Subgroups — C₁₂₇×#2950 #30635902

Order 16256.

456 conjugacy classes of subgroups.

The subgroup lattice has 146 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Q₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	#519×C₁₂₇	C₂	1	2	yes	maximal
3	#441×C₁₂₇	C₂	1	4	yes	maximal, complemented
4	C₄×Q₁₆×C₁₂₇	C₂	1	4	yes	maximal, complemented
5	#427×C₁₂₇	C₂	1	1	yes	maximal, characteristic
6	#419×C₁₂₇	C₂	1	2	yes	maximal
7	#405×C₁₂₇	C₂	1	1	yes	maximal, characteristic
8	C₂²×Q₈×C₁₂₇	C₂²	1	2	yes	—
9	C₂×Q₁₆×C₁₂₇	C₄	1	8	yes	complemented
10	C₂×Q₁₆×C₁₂₇	C₂²	1	2	yes	—
11	C₂×Q₁₆×C₁₂₇	C₂²	1	2	yes	—
12	C₂×M₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
14	C₄×Q₈×C₁₂₇	—	2	4	no	—
15	C₄×Q₈×C₁₂₇	C₂²	1	8	yes	—
16	#117×C₁₂₇	—	2	4	no	—
17	#117×C₁₂₇	C₂²	1	2	yes	—
18	#117×C₁₂₇	C₂²	1	1	yes	characteristic
19	#107×C₁₂₇	C₂²	1	2	yes	—
20	#106×C₁₂₇	C₂²	1	2	yes	—
21	#103×C₁₂₇	C₂²	1	8	yes	—
22	#97×C₁₂₇	C₂²	1	2	yes	—
23	C₄×C₈×C₁₂₇	C₂²	1	2	yes	—
24	C₂×Q₈×C₁₂₇	—	2	2	no	—
25	C₂×Q₈×C₁₂₇	—	2	4	no	—
26	C₂×Q₈×C₁₂₇	C₂×C₄	1	8	yes	—
27	C₂×Q₈×C₁₂₇	C₂³	1	4	yes	—
28	C₂²×C₄×C₁₂₇	—	2	2	no	—
29	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
30	Q₁₆×C₁₂₇	C₂×C₄	1	8	yes	—
31	Q₁₆×C₁₂₇	C₂×C₄	1	8	yes	—
32	M₁₆×C₁₂₇	—	2	2	no	—
33	C₂×C₈×C₁₂₇	C₂×C₄	1	4	yes	—
34	C₂×C₈×C₁₂₇	C₂³	1	2	yes	—
35	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
36	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
37	C₄⋊C₄×C₁₂₇	—	2	8	no	—
38	C₄⋊C₄×C₁₂₇	D₄	1	2	yes	—
39	C₄⋊C₄×C₁₂₇	C₂³	1	4	yes	—
40	C₂²⋊C₄×C₁₂₇	—	2	2	no	—
41	C₂²⋊C₄×C₁₂₇	D₄	1	2	yes	—
42	C₄²×C₁₂₇	—	2	8	no	—
43	C₄²×C₁₂₇	—	2	2	no	—
44	C₄²×C₁₂₇	C₂³	1	2	yes	—
45	C₂³×C₁₂₇	C₂×D₄	1	1	yes	characteristic
46	Q₈×C₁₂₇	—	2	8	no	—
47	Q₈×C₁₂₇	—	2	4	no	—
48	Q₈×C₁₂₇	C₂²×C₄	1	8	yes	—
49	C₂×C₄×C₁₂₇	—	2	8	no	—
50	C₂×C₄×C₁₂₇	—	2	4	no	—
51	C₂×C₄×C₁₂₇	—	2	4	no	—
52	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
53	C₂×C₄×C₁₂₇	C₂×D₄	1	4	yes	—
54	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₁₂₇	#41	1	2	yes	—
56	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
57	C₈×C₁₂₇	—	2	2	no	—
58	C₈×C₁₂₇	C₂²×C₄	1	2	yes	—
59	C₈×C₁₂₇	C₂²×C₄	1	2	yes	—
60	C₂²×C₁₂₇	—	2	2	no	—
61	C₂²×C₁₂₇	C₄×D₄	1	2	yes	—
62	C₂²×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
63	C₄×C₁₂₇	—	2	8	no	—
64	C₄×C₁₂₇	—	2	4	no	—
65	C₄×C₁₂₇	—	4	4	no	—
66	C₄×C₁₂₇	C₄×D₄	1	2	yes	—
67	C₄×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
68	C₄×C₁₂₇	C₂×#41	1	1	yes	characteristic
69	C₂×C₁₂₇	—	2	2	no	—
70	C₂×C₁₂₇	C₂×C₄×D₄	1	1	yes	characteristic
71	C₂×C₁₂₇	#578	1	2	yes	—
72	#2950	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
73	C₁₂₇	#2950	1	1	yes	characteristic, direct product factor, Sylow
74	C₂²×Q₁₆	C₂×C₁₂₇	1	1	yes	characteristic
75	#519	C₂×C₁₂₇	1	2	yes	—
76	#441	C₂×C₁₂₇	1	4	yes	complemented
77	C₄×Q₁₆	C₂×C₁₂₇	1	4	yes	complemented
78	#427	C₂×C₁₂₇	1	1	yes	characteristic
79	#419	C₂×C₁₂₇	1	2	yes	—
80	#405	C₂×C₁₂₇	1	1	yes	characteristic
81	C₂²×Q₈	C₂²×C₁₂₇	1	2	yes	—
82	C₂×Q₁₆	C₄×C₁₂₇	1	8	yes	complemented
83	C₂×Q₁₆	C₂²×C₁₂₇	1	2	yes	—
84	C₂×Q₁₆	C₂²×C₁₂₇	1	2	yes	—
85	C₂×M₁₆	C₂²×C₁₂₇	1	1	yes	characteristic
86	C₂²×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
87	C₄×Q₈	—	2	4	no	—
88	C₄×Q₈	C₂²×C₁₂₇	1	8	yes	—
89	#117	—	2	4	no	—
90	#117	C₂²×C₁₂₇	1	2	yes	—
91	#117	C₂²×C₁₂₇	1	1	yes	characteristic
92	#107	C₂²×C₁₂₇	1	2	yes	—
93	#106	C₂²×C₁₂₇	1	2	yes	—
94	#103	C₂²×C₁₂₇	1	8	yes	—
95	#97	C₂²×C₁₂₇	1	2	yes	—
96	C₄×C₈	C₂²×C₁₂₇	1	2	yes	—
97	C₂×Q₈	—	2	2	no	—
98	C₂×Q₈	—	2	4	no	—
99	C₂×Q₈	C₂×C₄×C₁₂₇	1	8	yes	—
100	C₂×Q₈	C₂³×C₁₂₇	1	4	yes	—
101	C₂²×C₄	—	2	2	no	—
102	C₂²×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
103	Q₁₆	C₂×C₄×C₁₂₇	1	8	yes	—
104	Q₁₆	C₂×C₄×C₁₂₇	1	8	yes	—
105	M₁₆	—	2	2	no	—
106	C₂×C₈	C₂×C₄×C₁₂₇	1	4	yes	—
107	C₂×C₈	C₂³×C₁₂₇	1	2	yes	—
108	C₂×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
109	C₂×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
110	C₄⋊C₄	—	2	8	no	—
111	C₄⋊C₄	D₄×C₁₂₇	1	2	yes	—
112	C₄⋊C₄	C₂³×C₁₂₇	1	4	yes	—
113	C₂²⋊C₄	—	2	2	no	—
114	C₂²⋊C₄	D₄×C₁₂₇	1	2	yes	—
115	C₄²	—	2	8	no	—
116	C₄²	—	2	2	no	—
117	C₄²	C₂³×C₁₂₇	1	2	yes	—
118	C₂³	C₂×D₄×C₁₂₇	1	1	yes	characteristic
119	Q₈	—	2	8	no	—
120	Q₈	—	2	4	no	—
121	Q₈	C₂²×C₄×C₁₂₇	1	8	yes	—
122	C₂×C₄	—	2	8	no	—
123	C₂×C₄	—	2	4	no	—
124	C₂×C₄	—	2	4	no	—
125	C₂×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
126	C₂×C₄	C₂×D₄×C₁₂₇	1	4	yes	—
127	C₂×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
128	C₂×C₄	#41×C₁₂₇	1	2	yes	—
129	C₂×C₄	C₂⁴×C₁₂₇	1	1	yes	characteristic
130	C₈	—	2	2	no	—
131	C₈	C₂²×C₄×C₁₂₇	1	2	yes	—
132	C₈	C₂²×C₄×C₁₂₇	1	2	yes	—
133	C₂²	—	2	2	no	—
134	C₂²	C₄×D₄×C₁₂₇	1	2	yes	—
135	C₂²	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
136	C₄	—	2	8	no	—
137	C₄	—	2	4	no	—
138	C₄	—	4	4	no	—
139	C₄	C₄×D₄×C₁₂₇	1	2	yes	—
140	C₄	C₂³×C₄×C₁₂₇	1	1	yes	characteristic
141	C₄	C₂×#41×C₁₂₇	1	1	yes	characteristic
142	C₂	—	2	2	no	—
143	C₂	C₂×C₄×D₄×C₁₂₇	1	1	yes	characteristic
144	C₂	#578×C₁₂₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₂₇	C₂²×D₄
commutator	C₄	C₂³×C₄×C₁₂₇
Frattini, cosocle	C₂×C₄	C₂⁴×C₁₂₇
prime radical	C₂³×C₁₂₇	C₂×D₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₂₇ ◁ #117×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. #441×C₁₂₇	C₂	semidirect
4. C₄×Q₁₆×C₁₂₇	C₂	semidirect
9. C₂×Q₁₆×C₁₂₇	C₄	semidirect
72. #2950	C₁₂₇	direct (N × H)
73. C₁₂₇	#2950	direct (N × H)
76. #441	C₂×C₁₂₇	semidirect
77. C₄×Q₁₆	C₂×C₁₂₇	semidirect
82. C₂×Q₁₆	C₄×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×Q₁₆×C₁₂₇	2
#519×C₁₂₇	2
#441×C₁₂₇	2
C₄×Q₁₆×C₁₂₇	2
#427×C₁₂₇	2
#419×C₁₂₇	2
#405×C₁₂₇	2
#2950	127

Subgroups — C127×#2950 #30635902

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#2950 #30635902