Subgroups — C₁₂₇×#20472 #44584145

Order 32512.

1050 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#431×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×M₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	C₄×#117×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₁₂₇×#1771	C₂	1	4	yes	maximal, complemented
5	C₁₂₇×#1753	C₂	1	4	yes	maximal
6	C₁₂₇×#1744	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₂₇×#1744	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#97×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄²×C₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×M₁₆×C₁₂₇	C₂²	1	2	yes	—
11	C₂×#117×C₁₂₇	C₄	1	2	yes	—
12	C₂²×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
13	#431×C₁₂₇	C₄	1	8	yes	complemented
14	C₂×C₄⋊C₈×C₁₂₇	C₂²	1	2	yes	—
15	C₂×C₂²⋊C₈×C₁₂₇	C₂²	1	2	yes	—
16	C₄×M₁₆×C₁₂₇	C₄	1	8	yes	complemented
17	C₂×#97×C₁₂₇	C₂²	1	4	yes	—
18	C₂×#97×C₁₂₇	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₁₂₇	C₂²	1	4	yes	—
20	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
24	C₄×C₄⋊C₄×C₁₂₇	C₄	1	2	yes	—
25	C₄×C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
26	C₄×C₂²⋊C₄×C₁₂₇	C₄	1	2	yes	—
27	C₄×C₂²⋊C₄×C₁₂₇	C₂²	1	2	yes	—
28	#376×C₁₂₇	C₄	1	2	yes	—
29	C₄³×C₁₂₇	C₂²	1	1	yes	characteristic
30	C₄³×C₁₂₇	C₂²	1	1	yes	characteristic
31	#336×C₁₂₇	C₂²	1	4	yes	—
32	#336×C₁₂₇	C₂²	1	4	yes	—
33	C₂³×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
34	C₂³×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
35	C₂³×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
36	C₂×M₁₆×C₁₂₇	—	2	8	no	—
37	C₂×M₁₆×C₁₂₇	C₂×C₄	1	8	yes	—
38	C₂²×C₈×C₁₂₇	C₂³	1	2	yes	—
39	C₂²×C₈×C₁₂₇	C₂³	1	2	yes	—
40	C₂²×C₈×C₁₂₇	C₂³	1	2	yes	—
41	C₂²×C₈×C₁₂₇	C₂³	1	2	yes	—
42	#117×C₁₂₇	—	2	8	no	—
43	C₂×C₄⋊C₄×C₁₂₇	C₂×C₄	1	4	yes	—
44	C₂×C₂²⋊C₄×C₁₂₇	C₂×C₄	1	4	yes	—
45	C₂×C₄²×C₁₂₇	C₂×C₄	1	2	yes	—
46	C₂×C₄²×C₁₂₇	C₂×C₄	1	2	yes	—
47	C₂×C₄²×C₁₂₇	C₂×C₄	1	2	yes	—
48	C₂×C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄²×C₁₂₇	C₂×C₄	1	4	yes	—
51	C₂×C₄²×C₁₂₇	C₂×C₄	1	4	yes	—
52	C₂×C₄²×C₁₂₇	C₂×C₄	1	2	yes	—
53	C₂×C₄²×C₁₂₇	C₂³	1	2	yes	—
54	C₂×C₄²×C₁₂₇	C₂³	1	2	yes	—
55	C₂×C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
56	C₂×C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
57	C₄⋊C₈×C₁₂₇	C₂×C₄	1	8	yes	—
58	C₂²⋊C₈×C₁₂₇	C₂×C₄	1	8	yes	—
59	#97×C₁₂₇	—	2	8	no	—
60	#97×C₁₂₇	C₂×C₄	1	8	yes	—
61	C₄×C₈×C₁₂₇	—	2	8	no	—
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	4	yes	—
65	C₄×C₈×C₁₂₇	C₂×C₄	1	4	yes	—
66	#95×C₁₂₇	C₂×C₄	1	4	yes	—
67	#95×C₁₂₇	C₂×C₄	1	4	yes	—
68	C₂⁴×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₂₇	—	2	4	no	—
70	C₂²×C₄×C₁₂₇	—	2	4	no	—
71	C₂²×C₄×C₁₂₇	—	2	4	no	—
72	C₂²×C₄×C₁₂₇	C₄²	1	4	yes	—
73	C₂²×C₄×C₁₂₇	C₄²	1	4	yes	—
74	C₂²×C₄×C₁₂₇	C₂²×C₄	1	4	yes	—
75	C₂²×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
79	C₂²×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
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	C₂⁴	1	1	yes	characteristic
84	M₁₆×C₁₂₇	—	2	16	no	—
85	C₂×C₈×C₁₂₇	—	2	4	no	—
86	C₂×C₈×C₁₂₇	—	2	4	no	—
87	C₂×C₈×C₁₂₇	—	2	4	no	—
88	C₂×C₈×C₁₂₇	C₂²×C₄	1	4	yes	—
89	C₂×C₈×C₁₂₇	C₂²×C₄	1	4	yes	—
90	C₂×C₈×C₁₂₇	C₂²×C₄	1	4	yes	—
91	C₂×C₈×C₁₂₇	C₂²×C₄	1	4	yes	—
92	C₂×C₈×C₁₂₇	#41	1	8	yes	—
93	C₄⋊C₄×C₁₂₇	—	2	8	no	—
94	C₂²⋊C₄×C₁₂₇	—	2	8	no	—
95	C₄²×C₁₂₇	—	2	4	no	—
96	C₄²×C₁₂₇	—	2	4	no	—
97	C₄²×C₁₂₇	—	2	4	no	—
98	C₄²×C₁₂₇	—	2	4	no	—
99	C₄²×C₁₂₇	—	2	4	no	—
100	C₄²×C₁₂₇	C₄²	1	4	yes	—
101	C₄²×C₁₂₇	C₄²	1	4	yes	—
102	C₄²×C₁₂₇	M₁₆	1	8	yes	complemented
103	C₄²×C₁₂₇	C₂²×C₄	1	2	yes	—
104	C₄²×C₁₂₇	C₂²×C₄	1	2	yes	—
105	C₄²×C₁₂₇	C₂²×C₄	1	2	yes	—
106	C₄²×C₁₂₇	C₂²×C₄	1	2	yes	—
107	C₂³×C₁₂₇	—	2	2	no	—
108	C₂³×C₁₂₇	—	2	2	no	—
109	C₂³×C₁₂₇	—	2	2	no	—
110	C₂³×C₁₂₇	C₂×C₄²	1	2	yes	—
111	C₂³×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
112	C₂×C₄×C₁₂₇	—	2	4	no	—
113	C₂×C₄×C₁₂₇	—	2	2	no	—
114	C₂×C₄×C₁₂₇	—	2	2	no	—
115	C₂×C₄×C₁₂₇	—	2	4	no	—
116	C₂×C₄×C₁₂₇	—	2	2	no	—
117	C₂×C₄×C₁₂₇	—	2	4	no	—
118	C₂×C₄×C₁₂₇	—	2	4	no	—
119	C₂×C₄×C₁₂₇	—	2	8	no	—
120	C₂×C₄×C₁₂₇	—	2	8	no	—
121	C₂×C₄×C₁₂₇	—	2	4	no	—
122	C₂×C₄×C₁₂₇	—	2	4	no	—
123	C₂×C₄×C₁₂₇	—	2	4	no	—
124	C₂×C₄×C₁₂₇	C₂×C₄²	1	2	yes	—
125	C₂×C₄×C₁₂₇	C₂×C₄²	1	2	yes	—
126	C₂×C₄×C₁₂₇	C₂×C₄²	1	2	yes	—
127	C₂×C₄×C₁₂₇	C₂×C₄²	1	2	yes	—
128	C₂×C₄×C₁₂₇	C₂×C₄²	1	2	yes	—
129	C₂×C₄×C₁₂₇	C₂×M₁₆	1	8	yes	—
130	C₂×C₄×C₁₂₇	C₂×M₁₆	1	4	yes	—
131	C₂×C₄×C₁₂₇	#131	1	4	yes	—
132	C₂×C₄×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
133	C₂×C₄×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
134	C₂×C₄×C₁₂₇	C₂×#41	1	4	yes	—
135	C₈×C₁₂₇	—	2	4	no	—
136	C₈×C₁₂₇	—	2	4	no	—
137	C₈×C₁₂₇	—	2	4	no	—
138	C₈×C₁₂₇	#117	1	8	yes	complemented
139	C₂²×C₁₂₇	—	2	2	no	—
140	C₂²×C₁₂₇	—	2	2	no	—
141	C₂²×C₁₂₇	—	2	2	no	—
142	C₂²×C₁₂₇	—	4	4	no	—
143	C₂²×C₁₂₇	C₂²×C₄²	1	1	yes	characteristic
144	C₂²×C₁₂₇	C₄×#41	1	2	yes	—
145	C₂²×C₁₂₇	C₂²×M₁₆	1	2	yes	—
146	C₂²×C₁₂₇	C₂×#131	1	2	yes	—
147	C₄×C₁₂₇	—	2	2	no	—
148	C₄×C₁₂₇	—	2	2	no	—
149	C₄×C₁₂₇	—	2	8	no	—
150	C₄×C₁₂₇	—	2	2	no	—
151	C₄×C₁₂₇	—	4	4	no	—
152	C₄×C₁₂₇	C₄×M₁₆	1	4	yes	—
153	C₄×C₁₂₇	#405	1	4	yes	—
154	C₄×C₁₂₇	C₂×#117	1	2	yes	—
155	C₄×C₁₂₇	C₄×#41	1	2	yes	—
156	#20472	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
157	C₂×C₁₂₇	—	4	2	no	—
158	C₂×C₁₂₇	C₄×#117	1	1	yes	characteristic
159	C₂×C₁₂₇	C₂×C₄×M₁₆	1	1	yes	characteristic
160	C₂×C₁₂₇	C₂×#405	1	1	yes	characteristic
161	C₂×C₁₂₇	#2965	1	4	yes	—
162	C₂×#431	C₂×C₁₂₇	1	1	yes	characteristic
163	C₂×C₄×M₁₆	C₂×C₁₂₇	1	1	yes	characteristic
164	C₄×#117	C₂×C₁₂₇	1	1	yes	characteristic
165	#1771	C₂×C₁₂₇	1	4	yes	complemented
166	#1753	C₂×C₁₂₇	1	4	yes	—
167	#1744	C₂×C₁₂₇	1	1	yes	characteristic, complemented
168	#1744	C₂×C₁₂₇	1	1	yes	characteristic, complemented
169	C₄×#97	C₂×C₁₂₇	1	1	yes	characteristic, complemented
170	C₄²×C₈	C₂×C₁₂₇	1	1	yes	characteristic, complemented
171	C₁₂₇	#20472	1	1	yes	characteristic, direct product factor, Sylow
172	C₂²×M₁₆	C₂²×C₁₂₇	1	2	yes	—
173	C₂×#117	C₄×C₁₂₇	1	2	yes	—
174	C₂²×C₄²	C₂²×C₁₂₇	1	1	yes	characteristic
175	#431	C₄×C₁₂₇	1	8	yes	complemented
176	C₂×C₄⋊C₈	C₂²×C₁₂₇	1	2	yes	—
177	C₂×C₂²⋊C₈	C₂²×C₁₂₇	1	2	yes	—
178	C₄×M₁₆	C₄×C₁₂₇	1	8	yes	complemented
179	C₂×#97	C₂²×C₁₂₇	1	4	yes	—
180	C₂×#97	C₂²×C₁₂₇	1	2	yes	—
181	C₂×C₄×C₈	C₂²×C₁₂₇	1	4	yes	—
182	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
183	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
184	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
185	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
186	C₄×C₄⋊C₄	C₄×C₁₂₇	1	2	yes	—
187	C₄×C₄⋊C₄	C₂²×C₁₂₇	1	2	yes	—
188	C₄×C₂²⋊C₄	C₄×C₁₂₇	1	2	yes	—
189	C₄×C₂²⋊C₄	C₂²×C₁₂₇	1	2	yes	—
190	#376	C₄×C₁₂₇	1	2	yes	—
191	C₄³	C₂²×C₁₂₇	1	1	yes	characteristic
192	C₄³	C₂²×C₁₂₇	1	1	yes	characteristic
193	#336	C₂²×C₁₂₇	1	4	yes	—
194	#336	C₂²×C₁₂₇	1	4	yes	—
195	C₂³×C₄	C₂×C₄×C₁₂₇	1	1	yes	characteristic
196	C₂³×C₄	C₂×C₄×C₁₂₇	1	1	yes	characteristic
197	C₂³×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
198	C₂×M₁₆	—	2	8	no	—
199	C₂×M₁₆	C₂×C₄×C₁₂₇	1	8	yes	—
200	C₂²×C₈	C₂³×C₁₂₇	1	2	yes	—
201	C₂²×C₈	C₂³×C₁₂₇	1	2	yes	—
202	C₂²×C₈	C₂³×C₁₂₇	1	2	yes	—
203	C₂²×C₈	C₂³×C₁₂₇	1	2	yes	—
204	#117	—	2	8	no	—
205	C₂×C₄⋊C₄	C₂×C₄×C₁₂₇	1	4	yes	—
206	C₂×C₂²⋊C₄	C₂×C₄×C₁₂₇	1	4	yes	—
207	C₂×C₄²	C₂×C₄×C₁₂₇	1	2	yes	—
208	C₂×C₄²	C₂×C₄×C₁₂₇	1	2	yes	—
209	C₂×C₄²	C₂×C₄×C₁₂₇	1	2	yes	—
210	C₂×C₄²	C₂×C₄×C₁₂₇	1	1	yes	characteristic
211	C₂×C₄²	C₂×C₄×C₁₂₇	1	1	yes	characteristic
212	C₂×C₄²	C₂×C₄×C₁₂₇	1	4	yes	—
213	C₂×C₄²	C₂×C₄×C₁₂₇	1	4	yes	—
214	C₂×C₄²	C₂×C₄×C₁₂₇	1	2	yes	—
215	C₂×C₄²	C₂³×C₁₂₇	1	2	yes	—
216	C₂×C₄²	C₂³×C₁₂₇	1	2	yes	—
217	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
218	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
219	C₄⋊C₈	C₂×C₄×C₁₂₇	1	8	yes	—
220	C₂²⋊C₈	C₂×C₄×C₁₂₇	1	8	yes	—
221	#97	—	2	8	no	—
222	#97	C₂×C₄×C₁₂₇	1	8	yes	—
223	C₄×C₈	—	2	8	no	—
224	C₄×C₈	C₂×C₄×C₁₂₇	1	4	yes	—
225	C₄×C₈	C₂×C₄×C₁₂₇	1	4	yes	—
226	C₄×C₈	C₂×C₄×C₁₂₇	1	4	yes	—
227	C₄×C₈	C₂×C₄×C₁₂₇	1	4	yes	—
228	#95	C₂×C₄×C₁₂₇	1	4	yes	—
229	#95	C₂×C₄×C₁₂₇	1	4	yes	—
230	C₂⁴	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
231	C₂²×C₄	—	2	4	no	—
232	C₂²×C₄	—	2	4	no	—
233	C₂²×C₄	—	2	4	no	—
234	C₂²×C₄	C₄²×C₁₂₇	1	4	yes	—
235	C₂²×C₄	C₄²×C₁₂₇	1	4	yes	—
236	C₂²×C₄	C₂²×C₄×C₁₂₇	1	4	yes	—
237	C₂²×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
238	C₂²×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
239	C₂²×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
240	C₂²×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
241	C₂²×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
242	C₂²×C₄	C₂²×C₄×C₁₂₇	1	4	yes	—
243	C₂²×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
244	C₂²×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
245	C₂²×C₄	C₂⁴×C₁₂₇	1	1	yes	characteristic
246	M₁₆	—	2	16	no	—
247	C₂×C₈	—	2	4	no	—
248	C₂×C₈	—	2	4	no	—
249	C₂×C₈	—	2	4	no	—
250	C₂×C₈	C₂²×C₄×C₁₂₇	1	4	yes	—
251	C₂×C₈	C₂²×C₄×C₁₂₇	1	4	yes	—
252	C₂×C₈	C₂²×C₄×C₁₂₇	1	4	yes	—
253	C₂×C₈	C₂²×C₄×C₁₂₇	1	4	yes	—
254	C₂×C₈	#41×C₁₂₇	1	8	yes	—
255	C₄⋊C₄	—	2	8	no	—
256	C₂²⋊C₄	—	2	8	no	—
257	C₄²	—	2	4	no	—
258	C₄²	—	2	4	no	—
259	C₄²	—	2	4	no	—
260	C₄²	—	2	4	no	—
261	C₄²	—	2	4	no	—
262	C₄²	C₄²×C₁₂₇	1	4	yes	—
263	C₄²	C₄²×C₁₂₇	1	4	yes	—
264	C₄²	M₁₆×C₁₂₇	1	8	yes	complemented
265	C₄²	C₂²×C₄×C₁₂₇	1	2	yes	—
266	C₄²	C₂²×C₄×C₁₂₇	1	2	yes	—
267	C₄²	C₂²×C₄×C₁₂₇	1	2	yes	—
268	C₄²	C₂²×C₄×C₁₂₇	1	2	yes	—
269	C₂³	—	2	2	no	—
270	C₂³	—	2	2	no	—
271	C₂³	—	2	2	no	—
272	C₂³	C₂×C₄²×C₁₂₇	1	2	yes	—
273	C₂³	C₂³×C₄×C₁₂₇	1	1	yes	characteristic
274	C₂×C₄	—	2	4	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	4	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	4	no	—
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	2	8	no	—
282	C₂×C₄	—	2	8	no	—
283	C₂×C₄	—	2	4	no	—
284	C₂×C₄	—	2	4	no	—
285	C₂×C₄	—	2	4	no	—
286	C₂×C₄	C₂×C₄²×C₁₂₇	1	2	yes	—
287	C₂×C₄	C₂×C₄²×C₁₂₇	1	2	yes	—
288	C₂×C₄	C₂×C₄²×C₁₂₇	1	2	yes	—
289	C₂×C₄	C₂×C₄²×C₁₂₇	1	2	yes	—
290	C₂×C₄	C₂×C₄²×C₁₂₇	1	2	yes	—
291	C₂×C₄	C₂×M₁₆×C₁₂₇	1	8	yes	—
292	C₂×C₄	C₂×M₁₆×C₁₂₇	1	4	yes	—
293	C₂×C₄	#131×C₁₂₇	1	4	yes	—
294	C₂×C₄	C₂³×C₄×C₁₂₇	1	1	yes	characteristic
295	C₂×C₄	C₂³×C₄×C₁₂₇	1	1	yes	characteristic
296	C₂×C₄	C₂×#41×C₁₂₇	1	4	yes	—
297	C₈	—	2	4	no	—
298	C₈	—	2	4	no	—
299	C₈	—	2	4	no	—
300	C₈	#117×C₁₂₇	1	8	yes	complemented
301	C₂²	—	2	2	no	—
302	C₂²	—	2	2	no	—
303	C₂²	—	2	2	no	—
304	C₂²	—	4	4	no	—
305	C₂²	C₂²×C₄²×C₁₂₇	1	1	yes	characteristic
306	C₂²	C₄×#41×C₁₂₇	1	2	yes	—
307	C₂²	C₂²×M₁₆×C₁₂₇	1	2	yes	—
308	C₂²	C₂×#131×C₁₂₇	1	2	yes	—
309	C₄	—	2	2	no	—
310	C₄	—	2	2	no	—
311	C₄	—	2	8	no	—
312	C₄	—	2	2	no	—
313	C₄	—	4	4	no	—
314	C₄	C₄×M₁₆×C₁₂₇	1	4	yes	—
315	C₄	#405×C₁₂₇	1	4	yes	—
316	C₄	C₂×#117×C₁₂₇	1	2	yes	—
317	C₄	C₄×#41×C₁₂₇	1	2	yes	—
318	C₂	—	4	2	no	—
319	C₂	C₄×#117×C₁₂₇	1	1	yes	characteristic
320	C₂	C₂×C₄×M₁₆×C₁₂₇	1	1	yes	characteristic
321	C₂	C₂×#405×C₁₂₇	1	1	yes	characteristic
322	C₂	C₁₂₇×#2965	1	4	yes	—

	subgroup	G/H
center	C₂×C₄²×C₁₂₇	C₂³
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 ◁ C₂×C₄²×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₁₂₇×#1771	C₂	semidirect
6. C₁₂₇×#1744	C₂	semidirect
7. C₁₂₇×#1744	C₂	semidirect
8. C₄×#97×C₁₂₇	C₂	semidirect
9. C₄²×C₈×C₁₂₇	C₂	semidirect
13. #431×C₁₂₇	C₄	semidirect
16. C₄×M₁₆×C₁₂₇	C₄	semidirect
102. C₄²×C₁₂₇	M₁₆	semidirect
138. C₈×C₁₂₇	#117	semidirect
156. #20472	C₁₂₇	direct (N × H)
165. #1771	C₂×C₁₂₇	semidirect
167. #1744	C₂×C₁₂₇	semidirect
168. #1744	C₂×C₁₂₇	semidirect
169. C₄×#97	C₂×C₁₂₇	semidirect
170. C₄²×C₈	C₂×C₁₂₇	semidirect
171. C₁₂₇	#20472	direct (N × H)
175. #431	C₄×C₁₂₇	semidirect
178. C₄×M₁₆	C₄×C₁₂₇	semidirect
264. C₄²	M₁₆×C₁₂₇	semidirect
300. C₈	#117×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#431×C₁₂₇	2
C₂×C₄×M₁₆×C₁₂₇	2
C₄×#117×C₁₂₇	2
C₁₂₇×#1771	2
C₁₂₇×#1753	2
C₁₂₇×#1744	2
C₄×#97×C₁₂₇	2
C₄²×C₈×C₁₂₇	2
#20472	127

Subgroups — C127×#20472 #44584145

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#20472 #44584145