Subgroups — C₁₂₇×#2966 #30635918

Order 16256.

380 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	#529×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	#446×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈×Q₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#436×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
5	#435×C₁₂₇	C₂	1	2	yes	maximal, complemented
6	C₈×D₄×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈×D₄×C₁₂₇	C₂	1	2	yes	maximal, complemented
8	#433×C₁₂₇	C₂	1	2	yes	maximal, complemented
9	#432×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
10	#405×C₁₂₇	C₂	1	2	yes	maximal, complemented
11	C₂×C₄×C₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×M₁₆×C₁₂₇	C₂²	1	2	yes	—
13	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
14	C₂²×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
15	C₂²×C₈×C₁₂₇	C₂²	1	2	yes	—
16	#126×C₁₂₇	C₄	1	2	yes	—
17	#125×C₁₂₇	C₄	1	1	yes	characteristic
18	#124×C₁₂₇	C₄	1	1	yes	characteristic
19	#123×C₁₂₇	C₄	1	2	yes	—
20	C₂²⋊Q₈×C₁₂₇	C₄	1	1	yes	characteristic
21	#121×C₁₂₇	C₄	1	1	yes	characteristic
22	C₄×Q₈×C₁₂₇	C₂²	1	1	yes	characteristic
23	C₄×D₄×C₁₂₇	C₂²	1	2	yes	—
24	C₄×D₄×C₁₂₇	C₂²	1	1	yes	characteristic
25	#117×C₁₂₇	C₂²	1	2	yes	—
26	C₂×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
27	C₄⋊C₈×C₁₂₇	C₂²	1	1	yes	characteristic
28	C₄⋊C₈×C₁₂₇	C₂²	1	1	yes	characteristic
29	C₄⋊C₈×C₁₂₇	C₂²	1	2	yes	—
30	C₄⋊C₈×C₁₂₇	C₂²	1	2	yes	—
31	C₂²⋊C₈×C₁₂₇	C₂²	1	2	yes	—
32	C₂²⋊C₈×C₁₂₇	C₂²	1	2	yes	—
33	C₂²⋊C₈×C₁₂₇	C₂²	1	2	yes	—
34	#97×C₁₂₇	C₂²	1	2	yes	—
35	#97×C₁₂₇	C₂²	1	2	yes	—
36	C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
37	C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
38	C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
39	C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
40	C₄×C₈×C₁₂₇	C₂²	1	2	yes	—
41	C₂×Q₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
42	C₂×D₄×C₁₂₇	C₂×C₄	1	2	yes	—
43	C₂×D₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
47	C₂²×C₄×C₁₂₇	C₂³	1	2	yes	—
48	M₁₆×C₁₂₇	—	2	4	no	—
49	C₂×C₈×C₁₂₇	—	2	2	no	—
50	C₂×C₈×C₁₂₇	—	2	4	no	—
51	C₂×C₈×C₁₂₇	—	2	2	no	—
52	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
53	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
54	C₂×C₈×C₁₂₇	C₂³	1	2	yes	—
55	C₂×C₈×C₁₂₇	C₂³	1	2	yes	—
56	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
57	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
58	C₄⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
59	C₄⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
60	C₄⋊C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
61	C₄⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
62	C₄⋊C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
63	C₄⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
64	C₂²⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
65	C₂²⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
66	C₂²⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
67	C₂²⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
68	C₂²⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
69	C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
70	C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
71	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
72	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
73	C₄²×C₁₂₇	C₂³	1	2	yes	—
74	C₂³×C₁₂₇	C₂²×C₄	1	2	yes	—
75	C₂³×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
76	Q₈×C₁₂₇	—	2	2	no	—
77	D₄×C₁₂₇	—	2	4	no	—
78	D₄×C₁₂₇	—	2	2	no	—
79	C₂×C₄×C₁₂₇	—	2	2	no	—
80	C₂×C₄×C₁₂₇	—	2	2	no	—
81	C₂×C₄×C₁₂₇	—	2	1	no	—
82	C₂×C₄×C₁₂₇	—	2	1	no	—
83	C₂×C₄×C₁₂₇	—	2	2	no	—
84	C₂×C₄×C₁₂₇	—	2	2	no	—
85	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
90	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
91	C₂×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
92	C₂×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
94	C₈×C₁₂₇	—	2	1	no	—
95	C₈×C₁₂₇	—	2	1	no	—
96	C₈×C₁₂₇	—	2	2	no	—
97	C₈×C₁₂₇	—	2	2	no	—
98	C₈×C₁₂₇	#41	1	2	yes	—
99	C₈×C₁₂₇	#41	1	2	yes	—
100	C₂²×C₁₂₇	—	2	1	no	—
101	C₂²×C₁₂₇	—	2	1	no	—
102	C₂²×C₁₂₇	—	2	2	no	—
103	C₂²×C₁₂₇	—	2	2	no	—
104	C₂²×C₁₂₇	—	2	2	no	—
105	C₂²×C₁₂₇	#131	1	2	yes	—
106	C₂²×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
107	C₄×C₁₂₇	—	2	1	no	—
108	C₄×C₁₂₇	—	2	2	no	—
109	C₄×C₁₂₇	—	2	1	no	—
110	C₄×C₁₂₇	—	2	1	no	—
111	C₄×C₁₂₇	—	2	1	no	—
112	C₄×C₁₂₇	—	2	2	no	—
113	C₄×C₁₂₇	—	2	2	no	—
114	C₄×C₁₂₇	#131	1	2	yes	—
115	C₄×C₁₂₇	C₂×#41	1	1	yes	characteristic
116	C₄×C₁₂₇	C₂×#41	1	1	yes	characteristic
117	C₂×C₁₂₇	—	2	2	no	—
118	C₂×C₁₂₇	—	4	2	no	—
119	C₂×C₁₂₇	C₄×#41	1	1	yes	characteristic
120	C₂×C₁₂₇	C₂×#131	1	1	yes	characteristic
121	C₂×C₁₂₇	C₂×#131	1	1	yes	characteristic
122	#2966	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
123	C₁₂₇	#2966	1	1	yes	characteristic, direct product factor, Sylow
124	#529	C₂×C₁₂₇	1	1	yes	characteristic
125	#446	C₂×C₁₂₇	1	1	yes	characteristic, complemented
126	C₈×Q₈	C₂×C₁₂₇	1	1	yes	characteristic, complemented
127	#436	C₂×C₁₂₇	1	1	yes	characteristic, complemented
128	#435	C₂×C₁₂₇	1	2	yes	complemented
129	C₈×D₄	C₂×C₁₂₇	1	1	yes	characteristic, complemented
130	C₈×D₄	C₂×C₁₂₇	1	2	yes	complemented
131	#433	C₂×C₁₂₇	1	2	yes	complemented
132	#432	C₂×C₁₂₇	1	1	yes	characteristic, complemented
133	#405	C₂×C₁₂₇	1	2	yes	complemented
134	C₂×C₄×C₈	C₂×C₁₂₇	1	1	yes	characteristic, complemented
135	C₂×M₁₆	C₂²×C₁₂₇	1	2	yes	—
136	C₂²×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
137	C₂²×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
138	C₂²×C₈	C₂²×C₁₂₇	1	2	yes	—
139	#126	C₄×C₁₂₇	1	2	yes	—
140	#125	C₄×C₁₂₇	1	1	yes	characteristic
141	#124	C₄×C₁₂₇	1	1	yes	characteristic
142	#123	C₄×C₁₂₇	1	2	yes	—
143	C₂²⋊Q₈	C₄×C₁₂₇	1	1	yes	characteristic
144	#121	C₄×C₁₂₇	1	1	yes	characteristic
145	C₄×Q₈	C₂²×C₁₂₇	1	1	yes	characteristic
146	C₄×D₄	C₂²×C₁₂₇	1	2	yes	—
147	C₄×D₄	C₂²×C₁₂₇	1	1	yes	characteristic
148	#117	C₂²×C₁₂₇	1	2	yes	—
149	C₂×C₄²	C₂²×C₁₂₇	1	1	yes	characteristic
150	C₄⋊C₈	C₂²×C₁₂₇	1	1	yes	characteristic
151	C₄⋊C₈	C₂²×C₁₂₇	1	1	yes	characteristic
152	C₄⋊C₈	C₂²×C₁₂₇	1	2	yes	—
153	C₄⋊C₈	C₂²×C₁₂₇	1	2	yes	—
154	C₂²⋊C₈	C₂²×C₁₂₇	1	2	yes	—
155	C₂²⋊C₈	C₂²×C₁₂₇	1	2	yes	—
156	C₂²⋊C₈	C₂²×C₁₂₇	1	2	yes	—
157	#97	C₂²×C₁₂₇	1	2	yes	—
158	#97	C₂²×C₁₂₇	1	2	yes	—
159	C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
160	C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
161	C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
162	C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
163	C₄×C₈	C₂²×C₁₂₇	1	2	yes	—
164	C₂×Q₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
165	C₂×D₄	C₂×C₄×C₁₂₇	1	2	yes	—
166	C₂×D₄	C₂×C₄×C₁₂₇	1	1	yes	characteristic
167	C₂²×C₄	C₂×C₄×C₁₂₇	1	1	yes	characteristic
168	C₂²×C₄	C₂×C₄×C₁₂₇	1	1	yes	characteristic
169	C₂²×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
170	C₂²×C₄	C₂³×C₁₂₇	1	2	yes	—
171	M₁₆	—	2	4	no	—
172	C₂×C₈	—	2	2	no	—
173	C₂×C₈	—	2	4	no	—
174	C₂×C₈	—	2	2	no	—
175	C₂×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
176	C₂×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
177	C₂×C₈	C₂³×C₁₂₇	1	2	yes	—
178	C₂×C₈	C₂³×C₁₂₇	1	2	yes	—
179	C₂×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
180	C₂×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
181	C₄⋊C₄	C₂×C₄×C₁₂₇	1	2	yes	—
182	C₄⋊C₄	C₂×C₄×C₁₂₇	1	2	yes	—
183	C₄⋊C₄	C₂×C₄×C₁₂₇	1	1	yes	characteristic
184	C₄⋊C₄	C₂×C₄×C₁₂₇	1	2	yes	—
185	C₄⋊C₄	C₂×C₄×C₁₂₇	1	1	yes	characteristic
186	C₄⋊C₄	C₂×C₄×C₁₂₇	1	2	yes	—
187	C₂²⋊C₄	C₂×C₄×C₁₂₇	1	2	yes	—
188	C₂²⋊C₄	C₂×C₄×C₁₂₇	1	2	yes	—
189	C₂²⋊C₄	C₂×C₄×C₁₂₇	1	2	yes	—
190	C₂²⋊C₄	C₂×C₄×C₁₂₇	1	2	yes	—
191	C₂²⋊C₄	C₂×C₄×C₁₂₇	1	2	yes	—
192	C₄²	C₂×C₄×C₁₂₇	1	1	yes	characteristic
193	C₄²	C₂×C₄×C₁₂₇	1	1	yes	characteristic
194	C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
195	C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
196	C₄²	C₂³×C₁₂₇	1	2	yes	—
197	C₂³	C₂²×C₄×C₁₂₇	1	2	yes	—
198	C₂³	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
199	Q₈	—	2	2	no	—
200	D₄	—	2	4	no	—
201	D₄	—	2	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
209	C₂×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
210	C₂×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
211	C₂×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
212	C₂×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
213	C₂×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
214	C₂×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
215	C₂×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
216	C₂×C₄	C₂⁴×C₁₂₇	1	1	yes	characteristic
217	C₈	—	2	1	no	—
218	C₈	—	2	1	no	—
219	C₈	—	2	2	no	—
220	C₈	—	2	2	no	—
221	C₈	#41×C₁₂₇	1	2	yes	—
222	C₈	#41×C₁₂₇	1	2	yes	—
223	C₂²	—	2	1	no	—
224	C₂²	—	2	1	no	—
225	C₂²	—	2	2	no	—
226	C₂²	—	2	2	no	—
227	C₂²	—	2	2	no	—
228	C₂²	#131×C₁₂₇	1	2	yes	—
229	C₂²	C₂³×C₄×C₁₂₇	1	1	yes	characteristic
230	C₄	—	2	1	no	—
231	C₄	—	2	2	no	—
232	C₄	—	2	1	no	—
233	C₄	—	2	1	no	—
234	C₄	—	2	1	no	—
235	C₄	—	2	2	no	—
236	C₄	—	2	2	no	—
237	C₄	#131×C₁₂₇	1	2	yes	—
238	C₄	C₂×#41×C₁₂₇	1	1	yes	characteristic
239	C₄	C₂×#41×C₁₂₇	1	1	yes	characteristic
240	C₂	—	2	2	no	—
241	C₂	—	4	2	no	—
242	C₂	C₄×#41×C₁₂₇	1	1	yes	characteristic
243	C₂	C₂×#131×C₁₂₇	1	1	yes	characteristic
244	C₂	C₂×#131×C₁₂₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈×C₁₂₇	C₂³
commutator	C₂²	C₂³×C₄×C₁₂₇
Frattini, cosocle	C₂×C₄	C₂⁴×C₁₂₇
socle	C₂²×C₁₂₇	C₂³×C₄
prime radical	#121×C₁₂₇	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. #446×C₁₂₇	C₂	semidirect
3. C₈×Q₈×C₁₂₇	C₂	semidirect
4. #436×C₁₂₇	C₂	semidirect
5. #435×C₁₂₇	C₂	semidirect
6. C₈×D₄×C₁₂₇	C₂	semidirect
7. C₈×D₄×C₁₂₇	C₂	semidirect
8. #433×C₁₂₇	C₂	semidirect
9. #432×C₁₂₇	C₂	semidirect
10. #405×C₁₂₇	C₂	semidirect
11. C₂×C₄×C₈×C₁₂₇	C₂	semidirect
122. #2966	C₁₂₇	direct (N × H)
123. C₁₂₇	#2966	direct (N × H)
125. #446	C₂×C₁₂₇	semidirect
126. C₈×Q₈	C₂×C₁₂₇	semidirect
127. #436	C₂×C₁₂₇	semidirect
128. #435	C₂×C₁₂₇	semidirect
129. C₈×D₄	C₂×C₁₂₇	semidirect
130. C₈×D₄	C₂×C₁₂₇	semidirect
131. #433	C₂×C₁₂₇	semidirect
132. #432	C₂×C₁₂₇	semidirect
133. #405	C₂×C₁₂₇	semidirect
134. C₂×C₄×C₈	C₂×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
#529×C₁₂₇	2
#446×C₁₂₇	2
C₈×Q₈×C₁₂₇	2
#436×C₁₂₇	2
#435×C₁₂₇	2
C₈×D₄×C₁₂₇	2
#433×C₁₂₇	2
#432×C₁₂₇	2
#405×C₁₂₇	2
C₂×C₄×C₈×C₁₂₇	2
#2966	127

Subgroups — C127×#2966 #30635918

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#2966 #30635918