇	#1406	1	1	yes	characteristic, complemented, Sylow
50	#496	—	127	1	no	—
51	#355	D₁₂₇	1	1	yes	characteristic, complemented
52	#354	—	127	1	no	—
53	C₂×SD₁₆	—	254	1	no	—
54	C₂×D₈	—	254	1	no	—
55	#127	—	127	1	no	—
56	#124	C₂×D₁₂₇	1	1	yes	characteristic
57	#101	—	2	1	no	—
58	#99	—	254	1	no	—
59	#97	—	127	1	no	—
60	C₂×Q₈	C₂²⋊D₁₂₇	1	1	yes	characteristic
61	C₂×D₄	—	508	1	no	—
62	C₂×D₄	—	254	1	no	—
63	C₂×D₄	D₅₀₈	1	1	yes	characteristic
64	SD₁₆	—	508	2	no	—
65	D₈	—	508	2	no	—
66	M₁₆	—	4	1	no	—
67	C₂×C₈	—	254	1	no	—
68	C₂²⋊C₄	—	4	1	no	—
69	C₂²⋊C₄	—	508	1	no	—
70	C₄²	C₄×D₁₂₇	1	1	yes	characteristic
71	C₂³	—	2	1	no	—
72	C₂³	—	508	1	no	—
73	Q₈	—	4	1	no	—
74	D₄	—	4	1	no	—
75	D₄	—	1016	2	no	—
76	D₄	—	508	1	no	—
77	D₄	—	508	1	no	—
78	C₂×C₄	—	2	1	no	—
79	C₂×C₄	—	2	1	no	—
80	C₂×C₄	—	1016	1	no	—
81	C₂×C₄	(C₂×D₁₂₇)⋊C₄	1	1	yes	characteristic
82	C₈	—	8	1	no	—
83	C₈	—	254	2	no	—
84	C₂²	—	4	1	no	—
85	C₂²	—	8	1	no	—
86	C₂²	—	1016	2	no	—
87	C₂²	—	1016	1	no	—
88	C₂²	#16861123	1	1	yes	characteristic
89	C₄	—	2	1	no	—
90	C₄	—	4	1	no	—
91	C₄	—	4	1	no	—
92	C₄	—	2032	1	no	—
93	C₂	—	2	1	no	—
94	C₂	—	8	1	no	—
95	C₂	—	2032	1	no	—
96	C₂	#22549668	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#22549668
commutator	C₄²×C₁₂₇	C₂×C₄
Frattini	#124	C₂×D₁₂₇
Fitting	#355×C₁₂₇	C₂
socle	C₂×C₁₂₇	#353
prime radical	#22550266	C₂
cosocle	#124×C₁₂₇	C₂²
hypercenter	#355	D₁₂₇
hypocenter	C₁₂₇	#1406
LCS	G ▷ C₄²×C₁₂₇ ▷ C₂×C₄×C₁₂₇ ▷ C₂²×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ #124 ◁ #355

Complements

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

N	H ≅ G/N	type
2. #22549671	C₂	semidirect
3. #355×C₁₂₇	C₂	semidirect
6. #16861206	C₄	semidirect
8. #16861120	C₄	semidirect
49. C₁₂₇	#1406	semidirect
51. #355	D₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
#22550266	2
#22549671	2
#355×C₁₂₇	2
#1406	127