Subgroups — C₂×#436×C₁₂₇ #30635880

Order 16256.

552 conjugacy classes of subgroups.

The subgroup lattice has 172 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, complemented
2	C₂×C₄×D₄×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	#436×C₁₂₇	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₄⋊C₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₂²⋊C₈×C₁₂₇	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×C₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₄×C₁₂₇	C₄	1	1	yes	characteristic
8	C₂³×C₄×C₁₂₇	C₂²	1	2	yes	—
9	C₂×M₁₆×C₁₂₇	—	2	8	no	—
10	C₂×M₁₆×C₁₂₇	C₂²	1	8	yes	complemented
11	C₂²×C₈×C₁₂₇	C₂²	1	2	yes	—
12	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
14	C₄×D₄×C₁₂₇	C₄	1	4	yes	—
15	C₄×D₄×C₁₂₇	C₂²	1	4	yes	—
16	C₂×C₄⋊C₄×C₁₂₇	C₄	1	1	yes	characteristic
17	C₂×C₂²⋊C₄×C₁₂₇	C₄	1	2	yes	—
18	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
19	C₄⋊C₈×C₁₂₇	C₂²	1	4	yes	complemented
20	C₂²⋊C₈×C₁₂₇	C₂²	1	8	yes	complemented
21	C₄×C₈×C₁₂₇	C₂²	1	4	yes	complemented
22	C₂⁴×C₁₂₇	C₂×C₄	1	2	yes	—
23	C₂×D₄×C₁₂₇	—	2	4	no	—
24	C₂×D₄×C₁₂₇	C₂×C₄	1	4	yes	—
25	C₂²×C₄×C₁₂₇	—	2	4	no	—
26	C₂²×C₄×C₁₂₇	—	2	4	no	—
27	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
29	C₂²×C₄×C₁₂₇	C₂×C₄	1	4	yes	—
30	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₁₂₇	C₂³	1	4	yes	—
32	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
33	M₁₆×C₁₂₇	—	2	16	no	—
34	C₂×C₈×C₁₂₇	—	2	4	no	—
35	C₂×C₈×C₁₂₇	—	2	2	no	—
36	C₂×C₈×C₁₂₇	D₄	1	4	yes	complemented
37	C₂×C₈×C₁₂₇	C₂³	1	2	yes	—
38	C₂×C₈×C₁₂₇	C₂³	1	4	yes	—
39	C₂×C₈×C₁₂₇	C₂³	1	2	yes	—
40	C₄⋊C₄×C₁₂₇	C₂×C₄	1	4	yes	—
41	C₂²⋊C₄×C₁₂₇	C₂×C₄	1	8	yes	—
42	C₄²×C₁₂₇	C₂×C₄	1	2	yes	—
43	C₄²×C₁₂₇	C₂³	1	2	yes	—
44	C₂³×C₁₂₇	—	2	4	no	—
45	C₂³×C₁₂₇	—	2	4	no	—
46	C₂³×C₁₂₇	—	2	4	no	—
47	C₂³×C₁₂₇	C₂²×C₄	1	4	yes	—
48	C₂³×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
49	D₄×C₁₂₇	—	2	8	no	—
50	C₂×C₄×C₁₂₇	—	2	4	no	—
51	C₂×C₄×C₁₂₇	—	2	2	no	—
52	C₂×C₄×C₁₂₇	—	2	8	no	—
53	C₂×C₄×C₁₂₇	—	2	4	no	—
54	C₂×C₄×C₁₂₇	—	2	4	no	—
55	C₂×C₄×C₁₂₇	M₁₆	1	4	yes	complemented
56	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
57	C₂×C₄×C₁₂₇	C₂²×C₄	1	4	yes	—
58	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
60	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
61	C₂×C₄×C₁₂₇	#41	1	2	yes	—
62	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
63	C₈×C₁₂₇	—	2	4	no	—
64	C₈×C₁₂₇	—	2	2	no	—
65	C₈×C₁₂₇	C₂×D₄	1	4	yes	complemented
66	C₂²×C₁₂₇	—	2	4	no	—
67	C₂²×C₁₂₇	—	2	4	no	—
68	C₂²×C₁₂₇	—	2	4	no	—
69	C₂²×C₁₂₇	—	4	8	no	—
70	C₂²×C₁₂₇	C₄×D₄	1	2	yes	—
71	C₂²×C₁₂₇	C₂×M₁₆	1	2	yes	—
72	C₂²×C₁₂₇	#131	1	2	yes	—
73	C₂²×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
74	C₄×C₁₂₇	—	2	4	no	—
75	C₄×C₁₂₇	—	2	2	no	—
76	C₄×C₁₂₇	C₄×D₄	1	2	yes	—
77	C₄×C₁₂₇	C₂×M₁₆	1	4	yes	complemented
78	C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
79	C₄×C₁₂₇	C₂×#41	1	1	yes	characteristic
80	C₂×C₁₂₇	—	4	4	no	—
81	C₂×C₁₂₇	#436	1	4	yes	direct product factor
82	C₂×C₁₂₇	C₂×C₄×D₄	1	1	yes	characteristic
83	C₂×C₁₂₇	C₂²×M₁₆	1	1	yes	characteristic
84	C₂×C₁₂₇	C₂×#131	1	1	yes	characteristic
85	C₂×#436	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
86	C₁₂₇	C₂×#436	1	1	yes	characteristic, direct product factor, Sylow
87	C₂²×M₁₆	C₂×C₁₂₇	1	2	yes	complemented
88	C₂×C₄×D₄	C₂×C₁₂₇	1	1	yes	characteristic
89	#436	C₂×C₁₂₇	1	8	yes	direct product factor
90	C₂×C₄⋊C₈	C₂×C₁₂₇	1	1	yes	characteristic, complemented
91	C₂×C₂²⋊C₈	C₂×C₁₂₇	1	2	yes	complemented
92	C₂×C₄×C₈	C₂×C₁₂₇	1	1	yes	characteristic, complemented
93	C₂²×D₄	C₄×C₁₂₇	1	1	yes	characteristic
94	C₂³×C₄	C₂²×C₁₂₇	1	2	yes	—
95	C₂×M₁₆	—	2	8	no	—
96	C₂×M₁₆	C₂²×C₁₂₇	1	8	yes	complemented
97	C₂²×C₈	C₂²×C₁₂₇	1	2	yes	—
98	C₂²×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
99	C₂²×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
100	C₄×D₄	C₄×C₁₂₇	1	4	yes	—
101	C₄×D₄	C₂²×C₁₂₇	1	4	yes	—
102	C₂×C₄⋊C₄	C₄×C₁₂₇	1	1	yes	characteristic
103	C₂×C₂²⋊C₄	C₄×C₁₂₇	1	2	yes	—
104	C₂×C₄²	C₂²×C₁₂₇	1	1	yes	characteristic
105	C₄⋊C₈	C₂²×C₁₂₇	1	4	yes	complemented
106	C₂²⋊C₈	C₂²×C₁₂₇	1	8	yes	complemented
107	C₄×C₈	C₂²×C₁₂₇	1	4	yes	complemented
108	C₂⁴	C₂×C₄×C₁₂₇	1	2	yes	—
109	C₂×D₄	—	2	4	no	—
110	C₂×D₄	C₂×C₄×C₁₂₇	1	4	yes	—
111	C₂²×C₄	—	2	4	no	—
112	C₂²×C₄	—	2	4	no	—
113	C₂²×C₄	C₂×C₄×C₁₂₇	1	1	yes	characteristic
114	C₂²×C₄	C₂×C₄×C₁₂₇	1	2	yes	—
115	C₂²×C₄	C₂×C₄×C₁₂₇	1	4	yes	—
116	C₂²×C₄	C₂×C₄×C₁₂₇	1	1	yes	characteristic
117	C₂²×C₄	C₂³×C₁₂₇	1	4	yes	—
118	C₂²×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
119	M₁₆	—	2	16	no	—
120	C₂×C₈	—	2	4	no	—
121	C₂×C₈	—	2	2	no	—
122	C₂×C₈	D₄×C₁₂₇	1	4	yes	complemented
123	C₂×C₈	C₂³×C₁₂₇	1	2	yes	—
124	C₂×C₈	C₂³×C₁₂₇	1	4	yes	—
125	C₂×C₈	C₂³×C₁₂₇	1	2	yes	—
126	C₄⋊C₄	C₂×C₄×C₁₂₇	1	4	yes	—
127	C₂²⋊C₄	C₂×C₄×C₁₂₇	1	8	yes	—
128	C₄²	C₂×C₄×C₁₂₇	1	2	yes	—
129	C₄²	C₂³×C₁₂₇	1	2	yes	—
130	C₂³	—	2	4	no	—
131	C₂³	—	2	4	no	—
132	C₂³	—	2	4	no	—
133	C₂³	C₂²×C₄×C₁₂₇	1	4	yes	—
134	C₂³	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
135	D₄	—	2	8	no	—
136	C₂×C₄	—	2	4	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	2	8	no	—
139	C₂×C₄	—	2	4	no	—
140	C₂×C₄	—	2	4	no	—
141	C₂×C₄	M₁₆×C₁₂₇	1	4	yes	complemented
142	C₂×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
143	C₂×C₄	C₂²×C₄×C₁₂₇	1	4	yes	—
144	C₂×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
145	C₂×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
146	C₂×C₄	C₂×D₄×C₁₂₇	1	2	yes	—
147	C₂×C₄	#41×C₁₂₇	1	2	yes	—
148	C₂×C₄	C₂⁴×C₁₂₇	1	1	yes	characteristic
149	C₈	—	2	4	no	—
150	C₈	—	2	2	no	—
151	C₈	C₂×D₄×C₁₂₇	1	4	yes	complemented
152	C₂²	—	2	4	no	—
153	C₂²	—	2	4	no	—
154	C₂²	—	2	4	no	—
155	C₂²	—	4	8	no	—
156	C₂²	C₄×D₄×C₁₂₇	1	2	yes	—
157	C₂²	C₂×M₁₆×C₁₂₇	1	2	yes	—
158	C₂²	#131×C₁₂₇	1	2	yes	—
159	C₂²	C₂³×C₄×C₁₂₇	1	1	yes	characteristic
160	C₄	—	2	4	no	—
161	C₄	—	2	2	no	—
162	C₄	C₄×D₄×C₁₂₇	1	2	yes	—
163	C₄	C₂×M₁₆×C₁₂₇	1	4	yes	complemented
164	C₄	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
165	C₄	C₂×#41×C₁₂₇	1	1	yes	characteristic
166	C₂	—	4	4	no	—
167	C₂	#436×C₁₂₇	1	4	yes	direct product factor
168	C₂	C₂×C₄×D₄×C₁₂₇	1	1	yes	characteristic
169	C₂	C₂²×M₁₆×C₁₂₇	1	1	yes	characteristic
170	C₂	C₂×#131×C₁₂₇	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₂²×M₁₆×C₁₂₇	C₂	semidirect
3. #436×C₁₂₇	C₂	direct (N × H)
4. C₂×C₄⋊C₈×C₁₂₇	C₂	semidirect
5. C₂×C₂²⋊C₈×C₁₂₇	C₂	semidirect
6. C₂×C₄×C₈×C₁₂₇	C₂	semidirect
10. C₂×M₁₆×C₁₂₇	C₂²	semidirect
19. C₄⋊C₈×C₁₂₇	C₂²	semidirect
20. C₂²⋊C₈×C₁₂₇	C₂²	semidirect
21. C₄×C₈×C₁₂₇	C₂²	semidirect
36. C₂×C₈×C₁₂₇	D₄	semidirect
55. C₂×C₄×C₁₂₇	M₁₆	semidirect
65. C₈×C₁₂₇	C₂×D₄	semidirect
77. C₄×C₁₂₇	C₂×M₁₆	semidirect
81. C₂×C₁₂₇	#436	direct (N × H)
85. C₂×#436	C₁₂₇	direct (N × H)
86. C₁₂₇	C₂×#436	direct (N × H)
87. C₂²×M₁₆	C₂×C₁₂₇	semidirect
89. #436	C₂×C₁₂₇	direct (N × H)
90. C₂×C₄⋊C₈	C₂×C₁₂₇	semidirect
91. C₂×C₂²⋊C₈	C₂×C₁₂₇	semidirect
92. C₂×C₄×C₈	C₂×C₁₂₇	semidirect
96. C₂×M₁₆	C₂²×C₁₂₇	semidirect
105. C₄⋊C₈	C₂²×C₁₂₇	semidirect
106. C₂²⋊C₈	C₂²×C₁₂₇	semidirect
107. C₄×C₈	C₂²×C₁₂₇	semidirect
122. C₂×C₈	D₄×C₁₂₇	semidirect
141. C₂×C₄	M₁₆×C₁₂₇	semidirect
151. C₈	C₂×D₄×C₁₂₇	semidirect
163. C₄	C₂×M₁₆×C₁₂₇	semidirect
167. C₂	#436×C₁₂₇	direct (N × H)

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×C₄×D₄×C₁₂₇	2
#436×C₁₂₇	2
C₂×C₄⋊C₈×C₁₂₇	2
C₂×C₂²⋊C₈×C₁₂₇	2
C₂×C₄×C₈×C₁₂₇	2
C₂×#436	127

Subgroups — C2×#436×C127 #30635880

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#436×C₁₂₇ #30635880