Subgroups — C₁₂₇×#21696 #44585369

Order 32512.

1110 conjugacy classes of subgroups.

The subgroup lattice has 428 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×#440×C₁₂₇	C₂	1	2	yes	maximal, complemented
3	C₂×C₄×Q₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₄×D₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic
5	C₁₂₇×#2326	C₂	1	2	yes	maximal, complemented
6	C₁₂₇×#1973	C₂	1	2	yes	maximal, complemented
7	C₁₂₇×#1970	C₂	1	1	yes	maximal, characteristic
8	C₁₂₇×#1969	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₂₇×#1959	C₂	1	2	yes	maximal, complemented
10	C₁₂₇×#1845	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₂₇×#1726	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	#440×C₁₂₇	—	2	8	no	—
20	C₄×Q₁₆×C₁₂₇	—	2	4	no	—
21	C₄×D₈×C₁₂₇	—	2	4	no	—
22	C₂×#107×C₁₂₇	C₂²	1	1	yes	characteristic
23	C₂×#107×C₁₂₇	C₂²	1	1	yes	characteristic
24	C₂×#107×C₁₂₇	C₂²	1	2	yes	—
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×C₄×C₈×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	#385×C₁₂₇	—	2	4	no	—
39	#384×C₁₂₇	C₂²	1	2	yes	—
40	#376×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	—	2	4	no	—
65	C₂×C₄⋊C₄×C₁₂₇	C₂³	1	2	yes	—
66	C₂×C₄⋊C₄×C₁₂₇	C₂³	1	2	yes	—
67	C₂×C₂²⋊C₄×C₁₂₇	—	2	4	no	—
68	C₂×C₂²⋊C₄×C₁₂₇	—	2	2	no	—
69	C₂×C₂²⋊C₄×C₁₂₇	—	2	4	no	—
70	C₂×C₄²×C₁₂₇	—	2	2	no	—
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	2	no	—
75	#107×C₁₂₇	—	2	2	no	—
76	#107×C₁₂₇	—	2	4	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	C₄×C₈×C₁₂₇	—	2	2	no	—
85	C₄×C₈×C₁₂₇	C₂×C₄	1	4	yes	complemented
86	C₄×C₈×C₁₂₇	D₄	1	4	yes	—
87	#95×C₁₂₇	—	2	4	no	—
88	#95×C₁₂₇	D₄	1	2	yes	—
89	#95×C₁₂₇	D₄	1	2	yes	—
90	C₂⁴×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	—	2	2	no	—
96	C₂×Q₈×C₁₂₇	C₂²×C₄	1	4	yes	—
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	—	2	2	no	—
102	C₂×D₄×C₁₂₇	C₂²×C₄	1	4	yes	—
103	C₂²×C₄×C₁₂₇	—	2	2	no	—
104	C₂²×C₄×C₁₂₇	—	2	2	no	—
105	C₂²×C₄×C₁₂₇	—	2	2	no	—
106	C₂²×C₄×C₁₂₇	—	4	4	no	—
107	C₂²×C₄×C₁₂₇	—	8	4	no	—
108	C₂²×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
109	C₂²×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
110	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
111	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
112	C₂²×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
113	Q₁₆×C₁₂₇	—	2	8	no	—
114	SD₁₆×C₁₂₇	—	2	16	no	—
115	D₈×C₁₂₇	—	2	8	no	—
116	C₂×C₈×C₁₂₇	—	2	1	no	—
117	C₂×C₈×C₁₂₇	—	2	1	no	—
118	C₂×C₈×C₁₂₇	—	2	1	no	—
119	C₂×C₈×C₁₂₇	—	2	1	no	—
120	C₂×C₈×C₁₂₇	C₂²×C₄	1	2	yes	—
121	C₂×C₈×C₁₂₇	C₂²×C₄	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	C₂×D₄	1	2	yes	—
125	C₂×C₈×C₁₂₇	C₂×D₄	1	2	yes	—
126	C₂×C₈×C₁₂₇	#41	1	2	yes	—
127	C₂×C₈×C₁₂₇	#41	1	2	yes	—
128	C₄⋊C₄×C₁₂₇	—	2	2	no	—
129	C₄⋊C₄×C₁₂₇	—	2	2	no	—
130	C₄⋊C₄×C₁₂₇	—	2	2	no	—
131	C₄⋊C₄×C₁₂₇	—	2	2	no	—
132	C₄⋊C₄×C₁₂₇	—	4	4	no	—
133	C₄⋊C₄×C₁₂₇	—	8	4	no	—
134	C₂²⋊C₄×C₁₂₇	—	2	16	no	—
135	C₂²⋊C₄×C₁₂₇	—	4	4	no	—
136	C₂²⋊C₄×C₁₂₇	—	8	4	no	—
137	C₄²×C₁₂₇	—	2	1	no	—
138	C₄²×C₁₂₇	—	2	1	no	—
139	C₄²×C₁₂₇	—	4	4	no	—
140	C₄²×C₁₂₇	C₂²×C₄	1	2	yes	—
141	C₄²×C₁₂₇	C₂×D₄	1	2	yes	—
142	C₄²×C₁₂₇	C₂×D₄	1	2	yes	—
143	C₄²×C₁₂₇	#41	1	2	yes	—
144	C₂³×C₁₂₇	—	2	4	no	—
145	C₂³×C₁₂₇	—	4	2	no	—
146	C₂³×C₁₂₇	—	4	2	no	—
147	C₂³×C₁₂₇	—	4	4	no	—
148	C₂³×C₁₂₇	—	4	4	no	—
149	C₂³×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
150	Q₈×C₁₂₇	—	2	4	no	—
151	Q₈×C₁₂₇	—	2	4	no	—
152	Q₈×C₁₂₇	—	4	2	no	—
153	Q₈×C₁₂₇	—	4	2	no	—
154	D₄×C₁₂₇	—	2	4	no	—
155	D₄×C₁₂₇	—	2	4	no	—
156	D₄×C₁₂₇	—	4	2	no	—
157	D₄×C₁₂₇	—	4	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	4	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	—	4	2	no	—
167	C₂×C₄×C₁₂₇	—	4	2	no	—
168	C₂×C₄×C₁₂₇	—	4	2	no	—
169	C₂×C₄×C₁₂₇	—	4	2	no	—
170	C₂×C₄×C₁₂₇	—	4	4	no	—
171	C₂×C₄×C₁₂₇	—	4	4	no	—
172	C₂×C₄×C₁₂₇	—	8	8	no	—
173	C₂×C₄×C₁₂₇	C₄×D₄	1	4	yes	—
174	C₂×C₄×C₁₂₇	#127	1	2	yes	—
175	C₂×C₄×C₁₂₇	#127	1	2	yes	—
176	C₂×C₄×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
177	C₂×C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
178	C₂×C₄×C₁₂₇	C₂²×D₄	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	C₂×C₄×C₁₂₇	C₂×#41	1	1	yes	characteristic
182	C₈×C₁₂₇	—	2	2	no	—
183	C₈×C₁₂₇	—	2	2	no	—
184	C₈×C₁₂₇	C₄×D₄	1	4	yes	—
185	C₈×C₁₂₇	C₄×D₄	1	4	yes	—
186	C₂²×C₁₂₇	—	4	4	no	—
187	C₂²×C₁₂₇	—	4	8	no	—
188	C₂²×C₁₂₇	—	8	2	no	—
189	C₂²×C₁₂₇	—	8	2	no	—
190	C₂²×C₁₂₇	—	8	4	no	—
191	C₂²×C₁₂₇	C₂×C₄×D₄	1	1	yes	characteristic
192	C₂²×C₁₂₇	C₂×#127	1	1	yes	characteristic
193	C₂²×C₁₂₇	#532	1	1	yes	characteristic
194	C₂²×C₁₂₇	#576	1	2	yes	—
195	C₂²×C₁₂₇	#578	1	2	yes	—
196	C₄×C₁₂₇	—	2	2	no	—
197	C₄×C₁₂₇	—	2	2	no	—
198	C₄×C₁₂₇	—	2	4	no	—
199	C₄×C₁₂₇	—	4	4	no	—
200	C₄×C₁₂₇	—	4	2	no	—
201	C₄×C₁₂₇	—	8	2	no	—
202	C₄×C₁₂₇	—	8	2	no	—
203	C₄×C₁₂₇	C₂×C₄×D₄	1	1	yes	characteristic
204	C₄×C₁₂₇	C₂×C₄×D₄	1	1	yes	characteristic
205	C₄×C₁₂₇	#532	1	1	yes	characteristic
206	C₄×C₁₂₇	#532	1	1	yes	characteristic
207	#21696	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
208	C₂×C₁₂₇	—	8	4	no	—
209	C₂×C₁₂₇	C₄×#127	1	1	yes	characteristic
210	C₂×C₁₂₇	#2951	1	2	yes	—
211	C₂×C₁₂₇	#3155	1	1	yes	characteristic
212	C₂×C₁₂₇	#3156	1	1	yes	characteristic
213	C₂×C₁₂₇	#3175	1	2	yes	—
214	C₂×#495	C₂×C₁₂₇	1	1	yes	characteristic
215	C₂×#440	C₂×C₁₂₇	1	2	yes	complemented
216	C₂×C₄×Q₁₆	C₂×C₁₂₇	1	1	yes	characteristic, complemented
217	C₂×C₄×D₈	C₂×C₁₂₇	1	1	yes	characteristic
218	#2326	C₂×C₁₂₇	1	2	yes	complemented
219	#1973	C₂×C₁₂₇	1	2	yes	complemented
220	#1970	C₂×C₁₂₇	1	1	yes	characteristic
221	#1969	C₂×C₁₂₇	1	1	yes	characteristic, complemented
222	#1959	C₂×C₁₂₇	1	2	yes	complemented
223	#1845	C₂×C₁₂₇	1	1	yes	characteristic, complemented
224	#1726	C₂×C₁₂₇	1	1	yes	characteristic, complemented
225	C₁₂₇	#21696	1	1	yes	characteristic, direct product factor, Sylow
226	C₂²×Q₁₆	C₂²×C₁₂₇	1	1	yes	characteristic
227	C₂²×SD₁₆	C₂²×C₁₂₇	1	2	yes	—
228	C₂²×D₈	C₂²×C₁₂₇	1	1	yes	characteristic
229	C₂×#124	C₂²×C₁₂₇	1	2	yes	—
230	C₂×C₄×Q₈	C₂²×C₁₂₇	1	2	yes	—
231	C₂×C₄×D₄	C₂²×C₁₂₇	1	2	yes	—
232	#495	C₄×C₁₂₇	1	8	yes	complemented
233	#440	—	2	8	no	—
234	C₄×Q₁₆	—	2	4	no	—
235	C₄×D₈	—	2	4	no	—
236	C₂×#107	C₂²×C₁₂₇	1	1	yes	characteristic
237	C₂×#107	C₂²×C₁₂₇	1	1	yes	characteristic
238	C₂×#107	C₂²×C₁₂₇	1	2	yes	—
239	C₂×#103	C₂²×C₁₂₇	1	2	yes	—
240	C₂×#103	C₂²×C₁₂₇	1	2	yes	—
241	C₂×#102	C₂²×C₁₂₇	1	2	yes	—
242	C₂×#102	C₂²×C₁₂₇	1	2	yes	—
243	C₂×#97	C₂²×C₁₂₇	1	1	yes	characteristic
244	C₂×#97	C₂²×C₁₂₇	1	1	yes	characteristic
245	C₂×#97	C₂²×C₁₂₇	1	1	yes	characteristic
246	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
247	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
248	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
249	#391	C₂²×C₁₂₇	1	2	yes	—
250	#390	C₂²×C₁₂₇	1	2	yes	—
251	#388	—	2	4	no	—
252	#385	—	2	4	no	—
253	#384	C₂²×C₁₂₇	1	2	yes	—
254	#376	C₂²×C₁₂₇	1	1	yes	characteristic
255	C₂²×Q₈	C₂³×C₁₂₇	1	2	yes	—
256	C₂²×D₄	C₂³×C₁₂₇	1	2	yes	—
257	C₂³×C₄	—	2	2	no	—
258	C₂×Q₁₆	—	2	2	no	—
259	C₂×Q₁₆	—	2	2	no	—
260	C₂×Q₁₆	C₂×C₄×C₁₂₇	1	4	yes	complemented
261	C₂×SD₁₆	—	2	4	no	—
262	C₂×SD₁₆	—	2	4	no	—
263	C₂×SD₁₆	C₂×C₄×C₁₂₇	1	8	yes	complemented
264	C₂×D₈	—	2	2	no	—
265	C₂×D₈	—	2	2	no	—
266	C₂×D₈	C₂×C₄×C₁₂₇	1	4	yes	—
267	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
268	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
269	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
270	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
271	#124	—	2	4	no	—
272	#124	C₂×C₄×C₁₂₇	1	8	yes	complemented
273	C₄×Q₈	—	2	4	no	—
274	C₄×Q₈	—	2	4	no	—
275	C₄×D₄	—	2	4	no	—
276	C₄×D₄	—	2	4	no	—
277	C₂×C₄⋊C₄	—	2	2	no	—
278	C₂×C₄⋊C₄	—	2	4	no	—
279	C₂×C₄⋊C₄	C₂³×C₁₂₇	1	2	yes	—
280	C₂×C₄⋊C₄	C₂³×C₁₂₇	1	2	yes	—
281	C₂×C₂²⋊C₄	—	2	4	no	—
282	C₂×C₂²⋊C₄	—	2	2	no	—
283	C₂×C₂²⋊C₄	—	2	4	no	—
284	C₂×C₄²	—	2	2	no	—
285	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
286	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
287	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
288	#107	—	2	2	no	—
289	#107	—	2	2	no	—
290	#107	—	2	4	no	—
291	#103	—	2	4	no	—
292	#103	—	2	4	no	—
293	#102	—	2	4	no	—
294	#102	—	2	4	no	—
295	#97	—	2	2	no	—
296	#97	—	2	2	no	—
297	#97	D₄×C₁₂₇	1	4	yes	—
298	C₄×C₈	—	2	2	no	—
299	C₄×C₈	C₂×C₄×C₁₂₇	1	4	yes	complemented
300	C₄×C₈	D₄×C₁₂₇	1	4	yes	—
301	#95	—	2	4	no	—
302	#95	D₄×C₁₂₇	1	2	yes	—
303	#95	D₄×C₁₂₇	1	2	yes	—
304	C₂⁴	—	2	2	no	—
305	C₂×Q₈	—	2	2	no	—
306	C₂×Q₈	—	2	2	no	—
307	C₂×Q₈	—	2	2	no	—
308	C₂×Q₈	—	2	2	no	—
309	C₂×Q₈	—	2	2	no	—
310	C₂×Q₈	C₂²×C₄×C₁₂₇	1	4	yes	—
311	C₂×D₄	—	2	2	no	—
312	C₂×D₄	—	2	2	no	—
313	C₂×D₄	—	2	2	no	—
314	C₂×D₄	—	2	2	no	—
315	C₂×D₄	—	2	2	no	—
316	C₂×D₄	C₂²×C₄×C₁₂₇	1	4	yes	—
317	C₂²×C₄	—	2	2	no	—
318	C₂²×C₄	—	2	2	no	—
319	C₂²×C₄	—	2	2	no	—
320	C₂²×C₄	—	4	4	no	—
321	C₂²×C₄	—	8	4	no	—
322	C₂²×C₄	C₂×D₄×C₁₂₇	1	2	yes	—
323	C₂²×C₄	C₂×D₄×C₁₂₇	1	2	yes	—
324	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
325	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
326	C₂²×C₄	C₂⁴×C₁₂₇	1	1	yes	characteristic
327	Q₁₆	—	2	8	no	—
328	SD₁₆	—	2	16	no	—
329	D₈	—	2	8	no	—
330	C₂×C₈	—	2	1	no	—
331	C₂×C₈	—	2	1	no	—
332	C₂×C₈	—	2	1	no	—
333	C₂×C₈	—	2	1	no	—
334	C₂×C₈	C₂²×C₄×C₁₂₇	1	2	yes	—
335	C₂×C₈	C₂²×C₄×C₁₂₇	1	2	yes	—
336	C₂×C₈	C₂×D₄×C₁₂₇	1	2	yes	—
337	C₂×C₈	C₂×D₄×C₁₂₇	1	2	yes	—
338	C₂×C₈	C₂×D₄×C₁₂₇	1	2	yes	—
339	C₂×C₈	C₂×D₄×C₁₂₇	1	2	yes	—
340	C₂×C₈	#41×C₁₂₇	1	2	yes	—
341	C₂×C₈	#41×C₁₂₇	1	2	yes	—
342	C₄⋊C₄	—	2	2	no	—
343	C₄⋊C₄	—	2	2	no	—
344	C₄⋊C₄	—	2	2	no	—
345	C₄⋊C₄	—	2	2	no	—
346	C₄⋊C₄	—	4	4	no	—
347	C₄⋊C₄	—	8	4	no	—
348	C₂²⋊C₄	—	2	16	no	—
349	C₂²⋊C₄	—	4	4	no	—
350	C₂²⋊C₄	—	8	4	no	—
351	C₄²	—	2	1	no	—
352	C₄²	—	2	1	no	—
353	C₄²	—	4	4	no	—
354	C₄²	C₂²×C₄×C₁₂₇	1	2	yes	—
355	C₄²	C₂×D₄×C₁₂₇	1	2	yes	—
356	C₄²	C₂×D₄×C₁₂₇	1	2	yes	—
357	C₄²	#41×C₁₂₇	1	2	yes	—
358	C₂³	—	2	4	no	—
359	C₂³	—	4	2	no	—
360	C₂³	—	4	2	no	—
361	C₂³	—	4	4	no	—
362	C₂³	—	4	4	no	—
363	C₂³	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
364	Q₈	—	2	4	no	—
365	Q₈	—	2	4	no	—
366	Q₈	—	4	2	no	—
367	Q₈	—	4	2	no	—
368	D₄	—	2	4	no	—
369	D₄	—	2	4	no	—
370	D₄	—	4	2	no	—
371	D₄	—	4	2	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	4	no	—
377	C₂×C₄	—	2	2	no	—
378	C₂×C₄	—	2	2	no	—
379	C₂×C₄	—	2	2	no	—
380	C₂×C₄	—	4	2	no	—
381	C₂×C₄	—	4	2	no	—
382	C₂×C₄	—	4	2	no	—
383	C₂×C₄	—	4	2	no	—
384	C₂×C₄	—	4	4	no	—
385	C₂×C₄	—	4	4	no	—
386	C₂×C₄	—	8	8	no	—
387	C₂×C₄	C₄×D₄×C₁₂₇	1	4	yes	—
388	C₂×C₄	#127×C₁₂₇	1	2	yes	—
389	C₂×C₄	#127×C₁₂₇	1	2	yes	—
390	C₂×C₄	C₂³×C₄×C₁₂₇	1	1	yes	characteristic
391	C₂×C₄	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
392	C₂×C₄	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
393	C₂×C₄	C₂×#41×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	C₈	—	2	2	no	—
397	C₈	—	2	2	no	—
398	C₈	C₄×D₄×C₁₂₇	1	4	yes	—
399	C₈	C₄×D₄×C₁₂₇	1	4	yes	—
400	C₂²	—	4	8	no	—
401	C₂²	—	4	4	no	—
402	C₂²	—	8	2	no	—
403	C₂²	—	8	2	no	—
404	C₂²	—	8	4	no	—
405	C₂²	C₂×C₄×D₄×C₁₂₇	1	1	yes	characteristic
406	C₂²	C₂×#127×C₁₂₇	1	1	yes	characteristic
407	C₂²	#532×C₁₂₇	1	1	yes	characteristic
408	C₂²	#576×C₁₂₇	1	2	yes	—
409	C₂²	#578×C₁₂₇	1	2	yes	—
410	C₄	—	2	2	no	—
411	C₄	—	2	2	no	—
412	C₄	—	2	4	no	—
413	C₄	—	4	2	no	—
414	C₄	—	4	4	no	—
415	C₄	—	8	2	no	—
416	C₄	—	8	2	no	—
417	C₄	C₂×C₄×D₄×C₁₂₇	1	1	yes	characteristic
418	C₄	C₂×C₄×D₄×C₁₂₇	1	1	yes	characteristic
419	C₄	#532×C₁₂₇	1	1	yes	characteristic
420	C₄	#532×C₁₂₇	1	1	yes	characteristic
421	C₂	—	8	4	no	—
422	C₂	C₄×#127×C₁₂₇	1	1	yes	characteristic
423	C₂	C₁₂₇×#2951	1	2	yes	—
424	C₂	C₁₂₇×#3155	1	1	yes	characteristic
425	C₂	C₁₂₇×#3156	1	1	yes	characteristic
426	C₂	C₁₂₇×#3175	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 ◁ #376×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#440×C₁₂₇	C₂	semidirect
3. C₂×C₄×Q₁₆×C₁₂₇	C₂	semidirect
5. C₁₂₇×#2326	C₂	semidirect
6. C₁₂₇×#1973	C₂	semidirect
8. C₁₂₇×#1969	C₂	semidirect
9. C₁₂₇×#1959	C₂	semidirect
10. C₁₂₇×#1845	C₂	semidirect
11. C₁₂₇×#1726	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
85. C₄×C₈×C₁₂₇	C₂×C₄	semidirect
207. #21696	C₁₂₇	direct (N × H)
215. C₂×#440	C₂×C₁₂₇	semidirect
216. C₂×C₄×Q₁₆	C₂×C₁₂₇	semidirect
218. #2326	C₂×C₁₂₇	semidirect
219. #1973	C₂×C₁₂₇	semidirect
221. #1969	C₂×C₁₂₇	semidirect
222. #1959	C₂×C₁₂₇	semidirect
223. #1845	C₂×C₁₂₇	semidirect
224. #1726	C₂×C₁₂₇	semidirect
225. C₁₂₇	#21696	direct (N × H)
232. #495	C₄×C₁₂₇	semidirect
260. C₂×Q₁₆	C₂×C₄×C₁₂₇	semidirect
263. C₂×SD₁₆	C₂×C₄×C₁₂₇	semidirect
272. #124	C₂×C₄×C₁₂₇	semidirect
299. 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×#440×C₁₂₇	2
C₂×C₄×Q₁₆×C₁₂₇	2
C₂×C₄×D₈×C₁₂₇	2
C₁₂₇×#2326	2
C₁₂₇×#1973	2
C₁₂₇×#1970	2
C₁₂₇×#1969	2
C₁₂₇×#1959	2
C₁₂₇×#1845	2
C₁₂₇×#1726	2
#21696	127

Subgroups — C127×#21696 #44585369

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#21696 #44585369