Subgroups — C₁₂₇×#21232 #44584905

Order 32512.

1470 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×Q₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#427×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#419×C₁₂₇	C₂	1	2	yes	maximal
4	C₂×#405×C₁₂₇	C₂	1	1	yes	maximal, characteristic
5	C₁₂₇×#2310	C₂	1	2	yes	maximal
6	C₁₂₇×#1971	C₂	1	4	yes	maximal, complemented
7	C₁₂₇×#1969	C₂	1	4	yes	maximal, complemented
8	C₂³×Q₈×C₁₂₇	C₂²	1	2	yes	—
9	C₂²×Q₁₆×C₁₂₇	—	2	4	no	—
10	C₂²×Q₁₆×C₁₂₇	—	2	4	no	—
11	C₂²×Q₁₆×C₁₂₇	C₄	1	8	yes	complemented
12	C₂²×Q₁₆×C₁₂₇	C₂²	1	2	yes	—
13	C₂²×Q₁₆×C₁₂₇	C₂²	1	2	yes	—
14	C₂²×M₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
15	C₂³×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
16	C₂×#117×C₁₂₇	C₂²	1	2	yes	—
17	C₂×#117×C₁₂₇	C₂²	1	1	yes	characteristic
18	#427×C₁₂₇	—	2	4	no	—
19	C₂×#107×C₁₂₇	C₂²	1	2	yes	—
20	C₂×#106×C₁₂₇	C₂²	1	2	yes	—
21	#419×C₁₂₇	—	2	8	no	—
22	C₂×#103×C₁₂₇	C₂²	1	8	yes	—
23	#405×C₁₂₇	—	2	4	no	—
24	C₂×#97×C₁₂₇	C₂²	1	2	yes	—
25	C₂×C₄×C₈×C₁₂₇	C₂²	1	2	yes	—
26	#391×C₁₂₇	—	2	4	no	—
27	#391×C₁₂₇	C₂²	1	8	yes	—
28	#381×C₁₂₇	—	2	4	no	—
29	C₂²×Q₈×C₁₂₇	—	2	2	no	—
30	C₂²×Q₈×C₁₂₇	—	2	4	no	—
31	C₂²×Q₈×C₁₂₇	—	2	4	no	—
32	C₂²×Q₈×C₁₂₇	—	2	4	no	—
33	C₂²×Q₈×C₁₂₇	—	2	4	no	—
34	C₂²×Q₈×C₁₂₇	—	2	4	no	—
35	C₂²×Q₈×C₁₂₇	C₂×C₄	1	8	yes	—
36	C₂²×Q₈×C₁₂₇	C₂³	1	4	yes	—
37	C₂³×C₄×C₁₂₇	—	2	2	no	—
38	C₂³×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
39	C₂×Q₁₆×C₁₂₇	—	2	32	no	—
40	C₂×Q₁₆×C₁₂₇	—	2	4	no	—
41	C₂×Q₁₆×C₁₂₇	—	2	4	no	—
42	C₂×Q₁₆×C₁₂₇	—	2	4	no	—
43	C₂×Q₁₆×C₁₂₇	—	2	4	no	—
44	C₂×Q₁₆×C₁₂₇	C₂×C₄	1	8	yes	—
45	C₂×Q₁₆×C₁₂₇	C₂×C₄	1	8	yes	—
46	C₂×M₁₆×C₁₂₇	—	2	2	no	—
47	C₂×M₁₆×C₁₂₇	—	2	2	no	—
48	C₂×M₁₆×C₁₂₇	D₄	1	4	yes	—
49	C₂²×C₈×C₁₂₇	C₂×C₄	1	4	yes	—
50	C₂²×C₈×C₁₂₇	D₄	1	4	yes	—
51	C₂²×C₈×C₁₂₇	D₄	1	4	yes	—
52	C₂²×C₈×C₁₂₇	C₂³	1	2	yes	—
53	C₂²×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
54	C₂²×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
55	#117×C₁₂₇	—	2	2	no	—
56	#117×C₁₂₇	—	2	2	no	—
57	#117×C₁₂₇	—	2	4	no	—
58	#117×C₁₂₇	—	2	4	no	—
59	C₂×C₄⋊C₄×C₁₂₇	D₄	1	2	yes	—
60	C₂×C₄⋊C₄×C₁₂₇	C₂³	1	4	yes	—
61	C₂×C₂²⋊C₄×C₁₂₇	—	2	2	no	—
62	C₂×C₂²⋊C₄×C₁₂₇	D₄	1	2	yes	—
63	C₂×C₄²×C₁₂₇	—	2	2	no	—
64	C₂×C₄²×C₁₂₇	C₂³	1	2	yes	—
65	#107×C₁₂₇	—	2	4	no	—
66	#106×C₁₂₇	—	2	4	no	—
67	#103×C₁₂₇	—	2	16	no	—
68	#97×C₁₂₇	—	2	4	no	—
69	C₄×C₈×C₁₂₇	—	2	4	no	—
70	#95×C₁₂₇	—	2	16	no	—
71	C₂⁴×C₁₂₇	C₂×D₄	1	1	yes	characteristic
72	C₂×Q₈×C₁₂₇	—	2	8	no	—
73	C₂×Q₈×C₁₂₇	—	2	4	no	—
74	C₂×Q₈×C₁₂₇	—	2	16	no	—
75	C₂×Q₈×C₁₂₇	—	2	16	no	—
76	C₂×Q₈×C₁₂₇	—	2	4	no	—
77	C₂×Q₈×C₁₂₇	—	2	4	no	—
78	C₂×Q₈×C₁₂₇	—	2	4	no	—
79	C₂×Q₈×C₁₂₇	—	2	4	no	—
80	C₂×Q₈×C₁₂₇	—	4	4	no	—
81	C₂×Q₈×C₁₂₇	—	4	8	no	—
82	C₂×Q₈×C₁₂₇	—	4	8	no	—
83	C₂×Q₈×C₁₂₇	—	4	4	no	—
84	C₂×Q₈×C₁₂₇	C₂²×C₄	1	8	yes	—
85	C₂²×C₄×C₁₂₇	—	2	8	no	—
86	C₂²×C₄×C₁₂₇	—	2	4	no	—
87	C₂²×C₄×C₁₂₇	—	2	4	no	—
88	C₂²×C₄×C₁₂₇	—	4	4	no	—
89	C₂²×C₄×C₁₂₇	—	4	4	no	—
90	C₂²×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
91	C₂²×C₄×C₁₂₇	C₂×D₄	1	4	yes	—
92	C₂²×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
93	C₂²×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
94	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
95	C₂²×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
96	C₂²×C₄×C₁₂₇	#41	1	2	yes	—
97	C₂²×C₄×C₁₂₇	#41	1	2	yes	—
98	C₂²×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
99	Q₁₆×C₁₂₇	—	2	16	no	—
100	Q₁₆×C₁₂₇	—	2	16	no	—
101	M₁₆×C₁₂₇	—	2	8	no	—
102	C₂×C₈×C₁₂₇	—	2	2	no	—
103	C₂×C₈×C₁₂₇	—	2	2	no	—
104	C₂×C₈×C₁₂₇	C₂²⋊C₄	1	16	yes	—
105	C₂×C₈×C₁₂₇	C₂²×C₄	1	2	yes	—
106	C₂×C₈×C₁₂₇	C₂²×C₄	1	2	yes	—
107	C₂×C₈×C₁₂₇	C₂×D₄	1	4	yes	—
108	C₂×C₈×C₁₂₇	C₂×D₄	1	2	yes	—
109	C₂×C₈×C₁₂₇	C₂×D₄	1	2	yes	—
110	C₂×C₈×C₁₂₇	C₂×D₄	1	2	yes	—
111	C₂×C₈×C₁₂₇	C₂×D₄	1	2	yes	—
112	C₄⋊C₄×C₁₂₇	—	2	4	no	—
113	C₄⋊C₄×C₁₂₇	—	2	4	no	—
114	C₄⋊C₄×C₁₂₇	—	2	4	no	—
115	C₂²⋊C₄×C₁₂₇	—	2	4	no	—
116	C₂²⋊C₄×C₁₂₇	—	4	4	no	—
117	C₄²×C₁₂₇	—	2	2	no	—
118	C₄²×C₁₂₇	—	2	2	no	—
119	C₄²×C₁₂₇	—	4	4	no	—
120	C₂³×C₁₂₇	—	2	4	no	—
121	C₂³×C₁₂₇	C₄×D₄	1	2	yes	—
122	C₂³×C₁₂₇	#121	1	2	yes	—
123	C₂³×C₁₂₇	#127	1	2	yes	—
124	C₂³×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
125	Q₈×C₁₂₇	—	2	8	no	—
126	Q₈×C₁₂₇	—	2	8	no	—
127	Q₈×C₁₂₇	—	4	4	no	—
128	Q₈×C₁₂₇	—	4	4	no	—
129	Q₈×C₁₂₇	—	4	16	no	—
130	C₂×C₄×C₁₂₇	—	2	8	no	—
131	C₂×C₄×C₁₂₇	—	2	4	no	—
132	C₂×C₄×C₁₂₇	—	2	8	no	—
133	C₂×C₄×C₁₂₇	—	4	4	no	—
134	C₂×C₄×C₁₂₇	—	4	4	no	—
135	C₂×C₄×C₁₂₇	—	4	16	no	—
136	C₂×C₄×C₁₂₇	—	4	4	no	—
137	C₂×C₄×C₁₂₇	—	4	8	no	—
138	C₂×C₄×C₁₂₇	C₂×C₂²⋊C₄	1	4	yes	—
139	C₂×C₄×C₁₂₇	C₄×D₄	1	4	yes	—
140	C₂×C₄×C₁₂₇	C₄×D₄	1	2	yes	—
141	C₂×C₄×C₁₂₇	#121	1	4	yes	—
142	C₂×C₄×C₁₂₇	#121	1	2	yes	—
143	C₂×C₄×C₁₂₇	#124	1	4	yes	—
144	C₂×C₄×C₁₂₇	#127	1	2	yes	—
145	C₂×C₄×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
146	C₂×C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
147	C₂×C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
148	C₂×C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
149	C₂×C₄×C₁₂₇	C₂×#41	1	1	yes	characteristic
150	C₂×C₄×C₁₂₇	C₂×#41	1	1	yes	characteristic
151	C₈×C₁₂₇	—	2	4	no	—
152	C₈×C₁₂₇	C₂×C₂²⋊C₄	1	4	yes	—
153	C₈×C₁₂₇	C₂×C₂²⋊C₄	1	4	yes	—
154	C₂²×C₁₂₇	—	2	4	no	—
155	C₂²×C₁₂₇	—	2	4	no	—
156	C₂²×C₁₂₇	—	2	4	no	—
157	C₂²×C₁₂₇	#390	1	4	yes	—
158	C₂²×C₁₂₇	C₂×C₄×D₄	1	1	yes	characteristic
159	C₂²×C₁₂₇	C₂×#121	1	1	yes	characteristic
160	C₂²×C₁₂₇	C₂×#127	1	1	yes	characteristic
161	C₂²×C₁₂₇	#578	1	4	yes	—
162	C₄×C₁₂₇	—	4	4	no	—
163	C₄×C₁₂₇	—	4	8	no	—
164	C₄×C₁₂₇	—	8	4	no	—
165	C₄×C₁₂₇	#390	1	4	yes	—
166	C₄×C₁₂₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
167	C₄×C₁₂₇	C₂×C₄×D₄	1	1	yes	characteristic
168	C₄×C₁₂₇	C₂×#121	1	1	yes	characteristic
169	C₄×C₁₂₇	C₂×#124	1	1	yes	characteristic
170	#21232	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
171	C₂×C₁₂₇	—	2	4	no	—
172	C₂×C₁₂₇	C₂×#390	1	1	yes	characteristic
173	C₂×C₁₂₇	#2950	1	2	yes	—
174	C₂×C₁₂₇	#3059	1	2	yes	—
175	C₂×C₁₂₇	#3156	1	2	yes	—
176	C₂³×Q₁₆	C₂×C₁₂₇	1	1	yes	characteristic
177	C₂×#427	C₂×C₁₂₇	1	1	yes	characteristic
178	C₂×#419	C₂×C₁₂₇	1	2	yes	—
179	C₂×#405	C₂×C₁₂₇	1	1	yes	characteristic
180	#2310	C₂×C₁₂₇	1	2	yes	—
181	#1971	C₂×C₁₂₇	1	4	yes	complemented
182	#1969	C₂×C₁₂₇	1	4	yes	complemented
183	C₁₂₇	#21232	1	1	yes	characteristic, direct product factor, Sylow
184	C₂³×Q₈	C₂²×C₁₂₇	1	2	yes	—
185	C₂²×Q₁₆	—	2	4	no	—
186	C₂²×Q₁₆	—	2	4	no	—
187	C₂²×Q₁₆	C₄×C₁₂₇	1	8	yes	complemented
188	C₂²×Q₁₆	C₂²×C₁₂₇	1	2	yes	—
189	C₂²×Q₁₆	C₂²×C₁₂₇	1	2	yes	—
190	C₂²×M₁₆	C₂²×C₁₂₇	1	1	yes	characteristic
191	C₂³×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
192	C₂×#117	C₂²×C₁₂₇	1	2	yes	—
193	C₂×#117	C₂²×C₁₂₇	1	1	yes	characteristic
194	#427	—	2	4	no	—
195	C₂×#107	C₂²×C₁₂₇	1	2	yes	—
196	C₂×#106	C₂²×C₁₂₇	1	2	yes	—
197	#419	—	2	8	no	—
198	C₂×#103	C₂²×C₁₂₇	1	8	yes	—
199	#405	—	2	4	no	—
200	C₂×#97	C₂²×C₁₂₇	1	2	yes	—
201	C₂×C₄×C₈	C₂²×C₁₂₇	1	2	yes	—
202	#391	—	2	4	no	—
203	#391	C₂²×C₁₂₇	1	8	yes	—
204	#381	—	2	4	no	—
205	C₂²×Q₈	—	2	2	no	—
206	C₂²×Q₈	—	2	4	no	—
207	C₂²×Q₈	—	2	4	no	—
208	C₂²×Q₈	—	2	4	no	—
209	C₂²×Q₈	—	2	4	no	—
210	C₂²×Q₈	—	2	4	no	—
211	C₂²×Q₈	C₂×C₄×C₁₂₇	1	8	yes	—
212	C₂²×Q₈	C₂³×C₁₂₇	1	4	yes	—
213	C₂³×C₄	—	2	2	no	—
214	C₂³×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
215	C₂×Q₁₆	—	2	32	no	—
216	C₂×Q₁₆	—	2	4	no	—
217	C₂×Q₁₆	—	2	4	no	—
218	C₂×Q₁₆	—	2	4	no	—
219	C₂×Q₁₆	—	2	4	no	—
220	C₂×Q₁₆	C₂×C₄×C₁₂₇	1	8	yes	—
221	C₂×Q₁₆	C₂×C₄×C₁₂₇	1	8	yes	—
222	C₂×M₁₆	—	2	2	no	—
223	C₂×M₁₆	—	2	2	no	—
224	C₂×M₁₆	D₄×C₁₂₇	1	4	yes	—
225	C₂²×C₈	C₂×C₄×C₁₂₇	1	4	yes	—
226	C₂²×C₈	D₄×C₁₂₇	1	4	yes	—
227	C₂²×C₈	D₄×C₁₂₇	1	4	yes	—
228	C₂²×C₈	C₂³×C₁₂₇	1	2	yes	—
229	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
230	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
231	#117	—	2	2	no	—
232	#117	—	2	2	no	—
233	#117	—	2	4	no	—
234	#117	—	2	4	no	—
235	C₂×C₄⋊C₄	D₄×C₁₂₇	1	2	yes	—
236	C₂×C₄⋊C₄	C₂³×C₁₂₇	1	4	yes	—
237	C₂×C₂²⋊C₄	—	2	2	no	—
238	C₂×C₂²⋊C₄	D₄×C₁₂₇	1	2	yes	—
239	C₂×C₄²	—	2	2	no	—
240	C₂×C₄²	C₂³×C₁₂₇	1	2	yes	—
241	#107	—	2	4	no	—
242	#106	—	2	4	no	—
243	#103	—	2	16	no	—
244	#97	—	2	4	no	—
245	C₄×C₈	—	2	4	no	—
246	#95	—	2	16	no	—
247	C₂⁴	C₂×D₄×C₁₂₇	1	1	yes	characteristic
248	C₂×Q₈	—	2	8	no	—
249	C₂×Q₈	—	2	4	no	—
250	C₂×Q₈	—	2	16	no	—
251	C₂×Q₈	—	2	16	no	—
252	C₂×Q₈	—	2	4	no	—
253	C₂×Q₈	—	2	4	no	—
254	C₂×Q₈	—	2	4	no	—
255	C₂×Q₈	—	2	4	no	—
256	C₂×Q₈	—	4	4	no	—
257	C₂×Q₈	—	4	8	no	—
258	C₂×Q₈	—	4	8	no	—
259	C₂×Q₈	—	4	4	no	—
260	C₂×Q₈	C₂²×C₄×C₁₂₇	1	8	yes	—
261	C₂²×C₄	—	2	8	no	—
262	C₂²×C₄	—	2	4	no	—
263	C₂²×C₄	—	2	4	no	—
264	C₂²×C₄	—	4	4	no	—
265	C₂²×C₄	—	4	4	no	—
266	C₂²×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
267	C₂²×C₄	C₂×D₄×C₁₂₇	1	4	yes	—
268	C₂²×C₄	C₂×D₄×C₁₂₇	1	2	yes	—
269	C₂²×C₄	C₂×D₄×C₁₂₇	1	2	yes	—
270	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
271	C₂²×C₄	C₂×D₄×C₁₂₇	1	2	yes	—
272	C₂²×C₄	#41×C₁₂₇	1	2	yes	—
273	C₂²×C₄	#41×C₁₂₇	1	2	yes	—
274	C₂²×C₄	C₂⁴×C₁₂₇	1	1	yes	characteristic
275	Q₁₆	—	2	16	no	—
276	Q₁₆	—	2	16	no	—
277	M₁₆	—	2	8	no	—
278	C₂×C₈	—	2	2	no	—
279	C₂×C₈	—	2	2	no	—
280	C₂×C₈	C₂²⋊C₄×C₁₂₇	1	16	yes	—
281	C₂×C₈	C₂²×C₄×C₁₂₇	1	2	yes	—
282	C₂×C₈	C₂²×C₄×C₁₂₇	1	2	yes	—
283	C₂×C₈	C₂×D₄×C₁₂₇	1	4	yes	—
284	C₂×C₈	C₂×D₄×C₁₂₇	1	2	yes	—
285	C₂×C₈	C₂×D₄×C₁₂₇	1	2	yes	—
286	C₂×C₈	C₂×D₄×C₁₂₇	1	2	yes	—
287	C₂×C₈	C₂×D₄×C₁₂₇	1	2	yes	—
288	C₄⋊C₄	—	2	4	no	—
289	C₄⋊C₄	—	2	4	no	—
290	C₄⋊C₄	—	2	4	no	—
291	C₂²⋊C₄	—	2	4	no	—
292	C₂²⋊C₄	—	4	4	no	—
293	C₄²	—	2	2	no	—
294	C₄²	—	2	2	no	—
295	C₄²	—	4	4	no	—
296	C₂³	—	2	4	no	—
297	C₂³	C₄×D₄×C₁₂₇	1	2	yes	—
298	C₂³	#121×C₁₂₇	1	2	yes	—
299	C₂³	#127×C₁₂₇	1	2	yes	—
300	C₂³	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
301	Q₈	—	2	8	no	—
302	Q₈	—	2	8	no	—
303	Q₈	—	4	4	no	—
304	Q₈	—	4	4	no	—
305	Q₈	—	4	16	no	—
306	C₂×C₄	—	2	8	no	—
307	C₂×C₄	—	2	4	no	—
308	C₂×C₄	—	2	8	no	—
309	C₂×C₄	—	4	4	no	—
310	C₂×C₄	—	4	4	no	—
311	C₂×C₄	—	4	16	no	—
312	C₂×C₄	—	4	4	no	—
313	C₂×C₄	—	4	8	no	—
314	C₂×C₄	C₂×C₂²⋊C₄×C₁₂₇	1	4	yes	—
315	C₂×C₄	C₄×D₄×C₁₂₇	1	4	yes	—
316	C₂×C₄	C₄×D₄×C₁₂₇	1	2	yes	—
317	C₂×C₄	#121×C₁₂₇	1	4	yes	—
318	C₂×C₄	#121×C₁₂₇	1	2	yes	—
319	C₂×C₄	#124×C₁₂₇	1	4	yes	—
320	C₂×C₄	#127×C₁₂₇	1	2	yes	—
321	C₂×C₄	C₂³×C₄×C₁₂₇	1	1	yes	characteristic
322	C₂×C₄	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
323	C₂×C₄	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
324	C₂×C₄	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
325	C₂×C₄	C₂×#41×C₁₂₇	1	1	yes	characteristic
326	C₂×C₄	C₂×#41×C₁₂₇	1	1	yes	characteristic
327	C₈	—	2	4	no	—
328	C₈	C₂×C₂²⋊C₄×C₁₂₇	1	4	yes	—
329	C₈	C₂×C₂²⋊C₄×C₁₂₇	1	4	yes	—
330	C₂²	—	2	4	no	—
331	C₂²	—	2	4	no	—
332	C₂²	—	2	4	no	—
333	C₂²	#390×C₁₂₇	1	4	yes	—
334	C₂²	C₂×C₄×D₄×C₁₂₇	1	1	yes	characteristic
335	C₂²	C₂×#121×C₁₂₇	1	1	yes	characteristic
336	C₂²	C₂×#127×C₁₂₇	1	1	yes	characteristic
337	C₂²	#578×C₁₂₇	1	4	yes	—
338	C₄	—	4	4	no	—
339	C₄	—	4	8	no	—
340	C₄	—	8	4	no	—
341	C₄	#390×C₁₂₇	1	4	yes	—
342	C₄	C₂²×C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
343	C₄	C₂×C₄×D₄×C₁₂₇	1	1	yes	characteristic
344	C₄	C₂×#121×C₁₂₇	1	1	yes	characteristic
345	C₄	C₂×#124×C₁₂₇	1	1	yes	characteristic
346	C₂	—	2	4	no	—
347	C₂	C₂×#390×C₁₂₇	1	1	yes	characteristic
348	C₂	C₁₂₇×#2950	1	2	yes	—
349	C₂	C₁₂₇×#3059	1	2	yes	—
350	C₂	C₁₂₇×#3156	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₂₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₁₂₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₂₇
prime radical	C₂⁴×C₁₂₇	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₂₇ ◁ C₂×#117×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
6. C₁₂₇×#1971	C₂	semidirect
7. C₁₂₇×#1969	C₂	semidirect
11. C₂²×Q₁₆×C₁₂₇	C₄	semidirect
170. #21232	C₁₂₇	direct (N × H)
181. #1971	C₂×C₁₂₇	semidirect
182. #1969	C₂×C₁₂₇	semidirect
183. C₁₂₇	#21232	direct (N × H)
187. C₂²×Q₁₆	C₄×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₂³×Q₁₆×C₁₂₇	2
C₂×#427×C₁₂₇	2
C₂×#419×C₁₂₇	2
C₂×#405×C₁₂₇	2
C₁₂₇×#2310	2
C₁₂₇×#1971	2
C₁₂₇×#1969	2
#21232	127

Subgroups — C127×#21232 #44584905

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#21232 #44584905