Subgroups — C₁₂₇×#21695 #44585368

Order 32512.

1126 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#492×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#440×C₁₂₇	C₂	1	2	yes	maximal, complemented
3	C₂×C₄×SD₁₆×C₁₂₇	C₂	1	2	yes	maximal, complemented
4	C₁₂₇×#2328	C₂	1	1	yes	maximal, characteristic
5	C₁₂₇×#2327	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₂₇×#1973	C₂	1	2	yes	maximal, complemented
7	C₁₂₇×#1968	C₂	1	2	yes	maximal, complemented
8	C₁₂₇×#1961	C₂	1	1	yes	maximal, characteristic
9	C₁₂₇×#1960	C₂	1	1	yes	maximal, characteristic, complemented
10	C₁₂₇×#1846	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₂₇×#1726	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×SD₁₆×C₁₂₇	C₂²	1	2	yes	—
13	C₂²×SD₁₆×C₁₂₇	C₂²	1	2	yes	—
14	C₂×C₄⋊Q₈×C₁₂₇	C₂²	1	1	yes	characteristic
15	C₂×#127×C₁₂₇	C₂²	1	1	yes	characteristic
16	C₂×C₄×Q₈×C₁₂₇	C₂²	1	2	yes	—
17	C₂×C₄×D₄×C₁₂₇	C₂²	1	2	yes	—
18	#492×C₁₂₇	C₄	1	8	yes	complemented
19	#440×C₁₂₇	—	2	8	no	—
20	C₄×SD₁₆×C₁₂₇	—	2	8	no	—
21	C₂×#107×C₁₂₇	C₂²	1	2	yes	—
22	C₂×#106×C₁₂₇	C₂²	1	2	yes	—
23	C₂×#103×C₁₂₇	C₂²	1	2	yes	—
24	C₂×#103×C₁₂₇	C₂²	1	2	yes	—
25	C₂×#102×C₁₂₇	C₂²	1	2	yes	—
26	C₂×#102×C₁₂₇	C₂²	1	2	yes	—
27	C₂×#97×C₁₂₇	C₂²	1	1	yes	characteristic
28	C₂×#97×C₁₂₇	C₂²	1	1	yes	characteristic
29	C₂×#97×C₁₂₇	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
31	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
32	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
33	#391×C₁₂₇	C₂²	1	2	yes	—
34	#390×C₁₂₇	C₂²	1	2	yes	—
35	#387×C₁₂₇	—	2	4	no	—
36	#386×C₁₂₇	—	2	4	no	—
37	#382×C₁₂₇	C₂²	1	1	yes	characteristic
38	#382×C₁₂₇	C₂²	1	1	yes	characteristic
39	#376×C₁₂₇	C₂²	1	1	yes	characteristic
40	C₂²×Q₈×C₁₂₇	C₂³	1	2	yes	—
41	C₂²×D₄×C₁₂₇	—	2	2	no	—
42	C₂²×D₄×C₁₂₇	C₂³	1	2	yes	—
43	C₂³×C₄×C₁₂₇	—	2	2	no	—
44	C₂×SD₁₆×C₁₂₇	—	2	4	no	—
45	C₂×SD₁₆×C₁₂₇	—	2	4	no	—
46	C₂×SD₁₆×C₁₂₇	—	2	4	no	—
47	C₂×SD₁₆×C₁₂₇	—	2	4	no	—
48	C₂×SD₁₆×C₁₂₇	C₂×C₄	1	8	yes	complemented
49	C₂×SD₁₆×C₁₂₇	C₂×C₄	1	8	yes	complemented
50	C₂²×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
51	C₂²×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
53	C₂²×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
54	C₄⋊Q₈×C₁₂₇	—	2	2	no	—
55	C₄⋊Q₈×C₁₂₇	C₂×C₄	1	4	yes	complemented
56	#127×C₁₂₇	—	2	2	no	—
57	#127×C₁₂₇	C₂×C₄	1	4	yes	—
58	C₄×Q₈×C₁₂₇	—	2	4	no	—
59	C₄×Q₈×C₁₂₇	—	2	4	no	—
60	C₄×D₄×C₁₂₇	—	2	4	no	—
61	C₄×D₄×C₁₂₇	—	2	4	no	—
62	C₂×C₄⋊C₄×C₁₂₇	—	2	2	no	—
63	C₂×C₄⋊C₄×C₁₂₇	—	2	2	no	—
64	C₂×C₄⋊C₄×C₁₂₇	C₂³	1	2	yes	—
65	C₂×C₄⋊C₄×C₁₂₇	C₂³	1	2	yes	—
66	C₂×C₂²⋊C₄×C₁₂₇	—	2	2	no	—
67	C₂×C₂²⋊C₄×C₁₂₇	—	2	4	no	—
68	C₂×C₄²×C₁₂₇	—	2	2	no	—
69	C₂×C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
70	C₂×C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
71	C₂×C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
72	#107×C₁₂₇	—	2	4	no	—
73	#106×C₁₂₇	—	2	4	no	—
74	#103×C₁₂₇	—	2	4	no	—
75	#103×C₁₂₇	—	2	4	no	—
76	#102×C₁₂₇	—	2	4	no	—
77	#102×C₁₂₇	—	2	4	no	—
78	#97×C₁₂₇	—	2	2	no	—
79	#97×C₁₂₇	—	2	2	no	—
80	#97×C₁₂₇	D₄	1	4	yes	complemented
81	C₄×C₈×C₁₂₇	—	2	2	no	—
82	C₄×C₈×C₁₂₇	C₂×C₄	1	4	yes	complemented
83	C₄×C₈×C₁₂₇	D₄	1	4	yes	complemented
84	#95×C₁₂₇	—	2	2	no	—
85	#95×C₁₂₇	—	2	2	no	—
86	#95×C₁₂₇	—	2	4	no	—
87	#95×C₁₂₇	D₄	1	4	yes	—
88	C₂⁴×C₁₂₇	—	2	2	no	—
89	C₂×Q₈×C₁₂₇	—	2	2	no	—
90	C₂×Q₈×C₁₂₇	—	2	2	no	—
91	C₂×Q₈×C₁₂₇	—	2	2	no	—
92	C₂×Q₈×C₁₂₇	—	2	2	no	—
93	C₂×Q₈×C₁₂₇	—	2	2	no	—
94	C₂×Q₈×C₁₂₇	C₂²×C₄	1	4	yes	—
95	C₂×D₄×C₁₂₇	—	2	8	no	—
96	C₂×D₄×C₁₂₇	—	2	2	no	—
97	C₂×D₄×C₁₂₇	—	2	2	no	—
98	C₂×D₄×C₁₂₇	—	2	2	no	—
99	C₂×D₄×C₁₂₇	—	2	2	no	—
100	C₂×D₄×C₁₂₇	—	2	2	no	—
101	C₂×D₄×C₁₂₇	—	8	4	no	—
102	C₂×D₄×C₁₂₇	C₂²×C₄	1	4	yes	—
103	C₂²×C₄×C₁₂₇	—	2	2	no	—
104	C₂²×C₄×C₁₂₇	—	2	2	no	—
105	C₂²×C₄×C₁₂₇	—	2	2	no	—
106	C₂²×C₄×C₁₂₇	—	4	4	no	—
107	C₂²×C₄×C₁₂₇	—	8	4	no	—
108	C₂²×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
109	C₂²×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
110	C₂²×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
111	C₂²×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
112	SD₁₆×C₁₂₇	—	2	16	no	—
113	SD₁₆×C₁₂₇	—	2	16	no	—
114	C₂×C₈×C₁₂₇	—	2	1	no	—
115	C₂×C₈×C₁₂₇	—	2	1	no	—
116	C₂×C₈×C₁₂₇	—	2	1	no	—
117	C₂×C₈×C₁₂₇	—	2	1	no	—
118	C₂×C₈×C₁₂₇	C₂²×C₄	1	2	yes	—
119	C₂×C₈×C₁₂₇	C₂²×C₄	1	2	yes	—
120	C₂×C₈×C₁₂₇	C₂×D₄	1	2	yes	—
121	C₂×C₈×C₁₂₇	C₂×D₄	1	2	yes	—
122	C₂×C₈×C₁₂₇	C₂×D₄	1	2	yes	—
123	C₂×C₈×C₁₂₇	C₂×D₄	1	2	yes	—
124	C₂×C₈×C₁₂₇	#41	1	2	yes	—
125	C₂×C₈×C₁₂₇	#41	1	2	yes	—
126	C₄⋊C₄×C₁₂₇	—	2	8	no	—
127	C₄⋊C₄×C₁₂₇	—	2	2	no	—
128	C₄⋊C₄×C₁₂₇	—	2	2	no	—
129	C₄⋊C₄×C₁₂₇	—	2	2	no	—
130	C₄⋊C₄×C₁₂₇	—	2	2	no	—
131	C₄⋊C₄×C₁₂₇	—	4	4	no	—
132	C₂²⋊C₄×C₁₂₇	—	4	4	no	—
133	C₂²⋊C₄×C₁₂₇	—	8	4	no	—
134	C₄²×C₁₂₇	—	2	1	no	—
135	C₄²×C₁₂₇	—	2	1	no	—
136	C₄²×C₁₂₇	—	4	4	no	—
137	C₄²×C₁₂₇	C₂²×C₄	1	2	yes	—
138	C₄²×C₁₂₇	C₂×D₄	1	2	yes	—
139	C₄²×C₁₂₇	C₂×D₄	1	2	yes	—
140	C₄²×C₁₂₇	#41	1	2	yes	—
141	C₂³×C₁₂₇	—	2	4	no	—
142	C₂³×C₁₂₇	—	4	2	no	—
143	C₂³×C₁₂₇	—	4	2	no	—
144	C₂³×C₁₂₇	—	4	4	no	—
145	C₂³×C₁₂₇	—	4	4	no	—
146	C₂³×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
147	Q₈×C₁₂₇	—	2	4	no	—
148	Q₈×C₁₂₇	—	2	4	no	—
149	Q₈×C₁₂₇	—	4	2	no	—
150	Q₈×C₁₂₇	—	4	2	no	—
151	D₄×C₁₂₇	—	2	4	no	—
152	D₄×C₁₂₇	—	2	4	no	—
153	D₄×C₁₂₇	—	4	2	no	—
154	D₄×C₁₂₇	—	4	2	no	—
155	D₄×C₁₂₇	—	8	8	no	—
156	C₂×C₄×C₁₂₇	—	2	4	no	—
157	C₂×C₄×C₁₂₇	—	2	4	no	—
158	C₂×C₄×C₁₂₇	—	2	4	no	—
159	C₂×C₄×C₁₂₇	—	2	2	no	—
160	C₂×C₄×C₁₂₇	—	2	4	no	—
161	C₂×C₄×C₁₂₇	—	4	2	no	—
162	C₂×C₄×C₁₂₇	—	4	2	no	—
163	C₂×C₄×C₁₂₇	—	4	2	no	—
164	C₂×C₄×C₁₂₇	—	4	2	no	—
165	C₂×C₄×C₁₂₇	—	4	4	no	—
166	C₂×C₄×C₁₂₇	—	4	4	no	—
167	C₂×C₄×C₁₂₇	—	8	8	no	—
168	C₂×C₄×C₁₂₇	C₄×D₄	1	4	yes	—
169	C₂×C₄×C₁₂₇	#127	1	4	yes	—
170	C₂×C₄×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
171	C₂×C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
172	C₂×C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
173	C₂×C₄×C₁₂₇	C₂×#41	1	1	yes	characteristic
174	C₂×C₄×C₁₂₇	C₂×#41	1	1	yes	characteristic
175	C₂×C₄×C₁₂₇	C₂×#41	1	1	yes	characteristic
176	C₈×C₁₂₇	—	2	2	no	—
177	C₈×C₁₂₇	—	2	2	no	—
178	C₈×C₁₂₇	C₄×D₄	1	4	yes	—
179	C₈×C₁₂₇	C₄×D₄	1	4	yes	—
180	C₂²×C₁₂₇	—	4	4	no	—
181	C₂²×C₁₂₇	—	4	8	no	—
182	C₂²×C₁₂₇	—	8	2	no	—
183	C₂²×C₁₂₇	—	8	2	no	—
184	C₂²×C₁₂₇	—	8	4	no	—
185	C₂²×C₁₂₇	C₂×C₄×D₄	1	1	yes	characteristic
186	C₂²×C₁₂₇	C₂×#127	1	1	yes	characteristic
187	C₂²×C₁₂₇	#532	1	1	yes	characteristic
188	C₂²×C₁₂₇	#577	1	4	yes	—
189	C₄×C₁₂₇	—	2	4	no	—
190	C₄×C₁₂₇	—	2	4	no	—
191	C₄×C₁₂₇	—	4	4	no	—
192	C₄×C₁₂₇	—	4	2	no	—
193	C₄×C₁₂₇	—	8	2	no	—
194	C₄×C₁₂₇	—	8	2	no	—
195	C₄×C₁₂₇	C₂×C₄×D₄	1	1	yes	characteristic
196	C₄×C₁₂₇	C₂×C₄×D₄	1	1	yes	characteristic
197	C₄×C₁₂₇	#532	1	1	yes	characteristic
198	C₄×C₁₂₇	#532	1	1	yes	characteristic
199	#21695	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
200	C₂×C₁₂₇	—	8	4	no	—
201	C₂×C₁₂₇	C₄×#127	1	1	yes	characteristic
202	C₂×C₁₂₇	#2949	1	2	yes	—
203	C₂×C₁₂₇	#3154	1	2	yes	—
204	C₂×C₁₂₇	#3174	1	2	yes	—
205	C₂×#492	C₂×C₁₂₇	1	1	yes	characteristic
206	C₂×#440	C₂×C₁₂₇	1	2	yes	complemented
207	C₂×C₄×SD₁₆	C₂×C₁₂₇	1	2	yes	complemented
208	#2328	C₂×C₁₂₇	1	1	yes	characteristic
209	#2327	C₂×C₁₂₇	1	1	yes	characteristic, complemented
210	#1973	C₂×C₁₂₇	1	2	yes	complemented
211	#1968	C₂×C₁₂₇	1	2	yes	complemented
212	#1961	C₂×C₁₂₇	1	1	yes	characteristic
213	#1960	C₂×C₁₂₇	1	1	yes	characteristic, complemented
214	#1846	C₂×C₁₂₇	1	1	yes	characteristic, complemented
215	#1726	C₂×C₁₂₇	1	1	yes	characteristic, complemented
216	C₁₂₇	#21695	1	1	yes	characteristic, direct product factor, Sylow
217	C₂²×SD₁₆	C₂²×C₁₂₇	1	2	yes	—
218	C₂²×SD₁₆	C₂²×C₁₂₇	1	2	yes	—
219	C₂×C₄⋊Q₈	C₂²×C₁₂₇	1	1	yes	characteristic
220	C₂×#127	C₂²×C₁₂₇	1	1	yes	characteristic
221	C₂×C₄×Q₈	C₂²×C₁₂₇	1	2	yes	—
222	C₂×C₄×D₄	C₂²×C₁₂₇	1	2	yes	—
223	#492	C₄×C₁₂₇	1	8	yes	complemented
224	#440	—	2	8	no	—
225	C₄×SD₁₆	—	2	8	no	—
226	C₂×#107	C₂²×C₁₂₇	1	2	yes	—
227	C₂×#106	C₂²×C₁₂₇	1	2	yes	—
228	C₂×#103	C₂²×C₁₂₇	1	2	yes	—
229	C₂×#103	C₂²×C₁₂₇	1	2	yes	—
230	C₂×#102	C₂²×C₁₂₇	1	2	yes	—
231	C₂×#102	C₂²×C₁₂₇	1	2	yes	—
232	C₂×#97	C₂²×C₁₂₇	1	1	yes	characteristic
233	C₂×#97	C₂²×C₁₂₇	1	1	yes	characteristic
234	C₂×#97	C₂²×C₁₂₇	1	1	yes	characteristic
235	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
236	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
237	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
238	#391	C₂²×C₁₂₇	1	2	yes	—
239	#390	C₂²×C₁₂₇	1	2	yes	—
240	#387	—	2	4	no	—
241	#386	—	2	4	no	—
242	#382	C₂²×C₁₂₇	1	1	yes	characteristic
243	#382	C₂²×C₁₂₇	1	1	yes	characteristic
244	#376	C₂²×C₁₂₇	1	1	yes	characteristic
245	C₂²×Q₈	C₂³×C₁₂₇	1	2	yes	—
246	C₂²×D₄	—	2	2	no	—
247	C₂²×D₄	C₂³×C₁₂₇	1	2	yes	—
248	C₂³×C₄	—	2	2	no	—
249	C₂×SD₁₆	—	2	4	no	—
250	C₂×SD₁₆	—	2	4	no	—
251	C₂×SD₁₆	—	2	4	no	—
252	C₂×SD₁₆	—	2	4	no	—
253	C₂×SD₁₆	C₂×C₄×C₁₂₇	1	8	yes	complemented
254	C₂×SD₁₆	C₂×C₄×C₁₂₇	1	8	yes	complemented
255	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
256	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
257	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
258	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
259	C₄⋊Q₈	—	2	2	no	—
260	C₄⋊Q₈	C₂×C₄×C₁₂₇	1	4	yes	complemented
261	#127	—	2	2	no	—
262	#127	C₂×C₄×C₁₂₇	1	4	yes	—
263	C₄×Q₈	—	2	4	no	—
264	C₄×Q₈	—	2	4	no	—
265	C₄×D₄	—	2	4	no	—
266	C₄×D₄	—	2	4	no	—
267	C₂×C₄⋊C₄	—	2	2	no	—
268	C₂×C₄⋊C₄	—	2	2	no	—
269	C₂×C₄⋊C₄	C₂³×C₁₂₇	1	2	yes	—
270	C₂×C₄⋊C₄	C₂³×C₁₂₇	1	2	yes	—
271	C₂×C₂²⋊C₄	—	2	2	no	—
272	C₂×C₂²⋊C₄	—	2	4	no	—
273	C₂×C₄²	—	2	2	no	—
274	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
275	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
276	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
277	#107	—	2	4	no	—
278	#106	—	2	4	no	—
279	#103	—	2	4	no	—
280	#103	—	2	4	no	—
281	#102	—	2	4	no	—
282	#102	—	2	4	no	—
283	#97	—	2	2	no	—
284	#97	—	2	2	no	—
285	#97	D₄×C₁₂₇	1	4	yes	complemented
286	C₄×C₈	—	2	2	no	—
287	C₄×C₈	C₂×C₄×C₁₂₇	1	4	yes	complemented
288	C₄×C₈	D₄×C₁₂₇	1	4	yes	complemented
289	#95	—	2	2	no	—
290	#95	—	2	2	no	—
291	#95	—	2	4	no	—
292	#95	D₄×C₁₂₇	1	4	yes	—
293	C₂⁴	—	2	2	no	—
294	C₂×Q₈	—	2	2	no	—
295	C₂×Q₈	—	2	2	no	—
296	C₂×Q₈	—	2	2	no	—
297	C₂×Q₈	—	2	2	no	—
298	C₂×Q₈	—	2	2	no	—
299	C₂×Q₈	C₂²×C₄×C₁₂₇	1	4	yes	—
300	C₂×D₄	—	2	8	no	—
301	C₂×D₄	—	2	2	no	—
302	C₂×D₄	—	2	2	no	—
303	C₂×D₄	—	2	2	no	—
304	C₂×D₄	—	2	2	no	—
305	C₂×D₄	—	2	2	no	—
306	C₂×D₄	—	8	4	no	—
307	C₂×D₄	C₂²×C₄×C₁₂₇	1	4	yes	—
308	C₂²×C₄	—	2	2	no	—
309	C₂²×C₄	—	2	2	no	—
310	C₂²×C₄	—	2	2	no	—
311	C₂²×C₄	—	4	4	no	—
312	C₂²×C₄	—	8	4	no	—
313	C₂²×C₄	C₂×D₄×C₁₂₇	1	2	yes	—
314	C₂²×C₄	C₂×D₄×C₁₂₇	1	2	yes	—
315	C₂²×C₄	C₂×D₄×C₁₂₇	1	2	yes	—
316	C₂²×C₄	C₂⁴×C₁₂₇	1	1	yes	characteristic
317	SD₁₆	—	2	16	no	—
318	SD₁₆	—	2	16	no	—
319	C₂×C₈	—	2	1	no	—
320	C₂×C₈	—	2	1	no	—
321	C₂×C₈	—	2	1	no	—
322	C₂×C₈	—	2	1	no	—
323	C₂×C₈	C₂²×C₄×C₁₂₇	1	2	yes	—
324	C₂×C₈	C₂²×C₄×C₁₂₇	1	2	yes	—
325	C₂×C₈	C₂×D₄×C₁₂₇	1	2	yes	—
326	C₂×C₈	C₂×D₄×C₁₂₇	1	2	yes	—
327	C₂×C₈	C₂×D₄×C₁₂₇	1	2	yes	—
328	C₂×C₈	C₂×D₄×C₁₂₇	1	2	yes	—
329	C₂×C₈	#41×C₁₂₇	1	2	yes	—
330	C₂×C₈	#41×C₁₂₇	1	2	yes	—
331	C₄⋊C₄	—	2	8	no	—
332	C₄⋊C₄	—	2	2	no	—
333	C₄⋊C₄	—	2	2	no	—
334	C₄⋊C₄	—	2	2	no	—
335	C₄⋊C₄	—	2	2	no	—
336	C₄⋊C₄	—	4	4	no	—
337	C₂²⋊C₄	—	4	4	no	—
338	C₂²⋊C₄	—	8	4	no	—
339	C₄²	—	2	1	no	—
340	C₄²	—	2	1	no	—
341	C₄²	—	4	4	no	—
342	C₄²	C₂²×C₄×C₁₂₇	1	2	yes	—
343	C₄²	C₂×D₄×C₁₂₇	1	2	yes	—
344	C₄²	C₂×D₄×C₁₂₇	1	2	yes	—
345	C₄²	#41×C₁₂₇	1	2	yes	—
346	C₂³	—	2	4	no	—
347	C₂³	—	4	2	no	—
348	C₂³	—	4	2	no	—
349	C₂³	—	4	4	no	—
350	C₂³	—	4	4	no	—
351	C₂³	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
352	Q₈	—	2	4	no	—
353	Q₈	—	2	4	no	—
354	Q₈	—	4	2	no	—
355	Q₈	—	4	2	no	—
356	D₄	—	2	4	no	—
357	D₄	—	2	4	no	—
358	D₄	—	4	2	no	—
359	D₄	—	4	2	no	—
360	D₄	—	8	8	no	—
361	C₂×C₄	—	2	4	no	—
362	C₂×C₄	—	2	4	no	—
363	C₂×C₄	—	2	4	no	—
364	C₂×C₄	—	2	2	no	—
365	C₂×C₄	—	2	4	no	—
366	C₂×C₄	—	4	2	no	—
367	C₂×C₄	—	4	2	no	—
368	C₂×C₄	—	4	2	no	—
369	C₂×C₄	—	4	2	no	—
370	C₂×C₄	—	4	4	no	—
371	C₂×C₄	—	4	4	no	—
372	C₂×C₄	—	8	8	no	—
373	C₂×C₄	C₄×D₄×C₁₂₇	1	4	yes	—
374	C₂×C₄	#127×C₁₂₇	1	4	yes	—
375	C₂×C₄	C₂³×C₄×C₁₂₇	1	1	yes	characteristic
376	C₂×C₄	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
377	C₂×C₄	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
378	C₂×C₄	C₂×#41×C₁₂₇	1	1	yes	characteristic
379	C₂×C₄	C₂×#41×C₁₂₇	1	1	yes	characteristic
380	C₂×C₄	C₂×#41×C₁₂₇	1	1	yes	characteristic
381	C₈	—	2	2	no	—
382	C₈	—	2	2	no	—
383	C₈	C₄×D₄×C₁₂₇	1	4	yes	—
384	C₈	C₄×D₄×C₁₂₇	1	4	yes	—
385	C₂²	—	4	8	no	—
386	C₂²	—	4	4	no	—
387	C₂²	—	8	2	no	—
388	C₂²	—	8	2	no	—
389	C₂²	—	8	4	no	—
390	C₂²	C₂×C₄×D₄×C₁₂₇	1	1	yes	characteristic
391	C₂²	C₂×#127×C₁₂₇	1	1	yes	characteristic
392	C₂²	#532×C₁₂₇	1	1	yes	characteristic
393	C₂²	#577×C₁₂₇	1	4	yes	—
394	C₄	—	2	4	no	—
395	C₄	—	2	4	no	—
396	C₄	—	4	2	no	—
397	C₄	—	4	4	no	—
398	C₄	—	8	2	no	—
399	C₄	—	8	2	no	—
400	C₄	C₂×C₄×D₄×C₁₂₇	1	1	yes	characteristic
401	C₄	C₂×C₄×D₄×C₁₂₇	1	1	yes	characteristic
402	C₄	#532×C₁₂₇	1	1	yes	characteristic
403	C₄	#532×C₁₂₇	1	1	yes	characteristic
404	C₂	—	8	4	no	—
405	C₂	C₄×#127×C₁₂₇	1	1	yes	characteristic
406	C₂	C₁₂₇×#2949	1	2	yes	—
407	C₂	C₁₂₇×#3154	1	2	yes	—
408	C₂	C₁₂₇×#3174	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	C₂×#127×C₁₂₇	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₂₇ ◁ #376×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#440×C₁₂₇	C₂	semidirect
3. C₂×C₄×SD₁₆×C₁₂₇	C₂	semidirect
5. C₁₂₇×#2327	C₂	semidirect
6. C₁₂₇×#1973	C₂	semidirect
7. C₁₂₇×#1968	C₂	semidirect
9. C₁₂₇×#1960	C₂	semidirect
10. C₁₂₇×#1846	C₂	semidirect
11. C₁₂₇×#1726	C₂	semidirect
18. #492×C₁₂₇	C₄	semidirect
48. C₂×SD₁₆×C₁₂₇	C₂×C₄	semidirect
49. C₂×SD₁₆×C₁₂₇	C₂×C₄	semidirect
55. C₄⋊Q₈×C₁₂₇	C₂×C₄	semidirect
80. #97×C₁₂₇	D₄	semidirect
82. C₄×C₈×C₁₂₇	C₂×C₄	semidirect
83. C₄×C₈×C₁₂₇	D₄	semidirect
199. #21695	C₁₂₇	direct (N × H)
206. C₂×#440	C₂×C₁₂₇	semidirect
207. C₂×C₄×SD₁₆	C₂×C₁₂₇	semidirect
209. #2327	C₂×C₁₂₇	semidirect
210. #1973	C₂×C₁₂₇	semidirect
211. #1968	C₂×C₁₂₇	semidirect
213. #1960	C₂×C₁₂₇	semidirect
214. #1846	C₂×C₁₂₇	semidirect
215. #1726	C₂×C₁₂₇	semidirect
216. C₁₂₇	#21695	direct (N × H)
223. #492	C₄×C₁₂₇	semidirect
253. C₂×SD₁₆	C₂×C₄×C₁₂₇	semidirect
254. C₂×SD₁₆	C₂×C₄×C₁₂₇	semidirect
260. C₄⋊Q₈	C₂×C₄×C₁₂₇	semidirect
285. #97	D₄×C₁₂₇	semidirect
287. C₄×C₈	C₂×C₄×C₁₂₇	semidirect
288. C₄×C₈	D₄×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#492×C₁₂₇	2
C₂×#440×C₁₂₇	2
C₂×C₄×SD₁₆×C₁₂₇	2
C₁₂₇×#2328	2
C₁₂₇×#2327	2
C₁₂₇×#1973	2
C₁₂₇×#1968	2
C₁₂₇×#1961	2
C₁₂₇×#1960	2
C₁₂₇×#1846	2
C₁₂₇×#1726	2
#21695	127

Subgroups — C127×#21695 #44585368

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#21695 #44585368