Subgroups — C₁₂₇×#21693 #44585366

Order 32512.

1126 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#495×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#442×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#441×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#440×C₁₂₇	C₂	1	2	yes	maximal, complemented
5	C₄×#124×C₁₂₇	C₂	1	2	yes	maximal, complemented
6	C₁₂₇×#1973	C₂	1	2	yes	maximal, complemented
7	C₁₂₇×#1972	C₂	1	1	yes	maximal, characteristic
8	C₁₂₇×#1971	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₂₇×#1958	C₂	1	2	yes	maximal, complemented
10	C₁₂₇×#1846	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#97×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×Q₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
13	C₂²×SD₁₆×C₁₂₇	C₂²	1	2	yes	—
14	C₂²×D₈×C₁₂₇	C₂²	1	1	yes	characteristic
15	C₂×#124×C₁₂₇	C₂²	1	2	yes	—
16	C₂×C₄×Q₈×C₁₂₇	C₂²	1	2	yes	—
17	C₂×C₄×D₄×C₁₂₇	C₂²	1	2	yes	—
18	#495×C₁₂₇	C₄	1	8	yes	complemented
19	#442×C₁₂₇	—	2	4	no	—
20	#441×C₁₂₇	—	2	4	no	—
21	#440×C₁₂₇	—	2	8	no	—
22	C₂×#107×C₁₂₇	C₂²	1	2	yes	—
23	C₂×#106×C₁₂₇	C₂²	1	1	yes	characteristic
24	C₂×#106×C₁₂₇	C₂²	1	1	yes	characteristic
25	C₂×#103×C₁₂₇	C₂²	1	2	yes	—
26	C₂×#103×C₁₂₇	C₂²	1	2	yes	—
27	C₂×#102×C₁₂₇	C₂²	1	2	yes	—
28	C₂×#102×C₁₂₇	C₂²	1	2	yes	—
29	C₂×#97×C₁₂₇	C₂²	1	1	yes	characteristic
30	C₂×#97×C₁₂₇	C₂²	1	1	yes	characteristic
31	C₂×#97×C₁₂₇	C₂²	1	1	yes	characteristic
32	C₂×#97×C₁₂₇	C₂²	1	1	yes	characteristic
33	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
34	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
35	#391×C₁₂₇	C₂²	1	2	yes	—
36	#390×C₁₂₇	C₂²	1	2	yes	—
37	#388×C₁₂₇	—	2	4	no	—
38	#382×C₁₂₇	C₂²	1	2	yes	—
39	C₄×C₂²⋊C₄×C₁₂₇	—	2	4	no	—
40	C₄³×C₁₂₇	C₂²	1	1	yes	characteristic
41	C₂²×Q₈×C₁₂₇	C₂³	1	2	yes	—
42	C₂²×D₄×C₁₂₇	C₂³	1	2	yes	—
43	C₂³×C₄×C₁₂₇	—	2	2	no	—
44	C₂×Q₁₆×C₁₂₇	—	2	2	no	—
45	C₂×Q₁₆×C₁₂₇	—	2	2	no	—
46	C₂×Q₁₆×C₁₂₇	C₂×C₄	1	4	yes	complemented
47	C₂×SD₁₆×C₁₂₇	—	2	4	no	—
48	C₂×SD₁₆×C₁₂₇	—	2	4	no	—
49	C₂×SD₁₆×C₁₂₇	C₂×C₄	1	8	yes	complemented
50	C₂×D₈×C₁₂₇	—	2	2	no	—
51	C₂×D₈×C₁₂₇	—	2	2	no	—
52	C₂×D₈×C₁₂₇	C₂×C₄	1	4	yes	—
53	C₂²×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
54	C₂²×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
55	C₂²×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
56	C₂²×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
57	#124×C₁₂₇	—	2	4	no	—
58	#124×C₁₂₇	C₂×C₄	1	8	yes	complemented
59	C₄×Q₈×C₁₂₇	—	2	4	no	—
60	C₄×Q₈×C₁₂₇	—	2	4	no	—
61	C₄×D₄×C₁₂₇	—	2	4	no	—
62	C₄×D₄×C₁₂₇	—	2	4	no	—
63	C₂×C₄⋊C₄×C₁₂₇	—	2	2	no	—
64	C₂×C₄⋊C₄×C₁₂₇	C₂³	1	2	yes	—
65	C₂×C₄⋊C₄×C₁₂₇	C₂³	1	2	yes	—
66	C₂×C₂²⋊C₄×C₁₂₇	—	2	4	no	—
67	C₂×C₂²⋊C₄×C₁₂₇	—	2	2	no	—
68	C₂×C₂²⋊C₄×C₁₂₇	—	2	4	no	—
69	C₂×C₄²×C₁₂₇	—	2	2	no	—
70	C₂×C₄²×C₁₂₇	D₄	1	4	yes	—
71	C₂×C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
72	C₂×C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
73	C₂×C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
74	#107×C₁₂₇	—	2	4	no	—
75	#106×C₁₂₇	—	2	2	no	—
76	#106×C₁₂₇	—	2	2	no	—
77	#103×C₁₂₇	—	2	4	no	—
78	#103×C₁₂₇	—	2	4	no	—
79	#102×C₁₂₇	—	2	4	no	—
80	#102×C₁₂₇	—	2	4	no	—
81	#97×C₁₂₇	—	2	2	no	—
82	#97×C₁₂₇	—	2	2	no	—
83	#97×C₁₂₇	D₄	1	4	yes	—
84	#97×C₁₂₇	D₄	1	4	yes	—
85	C₄×C₈×C₁₂₇	—	2	2	no	—
86	C₄×C₈×C₁₂₇	C₂×C₄	1	4	yes	complemented
87	#95×C₁₂₇	—	2	4	no	—
88	#95×C₁₂₇	—	2	4	no	—
89	C₂⁴×C₁₂₇	—	2	2	no	—
90	C₂×Q₈×C₁₂₇	—	2	2	no	—
91	C₂×Q₈×C₁₂₇	—	2	2	no	—
92	C₂×Q₈×C₁₂₇	—	2	2	no	—
93	C₂×Q₈×C₁₂₇	—	2	2	no	—
94	C₂×Q₈×C₁₂₇	—	2	2	no	—
95	C₂×Q₈×C₁₂₇	C₂²×C₄	1	4	yes	—
96	C₂×D₄×C₁₂₇	—	2	2	no	—
97	C₂×D₄×C₁₂₇	—	2	2	no	—
98	C₂×D₄×C₁₂₇	—	2	2	no	—
99	C₂×D₄×C₁₂₇	—	2	2	no	—
100	C₂×D₄×C₁₂₇	—	2	2	no	—
101	C₂×D₄×C₁₂₇	C₂²×C₄	1	4	yes	—
102	C₂²×C₄×C₁₂₇	—	2	2	no	—
103	C₂²×C₄×C₁₂₇	—	2	2	no	—
104	C₂²×C₄×C₁₂₇	—	2	2	no	—
105	C₂²×C₄×C₁₂₇	—	4	8	no	—
106	C₂²×C₄×C₁₂₇	—	4	4	no	—
107	C₂²×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
108	C₂²×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
109	C₂²×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
110	C₂²×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
111	Q₁₆×C₁₂₇	—	2	8	no	—
112	SD₁₆×C₁₂₇	—	2	16	no	—
113	D₈×C₁₂₇	—	2	8	no	—
114	C₂×C₈×C₁₂₇	—	2	1	no	—
115	C₂×C₈×C₁₂₇	—	2	1	no	—
116	C₂×C₈×C₁₂₇	—	2	1	no	—
117	C₂×C₈×C₁₂₇	—	2	1	no	—
118	C₂×C₈×C₁₂₇	C₂²×C₄	1	2	yes	—
119	C₂×C₈×C₁₂₇	C₂²×C₄	1	2	yes	—
120	C₂×C₈×C₁₂₇	C₂×D₄	1	2	yes	—
121	C₂×C₈×C₁₂₇	C₂×D₄	1	2	yes	—
122	C₂×C₈×C₁₂₇	C₂×D₄	1	2	yes	—
123	C₂×C₈×C₁₂₇	C₂×D₄	1	2	yes	—
124	C₂×C₈×C₁₂₇	#41	1	2	yes	—
125	C₂×C₈×C₁₂₇	#41	1	2	yes	—
126	C₄⋊C₄×C₁₂₇	—	2	2	no	—
127	C₄⋊C₄×C₁₂₇	—	2	2	no	—
128	C₄⋊C₄×C₁₂₇	—	2	2	no	—
129	C₄⋊C₄×C₁₂₇	—	2	2	no	—
130	C₄⋊C₄×C₁₂₇	—	4	4	no	—
131	C₂²⋊C₄×C₁₂₇	—	2	16	no	—
132	C₂²⋊C₄×C₁₂₇	—	4	8	no	—
133	C₂²⋊C₄×C₁₂₇	—	4	4	no	—
134	C₄²×C₁₂₇	—	2	1	no	—
135	C₄²×C₁₂₇	—	2	1	no	—
136	C₄²×C₁₂₇	—	4	4	no	—
137	C₄²×C₁₂₇	—	4	4	no	—
138	C₄²×C₁₂₇	C₂²×C₄	1	2	yes	—
139	C₄²×C₁₂₇	C₂×D₄	1	2	yes	—
140	C₄²×C₁₂₇	C₂×D₄	1	2	yes	—
141	C₄²×C₁₂₇	#41	1	2	yes	—
142	C₂³×C₁₂₇	—	2	4	no	—
143	C₂³×C₁₂₇	—	4	2	no	—
144	C₂³×C₁₂₇	—	4	2	no	—
145	C₂³×C₁₂₇	—	4	4	no	—
146	C₂³×C₁₂₇	—	4	4	no	—
147	C₂³×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
148	Q₈×C₁₂₇	—	2	4	no	—
149	Q₈×C₁₂₇	—	2	4	no	—
150	Q₈×C₁₂₇	—	4	2	no	—
151	Q₈×C₁₂₇	—	4	2	no	—
152	D₄×C₁₂₇	—	2	4	no	—
153	D₄×C₁₂₇	—	2	4	no	—
154	D₄×C₁₂₇	—	4	2	no	—
155	D₄×C₁₂₇	—	4	2	no	—
156	C₂×C₄×C₁₂₇	—	2	4	no	—
157	C₂×C₄×C₁₂₇	—	2	4	no	—
158	C₂×C₄×C₁₂₇	—	2	4	no	—
159	C₂×C₄×C₁₂₇	—	2	2	no	—
160	C₂×C₄×C₁₂₇	—	2	4	no	—
161	C₂×C₄×C₁₂₇	—	4	2	no	—
162	C₂×C₄×C₁₂₇	—	4	2	no	—
163	C₂×C₄×C₁₂₇	—	4	2	no	—
164	C₂×C₄×C₁₂₇	—	4	2	no	—
165	C₂×C₄×C₁₂₇	—	4	4	no	—
166	C₂×C₄×C₁₂₇	—	4	4	no	—
167	C₂×C₄×C₁₂₇	—	8	8	no	—
168	C₂×C₄×C₁₂₇	C₄×D₄	1	4	yes	—
169	C₂×C₄×C₁₂₇	#127	1	4	yes	—
170	C₂×C₄×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
171	C₂×C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
172	C₂×C₄×C₁₂₇	C₂²×D₄	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	C₈×C₁₂₇	—	2	2	no	—
177	C₈×C₁₂₇	—	2	2	no	—
178	C₈×C₁₂₇	C₄×D₄	1	4	yes	—
179	C₈×C₁₂₇	C₄×D₄	1	4	yes	—
180	C₂²×C₁₂₇	—	4	4	no	—
181	C₂²×C₁₂₇	—	4	8	no	—
182	C₂²×C₁₂₇	—	8	2	no	—
183	C₂²×C₁₂₇	—	8	2	no	—
184	C₂²×C₁₂₇	—	8	4	no	—
185	C₂²×C₁₂₇	C₂×C₄×D₄	1	1	yes	characteristic
186	C₂²×C₁₂₇	C₂×#127	1	1	yes	characteristic
187	C₂²×C₁₂₇	#532	1	1	yes	characteristic
188	C₂²×C₁₂₇	#575	1	4	yes	—
189	C₄×C₁₂₇	—	2	4	no	—
190	C₄×C₁₂₇	—	2	4	no	—
191	C₄×C₁₂₇	—	4	4	no	—
192	C₄×C₁₂₇	—	4	2	no	—
193	C₄×C₁₂₇	—	8	2	no	—
194	C₄×C₁₂₇	—	8	2	no	—
195	C₄×C₁₂₇	C₂×C₄×D₄	1	1	yes	characteristic
196	C₄×C₁₂₇	C₂×C₄×D₄	1	1	yes	characteristic
197	C₄×C₁₂₇	#532	1	1	yes	characteristic
198	C₄×C₁₂₇	#532	1	1	yes	characteristic
199	#21693	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
200	C₂×C₁₂₇	—	8	4	no	—
201	C₂×C₁₂₇	C₄×#127	1	1	yes	characteristic
202	C₂×C₁₂₇	#2943	1	2	yes	—
203	C₂×C₁₂₇	#3153	1	2	yes	—
204	C₂×C₁₂₇	#3183	1	2	yes	—
205	C₂×#495	C₂×C₁₂₇	1	1	yes	characteristic
206	C₂×#442	C₂×C₁₂₇	1	1	yes	characteristic
207	C₂×#441	C₂×C₁₂₇	1	1	yes	characteristic, complemented
208	C₂×#440	C₂×C₁₂₇	1	2	yes	complemented
209	C₄×#124	C₂×C₁₂₇	1	2	yes	complemented
210	#1973	C₂×C₁₂₇	1	2	yes	complemented
211	#1972	C₂×C₁₂₇	1	1	yes	characteristic
212	#1971	C₂×C₁₂₇	1	1	yes	characteristic, complemented
213	#1958	C₂×C₁₂₇	1	2	yes	complemented
214	#1846	C₂×C₁₂₇	1	1	yes	characteristic, complemented
215	C₄×#97	C₂×C₁₂₇	1	1	yes	characteristic, complemented
216	C₁₂₇	#21693	1	1	yes	characteristic, direct product factor, Sylow
217	C₂²×Q₁₆	C₂²×C₁₂₇	1	1	yes	characteristic
218	C₂²×SD₁₆	C₂²×C₁₂₇	1	2	yes	—
219	C₂²×D₈	C₂²×C₁₂₇	1	1	yes	characteristic
220	C₂×#124	C₂²×C₁₂₇	1	2	yes	—
221	C₂×C₄×Q₈	C₂²×C₁₂₇	1	2	yes	—
222	C₂×C₄×D₄	C₂²×C₁₂₇	1	2	yes	—
223	#495	C₄×C₁₂₇	1	8	yes	complemented
224	#442	—	2	4	no	—
225	#441	—	2	4	no	—
226	#440	—	2	8	no	—
227	C₂×#107	C₂²×C₁₂₇	1	2	yes	—
228	C₂×#106	C₂²×C₁₂₇	1	1	yes	characteristic
229	C₂×#106	C₂²×C₁₂₇	1	1	yes	characteristic
230	C₂×#103	C₂²×C₁₂₇	1	2	yes	—
231	C₂×#103	C₂²×C₁₂₇	1	2	yes	—
232	C₂×#102	C₂²×C₁₂₇	1	2	yes	—
233	C₂×#102	C₂²×C₁₂₇	1	2	yes	—
234	C₂×#97	C₂²×C₁₂₇	1	1	yes	characteristic
235	C₂×#97	C₂²×C₁₂₇	1	1	yes	characteristic
236	C₂×#97	C₂²×C₁₂₇	1	1	yes	characteristic
237	C₂×#97	C₂²×C₁₂₇	1	1	yes	characteristic
238	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
239	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
240	#391	C₂²×C₁₂₇	1	2	yes	—
241	#390	C₂²×C₁₂₇	1	2	yes	—
242	#388	—	2	4	no	—
243	#382	C₂²×C₁₂₇	1	2	yes	—
244	C₄×C₂²⋊C₄	—	2	4	no	—
245	C₄³	C₂²×C₁₂₇	1	1	yes	characteristic
246	C₂²×Q₈	C₂³×C₁₂₇	1	2	yes	—
247	C₂²×D₄	C₂³×C₁₂₇	1	2	yes	—
248	C₂³×C₄	—	2	2	no	—
249	C₂×Q₁₆	—	2	2	no	—
250	C₂×Q₁₆	—	2	2	no	—
251	C₂×Q₁₆	C₂×C₄×C₁₂₇	1	4	yes	complemented
252	C₂×SD₁₆	—	2	4	no	—
253	C₂×SD₁₆	—	2	4	no	—
254	C₂×SD₁₆	C₂×C₄×C₁₂₇	1	8	yes	complemented
255	C₂×D₈	—	2	2	no	—
256	C₂×D₈	—	2	2	no	—
257	C₂×D₈	C₂×C₄×C₁₂₇	1	4	yes	—
258	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
259	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
260	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
261	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
262	#124	—	2	4	no	—
263	#124	C₂×C₄×C₁₂₇	1	8	yes	complemented
264	C₄×Q₈	—	2	4	no	—
265	C₄×Q₈	—	2	4	no	—
266	C₄×D₄	—	2	4	no	—
267	C₄×D₄	—	2	4	no	—
268	C₂×C₄⋊C₄	—	2	2	no	—
269	C₂×C₄⋊C₄	C₂³×C₁₂₇	1	2	yes	—
270	C₂×C₄⋊C₄	C₂³×C₁₂₇	1	2	yes	—
271	C₂×C₂²⋊C₄	—	2	4	no	—
272	C₂×C₂²⋊C₄	—	2	2	no	—
273	C₂×C₂²⋊C₄	—	2	4	no	—
274	C₂×C₄²	—	2	2	no	—
275	C₂×C₄²	D₄×C₁₂₇	1	4	yes	—
276	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
277	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
278	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
279	#107	—	2	4	no	—
280	#106	—	2	2	no	—
281	#106	—	2	2	no	—
282	#103	—	2	4	no	—
283	#103	—	2	4	no	—
284	#102	—	2	4	no	—
285	#102	—	2	4	no	—
286	#97	—	2	2	no	—
287	#97	—	2	2	no	—
288	#97	D₄×C₁₂₇	1	4	yes	—
289	#97	D₄×C₁₂₇	1	4	yes	—
290	C₄×C₈	—	2	2	no	—
291	C₄×C₈	C₂×C₄×C₁₂₇	1	4	yes	complemented
292	#95	—	2	4	no	—
293	#95	—	2	4	no	—
294	C₂⁴	—	2	2	no	—
295	C₂×Q₈	—	2	2	no	—
296	C₂×Q₈	—	2	2	no	—
297	C₂×Q₈	—	2	2	no	—
298	C₂×Q₈	—	2	2	no	—
299	C₂×Q₈	—	2	2	no	—
300	C₂×Q₈	C₂²×C₄×C₁₂₇	1	4	yes	—
301	C₂×D₄	—	2	2	no	—
302	C₂×D₄	—	2	2	no	—
303	C₂×D₄	—	2	2	no	—
304	C₂×D₄	—	2	2	no	—
305	C₂×D₄	—	2	2	no	—
306	C₂×D₄	C₂²×C₄×C₁₂₇	1	4	yes	—
307	C₂²×C₄	—	2	2	no	—
308	C₂²×C₄	—	2	2	no	—
309	C₂²×C₄	—	2	2	no	—
310	C₂²×C₄	—	4	8	no	—
311	C₂²×C₄	—	4	4	no	—
312	C₂²×C₄	C₂×D₄×C₁₂₇	1	2	yes	—
313	C₂²×C₄	C₂×D₄×C₁₂₇	1	2	yes	—
314	C₂²×C₄	C₂×D₄×C₁₂₇	1	2	yes	—
315	C₂²×C₄	C₂⁴×C₁₂₇	1	1	yes	characteristic
316	Q₁₆	—	2	8	no	—
317	SD₁₆	—	2	16	no	—
318	D₈	—	2	8	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	C₂²×C₄×C₁₂₇	1	2	yes	—
324	C₂×C₈	C₂²×C₄×C₁₂₇	1	2	yes	—
325	C₂×C₈	C₂×D₄×C₁₂₇	1	2	yes	—
326	C₂×C₈	C₂×D₄×C₁₂₇	1	2	yes	—
327	C₂×C₈	C₂×D₄×C₁₂₇	1	2	yes	—
328	C₂×C₈	C₂×D₄×C₁₂₇	1	2	yes	—
329	C₂×C₈	#41×C₁₂₇	1	2	yes	—
330	C₂×C₈	#41×C₁₂₇	1	2	yes	—
331	C₄⋊C₄	—	2	2	no	—
332	C₄⋊C₄	—	2	2	no	—
333	C₄⋊C₄	—	2	2	no	—
334	C₄⋊C₄	—	2	2	no	—
335	C₄⋊C₄	—	4	4	no	—
336	C₂²⋊C₄	—	2	16	no	—
337	C₂²⋊C₄	—	4	8	no	—
338	C₂²⋊C₄	—	4	4	no	—
339	C₄²	—	2	1	no	—
340	C₄²	—	2	1	no	—
341	C₄²	—	4	4	no	—
342	C₄²	—	4	4	no	—
343	C₄²	C₂²×C₄×C₁₂₇	1	2	yes	—
344	C₄²	C₂×D₄×C₁₂₇	1	2	yes	—
345	C₄²	C₂×D₄×C₁₂₇	1	2	yes	—
346	C₄²	#41×C₁₂₇	1	2	yes	—
347	C₂³	—	2	4	no	—
348	C₂³	—	4	2	no	—
349	C₂³	—	4	2	no	—
350	C₂³	—	4	4	no	—
351	C₂³	—	4	4	no	—
352	C₂³	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
353	Q₈	—	2	4	no	—
354	Q₈	—	2	4	no	—
355	Q₈	—	4	2	no	—
356	Q₈	—	4	2	no	—
357	D₄	—	2	4	no	—
358	D₄	—	2	4	no	—
359	D₄	—	4	2	no	—
360	D₄	—	4	2	no	—
361	C₂×C₄	—	2	4	no	—
362	C₂×C₄	—	2	4	no	—
363	C₂×C₄	—	2	4	no	—
364	C₂×C₄	—	2	2	no	—
365	C₂×C₄	—	2	4	no	—
366	C₂×C₄	—	4	2	no	—
367	C₂×C₄	—	4	2	no	—
368	C₂×C₄	—	4	2	no	—
369	C₂×C₄	—	4	2	no	—
370	C₂×C₄	—	4	4	no	—
371	C₂×C₄	—	4	4	no	—
372	C₂×C₄	—	8	8	no	—
373	C₂×C₄	C₄×D₄×C₁₂₇	1	4	yes	—
374	C₂×C₄	#127×C₁₂₇	1	4	yes	—
375	C₂×C₄	C₂³×C₄×C₁₂₇	1	1	yes	characteristic
376	C₂×C₄	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
377	C₂×C₄	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
378	C₂×C₄	C₂×#41×C₁₂₇	1	1	yes	characteristic
379	C₂×C₄	C₂×#41×C₁₂₇	1	1	yes	characteristic
380	C₂×C₄	C₂×#41×C₁₂₇	1	1	yes	characteristic
381	C₈	—	2	2	no	—
382	C₈	—	2	2	no	—
383	C₈	C₄×D₄×C₁₂₇	1	4	yes	—
384	C₈	C₄×D₄×C₁₂₇	1	4	yes	—
385	C₂²	—	4	8	no	—
386	C₂²	—	4	4	no	—
387	C₂²	—	8	2	no	—
388	C₂²	—	8	2	no	—
389	C₂²	—	8	4	no	—
390	C₂²	C₂×C₄×D₄×C₁₂₇	1	1	yes	characteristic
391	C₂²	C₂×#127×C₁₂₇	1	1	yes	characteristic
392	C₂²	#532×C₁₂₇	1	1	yes	characteristic
393	C₂²	#575×C₁₂₇	1	4	yes	—
394	C₄	—	2	4	no	—
395	C₄	—	2	4	no	—
396	C₄	—	4	2	no	—
397	C₄	—	4	4	no	—
398	C₄	—	8	2	no	—
399	C₄	—	8	2	no	—
400	C₄	C₂×C₄×D₄×C₁₂₇	1	1	yes	characteristic
401	C₄	C₂×C₄×D₄×C₁₂₇	1	1	yes	characteristic
402	C₄	#532×C₁₂₇	1	1	yes	characteristic
403	C₄	#532×C₁₂₇	1	1	yes	characteristic
404	C₂	—	8	4	no	—
405	C₂	C₄×#127×C₁₂₇	1	1	yes	characteristic
406	C₂	C₁₂₇×#2943	1	2	yes	—
407	C₂	C₁₂₇×#3153	1	2	yes	—
408	C₂	C₁₂₇×#3183	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₂₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₁₂₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₂₇
prime radical	C₂²×D₈×C₁₂₇	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₂₇ ◁ C₄³×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×#441×C₁₂₇	C₂	semidirect
4. C₂×#440×C₁₂₇	C₂	semidirect
5. C₄×#124×C₁₂₇	C₂	semidirect
6. C₁₂₇×#1973	C₂	semidirect
8. C₁₂₇×#1971	C₂	semidirect
9. C₁₂₇×#1958	C₂	semidirect
10. C₁₂₇×#1846	C₂	semidirect
11. C₄×#97×C₁₂₇	C₂	semidirect
18. #495×C₁₂₇	C₄	semidirect
46. C₂×Q₁₆×C₁₂₇	C₂×C₄	semidirect
49. C₂×SD₁₆×C₁₂₇	C₂×C₄	semidirect
58. #124×C₁₂₇	C₂×C₄	semidirect
86. C₄×C₈×C₁₂₇	C₂×C₄	semidirect
199. #21693	C₁₂₇	direct (N × H)
207. C₂×#441	C₂×C₁₂₇	semidirect
208. C₂×#440	C₂×C₁₂₇	semidirect
209. C₄×#124	C₂×C₁₂₇	semidirect
210. #1973	C₂×C₁₂₇	semidirect
212. #1971	C₂×C₁₂₇	semidirect
213. #1958	C₂×C₁₂₇	semidirect
214. #1846	C₂×C₁₂₇	semidirect
215. C₄×#97	C₂×C₁₂₇	semidirect
216. C₁₂₇	#21693	direct (N × H)
223. #495	C₄×C₁₂₇	semidirect
251. C₂×Q₁₆	C₂×C₄×C₁₂₇	semidirect
254. C₂×SD₁₆	C₂×C₄×C₁₂₇	semidirect
263. #124	C₂×C₄×C₁₂₇	semidirect
291. C₄×C₈	C₂×C₄×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#495×C₁₂₇	2
C₂×#442×C₁₂₇	2
C₂×#441×C₁₂₇	2
C₂×#440×C₁₂₇	2
C₄×#124×C₁₂₇	2
C₁₂₇×#1973	2
C₁₂₇×#1972	2
C₁₂₇×#1971	2
C₁₂₇×#1958	2
C₁₂₇×#1846	2
C₄×#97×C₁₂₇	2
#21693	127

Subgroups — C127×#21693 #44585366

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#21693 #44585366