Subgroups — C₂×#407×C₁₂₇ #30635829

Order 16256.

1008 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	C₂⁴×C₄×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×M₁₆×C₁₂₇	C₂	1	2	yes	maximal, complemented
3	#407×C₁₂₇	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₂²⋊C₈×C₁₂₇	C₂	1	4	yes	maximal, complemented
5	C₂⁵×C₁₂₇	C₄	1	1	yes	characteristic
6	C₂³×C₄×C₁₂₇	—	2	8	no	—
7	C₂³×C₄×C₁₂₇	C₄	1	1	yes	characteristic
8	C₂³×C₄×C₁₂₇	C₄	1	4	yes	—
9	C₂³×C₄×C₁₂₇	C₄	1	2	yes	—
10	C₂³×C₄×C₁₂₇	C₂²	1	4	yes	—
11	C₂³×C₄×C₁₂₇	C₂²	1	2	yes	—
12	C₂³×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
13	C₂×M₁₆×C₁₂₇	—	2	8	no	—
14	C₂×M₁₆×C₁₂₇	C₂²	1	8	yes	complemented
15	C₂²×C₈×C₁₂₇	C₂²	1	4	yes	complemented
16	C₂²⋊C₈×C₁₂₇	C₂²	1	16	yes	complemented
17	C₂⁴×C₁₂₇	—	2	4	no	—
18	C₂⁴×C₁₂₇	—	2	8	no	—
19	C₂⁴×C₁₂₇	C₂×C₄	1	4	yes	—
20	C₂⁴×C₁₂₇	C₂×C₄	1	1	yes	characteristic
21	C₂⁴×C₁₂₇	C₂×C₄	1	2	yes	—
22	C₂²×C₄×C₁₂₇	—	2	4	no	—
23	C₂²×C₄×C₁₂₇	—	2	16	no	—
24	C₂²×C₄×C₁₂₇	—	2	8	no	—
25	C₂²×C₄×C₁₂₇	—	2	8	no	—
26	C₂²×C₄×C₁₂₇	—	2	16	no	—
27	C₂²×C₄×C₁₂₇	C₂×C₄	1	4	yes	—
28	C₂²×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
29	C₂²×C₄×C₁₂₇	C₂×C₄	1	8	yes	—
30	C₂²×C₄×C₁₂₇	C₂×C₄	1	4	yes	—
31	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
33	C₂²×C₄×C₁₂₇	D₄	1	8	yes	—
34	C₂²×C₄×C₁₂₇	C₂³	1	4	yes	—
35	C₂²×C₄×C₁₂₇	C₂³	1	2	yes	—
36	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
37	M₁₆×C₁₂₇	—	2	16	no	—
38	C₂×C₈×C₁₂₇	—	2	8	no	—
39	C₂×C₈×C₁₂₇	C₂³	1	8	yes	complemented
40	C₂³×C₁₂₇	—	2	32	no	—
41	C₂³×C₁₂₇	—	2	8	no	—
42	C₂³×C₁₂₇	—	2	4	no	—
43	C₂³×C₁₂₇	—	2	4	no	—
44	C₂³×C₁₂₇	—	2	4	no	—
45	C₂³×C₁₂₇	—	2	16	no	—
46	C₂³×C₁₂₇	C₂²⋊C₄	1	4	yes	—
47	C₂³×C₁₂₇	M₁₆	1	8	yes	complemented
48	C₂³×C₁₂₇	C₂²×C₄	1	2	yes	—
49	C₂³×C₁₂₇	C₂²×C₄	1	4	yes	—
50	C₂³×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₁₂₇	—	2	8	no	—
52	C₂×C₄×C₁₂₇	—	2	4	no	—
53	C₂×C₄×C₁₂₇	—	2	16	no	—
54	C₂×C₄×C₁₂₇	—	2	8	no	—
55	C₂×C₄×C₁₂₇	—	2	8	no	—
56	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	8	yes	—
57	C₂×C₄×C₁₂₇	C₂²⋊C₄	1	4	yes	—
58	C₂×C₄×C₁₂₇	C₂²×C₄	1	4	yes	—
59	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
60	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₄×C₁₂₇	C₂×D₄	1	4	yes	—
62	C₂×C₄×C₁₂₇	C₂×D₄	1	8	yes	—
63	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
64	C₈×C₁₂₇	—	2	8	no	—
65	C₂²×C₁₂₇	—	2	32	no	—
66	C₂²×C₁₂₇	—	2	4	no	—
67	C₂²×C₁₂₇	—	2	4	no	—
68	C₂²×C₁₂₇	—	2	4	no	—
69	C₂²×C₁₂₇	—	2	8	no	—
70	C₂²×C₁₂₇	—	2	16	no	—
71	C₂²×C₁₂₇	C₂×C₂²⋊C₄	1	4	yes	—
72	C₂²×C₁₂₇	C₂×C₂²⋊C₄	1	2	yes	—
73	C₂²×C₁₂₇	C₂×M₁₆	1	4	yes	—
74	C₂²×C₁₂₇	C₂×M₁₆	1	8	yes	complemented
75	C₂²×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
76	C₄×C₁₂₇	—	2	4	no	—
77	C₄×C₁₂₇	C₂×C₂²⋊C₄	1	4	yes	—
78	C₄×C₁₂₇	C₂×C₂²⋊C₄	1	2	yes	—
79	C₄×C₁₂₇	C₂²×D₄	1	2	yes	—
80	C₂×C₁₂₇	—	2	4	no	—
81	C₂×C₁₂₇	—	2	8	no	—
82	C₂×C₁₂₇	#407	1	4	yes	direct product factor
83	C₂×C₁₂₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₁₂₇	C₂²×M₁₆	1	2	yes	—
85	C₂×#407	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
86	C₁₂₇	C₂×#407	1	1	yes	characteristic, direct product factor, Sylow
87	C₂⁴×C₄	C₂×C₁₂₇	1	1	yes	characteristic
88	C₂²×M₁₆	C₂×C₁₂₇	1	2	yes	complemented
89	#407	C₂×C₁₂₇	1	8	yes	direct product factor
90	C₂×C₂²⋊C₈	C₂×C₁₂₇	1	4	yes	complemented
91	C₂⁵	C₄×C₁₂₇	1	1	yes	characteristic
92	C₂³×C₄	—	2	8	no	—
93	C₂³×C₄	C₄×C₁₂₇	1	1	yes	characteristic
94	C₂³×C₄	C₄×C₁₂₇	1	4	yes	—
95	C₂³×C₄	C₄×C₁₂₇	1	2	yes	—
96	C₂³×C₄	C₂²×C₁₂₇	1	2	yes	—
97	C₂³×C₄	C₂²×C₁₂₇	1	4	yes	—
98	C₂³×C₄	C₂²×C₁₂₇	1	1	yes	characteristic
99	C₂×M₁₆	—	2	8	no	—
100	C₂×M₁₆	C₂²×C₁₂₇	1	8	yes	complemented
101	C₂²×C₈	C₂²×C₁₂₇	1	4	yes	complemented
102	C₂²⋊C₈	C₂²×C₁₂₇	1	16	yes	complemented
103	C₂⁴	—	2	4	no	—
104	C₂⁴	—	2	8	no	—
105	C₂⁴	C₂×C₄×C₁₂₇	1	4	yes	—
106	C₂⁴	C₂×C₄×C₁₂₇	1	1	yes	characteristic
107	C₂⁴	C₂×C₄×C₁₂₇	1	2	yes	—
108	C₂²×C₄	—	2	4	no	—
109	C₂²×C₄	—	2	16	no	—
110	C₂²×C₄	—	2	8	no	—
111	C₂²×C₄	—	2	8	no	—
112	C₂²×C₄	—	2	16	no	—
113	C₂²×C₄	C₂×C₄×C₁₂₇	1	4	yes	—
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	8	yes	—
117	C₂²×C₄	C₂×C₄×C₁₂₇	1	2	yes	—
118	C₂²×C₄	C₂×C₄×C₁₂₇	1	1	yes	characteristic
119	C₂²×C₄	D₄×C₁₂₇	1	8	yes	—
120	C₂²×C₄	C₂³×C₁₂₇	1	4	yes	—
121	C₂²×C₄	C₂³×C₁₂₇	1	2	yes	—
122	C₂²×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
123	M₁₆	—	2	16	no	—
124	C₂×C₈	—	2	8	no	—
125	C₂×C₈	C₂³×C₁₂₇	1	8	yes	complemented
126	C₂³	—	2	32	no	—
127	C₂³	—	2	8	no	—
128	C₂³	—	2	4	no	—
129	C₂³	—	2	4	no	—
130	C₂³	—	2	4	no	—
131	C₂³	—	2	16	no	—
132	C₂³	C₂²⋊C₄×C₁₂₇	1	4	yes	—
133	C₂³	M₁₆×C₁₂₇	1	8	yes	complemented
134	C₂³	C₂²×C₄×C₁₂₇	1	2	yes	—
135	C₂³	C₂²×C₄×C₁₂₇	1	4	yes	—
136	C₂³	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
137	C₂×C₄	—	2	8	no	—
138	C₂×C₄	—	2	4	no	—
139	C₂×C₄	—	2	8	no	—
140	C₂×C₄	—	2	16	no	—
141	C₂×C₄	—	2	8	no	—
142	C₂×C₄	C₂²⋊C₄×C₁₂₇	1	8	yes	—
143	C₂×C₄	C₂²⋊C₄×C₁₂₇	1	4	yes	—
144	C₂×C₄	C₂²×C₄×C₁₂₇	1	4	yes	—
145	C₂×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
146	C₂×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
147	C₂×C₄	C₂×D₄×C₁₂₇	1	4	yes	—
148	C₂×C₄	C₂×D₄×C₁₂₇	1	8	yes	—
149	C₂×C₄	C₂⁴×C₁₂₇	1	1	yes	characteristic
150	C₈	—	2	8	no	—
151	C₂²	—	2	32	no	—
152	C₂²	—	2	4	no	—
153	C₂²	—	2	4	no	—
154	C₂²	—	2	4	no	—
155	C₂²	—	2	8	no	—
156	C₂²	—	2	16	no	—
157	C₂²	C₂×C₂²⋊C₄×C₁₂₇	1	4	yes	—
158	C₂²	C₂×C₂²⋊C₄×C₁₂₇	1	2	yes	—
159	C₂²	C₂×M₁₆×C₁₂₇	1	4	yes	—
160	C₂²	C₂×M₁₆×C₁₂₇	1	8	yes	complemented
161	C₂²	C₂³×C₄×C₁₂₇	1	1	yes	characteristic
162	C₄	—	2	4	no	—
163	C₄	C₂×C₂²⋊C₄×C₁₂₇	1	2	yes	—
164	C₄	C₂×C₂²⋊C₄×C₁₂₇	1	4	yes	—
165	C₄	C₂²×D₄×C₁₂₇	1	2	yes	—
166	C₂	—	2	4	no	—
167	C₂	—	2	8	no	—
168	C₂	#407×C₁₂₇	1	4	yes	direct product factor
169	C₂	C₂²×C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
170	C₂	C₂²×M₁₆×C₁₂₇	1	2	yes	—

	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	C₂⁵×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
2. C₂²×M₁₆×C₁₂₇	C₂	semidirect
3. #407×C₁₂₇	C₂	direct (N × H)
4. C₂×C₂²⋊C₈×C₁₂₇	C₂	semidirect
14. C₂×M₁₆×C₁₂₇	C₂²	semidirect
15. C₂²×C₈×C₁₂₇	C₂²	semidirect
16. C₂²⋊C₈×C₁₂₇	C₂²	semidirect
39. C₂×C₈×C₁₂₇	C₂³	semidirect
47. C₂³×C₁₂₇	M₁₆	semidirect
74. C₂²×C₁₂₇	C₂×M₁₆	semidirect
82. C₂×C₁₂₇	#407	direct (N × H)
85. C₂×#407	C₁₂₇	direct (N × H)
86. C₁₂₇	C₂×#407	direct (N × H)
88. C₂²×M₁₆	C₂×C₁₂₇	semidirect
89. #407	C₂×C₁₂₇	direct (N × H)
90. C₂×C₂²⋊C₈	C₂×C₁₂₇	semidirect
100. C₂×M₁₆	C₂²×C₁₂₇	semidirect
101. C₂²×C₈	C₂²×C₁₂₇	semidirect
102. C₂²⋊C₈	C₂²×C₁₂₇	semidirect
125. C₂×C₈	C₂³×C₁₂₇	semidirect
133. C₂³	M₁₆×C₁₂₇	semidirect
160. C₂²	C₂×M₁₆×C₁₂₇	semidirect
168. C₂	#407×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
C₂⁴×C₄×C₁₂₇	2
C₂²×M₁₆×C₁₂₇	2
#407×C₁₂₇	2
C₂×C₂²⋊C₈×C₁₂₇	2
C₂×#407	127

Subgroups — C2×#407×C127 #30635829

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#407×C₁₂₇ #30635829