Subgroups — C₁₂₇×#3361 #30636313

Order 16256.

362 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	#559×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
2	#554×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#488×C₁₂₇	C₂	1	2	yes	maximal, complemented
4	#486×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
5	#478×C₁₂₇	C₂	1	2	yes	maximal, complemented
6	#477×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
7	#463×C₁₂₇	C₂	1	2	yes	maximal, complemented
8	#459×C₁₂₇	C₂	1	1	yes	maximal, characteristic
9	C₈×Q₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×SD₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×D₈×C₁₂₇	C₂	1	2	yes	maximal, complemented
12	C₂×SD₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
13	C₂×D₈×C₁₂₇	C₂²	1	2	yes	—
14	C₄⋊Q₈×C₁₂₇	C₂²	1	1	yes	characteristic
15	#127×C₁₂₇	C₂²	1	1	yes	characteristic
16	#125×C₁₂₇	C₂²	1	2	yes	—
17	#124×C₁₂₇	—	2	1	no	—
18	#124×C₁₂₇	C₂²	1	2	yes	—
19	#123×C₁₂₇	—	2	2	no	—
20	C₂²⋊Q₈×C₁₂₇	—	2	1	no	—
21	C₂²⋊Q₈×C₁₂₇	—	2	2	no	—
22	C₄×Q₈×C₁₂₇	C₂²	1	1	yes	characteristic
23	C₄×Q₈×C₁₂₇	C₂²	1	1	yes	characteristic
24	C₄×D₄×C₁₂₇	—	2	1	no	—
25	C₄×D₄×C₁₂₇	C₂²	1	2	yes	—
26	C₄×D₄×C₁₂₇	C₂²	1	1	yes	characteristic
27	C₄×D₄×C₁₂₇	C₂²	1	2	yes	—
28	C₂×C₄⋊C₄×C₁₂₇	—	2	1	no	—
29	#107×C₁₂₇	C₂²	1	2	yes	—
30	#106×C₁₂₇	C₂²	1	1	yes	characteristic
31	C₄⋊C₈×C₁₂₇	C₂²	1	1	yes	characteristic
32	C₄⋊C₈×C₁₂₇	C₂²	1	2	yes	—
33	#103×C₁₂₇	C₂²	1	1	yes	characteristic
34	#103×C₁₂₇	C₂²	1	2	yes	—
35	#102×C₁₂₇	C₂²	1	1	yes	characteristic
36	#102×C₁₂₇	C₂²	1	2	yes	—
37	#102×C₁₂₇	C₂²	1	2	yes	—
38	#102×C₁₂₇	C₂²	1	2	yes	—
39	C₄×C₈×C₁₂₇	C₂²	1	2	yes	—
40	C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
41	C₂×Q₈×C₁₂₇	—	2	1	no	—
42	C₂×Q₈×C₁₂₇	D₄	1	1	yes	characteristic
43	C₂×Q₈×C₁₂₇	C₂³	1	1	yes	characteristic
44	C₂×D₄×C₁₂₇	—	2	1	no	—
45	C₂×D₄×C₁₂₇	—	2	1	no	—
46	C₂×D₄×C₁₂₇	C₂³	1	2	yes	—
47	C₂×D₄×C₁₂₇	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₁₂₇	—	2	1	no	—
49	C₂²×C₄×C₁₂₇	—	2	2	no	—
50	C₂²×C₄×C₁₂₇	—	2	2	no	—
51	SD₁₆×C₁₂₇	—	2	2	no	—
52	D₈×C₁₂₇	—	2	4	no	—
53	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
54	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
55	C₂×C₈×C₁₂₇	C₂³	1	2	yes	—
56	C₄⋊C₄×C₁₂₇	—	2	2	no	—
57	C₄⋊C₄×C₁₂₇	—	2	1	no	—
58	C₄⋊C₄×C₁₂₇	—	2	1	no	—
59	C₄⋊C₄×C₁₂₇	—	2	2	no	—
60	C₄⋊C₄×C₁₂₇	D₄	1	1	yes	characteristic
61	C₄⋊C₄×C₁₂₇	D₄	1	2	yes	—
62	C₄⋊C₄×C₁₂₇	C₂³	1	1	yes	characteristic
63	C₄⋊C₄×C₁₂₇	C₂³	1	2	yes	—
64	C₄⋊C₄×C₁₂₇	C₂³	1	1	yes	characteristic
65	C₂²⋊C₄×C₁₂₇	—	2	2	no	—
66	C₂²⋊C₄×C₁₂₇	—	2	1	no	—
67	C₂²⋊C₄×C₁₂₇	—	2	2	no	—
68	C₂²⋊C₄×C₁₂₇	—	2	2	no	—
69	C₂²⋊C₄×C₁₂₇	—	2	2	no	—
70	C₄²×C₁₂₇	—	2	1	no	—
71	C₄²×C₁₂₇	C₂³	1	2	yes	—
72	C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
73	C₂³×C₁₂₇	—	2	1	no	—
74	C₂³×C₁₂₇	—	2	2	no	—
75	Q₈×C₁₂₇	—	2	2	no	—
76	Q₈×C₁₂₇	—	2	1	no	—
77	Q₈×C₁₂₇	—	2	1	no	—
78	Q₈×C₁₂₇	—	8	1	no	—
79	D₄×C₁₂₇	—	2	2	no	—
80	D₄×C₁₂₇	—	2	1	no	—
81	D₄×C₁₂₇	—	2	2	no	—
82	D₄×C₁₂₇	—	4	2	no	—
83	D₄×C₁₂₇	—	8	1	no	—
84	D₄×C₁₂₇	#41	1	2	yes	—
85	C₂×C₄×C₁₂₇	—	2	1	no	—
86	C₂×C₄×C₁₂₇	—	2	1	no	—
87	C₂×C₄×C₁₂₇	—	2	1	no	—
88	C₂×C₄×C₁₂₇	—	2	2	no	—
89	C₂×C₄×C₁₂₇	—	4	2	no	—
90	C₂×C₄×C₁₂₇	—	8	2	no	—
91	C₂×C₄×C₁₂₇	—	8	2	no	—
92	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
93	C₂×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
94	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
95	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
96	C₂×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
97	C₈×C₁₂₇	—	2	2	no	—
98	C₈×C₁₂₇	—	2	1	no	—
99	C₈×C₁₂₇	—	2	1	no	—
100	C₂²×C₁₂₇	—	2	2	no	—
101	C₂²×C₁₂₇	—	4	1	no	—
102	C₂²×C₁₂₇	—	4	1	no	—
103	C₂²×C₁₂₇	—	4	2	no	—
104	C₂²×C₁₂₇	—	4	2	no	—
105	C₂²×C₁₂₇	—	4	2	no	—
106	C₂²×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
107	C₄×C₁₂₇	—	2	2	no	—
108	C₄×C₁₂₇	—	2	2	no	—
109	C₄×C₁₂₇	—	2	1	no	—
110	C₄×C₁₂₇	—	4	1	no	—
111	C₄×C₁₂₇	—	4	1	no	—
112	C₄×C₁₂₇	—	4	1	no	—
113	C₄×C₁₂₇	—	4	2	no	—
114	C₄×C₁₂₇	#135	1	2	yes	—
115	C₄×C₁₂₇	C₂×#41	1	1	yes	characteristic
116	C₄×C₁₂₇	2₋¹⁺⁴	1	1	yes	characteristic
117	C₂×C₁₂₇	—	4	2	no	—
118	C₂×C₁₂₇	—	8	2	no	—
119	C₂×C₁₂₇	#548	1	1	yes	characteristic
120	C₂×C₁₂₇	C₂×#135	1	1	yes	characteristic
121	C₂×C₁₂₇	#576	1	1	yes	characteristic
122	#3361	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
123	C₁₂₇	#3361	1	1	yes	characteristic, direct product factor, Sylow
124	#559	C₂×C₁₂₇	1	1	yes	characteristic, complemented
125	#554	C₂×C₁₂₇	1	1	yes	characteristic, complemented
126	#488	C₂×C₁₂₇	1	2	yes	complemented
127	#486	C₂×C₁₂₇	1	1	yes	characteristic, complemented
128	#478	C₂×C₁₂₇	1	2	yes	complemented
129	#477	C₂×C₁₂₇	1	1	yes	characteristic, complemented
130	#463	C₂×C₁₂₇	1	2	yes	complemented
131	#459	C₂×C₁₂₇	1	1	yes	characteristic
132	C₈×Q₈	C₂×C₁₂₇	1	1	yes	characteristic, complemented
133	C₄×SD₁₆	C₂×C₁₂₇	1	1	yes	characteristic, complemented
134	C₄×D₈	C₂×C₁₂₇	1	2	yes	complemented
135	C₂×SD₁₆	C₂²×C₁₂₇	1	1	yes	characteristic
136	C₂×D₈	C₂²×C₁₂₇	1	2	yes	—
137	C₄⋊Q₈	C₂²×C₁₂₇	1	1	yes	characteristic
138	#127	C₂²×C₁₂₇	1	1	yes	characteristic
139	#125	C₂²×C₁₂₇	1	2	yes	—
140	#124	—	2	1	no	—
141	#124	C₂²×C₁₂₇	1	2	yes	—
142	#123	—	2	2	no	—
143	C₂²⋊Q₈	—	2	1	no	—
144	C₂²⋊Q₈	—	2	2	no	—
145	C₄×Q₈	C₂²×C₁₂₇	1	1	yes	characteristic
146	C₄×Q₈	C₂²×C₁₂₇	1	1	yes	characteristic
147	C₄×D₄	—	2	1	no	—
148	C₄×D₄	C₂²×C₁₂₇	1	2	yes	—
149	C₄×D₄	C₂²×C₁₂₇	1	1	yes	characteristic
150	C₄×D₄	C₂²×C₁₂₇	1	2	yes	—
151	C₂×C₄⋊C₄	—	2	1	no	—
152	#107	C₂²×C₁₂₇	1	2	yes	—
153	#106	C₂²×C₁₂₇	1	1	yes	characteristic
154	C₄⋊C₈	C₂²×C₁₂₇	1	1	yes	characteristic
155	C₄⋊C₈	C₂²×C₁₂₇	1	2	yes	—
156	#103	C₂²×C₁₂₇	1	1	yes	characteristic
157	#103	C₂²×C₁₂₇	1	2	yes	—
158	#102	C₂²×C₁₂₇	1	1	yes	characteristic
159	#102	C₂²×C₁₂₇	1	2	yes	—
160	#102	C₂²×C₁₂₇	1	2	yes	—
161	#102	C₂²×C₁₂₇	1	2	yes	—
162	C₄×C₈	C₂²×C₁₂₇	1	2	yes	—
163	C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
164	C₂×Q₈	—	2	1	no	—
165	C₂×Q₈	D₄×C₁₂₇	1	1	yes	characteristic
166	C₂×Q₈	C₂³×C₁₂₇	1	1	yes	characteristic
167	C₂×D₄	—	2	1	no	—
168	C₂×D₄	—	2	1	no	—
169	C₂×D₄	C₂³×C₁₂₇	1	2	yes	—
170	C₂×D₄	C₂³×C₁₂₇	1	1	yes	characteristic
171	C₂²×C₄	—	2	1	no	—
172	C₂²×C₄	—	2	2	no	—
173	C₂²×C₄	—	2	2	no	—
174	SD₁₆	—	2	2	no	—
175	D₈	—	2	4	no	—
176	C₂×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
177	C₂×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
178	C₂×C₈	C₂³×C₁₂₇	1	2	yes	—
179	C₄⋊C₄	—	2	2	no	—
180	C₄⋊C₄	—	2	1	no	—
181	C₄⋊C₄	—	2	1	no	—
182	C₄⋊C₄	—	2	2	no	—
183	C₄⋊C₄	D₄×C₁₂₇	1	1	yes	characteristic
184	C₄⋊C₄	D₄×C₁₂₇	1	2	yes	—
185	C₄⋊C₄	C₂³×C₁₂₇	1	1	yes	characteristic
186	C₄⋊C₄	C₂³×C₁₂₇	1	2	yes	—
187	C₄⋊C₄	C₂³×C₁₂₇	1	1	yes	characteristic
188	C₂²⋊C₄	—	2	2	no	—
189	C₂²⋊C₄	—	2	1	no	—
190	C₂²⋊C₄	—	2	2	no	—
191	C₂²⋊C₄	—	2	2	no	—
192	C₂²⋊C₄	—	2	2	no	—
193	C₄²	—	2	1	no	—
194	C₄²	C₂³×C₁₂₇	1	2	yes	—
195	C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
196	C₂³	—	2	1	no	—
197	C₂³	—	2	2	no	—
198	Q₈	—	2	2	no	—
199	Q₈	—	2	1	no	—
200	Q₈	—	2	1	no	—
201	Q₈	—	8	1	no	—
202	D₄	—	2	2	no	—
203	D₄	—	2	1	no	—
204	D₄	—	2	2	no	—
205	D₄	—	4	2	no	—
206	D₄	—	8	1	no	—
207	D₄	#41×C₁₂₇	1	2	yes	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	4	2	no	—
213	C₂×C₄	—	8	2	no	—
214	C₂×C₄	—	8	2	no	—
215	C₂×C₄	C₂×D₄×C₁₂₇	1	2	yes	—
216	C₂×C₄	C₂×D₄×C₁₂₇	1	2	yes	—
217	C₂×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
218	C₂×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
219	C₂×C₄	C₂⁴×C₁₂₇	1	1	yes	characteristic
220	C₈	—	2	2	no	—
221	C₈	—	2	1	no	—
222	C₈	—	2	1	no	—
223	C₂²	—	2	2	no	—
224	C₂²	—	4	1	no	—
225	C₂²	—	4	1	no	—
226	C₂²	—	4	2	no	—
227	C₂²	—	4	2	no	—
228	C₂²	—	4	2	no	—
229	C₂²	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
230	C₄	—	2	2	no	—
231	C₄	—	2	2	no	—
232	C₄	—	2	1	no	—
233	C₄	—	4	1	no	—
234	C₄	—	4	1	no	—
235	C₄	—	4	1	no	—
236	C₄	—	4	2	no	—
237	C₄	#135×C₁₂₇	1	2	yes	—
238	C₄	C₂×#41×C₁₂₇	1	1	yes	characteristic
239	C₄	2₋¹⁺⁴×C₁₂₇	1	1	yes	characteristic
240	C₂	—	4	2	no	—
241	C₂	—	8	2	no	—
242	C₂	#548×C₁₂₇	1	1	yes	characteristic
243	C₂	C₂×#135×C₁₂₇	1	1	yes	characteristic
244	C₂	#576×C₁₂₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₂₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₂₇
prime radical	#459×C₁₂₇	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₂₇ ◁ C₄×Q₈×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. #559×C₁₂₇	C₂	semidirect
2. #554×C₁₂₇	C₂	semidirect
3. #488×C₁₂₇	C₂	semidirect
4. #486×C₁₂₇	C₂	semidirect
5. #478×C₁₂₇	C₂	semidirect
6. #477×C₁₂₇	C₂	semidirect
7. #463×C₁₂₇	C₂	semidirect
9. C₈×Q₈×C₁₂₇	C₂	semidirect
10. C₄×SD₁₆×C₁₂₇	C₂	semidirect
11. C₄×D₈×C₁₂₇	C₂	semidirect
122. #3361	C₁₂₇	direct (N × H)
123. C₁₂₇	#3361	direct (N × H)
124. #559	C₂×C₁₂₇	semidirect
125. #554	C₂×C₁₂₇	semidirect
126. #488	C₂×C₁₂₇	semidirect
127. #486	C₂×C₁₂₇	semidirect
128. #478	C₂×C₁₂₇	semidirect
129. #477	C₂×C₁₂₇	semidirect
130. #463	C₂×C₁₂₇	semidirect
132. C₈×Q₈	C₂×C₁₂₇	semidirect
133. C₄×SD₁₆	C₂×C₁₂₇	semidirect
134. C₄×D₈	C₂×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
#559×C₁₂₇	2
#554×C₁₂₇	2
#488×C₁₂₇	2
#486×C₁₂₇	2
#478×C₁₂₇	2
#477×C₁₂₇	2
#463×C₁₂₇	2
#459×C₁₂₇	2
C₈×Q₈×C₁₂₇	2
C₄×SD₁₆×C₁₂₇	2
C₄×D₈×C₁₂₇	2
#3361	127

Subgroups — C127×#3361 #30636313

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#3361 #30636313