Subgroups — C₂×#534×C₁₂₇ #30636397

Order 16256.

1984 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₄×C₁₂₇	C₂	1	2	yes	maximal, complemented
2	#534×C₁₂₇	C₂	1	16	yes	maximal, direct product factor
3	C₂×#123×C₁₂₇	C₂	1	4	yes	maximal, complemented
4	C₂×#121×C₁₂₇	C₂	1	4	yes	maximal, complemented
5	C₂×#120×C₁₂₇	C₂	1	4	yes	maximal, complemented
6	C₂²×C₂²⋊C₄×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂⁵×C₁₂₇	C₂²	1	1	yes	characteristic
8	C₂⁵×C₁₂₇	C₂²	1	2	yes	complemented
9	C₂²×D₄×C₁₂₇	—	2	16	no	—
10	C₂²×D₄×C₁₂₇	C₂²	1	4	yes	—
11	C₂²×D₄×C₁₂₇	C₂²	1	4	yes	complemented
12	C₂²×D₄×C₁₂₇	C₂²	1	4	yes	complemented
13	C₂²×D₄×C₁₂₇	C₂²	1	16	yes	complemented
14	C₂³×C₄×C₁₂₇	C₂²	1	2	yes	complemented
15	C₂³×C₄×C₁₂₇	C₂²	1	2	yes	complemented
16	#123×C₁₂₇	C₂²	1	32	yes	complemented
17	#121×C₁₂₇	C₂²	1	32	yes	complemented
18	#120×C₁₂₇	C₂²	1	32	yes	complemented
19	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	4	yes	complemented
20	C₂×C₂²⋊C₄×C₁₂₇	C₂²	1	4	yes	complemented
21	C₂×C₂²⋊C₄×C₁₂₇	C₂²	1	8	yes	complemented
22	C₂×C₂²⋊C₄×C₁₂₇	C₂²	1	8	yes	complemented
23	C₂⁴×C₁₂₇	—	2	8	no	—
24	C₂⁴×C₁₂₇	—	4	4	no	—
25	C₂⁴×C₁₂₇	—	4	8	no	—
26	C₂⁴×C₁₂₇	D₄	1	8	yes	complemented
27	C₂⁴×C₁₂₇	C₂³	1	4	yes	—
28	C₂⁴×C₁₂₇	C₂³	1	2	yes	—
29	C₂⁴×C₁₂₇	C₂³	1	1	yes	characteristic
30	C₂⁴×C₁₂₇	C₂³	1	4	yes	—
31	C₂⁴×C₁₂₇	C₂³	1	8	yes	complemented
32	C₂×D₄×C₁₂₇	—	2	16	no	—
33	C₂×D₄×C₁₂₇	—	2	16	no	—
34	C₂×D₄×C₁₂₇	—	2	64	no	—
35	C₂×D₄×C₁₂₇	—	4	8	no	—
36	C₂×D₄×C₁₂₇	C₂³	1	16	yes	—
37	C₂×D₄×C₁₂₇	C₂³	1	16	yes	complemented
38	C₂×D₄×C₁₂₇	C₂³	1	16	yes	complemented
39	C₂²×C₄×C₁₂₇	—	2	8	no	—
40	C₂²×C₄×C₁₂₇	—	4	4	no	—
41	C₂²×C₄×C₁₂₇	C₂³	1	4	yes	—
42	C₂²×C₄×C₁₂₇	C₂³	1	4	yes	—
43	C₂²×C₄×C₁₂₇	C₂³	1	8	yes	complemented
44	C₂²×C₄×C₁₂₇	C₂³	1	8	yes	complemented
45	C₄⋊C₄×C₁₂₇	C₂³	1	16	yes	complemented
46	C₂²⋊C₄×C₁₂₇	C₂³	1	16	yes	complemented
47	C₂²⋊C₄×C₁₂₇	C₂³	1	32	yes	complemented
48	C₂³×C₁₂₇	—	2	4	no	—
49	C₂³×C₁₂₇	—	2	8	no	—
50	C₂³×C₁₂₇	—	2	8	no	—
51	C₂³×C₁₂₇	—	2	16	no	—
52	C₂³×C₁₂₇	—	2	16	no	—
53	C₂³×C₁₂₇	—	4	16	no	—
54	C₂³×C₁₂₇	—	4	8	no	—
55	C₂³×C₁₂₇	—	4	16	no	—
56	C₂³×C₁₂₇	—	4	32	no	—
57	C₂³×C₁₂₇	C₂×D₄	1	8	yes	—
58	C₂³×C₁₂₇	C₂×D₄	1	4	yes	—
59	C₂³×C₁₂₇	C₂×D₄	1	16	yes	complemented
60	C₂³×C₁₂₇	C₂⁴	1	8	yes	—
61	C₂³×C₁₂₇	C₂⁴	1	4	yes	—
62	C₂³×C₁₂₇	C₂⁴	1	2	yes	—
63	C₂³×C₁₂₇	C₂⁴	1	1	yes	characteristic
64	D₄×C₁₂₇	—	2	32	no	—
65	D₄×C₁₂₇	—	2	32	no	—
66	D₄×C₁₂₇	—	4	16	no	—
67	C₂×C₄×C₁₂₇	—	2	8	no	—
68	C₂×C₄×C₁₂₇	—	2	16	no	—
69	C₂×C₄×C₁₂₇	—	2	8	no	—
70	C₂×C₄×C₁₂₇	—	4	8	no	—
71	C₂×C₄×C₁₂₇	C₂⁴	1	8	yes	—
72	C₂×C₄×C₁₂₇	C₂⁴	1	8	yes	—
73	C₂²×C₁₂₇	—	2	16	no	—
74	C₂²×C₁₂₇	—	2	8	no	—
75	C₂²×C₁₂₇	—	2	4	no	—
76	C₂²×C₁₂₇	—	2	8	no	—
77	C₂²×C₁₂₇	—	2	8	no	—
78	C₂²×C₁₂₇	—	2	16	no	—
79	C₂²×C₁₂₇	—	4	16	no	—
80	C₂²×C₁₂₇	—	4	32	no	—
81	C₂²×C₁₂₇	—	4	16	no	—
82	C₂²×C₁₂₇	C₂²×D₄	1	8	yes	—
83	C₂²×C₁₂₇	C₂²×D₄	1	4	yes	—
84	C₂²×C₁₂₇	C₂²×D₄	1	2	yes	—
85	C₂²×C₁₂₇	H(5,𝔽₂)	1	4	yes	—
86	C₂²×C₁₂₇	C₂⁵	1	1	yes	characteristic
87	C₄×C₁₂₇	—	2	8	no	—
88	C₄×C₁₂₇	—	2	8	no	—
89	C₂×C₁₂₇	—	2	8	no	—
90	C₂×C₁₂₇	—	2	4	no	—
91	C₂×C₁₂₇	—	4	8	no	—
92	C₂×C₁₂₇	#534	1	4	yes	direct product factor
93	C₂×C₁₂₇	C₂³×D₄	1	1	yes	characteristic
94	C₂×C₁₂₇	C₂×H(5,𝔽₂)	1	2	yes	—
95	C₂×#534	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
96	C₁₂₇	C₂×#534	1	1	yes	characteristic, direct product factor, Sylow
97	C₂³×D₄	C₂×C₁₂₇	1	2	yes	complemented
98	#534	C₂×C₁₂₇	1	16	yes	direct product factor
99	C₂×#123	C₂×C₁₂₇	1	4	yes	complemented
100	C₂×#121	C₂×C₁₂₇	1	4	yes	complemented
101	C₂×#120	C₂×C₁₂₇	1	4	yes	complemented
102	C₂²×C₂²⋊C₄	C₂×C₁₂₇	1	1	yes	characteristic, complemented
103	C₂⁵	C₂²×C₁₂₇	1	1	yes	characteristic
104	C₂⁵	C₂²×C₁₂₇	1	2	yes	complemented
105	C₂²×D₄	—	2	16	no	—
106	C₂²×D₄	C₂²×C₁₂₇	1	4	yes	—
107	C₂²×D₄	C₂²×C₁₂₇	1	4	yes	complemented
108	C₂²×D₄	C₂²×C₁₂₇	1	4	yes	complemented
109	C₂²×D₄	C₂²×C₁₂₇	1	16	yes	complemented
110	C₂³×C₄	C₂²×C₁₂₇	1	2	yes	complemented
111	C₂³×C₄	C₂²×C₁₂₇	1	2	yes	complemented
112	#123	C₂²×C₁₂₇	1	32	yes	complemented
113	#121	C₂²×C₁₂₇	1	32	yes	complemented
114	#120	C₂²×C₁₂₇	1	32	yes	complemented
115	C₂×C₄⋊C₄	C₂²×C₁₂₇	1	4	yes	complemented
116	C₂×C₂²⋊C₄	C₂²×C₁₂₇	1	4	yes	complemented
117	C₂×C₂²⋊C₄	C₂²×C₁₂₇	1	8	yes	complemented
118	C₂×C₂²⋊C₄	C₂²×C₁₂₇	1	8	yes	complemented
119	C₂⁴	—	2	8	no	—
120	C₂⁴	—	4	4	no	—
121	C₂⁴	—	4	8	no	—
122	C₂⁴	D₄×C₁₂₇	1	8	yes	complemented
123	C₂⁴	C₂³×C₁₂₇	1	4	yes	—
124	C₂⁴	C₂³×C₁₂₇	1	2	yes	—
125	C₂⁴	C₂³×C₁₂₇	1	1	yes	characteristic
126	C₂⁴	C₂³×C₁₂₇	1	4	yes	—
127	C₂⁴	C₂³×C₁₂₇	1	8	yes	complemented
128	C₂×D₄	—	2	16	no	—
129	C₂×D₄	—	2	16	no	—
130	C₂×D₄	—	2	64	no	—
131	C₂×D₄	—	4	8	no	—
132	C₂×D₄	C₂³×C₁₂₇	1	16	yes	—
133	C₂×D₄	C₂³×C₁₂₇	1	16	yes	complemented
134	C₂×D₄	C₂³×C₁₂₇	1	16	yes	complemented
135	C₂²×C₄	—	2	8	no	—
136	C₂²×C₄	—	4	4	no	—
137	C₂²×C₄	C₂³×C₁₂₇	1	4	yes	—
138	C₂²×C₄	C₂³×C₁₂₇	1	4	yes	—
139	C₂²×C₄	C₂³×C₁₂₇	1	8	yes	complemented
140	C₂²×C₄	C₂³×C₁₂₇	1	8	yes	complemented
141	C₄⋊C₄	C₂³×C₁₂₇	1	16	yes	complemented
142	C₂²⋊C₄	C₂³×C₁₂₇	1	16	yes	complemented
143	C₂²⋊C₄	C₂³×C₁₂₇	1	32	yes	complemented
144	C₂³	—	2	4	no	—
145	C₂³	—	2	8	no	—
146	C₂³	—	2	8	no	—
147	C₂³	—	2	16	no	—
148	C₂³	—	2	16	no	—
149	C₂³	—	4	16	no	—
150	C₂³	—	4	8	no	—
151	C₂³	—	4	16	no	—
152	C₂³	—	4	32	no	—
153	C₂³	C₂×D₄×C₁₂₇	1	8	yes	—
154	C₂³	C₂×D₄×C₁₂₇	1	4	yes	—
155	C₂³	C₂×D₄×C₁₂₇	1	16	yes	complemented
156	C₂³	C₂⁴×C₁₂₇	1	8	yes	—
157	C₂³	C₂⁴×C₁₂₇	1	4	yes	—
158	C₂³	C₂⁴×C₁₂₇	1	2	yes	—
159	C₂³	C₂⁴×C₁₂₇	1	1	yes	characteristic
160	D₄	—	2	32	no	—
161	D₄	—	2	32	no	—
162	D₄	—	4	16	no	—
163	C₂×C₄	—	2	8	no	—
164	C₂×C₄	—	2	16	no	—
165	C₂×C₄	—	2	8	no	—
166	C₂×C₄	—	4	8	no	—
167	C₂×C₄	C₂⁴×C₁₂₇	1	8	yes	—
168	C₂×C₄	C₂⁴×C₁₂₇	1	8	yes	—
169	C₂²	—	2	16	no	—
170	C₂²	—	2	8	no	—
171	C₂²	—	2	4	no	—
172	C₂²	—	2	8	no	—
173	C₂²	—	2	8	no	—
174	C₂²	—	2	16	no	—
175	C₂²	—	4	16	no	—
176	C₂²	—	4	32	no	—
177	C₂²	—	4	16	no	—
178	C₂²	C₂²×D₄×C₁₂₇	1	8	yes	—
179	C₂²	C₂²×D₄×C₁₂₇	1	4	yes	—
180	C₂²	C₂²×D₄×C₁₂₇	1	2	yes	—
181	C₂²	H(5,𝔽₂)×C₁₂₇	1	4	yes	—
182	C₂²	C₂⁵×C₁₂₇	1	1	yes	characteristic
183	C₄	—	2	8	no	—
184	C₄	—	2	8	no	—
185	C₂	—	2	8	no	—
186	C₂	—	2	4	no	—
187	C₂	—	4	8	no	—
188	C₂	#534×C₁₂₇	1	4	yes	direct product factor
189	C₂	C₂³×D₄×C₁₂₇	1	1	yes	characteristic
190	C₂	C₂×H(5,𝔽₂)×C₁₂₇	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
1. C₂³×D₄×C₁₂₇	C₂	semidirect
2. #534×C₁₂₇	C₂	direct (N × H)
3. C₂×#123×C₁₂₇	C₂	semidirect
4. C₂×#121×C₁₂₇	C₂	semidirect
5. C₂×#120×C₁₂₇	C₂	semidirect
6. C₂²×C₂²⋊C₄×C₁₂₇	C₂	semidirect
8. C₂⁵×C₁₂₇	C₂²	semidirect
11. C₂²×D₄×C₁₂₇	C₂²	semidirect
12. C₂²×D₄×C₁₂₇	C₂²	semidirect
13. C₂²×D₄×C₁₂₇	C₂²	semidirect
14. C₂³×C₄×C₁₂₇	C₂²	semidirect
15. C₂³×C₄×C₁₂₇	C₂²	semidirect
16. #123×C₁₂₇	C₂²	semidirect
17. #121×C₁₂₇	C₂²	semidirect
18. #120×C₁₂₇	C₂²	semidirect
19. C₂×C₄⋊C₄×C₁₂₇	C₂²	semidirect
20. C₂×C₂²⋊C₄×C₁₂₇	C₂²	semidirect
21. C₂×C₂²⋊C₄×C₁₂₇	C₂²	semidirect
22. C₂×C₂²⋊C₄×C₁₂₇	C₂²	semidirect
26. C₂⁴×C₁₂₇	D₄	semidirect
31. C₂⁴×C₁₂₇	C₂³	semidirect
37. C₂×D₄×C₁₂₇	C₂³	semidirect
38. C₂×D₄×C₁₂₇	C₂³	semidirect
43. C₂²×C₄×C₁₂₇	C₂³	semidirect
44. C₂²×C₄×C₁₂₇	C₂³	semidirect
45. C₄⋊C₄×C₁₂₇	C₂³	semidirect
46. C₂²⋊C₄×C₁₂₇	C₂³	semidirect
47. C₂²⋊C₄×C₁₂₇	C₂³	semidirect
59. C₂³×C₁₂₇	C₂×D₄	semidirect
92. C₂×C₁₂₇	#534	direct (N × H)
95. C₂×#534	C₁₂₇	direct (N × H)
96. C₁₂₇	C₂×#534	direct (N × H)
97. C₂³×D₄	C₂×C₁₂₇	semidirect
98. #534	C₂×C₁₂₇	direct (N × H)
99. C₂×#123	C₂×C₁₂₇	semidirect
100. C₂×#121	C₂×C₁₂₇	semidirect
101. C₂×#120	C₂×C₁₂₇	semidirect
102. C₂²×C₂²⋊C₄	C₂×C₁₂₇	semidirect
104. C₂⁵	C₂²×C₁₂₇	semidirect
107. C₂²×D₄	C₂²×C₁₂₇	semidirect
108. C₂²×D₄	C₂²×C₁₂₇	semidirect
109. C₂²×D₄	C₂²×C₁₂₇	semidirect
110. C₂³×C₄	C₂²×C₁₂₇	semidirect
111. C₂³×C₄	C₂²×C₁₂₇	semidirect
112. #123	C₂²×C₁₂₇	semidirect
113. #121	C₂²×C₁₂₇	semidirect
114. #120	C₂²×C₁₂₇	semidirect
115. C₂×C₄⋊C₄	C₂²×C₁₂₇	semidirect
116. C₂×C₂²⋊C₄	C₂²×C₁₂₇	semidirect
117. C₂×C₂²⋊C₄	C₂²×C₁₂₇	semidirect
118. C₂×C₂²⋊C₄	C₂²×C₁₂₇	semidirect
122. C₂⁴	D₄×C₁₂₇	semidirect
127. C₂⁴	C₂³×C₁₂₇	semidirect
133. C₂×D₄	C₂³×C₁₂₇	semidirect
134. C₂×D₄	C₂³×C₁₂₇	semidirect
139. C₂²×C₄	C₂³×C₁₂₇	semidirect
140. C₂²×C₄	C₂³×C₁₂₇	semidirect
141. C₄⋊C₄	C₂³×C₁₂₇	semidirect
142. C₂²⋊C₄	C₂³×C₁₂₇	semidirect
143. C₂²⋊C₄	C₂³×C₁₂₇	semidirect
155. C₂³	C₂×D₄×C₁₂₇	semidirect
188. C₂	#534×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
C₂³×D₄×C₁₂₇	2
#534×C₁₂₇	2
C₂×#123×C₁₂₇	2
C₂×#121×C₁₂₇	2
C₂×#120×C₁₂₇	2
C₂²×C₂²⋊C₄×C₁₂₇	2
C₂×#534	127

Subgroups — C2×#534×C127 #30636397

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#534×C₁₂₇ #30636397