Subgroups — C₁₂₇×#2905 #30635857

Order 16256.

484 conjugacy classes of subgroups.

The subgroup lattice has 128 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×M₁₆×C₁₂₇	C₂	1	2	yes	maximal
2	C₂×#117×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	#424×C₁₂₇	C₂	1	8	yes	maximal, complemented
4	#423×C₁₂₇	C₂	1	4	yes	maximal, complemented
5	C₂³×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
6	C₂×M₁₆×C₁₂₇	—	2	8	no	—
7	C₂×M₁₆×C₁₂₇	C₂²	1	4	yes	—
8	C₂×M₁₆×C₁₂₇	C₂²	1	4	yes	—
9	C₂²×C₈×C₁₂₇	C₂²	1	4	yes	—
10	#117×C₁₂₇	C₄	1	4	yes	—
11	#117×C₁₂₇	C₂²	1	4	yes	—
12	C₂×C₄⋊C₄×C₁₂₇	C₄	1	2	yes	—
13	C₂×C₂²⋊C₄×C₁₂₇	C₄	1	2	yes	—
14	C₂×C₄²×C₁₂₇	C₂²	1	2	yes	—
15	C₄⋊C₈×C₁₂₇	C₂²	1	16	yes	complemented
16	C₂⁴×C₁₂₇	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₄×C₁₂₇	C₂×C₄	1	4	yes	—
18	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
20	C₂²×C₄×C₁₂₇	D₄	1	4	yes	—
21	C₂²×C₄×C₁₂₇	Q₈	1	4	yes	—
22	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
23	C₂²×C₄×C₁₂₇	C₂³	1	2	yes	—
24	M₁₆×C₁₂₇	—	2	8	no	—
25	M₁₆×C₁₂₇	—	2	8	no	—
26	C₂×C₈×C₁₂₇	—	2	8	no	—
27	C₂×C₈×C₁₂₇	C₂³	1	8	yes	—
28	C₄⋊C₄×C₁₂₇	C₂×C₄	1	8	yes	—
29	C₂²⋊C₄×C₁₂₇	C₂×C₄	1	8	yes	—
30	C₄²×C₁₂₇	C₂×C₄	1	4	yes	—
31	C₄²×C₁₂₇	C₂³	1	4	yes	—
32	C₂³×C₁₂₇	—	4	2	no	—
33	C₂³×C₁₂₇	C₄⋊C₄	1	4	yes	—
34	C₂³×C₁₂₇	C₂²×C₄	1	2	yes	—
35	C₂³×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₁₂₇	—	4	4	no	—
37	C₂×C₄×C₁₂₇	C₄⋊C₄	1	4	yes	—
38	C₂×C₄×C₁₂₇	C₄⋊C₄	1	8	yes	—
39	C₂×C₄×C₁₂₇	C₂²×C₄	1	8	yes	—
40	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
42	C₂×C₄×C₁₂₇	C₂×D₄	1	4	yes	—
43	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₁₂₇	C₂×Q₈	1	2	yes	—
45	C₂×C₄×C₁₂₇	C₂×Q₈	1	4	yes	—
46	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
47	C₈×C₁₂₇	—	2	8	no	—
48	C₂²×C₁₂₇	—	2	2	no	—
49	C₂²×C₁₂₇	—	2	4	no	—
50	C₂²×C₁₂₇	—	4	4	no	—
51	C₂²×C₁₂₇	C₂×C₄⋊C₄	1	4	yes	—
52	C₂²×C₁₂₇	C₂×C₄⋊C₄	1	2	yes	—
53	C₂²×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
54	C₄×C₁₂₇	—	2	8	no	—
55	C₄×C₁₂₇	C₂×C₄⋊C₄	1	2	yes	—
56	C₄×C₁₂₇	C₂×C₄⋊C₄	1	4	yes	—
57	C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
58	C₄×C₁₂₇	C₂²×Q₈	1	1	yes	characteristic
59	C₂×C₁₂₇	—	2	2	no	—
60	C₂×C₁₂₇	—	2	4	no	—
61	C₂×C₁₂₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
62	C₂×C₁₂₇	#568	1	2	yes	—
63	#2905	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
64	C₁₂₇	#2905	1	1	yes	characteristic, direct product factor, Sylow
65	C₂²×M₁₆	C₂×C₁₂₇	1	2	yes	—
66	C₂×#117	C₂×C₁₂₇	1	1	yes	characteristic
67	#424	C₂×C₁₂₇	1	8	yes	complemented
68	#423	C₂×C₁₂₇	1	4	yes	complemented
69	C₂³×C₄	C₂²×C₁₂₇	1	1	yes	characteristic
70	C₂×M₁₆	—	2	8	no	—
71	C₂×M₁₆	C₂²×C₁₂₇	1	4	yes	—
72	C₂×M₁₆	C₂²×C₁₂₇	1	4	yes	—
73	C₂²×C₈	C₂²×C₁₂₇	1	4	yes	—
74	#117	C₄×C₁₂₇	1	4	yes	—
75	#117	C₂²×C₁₂₇	1	4	yes	—
76	C₂×C₄⋊C₄	C₄×C₁₂₇	1	2	yes	—
77	C₂×C₂²⋊C₄	C₄×C₁₂₇	1	2	yes	—
78	C₂×C₄²	C₂²×C₁₂₇	1	2	yes	—
79	C₄⋊C₈	C₂²×C₁₂₇	1	16	yes	complemented
80	C₂⁴	C₂×C₄×C₁₂₇	1	1	yes	characteristic
81	C₂²×C₄	C₂×C₄×C₁₂₇	1	4	yes	—
82	C₂²×C₄	C₂×C₄×C₁₂₇	1	1	yes	characteristic
83	C₂²×C₄	C₂×C₄×C₁₂₇	1	2	yes	—
84	C₂²×C₄	D₄×C₁₂₇	1	4	yes	—
85	C₂²×C₄	Q₈×C₁₂₇	1	4	yes	—
86	C₂²×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
87	C₂²×C₄	C₂³×C₁₂₇	1	2	yes	—
88	M₁₆	—	2	8	no	—
89	M₁₆	—	2	8	no	—
90	C₂×C₈	—	2	8	no	—
91	C₂×C₈	C₂³×C₁₂₇	1	8	yes	—
92	C₄⋊C₄	C₂×C₄×C₁₂₇	1	8	yes	—
93	C₂²⋊C₄	C₂×C₄×C₁₂₇	1	8	yes	—
94	C₄²	C₂×C₄×C₁₂₇	1	4	yes	—
95	C₄²	C₂³×C₁₂₇	1	4	yes	—
96	C₂³	—	4	2	no	—
97	C₂³	C₄⋊C₄×C₁₂₇	1	4	yes	—
98	C₂³	C₂²×C₄×C₁₂₇	1	2	yes	—
99	C₂³	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
100	C₂×C₄	—	4	4	no	—
101	C₂×C₄	C₄⋊C₄×C₁₂₇	1	4	yes	—
102	C₂×C₄	C₄⋊C₄×C₁₂₇	1	8	yes	—
103	C₂×C₄	C₂²×C₄×C₁₂₇	1	8	yes	—
104	C₂×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
105	C₂×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
106	C₂×C₄	C₂×D₄×C₁₂₇	1	4	yes	—
107	C₂×C₄	C₂×D₄×C₁₂₇	1	2	yes	—
108	C₂×C₄	C₂×Q₈×C₁₂₇	1	2	yes	—
109	C₂×C₄	C₂×Q₈×C₁₂₇	1	4	yes	—
110	C₂×C₄	C₂⁴×C₁₂₇	1	1	yes	characteristic
111	C₈	—	2	8	no	—
112	C₂²	—	2	2	no	—
113	C₂²	—	2	4	no	—
114	C₂²	—	4	4	no	—
115	C₂²	C₂×C₄⋊C₄×C₁₂₇	1	4	yes	—
116	C₂²	C₂×C₄⋊C₄×C₁₂₇	1	2	yes	—
117	C₂²	C₂³×C₄×C₁₂₇	1	1	yes	characteristic
118	C₄	—	2	8	no	—
119	C₄	C₂×C₄⋊C₄×C₁₂₇	1	2	yes	—
120	C₄	C₂×C₄⋊C₄×C₁₂₇	1	4	yes	—
121	C₄	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
122	C₄	C₂²×Q₈×C₁₂₇	1	1	yes	characteristic
123	C₂	—	2	2	no	—
124	C₂	—	2	4	no	—
125	C₂	C₂²×C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
126	C₂	#568×C₁₂₇	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₁₂₇	C₂⁴
commutator	C₂²	C₂³×C₄×C₁₂₇
Frattini, cosocle	C₂×C₄	C₂⁴×C₁₂₇
socle	C₂²×C₁₂₇	C₂³×C₄
prime radical	C₂⁴×C₁₂₇	C₂×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. #424×C₁₂₇	C₂	semidirect
4. #423×C₁₂₇	C₂	semidirect
15. C₄⋊C₈×C₁₂₇	C₂²	semidirect
63. #2905	C₁₂₇	direct (N × H)
64. C₁₂₇	#2905	direct (N × H)
67. #424	C₂×C₁₂₇	semidirect
68. #423	C₂×C₁₂₇	semidirect
79. C₄⋊C₈	C₂²×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×M₁₆×C₁₂₇	2
C₂×#117×C₁₂₇	2
#424×C₁₂₇	2
#423×C₁₂₇	2
#2905	127

Subgroups — C127×#2905 #30635857

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#2905 #30635857