Subgroups — C₁₂₇×#21023 #44584696

Order 32512.

1018 conjugacy classes of subgroups.

The subgroup lattice has 418 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	C₂×C₄×M₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₄⋊C₄×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₁₂₇×#1917	C₂	1	4	yes	maximal, complemented
5	C₁₂₇×#1916	C₂	1	4	yes	maximal, complemented
6	C₁₂₇×#1728	C₂	1	1	yes	maximal, characteristic
7	C₁₂₇×#1728	C₂	1	1	yes	maximal, characteristic
8	C₁₂₇×#1728	C₂	1	1	yes	maximal, characteristic
9	C₁₂₇×#1728	C₂	1	1	yes	maximal, characteristic
10	C₂²×M₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
11	C₂²×M₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
12	C₂²×M₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
13	C₂²×M₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
14	C₂²×C₄⋊C₄×C₁₂₇	C₄	1	2	yes	—
15	C₂²×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
16	C₂²×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
17	C₂²×C₄²×C₁₂₇	C₂²	1	1	yes	characteristic
18	#423×C₁₂₇	—	2	4	no	—
19	C₂×C₄⋊C₈×C₁₂₇	C₂²	1	4	yes	—
20	C₄×M₁₆×C₁₂₇	—	2	4	no	—
21	C₂×#97×C₁₂₇	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₁₂₇	C₂²	1	2	yes	—
23	C₄×C₄⋊C₄×C₁₂₇	C₄	1	2	yes	—
24	C₄×C₄⋊C₄×C₁₂₇	C₄	1	2	yes	—
25	C₄×C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
26	C₄×C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
27	C₂×#95×C₁₂₇	C₄	1	2	yes	—
28	#336×C₁₂₇	C₂²	1	2	yes	—
29	#336×C₁₂₇	C₂²	1	4	yes	—
30	#336×C₁₂₇	C₂²	1	4	yes	—
31	#336×C₁₂₇	C₂²	1	4	yes	—
32	#336×C₁₂₇	C₂²	1	2	yes	—
33	C₂³×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
34	C₂³×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
35	C₂³×C₄×C₁₂₇	C₂×C₄	1	2	yes	—
36	C₂³×C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
37	C₂³×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
38	C₂×M₁₆×C₁₂₇	—	2	2	no	—
39	C₂×M₁₆×C₁₂₇	—	2	2	no	—
40	C₂×M₁₆×C₁₂₇	—	2	2	no	—
41	C₂×M₁₆×C₁₂₇	—	2	2	no	—
42	C₂×M₁₆×C₁₂₇	—	2	2	no	—
43	C₂×M₁₆×C₁₂₇	—	2	2	no	—
44	C₂×M₁₆×C₁₂₇	—	2	2	no	—
45	C₂×M₁₆×C₁₂₇	—	2	2	no	—
46	C₂×M₁₆×C₁₂₇	—	2	2	no	—
47	C₂×M₁₆×C₁₂₇	—	2	2	no	—
48	C₂×M₁₆×C₁₂₇	D₄	1	4	yes	—
49	C₂×M₁₆×C₁₂₇	Q₈	1	4	yes	—
50	C₂²×C₈×C₁₂₇	C₂³	1	2	yes	—
51	C₂²×C₈×C₁₂₇	C₂³	1	2	yes	—
52	C₂²×C₈×C₁₂₇	C₂³	1	2	yes	—
53	C₂²×C₈×C₁₂₇	C₂³	1	2	yes	—
54	C₂×C₄⋊C₄×C₁₂₇	—	2	8	no	—
55	C₂×C₄⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
56	C₂×C₄⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
57	C₂×C₄⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
58	C₂×C₄⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
59	C₂×C₄²×C₁₂₇	—	2	2	no	—
60	C₂×C₄²×C₁₂₇	—	2	2	no	—
61	C₂×C₄²×C₁₂₇	—	2	2	no	—
62	C₂×C₄²×C₁₂₇	—	2	2	no	—
63	C₂×C₄²×C₁₂₇	—	2	2	no	—
64	C₂×C₄²×C₁₂₇	—	2	2	no	—
65	C₂×C₄²×C₁₂₇	C₂×C₄	1	2	yes	—
66	C₂×C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
67	C₂×C₄²×C₁₂₇	C₂×C₄	1	2	yes	—
68	C₂×C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
69	C₂×C₄²×C₁₂₇	C₂³	1	2	yes	—
70	C₂×C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
71	C₂×C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
72	C₂×C₄²×C₁₂₇	C₂³	1	2	yes	—
73	C₄⋊C₈×C₁₂₇	—	2	8	no	—
74	#97×C₁₂₇	—	2	4	no	—
75	C₄×C₈×C₁₂₇	—	2	4	no	—
76	#95×C₁₂₇	C₂×C₄	1	2	yes	—
77	#95×C₁₂₇	C₂×C₄	1	2	yes	—
78	#95×C₁₂₇	C₂×C₄	1	2	yes	—
79	#95×C₁₂₇	C₂×C₄	1	2	yes	—
80	C₂⁴×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
81	C₂²×C₄×C₁₂₇	—	2	2	no	—
82	C₂²×C₄×C₁₂₇	—	2	4	no	—
83	C₂²×C₄×C₁₂₇	—	2	2	no	—
84	C₂²×C₄×C₁₂₇	—	2	4	no	—
85	C₂²×C₄×C₁₂₇	—	2	4	no	—
86	C₂²×C₄×C₁₂₇	—	2	4	no	—
87	C₂²×C₄×C₁₂₇	—	2	4	no	—
88	C₂²×C₄×C₁₂₇	—	4	2	no	—
89	C₂²×C₄×C₁₂₇	—	4	2	no	—
90	C₂²×C₄×C₁₂₇	—	4	2	no	—
91	C₂²×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
92	C₂²×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
93	C₂²×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
94	C₂²×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
95	C₂²×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
96	C₂²×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
97	C₂²×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
98	C₂²×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
99	C₂²×C₄×C₁₂₇	C₂×Q₈	1	2	yes	—
100	C₂²×C₄×C₁₂₇	#41	1	2	yes	—
101	C₂²×C₄×C₁₂₇	#41	1	2	yes	—
102	C₂²×C₄×C₁₂₇	#41	1	2	yes	—
103	C₂²×C₄×C₁₂₇	#41	1	2	yes	—
104	C₂²×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
105	M₁₆×C₁₂₇	—	2	8	no	—
106	M₁₆×C₁₂₇	—	2	8	no	—
107	M₁₆×C₁₂₇	—	4	4	no	—
108	M₁₆×C₁₂₇	—	4	4	no	—
109	C₂×C₈×C₁₂₇	—	2	2	no	—
110	C₂×C₈×C₁₂₇	—	2	2	no	—
111	C₂×C₈×C₁₂₇	—	2	2	no	—
112	C₂×C₈×C₁₂₇	—	2	2	no	—
113	C₂×C₈×C₁₂₇	—	2	2	no	—
114	C₂×C₈×C₁₂₇	—	2	2	no	—
115	C₂×C₈×C₁₂₇	—	2	2	no	—
116	C₂×C₈×C₁₂₇	—	2	2	no	—
117	C₂×C₈×C₁₂₇	—	2	2	no	—
118	C₂×C₈×C₁₂₇	—	2	2	no	—
119	C₂×C₈×C₁₂₇	C₂×D₄	1	4	yes	—
120	C₂×C₈×C₁₂₇	C₂×Q₈	1	4	yes	—
121	C₄⋊C₄×C₁₂₇	—	2	4	no	—
122	C₄⋊C₄×C₁₂₇	—	2	4	no	—
123	C₄⋊C₄×C₁₂₇	—	2	4	no	—
124	C₄⋊C₄×C₁₂₇	M₁₆	1	8	yes	complemented
125	C₄²×C₁₂₇	—	2	4	no	—
126	C₄²×C₁₂₇	—	2	2	no	—
127	C₄²×C₁₂₇	—	2	4	no	—
128	C₄²×C₁₂₇	—	2	2	no	—
129	C₄²×C₁₂₇	—	2	1	no	—
130	C₄²×C₁₂₇	—	2	1	no	—
131	C₄²×C₁₂₇	—	2	2	no	—
132	C₄²×C₁₂₇	—	2	2	no	—
133	C₄²×C₁₂₇	—	2	2	no	—
134	C₄²×C₁₂₇	—	2	1	no	—
135	C₄²×C₁₂₇	—	2	2	no	—
136	C₄²×C₁₂₇	—	2	1	no	—
137	C₂³×C₁₂₇	—	2	2	no	—
138	C₂³×C₁₂₇	—	2	2	no	—
139	C₂³×C₁₂₇	#117	1	2	yes	—
140	C₂³×C₁₂₇	C₄×D₄	1	2	yes	—
141	C₂³×C₁₂₇	C₄×Q₈	1	2	yes	—
142	C₂³×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
143	C₂×C₄×C₁₂₇	—	2	8	no	—
144	C₂×C₄×C₁₂₇	—	2	2	no	—
145	C₂×C₄×C₁₂₇	—	2	2	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	2	no	—
154	C₂×C₄×C₁₂₇	—	2	2	no	—
155	C₂×C₄×C₁₂₇	—	2	4	no	—
156	C₂×C₄×C₁₂₇	—	2	2	no	—
157	C₂×C₄×C₁₂₇	—	2	2	no	—
158	C₂×C₄×C₁₂₇	—	4	8	no	—
159	C₂×C₄×C₁₂₇	—	4	8	no	—
160	C₂×C₄×C₁₂₇	—	4	4	no	—
161	C₂×C₄×C₁₂₇	#117	1	2	yes	—
162	C₂×C₄×C₁₂₇	C₄×D₄	1	2	yes	—
163	C₂×C₄×C₁₂₇	C₄×Q₈	1	2	yes	—
164	C₂×C₄×C₁₂₇	C₂²⋊Q₈	1	4	yes	—
165	C₂×C₄×C₁₂₇	#123	1	4	yes	—
166	C₂×C₄×C₁₂₇	#125	1	4	yes	—
167	C₂×C₄×C₁₂₇	#126	1	4	yes	—
168	C₂×C₄×C₁₂₇	C₂×M₁₆	1	4	yes	—
169	C₂×C₄×C₁₂₇	C₂×M₁₆	1	4	yes	—
170	C₂×C₄×C₁₂₇	C₂×M₁₆	1	4	yes	—
171	C₂×C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
172	C₂×C₄×C₁₂₇	C₂²×Q₈	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	C₂×C₄×C₁₂₇	C₂×#41	1	1	yes	characteristic
177	C₈×C₁₂₇	—	2	4	no	—
178	C₈×C₁₂₇	—	2	4	no	—
179	C₈×C₁₂₇	—	4	2	no	—
180	C₈×C₁₂₇	—	4	2	no	—
181	C₂²×C₁₂₇	—	2	2	no	—
182	C₂²×C₁₂₇	—	2	2	no	—
183	C₂²×C₁₂₇	—	2	4	no	—
184	C₂²×C₁₂₇	—	2	4	no	—
185	C₂²×C₁₂₇	#387	1	4	yes	—
186	C₂²×C₁₂₇	C₂×#117	1	1	yes	characteristic
187	C₂²×C₁₂₇	C₂×C₄×D₄	1	1	yes	characteristic
188	C₂²×C₁₂₇	C₂×C₄×Q₈	1	1	yes	characteristic
189	C₂²×C₁₂₇	C₂²×M₁₆	1	2	yes	—
190	C₂²×C₁₂₇	#568	1	2	yes	—
191	C₄×C₁₂₇	—	2	4	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	—	4	2	no	—
197	C₄×C₁₂₇	#387	1	4	yes	—
198	C₄×C₁₂₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
199	C₄×C₁₂₇	C₂×#123	1	1	yes	characteristic
200	C₄×C₁₂₇	C₂×#125	1	1	yes	characteristic
201	C₄×C₁₂₇	C₂×#126	1	1	yes	characteristic
202	#21023	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
203	C₂×C₁₂₇	—	2	4	no	—
204	C₂×C₁₂₇	C₂×#387	1	1	yes	characteristic
205	C₂×C₁₂₇	#2920	1	1	yes	characteristic
206	C₂×C₁₂₇	#2921	1	1	yes	characteristic
207	C₂×C₁₂₇	D₄×M₁₆	1	2	yes	—
208	C₂×C₁₂₇	Q₈×M₁₆	1	2	yes	—
209	C₂×#423	C₂×C₁₂₇	1	1	yes	characteristic
210	C₂×C₄×M₁₆	C₂×C₁₂₇	1	1	yes	characteristic
211	C₂×C₄×C₄⋊C₄	C₂×C₁₂₇	1	1	yes	characteristic
212	#1917	C₂×C₁₂₇	1	4	yes	complemented
213	#1916	C₂×C₁₂₇	1	4	yes	complemented
214	#1728	C₂×C₁₂₇	1	1	yes	characteristic
215	#1728	C₂×C₁₂₇	1	1	yes	characteristic
216	#1728	C₂×C₁₂₇	1	1	yes	characteristic
217	#1728	C₂×C₁₂₇	1	1	yes	characteristic
218	C₁₂₇	#21023	1	1	yes	characteristic, direct product factor, Sylow
219	C₂²×M₁₆	C₂²×C₁₂₇	1	1	yes	characteristic
220	C₂²×M₁₆	C₂²×C₁₂₇	1	1	yes	characteristic
221	C₂²×M₁₆	C₂²×C₁₂₇	1	1	yes	characteristic
222	C₂²×M₁₆	C₂²×C₁₂₇	1	1	yes	characteristic
223	C₂²×C₄⋊C₄	C₄×C₁₂₇	1	2	yes	—
224	C₂²×C₄²	C₂²×C₁₂₇	1	1	yes	characteristic
225	C₂²×C₄²	C₂²×C₁₂₇	1	1	yes	characteristic
226	C₂²×C₄²	C₂²×C₁₂₇	1	1	yes	characteristic
227	#423	—	2	4	no	—
228	C₂×C₄⋊C₈	C₂²×C₁₂₇	1	4	yes	—
229	C₄×M₁₆	—	2	4	no	—
230	C₂×#97	C₂²×C₁₂₇	1	2	yes	—
231	C₂×C₄×C₈	C₂²×C₁₂₇	1	2	yes	—
232	C₄×C₄⋊C₄	C₄×C₁₂₇	1	2	yes	—
233	C₄×C₄⋊C₄	C₄×C₁₂₇	1	2	yes	—
234	C₄×C₄⋊C₄	C₂²×C₁₂₇	1	2	yes	—
235	C₄×C₄⋊C₄	C₂²×C₁₂₇	1	2	yes	—
236	C₂×#95	C₄×C₁₂₇	1	2	yes	—
237	#336	C₂²×C₁₂₇	1	2	yes	—
238	#336	C₂²×C₁₂₇	1	4	yes	—
239	#336	C₂²×C₁₂₇	1	4	yes	—
240	#336	C₂²×C₁₂₇	1	4	yes	—
241	#336	C₂²×C₁₂₇	1	2	yes	—
242	C₂³×C₄	C₂×C₄×C₁₂₇	1	1	yes	characteristic
243	C₂³×C₄	C₂×C₄×C₁₂₇	1	2	yes	—
244	C₂³×C₄	C₂×C₄×C₁₂₇	1	2	yes	—
245	C₂³×C₄	C₂×C₄×C₁₂₇	1	1	yes	characteristic
246	C₂³×C₄	C₂³×C₁₂₇	1	1	yes	characteristic
247	C₂×M₁₆	—	2	2	no	—
248	C₂×M₁₆	—	2	2	no	—
249	C₂×M₁₆	—	2	2	no	—
250	C₂×M₁₆	—	2	2	no	—
251	C₂×M₁₆	—	2	2	no	—
252	C₂×M₁₆	—	2	2	no	—
253	C₂×M₁₆	—	2	2	no	—
254	C₂×M₁₆	—	2	2	no	—
255	C₂×M₁₆	—	2	2	no	—
256	C₂×M₁₆	—	2	2	no	—
257	C₂×M₁₆	D₄×C₁₂₇	1	4	yes	—
258	C₂×M₁₆	Q₈×C₁₂₇	1	4	yes	—
259	C₂²×C₈	C₂³×C₁₂₇	1	2	yes	—
260	C₂²×C₈	C₂³×C₁₂₇	1	2	yes	—
261	C₂²×C₈	C₂³×C₁₂₇	1	2	yes	—
262	C₂²×C₈	C₂³×C₁₂₇	1	2	yes	—
263	C₂×C₄⋊C₄	—	2	8	no	—
264	C₂×C₄⋊C₄	C₂×C₄×C₁₂₇	1	2	yes	—
265	C₂×C₄⋊C₄	C₂×C₄×C₁₂₇	1	2	yes	—
266	C₂×C₄⋊C₄	C₂×C₄×C₁₂₇	1	2	yes	—
267	C₂×C₄⋊C₄	C₂×C₄×C₁₂₇	1	2	yes	—
268	C₂×C₄²	—	2	2	no	—
269	C₂×C₄²	—	2	2	no	—
270	C₂×C₄²	—	2	2	no	—
271	C₂×C₄²	—	2	2	no	—
272	C₂×C₄²	—	2	2	no	—
273	C₂×C₄²	—	2	2	no	—
274	C₂×C₄²	C₂×C₄×C₁₂₇	1	2	yes	—
275	C₂×C₄²	C₂×C₄×C₁₂₇	1	1	yes	characteristic
276	C₂×C₄²	C₂×C₄×C₁₂₇	1	2	yes	—
277	C₂×C₄²	C₂×C₄×C₁₂₇	1	1	yes	characteristic
278	C₂×C₄²	C₂³×C₁₂₇	1	2	yes	—
279	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
280	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
281	C₂×C₄²	C₂³×C₁₂₇	1	2	yes	—
282	C₄⋊C₈	—	2	8	no	—
283	#97	—	2	4	no	—
284	C₄×C₈	—	2	4	no	—
285	#95	C₂×C₄×C₁₂₇	1	2	yes	—
286	#95	C₂×C₄×C₁₂₇	1	2	yes	—
287	#95	C₂×C₄×C₁₂₇	1	2	yes	—
288	#95	C₂×C₄×C₁₂₇	1	2	yes	—
289	C₂⁴	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
290	C₂²×C₄	—	2	2	no	—
291	C₂²×C₄	—	2	4	no	—
292	C₂²×C₄	—	2	2	no	—
293	C₂²×C₄	—	2	4	no	—
294	C₂²×C₄	—	2	4	no	—
295	C₂²×C₄	—	2	4	no	—
296	C₂²×C₄	—	2	4	no	—
297	C₂²×C₄	—	4	2	no	—
298	C₂²×C₄	—	4	2	no	—
299	C₂²×C₄	—	4	2	no	—
300	C₂²×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
301	C₂²×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
302	C₂²×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
303	C₂²×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
304	C₂²×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
305	C₂²×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
306	C₂²×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
307	C₂²×C₄	C₂×D₄×C₁₂₇	1	2	yes	—
308	C₂²×C₄	C₂×Q₈×C₁₂₇	1	2	yes	—
309	C₂²×C₄	#41×C₁₂₇	1	2	yes	—
310	C₂²×C₄	#41×C₁₂₇	1	2	yes	—
311	C₂²×C₄	#41×C₁₂₇	1	2	yes	—
312	C₂²×C₄	#41×C₁₂₇	1	2	yes	—
313	C₂²×C₄	C₂⁴×C₁₂₇	1	1	yes	characteristic
314	M₁₆	—	2	8	no	—
315	M₁₆	—	2	8	no	—
316	M₁₆	—	4	4	no	—
317	M₁₆	—	4	4	no	—
318	C₂×C₈	—	2	2	no	—
319	C₂×C₈	—	2	2	no	—
320	C₂×C₈	—	2	2	no	—
321	C₂×C₈	—	2	2	no	—
322	C₂×C₈	—	2	2	no	—
323	C₂×C₈	—	2	2	no	—
324	C₂×C₈	—	2	2	no	—
325	C₂×C₈	—	2	2	no	—
326	C₂×C₈	—	2	2	no	—
327	C₂×C₈	—	2	2	no	—
328	C₂×C₈	C₂×D₄×C₁₂₇	1	4	yes	—
329	C₂×C₈	C₂×Q₈×C₁₂₇	1	4	yes	—
330	C₄⋊C₄	—	2	4	no	—
331	C₄⋊C₄	—	2	4	no	—
332	C₄⋊C₄	—	2	4	no	—
333	C₄⋊C₄	M₁₆×C₁₂₇	1	8	yes	complemented
334	C₄²	—	2	4	no	—
335	C₄²	—	2	2	no	—
336	C₄²	—	2	4	no	—
337	C₄²	—	2	2	no	—
338	C₄²	—	2	1	no	—
339	C₄²	—	2	1	no	—
340	C₄²	—	2	2	no	—
341	C₄²	—	2	2	no	—
342	C₄²	—	2	2	no	—
343	C₄²	—	2	1	no	—
344	C₄²	—	2	2	no	—
345	C₄²	—	2	1	no	—
346	C₂³	—	2	2	no	—
347	C₂³	—	2	2	no	—
348	C₂³	#117×C₁₂₇	1	2	yes	—
349	C₂³	C₄×D₄×C₁₂₇	1	2	yes	—
350	C₂³	C₄×Q₈×C₁₂₇	1	2	yes	—
351	C₂³	C₂³×C₄×C₁₂₇	1	1	yes	characteristic
352	C₂×C₄	—	2	8	no	—
353	C₂×C₄	—	2	2	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	2	no	—
362	C₂×C₄	—	2	2	no	—
363	C₂×C₄	—	2	2	no	—
364	C₂×C₄	—	2	4	no	—
365	C₂×C₄	—	2	2	no	—
366	C₂×C₄	—	2	2	no	—
367	C₂×C₄	—	4	8	no	—
368	C₂×C₄	—	4	8	no	—
369	C₂×C₄	—	4	4	no	—
370	C₂×C₄	#117×C₁₂₇	1	2	yes	—
371	C₂×C₄	C₄×D₄×C₁₂₇	1	2	yes	—
372	C₂×C₄	C₄×Q₈×C₁₂₇	1	2	yes	—
373	C₂×C₄	C₂²⋊Q₈×C₁₂₇	1	4	yes	—
374	C₂×C₄	#123×C₁₂₇	1	4	yes	—
375	C₂×C₄	#125×C₁₂₇	1	4	yes	—
376	C₂×C₄	#126×C₁₂₇	1	4	yes	—
377	C₂×C₄	C₂×M₁₆×C₁₂₇	1	4	yes	—
378	C₂×C₄	C₂×M₁₆×C₁₂₇	1	4	yes	—
379	C₂×C₄	C₂×M₁₆×C₁₂₇	1	4	yes	—
380	C₂×C₄	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
381	C₂×C₄	C₂²×Q₈×C₁₂₇	1	1	yes	characteristic
382	C₂×C₄	C₂×#41×C₁₂₇	1	1	yes	characteristic
383	C₂×C₄	C₂×#41×C₁₂₇	1	1	yes	characteristic
384	C₂×C₄	C₂×#41×C₁₂₇	1	1	yes	characteristic
385	C₂×C₄	C₂×#41×C₁₂₇	1	1	yes	characteristic
386	C₈	—	2	4	no	—
387	C₈	—	2	4	no	—
388	C₈	—	4	2	no	—
389	C₈	—	4	2	no	—
390	C₂²	—	2	2	no	—
391	C₂²	—	2	2	no	—
392	C₂²	—	2	4	no	—
393	C₂²	—	2	4	no	—
394	C₂²	#387×C₁₂₇	1	4	yes	—
395	C₂²	C₂×#117×C₁₂₇	1	1	yes	characteristic
396	C₂²	C₂×C₄×D₄×C₁₂₇	1	1	yes	characteristic
397	C₂²	C₂×C₄×Q₈×C₁₂₇	1	1	yes	characteristic
398	C₂²	C₂²×M₁₆×C₁₂₇	1	2	yes	—
399	C₂²	#568×C₁₂₇	1	2	yes	—
400	C₄	—	2	4	no	—
401	C₄	—	2	4	no	—
402	C₄	—	2	4	no	—
403	C₄	—	4	2	no	—
404	C₄	—	4	2	no	—
405	C₄	—	4	2	no	—
406	C₄	#387×C₁₂₇	1	4	yes	—
407	C₄	C₂×C₂²⋊Q₈×C₁₂₇	1	1	yes	characteristic
408	C₄	C₂×#123×C₁₂₇	1	1	yes	characteristic
409	C₄	C₂×#125×C₁₂₇	1	1	yes	characteristic
410	C₄	C₂×#126×C₁₂₇	1	1	yes	characteristic
411	C₂	—	2	4	no	—
412	C₂	C₂×#387×C₁₂₇	1	1	yes	characteristic
413	C₂	C₁₂₇×#2920	1	1	yes	characteristic
414	C₂	C₁₂₇×#2921	1	1	yes	characteristic
415	C₂	D₄×M₁₆×C₁₂₇	1	2	yes	—
416	C₂	Q₈×M₁₆×C₁₂₇	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	C₂⁴×C₁₂₇	C₂²×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
4. C₁₂₇×#1917	C₂	semidirect
5. C₁₂₇×#1916	C₂	semidirect
124. C₄⋊C₄×C₁₂₇	M₁₆	semidirect
202. #21023	C₁₂₇	direct (N × H)
212. #1917	C₂×C₁₂₇	semidirect
213. #1916	C₂×C₁₂₇	semidirect
218. C₁₂₇	#21023	direct (N × H)
333. C₄⋊C₄	M₁₆×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
C₂×C₄×M₁₆×C₁₂₇	2
C₂×C₄×C₄⋊C₄×C₁₂₇	2
C₁₂₇×#1917	2
C₁₂₇×#1916	2
C₁₂₇×#1728	2
#21023	127

Subgroups — C127×#21023 #44584696

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#21023 #44584696