Subgroups — C₂×#397×C₁₂₇ #30635342

Order 16256.

820 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊C₄×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₂²⋊C₄×C₁₂₇	C₂	1	2	yes	maximal
3	C₂²×C₂²⋊C₄×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	#397×C₁₂₇	C₂	1	8	yes	maximal, direct product factor
5	C₂×#95×C₁₂₇	C₂	1	2	yes	maximal, complemented
6	C₂×#95×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂⁵×C₁₂₇	C₂²	1	1	yes	characteristic
8	C₂³×C₄×C₁₂₇	C₂²	1	2	yes	—
9	C₂³×C₄×C₁₂₇	C₂²	1	2	yes	—
10	C₂³×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
11	C₂³×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊C₄×C₁₂₇	—	2	4	no	—
13	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	4	yes	complemented
14	C₂×C₂²⋊C₄×C₁₂₇	—	2	8	no	—
15	C₂×C₂²⋊C₄×C₁₂₇	—	2	4	no	—
16	C₂×C₂²⋊C₄×C₁₂₇	C₂²	1	8	yes	—
17	C₂×C₂²⋊C₄×C₁₂₇	C₂²	1	4	yes	—
18	#95×C₁₂₇	C₂²	1	8	yes	complemented
19	#95×C₁₂₇	C₂²	1	4	yes	complemented
20	C₂⁴×C₁₂₇	—	2	2	no	—
21	C₂⁴×C₁₂₇	—	2	2	no	—
22	C₂⁴×C₁₂₇	—	2	4	no	—
23	C₂⁴×C₁₂₇	—	2	4	no	—
24	C₂⁴×C₁₂₇	D₄	1	4	yes	—
25	C₂⁴×C₁₂₇	C₂³	1	2	yes	—
26	C₂⁴×C₁₂₇	C₂³	1	1	yes	characteristic
27	C₂²×C₄×C₁₂₇	—	2	4	no	—
28	C₂²×C₄×C₁₂₇	—	2	4	no	—
29	C₂²×C₄×C₁₂₇	—	2	4	no	—
30	C₂²×C₄×C₁₂₇	—	2	2	no	—
31	C₂²×C₄×C₁₂₇	—	2	4	no	—
32	C₂²×C₄×C₁₂₇	—	2	4	no	—
33	C₂²×C₄×C₁₂₇	—	2	4	no	—
34	C₂²×C₄×C₁₂₇	—	2	4	no	—
35	C₂²×C₄×C₁₂₇	—	2	4	no	—
36	C₂²×C₄×C₁₂₇	D₄	1	4	yes	—
37	C₂²×C₄×C₁₂₇	C₂³	1	4	yes	—
38	C₂²×C₄×C₁₂₇	C₂³	1	2	yes	—
39	C₂²×C₄×C₁₂₇	C₂³	1	4	yes	—
40	C₂²×C₄×C₁₂₇	C₂³	1	2	yes	—
41	C₄⋊C₄×C₁₂₇	—	2	8	no	—
42	C₂²⋊C₄×C₁₂₇	—	2	16	no	—
43	C₂²⋊C₄×C₁₂₇	—	2	8	no	—
44	C₂³×C₁₂₇	—	2	8	no	—
45	C₂³×C₁₂₇	—	2	16	no	—
46	C₂³×C₁₂₇	—	2	2	no	—
47	C₂³×C₁₂₇	—	2	2	no	—
48	C₂³×C₁₂₇	—	2	4	no	—
49	C₂³×C₁₂₇	—	2	4	no	—
50	C₂³×C₁₂₇	—	4	4	no	—
51	C₂³×C₁₂₇	—	4	4	no	—
52	C₂³×C₁₂₇	—	4	8	no	—
53	C₂³×C₁₂₇	C₂×D₄	1	4	yes	—
54	C₂³×C₁₂₇	C₂×D₄	1	2	yes	—
55	C₂³×C₁₂₇	C₂×D₄	1	2	yes	—
56	C₂³×C₁₂₇	#41	1	2	yes	—
57	C₂³×C₁₂₇	#41	1	4	yes	—
58	C₂³×C₁₂₇	#41	1	4	yes	—
59	C₂³×C₁₂₇	C₂⁴	1	1	yes	characteristic
60	C₂×C₄×C₁₂₇	—	2	8	no	—
61	C₂×C₄×C₁₂₇	—	2	4	no	—
62	C₂×C₄×C₁₂₇	—	2	4	no	—
63	C₂×C₄×C₁₂₇	—	2	4	no	—
64	C₂×C₄×C₁₂₇	—	2	2	no	—
65	C₂×C₄×C₁₂₇	—	2	4	no	—
66	C₂×C₄×C₁₂₇	—	2	4	no	—
67	C₂×C₄×C₁₂₇	—	2	4	no	—
68	C₂×C₄×C₁₂₇	—	2	4	no	—
69	C₂×C₄×C₁₂₇	—	2	4	no	—
70	C₂×C₄×C₁₂₇	—	4	8	no	—
71	C₂×C₄×C₁₂₇	—	4	8	no	—
72	C₂×C₄×C₁₂₇	—	4	4	no	—
73	C₂×C₄×C₁₂₇	C₂×D₄	1	4	yes	—
74	C₂²×C₁₂₇	—	2	4	no	—
75	C₂²×C₁₂₇	—	2	8	no	—
76	C₂²×C₁₂₇	—	4	2	no	—
77	C₂²×C₁₂₇	—	4	2	no	—
78	C₂²×C₁₂₇	—	4	4	no	—
79	C₂²×C₁₂₇	—	4	16	no	—
80	C₂²×C₁₂₇	#121	1	4	yes	—
81	C₂²×C₁₂₇	#123	1	4	yes	—
82	C₂²×C₁₂₇	#123	1	8	yes	—
83	C₂²×C₁₂₇	#124	1	4	yes	—
84	C₂²×C₁₂₇	#126	1	8	yes	—
85	C₂²×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
86	C₂²×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
87	C₂²×C₁₂₇	C₂×#41	1	1	yes	characteristic
88	C₂²×C₁₂₇	C₂×#41	1	2	yes	—
89	C₂²×C₁₂₇	C₂×#41	1	2	yes	—
90	C₄×C₁₂₇	—	2	4	no	—
91	C₄×C₁₂₇	—	4	4	no	—
92	C₄×C₁₂₇	—	4	4	no	—
93	C₄×C₁₂₇	—	4	2	no	—
94	C₂×C₁₂₇	—	4	4	no	—
95	C₂×C₁₂₇	#397	1	8	yes	direct product factor
96	C₂×C₁₂₇	C₂×#121	1	1	yes	characteristic
97	C₂×C₁₂₇	C₂×#123	1	1	yes	characteristic
98	C₂×C₁₂₇	C₂×#123	1	2	yes	—
99	C₂×C₁₂₇	C₂×#124	1	1	yes	characteristic
100	C₂×C₁₂₇	C₂×#126	1	2	yes	—
101	C₂×#397	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
102	C₁₂₇	C₂×#397	1	1	yes	characteristic, direct product factor, Sylow
103	C₂²×C₄⋊C₄	C₂×C₁₂₇	1	1	yes	characteristic, complemented
104	C₂²×C₂²⋊C₄	C₂×C₁₂₇	1	2	yes	—
105	C₂²×C₂²⋊C₄	C₂×C₁₂₇	1	1	yes	characteristic
106	#397	C₂×C₁₂₇	1	8	yes	direct product factor
107	C₂×#95	C₂×C₁₂₇	1	2	yes	complemented
108	C₂×#95	C₂×C₁₂₇	1	1	yes	characteristic, complemented
109	C₂⁵	C₂²×C₁₂₇	1	1	yes	characteristic
110	C₂³×C₄	C₂²×C₁₂₇	1	2	yes	—
111	C₂³×C₄	C₂²×C₁₂₇	1	2	yes	—
112	C₂³×C₄	C₂²×C₁₂₇	1	1	yes	characteristic
113	C₂³×C₄	C₂²×C₁₂₇	1	1	yes	characteristic
114	C₂×C₄⋊C₄	—	2	4	no	—
115	C₂×C₄⋊C₄	C₂²×C₁₂₇	1	4	yes	complemented
116	C₂×C₂²⋊C₄	—	2	8	no	—
117	C₂×C₂²⋊C₄	—	2	4	no	—
118	C₂×C₂²⋊C₄	C₂²×C₁₂₇	1	8	yes	—
119	C₂×C₂²⋊C₄	C₂²×C₁₂₇	1	4	yes	—
120	#95	C₂²×C₁₂₇	1	8	yes	complemented
121	#95	C₂²×C₁₂₇	1	4	yes	complemented
122	C₂⁴	—	2	2	no	—
123	C₂⁴	—	2	2	no	—
124	C₂⁴	—	2	4	no	—
125	C₂⁴	—	2	4	no	—
126	C₂⁴	D₄×C₁₂₇	1	4	yes	—
127	C₂⁴	C₂³×C₁₂₇	1	2	yes	—
128	C₂⁴	C₂³×C₁₂₇	1	1	yes	characteristic
129	C₂²×C₄	—	2	4	no	—
130	C₂²×C₄	—	2	4	no	—
131	C₂²×C₄	—	2	4	no	—
132	C₂²×C₄	—	2	2	no	—
133	C₂²×C₄	—	2	4	no	—
134	C₂²×C₄	—	2	4	no	—
135	C₂²×C₄	—	2	4	no	—
136	C₂²×C₄	—	2	4	no	—
137	C₂²×C₄	—	2	4	no	—
138	C₂²×C₄	D₄×C₁₂₇	1	4	yes	—
139	C₂²×C₄	C₂³×C₁₂₇	1	4	yes	—
140	C₂²×C₄	C₂³×C₁₂₇	1	2	yes	—
141	C₂²×C₄	C₂³×C₁₂₇	1	4	yes	—
142	C₂²×C₄	C₂³×C₁₂₇	1	2	yes	—
143	C₄⋊C₄	—	2	8	no	—
144	C₂²⋊C₄	—	2	16	no	—
145	C₂²⋊C₄	—	2	8	no	—
146	C₂³	—	2	8	no	—
147	C₂³	—	2	16	no	—
148	C₂³	—	2	2	no	—
149	C₂³	—	2	2	no	—
150	C₂³	—	2	4	no	—
151	C₂³	—	2	4	no	—
152	C₂³	—	4	4	no	—
153	C₂³	—	4	4	no	—
154	C₂³	—	4	8	no	—
155	C₂³	C₂×D₄×C₁₂₇	1	4	yes	—
156	C₂³	C₂×D₄×C₁₂₇	1	2	yes	—
157	C₂³	C₂×D₄×C₁₂₇	1	2	yes	—
158	C₂³	#41×C₁₂₇	1	2	yes	—
159	C₂³	#41×C₁₂₇	1	4	yes	—
160	C₂³	#41×C₁₂₇	1	4	yes	—
161	C₂³	C₂⁴×C₁₂₇	1	1	yes	characteristic
162	C₂×C₄	—	2	8	no	—
163	C₂×C₄	—	2	4	no	—
164	C₂×C₄	—	2	4	no	—
165	C₂×C₄	—	2	4	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	4	no	—
168	C₂×C₄	—	2	4	no	—
169	C₂×C₄	—	2	4	no	—
170	C₂×C₄	—	2	4	no	—
171	C₂×C₄	—	2	4	no	—
172	C₂×C₄	—	4	8	no	—
173	C₂×C₄	—	4	8	no	—
174	C₂×C₄	—	4	4	no	—
175	C₂×C₄	C₂×D₄×C₁₂₇	1	4	yes	—
176	C₂²	—	2	4	no	—
177	C₂²	—	2	8	no	—
178	C₂²	—	4	2	no	—
179	C₂²	—	4	2	no	—
180	C₂²	—	4	4	no	—
181	C₂²	—	4	16	no	—
182	C₂²	#121×C₁₂₇	1	4	yes	—
183	C₂²	#123×C₁₂₇	1	4	yes	—
184	C₂²	#123×C₁₂₇	1	8	yes	—
185	C₂²	#124×C₁₂₇	1	4	yes	—
186	C₂²	#126×C₁₂₇	1	8	yes	—
187	C₂²	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
188	C₂²	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
189	C₂²	C₂×#41×C₁₂₇	1	1	yes	characteristic
190	C₂²	C₂×#41×C₁₂₇	1	2	yes	—
191	C₂²	C₂×#41×C₁₂₇	1	2	yes	—
192	C₄	—	2	4	no	—
193	C₄	—	4	4	no	—
194	C₄	—	4	4	no	—
195	C₄	—	4	2	no	—
196	C₂	—	4	4	no	—
197	C₂	#397×C₁₂₇	1	8	yes	direct product factor
198	C₂	C₂×#121×C₁₂₇	1	1	yes	characteristic
199	C₂	C₂×#123×C₁₂₇	1	1	yes	characteristic
200	C₂	C₂×#123×C₁₂₇	1	2	yes	—
201	C₂	C₂×#124×C₁₂₇	1	1	yes	characteristic
202	C₂	C₂×#126×C₁₂₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₁₂₇	C₂³
commutator, Frattini, cosocle	C₂³	C₂⁴×C₁₂₇
prime radical	C₂⁵×C₁₂₇	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₄⋊C₄×C₁₂₇	C₂	semidirect
4. #397×C₁₂₇	C₂	direct (N × H)
5. C₂×#95×C₁₂₇	C₂	semidirect
6. C₂×#95×C₁₂₇	C₂	semidirect
13. C₂×C₄⋊C₄×C₁₂₇	C₂²	semidirect
18. #95×C₁₂₇	C₂²	semidirect
19. #95×C₁₂₇	C₂²	semidirect
95. C₂×C₁₂₇	#397	direct (N × H)
101. C₂×#397	C₁₂₇	direct (N × H)
102. C₁₂₇	C₂×#397	direct (N × H)
103. C₂²×C₄⋊C₄	C₂×C₁₂₇	semidirect
106. #397	C₂×C₁₂₇	direct (N × H)
107. C₂×#95	C₂×C₁₂₇	semidirect
108. C₂×#95	C₂×C₁₂₇	semidirect
115. C₂×C₄⋊C₄	C₂²×C₁₂₇	semidirect
120. #95	C₂²×C₁₂₇	semidirect
121. #95	C₂²×C₁₂₇	semidirect
197. C₂	#397×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×C₄⋊C₄×C₁₂₇	2
C₂²×C₂²⋊C₄×C₁₂₇	2
#397×C₁₂₇	2
C₂×#95×C₁₂₇	2
C₂×#397	127

Subgroups — C2×#397×C127 #30635342

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#397×C₁₂₇ #30635342