Subgroups — C₁₂₇×#20431 #44584104

Order 32512.

1146 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#423×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊C₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#405×C₁₂₇	C₂	1	2	yes	maximal
4	C₄×#117×C₁₂₇	C₂	1	1	yes	maximal, characteristic
5	C₁₂₇×#1768	C₂	1	4	yes	maximal, complemented
6	C₁₂₇×#1767	C₂	1	2	yes	maximal, complemented
7	C₄×C₄⋊C₈×C₁₂₇	C₂	1	2	yes	maximal, complemented
8	C₁₂₇×#1729	C₂	1	2	yes	maximal
9	C₂²×M₁₆×C₁₂₇	C₂²	1	2	yes	—
10	C₂³×C₈×C₁₂₇	C₂²	1	2	yes	—
11	C₂×#117×C₁₂₇	C₂²	1	1	yes	characteristic
12	C₂×#117×C₁₂₇	C₂²	1	1	yes	characteristic
13	C₂²×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
14	#423×C₁₂₇	C₄	1	8	yes	complemented
15	C₂×C₄⋊C₈×C₁₂₇	C₄	1	8	yes	complemented
16	C₂×C₄⋊C₈×C₁₂₇	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₈×C₁₂₇	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₈×C₁₂₇	C₂²	1	2	yes	—
19	C₂×C₄⋊C₈×C₁₂₇	C₂²	1	2	yes	—
20	C₂×C₄⋊C₈×C₁₂₇	C₂²	1	2	yes	—
21	#405×C₁₂₇	—	2	8	no	—
22	C₂×#97×C₁₂₇	C₂²	1	4	yes	—
23	C₂×C₄×C₈×C₁₂₇	C₂²	1	4	yes	—
24	C₄×C₄⋊C₄×C₁₂₇	C₄	1	2	yes	—
25	C₄×C₄⋊C₄×C₁₂₇	C₄	1	2	yes	—
26	C₄×C₂²⋊C₄×C₁₂₇	C₄	1	2	yes	—
27	C₄×C₂²⋊C₄×C₁₂₇	C₄	1	2	yes	—
28	#376×C₁₂₇	C₂²	1	2	yes	—
29	C₄³×C₁₂₇	C₂²	1	2	yes	—
30	#336×C₁₂₇	C₂²	1	4	yes	—
31	#336×C₁₂₇	C₂²	1	4	yes	—
32	C₂³×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
33	C₂³×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
34	C₂³×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
35	C₂×M₁₆×C₁₂₇	—	2	4	no	—
36	C₂×M₁₆×C₁₂₇	—	2	4	no	—
37	C₂×M₁₆×C₁₂₇	C₂×C₄	1	8	yes	—
38	C₂²×C₈×C₁₂₇	—	2	4	no	—
39	C₂²×C₈×C₁₂₇	—	2	4	no	—
40	C₂²×C₈×C₁₂₇	C₂×C₄	1	8	yes	—
41	C₂²×C₈×C₁₂₇	C₂³	1	2	yes	—
42	C₂²×C₈×C₁₂₇	C₂³	1	2	yes	—
43	C₂²×C₈×C₁₂₇	C₂³	1	4	yes	—
44	#117×C₁₂₇	—	2	2	no	—
45	#117×C₁₂₇	—	2	2	no	—
46	#117×C₁₂₇	D₄	1	4	yes	—
47	#117×C₁₂₇	Q₈	1	4	yes	—
48	C₂×C₄⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
49	C₂×C₄⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
50	C₂×C₂²⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
51	C₂×C₂²⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
52	C₂×C₄²×C₁₂₇	C₂×C₄	1	4	yes	—
53	C₂×C₄²×C₁₂₇	C₂×C₄	1	4	yes	—
54	C₂×C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
56	C₂×C₄²×C₁₂₇	C₂×C₄	1	4	yes	—
57	C₂×C₄²×C₁₂₇	C₂×C₄	1	4	yes	—
58	C₂×C₄²×C₁₂₇	C₂³	1	2	yes	—
59	C₂×C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
60	C₂×C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
61	C₂×C₄²×C₁₂₇	C₂³	1	2	yes	—
62	C₄⋊C₈×C₁₂₇	C₂×C₄	1	4	yes	—
63	C₄⋊C₈×C₁₂₇	C₂×C₄	1	4	yes	—
64	C₄⋊C₈×C₁₂₇	C₂×C₄	1	8	yes	—
65	C₄⋊C₈×C₁₂₇	C₂×C₄	1	8	yes	—
66	C₄⋊C₈×C₁₂₇	C₂×C₄	1	8	yes	—
67	#97×C₁₂₇	—	2	8	no	—
68	C₄×C₈×C₁₂₇	—	2	8	no	—
69	#95×C₁₂₇	C₂×C₄	1	4	yes	—
70	#95×C₁₂₇	C₂×C₄	1	4	yes	—
71	C₂⁴×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₂₇	—	2	4	no	—
73	C₂²×C₄×C₁₂₇	—	4	2	no	—
74	C₂²×C₄×C₁₂₇	C₄²	1	4	yes	—
75	C₂²×C₄×C₁₂₇	C₄²	1	4	yes	—
76	C₂²×C₄×C₁₂₇	C₂²×C₄	1	4	yes	—
77	C₂²×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₄×C₁₂₇	C₂²×C₄	1	4	yes	—
81	C₂²×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
84	C₂²×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
85	C₂²×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
86	C₂²×C₄×C₁₂₇	C₂×Q₈	1	2	yes	—
87	C₂²×C₄×C₁₂₇	#41	1	2	yes	—
88	C₂²×C₄×C₁₂₇	#41	1	2	yes	—
89	C₂²×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
90	M₁₆×C₁₂₇	—	2	16	no	—
91	C₂×C₈×C₁₂₇	—	2	16	no	—
92	C₂×C₈×C₁₂₇	—	2	2	no	—
93	C₂×C₈×C₁₂₇	—	2	4	no	—
94	C₂×C₈×C₁₂₇	—	2	2	no	—
95	C₂×C₈×C₁₂₇	—	2	2	no	—
96	C₂×C₈×C₁₂₇	—	2	2	no	—
97	C₂×C₈×C₁₂₇	—	2	4	no	—
98	C₂×C₈×C₁₂₇	C₂²×C₄	1	4	yes	—
99	C₂×C₈×C₁₂₇	C₂²×C₄	1	4	yes	—
100	C₂×C₈×C₁₂₇	C₂²×C₄	1	8	yes	—
101	C₄⋊C₄×C₁₂₇	—	2	4	no	—
102	C₄⋊C₄×C₁₂₇	C₄⋊C₄	1	8	yes	—
103	C₂²⋊C₄×C₁₂₇	—	2	4	no	—
104	C₂²⋊C₄×C₁₂₇	C₄⋊C₄	1	8	yes	—
105	C₄²×C₁₂₇	—	2	8	no	—
106	C₄²×C₁₂₇	—	2	2	no	—
107	C₄²×C₁₂₇	—	2	2	no	—
108	C₄²×C₁₂₇	—	4	4	no	—
109	C₄²×C₁₂₇	C₄²	1	4	yes	—
110	C₄²×C₁₂₇	C₄²	1	4	yes	—
111	C₄²×C₁₂₇	C₂²×C₄	1	2	yes	—
112	C₄²×C₁₂₇	C₂²×C₄	1	2	yes	—
113	C₄²×C₁₂₇	C₂²×C₄	1	2	yes	—
114	C₄²×C₁₂₇	C₂²×C₄	1	2	yes	—
115	C₄²×C₁₂₇	C₂×D₄	1	4	yes	—
116	C₄²×C₁₂₇	C₂×Q₈	1	4	yes	—
117	C₂³×C₁₂₇	—	2	2	no	—
118	C₂³×C₁₂₇	—	2	2	no	—
119	C₂³×C₁₂₇	C₂×C₄²	1	2	yes	—
120	C₂³×C₁₂₇	C₂×C₄⋊C₄	1	2	yes	—
121	C₂³×C₁₂₇	#117	1	2	yes	—
122	C₂³×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
123	C₂×C₄×C₁₂₇	—	2	4	no	—
124	C₂×C₄×C₁₂₇	—	2	2	no	—
125	C₂×C₄×C₁₂₇	—	2	2	no	—
126	C₂×C₄×C₁₂₇	—	2	4	no	—
127	C₂×C₄×C₁₂₇	—	2	2	no	—
128	C₂×C₄×C₁₂₇	—	2	8	no	—
129	C₂×C₄×C₁₂₇	—	2	4	no	—
130	C₂×C₄×C₁₂₇	—	2	4	no	—
131	C₂×C₄×C₁₂₇	—	2	4	no	—
132	C₂×C₄×C₁₂₇	—	4	4	no	—
133	C₂×C₄×C₁₂₇	C₂×C₄²	1	2	yes	—
134	C₂×C₄×C₁₂₇	C₂×C₄²	1	2	yes	—
135	C₂×C₄×C₁₂₇	C₂×C₄²	1	2	yes	—
136	C₂×C₄×C₁₂₇	C₂×C₄²	1	2	yes	—
137	C₂×C₄×C₁₂₇	C₂×C₄²	1	2	yes	—
138	C₂×C₄×C₁₂₇	C₂×C₄⋊C₄	1	8	yes	—
139	C₂×C₄×C₁₂₇	C₂×C₄⋊C₄	1	2	yes	—
140	C₂×C₄×C₁₂₇	#117	1	2	yes	—
141	C₂×C₄×C₁₂₇	C₄×D₄	1	4	yes	—
142	C₂×C₄×C₁₂₇	C₄×D₄	1	4	yes	—
143	C₂×C₄×C₁₂₇	C₄×Q₈	1	4	yes	—
144	C₂×C₄×C₁₂₇	C₄×Q₈	1	4	yes	—
145	C₂×C₄×C₁₂₇	#131	1	4	yes	—
146	C₂×C₄×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
147	C₂×C₄×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
148	C₂×C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
149	C₂×C₄×C₁₂₇	C₂²×Q₈	1	1	yes	characteristic
150	C₂×C₄×C₁₂₇	C₂×#41	1	1	yes	characteristic
151	C₂×C₄×C₁₂₇	C₂×#41	1	1	yes	characteristic
152	C₈×C₁₂₇	—	2	4	no	—
153	C₈×C₁₂₇	—	2	4	no	—
154	C₈×C₁₂₇	—	2	8	no	—
155	C₂²×C₁₂₇	—	2	2	no	—
156	C₂²×C₁₂₇	—	2	2	no	—
157	C₂²×C₁₂₇	—	2	4	no	—
158	C₂²×C₁₂₇	—	2	4	no	—
159	C₂²×C₁₂₇	C₄×C₄⋊C₄	1	4	yes	—
160	C₂²×C₁₂₇	C₂²×C₄²	1	1	yes	characteristic
161	C₂²×C₁₂₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
162	C₂²×C₁₂₇	C₂×#117	1	1	yes	characteristic
163	C₂²×C₁₂₇	C₂×#131	1	2	yes	—
164	C₂²×C₁₂₇	#568	1	2	yes	—
165	C₄×C₁₂₇	—	2	2	no	—
166	C₄×C₁₂₇	—	2	2	no	—
167	C₄×C₁₂₇	—	2	2	no	—
168	C₄×C₁₂₇	—	2	8	no	—
169	C₄×C₁₂₇	—	4	4	no	—
170	C₄×C₁₂₇	C₄×C₄⋊C₄	1	4	yes	—
171	C₄×C₁₂₇	#405	1	4	yes	complemented
172	C₄×C₁₂₇	C₂×C₄×D₄	1	1	yes	characteristic
173	C₄×C₁₂₇	C₂×C₄×D₄	1	1	yes	characteristic
174	C₄×C₁₂₇	C₂×C₄×Q₈	1	1	yes	characteristic
175	C₄×C₁₂₇	C₂×C₄×Q₈	1	1	yes	characteristic
176	#20431	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
177	C₂×C₁₂₇	—	2	4	no	—
178	C₂×C₁₂₇	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
179	C₂×C₁₂₇	C₂×#405	1	1	yes	characteristic
180	C₂×C₁₂₇	#2873	1	1	yes	characteristic
181	C₂×C₁₂₇	#2906	1	2	yes	—
182	C₂×C₁₂₇	#2925	1	2	yes	—
183	C₂×#423	C₂×C₁₂₇	1	1	yes	characteristic
184	C₂²×C₄⋊C₈	C₂×C₁₂₇	1	1	yes	characteristic
185	C₂×#405	C₂×C₁₂₇	1	2	yes	—
186	C₄×#117	C₂×C₁₂₇	1	1	yes	characteristic
187	#1768	C₂×C₁₂₇	1	4	yes	complemented
188	#1767	C₂×C₁₂₇	1	2	yes	complemented
189	C₄×C₄⋊C₈	C₂×C₁₂₇	1	2	yes	complemented
190	#1729	C₂×C₁₂₇	1	2	yes	—
191	C₁₂₇	#20431	1	1	yes	characteristic, direct product factor, Sylow
192	C₂²×M₁₆	C₂²×C₁₂₇	1	2	yes	—
193	C₂³×C₈	C₂²×C₁₂₇	1	2	yes	—
194	C₂×#117	C₂²×C₁₂₇	1	1	yes	characteristic
195	C₂×#117	C₂²×C₁₂₇	1	1	yes	characteristic
196	C₂²×C₄²	C₂²×C₁₂₇	1	1	yes	characteristic
197	#423	C₄×C₁₂₇	1	8	yes	complemented
198	C₂×C₄⋊C₈	C₄×C₁₂₇	1	8	yes	complemented
199	C₂×C₄⋊C₈	C₂²×C₁₂₇	1	1	yes	characteristic
200	C₂×C₄⋊C₈	C₂²×C₁₂₇	1	1	yes	characteristic
201	C₂×C₄⋊C₈	C₂²×C₁₂₇	1	2	yes	—
202	C₂×C₄⋊C₈	C₂²×C₁₂₇	1	2	yes	—
203	C₂×C₄⋊C₈	C₂²×C₁₂₇	1	2	yes	—
204	#405	—	2	8	no	—
205	C₂×#97	C₂²×C₁₂₇	1	4	yes	—
206	C₂×C₄×C₈	C₂²×C₁₂₇	1	4	yes	—
207	C₄×C₄⋊C₄	C₄×C₁₂₇	1	2	yes	—
208	C₄×C₄⋊C₄	C₄×C₁₂₇	1	2	yes	—
209	C₄×C₂²⋊C₄	C₄×C₁₂₇	1	2	yes	—
210	C₄×C₂²⋊C₄	C₄×C₁₂₇	1	2	yes	—
211	#376	C₂²×C₁₂₇	1	2	yes	—
212	C₄³	C₂²×C₁₂₇	1	2	yes	—
213	#336	C₂²×C₁₂₇	1	4	yes	—
214	#336	C₂²×C₁₂₇	1	4	yes	—
215	C₂³×C₄	C₂×C₄×C₁₂₇	1	1	yes	characteristic
216	C₂³×C₄	C₂×C₄×C₁₂₇	1	1	yes	characteristic
217	C₂³×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
218	C₂×M₁₆	—	2	4	no	—
219	C₂×M₁₆	—	2	4	no	—
220	C₂×M₁₆	C₂×C₄×C₁₂₇	1	8	yes	—
221	C₂²×C₈	—	2	4	no	—
222	C₂²×C₈	—	2	4	no	—
223	C₂²×C₈	C₂×C₄×C₁₂₇	1	8	yes	—
224	C₂²×C₈	C₂³×C₁₂₇	1	2	yes	—
225	C₂²×C₈	C₂³×C₁₂₇	1	2	yes	—
226	C₂²×C₈	C₂³×C₁₂₇	1	4	yes	—
227	#117	—	2	2	no	—
228	#117	—	2	2	no	—
229	#117	D₄×C₁₂₇	1	4	yes	—
230	#117	Q₈×C₁₂₇	1	4	yes	—
231	C₂×C₄⋊C₄	C₂×C₄×C₁₂₇	1	2	yes	—
232	C₂×C₄⋊C₄	C₂×C₄×C₁₂₇	1	2	yes	—
233	C₂×C₂²⋊C₄	C₂×C₄×C₁₂₇	1	2	yes	—
234	C₂×C₂²⋊C₄	C₂×C₄×C₁₂₇	1	2	yes	—
235	C₂×C₄²	C₂×C₄×C₁₂₇	1	4	yes	—
236	C₂×C₄²	C₂×C₄×C₁₂₇	1	4	yes	—
237	C₂×C₄²	C₂×C₄×C₁₂₇	1	1	yes	characteristic
238	C₂×C₄²	C₂×C₄×C₁₂₇	1	1	yes	characteristic
239	C₂×C₄²	C₂×C₄×C₁₂₇	1	4	yes	—
240	C₂×C₄²	C₂×C₄×C₁₂₇	1	4	yes	—
241	C₂×C₄²	C₂³×C₁₂₇	1	2	yes	—
242	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
243	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
244	C₂×C₄²	C₂³×C₁₂₇	1	2	yes	—
245	C₄⋊C₈	C₂×C₄×C₁₂₇	1	4	yes	—
246	C₄⋊C₈	C₂×C₄×C₁₂₇	1	4	yes	—
247	C₄⋊C₈	C₂×C₄×C₁₂₇	1	8	yes	—
248	C₄⋊C₈	C₂×C₄×C₁₂₇	1	8	yes	—
249	C₄⋊C₈	C₂×C₄×C₁₂₇	1	8	yes	—
250	#97	—	2	8	no	—
251	C₄×C₈	—	2	8	no	—
252	#95	C₂×C₄×C₁₂₇	1	4	yes	—
253	#95	C₂×C₄×C₁₂₇	1	4	yes	—
254	C₂⁴	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
255	C₂²×C₄	—	2	4	no	—
256	C₂²×C₄	—	4	2	no	—
257	C₂²×C₄	C₄²×C₁₂₇	1	4	yes	—
258	C₂²×C₄	C₄²×C₁₂₇	1	4	yes	—
259	C₂²×C₄	C₂²×C₄×C₁₂₇	1	4	yes	—
260	C₂²×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
261	C₂²×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
262	C₂²×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
263	C₂²×C₄	C₂²×C₄×C₁₂₇	1	4	yes	—
264	C₂²×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
265	C₂²×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
266	C₂²×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
267	C₂²×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
268	C₂²×C₄	C₂×D₄×C₁₂₇	1	2	yes	—
269	C₂²×C₄	C₂×Q₈×C₁₂₇	1	2	yes	—
270	C₂²×C₄	#41×C₁₂₇	1	2	yes	—
271	C₂²×C₄	#41×C₁₂₇	1	2	yes	—
272	C₂²×C₄	C₂⁴×C₁₂₇	1	1	yes	characteristic
273	M₁₆	—	2	16	no	—
274	C₂×C₈	—	2	16	no	—
275	C₂×C₈	—	2	2	no	—
276	C₂×C₈	—	2	4	no	—
277	C₂×C₈	—	2	2	no	—
278	C₂×C₈	—	2	2	no	—
279	C₂×C₈	—	2	2	no	—
280	C₂×C₈	—	2	4	no	—
281	C₂×C₈	C₂²×C₄×C₁₂₇	1	4	yes	—
282	C₂×C₈	C₂²×C₄×C₁₂₇	1	4	yes	—
283	C₂×C₈	C₂²×C₄×C₁₂₇	1	8	yes	—
284	C₄⋊C₄	—	2	4	no	—
285	C₄⋊C₄	C₄⋊C₄×C₁₂₇	1	8	yes	—
286	C₂²⋊C₄	—	2	4	no	—
287	C₂²⋊C₄	C₄⋊C₄×C₁₂₇	1	8	yes	—
288	C₄²	—	2	8	no	—
289	C₄²	—	2	2	no	—
290	C₄²	—	2	2	no	—
291	C₄²	—	4	4	no	—
292	C₄²	C₄²×C₁₂₇	1	4	yes	—
293	C₄²	C₄²×C₁₂₇	1	4	yes	—
294	C₄²	C₂²×C₄×C₁₂₇	1	2	yes	—
295	C₄²	C₂²×C₄×C₁₂₇	1	2	yes	—
296	C₄²	C₂²×C₄×C₁₂₇	1	2	yes	—
297	C₄²	C₂²×C₄×C₁₂₇	1	2	yes	—
298	C₄²	C₂×D₄×C₁₂₇	1	4	yes	—
299	C₄²	C₂×Q₈×C₁₂₇	1	4	yes	—
300	C₂³	—	2	2	no	—
301	C₂³	—	2	2	no	—
302	C₂³	C₂×C₄²×C₁₂₇	1	2	yes	—
303	C₂³	C₂×C₄⋊C₄×C₁₂₇	1	2	yes	—
304	C₂³	#117×C₁₂₇	1	2	yes	—
305	C₂³	C₂³×C₄×C₁₂₇	1	1	yes	characteristic
306	C₂×C₄	—	2	4	no	—
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	2	4	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	2	8	no	—
312	C₂×C₄	—	2	4	no	—
313	C₂×C₄	—	2	4	no	—
314	C₂×C₄	—	2	4	no	—
315	C₂×C₄	—	4	4	no	—
316	C₂×C₄	C₂×C₄²×C₁₂₇	1	2	yes	—
317	C₂×C₄	C₂×C₄²×C₁₂₇	1	2	yes	—
318	C₂×C₄	C₂×C₄²×C₁₂₇	1	2	yes	—
319	C₂×C₄	C₂×C₄²×C₁₂₇	1	2	yes	—
320	C₂×C₄	C₂×C₄²×C₁₂₇	1	2	yes	—
321	C₂×C₄	C₂×C₄⋊C₄×C₁₂₇	1	8	yes	—
322	C₂×C₄	C₂×C₄⋊C₄×C₁₂₇	1	2	yes	—
323	C₂×C₄	#117×C₁₂₇	1	2	yes	—
324	C₂×C₄	C₄×D₄×C₁₂₇	1	4	yes	—
325	C₂×C₄	C₄×D₄×C₁₂₇	1	4	yes	—
326	C₂×C₄	C₄×Q₈×C₁₂₇	1	4	yes	—
327	C₂×C₄	C₄×Q₈×C₁₂₇	1	4	yes	—
328	C₂×C₄	#131×C₁₂₇	1	4	yes	—
329	C₂×C₄	C₂³×C₄×C₁₂₇	1	1	yes	characteristic
330	C₂×C₄	C₂³×C₄×C₁₂₇	1	1	yes	characteristic
331	C₂×C₄	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
332	C₂×C₄	C₂²×Q₈×C₁₂₇	1	1	yes	characteristic
333	C₂×C₄	C₂×#41×C₁₂₇	1	1	yes	characteristic
334	C₂×C₄	C₂×#41×C₁₂₇	1	1	yes	characteristic
335	C₈	—	2	4	no	—
336	C₈	—	2	4	no	—
337	C₈	—	2	8	no	—
338	C₂²	—	2	2	no	—
339	C₂²	—	2	2	no	—
340	C₂²	—	2	4	no	—
341	C₂²	—	2	4	no	—
342	C₂²	C₄×C₄⋊C₄×C₁₂₇	1	4	yes	—
343	C₂²	C₂²×C₄²×C₁₂₇	1	1	yes	characteristic
344	C₂²	C₂²×C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
345	C₂²	C₂×#117×C₁₂₇	1	1	yes	characteristic
346	C₂²	C₂×#131×C₁₂₇	1	2	yes	—
347	C₂²	#568×C₁₂₇	1	2	yes	—
348	C₄	—	2	2	no	—
349	C₄	—	2	2	no	—
350	C₄	—	2	2	no	—
351	C₄	—	2	8	no	—
352	C₄	—	4	4	no	—
353	C₄	C₄×C₄⋊C₄×C₁₂₇	1	4	yes	—
354	C₄	#405×C₁₂₇	1	4	yes	complemented
355	C₄	C₂×C₄×D₄×C₁₂₇	1	1	yes	characteristic
356	C₄	C₂×C₄×D₄×C₁₂₇	1	1	yes	characteristic
357	C₄	C₂×C₄×Q₈×C₁₂₇	1	1	yes	characteristic
358	C₄	C₂×C₄×Q₈×C₁₂₇	1	1	yes	characteristic
359	C₂	—	2	4	no	—
360	C₂	C₂×C₄×C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
361	C₂	C₂×#405×C₁₂₇	1	1	yes	characteristic
362	C₂	C₁₂₇×#2873	1	1	yes	characteristic
363	C₂	C₁₂₇×#2906	1	2	yes	—
364	C₂	C₁₂₇×#2925	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₁₂₇	C₂⁴
commutator	C₂²	C₂²×C₄²×C₁₂₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₂₇
socle	C₂³×C₁₂₇	C₂³×C₄
prime radical	C₂⁴×C₁₂₇	C₂²×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
5. C₁₂₇×#1768	C₂	semidirect
6. C₁₂₇×#1767	C₂	semidirect
7. C₄×C₄⋊C₈×C₁₂₇	C₂	semidirect
14. #423×C₁₂₇	C₄	semidirect
15. C₂×C₄⋊C₈×C₁₂₇	C₄	semidirect
171. C₄×C₁₂₇	#405	semidirect
176. #20431	C₁₂₇	direct (N × H)
187. #1768	C₂×C₁₂₇	semidirect
188. #1767	C₂×C₁₂₇	semidirect
189. C₄×C₄⋊C₈	C₂×C₁₂₇	semidirect
191. C₁₂₇	#20431	direct (N × H)
197. #423	C₄×C₁₂₇	semidirect
198. C₂×C₄⋊C₈	C₄×C₁₂₇	semidirect
354. C₄	#405×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#423×C₁₂₇	2
C₂²×C₄⋊C₈×C₁₂₇	2
C₂×#405×C₁₂₇	2
C₄×#117×C₁₂₇	2
C₁₂₇×#1768	2
C₁₂₇×#1767	2
C₄×C₄⋊C₈×C₁₂₇	2
C₁₂₇×#1729	2
#20431	127

Subgroups — C127×#20431 #44584104

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#20431 #44584104