Subgroups — C₁₂₇×#21713 #44585386

Order 32512.

802 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#446×C₁₂₇	C₂	1	2	yes	maximal
2	C₄×C₄⋊Q₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	C₁₂₇×#1985	C₂	1	2	yes	maximal
4	C₁₂₇×#1940	C₂	1	2	yes	maximal
5	C₁₂₇×#1939	C₂	1	1	yes	maximal, characteristic
6	C₁₂₇×#1939	C₂	1	1	yes	maximal, characteristic
7	C₁₂₇×#1842	C₂	1	1	yes	maximal, characteristic
8	C₁₂₇×#1771	C₂	1	2	yes	maximal
9	C₁₂₇×#1770	C₂	1	1	yes	maximal, characteristic
10	C₈×C₄⋊C₄×C₁₂₇	C₂	1	1	yes	maximal, characteristic
11	C₄×#97×C₁₂₇	C₂	1	1	yes	maximal, characteristic
12	C₂×C₄⋊Q₈×C₁₂₇	C₄	1	1	yes	characteristic
13	C₂×C₄×Q₈×C₁₂₇	C₂²	1	2	yes	—
14	#446×C₁₂₇	—	2	8	no	—
15	C₂×C₄⋊C₈×C₁₂₇	C₂²	1	2	yes	—
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	C₂×#97×C₁₂₇	C₂²	1	2	yes	—
22	C₂×#97×C₁₂₇	C₂²	1	1	yes	characteristic
23	C₂×#97×C₁₂₇	C₂²	1	1	yes	characteristic
24	C₂×#97×C₁₂₇	C₂²	1	1	yes	characteristic
25	C₂×#97×C₁₂₇	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₁₂₇	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
29	#391×C₁₂₇	C₄	1	2	yes	—
30	#389×C₁₂₇	C₄	1	1	yes	characteristic
31	#389×C₁₂₇	C₄	1	1	yes	characteristic
32	#389×C₁₂₇	C₄	1	2	yes	—
33	#384×C₁₂₇	C₄	1	1	yes	characteristic
34	C₄×C₄⋊C₄×C₁₂₇	C₂²	1	1	yes	characteristic
35	C₄×C₄⋊C₄×C₁₂₇	C₂²	1	1	yes	characteristic
36	C₄×C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	—
37	C₄³×C₁₂₇	C₂²	1	1	yes	characteristic
38	#336×C₁₂₇	C₂²	1	2	yes	—
39	#336×C₁₂₇	C₂²	1	2	yes	—
40	#336×C₁₂₇	C₂²	1	2	yes	—
41	#336×C₁₂₇	C₂²	1	2	yes	—
42	C₂²×Q₈×C₁₂₇	C₂×C₄	1	2	yes	—
43	C₂²×C₈×C₁₂₇	C₂³	1	2	yes	—
44	C₂²×C₈×C₁₂₇	C₂³	1	2	yes	—
45	C₂²×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
46	C₂²×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
47	C₂²×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
48	C₂²×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
49	C₄⋊Q₈×C₁₂₇	—	2	4	no	—
50	C₄×Q₈×C₁₂₇	—	2	4	no	—
51	C₄×Q₈×C₁₂₇	—	2	4	no	—
52	C₂×C₄⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
53	C₂×C₄⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
54	C₂×C₄⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
55	C₂×C₄⋊C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
56	C₂×C₄⋊C₄×C₁₂₇	C₂×C₄	1	2	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×C₄⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
60	C₂×C₄⋊C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
61	C₂×C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
62	C₂×C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
63	C₂×C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
64	C₂×C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
65	C₂×C₄²×C₁₂₇	C₂³	1	2	yes	—
66	C₂×C₄²×C₁₂₇	C₂³	1	2	yes	—
67	C₂×C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
68	C₂×C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
69	C₂×C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
70	C₄⋊C₈×C₁₂₇	—	2	4	no	—
71	C₄⋊C₈×C₁₂₇	—	2	4	no	—
72	C₄⋊C₈×C₁₂₇	—	2	4	no	—
73	C₄⋊C₈×C₁₂₇	—	2	2	no	—
74	C₄⋊C₈×C₁₂₇	—	2	2	no	—
75	C₄⋊C₈×C₁₂₇	—	2	4	no	—
76	#97×C₁₂₇	—	2	4	no	—
77	#97×C₁₂₇	—	2	2	no	—
78	#97×C₁₂₇	—	2	2	no	—
79	#97×C₁₂₇	D₄	1	4	yes	—
80	#97×C₁₂₇	Q₈	1	4	yes	complemented
81	C₄×C₈×C₁₂₇	—	2	4	no	—
82	C₄×C₈×C₁₂₇	—	2	2	no	—
83	C₄×C₈×C₁₂₇	Q₈	1	4	yes	complemented
84	#95×C₁₂₇	C₂×C₄	1	2	yes	—
85	#95×C₁₂₇	C₂×C₄	1	2	yes	—
86	#95×C₁₂₇	C₂×C₄	1	2	yes	—
87	#95×C₁₂₇	C₂×C₄	1	2	yes	—
88	C₂×Q₈×C₁₂₇	—	2	4	no	—
89	C₂×Q₈×C₁₂₇	—	2	4	no	—
90	C₂×Q₈×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	1	yes	characteristic
95	C₂²×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
96	C₂²×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
97	C₂²×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
98	C₂²×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
99	C₂²×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
100	C₂²×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
101	C₂²×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
102	C₂×C₈×C₁₂₇	—	2	1	no	—
103	C₂×C₈×C₁₂₇	—	2	1	no	—
104	C₂×C₈×C₁₂₇	—	2	1	no	—
105	C₂×C₈×C₁₂₇	—	2	1	no	—
106	C₂×C₈×C₁₂₇	—	2	1	no	—
107	C₂×C₈×C₁₂₇	—	2	1	no	—
108	C₂×C₈×C₁₂₇	—	2	2	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	C₂×D₄	1	2	yes	—
115	C₂×C₈×C₁₂₇	C₂×D₄	1	2	yes	—
116	C₂×C₈×C₁₂₇	C₂×Q₈	1	2	yes	—
117	C₂×C₈×C₁₂₇	C₂×Q₈	1	2	yes	—
118	C₂×C₈×C₁₂₇	C₂×Q₈	1	2	yes	—
119	C₂×C₈×C₁₂₇	C₂×Q₈	1	2	yes	—
120	C₄⋊C₄×C₁₂₇	—	2	4	no	—
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	—	2	2	no	—
125	C₄⋊C₄×C₁₂₇	—	2	2	no	—
126	C₄⋊C₄×C₁₂₇	—	2	2	no	—
127	C₄⋊C₄×C₁₂₇	—	4	2	no	—
128	C₄⋊C₄×C₁₂₇	M₁₆	1	4	yes	—
129	C₄²×C₁₂₇	—	2	2	no	—
130	C₄²×C₁₂₇	—	2	2	no	—
131	C₄²×C₁₂₇	—	2	2	no	—
132	C₄²×C₁₂₇	—	2	2	no	—
133	C₄²×C₁₂₇	—	2	2	no	—
134	C₄²×C₁₂₇	—	2	2	no	—
135	C₄²×C₁₂₇	—	4	1	no	—
136	C₄²×C₁₂₇	—	4	1	no	—
137	C₄²×C₁₂₇	C₂×D₄	1	2	yes	—
138	C₄²×C₁₂₇	C₂×Q₈	1	2	yes	—
139	C₄²×C₁₂₇	C₂×Q₈	1	2	yes	—
140	C₄²×C₁₂₇	#41	1	2	yes	—
141	C₄²×C₁₂₇	#41	1	2	yes	—
142	C₄²×C₁₂₇	#41	1	2	yes	—
143	C₂³×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
144	Q₈×C₁₂₇	—	4	8	no	—
145	C₂×C₄×C₁₂₇	—	2	2	no	—
146	C₂×C₄×C₁₂₇	—	2	1	no	—
147	C₂×C₄×C₁₂₇	—	2	1	no	—
148	C₂×C₄×C₁₂₇	—	2	1	no	—
149	C₂×C₄×C₁₂₇	—	2	1	no	—
150	C₂×C₄×C₁₂₇	—	2	1	no	—
151	C₂×C₄×C₁₂₇	—	2	1	no	—
152	C₂×C₄×C₁₂₇	—	2	1	no	—
153	C₂×C₄×C₁₂₇	—	2	1	no	—
154	C₂×C₄×C₁₂₇	—	2	1	no	—
155	C₂×C₄×C₁₂₇	—	2	1	no	—
156	C₂×C₄×C₁₂₇	—	2	2	no	—
157	C₂×C₄×C₁₂₇	—	2	2	no	—
158	C₂×C₄×C₁₂₇	—	2	2	no	—
159	C₂×C₄×C₁₂₇	—	2	2	no	—
160	C₂×C₄×C₁₂₇	—	2	2	no	—
161	C₂×C₄×C₁₂₇	—	2	2	no	—
162	C₂×C₄×C₁₂₇	—	2	2	no	—
163	C₂×C₄×C₁₂₇	—	2	2	no	—
164	C₂×C₄×C₁₂₇	—	2	2	no	—
165	C₂×C₄×C₁₂₇	—	2	2	no	—
166	C₂×C₄×C₁₂₇	C₄×D₄	1	2	yes	—
167	C₂×C₄×C₁₂₇	C₄×D₄	1	2	yes	—
168	C₂×C₄×C₁₂₇	C₄×Q₈	1	2	yes	—
169	C₂×C₄×C₁₂₇	C₄×Q₈	1	2	yes	—
170	C₂×C₄×C₁₂₇	C₄×Q₈	1	2	yes	—
171	C₂×C₄×C₁₂₇	C₄×Q₈	1	2	yes	—
172	C₂×C₄×C₁₂₇	C₂×M₁₆	1	4	yes	—
173	C₂×C₄×C₁₂₇	C₂×M₁₆	1	2	yes	—
174	C₂×C₄×C₁₂₇	#131	1	2	yes	—
175	C₂×C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
176	C₂×C₄×C₁₂₇	C₂²×Q₈	1	1	yes	characteristic
177	C₂×C₄×C₁₂₇	C₂²×Q₈	1	1	yes	characteristic
178	C₂×C₄×C₁₂₇	C₂×#41	1	1	yes	characteristic
179	C₂×C₄×C₁₂₇	C₂×#41	1	1	yes	characteristic
180	C₂×C₄×C₁₂₇	C₂×#41	1	1	yes	characteristic
181	C₈×C₁₂₇	—	2	2	no	—
182	C₈×C₁₂₇	—	2	2	no	—
183	C₈×C₁₂₇	—	2	2	no	—
184	C₈×C₁₂₇	—	4	2	no	—
185	C₈×C₁₂₇	—	4	2	no	—
186	C₈×C₁₂₇	C₄⋊Q₈	1	4	yes	complemented
187	C₂²×C₁₂₇	C₂×C₄×D₄	1	1	yes	characteristic
188	C₂²×C₁₂₇	C₂×C₄×Q₈	1	1	yes	characteristic
189	C₂²×C₁₂₇	C₂×C₄×Q₈	1	1	yes	characteristic
190	C₂²×C₁₂₇	C₂²×M₁₆	1	1	yes	characteristic
191	C₂²×C₁₂₇	C₂×#131	1	1	yes	characteristic
192	C₂²×C₁₂₇	#568	1	1	yes	characteristic
193	C₂²×C₁₂₇	#568	1	1	yes	characteristic
194	C₄×C₁₂₇	—	2	2	no	—
195	C₄×C₁₂₇	—	2	4	no	—
196	C₄×C₁₂₇	—	2	2	no	—
197	C₄×C₁₂₇	—	2	2	no	—
198	C₄×C₁₂₇	—	2	2	no	—
199	C₄×C₁₂₇	—	2	2	no	—
200	C₄×C₁₂₇	—	4	2	no	—
201	C₄×C₁₂₇	—	4	2	no	—
202	C₄×C₁₂₇	—	4	2	no	—
203	C₄×C₁₂₇	#446	1	4	yes	complemented
204	C₄×C₁₂₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
205	C₄×C₁₂₇	#532	1	1	yes	characteristic
206	C₄×C₁₂₇	#533	1	1	yes	characteristic
207	C₄×C₁₂₇	#533	1	1	yes	characteristic
208	#21713	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
209	C₂×C₁₂₇	C₄×C₄⋊Q₈	1	1	yes	characteristic
210	C₂×C₁₂₇	D₄×M₁₆	1	1	yes	characteristic
211	C₂×C₁₂₇	#2935	1	1	yes	characteristic
212	C₂×C₁₂₇	C₂×#446	1	1	yes	characteristic
213	C₂×C₁₂₇	#2961	1	1	yes	characteristic
214	C₂×C₁₂₇	#2962	1	1	yes	characteristic
215	C₂×C₁₂₇	Q₈×M₁₆	1	1	yes	characteristic
216	C₂×#446	C₂×C₁₂₇	1	2	yes	—
217	C₄×C₄⋊Q₈	C₂×C₁₂₇	1	1	yes	characteristic
218	#1985	C₂×C₁₂₇	1	2	yes	—
219	#1940	C₂×C₁₂₇	1	2	yes	—
220	#1939	C₂×C₁₂₇	1	1	yes	characteristic
221	#1939	C₂×C₁₂₇	1	1	yes	characteristic
222	#1842	C₂×C₁₂₇	1	1	yes	characteristic
223	#1771	C₂×C₁₂₇	1	2	yes	—
224	#1770	C₂×C₁₂₇	1	1	yes	characteristic
225	C₈×C₄⋊C₄	C₂×C₁₂₇	1	1	yes	characteristic
226	C₄×#97	C₂×C₁₂₇	1	1	yes	characteristic
227	C₁₂₇	#21713	1	1	yes	characteristic, direct product factor, Sylow
228	C₂×C₄⋊Q₈	C₄×C₁₂₇	1	1	yes	characteristic
229	C₂×C₄×Q₈	C₂²×C₁₂₇	1	2	yes	—
230	#446	—	2	8	no	—
231	C₂×C₄⋊C₈	C₂²×C₁₂₇	1	2	yes	—
232	C₂×C₄⋊C₈	C₂²×C₁₂₇	1	1	yes	characteristic
233	C₂×C₄⋊C₈	C₂²×C₁₂₇	1	1	yes	characteristic
234	C₂×C₄⋊C₈	C₂²×C₁₂₇	1	2	yes	—
235	C₂×C₄⋊C₈	C₂²×C₁₂₇	1	2	yes	—
236	C₂×C₄⋊C₈	C₂²×C₁₂₇	1	2	yes	—
237	C₂×#97	C₂²×C₁₂₇	1	2	yes	—
238	C₂×#97	C₂²×C₁₂₇	1	1	yes	characteristic
239	C₂×#97	C₂²×C₁₂₇	1	1	yes	characteristic
240	C₂×#97	C₂²×C₁₂₇	1	1	yes	characteristic
241	C₂×#97	C₂²×C₁₂₇	1	1	yes	characteristic
242	C₂×C₄×C₈	C₂²×C₁₂₇	1	2	yes	—
243	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
244	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
245	#391	C₄×C₁₂₇	1	2	yes	—
246	#389	C₄×C₁₂₇	1	1	yes	characteristic
247	#389	C₄×C₁₂₇	1	1	yes	characteristic
248	#389	C₄×C₁₂₇	1	2	yes	—
249	#384	C₄×C₁₂₇	1	1	yes	characteristic
250	C₄×C₄⋊C₄	C₂²×C₁₂₇	1	1	yes	characteristic
251	C₄×C₄⋊C₄	C₂²×C₁₂₇	1	1	yes	characteristic
252	C₄×C₄⋊C₄	C₂²×C₁₂₇	1	2	yes	—
253	C₄³	C₂²×C₁₂₇	1	1	yes	characteristic
254	#336	C₂²×C₁₂₇	1	2	yes	—
255	#336	C₂²×C₁₂₇	1	2	yes	—
256	#336	C₂²×C₁₂₇	1	2	yes	—
257	#336	C₂²×C₁₂₇	1	2	yes	—
258	C₂²×Q₈	C₂×C₄×C₁₂₇	1	2	yes	—
259	C₂²×C₈	C₂³×C₁₂₇	1	2	yes	—
260	C₂²×C₈	C₂³×C₁₂₇	1	2	yes	—
261	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
262	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
263	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
264	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
265	C₄⋊Q₈	—	2	4	no	—
266	C₄×Q₈	—	2	4	no	—
267	C₄×Q₈	—	2	4	no	—
268	C₂×C₄⋊C₄	C₂×C₄×C₁₂₇	1	2	yes	—
269	C₂×C₄⋊C₄	C₂×C₄×C₁₂₇	1	2	yes	—
270	C₂×C₄⋊C₄	C₂×C₄×C₁₂₇	1	2	yes	—
271	C₂×C₄⋊C₄	C₂×C₄×C₁₂₇	1	1	yes	characteristic
272	C₂×C₄⋊C₄	C₂×C₄×C₁₂₇	1	2	yes	—
273	C₂×C₄⋊C₄	C₂×C₄×C₁₂₇	1	1	yes	characteristic
274	C₂×C₄⋊C₄	C₂×C₄×C₁₂₇	1	1	yes	characteristic
275	C₂×C₄⋊C₄	C₂×C₄×C₁₂₇	1	2	yes	—
276	C₂×C₄⋊C₄	C₂×C₄×C₁₂₇	1	1	yes	characteristic
277	C₂×C₄²	C₂×C₄×C₁₂₇	1	1	yes	characteristic
278	C₂×C₄²	C₂×C₄×C₁₂₇	1	1	yes	characteristic
279	C₂×C₄²	C₂×C₄×C₁₂₇	1	1	yes	characteristic
280	C₂×C₄²	C₂×C₄×C₁₂₇	1	1	yes	characteristic
281	C₂×C₄²	C₂³×C₁₂₇	1	2	yes	—
282	C₂×C₄²	C₂³×C₁₂₇	1	2	yes	—
283	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
284	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
285	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
286	C₄⋊C₈	—	2	4	no	—
287	C₄⋊C₈	—	2	4	no	—
288	C₄⋊C₈	—	2	4	no	—
289	C₄⋊C₈	—	2	2	no	—
290	C₄⋊C₈	—	2	2	no	—
291	C₄⋊C₈	—	2	4	no	—
292	#97	—	2	4	no	—
293	#97	—	2	2	no	—
294	#97	—	2	2	no	—
295	#97	D₄×C₁₂₇	1	4	yes	—
296	#97	Q₈×C₁₂₇	1	4	yes	complemented
297	C₄×C₈	—	2	4	no	—
298	C₄×C₈	—	2	2	no	—
299	C₄×C₈	Q₈×C₁₂₇	1	4	yes	complemented
300	#95	C₂×C₄×C₁₂₇	1	2	yes	—
301	#95	C₂×C₄×C₁₂₇	1	2	yes	—
302	#95	C₂×C₄×C₁₂₇	1	2	yes	—
303	#95	C₂×C₄×C₁₂₇	1	2	yes	—
304	C₂×Q₈	—	2	4	no	—
305	C₂×Q₈	—	2	4	no	—
306	C₂×Q₈	—	4	2	no	—
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	2	yes	—
310	C₂²×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
311	C₂²×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
312	C₂²×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
313	C₂²×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
314	C₂²×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
315	C₂²×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
316	C₂²×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
317	C₂²×C₄	C₂⁴×C₁₂₇	1	1	yes	characteristic
318	C₂×C₈	—	2	1	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	—	2	1	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	—	2	2	no	—
329	C₂×C₈	—	2	2	no	—
330	C₂×C₈	C₂×D₄×C₁₂₇	1	2	yes	—
331	C₂×C₈	C₂×D₄×C₁₂₇	1	2	yes	—
332	C₂×C₈	C₂×Q₈×C₁₂₇	1	2	yes	—
333	C₂×C₈	C₂×Q₈×C₁₂₇	1	2	yes	—
334	C₂×C₈	C₂×Q₈×C₁₂₇	1	2	yes	—
335	C₂×C₈	C₂×Q₈×C₁₂₇	1	2	yes	—
336	C₄⋊C₄	—	2	4	no	—
337	C₄⋊C₄	—	2	4	no	—
338	C₄⋊C₄	—	2	4	no	—
339	C₄⋊C₄	—	2	4	no	—
340	C₄⋊C₄	—	2	2	no	—
341	C₄⋊C₄	—	2	2	no	—
342	C₄⋊C₄	—	2	2	no	—
343	C₄⋊C₄	—	4	2	no	—
344	C₄⋊C₄	M₁₆×C₁₂₇	1	4	yes	—
345	C₄²	—	2	2	no	—
346	C₄²	—	2	2	no	—
347	C₄²	—	2	2	no	—
348	C₄²	—	2	2	no	—
349	C₄²	—	2	2	no	—
350	C₄²	—	2	2	no	—
351	C₄²	—	4	1	no	—
352	C₄²	—	4	1	no	—
353	C₄²	C₂×D₄×C₁₂₇	1	2	yes	—
354	C₄²	C₂×Q₈×C₁₂₇	1	2	yes	—
355	C₄²	C₂×Q₈×C₁₂₇	1	2	yes	—
356	C₄²	#41×C₁₂₇	1	2	yes	—
357	C₄²	#41×C₁₂₇	1	2	yes	—
358	C₄²	#41×C₁₂₇	1	2	yes	—
359	C₂³	C₂³×C₄×C₁₂₇	1	1	yes	characteristic
360	Q₈	—	4	8	no	—
361	C₂×C₄	—	2	2	no	—
362	C₂×C₄	—	2	1	no	—
363	C₂×C₄	—	2	1	no	—
364	C₂×C₄	—	2	1	no	—
365	C₂×C₄	—	2	1	no	—
366	C₂×C₄	—	2	1	no	—
367	C₂×C₄	—	2	1	no	—
368	C₂×C₄	—	2	1	no	—
369	C₂×C₄	—	2	1	no	—
370	C₂×C₄	—	2	1	no	—
371	C₂×C₄	—	2	1	no	—
372	C₂×C₄	—	2	2	no	—
373	C₂×C₄	—	2	2	no	—
374	C₂×C₄	—	2	2	no	—
375	C₂×C₄	—	2	2	no	—
376	C₂×C₄	—	2	2	no	—
377	C₂×C₄	—	2	2	no	—
378	C₂×C₄	—	2	2	no	—
379	C₂×C₄	—	2	2	no	—
380	C₂×C₄	—	2	2	no	—
381	C₂×C₄	—	2	2	no	—
382	C₂×C₄	C₄×D₄×C₁₂₇	1	2	yes	—
383	C₂×C₄	C₄×D₄×C₁₂₇	1	2	yes	—
384	C₂×C₄	C₄×Q₈×C₁₂₇	1	2	yes	—
385	C₂×C₄	C₄×Q₈×C₁₂₇	1	2	yes	—
386	C₂×C₄	C₄×Q₈×C₁₂₇	1	2	yes	—
387	C₂×C₄	C₄×Q₈×C₁₂₇	1	2	yes	—
388	C₂×C₄	C₂×M₁₆×C₁₂₇	1	4	yes	—
389	C₂×C₄	C₂×M₁₆×C₁₂₇	1	2	yes	—
390	C₂×C₄	#131×C₁₂₇	1	2	yes	—
391	C₂×C₄	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
392	C₂×C₄	C₂²×Q₈×C₁₂₇	1	1	yes	characteristic
393	C₂×C₄	C₂²×Q₈×C₁₂₇	1	1	yes	characteristic
394	C₂×C₄	C₂×#41×C₁₂₇	1	1	yes	characteristic
395	C₂×C₄	C₂×#41×C₁₂₇	1	1	yes	characteristic
396	C₂×C₄	C₂×#41×C₁₂₇	1	1	yes	characteristic
397	C₈	—	2	2	no	—
398	C₈	—	2	2	no	—
399	C₈	—	2	2	no	—
400	C₈	—	4	2	no	—
401	C₈	—	4	2	no	—
402	C₈	C₄⋊Q₈×C₁₂₇	1	4	yes	complemented
403	C₂²	C₂×C₄×D₄×C₁₂₇	1	1	yes	characteristic
404	C₂²	C₂×C₄×Q₈×C₁₂₇	1	1	yes	characteristic
405	C₂²	C₂×C₄×Q₈×C₁₂₇	1	1	yes	characteristic
406	C₂²	C₂²×M₁₆×C₁₂₇	1	1	yes	characteristic
407	C₂²	C₂×#131×C₁₂₇	1	1	yes	characteristic
408	C₂²	#568×C₁₂₇	1	1	yes	characteristic
409	C₂²	#568×C₁₂₇	1	1	yes	characteristic
410	C₄	—	2	2	no	—
411	C₄	—	2	4	no	—
412	C₄	—	2	2	no	—
413	C₄	—	2	2	no	—
414	C₄	—	2	2	no	—
415	C₄	—	2	2	no	—
416	C₄	—	4	2	no	—
417	C₄	—	4	2	no	—
418	C₄	—	4	2	no	—
419	C₄	#446×C₁₂₇	1	4	yes	complemented
420	C₄	C₂×C₄⋊Q₈×C₁₂₇	1	1	yes	characteristic
421	C₄	#532×C₁₂₇	1	1	yes	characteristic
422	C₄	#533×C₁₂₇	1	1	yes	characteristic
423	C₄	#533×C₁₂₇	1	1	yes	characteristic
424	C₂	C₄×C₄⋊Q₈×C₁₂₇	1	1	yes	characteristic
425	C₂	D₄×M₁₆×C₁₂₇	1	1	yes	characteristic
426	C₂	C₁₂₇×#2935	1	1	yes	characteristic
427	C₂	C₂×#446×C₁₂₇	1	1	yes	characteristic
428	C₂	C₁₂₇×#2961	1	1	yes	characteristic
429	C₂	C₁₂₇×#2962	1	1	yes	characteristic
430	C₂	Q₈×M₁₆×C₁₂₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₁₂₇	C₂⁴
commutator	C₂³	C₂³×C₄×C₁₂₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₂₇
socle, prime radical	C₂³×C₁₂₇	C₂³×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
80. #97×C₁₂₇	Q₈	semidirect
83. C₄×C₈×C₁₂₇	Q₈	semidirect
186. C₈×C₁₂₇	C₄⋊Q₈	semidirect
203. C₄×C₁₂₇	#446	semidirect
208. #21713	C₁₂₇	direct (N × H)
227. C₁₂₇	#21713	direct (N × H)
296. #97	Q₈×C₁₂₇	semidirect
299. C₄×C₈	Q₈×C₁₂₇	semidirect
402. C₈	C₄⋊Q₈×C₁₂₇	semidirect
419. C₄	#446×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#446×C₁₂₇	2
C₄×C₄⋊Q₈×C₁₂₇	2
C₁₂₇×#1985	2
C₁₂₇×#1940	2
C₁₂₇×#1939	2
C₁₂₇×#1842	2
C₁₂₇×#1771	2
C₁₂₇×#1770	2
C₈×C₄⋊C₄×C₁₂₇	2
C₄×#97×C₁₂₇	2
#21713	127

Subgroups — C127×#21713 #44585386

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#21713 #44585386