Subgroups — C₈×#123×C₁₂₇ #44585233

Order 32512.

1034 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×D₄×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×C₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#123×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₁₂₇×#1923	C₂	1	2	yes	maximal, complemented
5	C₁₂₇×#1916	C₂	1	2	yes	maximal, complemented
6	C₁₂₇×#1869	C₂	1	1	yes	maximal, characteristic
7	C₁₂₇×#1850	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₂₇×#1817	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈×C₄⋊C₄×C₁₂₇	C₂	1	2	yes	maximal, complemented
10	C₈×C₂²⋊C₄×C₁₂₇	C₂	1	2	yes	maximal, complemented
11	C₈×C₂²⋊C₄×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂³×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
13	C₂³×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
14	C₂³×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
15	C₂×#123×C₁₂₇	C₄	1	1	yes	characteristic
16	C₂×C₄×D₄×C₁₂₇	C₂²	1	1	yes	characteristic
17	C₂²×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
18	C₈×D₄×C₁₂₇	—	2	4	no	—
19	C₂×C₄⋊C₈×C₁₂₇	C₂²	1	2	yes	—
20	C₂×C₄⋊C₈×C₁₂₇	C₂²	1	1	yes	characteristic
21	C₂×C₂²⋊C₈×C₁₂₇	C₂²	1	1	yes	characteristic
22	C₂×C₂²⋊C₈×C₁₂₇	C₂²	1	1	yes	characteristic
23	C₂×C₂²⋊C₈×C₁₂₇	C₂²	1	2	yes	—
24	C₂×C₂²⋊C₈×C₁₂₇	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₁₂₇	—	2	4	no	—
26	C₂×C₄×C₈×C₁₂₇	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₁₂₇	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
29	C₂×C₄×C₈×C₁₂₇	C₂²	1	2	yes	—
30	#388×C₁₂₇	C₄	1	2	yes	—
31	#387×C₁₂₇	C₄	1	2	yes	—
32	#386×C₁₂₇	C₄	1	1	yes	characteristic
33	#385×C₁₂₇	C₄	1	1	yes	characteristic
34	#381×C₁₂₇	C₄	1	1	yes	characteristic
35	C₄×C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
36	C₄×C₂²⋊C₄×C₁₂₇	C₂²	1	1	yes	characteristic
37	C₄×C₂²⋊C₄×C₁₂₇	C₂²	1	2	yes	—
38	#336×C₁₂₇	C₂²	1	2	yes	—
39	#336×C₁₂₇	C₂²	1	2	yes	—
40	#336×C₁₂₇	C₂²	1	2	yes	—
41	#336×C₁₂₇	C₂²	1	2	yes	—
42	#336×C₁₂₇	C₂²	1	2	yes	—
43	C₂²×D₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
44	C₂³×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
45	C₂³×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
46	C₂³×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
47	C₂³×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
48	C₂²×C₈×C₁₂₇	—	2	2	no	—
49	C₂²×C₈×C₁₂₇	—	2	4	no	—
50	C₂²×C₈×C₁₂₇	—	2	4	no	—
51	C₂²×C₈×C₁₂₇	—	2	2	no	—
52	C₂²×C₈×C₁₂₇	—	2	4	no	—
53	C₂²×C₈×C₁₂₇	D₄	1	4	yes	—
54	C₂²×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
55	C₂²×C₈×C₁₂₇	C₂³	1	2	yes	—
56	C₂²×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
57	C₂²×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
58	C₂²×C₈×C₁₂₇	C₂³	1	2	yes	—
59	C₂²×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
60	#123×C₁₂₇	C₈	1	8	yes	direct product factor
61	C₄×D₄×C₁₂₇	—	2	2	no	—
62	C₄×D₄×C₁₂₇	—	2	2	no	—
63	C₂×C₄⋊C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
64	C₂×C₄⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
65	C₂×C₄⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
66	C₂×C₂²⋊C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
67	C₂×C₂²⋊C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
68	C₂×C₂²⋊C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
69	C₂×C₂²⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
70	C₂×C₂²⋊C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
71	C₂×C₂²⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
72	C₂×C₄²×C₁₂₇	—	2	2	no	—
73	C₂×C₄²×C₁₂₇	—	2	2	no	—
74	C₂×C₄²×C₁₂₇	C₂×C₄	1	2	yes	—
75	C₂×C₄²×C₁₂₇	C₂³	1	2	yes	—
76	C₂×C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
77	C₂×C₄²×C₁₂₇	C₂³	1	2	yes	—
78	C₄⋊C₈×C₁₂₇	—	2	2	no	—
79	C₄⋊C₈×C₁₂₇	—	2	4	no	—
80	C₂²⋊C₈×C₁₂₇	—	2	2	no	—
81	C₂²⋊C₈×C₁₂₇	—	2	2	no	—
82	C₂²⋊C₈×C₁₂₇	—	2	2	no	—
83	C₂²⋊C₈×C₁₂₇	—	2	4	no	—
84	C₄×C₈×C₁₂₇	—	2	4	no	—
85	C₄×C₈×C₁₂₇	—	2	4	no	—
86	C₄×C₈×C₁₂₇	—	2	2	no	—
87	C₄×C₈×C₁₂₇	—	2	4	no	—
88	#95×C₁₂₇	C₂×C₄	1	2	yes	—
89	#95×C₁₂₇	C₂×C₄	1	2	yes	—
90	#95×C₁₂₇	C₂×C₄	1	2	yes	—
91	#95×C₁₂₇	C₂×C₄	1	2	yes	—
92	#95×C₁₂₇	C₂×C₄	1	2	yes	—
93	C₂⁴×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
94	C₂⁴×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
95	C₂×D₄×C₁₂₇	—	2	2	no	—
96	C₂×D₄×C₁₂₇	—	2	2	no	—
97	C₂×D₄×C₁₂₇	C₂×C₈	1	4	yes	—
98	C₂²×C₄×C₁₂₇	—	2	2	no	—
99	C₂²×C₄×C₁₂₇	—	2	2	no	—
100	C₂²×C₄×C₁₂₇	—	2	2	no	—
101	C₂²×C₄×C₁₂₇	—	2	4	no	—
102	C₂²×C₄×C₁₂₇	—	2	2	no	—
103	C₂²×C₄×C₁₂₇	—	2	2	no	—
104	C₂²×C₄×C₁₂₇	—	2	1	no	—
105	C₂²×C₄×C₁₂₇	—	2	2	no	—
106	C₂²×C₄×C₁₂₇	—	2	1	no	—
107	C₂²×C₄×C₁₂₇	—	2	2	no	—
108	C₂²×C₄×C₁₂₇	C₂×C₈	1	4	yes	complemented
109	C₂²×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
110	C₂²×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
111	C₂²×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
112	C₂²×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
113	C₂²×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
114	C₂²×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
115	C₂²×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
116	C₂²×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
117	C₂²×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
118	C₂²×C₄×C₁₂₇	#41	1	2	yes	—
119	C₂²×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
120	C₂×C₈×C₁₂₇	—	2	8	no	—
121	C₂×C₈×C₁₂₇	—	2	2	no	—
122	C₂×C₈×C₁₂₇	—	2	2	no	—
123	C₂×C₈×C₁₂₇	—	2	2	no	—
124	C₂×C₈×C₁₂₇	—	2	8	no	—
125	C₂×C₈×C₁₂₇	—	2	1	no	—
126	C₂×C₈×C₁₂₇	—	2	1	no	—
127	C₂×C₈×C₁₂₇	—	2	2	no	—
128	C₂×C₈×C₁₂₇	—	2	2	no	—
129	C₂×C₈×C₁₂₇	—	2	2	no	—
130	C₂×C₈×C₁₂₇	—	2	2	no	—
131	C₂×C₈×C₁₂₇	—	2	2	no	—
132	C₂×C₈×C₁₂₇	—	4	4	no	—
133	C₂×C₈×C₁₂₇	C₂×D₄	1	2	yes	—
134	C₂×C₈×C₁₂₇	C₂×D₄	1	2	yes	—
135	C₂×C₈×C₁₂₇	#41	1	4	yes	—
136	C₂×C₈×C₁₂₇	#41	1	4	yes	—
137	C₄⋊C₄×C₁₂₇	—	2	2	no	—
138	C₄⋊C₄×C₁₂₇	—	2	4	no	—
139	C₄⋊C₄×C₁₂₇	C₂×C₈	1	8	yes	complemented
140	C₂²⋊C₄×C₁₂₇	—	2	2	no	—
141	C₂²⋊C₄×C₁₂₇	—	2	2	no	—
142	C₂²⋊C₄×C₁₂₇	—	2	2	no	—
143	C₂²⋊C₄×C₁₂₇	—	2	4	no	—
144	C₂²⋊C₄×C₁₂₇	C₂×C₈	1	4	yes	complemented
145	C₂²⋊C₄×C₁₂₇	C₂×C₈	1	8	yes	complemented
146	C₄²×C₁₂₇	—	2	4	no	—
147	C₄²×C₁₂₇	—	2	2	no	—
148	C₄²×C₁₂₇	—	2	1	no	—
149	C₄²×C₁₂₇	—	2	2	no	—
150	C₄²×C₁₂₇	—	2	2	no	—
151	C₄²×C₁₂₇	—	2	2	no	—
152	C₄²×C₁₂₇	—	2	1	no	—
153	C₂³×C₁₂₇	—	2	2	no	—
154	C₂³×C₁₂₇	—	2	2	no	—
155	C₂³×C₁₂₇	—	2	1	no	—
156	C₂³×C₁₂₇	—	2	2	no	—
157	C₂³×C₁₂₇	—	2	2	no	—
158	C₂³×C₁₂₇	—	2	1	no	—
159	C₂³×C₁₂₇	C₄×D₄	1	2	yes	—
160	C₂³×C₁₂₇	C₂²×C₈	1	2	yes	—
161	C₂³×C₁₂₇	C₂²×C₈	1	2	yes	—
162	C₂³×C₁₂₇	#131	1	2	yes	—
163	C₂³×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
164	D₄×C₁₂₇	—	2	8	no	—
165	C₂×C₄×C₁₂₇	—	2	8	no	—
166	C₂×C₄×C₁₂₇	—	2	2	no	—
167	C₂×C₄×C₁₂₇	—	2	2	no	—
168	C₂×C₄×C₁₂₇	—	2	1	no	—
169	C₂×C₄×C₁₂₇	—	2	2	no	—
170	C₂×C₄×C₁₂₇	—	2	2	no	—
171	C₂×C₄×C₁₂₇	—	2	2	no	—
172	C₂×C₄×C₁₂₇	—	2	2	no	—
173	C₂×C₄×C₁₂₇	—	2	2	no	—
174	C₂×C₄×C₁₂₇	—	2	1	no	—
175	C₂×C₄×C₁₂₇	—	2	8	no	—
176	C₂×C₄×C₁₂₇	—	2	2	no	—
177	C₂×C₄×C₁₂₇	—	2	2	no	—
178	C₂×C₄×C₁₂₇	—	2	4	no	—
179	C₂×C₄×C₁₂₇	—	2	1	no	—
180	C₂×C₄×C₁₂₇	—	2	1	no	—
181	C₂×C₄×C₁₂₇	—	2	2	no	—
182	C₂×C₄×C₁₂₇	—	2	2	no	—
183	C₂×C₄×C₁₂₇	—	2	2	no	—
184	C₂×C₄×C₁₂₇	—	2	2	no	—
185	C₂×C₄×C₁₂₇	—	2	2	no	—
186	C₂×C₄×C₁₂₇	—	4	1	no	—
187	C₂×C₄×C₁₂₇	—	4	1	no	—
188	C₂×C₄×C₁₂₇	—	4	2	no	—
189	C₂×C₄×C₁₂₇	C₄×D₄	1	2	yes	—
190	C₂×C₄×C₁₂₇	C₂²×C₈	1	4	yes	—
191	C₂×C₄×C₁₂₇	C₂²×C₈	1	4	yes	—
192	C₂×C₄×C₁₂₇	C₂²×C₈	1	2	yes	—
193	C₂×C₄×C₁₂₇	#131	1	4	yes	—
194	C₂×C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
195	C₂×C₄×C₁₂₇	C₂×#41	1	2	yes	—
196	C₂×C₄×C₁₂₇	C₂×#41	1	2	yes	—
197	C₂×C₄×C₁₂₇	C₂×#41	1	1	yes	characteristic
198	C₈×C₁₂₇	—	2	2	no	—
199	C₈×C₁₂₇	—	2	4	no	—
200	C₈×C₁₂₇	—	4	1	no	—
201	C₈×C₁₂₇	—	4	1	no	—
202	C₈×C₁₂₇	—	4	2	no	—
203	C₈×C₁₂₇	#123	1	4	yes	direct product factor
204	C₂²×C₁₂₇	—	2	2	no	—
205	C₂²×C₁₂₇	—	2	4	no	—
206	C₂²×C₁₂₇	—	2	4	no	—
207	C₂²×C₁₂₇	—	2	2	no	—
208	C₂²×C₁₂₇	—	2	2	no	—
209	C₂²×C₁₂₇	—	2	4	no	—
210	C₂²×C₁₂₇	—	4	1	no	—
211	C₂²×C₁₂₇	—	4	2	no	—
212	C₂²×C₁₂₇	—	4	1	no	—
213	C₂²×C₁₂₇	C₈×D₄	1	4	yes	—
214	C₂²×C₁₂₇	C₂×C₄×D₄	1	1	yes	characteristic
215	C₂²×C₁₂₇	C₄×#41	1	2	yes	—
216	C₂²×C₁₂₇	C₂³×C₈	1	1	yes	characteristic
217	C₂²×C₁₂₇	C₂×#131	1	1	yes	characteristic
218	C₂²×C₁₂₇	C₂×#131	1	2	yes	—
219	C₄×C₁₂₇	—	2	4	no	—
220	C₄×C₁₂₇	—	2	2	no	—
221	C₄×C₁₂₇	—	2	4	no	—
222	C₄×C₁₂₇	—	2	4	no	—
223	C₄×C₁₂₇	—	2	2	no	—
224	C₄×C₁₂₇	—	4	1	no	—
225	C₄×C₁₂₇	—	4	2	no	—
226	C₄×C₁₂₇	—	4	1	no	—
227	C₄×C₁₂₇	C₂×#123	1	1	yes	characteristic
228	C₄×C₁₂₇	#525	1	1	yes	characteristic
229	C₄×C₁₂₇	#529	1	2	yes	—
230	C₈×#123	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
231	C₂×C₁₂₇	—	2	4	no	—
232	C₂×C₁₂₇	—	4	2	no	—
233	C₂×C₁₂₇	C₄×#123	1	1	yes	characteristic
234	C₂×C₁₂₇	C₂×C₈×D₄	1	1	yes	characteristic
235	C₂×C₁₂₇	#2929	1	1	yes	characteristic
236	C₂×C₁₂₇	C₈×#41	1	2	yes	—
237	C₂×C₁₂₇	#2966	1	2	yes	—
238	C₂×C₈×D₄	C₂×C₁₂₇	1	1	yes	characteristic
239	C₂²×C₄×C₈	C₂×C₁₂₇	1	1	yes	characteristic, complemented
240	C₄×#123	C₂×C₁₂₇	1	1	yes	characteristic
241	#1923	C₂×C₁₂₇	1	2	yes	complemented
242	#1916	C₂×C₁₂₇	1	2	yes	complemented
243	#1869	C₂×C₁₂₇	1	1	yes	characteristic
244	#1850	C₂×C₁₂₇	1	1	yes	characteristic, complemented
245	#1817	C₂×C₁₂₇	1	1	yes	characteristic, complemented
246	C₈×C₄⋊C₄	C₂×C₁₂₇	1	2	yes	complemented
247	C₈×C₂²⋊C₄	C₂×C₁₂₇	1	2	yes	complemented
248	C₈×C₂²⋊C₄	C₂×C₁₂₇	1	1	yes	characteristic, complemented
249	C₁₂₇	C₈×#123	1	1	yes	characteristic, direct product factor, Sylow
250	C₂³×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
251	C₂³×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
252	C₂³×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
253	C₂×#123	C₄×C₁₂₇	1	1	yes	characteristic
254	C₂×C₄×D₄	C₂²×C₁₂₇	1	1	yes	characteristic
255	C₂²×C₄²	C₂²×C₁₂₇	1	1	yes	characteristic
256	C₈×D₄	—	2	4	no	—
257	C₂×C₄⋊C₈	C₂²×C₁₂₇	1	2	yes	—
258	C₂×C₄⋊C₈	C₂²×C₁₂₇	1	1	yes	characteristic
259	C₂×C₂²⋊C₈	C₂²×C₁₂₇	1	1	yes	characteristic
260	C₂×C₂²⋊C₈	C₂²×C₁₂₇	1	1	yes	characteristic
261	C₂×C₂²⋊C₈	C₂²×C₁₂₇	1	2	yes	—
262	C₂×C₂²⋊C₈	C₂²×C₁₂₇	1	1	yes	characteristic
263	C₂×C₄×C₈	—	2	4	no	—
264	C₂×C₄×C₈	C₂²×C₁₂₇	1	2	yes	—
265	C₂×C₄×C₈	C₂²×C₁₂₇	1	2	yes	—
266	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
267	C₂×C₄×C₈	C₂²×C₁₂₇	1	2	yes	—
268	#388	C₄×C₁₂₇	1	2	yes	—
269	#387	C₄×C₁₂₇	1	2	yes	—
270	#386	C₄×C₁₂₇	1	1	yes	characteristic
271	#385	C₄×C₁₂₇	1	1	yes	characteristic
272	#381	C₄×C₁₂₇	1	1	yes	characteristic
273	C₄×C₄⋊C₄	C₂²×C₁₂₇	1	2	yes	—
274	C₄×C₂²⋊C₄	C₂²×C₁₂₇	1	1	yes	characteristic
275	C₄×C₂²⋊C₄	C₂²×C₁₂₇	1	2	yes	—
276	#336	C₂²×C₁₂₇	1	2	yes	—
277	#336	C₂²×C₁₂₇	1	2	yes	—
278	#336	C₂²×C₁₂₇	1	2	yes	—
279	#336	C₂²×C₁₂₇	1	2	yes	—
280	#336	C₂²×C₁₂₇	1	2	yes	—
281	C₂²×D₄	C₂×C₄×C₁₂₇	1	1	yes	characteristic
282	C₂³×C₄	C₂×C₄×C₁₂₇	1	1	yes	characteristic
283	C₂³×C₄	C₂×C₄×C₁₂₇	1	1	yes	characteristic
284	C₂³×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
285	C₂³×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
286	C₂²×C₈	—	2	2	no	—
287	C₂²×C₈	—	2	4	no	—
288	C₂²×C₈	—	2	4	no	—
289	C₂²×C₈	—	2	2	no	—
290	C₂²×C₈	—	2	4	no	—
291	C₂²×C₈	D₄×C₁₂₇	1	4	yes	—
292	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
293	C₂²×C₈	C₂³×C₁₂₇	1	2	yes	—
294	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
295	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
296	C₂²×C₈	C₂³×C₁₂₇	1	2	yes	—
297	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
298	#123	C₈×C₁₂₇	1	8	yes	direct product factor
299	C₄×D₄	—	2	2	no	—
300	C₄×D₄	—	2	2	no	—
301	C₂×C₄⋊C₄	C₂×C₄×C₁₂₇	1	1	yes	characteristic
302	C₂×C₄⋊C₄	C₂×C₄×C₁₂₇	1	2	yes	—
303	C₂×C₄⋊C₄	C₂×C₄×C₁₂₇	1	2	yes	—
304	C₂×C₂²⋊C₄	C₂×C₄×C₁₂₇	1	1	yes	characteristic
305	C₂×C₂²⋊C₄	C₂×C₄×C₁₂₇	1	1	yes	characteristic
306	C₂×C₂²⋊C₄	C₂×C₄×C₁₂₇	1	1	yes	characteristic
307	C₂×C₂²⋊C₄	C₂×C₄×C₁₂₇	1	2	yes	—
308	C₂×C₂²⋊C₄	C₂×C₄×C₁₂₇	1	1	yes	characteristic
309	C₂×C₂²⋊C₄	C₂×C₄×C₁₂₇	1	2	yes	—
310	C₂×C₄²	—	2	2	no	—
311	C₂×C₄²	—	2	2	no	—
312	C₂×C₄²	C₂×C₄×C₁₂₇	1	2	yes	—
313	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
314	C₂×C₄²	C₂³×C₁₂₇	1	2	yes	—
315	C₂×C₄²	C₂³×C₁₂₇	1	2	yes	—
316	C₄⋊C₈	—	2	2	no	—
317	C₄⋊C₈	—	2	4	no	—
318	C₂²⋊C₈	—	2	2	no	—
319	C₂²⋊C₈	—	2	2	no	—
320	C₂²⋊C₈	—	2	2	no	—
321	C₂²⋊C₈	—	2	4	no	—
322	C₄×C₈	—	2	4	no	—
323	C₄×C₈	—	2	4	no	—
324	C₄×C₈	—	2	2	no	—
325	C₄×C₈	—	2	4	no	—
326	#95	C₂×C₄×C₁₂₇	1	2	yes	—
327	#95	C₂×C₄×C₁₂₇	1	2	yes	—
328	#95	C₂×C₄×C₁₂₇	1	2	yes	—
329	#95	C₂×C₄×C₁₂₇	1	2	yes	—
330	#95	C₂×C₄×C₁₂₇	1	2	yes	—
331	C₂⁴	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
332	C₂⁴	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
333	C₂×D₄	—	2	2	no	—
334	C₂×D₄	—	2	2	no	—
335	C₂×D₄	C₂×C₈×C₁₂₇	1	4	yes	—
336	C₂²×C₄	—	2	2	no	—
337	C₂²×C₄	—	2	2	no	—
338	C₂²×C₄	—	2	2	no	—
339	C₂²×C₄	—	2	4	no	—
340	C₂²×C₄	—	2	2	no	—
341	C₂²×C₄	—	2	2	no	—
342	C₂²×C₄	—	2	1	no	—
343	C₂²×C₄	—	2	2	no	—
344	C₂²×C₄	—	2	1	no	—
345	C₂²×C₄	—	2	2	no	—
346	C₂²×C₄	C₂×C₈×C₁₂₇	1	4	yes	complemented
347	C₂²×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
348	C₂²×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
349	C₂²×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
350	C₂²×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
351	C₂²×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
352	C₂²×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
353	C₂²×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
354	C₂²×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
355	C₂²×C₄	C₂×D₄×C₁₂₇	1	2	yes	—
356	C₂²×C₄	#41×C₁₂₇	1	2	yes	—
357	C₂²×C₄	C₂⁴×C₁₂₇	1	1	yes	characteristic
358	C₂×C₈	—	2	8	no	—
359	C₂×C₈	—	2	2	no	—
360	C₂×C₈	—	2	2	no	—
361	C₂×C₈	—	2	2	no	—
362	C₂×C₈	—	2	8	no	—
363	C₂×C₈	—	2	1	no	—
364	C₂×C₈	—	2	1	no	—
365	C₂×C₈	—	2	2	no	—
366	C₂×C₈	—	2	2	no	—
367	C₂×C₈	—	2	2	no	—
368	C₂×C₈	—	2	2	no	—
369	C₂×C₈	—	2	2	no	—
370	C₂×C₈	—	4	4	no	—
371	C₂×C₈	C₂×D₄×C₁₂₇	1	2	yes	—
372	C₂×C₈	C₂×D₄×C₁₂₇	1	2	yes	—
373	C₂×C₈	#41×C₁₂₇	1	4	yes	—
374	C₂×C₈	#41×C₁₂₇	1	4	yes	—
375	C₄⋊C₄	—	2	2	no	—
376	C₄⋊C₄	—	2	4	no	—
377	C₄⋊C₄	C₂×C₈×C₁₂₇	1	8	yes	complemented
378	C₂²⋊C₄	—	2	2	no	—
379	C₂²⋊C₄	—	2	2	no	—
380	C₂²⋊C₄	—	2	2	no	—
381	C₂²⋊C₄	—	2	4	no	—
382	C₂²⋊C₄	C₂×C₈×C₁₂₇	1	4	yes	complemented
383	C₂²⋊C₄	C₂×C₈×C₁₂₇	1	8	yes	complemented
384	C₄²	—	2	4	no	—
385	C₄²	—	2	2	no	—
386	C₄²	—	2	1	no	—
387	C₄²	—	2	2	no	—
388	C₄²	—	2	2	no	—
389	C₄²	—	2	2	no	—
390	C₄²	—	2	1	no	—
391	C₂³	—	2	2	no	—
392	C₂³	—	2	2	no	—
393	C₂³	—	2	1	no	—
394	C₂³	—	2	2	no	—
395	C₂³	—	2	2	no	—
396	C₂³	—	2	1	no	—
397	C₂³	C₄×D₄×C₁₂₇	1	2	yes	—
398	C₂³	C₂²×C₈×C₁₂₇	1	2	yes	—
399	C₂³	C₂²×C₈×C₁₂₇	1	2	yes	—
400	C₂³	#131×C₁₂₇	1	2	yes	—
401	C₂³	C₂³×C₄×C₁₂₇	1	1	yes	characteristic
402	D₄	—	2	8	no	—
403	C₂×C₄	—	2	8	no	—
404	C₂×C₄	—	2	2	no	—
405	C₂×C₄	—	2	2	no	—
406	C₂×C₄	—	2	1	no	—
407	C₂×C₄	—	2	2	no	—
408	C₂×C₄	—	2	2	no	—
409	C₂×C₄	—	2	2	no	—
410	C₂×C₄	—	2	2	no	—
411	C₂×C₄	—	2	2	no	—
412	C₂×C₄	—	2	1	no	—
413	C₂×C₄	—	2	8	no	—
414	C₂×C₄	—	2	2	no	—
415	C₂×C₄	—	2	2	no	—
416	C₂×C₄	—	2	4	no	—
417	C₂×C₄	—	2	1	no	—
418	C₂×C₄	—	2	1	no	—
419	C₂×C₄	—	2	2	no	—
420	C₂×C₄	—	2	2	no	—
421	C₂×C₄	—	2	2	no	—
422	C₂×C₄	—	2	2	no	—
423	C₂×C₄	—	2	2	no	—
424	C₂×C₄	—	4	1	no	—
425	C₂×C₄	—	4	1	no	—
426	C₂×C₄	—	4	2	no	—
427	C₂×C₄	C₄×D₄×C₁₂₇	1	2	yes	—
428	C₂×C₄	C₂²×C₈×C₁₂₇	1	4	yes	—
429	C₂×C₄	C₂²×C₈×C₁₂₇	1	4	yes	—
430	C₂×C₄	C₂²×C₈×C₁₂₇	1	2	yes	—
431	C₂×C₄	#131×C₁₂₇	1	4	yes	—
432	C₂×C₄	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
433	C₂×C₄	C₂×#41×C₁₂₇	1	2	yes	—
434	C₂×C₄	C₂×#41×C₁₂₇	1	2	yes	—
435	C₂×C₄	C₂×#41×C₁₂₇	1	1	yes	characteristic
436	C₈	—	2	2	no	—
437	C₈	—	2	4	no	—
438	C₈	—	4	1	no	—
439	C₈	—	4	1	no	—
440	C₈	—	4	2	no	—
441	C₈	#123×C₁₂₇	1	4	yes	direct product factor
442	C₂²	—	2	2	no	—
443	C₂²	—	2	4	no	—
444	C₂²	—	2	4	no	—
445	C₂²	—	2	2	no	—
446	C₂²	—	2	2	no	—
447	C₂²	—	2	4	no	—
448	C₂²	—	4	1	no	—
449	C₂²	—	4	2	no	—
450	C₂²	—	4	1	no	—
451	C₂²	C₈×D₄×C₁₂₇	1	4	yes	—
452	C₂²	C₂×C₄×D₄×C₁₂₇	1	1	yes	characteristic
453	C₂²	C₄×#41×C₁₂₇	1	2	yes	—
454	C₂²	C₂³×C₈×C₁₂₇	1	1	yes	characteristic
455	C₂²	C₂×#131×C₁₂₇	1	1	yes	characteristic
456	C₂²	C₂×#131×C₁₂₇	1	2	yes	—
457	C₄	—	2	4	no	—
458	C₄	—	2	2	no	—
459	C₄	—	2	4	no	—
460	C₄	—	2	4	no	—
461	C₄	—	2	2	no	—
462	C₄	—	4	1	no	—
463	C₄	—	4	2	no	—
464	C₄	—	4	1	no	—
465	C₄	C₂×#123×C₁₂₇	1	1	yes	characteristic
466	C₄	#525×C₁₂₇	1	1	yes	characteristic
467	C₄	#529×C₁₂₇	1	2	yes	—
468	C₂	—	2	4	no	—
469	C₂	—	4	2	no	—
470	C₂	C₄×#123×C₁₂₇	1	1	yes	characteristic
471	C₂	C₂×C₈×D₄×C₁₂₇	1	1	yes	characteristic
472	C₂	C₁₂₇×#2929	1	1	yes	characteristic
473	C₂	C₈×#41×C₁₂₇	1	2	yes	—
474	C₂	C₁₂₇×#2966	1	2	yes	—

	subgroup	G/H
center	C₂²×C₈×C₁₂₇	C₂³
commutator	C₂²	C₂³×C₈×C₁₂₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₂₇
socle	C₂³×C₁₂₇	C₂³×C₄
prime radical	C₂²×D₄×C₁₂₇	C₂×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₈×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₄×C₈×C₁₂₇	C₂	semidirect
4. C₁₂₇×#1923	C₂	semidirect
5. C₁₂₇×#1916	C₂	semidirect
7. C₁₂₇×#1850	C₂	semidirect
8. C₁₂₇×#1817	C₂	semidirect
9. C₈×C₄⋊C₄×C₁₂₇	C₂	semidirect
10. C₈×C₂²⋊C₄×C₁₂₇	C₂	semidirect
11. C₈×C₂²⋊C₄×C₁₂₇	C₂	semidirect
60. #123×C₁₂₇	C₈	direct (N × H)
108. C₂²×C₄×C₁₂₇	C₂×C₈	semidirect
139. C₄⋊C₄×C₁₂₇	C₂×C₈	semidirect
144. C₂²⋊C₄×C₁₂₇	C₂×C₈	semidirect
145. C₂²⋊C₄×C₁₂₇	C₂×C₈	semidirect
203. C₈×C₁₂₇	#123	direct (N × H)
230. C₈×#123	C₁₂₇	direct (N × H)
239. C₂²×C₄×C₈	C₂×C₁₂₇	semidirect
241. #1923	C₂×C₁₂₇	semidirect
242. #1916	C₂×C₁₂₇	semidirect
244. #1850	C₂×C₁₂₇	semidirect
245. #1817	C₂×C₁₂₇	semidirect
246. C₈×C₄⋊C₄	C₂×C₁₂₇	semidirect
247. C₈×C₂²⋊C₄	C₂×C₁₂₇	semidirect
248. C₈×C₂²⋊C₄	C₂×C₁₂₇	semidirect
249. C₁₂₇	C₈×#123	direct (N × H)
298. #123	C₈×C₁₂₇	direct (N × H)
346. C₂²×C₄	C₂×C₈×C₁₂₇	semidirect
377. C₄⋊C₄	C₂×C₈×C₁₂₇	semidirect
382. C₂²⋊C₄	C₂×C₈×C₁₂₇	semidirect
383. C₂²⋊C₄	C₂×C₈×C₁₂₇	semidirect
441. C₈	#123×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
C₂×C₈×D₄×C₁₂₇	2
C₂²×C₄×C₈×C₁₂₇	2
C₄×#123×C₁₂₇	2
C₁₂₇×#1923	2
C₁₂₇×#1916	2
C₁₂₇×#1869	2
C₁₂₇×#1850	2
C₁₂₇×#1817	2
C₈×C₄⋊C₄×C₁₂₇	2
C₈×C₂²⋊C₄×C₁₂₇	2
C₈×#123	127

Subgroups — C8×#123×C127 #44585233

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×#123×C₁₂₇ #44585233