Subgroups — C₂×#389×C₁₂₇ #30635243

Order 16256.

800 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
2	C₂²×C₄⋊C₄×C₁₂₇	C₂	1	2	yes	maximal
3	C₂²×C₄⋊C₄×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₄²×C₁₂₇	C₂	1	1	yes	maximal, characteristic
5	#389×C₁₂₇	C₂	1	8	yes	maximal, direct product factor
6	C₂×#95×C₁₂₇	C₂	1	2	yes	maximal
7	C₂³×C₄×C₁₂₇	C₂²	1	2	yes	—
8	C₂³×C₄×C₁₂₇	C₂²	1	2	yes	—
9	C₂³×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
10	C₂³×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
11	C₂³×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊C₄×C₁₂₇	—	2	8	no	—
13	C₂×C₄⋊C₄×C₁₂₇	—	2	4	no	—
14	C₂×C₄⋊C₄×C₁₂₇	C₄	1	8	yes	complemented
15	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	4	yes	—
16	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	8	yes	—
17	C₂×C₄⋊C₄×C₁₂₇	C₂²	1	4	yes	—
18	C₂×C₄²×C₁₂₇	—	2	4	no	—
19	C₂×C₄²×C₁₂₇	C₂²	1	4	yes	—
20	#95×C₁₂₇	C₂²	1	8	yes	—
21	C₂⁴×C₁₂₇	C₂³	1	1	yes	characteristic
22	C₂²×C₄×C₁₂₇	—	2	2	no	—
23	C₂²×C₄×C₁₂₇	—	2	2	no	—
24	C₂²×C₄×C₁₂₇	—	2	4	no	—
25	C₂²×C₄×C₁₂₇	—	2	4	no	—
26	C₂²×C₄×C₁₂₇	—	2	2	no	—
27	C₂²×C₄×C₁₂₇	—	2	2	no	—
28	C₂²×C₄×C₁₂₇	—	2	4	no	—
29	C₂²×C₄×C₁₂₇	—	2	4	no	—
30	C₂²×C₄×C₁₂₇	—	2	4	no	—
31	C₂²×C₄×C₁₂₇	C₂×C₄	1	8	yes	—
32	C₂²×C₄×C₁₂₇	C₂×C₄	1	4	yes	—
33	C₂²×C₄×C₁₂₇	D₄	1	4	yes	—
34	C₂²×C₄×C₁₂₇	D₄	1	4	yes	—
35	C₂²×C₄×C₁₂₇	Q₈	1	4	yes	—
36	C₂²×C₄×C₁₂₇	Q₈	1	4	yes	—
37	C₂²×C₄×C₁₂₇	C₂³	1	4	yes	—
38	C₂²×C₄×C₁₂₇	C₂³	1	4	yes	—
39	C₂²×C₄×C₁₂₇	C₂³	1	2	yes	—
40	C₂²×C₄×C₁₂₇	C₂³	1	2	yes	—
41	C₂²×C₄×C₁₂₇	C₂³	1	2	yes	—
42	C₄⋊C₄×C₁₂₇	—	2	16	no	—
43	C₄⋊C₄×C₁₂₇	—	2	8	no	—
44	C₄⋊C₄×C₁₂₇	C₂×C₄	1	16	yes	complemented
45	C₄²×C₁₂₇	—	2	8	no	—
46	C₂³×C₁₂₇	C₂²×C₄	1	2	yes	—
47	C₂³×C₁₂₇	C₂×D₄	1	2	yes	—
48	C₂³×C₁₂₇	C₂×D₄	1	2	yes	—
49	C₂³×C₁₂₇	C₂×Q₈	1	2	yes	—
50	C₂³×C₁₂₇	C₂×Q₈	1	2	yes	—
51	C₂³×C₁₂₇	#41	1	2	yes	—
52	C₂³×C₁₂₇	#41	1	2	yes	—
53	C₂³×C₁₂₇	C₂⁴	1	1	yes	characteristic
54	C₂×C₄×C₁₂₇	—	2	16	no	—
55	C₂×C₄×C₁₂₇	—	2	8	no	—
56	C₂×C₄×C₁₂₇	—	2	8	no	—
57	C₂×C₄×C₁₂₇	—	2	2	no	—
58	C₂×C₄×C₁₂₇	—	2	2	no	—
59	C₂×C₄×C₁₂₇	—	2	4	no	—
60	C₂×C₄×C₁₂₇	—	2	4	no	—
61	C₂×C₄×C₁₂₇	—	2	2	no	—
62	C₂×C₄×C₁₂₇	—	2	2	no	—
63	C₂×C₄×C₁₂₇	—	2	4	no	—
64	C₂×C₄×C₁₂₇	—	2	4	no	—
65	C₂×C₄×C₁₂₇	—	2	4	no	—
66	C₂×C₄×C₁₂₇	—	4	8	no	—
67	C₂×C₄×C₁₂₇	C₄⋊C₄	1	16	yes	complemented
68	C₂×C₄×C₁₂₇	C₂²×C₄	1	8	yes	—
69	C₂×C₄×C₁₂₇	C₂²×C₄	1	4	yes	—
70	C₂×C₄×C₁₂₇	C₂×D₄	1	4	yes	—
71	C₂×C₄×C₁₂₇	C₂×D₄	1	4	yes	—
72	C₂×C₄×C₁₂₇	C₂×Q₈	1	4	yes	—
73	C₂×C₄×C₁₂₇	C₂×Q₈	1	4	yes	—
74	C₂²×C₁₂₇	C₂×C₄⋊C₄	1	4	yes	—
75	C₂²×C₁₂₇	C₄×D₄	1	4	yes	—
76	C₂²×C₁₂₇	C₄×Q₈	1	4	yes	—
77	C₂²×C₁₂₇	#121	1	4	yes	—
78	C₂²×C₁₂₇	C₂²⋊Q₈	1	4	yes	—
79	C₂²×C₁₂₇	#125	1	4	yes	—
80	C₂²×C₁₂₇	C₄⋊Q₈	1	4	yes	—
81	C₂²×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
82	C₂²×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
83	C₂²×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
84	C₂²×C₁₂₇	C₂²×Q₈	1	1	yes	characteristic
85	C₂²×C₁₂₇	C₂²×Q₈	1	1	yes	characteristic
86	C₂²×C₁₂₇	C₂×#41	1	1	yes	characteristic
87	C₂²×C₁₂₇	C₂×#41	1	1	yes	characteristic
88	C₄×C₁₂₇	—	2	8	no	—
89	C₄×C₁₂₇	—	2	4	no	—
90	C₄×C₁₂₇	—	2	4	no	—
91	C₄×C₁₂₇	—	4	4	no	—
92	C₄×C₁₂₇	C₂×C₄⋊C₄	1	8	yes	complemented
93	C₂×C₁₂₇	#389	1	8	yes	direct product factor
94	C₂×C₁₂₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
95	C₂×C₁₂₇	C₂×C₄×D₄	1	1	yes	characteristic
96	C₂×C₁₂₇	C₂×C₄×Q₈	1	1	yes	characteristic
97	C₂×C₁₂₇	C₂×#121	1	1	yes	characteristic
98	C₂×C₁₂₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
99	C₂×C₁₂₇	C₂×#125	1	1	yes	characteristic
100	C₂×C₁₂₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
101	C₂×#389	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
102	C₁₂₇	C₂×#389	1	1	yes	characteristic, direct product factor, Sylow
103	C₂²×C₄⋊C₄	C₂×C₁₂₇	1	1	yes	characteristic
104	C₂²×C₄⋊C₄	C₂×C₁₂₇	1	2	yes	—
105	C₂²×C₄⋊C₄	C₂×C₁₂₇	1	1	yes	characteristic
106	C₂²×C₄²	C₂×C₁₂₇	1	1	yes	characteristic
107	#389	C₂×C₁₂₇	1	8	yes	direct product factor
108	C₂×#95	C₂×C₁₂₇	1	2	yes	—
109	C₂³×C₄	C₂²×C₁₂₇	1	2	yes	—
110	C₂³×C₄	C₂²×C₁₂₇	1	2	yes	—
111	C₂³×C₄	C₂²×C₁₂₇	1	1	yes	characteristic
112	C₂³×C₄	C₂²×C₁₂₇	1	1	yes	characteristic
113	C₂³×C₄	C₂²×C₁₂₇	1	1	yes	characteristic
114	C₂×C₄⋊C₄	—	2	8	no	—
115	C₂×C₄⋊C₄	—	2	4	no	—
116	C₂×C₄⋊C₄	C₄×C₁₂₇	1	8	yes	complemented
117	C₂×C₄⋊C₄	C₂²×C₁₂₇	1	4	yes	—
118	C₂×C₄⋊C₄	C₂²×C₁₂₇	1	8	yes	—
119	C₂×C₄⋊C₄	C₂²×C₁₂₇	1	4	yes	—
120	C₂×C₄²	—	2	4	no	—
121	C₂×C₄²	C₂²×C₁₂₇	1	4	yes	—
122	#95	C₂²×C₁₂₇	1	8	yes	—
123	C₂⁴	C₂³×C₁₂₇	1	1	yes	characteristic
124	C₂²×C₄	—	2	2	no	—
125	C₂²×C₄	—	2	2	no	—
126	C₂²×C₄	—	2	4	no	—
127	C₂²×C₄	—	2	4	no	—
128	C₂²×C₄	—	2	2	no	—
129	C₂²×C₄	—	2	2	no	—
130	C₂²×C₄	—	2	4	no	—
131	C₂²×C₄	—	2	4	no	—
132	C₂²×C₄	—	2	4	no	—
133	C₂²×C₄	C₂×C₄×C₁₂₇	1	8	yes	—
134	C₂²×C₄	C₂×C₄×C₁₂₇	1	4	yes	—
135	C₂²×C₄	D₄×C₁₂₇	1	4	yes	—
136	C₂²×C₄	D₄×C₁₂₇	1	4	yes	—
137	C₂²×C₄	Q₈×C₁₂₇	1	4	yes	—
138	C₂²×C₄	Q₈×C₁₂₇	1	4	yes	—
139	C₂²×C₄	C₂³×C₁₂₇	1	4	yes	—
140	C₂²×C₄	C₂³×C₁₂₇	1	4	yes	—
141	C₂²×C₄	C₂³×C₁₂₇	1	2	yes	—
142	C₂²×C₄	C₂³×C₁₂₇	1	2	yes	—
143	C₂²×C₄	C₂³×C₁₂₇	1	2	yes	—
144	C₄⋊C₄	—	2	16	no	—
145	C₄⋊C₄	—	2	8	no	—
146	C₄⋊C₄	C₂×C₄×C₁₂₇	1	16	yes	complemented
147	C₄²	—	2	8	no	—
148	C₂³	C₂²×C₄×C₁₂₇	1	2	yes	—
149	C₂³	C₂×D₄×C₁₂₇	1	2	yes	—
150	C₂³	C₂×D₄×C₁₂₇	1	2	yes	—
151	C₂³	C₂×Q₈×C₁₂₇	1	2	yes	—
152	C₂³	C₂×Q₈×C₁₂₇	1	2	yes	—
153	C₂³	#41×C₁₂₇	1	2	yes	—
154	C₂³	#41×C₁₂₇	1	2	yes	—
155	C₂³	C₂⁴×C₁₂₇	1	1	yes	characteristic
156	C₂×C₄	—	2	16	no	—
157	C₂×C₄	—	2	8	no	—
158	C₂×C₄	—	2	8	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	4	no	—
162	C₂×C₄	—	2	4	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	4	no	—
166	C₂×C₄	—	2	4	no	—
167	C₂×C₄	—	2	4	no	—
168	C₂×C₄	—	4	8	no	—
169	C₂×C₄	C₄⋊C₄×C₁₂₇	1	16	yes	complemented
170	C₂×C₄	C₂²×C₄×C₁₂₇	1	8	yes	—
171	C₂×C₄	C₂²×C₄×C₁₂₇	1	4	yes	—
172	C₂×C₄	C₂×D₄×C₁₂₇	1	4	yes	—
173	C₂×C₄	C₂×D₄×C₁₂₇	1	4	yes	—
174	C₂×C₄	C₂×Q₈×C₁₂₇	1	4	yes	—
175	C₂×C₄	C₂×Q₈×C₁₂₇	1	4	yes	—
176	C₂²	C₂×C₄⋊C₄×C₁₂₇	1	4	yes	—
177	C₂²	C₄×D₄×C₁₂₇	1	4	yes	—
178	C₂²	C₄×Q₈×C₁₂₇	1	4	yes	—
179	C₂²	#121×C₁₂₇	1	4	yes	—
180	C₂²	C₂²⋊Q₈×C₁₂₇	1	4	yes	—
181	C₂²	#125×C₁₂₇	1	4	yes	—
182	C₂²	C₄⋊Q₈×C₁₂₇	1	4	yes	—
183	C₂²	C₂³×C₄×C₁₂₇	1	1	yes	characteristic
184	C₂²	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
185	C₂²	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
186	C₂²	C₂²×Q₈×C₁₂₇	1	1	yes	characteristic
187	C₂²	C₂²×Q₈×C₁₂₇	1	1	yes	characteristic
188	C₂²	C₂×#41×C₁₂₇	1	1	yes	characteristic
189	C₂²	C₂×#41×C₁₂₇	1	1	yes	characteristic
190	C₄	—	2	8	no	—
191	C₄	—	2	4	no	—
192	C₄	—	2	4	no	—
193	C₄	—	4	4	no	—
194	C₄	C₂×C₄⋊C₄×C₁₂₇	1	8	yes	complemented
195	C₂	#389×C₁₂₇	1	8	yes	direct product factor
196	C₂	C₂²×C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
197	C₂	C₂×C₄×D₄×C₁₂₇	1	1	yes	characteristic
198	C₂	C₂×C₄×Q₈×C₁₂₇	1	1	yes	characteristic
199	C₂	C₂×#121×C₁₂₇	1	1	yes	characteristic
200	C₂	C₂×C₂²⋊Q₈×C₁₂₇	1	1	yes	characteristic
201	C₂	C₂×#125×C₁₂₇	1	1	yes	characteristic
202	C₂	C₂×C₄⋊Q₈×C₁₂₇	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
5. #389×C₁₂₇	C₂	direct (N × H)
14. C₂×C₄⋊C₄×C₁₂₇	C₄	semidirect
44. C₄⋊C₄×C₁₂₇	C₂×C₄	semidirect
67. C₂×C₄×C₁₂₇	C₄⋊C₄	semidirect
92. C₄×C₁₂₇	C₂×C₄⋊C₄	semidirect
93. C₂×C₁₂₇	#389	direct (N × H)
101. C₂×#389	C₁₂₇	direct (N × H)
102. C₁₂₇	C₂×#389	direct (N × H)
107. #389	C₂×C₁₂₇	direct (N × H)
116. C₂×C₄⋊C₄	C₄×C₁₂₇	semidirect
146. C₄⋊C₄	C₂×C₄×C₁₂₇	semidirect
169. C₂×C₄	C₄⋊C₄×C₁₂₇	semidirect
194. C₄	C₂×C₄⋊C₄×C₁₂₇	semidirect
195. C₂	#389×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×C₄²×C₁₂₇	2
#389×C₁₂₇	2
C₂×#95×C₁₂₇	2
C₂×#389	127

Subgroups — C2×#389×C127 #30635243

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#389×C₁₂₇ #30635243