Subgroups — C₁₂₇×#21682 #44585355

Order 32512.

1230 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#436×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#435×C₁₂₇	C₂	1	4	yes	maximal, complemented
3	C₂×C₈×D₄×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×#127×C₁₂₇	C₂	1	1	yes	maximal, characteristic
5	C₁₂₇×#1955	C₂	1	4	yes	maximal, complemented
6	C₁₂₇×#1954	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₂₇×#1954	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₂₇×#1842	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#97×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×M₁₆×C₁₂₇	C₂²	1	2	yes	—
11	C₂³×C₈×C₁₂₇	C₂²	1	2	yes	—
12	C₂×#127×C₁₂₇	C₄	1	1	yes	characteristic
13	C₂×C₄×D₄×C₁₂₇	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₄×C₁₂₇	C₂²	1	4	yes	—
15	C₂×C₄×D₄×C₁₂₇	C₂²	1	1	yes	characteristic
16	#436×C₁₂₇	—	2	4	no	—
17	#435×C₁₂₇	—	2	16	no	—
18	C₈×D₄×C₁₂₇	—	2	4	no	—
19	C₂×C₄⋊C₈×C₁₂₇	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₈×C₁₂₇	C₂²	1	4	yes	—
21	C₂×C₄⋊C₈×C₁₂₇	C₂²	1	1	yes	characteristic
22	C₂×C₂²⋊C₈×C₁₂₇	C₂²	1	2	yes	—
23	C₂×C₂²⋊C₈×C₁₂₇	C₂²	1	4	yes	—
24	C₂×C₂²⋊C₈×C₁₂₇	C₂²	1	4	yes	—
25	C₂×C₂²⋊C₈×C₁₂₇	C₂²	1	2	yes	—
26	C₂×#97×C₁₂₇	C₂²	1	2	yes	—
27	C₂×#97×C₁₂₇	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
29	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
30	#390×C₁₂₇	C₄	1	1	yes	characteristic
31	#390×C₁₂₇	C₄	1	1	yes	characteristic
32	#390×C₁₂₇	C₄	1	4	yes	—
33	#384×C₁₂₇	C₄	1	1	yes	characteristic
34	C₄³×C₁₂₇	C₂²	1	1	yes	characteristic
35	C₂²×D₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
36	C₂²×D₄×C₁₂₇	C₂×C₄	1	4	yes	—
37	C₂²×D₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
38	C₂³×C₄×C₁₂₇	C₂³	1	2	yes	—
39	C₂³×C₄×C₁₂₇	C₂³	1	2	yes	—
40	C₂×M₁₆×C₁₂₇	—	2	8	no	—
41	C₂×M₁₆×C₁₂₇	—	2	4	no	—
42	C₂²×C₈×C₁₂₇	—	2	4	no	—
43	C₂²×C₈×C₁₂₇	—	2	8	no	—
44	C₂²×C₈×C₁₂₇	C₂³	1	2	yes	—
45	C₂²×C₈×C₁₂₇	C₂³	1	2	yes	—
46	C₂²×C₈×C₁₂₇	C₂³	1	2	yes	—
47	C₂²×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
48	C₂²×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
49	#127×C₁₂₇	—	2	4	no	—
50	C₄×D₄×C₁₂₇	—	2	2	no	—
51	C₄×D₄×C₁₂₇	—	2	8	no	—
52	C₄×D₄×C₁₂₇	—	2	2	no	—
53	C₄×D₄×C₁₂₇	—	2	2	no	—
54	C₄×D₄×C₁₂₇	—	2	2	no	—
55	C₄×D₄×C₁₂₇	—	2	8	no	—
56	C₂×C₄⋊C₄×C₁₂₇	C₂×C₄	1	4	yes	—
57	C₂×C₄⋊C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄⋊C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
59	C₂×C₂²⋊C₄×C₁₂₇	C₂×C₄	1	4	yes	—
60	C₂×C₂²⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
61	C₂×C₂²⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
62	C₂×C₂²⋊C₄×C₁₂₇	C₂×C₄	1	4	yes	—
63	C₂×C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
64	C₂×C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
65	C₂×C₄²×C₁₂₇	C₂×C₄	1	2	yes	—
66	C₂×C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
67	C₂×C₄²×C₁₂₇	C₂³	1	2	yes	—
68	C₄⋊C₈×C₁₂₇	—	2	2	no	—
69	C₄⋊C₈×C₁₂₇	—	2	8	no	—
70	C₄⋊C₈×C₁₂₇	—	2	2	no	—
71	C₂²⋊C₈×C₁₂₇	—	2	4	no	—
72	C₂²⋊C₈×C₁₂₇	—	2	8	no	—
73	C₂²⋊C₈×C₁₂₇	—	2	8	no	—
74	C₂²⋊C₈×C₁₂₇	—	2	4	no	—
75	#97×C₁₂₇	—	2	4	no	—
76	#97×C₁₂₇	D₄	1	8	yes	complemented
77	C₄×C₈×C₁₂₇	—	2	2	no	—
78	C₄×C₈×C₁₂₇	D₄	1	4	yes	complemented
79	C₂⁴×C₁₂₇	C₂²×C₄	1	2	yes	—
80	C₂⁴×C₁₂₇	C₂²×C₄	1	2	yes	—
81	C₂×D₄×C₁₂₇	—	2	8	no	—
82	C₂×D₄×C₁₂₇	—	2	2	no	—
83	C₂×D₄×C₁₂₇	—	2	8	no	—
84	C₂×D₄×C₁₂₇	—	2	2	no	—
85	C₂×D₄×C₁₂₇	—	2	2	no	—
86	C₂×D₄×C₁₂₇	—	2	2	no	—
87	C₂×D₄×C₁₂₇	—	2	2	no	—
88	C₂×D₄×C₁₂₇	—	4	4	no	—
89	C₂×D₄×C₁₂₇	M₁₆	1	4	yes	complemented
90	C₂²×C₄×C₁₂₇	—	2	2	no	—
91	C₂²×C₄×C₁₂₇	—	2	4	no	—
92	C₂²×C₄×C₁₂₇	—	2	4	no	—
93	C₂²×C₄×C₁₂₇	—	2	2	no	—
94	C₂²×C₄×C₁₂₇	—	2	2	no	—
95	C₂²×C₄×C₁₂₇	—	2	4	no	—
96	C₂²×C₄×C₁₂₇	—	2	2	no	—
97	C₂²×C₄×C₁₂₇	—	2	4	no	—
98	C₂²×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
99	C₂²×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
100	C₂²×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
101	C₂²×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
102	C₂²×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
103	C₂²×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
104	C₂²×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
105	M₁₆×C₁₂₇	—	4	8	no	—
106	C₂×C₈×C₁₂₇	—	2	2	no	—
107	C₂×C₈×C₁₂₇	—	2	2	no	—
108	C₂×C₈×C₁₂₇	—	2	2	no	—
109	C₂×C₈×C₁₂₇	—	2	2	no	—
110	C₂×C₈×C₁₂₇	—	2	4	no	—
111	C₂×C₈×C₁₂₇	—	2	4	no	—
112	C₂×C₈×C₁₂₇	—	2	2	no	—
113	C₂×C₈×C₁₂₇	—	4	8	no	—
114	C₂×C₈×C₁₂₇	C₂×D₄	1	2	yes	—
115	C₂×C₈×C₁₂₇	C₂×D₄	1	4	yes	—
116	C₂×C₈×C₁₂₇	C₂×D₄	1	2	yes	—
117	C₂×C₈×C₁₂₇	C₂×D₄	1	4	yes	—
118	C₄⋊C₄×C₁₂₇	—	2	8	no	—
119	C₄⋊C₄×C₁₂₇	—	2	2	no	—
120	C₄⋊C₄×C₁₂₇	—	2	2	no	—
121	C₂²⋊C₄×C₁₂₇	—	2	4	no	—
122	C₂²⋊C₄×C₁₂₇	—	2	8	no	—
123	C₂²⋊C₄×C₁₂₇	—	2	8	no	—
124	C₂²⋊C₄×C₁₂₇	—	2	4	no	—
125	C₄²×C₁₂₇	—	2	2	no	—
126	C₄²×C₁₂₇	—	2	2	no	—
127	C₄²×C₁₂₇	—	4	2	no	—
128	C₄²×C₁₂₇	C₂×D₄	1	2	yes	—
129	C₄²×C₁₂₇	C₂×D₄	1	4	yes	—
130	C₄²×C₁₂₇	#41	1	2	yes	—
131	C₄²×C₁₂₇	#41	1	4	yes	—
132	C₂³×C₁₂₇	—	2	2	no	—
133	C₂³×C₁₂₇	—	2	4	no	—
134	C₂³×C₁₂₇	—	2	2	no	—
135	C₂³×C₁₂₇	—	2	4	no	—
136	C₂³×C₁₂₇	—	2	2	no	—
137	C₂³×C₁₂₇	—	2	4	no	—
138	C₂³×C₁₂₇	—	2	2	no	—
139	C₂³×C₁₂₇	—	2	2	no	—
140	C₂³×C₁₂₇	—	2	4	no	—
141	C₂³×C₁₂₇	C₂×M₁₆	1	4	yes	—
142	C₂³×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
143	D₄×C₁₂₇	—	2	8	no	—
144	D₄×C₁₂₇	—	4	16	no	—
145	D₄×C₁₂₇	—	4	4	no	—
146	C₂×C₄×C₁₂₇	—	2	2	no	—
147	C₂×C₄×C₁₂₇	—	2	2	no	—
148	C₂×C₄×C₁₂₇	—	2	2	no	—
149	C₂×C₄×C₁₂₇	—	2	2	no	—
150	C₂×C₄×C₁₂₇	—	2	2	no	—
151	C₂×C₄×C₁₂₇	—	2	2	no	—
152	C₂×C₄×C₁₂₇	—	2	2	no	—
153	C₂×C₄×C₁₂₇	—	2	4	no	—
154	C₂×C₄×C₁₂₇	—	2	4	no	—
155	C₂×C₄×C₁₂₇	—	4	2	no	—
156	C₂×C₄×C₁₂₇	—	4	2	no	—
157	C₂×C₄×C₁₂₇	—	4	2	no	—
158	C₂×C₄×C₁₂₇	—	4	4	no	—
159	C₂×C₄×C₁₂₇	—	4	4	no	—
160	C₂×C₄×C₁₂₇	—	4	2	no	—
161	C₂×C₄×C₁₂₇	C₄×D₄	1	2	yes	—
162	C₂×C₄×C₁₂₇	C₄×D₄	1	4	yes	—
163	C₂×C₄×C₁₂₇	C₄×D₄	1	4	yes	—
164	C₂×C₄×C₁₂₇	C₄×D₄	1	2	yes	—
165	C₂×C₄×C₁₂₇	C₂×M₁₆	1	2	yes	—
166	C₂×C₄×C₁₂₇	#131	1	2	yes	—
167	C₂×C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
168	C₂×C₄×C₁₂₇	C₂²×D₄	1	2	yes	—
169	C₂×C₄×C₁₂₇	C₂×#41	1	1	yes	characteristic
170	C₂×C₄×C₁₂₇	C₂×#41	1	2	yes	—
171	C₈×C₁₂₇	—	2	4	no	—
172	C₈×C₁₂₇	—	2	2	no	—
173	C₈×C₁₂₇	—	4	2	no	—
174	C₈×C₁₂₇	—	4	2	no	—
175	C₈×C₁₂₇	#127	1	4	yes	complemented
176	C₂²×C₁₂₇	—	2	4	no	—
177	C₂²×C₁₂₇	—	2	8	no	—
178	C₂²×C₁₂₇	—	4	2	no	—
179	C₂²×C₁₂₇	—	4	2	no	—
180	C₂²×C₁₂₇	—	4	4	no	—
181	C₂²×C₁₂₇	—	4	2	no	—
182	C₂²×C₁₂₇	—	4	2	no	—
183	C₂²×C₁₂₇	—	4	2	no	—
184	C₂²×C₁₂₇	—	4	4	no	—
185	C₂²×C₁₂₇	—	4	4	no	—
186	C₂²×C₁₂₇	C₂×C₄×D₄	1	1	yes	characteristic
187	C₂²×C₁₂₇	C₂×C₄×D₄	1	2	yes	—
188	C₂²×C₁₂₇	C₂²×M₁₆	1	1	yes	characteristic
189	C₂²×C₁₂₇	C₂×#131	1	1	yes	characteristic
190	C₂²×C₁₂₇	#568	1	2	yes	—
191	C₄×C₁₂₇	—	2	2	no	—
192	C₄×C₁₂₇	—	2	4	no	—
193	C₄×C₁₂₇	—	2	4	no	—
194	C₄×C₁₂₇	—	4	2	no	—
195	C₄×C₁₂₇	—	4	2	no	—
196	C₄×C₁₂₇	#435	1	4	yes	complemented
197	C₄×C₁₂₇	C₂×#127	1	1	yes	characteristic
198	C₄×C₁₂₇	#532	1	2	yes	—
199	C₄×C₁₂₇	#532	1	1	yes	characteristic
200	#21682	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
201	C₂×C₁₂₇	—	4	4	no	—
202	C₂×C₁₂₇	—	8	2	no	—
203	C₂×C₁₂₇	C₄×#127	1	1	yes	characteristic
204	C₂×C₁₂₇	C₂×#435	1	1	yes	characteristic
205	C₂×C₁₂₇	#2932	1	1	yes	characteristic
206	C₂×C₁₂₇	D₄×M₁₆	1	2	yes	—
207	C₂×C₁₂₇	#2935	1	2	yes	—
208	C₂×#436	C₂×C₁₂₇	1	1	yes	characteristic, complemented
209	C₂×#435	C₂×C₁₂₇	1	4	yes	complemented
210	C₂×C₈×D₄	C₂×C₁₂₇	1	1	yes	characteristic, complemented
211	C₄×#127	C₂×C₁₂₇	1	1	yes	characteristic
212	#1955	C₂×C₁₂₇	1	4	yes	complemented
213	#1954	C₂×C₁₂₇	1	1	yes	characteristic, complemented
214	#1954	C₂×C₁₂₇	1	1	yes	characteristic, complemented
215	#1842	C₂×C₁₂₇	1	1	yes	characteristic, complemented
216	C₄×#97	C₂×C₁₂₇	1	1	yes	characteristic, complemented
217	C₁₂₇	#21682	1	1	yes	characteristic, direct product factor, Sylow
218	C₂²×M₁₆	C₂²×C₁₂₇	1	2	yes	—
219	C₂³×C₈	C₂²×C₁₂₇	1	2	yes	—
220	C₂×#127	C₄×C₁₂₇	1	1	yes	characteristic
221	C₂×C₄×D₄	C₂²×C₁₂₇	1	1	yes	characteristic
222	C₂×C₄×D₄	C₂²×C₁₂₇	1	4	yes	—
223	C₂×C₄×D₄	C₂²×C₁₂₇	1	1	yes	characteristic
224	#436	—	2	4	no	—
225	#435	—	2	16	no	—
226	C₈×D₄	—	2	4	no	—
227	C₂×C₄⋊C₈	C₂²×C₁₂₇	1	1	yes	characteristic
228	C₂×C₄⋊C₈	C₂²×C₁₂₇	1	4	yes	—
229	C₂×C₄⋊C₈	C₂²×C₁₂₇	1	1	yes	characteristic
230	C₂×C₂²⋊C₈	C₂²×C₁₂₇	1	2	yes	—
231	C₂×C₂²⋊C₈	C₂²×C₁₂₇	1	4	yes	—
232	C₂×C₂²⋊C₈	C₂²×C₁₂₇	1	4	yes	—
233	C₂×C₂²⋊C₈	C₂²×C₁₂₇	1	2	yes	—
234	C₂×#97	C₂²×C₁₂₇	1	2	yes	—
235	C₂×#97	C₂²×C₁₂₇	1	2	yes	—
236	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
237	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
238	#390	C₄×C₁₂₇	1	1	yes	characteristic
239	#390	C₄×C₁₂₇	1	1	yes	characteristic
240	#390	C₄×C₁₂₇	1	4	yes	—
241	#384	C₄×C₁₂₇	1	1	yes	characteristic
242	C₄³	C₂²×C₁₂₇	1	1	yes	characteristic
243	C₂²×D₄	C₂×C₄×C₁₂₇	1	1	yes	characteristic
244	C₂²×D₄	C₂×C₄×C₁₂₇	1	4	yes	—
245	C₂²×D₄	C₂×C₄×C₁₂₇	1	1	yes	characteristic
246	C₂³×C₄	C₂³×C₁₂₇	1	2	yes	—
247	C₂³×C₄	C₂³×C₁₂₇	1	2	yes	—
248	C₂×M₁₆	—	2	8	no	—
249	C₂×M₁₆	—	2	4	no	—
250	C₂²×C₈	—	2	4	no	—
251	C₂²×C₈	—	2	8	no	—
252	C₂²×C₈	C₂³×C₁₂₇	1	2	yes	—
253	C₂²×C₈	C₂³×C₁₂₇	1	2	yes	—
254	C₂²×C₈	C₂³×C₁₂₇	1	2	yes	—
255	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
256	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
257	#127	—	2	4	no	—
258	C₄×D₄	—	2	2	no	—
259	C₄×D₄	—	2	8	no	—
260	C₄×D₄	—	2	2	no	—
261	C₄×D₄	—	2	2	no	—
262	C₄×D₄	—	2	2	no	—
263	C₄×D₄	—	2	8	no	—
264	C₂×C₄⋊C₄	C₂×C₄×C₁₂₇	1	4	yes	—
265	C₂×C₄⋊C₄	C₂×C₄×C₁₂₇	1	1	yes	characteristic
266	C₂×C₄⋊C₄	C₂×C₄×C₁₂₇	1	1	yes	characteristic
267	C₂×C₂²⋊C₄	C₂×C₄×C₁₂₇	1	4	yes	—
268	C₂×C₂²⋊C₄	C₂×C₄×C₁₂₇	1	2	yes	—
269	C₂×C₂²⋊C₄	C₂×C₄×C₁₂₇	1	2	yes	—
270	C₂×C₂²⋊C₄	C₂×C₄×C₁₂₇	1	4	yes	—
271	C₂×C₄²	C₂×C₄×C₁₂₇	1	1	yes	characteristic
272	C₂×C₄²	C₂×C₄×C₁₂₇	1	1	yes	characteristic
273	C₂×C₄²	C₂×C₄×C₁₂₇	1	2	yes	—
274	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
275	C₂×C₄²	C₂³×C₁₂₇	1	2	yes	—
276	C₄⋊C₈	—	2	2	no	—
277	C₄⋊C₈	—	2	8	no	—
278	C₄⋊C₈	—	2	2	no	—
279	C₂²⋊C₈	—	2	4	no	—
280	C₂²⋊C₈	—	2	8	no	—
281	C₂²⋊C₈	—	2	8	no	—
282	C₂²⋊C₈	—	2	4	no	—
283	#97	—	2	4	no	—
284	#97	D₄×C₁₂₇	1	8	yes	complemented
285	C₄×C₈	—	2	2	no	—
286	C₄×C₈	D₄×C₁₂₇	1	4	yes	complemented
287	C₂⁴	C₂²×C₄×C₁₂₇	1	2	yes	—
288	C₂⁴	C₂²×C₄×C₁₂₇	1	2	yes	—
289	C₂×D₄	—	2	8	no	—
290	C₂×D₄	—	2	2	no	—
291	C₂×D₄	—	2	8	no	—
292	C₂×D₄	—	2	2	no	—
293	C₂×D₄	—	2	2	no	—
294	C₂×D₄	—	2	2	no	—
295	C₂×D₄	—	2	2	no	—
296	C₂×D₄	—	4	4	no	—
297	C₂×D₄	M₁₆×C₁₂₇	1	4	yes	complemented
298	C₂²×C₄	—	2	2	no	—
299	C₂²×C₄	—	2	4	no	—
300	C₂²×C₄	—	2	4	no	—
301	C₂²×C₄	—	2	2	no	—
302	C₂²×C₄	—	2	2	no	—
303	C₂²×C₄	—	2	4	no	—
304	C₂²×C₄	—	2	2	no	—
305	C₂²×C₄	—	2	4	no	—
306	C₂²×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
307	C₂²×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
308	C₂²×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
309	C₂²×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
310	C₂²×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
311	C₂²×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
312	C₂²×C₄	C₂⁴×C₁₂₇	1	1	yes	characteristic
313	M₁₆	—	4	8	no	—
314	C₂×C₈	—	2	2	no	—
315	C₂×C₈	—	2	2	no	—
316	C₂×C₈	—	2	2	no	—
317	C₂×C₈	—	2	2	no	—
318	C₂×C₈	—	2	4	no	—
319	C₂×C₈	—	2	4	no	—
320	C₂×C₈	—	2	2	no	—
321	C₂×C₈	—	4	8	no	—
322	C₂×C₈	C₂×D₄×C₁₂₇	1	2	yes	—
323	C₂×C₈	C₂×D₄×C₁₂₇	1	4	yes	—
324	C₂×C₈	C₂×D₄×C₁₂₇	1	2	yes	—
325	C₂×C₈	C₂×D₄×C₁₂₇	1	4	yes	—
326	C₄⋊C₄	—	2	8	no	—
327	C₄⋊C₄	—	2	2	no	—
328	C₄⋊C₄	—	2	2	no	—
329	C₂²⋊C₄	—	2	4	no	—
330	C₂²⋊C₄	—	2	8	no	—
331	C₂²⋊C₄	—	2	8	no	—
332	C₂²⋊C₄	—	2	4	no	—
333	C₄²	—	2	2	no	—
334	C₄²	—	2	2	no	—
335	C₄²	—	4	2	no	—
336	C₄²	C₂×D₄×C₁₂₇	1	2	yes	—
337	C₄²	C₂×D₄×C₁₂₇	1	4	yes	—
338	C₄²	#41×C₁₂₇	1	2	yes	—
339	C₄²	#41×C₁₂₇	1	4	yes	—
340	C₂³	—	2	2	no	—
341	C₂³	—	2	4	no	—
342	C₂³	—	2	2	no	—
343	C₂³	—	2	4	no	—
344	C₂³	—	2	2	no	—
345	C₂³	—	2	4	no	—
346	C₂³	—	2	2	no	—
347	C₂³	—	2	2	no	—
348	C₂³	—	2	4	no	—
349	C₂³	C₂×M₁₆×C₁₂₇	1	4	yes	—
350	C₂³	C₂³×C₄×C₁₂₇	1	1	yes	characteristic
351	D₄	—	2	8	no	—
352	D₄	—	4	16	no	—
353	D₄	—	4	4	no	—
354	C₂×C₄	—	2	2	no	—
355	C₂×C₄	—	2	2	no	—
356	C₂×C₄	—	2	2	no	—
357	C₂×C₄	—	2	2	no	—
358	C₂×C₄	—	2	2	no	—
359	C₂×C₄	—	2	2	no	—
360	C₂×C₄	—	2	2	no	—
361	C₂×C₄	—	2	4	no	—
362	C₂×C₄	—	2	4	no	—
363	C₂×C₄	—	4	2	no	—
364	C₂×C₄	—	4	2	no	—
365	C₂×C₄	—	4	2	no	—
366	C₂×C₄	—	4	4	no	—
367	C₂×C₄	—	4	4	no	—
368	C₂×C₄	—	4	2	no	—
369	C₂×C₄	C₄×D₄×C₁₂₇	1	2	yes	—
370	C₂×C₄	C₄×D₄×C₁₂₇	1	4	yes	—
371	C₂×C₄	C₄×D₄×C₁₂₇	1	4	yes	—
372	C₂×C₄	C₄×D₄×C₁₂₇	1	2	yes	—
373	C₂×C₄	C₂×M₁₆×C₁₂₇	1	2	yes	—
374	C₂×C₄	#131×C₁₂₇	1	2	yes	—
375	C₂×C₄	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
376	C₂×C₄	C₂²×D₄×C₁₂₇	1	2	yes	—
377	C₂×C₄	C₂×#41×C₁₂₇	1	1	yes	characteristic
378	C₂×C₄	C₂×#41×C₁₂₇	1	2	yes	—
379	C₈	—	2	4	no	—
380	C₈	—	2	2	no	—
381	C₈	—	4	2	no	—
382	C₈	—	4	2	no	—
383	C₈	#127×C₁₂₇	1	4	yes	complemented
384	C₂²	—	2	4	no	—
385	C₂²	—	2	8	no	—
386	C₂²	—	4	2	no	—
387	C₂²	—	4	4	no	—
388	C₂²	—	4	2	no	—
389	C₂²	—	4	2	no	—
390	C₂²	—	4	2	no	—
391	C₂²	—	4	4	no	—
392	C₂²	—	4	4	no	—
393	C₂²	—	4	2	no	—
394	C₂²	C₂×C₄×D₄×C₁₂₇	1	1	yes	characteristic
395	C₂²	C₂×C₄×D₄×C₁₂₇	1	2	yes	—
396	C₂²	C₂²×M₁₆×C₁₂₇	1	1	yes	characteristic
397	C₂²	C₂×#131×C₁₂₇	1	1	yes	characteristic
398	C₂²	#568×C₁₂₇	1	2	yes	—
399	C₄	—	2	2	no	—
400	C₄	—	2	4	no	—
401	C₄	—	2	4	no	—
402	C₄	—	4	2	no	—
403	C₄	—	4	2	no	—
404	C₄	#435×C₁₂₇	1	4	yes	complemented
405	C₄	C₂×#127×C₁₂₇	1	1	yes	characteristic
406	C₄	#532×C₁₂₇	1	2	yes	—
407	C₄	#532×C₁₂₇	1	1	yes	characteristic
408	C₂	—	4	4	no	—
409	C₂	—	8	2	no	—
410	C₂	C₄×#127×C₁₂₇	1	1	yes	characteristic
411	C₂	C₂×#435×C₁₂₇	1	1	yes	characteristic
412	C₂	C₁₂₇×#2932	1	1	yes	characteristic
413	C₂	D₄×M₁₆×C₁₂₇	1	2	yes	—
414	C₂	C₁₂₇×#2935	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₁₂₇	C₂⁴
commutator	C₂³	C₂³×C₄×C₁₂₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₂₇
socle	C₂³×C₁₂₇	C₂³×C₄
prime radical	C₂×#127×C₁₂₇	C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#436×C₁₂₇	C₂	semidirect
2. C₂×#435×C₁₂₇	C₂	semidirect
3. C₂×C₈×D₄×C₁₂₇	C₂	semidirect
5. C₁₂₇×#1955	C₂	semidirect
6. C₁₂₇×#1954	C₂	semidirect
7. C₁₂₇×#1954	C₂	semidirect
8. C₁₂₇×#1842	C₂	semidirect
9. C₄×#97×C₁₂₇	C₂	semidirect
76. #97×C₁₂₇	D₄	semidirect
78. C₄×C₈×C₁₂₇	D₄	semidirect
89. C₂×D₄×C₁₂₇	M₁₆	semidirect
175. C₈×C₁₂₇	#127	semidirect
196. C₄×C₁₂₇	#435	semidirect
200. #21682	C₁₂₇	direct (N × H)
208. C₂×#436	C₂×C₁₂₇	semidirect
209. C₂×#435	C₂×C₁₂₇	semidirect
210. C₂×C₈×D₄	C₂×C₁₂₇	semidirect
212. #1955	C₂×C₁₂₇	semidirect
213. #1954	C₂×C₁₂₇	semidirect
214. #1954	C₂×C₁₂₇	semidirect
215. #1842	C₂×C₁₂₇	semidirect
216. C₄×#97	C₂×C₁₂₇	semidirect
217. C₁₂₇	#21682	direct (N × H)
284. #97	D₄×C₁₂₇	semidirect
286. C₄×C₈	D₄×C₁₂₇	semidirect
297. C₂×D₄	M₁₆×C₁₂₇	semidirect
383. C₈	#127×C₁₂₇	semidirect
404. C₄	#435×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#436×C₁₂₇	2
C₂×#435×C₁₂₇	2
C₂×C₈×D₄×C₁₂₇	2
C₄×#127×C₁₂₇	2
C₁₂₇×#1955	2
C₁₂₇×#1954	2
C₁₂₇×#1842	2
C₄×#97×C₁₂₇	2
#21682	127

Subgroups — C127×#21682 #44585355

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#21682 #44585355