Subgroups — #30638019

Order 16256.

262 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22550266	C₂	1	3	yes	maximal, complemented
2	#22549616	C₂	1	3	yes	maximal, complemented
3	H(3,𝔽₄)×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#14523235	—	2	3	no	—
5	C₂×#14523233	—	2	3	no	—
6	#16861206	C₂²	1	1	yes	characteristic, complemented
7	#16861120	C₂²	1	3	yes	complemented
8	#124×C₁₂₇	—	2	3	no	—
9	#124×C₁₂₇	C₂²	1	3	yes	—
10	#120×C₁₂₇	—	2	3	no	—
11	C₂×D₅₀₈	—	2	3	no	—
12	#14523235	—	4	6	no	—
13	#14523233	—	4	6	no	—
14	C₂×C₁₂₇⋊C₈	—	2	3	no	—
15	C₂⁴×C₁₂₇	—	2	1	no	—
16	C₂×Q₈×C₁₂₇	D₄	1	3	yes	—
17	C₂×D₄×C₁₂₇	—	2	3	no	—
18	C₂×D₄×C₁₂₇	D₄	1	3	yes	—
19	C₂²⋊C₄×C₁₂₇	—	2	3	no	—
20	C₂²⋊C₄×C₁₂₇	—	2	6	no	—
21	C₄²×C₁₂₇	—	2	1	no	—
22	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
23	C₂²×D₁₂₇	—	4	1	no	—
24	D₅₀₈	—	4	3	no	—
25	D₅₀₈	—	4	3	no	—
26	C₁₂₇⋊C₈	—	2	6	no	—
27	C₂³×C₁₂₇	—	2	3	no	—
28	C₂³×C₁₂₇	—	8	3	no	—
29	Q₈×C₁₂₇	—	4	3	no	—
30	D₄×C₁₂₇	—	4	3	no	—
31	D₄×C₁₂₇	—	8	3	no	—
32	C₂×C₄×C₁₂₇	—	2	3	no	—
33	C₂×C₄×C₁₂₇	C₂×D₄	1	3	yes	—
34	C₂²×C₁₂₇	—	4	3	no	—
35	C₂²×C₁₂₇	—	4	6	no	—
36	C₂²×C₁₂₇	—	8	4	no	—
37	C₂²×C₁₂₇	#120	1	1	yes	characteristic
38	C₂×D₁₂₇	—	8	3	no	—
39	C₄×C₁₂₇	—	2	3	no	—
40	C₄×C₁₂₇	—	4	3	no	—
41	C₂×C₁₂₇	—	8	3	no	—
42	C₂×C₁₂₇	#453	1	3	yes	—
43	D₁₂₇	—	16	1	no	—
44	#1655	—	127	1	no	maximal, Sylow
45	C₁₂₇	#1655	1	1	yes	characteristic, complemented, Sylow
46	H(3,𝔽₄)	D₁₂₇	1	1	yes	characteristic, complemented
47	#496	—	127	3	no	—
48	#329	—	127	3	no	—
49	C₂×SD₁₆	—	254	3	no	—
50	C₂×D₈	—	254	3	no	—
51	#127	—	127	1	no	—
52	#124	—	2	3	no	—
53	#124	C₂×D₁₂₇	1	3	yes	—
54	#120	—	2	3	no	—
55	#97	—	127	3	no	—
56	C₂⁴	—	2	1	no	—
57	C₂×Q₈	C₂²⋊D₁₂₇	1	3	yes	—
58	C₂×D₄	—	2	3	no	—
59	C₂×D₄	—	254	3	no	—
60	C₂×D₄	C₂²⋊D₁₂₇	1	3	yes	—
61	SD₁₆	—	508	6	no	—
62	D₈	—	508	6	no	—
63	C₂×C₈	—	254	3	no	—
64	C₂²⋊C₄	—	2	3	no	—
65	C₂²⋊C₄	—	2	6	no	—
66	C₄²	—	2	1	no	—
67	C₄²	C₂²×D₁₂₇	1	1	yes	characteristic
68	C₂³	—	2	3	no	—
69	C₂³	—	8	3	no	—
70	C₂³	—	508	1	no	—
71	Q₈	—	4	3	no	—
72	D₄	—	4	3	no	—
73	D₄	—	8	3	no	—
74	D₄	—	508	3	no	—
75	D₄	—	508	3	no	—
76	C₂×C₄	—	2	3	no	—
77	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	3	yes	—
78	C₈	—	254	6	no	—
79	C₂²	—	4	3	no	—
80	C₂²	—	4	6	no	—
81	C₂²	—	8	4	no	—
82	C₂²	—	1016	3	no	—
83	C₂²	#16861178	1	1	yes	characteristic
84	C₄	—	2	3	no	—
85	C₄	—	4	3	no	—
86	C₂	—	8	3	no	—
87	C₂	—	2032	1	no	—
88	C₂	#22550044	1	3	yes	—

	subgroup	G/H
center	C₂²	#16861178
commutator, cosocle	C₄²×C₁₂₇	C₂³
Frattini	C₄²	C₂²×D₁₂₇
Fitting	H(3,𝔽₄)×C₁₂₇	C₂
socle	C₂²×C₁₂₇	#120
hypercenter	H(3,𝔽₄)	D₁₂₇
hypocenter	C₁₂₇	#1655
LCS	G ▷ C₄²×C₁₂₇ ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ H(3,𝔽₄)

Complements

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

N	H ≅ G/N	type
1. #22550266	C₂	semidirect
2. #22549616	C₂	semidirect
3. H(3,𝔽₄)×C₁₂₇	C₂	semidirect
6. #16861206	C₂²	semidirect
7. #16861120	C₂²	semidirect
45. C₁₂₇	#1655	semidirect
46. H(3,𝔽₄)	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
#22549616	2
H(3,𝔽₄)×C₁₂₇	2
#1655	127