Subgroups — C₁₂₇×#1923 #30634875

Order 16256.

292 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊C₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂²⋊C₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂²⋊C₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×C₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×C₂²⋊C₄×C₁₂₇	C₂	1	1	yes	maximal, characteristic
6	#336×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
7	#336×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
9	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
10	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
11	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
12	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
13	C₂×C₂²⋊C₄×C₁₂₇	C₄	1	1	yes	characteristic
14	C₂×C₂²⋊C₄×C₁₂₇	C₄	1	1	yes	characteristic
15	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
16	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
17	C₄⋊C₈×C₁₂₇	—	2	2	no	—
18	C₂²⋊C₈×C₁₂₇	—	2	2	no	—
19	C₂²⋊C₈×C₁₂₇	—	2	2	no	—
20	C₄×C₈×C₁₂₇	—	2	2	no	—
21	#95×C₁₂₇	C₄	1	1	yes	characteristic
22	#95×C₁₂₇	C₄	1	1	yes	characteristic
23	C₂⁴×C₁₂₇	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₁₂₇	—	2	1	no	—
25	C₂²×C₄×C₁₂₇	—	2	1	no	—
26	C₂²×C₄×C₁₂₇	—	2	1	no	—
27	C₂²×C₄×C₁₂₇	—	2	1	no	—
28	C₂²×C₄×C₁₂₇	—	2	1	no	—
29	C₂²×C₄×C₁₂₇	—	2	1	no	—
30	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
36	C₂×C₈×C₁₂₇	—	2	1	no	—
37	C₂×C₈×C₁₂₇	—	2	1	no	—
38	C₂×C₈×C₁₂₇	—	2	1	no	—
39	C₂×C₈×C₁₂₇	—	2	1	no	—
40	C₂×C₈×C₁₂₇	—	2	1	no	—
41	C₂×C₈×C₁₂₇	—	2	1	no	—
42	C₂×C₈×C₁₂₇	—	2	1	no	—
43	C₂×C₈×C₁₂₇	—	2	1	no	—
44	C₂×C₈×C₁₂₇	—	2	1	no	—
45	C₂×C₈×C₁₂₇	—	2	1	no	—
46	C₂×C₈×C₁₂₇	D₄	1	2	yes	—
47	C₂×C₈×C₁₂₇	D₄	1	2	yes	—
48	C₂²⋊C₄×C₁₂₇	—	2	2	no	—
49	C₂²⋊C₄×C₁₂₇	C₈	1	4	yes	complemented
50	C₄²×C₁₂₇	—	2	1	no	—
51	C₄²×C₁₂₇	—	2	1	no	—
52	C₄²×C₁₂₇	—	2	1	no	—
53	C₄²×C₁₂₇	—	2	1	no	—
54	C₂³×C₁₂₇	—	2	1	no	—
55	C₂³×C₁₂₇	—	2	1	no	—
56	C₂³×C₁₂₇	—	2	1	no	—
57	C₂³×C₁₂₇	—	2	1	no	—
58	C₂³×C₁₂₇	—	2	1	no	—
59	C₂³×C₁₂₇	—	2	1	no	—
60	C₂³×C₁₂₇	C₂×C₈	1	2	yes	—
61	C₂³×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₄×C₁₂₇	—	2	1	no	—
63	C₂×C₄×C₁₂₇	—	2	1	no	—
64	C₂×C₄×C₁₂₇	—	2	1	no	—
65	C₂×C₄×C₁₂₇	—	2	1	no	—
66	C₂×C₄×C₁₂₇	—	2	1	no	—
67	C₂×C₄×C₁₂₇	—	2	1	no	—
68	C₂×C₄×C₁₂₇	—	2	1	no	—
69	C₂×C₄×C₁₂₇	—	2	1	no	—
70	C₂×C₄×C₁₂₇	—	2	1	no	—
71	C₂×C₄×C₁₂₇	—	2	1	no	—
72	C₂×C₄×C₁₂₇	—	2	1	no	—
73	C₂×C₄×C₁₂₇	—	2	1	no	—
74	C₂×C₄×C₁₂₇	—	4	1	no	—
75	C₂×C₄×C₁₂₇	—	4	1	no	—
76	C₂×C₄×C₁₂₇	—	4	1	no	—
77	C₂×C₄×C₁₂₇	—	4	1	no	—
78	C₂×C₄×C₁₂₇	C₂×C₈	1	2	yes	—
79	C₂×C₄×C₁₂₇	C₂×C₈	1	2	yes	—
80	C₂×C₄×C₁₂₇	M₁₆	1	2	yes	—
81	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
82	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
83	C₂×C₄×C₁₂₇	#41	1	1	yes	characteristic
84	C₂×C₄×C₁₂₇	#41	1	1	yes	characteristic
85	C₂×C₄×C₁₂₇	#41	1	1	yes	characteristic
86	C₂×C₄×C₁₂₇	#41	1	1	yes	characteristic
87	C₈×C₁₂₇	—	2	2	no	—
88	C₈×C₁₂₇	—	2	2	no	—
89	C₈×C₁₂₇	—	4	1	no	—
90	C₈×C₁₂₇	—	4	1	no	—
91	C₂²×C₁₂₇	—	2	2	no	—
92	C₂²×C₁₂₇	—	2	2	no	—
93	C₂²×C₁₂₇	—	2	2	no	—
94	C₂²×C₁₂₇	—	4	1	no	—
95	C₂²×C₁₂₇	—	4	1	no	—
96	C₂²×C₁₂₇	—	4	1	no	—
97	C₂²×C₁₂₇	—	4	1	no	—
98	C₂²×C₁₂₇	#117	1	1	yes	characteristic
99	C₂²×C₁₂₇	C₄×D₄	1	1	yes	characteristic
100	C₂²×C₁₂₇	C₄×D₄	1	1	yes	characteristic
101	C₂²×C₁₂₇	C₂²×C₈	1	1	yes	characteristic
102	C₂²×C₁₂₇	C₂×M₁₆	1	1	yes	characteristic
103	C₂²×C₁₂₇	#131	1	1	yes	characteristic
104	C₂²×C₁₂₇	#131	1	1	yes	characteristic
105	C₄×C₁₂₇	—	2	2	no	—
106	C₄×C₁₂₇	—	2	2	no	—
107	C₄×C₁₂₇	—	2	2	no	—
108	C₄×C₁₂₇	—	4	1	no	—
109	C₄×C₁₂₇	—	4	1	no	—
110	C₄×C₁₂₇	#121	1	1	yes	characteristic
111	C₄×C₁₂₇	#123	1	1	yes	characteristic
112	C₄×C₁₂₇	#124	1	1	yes	characteristic
113	C₄×C₁₂₇	#126	1	1	yes	characteristic
114	C₂×C₁₂₇	—	4	2	no	—
115	C₂×C₁₂₇	#388	1	1	yes	characteristic
116	C₂×C₁₂₇	#431	1	1	yes	characteristic
117	C₂×C₁₂₇	#433	1	1	yes	characteristic
118	C₂×C₁₂₇	C₈×D₄	1	1	yes	characteristic
119	C₂×C₁₂₇	C₈×D₄	1	1	yes	characteristic
120	C₂×C₁₂₇	#435	1	1	yes	characteristic
121	C₂×C₁₂₇	#436	1	1	yes	characteristic
122	#1923	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
123	C₁₂₇	#1923	1	1	yes	characteristic, direct product factor, Sylow
124	C₂×C₄⋊C₈	C₂×C₁₂₇	1	1	yes	characteristic, complemented
125	C₂×C₂²⋊C₈	C₂×C₁₂₇	1	1	yes	characteristic
126	C₂×C₂²⋊C₈	C₂×C₁₂₇	1	1	yes	characteristic
127	C₂×C₄×C₈	C₂×C₁₂₇	1	1	yes	characteristic, complemented
128	C₄×C₂²⋊C₄	C₂×C₁₂₇	1	1	yes	characteristic
129	#336	C₂×C₁₂₇	1	1	yes	characteristic, complemented
130	#336	C₂×C₁₂₇	1	1	yes	characteristic, complemented
131	C₂³×C₄	C₂²×C₁₂₇	1	1	yes	characteristic
132	C₂²×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
133	C₂²×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
134	C₂²×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
135	C₂²×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
136	C₂×C₂²⋊C₄	C₄×C₁₂₇	1	1	yes	characteristic
137	C₂×C₂²⋊C₄	C₄×C₁₂₇	1	1	yes	characteristic
138	C₂×C₄²	C₂²×C₁₂₇	1	1	yes	characteristic
139	C₂×C₄²	C₂²×C₁₂₇	1	1	yes	characteristic
140	C₄⋊C₈	—	2	2	no	—
141	C₂²⋊C₈	—	2	2	no	—
142	C₂²⋊C₈	—	2	2	no	—
143	C₄×C₈	—	2	2	no	—
144	#95	C₄×C₁₂₇	1	1	yes	characteristic
145	#95	C₄×C₁₂₇	1	1	yes	characteristic
146	C₂⁴	C₂×C₄×C₁₂₇	1	1	yes	characteristic
147	C₂²×C₄	—	2	1	no	—
148	C₂²×C₄	—	2	1	no	—
149	C₂²×C₄	—	2	1	no	—
150	C₂²×C₄	—	2	1	no	—
151	C₂²×C₄	—	2	1	no	—
152	C₂²×C₄	—	2	1	no	—
153	C₂²×C₄	C₂×C₄×C₁₂₇	1	1	yes	characteristic
154	C₂²×C₄	C₂×C₄×C₁₂₇	1	1	yes	characteristic
155	C₂²×C₄	C₂×C₄×C₁₂₇	1	1	yes	characteristic
156	C₂²×C₄	C₂×C₄×C₁₂₇	1	1	yes	characteristic
157	C₂²×C₄	C₂×C₄×C₁₂₇	1	1	yes	characteristic
158	C₂²×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
159	C₂×C₈	—	2	1	no	—
160	C₂×C₈	—	2	1	no	—
161	C₂×C₈	—	2	1	no	—
162	C₂×C₈	—	2	1	no	—
163	C₂×C₈	—	2	1	no	—
164	C₂×C₈	—	2	1	no	—
165	C₂×C₈	—	2	1	no	—
166	C₂×C₈	—	2	1	no	—
167	C₂×C₈	—	2	1	no	—
168	C₂×C₈	—	2	1	no	—
169	C₂×C₈	D₄×C₁₂₇	1	2	yes	—
170	C₂×C₈	D₄×C₁₂₇	1	2	yes	—
171	C₂²⋊C₄	—	2	2	no	—
172	C₂²⋊C₄	C₈×C₁₂₇	1	4	yes	complemented
173	C₄²	—	2	1	no	—
174	C₄²	—	2	1	no	—
175	C₄²	—	2	1	no	—
176	C₄²	—	2	1	no	—
177	C₂³	—	2	1	no	—
178	C₂³	—	2	1	no	—
179	C₂³	—	2	1	no	—
180	C₂³	—	2	1	no	—
181	C₂³	—	2	1	no	—
182	C₂³	—	2	1	no	—
183	C₂³	C₂×C₈×C₁₂₇	1	2	yes	—
184	C₂³	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
185	C₂×C₄	—	2	1	no	—
186	C₂×C₄	—	2	1	no	—
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	—	4	1	no	—
198	C₂×C₄	—	4	1	no	—
199	C₂×C₄	—	4	1	no	—
200	C₂×C₄	—	4	1	no	—
201	C₂×C₄	C₂×C₈×C₁₂₇	1	2	yes	—
202	C₂×C₄	C₂×C₈×C₁₂₇	1	2	yes	—
203	C₂×C₄	M₁₆×C₁₂₇	1	2	yes	—
204	C₂×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
205	C₂×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
206	C₂×C₄	#41×C₁₂₇	1	1	yes	characteristic
207	C₂×C₄	#41×C₁₂₇	1	1	yes	characteristic
208	C₂×C₄	#41×C₁₂₇	1	1	yes	characteristic
209	C₂×C₄	#41×C₁₂₇	1	1	yes	characteristic
210	C₈	—	2	2	no	—
211	C₈	—	2	2	no	—
212	C₈	—	4	1	no	—
213	C₈	—	4	1	no	—
214	C₂²	—	2	2	no	—
215	C₂²	—	2	2	no	—
216	C₂²	—	2	2	no	—
217	C₂²	—	4	1	no	—
218	C₂²	—	4	1	no	—
219	C₂²	—	4	1	no	—
220	C₂²	—	4	1	no	—
221	C₂²	#117×C₁₂₇	1	1	yes	characteristic
222	C₂²	C₄×D₄×C₁₂₇	1	1	yes	characteristic
223	C₂²	C₄×D₄×C₁₂₇	1	1	yes	characteristic
224	C₂²	C₂²×C₈×C₁₂₇	1	1	yes	characteristic
225	C₂²	C₂×M₁₆×C₁₂₇	1	1	yes	characteristic
226	C₂²	#131×C₁₂₇	1	1	yes	characteristic
227	C₂²	#131×C₁₂₇	1	1	yes	characteristic
228	C₄	—	2	2	no	—
229	C₄	—	2	2	no	—
230	C₄	—	2	2	no	—
231	C₄	—	4	1	no	—
232	C₄	—	4	1	no	—
233	C₄	#121×C₁₂₇	1	1	yes	characteristic
234	C₄	#123×C₁₂₇	1	1	yes	characteristic
235	C₄	#124×C₁₂₇	1	1	yes	characteristic
236	C₄	#126×C₁₂₇	1	1	yes	characteristic
237	C₂	—	4	2	no	—
238	C₂	#388×C₁₂₇	1	1	yes	characteristic
239	C₂	#431×C₁₂₇	1	1	yes	characteristic
240	C₂	#433×C₁₂₇	1	1	yes	characteristic
241	C₂	C₈×D₄×C₁₂₇	1	1	yes	characteristic
242	C₂	C₈×D₄×C₁₂₇	1	1	yes	characteristic
243	C₂	#435×C₁₂₇	1	1	yes	characteristic
244	C₂	#436×C₁₂₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₁₂₇	C₂³
commutator	C₂²	C₂²×C₈×C₁₂₇
Frattini, cosocle	C₂²×C₄	C₂³×C₁₂₇
socle	C₂³×C₁₂₇	C₂²×C₄
prime radical	C₂⁴×C₁₂₇	C₂×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. C₂×C₄⋊C₈×C₁₂₇	C₂	semidirect
4. C₂×C₄×C₈×C₁₂₇	C₂	semidirect
6. #336×C₁₂₇	C₂	semidirect
7. #336×C₁₂₇	C₂	semidirect
49. C₂²⋊C₄×C₁₂₇	C₈	semidirect
122. #1923	C₁₂₇	direct (N × H)
123. C₁₂₇	#1923	direct (N × H)
124. C₂×C₄⋊C₈	C₂×C₁₂₇	semidirect
127. C₂×C₄×C₈	C₂×C₁₂₇	semidirect
129. #336	C₂×C₁₂₇	semidirect
130. #336	C₂×C₁₂₇	semidirect
172. C₂²⋊C₄	C₈×C₁₂₇	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₄⋊C₈×C₁₂₇	2
C₂×C₂²⋊C₈×C₁₂₇	2
C₂×C₄×C₈×C₁₂₇	2
C₄×C₂²⋊C₄×C₁₂₇	2
#336×C₁₂₇	2
#1923	127

Subgroups — C127×#1923 #30634875

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#1923 #30634875