Subgroups — C₁₂₇×#21702 #44585375

Order 32512.

1094 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#497×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#441×C₁₂₇	C₂	1	2	yes	maximal, complemented
3	C₂×#440×C₁₂₇	C₂	1	2	yes	maximal
4	C₄×C₄⋊Q₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×#124×C₁₂₇	C₂	1	1	yes	maximal, characteristic
6	C₁₂₇×#1969	C₂	1	2	yes	maximal, complemented
7	C₁₂₇×#1968	C₂	1	2	yes	maximal
8	C₁₂₇×#1960	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₂₇×#1959	C₂	1	1	yes	maximal, characteristic
10	C₁₂₇×#1847	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#97×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×Q₁₆×C₁₂₇	C₂²	1	2	yes	—
13	C₂²×SD₁₆×C₁₂₇	C₂²	1	2	yes	—
14	C₂×C₄⋊Q₈×C₁₂₇	C₂²	1	1	yes	characteristic
15	C₂×#124×C₁₂₇	C₂²	1	1	yes	characteristic
16	C₂×C₄×Q₈×C₁₂₇	C₂²	1	2	yes	—
17	C₂×C₄×Q₈×C₁₂₇	C₂²	1	1	yes	characteristic
18	C₂×C₄×D₄×C₁₂₇	C₂²	1	1	yes	characteristic
19	#497×C₁₂₇	C₄	1	8	yes	complemented
20	#441×C₁₂₇	—	2	8	no	—
21	#440×C₁₂₇	—	2	8	no	—
22	C₂×#107×C₁₂₇	C₂²	1	2	yes	—
23	C₂×#106×C₁₂₇	C₂²	1	2	yes	—
24	C₂×#103×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×#97×C₁₂₇	C₂²	1	1	yes	characteristic
29	C₂×#97×C₁₂₇	C₂²	1	1	yes	characteristic
30	C₂×#97×C₁₂₇	C₂²	1	2	yes	—
31	C₂×C₄×C₈×C₁₂₇	C₂²	1	2	yes	—
32	#391×C₁₂₇	C₂²	1	2	yes	—
33	#391×C₁₂₇	C₂²	1	1	yes	characteristic
34	#390×C₁₂₇	C₂²	1	1	yes	characteristic
35	#389×C₁₂₇	—	2	2	no	—
36	#388×C₁₂₇	—	2	2	no	—
37	#384×C₁₂₇	C₂²	1	1	yes	characteristic
38	#382×C₁₂₇	C₂²	1	1	yes	characteristic
39	C₄×C₄⋊C₄×C₁₂₇	—	2	2	no	—
40	C₄×C₂²⋊C₄×C₁₂₇	—	2	2	no	—
41	C₄³×C₁₂₇	C₂²	1	1	yes	characteristic
42	C₂²×Q₈×C₁₂₇	C₂³	1	1	yes	characteristic
43	C₂²×Q₈×C₁₂₇	C₂³	1	2	yes	—
44	C₂²×D₄×C₁₂₇	C₂³	1	1	yes	characteristic
45	C₂³×C₄×C₁₂₇	—	2	1	no	—
46	C₂×Q₁₆×C₁₂₇	—	2	4	no	—
47	C₂×Q₁₆×C₁₂₇	—	2	4	no	—
48	C₂×Q₁₆×C₁₂₇	C₂×C₄	1	8	yes	complemented
49	C₂×SD₁₆×C₁₂₇	—	2	4	no	—
50	C₂×SD₁₆×C₁₂₇	—	2	4	no	—
51	C₂×SD₁₆×C₁₂₇	C₂×C₄	1	8	yes	—
52	C₂²×C₈×C₁₂₇	C₂³	1	2	yes	—
53	C₂²×C₈×C₁₂₇	C₂³	1	2	yes	—
54	C₄⋊Q₈×C₁₂₇	—	2	2	no	—
55	C₄⋊Q₈×C₁₂₇	C₂×C₄	1	4	yes	complemented
56	#124×C₁₂₇	—	2	2	no	—
57	#124×C₁₂₇	C₂×C₄	1	4	yes	—
58	C₄×Q₈×C₁₂₇	—	2	4	no	—
59	C₄×Q₈×C₁₂₇	—	2	4	no	—
60	C₄×Q₈×C₁₂₇	—	2	4	no	—
61	C₄×D₄×C₁₂₇	—	2	4	no	—
62	C₂×C₄⋊C₄×C₁₂₇	—	2	2	no	—
63	C₂×C₄⋊C₄×C₁₂₇	—	2	1	no	—
64	C₂×C₄⋊C₄×C₁₂₇	—	2	2	no	—
65	C₂×C₄⋊C₄×C₁₂₇	—	2	2	no	—
66	C₂×C₄⋊C₄×C₁₂₇	C₂³	1	1	yes	characteristic
67	C₂×C₄⋊C₄×C₁₂₇	C₂³	1	1	yes	characteristic
68	C₂×C₄⋊C₄×C₁₂₇	C₂³	1	2	yes	—
69	C₂×C₂²⋊C₄×C₁₂₇	—	2	2	no	—
70	C₂×C₂²⋊C₄×C₁₂₇	—	2	1	no	—
71	C₂×C₂²⋊C₄×C₁₂₇	—	2	2	no	—
72	C₂×C₄²×C₁₂₇	—	2	1	no	—
73	C₂×C₄²×C₁₂₇	—	2	2	no	—
74	C₂×C₄²×C₁₂₇	D₄	1	2	yes	—
75	C₂×C₄²×C₁₂₇	D₄	1	2	yes	—
76	C₂×C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
77	C₂×C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
78	C₂×C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
79	#107×C₁₂₇	—	2	4	no	—
80	#106×C₁₂₇	—	2	4	no	—
81	#103×C₁₂₇	—	2	4	no	—
82	#103×C₁₂₇	—	2	4	no	—
83	#103×C₁₂₇	—	2	4	no	—
84	#102×C₁₂₇	—	2	4	no	—
85	#97×C₁₂₇	—	2	2	no	—
86	#97×C₁₂₇	C₂×C₄	1	4	yes	complemented
87	#97×C₁₂₇	D₄	1	8	yes	—
88	C₄×C₈×C₁₂₇	—	2	4	no	—
89	#95×C₁₂₇	—	2	2	no	—
90	#95×C₁₂₇	—	2	2	no	—
91	#95×C₁₂₇	—	2	4	no	—
92	C₂⁴×C₁₂₇	—	2	1	no	—
93	C₂×Q₈×C₁₂₇	—	2	1	no	—
94	C₂×Q₈×C₁₂₇	—	2	2	no	—
95	C₂×Q₈×C₁₂₇	—	2	2	no	—
96	C₂×Q₈×C₁₂₇	—	2	2	no	—
97	C₂×Q₈×C₁₂₇	—	2	2	no	—
98	C₂×Q₈×C₁₂₇	—	2	2	no	—
99	C₂×Q₈×C₁₂₇	—	2	2	no	—
100	C₂×Q₈×C₁₂₇	—	2	2	no	—
101	C₂×Q₈×C₁₂₇	C₂²×C₄	1	2	yes	—
102	C₂×Q₈×C₁₂₇	C₂²×C₄	1	4	yes	—
103	C₂×D₄×C₁₂₇	—	2	2	no	—
104	C₂×D₄×C₁₂₇	—	2	2	no	—
105	C₂×D₄×C₁₂₇	—	2	1	no	—
106	C₂×D₄×C₁₂₇	C₂²×C₄	1	2	yes	—
107	C₂²×C₄×C₁₂₇	—	2	1	no	—
108	C₂²×C₄×C₁₂₇	—	2	2	no	—
109	C₂²×C₄×C₁₂₇	—	2	1	no	—
110	C₂²×C₄×C₁₂₇	—	2	1	no	—
111	C₂²×C₄×C₁₂₇	—	2	2	no	—
112	C₂²×C₄×C₁₂₇	—	4	4	no	—
113	C₂²×C₄×C₁₂₇	—	4	2	no	—
114	C₂²×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
115	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
116	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
117	C₂²×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
118	C₂²×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
119	Q₁₆×C₁₂₇	—	2	16	no	—
120	SD₁₆×C₁₂₇	—	2	16	no	—
121	C₂×C₈×C₁₂₇	—	2	2	no	—
122	C₂×C₈×C₁₂₇	—	2	2	no	—
123	C₂×C₈×C₁₂₇	C₂²×C₄	1	4	yes	—
124	C₂×C₈×C₁₂₇	C₂×D₄	1	4	yes	—
125	C₂×C₈×C₁₂₇	C₂×D₄	1	4	yes	—
126	C₂×C₈×C₁₂₇	#41	1	4	yes	—
127	C₄⋊C₄×C₁₂₇	—	2	8	no	—
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	—	4	2	no	—
134	C₄⋊C₄×C₁₂₇	—	4	4	no	—
135	C₂²⋊C₄×C₁₂₇	—	2	8	no	—
136	C₂²⋊C₄×C₁₂₇	—	4	2	no	—
137	C₂²⋊C₄×C₁₂₇	—	4	4	no	—
138	C₄²×C₁₂₇	—	2	4	no	—
139	C₄²×C₁₂₇	—	2	2	no	—
140	C₄²×C₁₂₇	—	4	2	no	—
141	C₄²×C₁₂₇	—	4	4	no	—
142	C₄²×C₁₂₇	—	4	2	no	—
143	C₄²×C₁₂₇	C₂²×C₄	1	2	yes	—
144	C₄²×C₁₂₇	C₂×D₄	1	4	yes	—
145	C₄²×C₁₂₇	#41	1	2	yes	—
146	C₂³×C₁₂₇	—	2	2	no	—
147	C₂³×C₁₂₇	—	4	1	no	—
148	C₂³×C₁₂₇	—	4	1	no	—
149	C₂³×C₁₂₇	—	4	2	no	—
150	C₂³×C₁₂₇	—	4	2	no	—
151	C₂³×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
152	Q₈×C₁₂₇	—	2	4	no	—
153	Q₈×C₁₂₇	—	2	4	no	—
154	Q₈×C₁₂₇	—	2	4	no	—
155	Q₈×C₁₂₇	—	4	2	no	—
156	Q₈×C₁₂₇	—	4	2	no	—
157	Q₈×C₁₂₇	—	4	2	no	—
158	D₄×C₁₂₇	—	2	4	no	—
159	D₄×C₁₂₇	—	4	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	—	2	2	no	—
167	C₂×C₄×C₁₂₇	—	2	4	no	—
168	C₂×C₄×C₁₂₇	—	4	1	no	—
169	C₂×C₄×C₁₂₇	—	4	2	no	—
170	C₂×C₄×C₁₂₇	—	4	1	no	—
171	C₂×C₄×C₁₂₇	—	4	2	no	—
172	C₂×C₄×C₁₂₇	—	4	1	no	—
173	C₂×C₄×C₁₂₇	—	4	2	no	—
174	C₂×C₄×C₁₂₇	—	4	1	no	—
175	C₂×C₄×C₁₂₇	—	4	2	no	—
176	C₂×C₄×C₁₂₇	—	4	4	no	—
177	C₂×C₄×C₁₂₇	—	4	2	no	—
178	C₂×C₄×C₁₂₇	—	8	4	no	—
179	C₂×C₄×C₁₂₇	C₄×D₄	1	4	yes	—
180	C₂×C₄×C₁₂₇	#127	1	2	yes	—
181	C₂×C₄×C₁₂₇	#127	1	2	yes	—
182	C₂×C₄×C₁₂₇	#137	1	4	yes	—
183	C₂×C₄×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
184	C₂×C₄×C₁₂₇	C₂²×D₄	1	2	yes	—
185	C₂×C₄×C₁₂₇	C₂×#41	1	2	yes	—
186	C₂×C₄×C₁₂₇	C₂×#41	1	1	yes	characteristic
187	C₈×C₁₂₇	—	2	4	no	—
188	C₈×C₁₂₇	C₄×D₄	1	8	yes	—
189	C₂²×C₁₂₇	—	4	2	no	—
190	C₂²×C₁₂₇	—	4	4	no	—
191	C₂²×C₁₂₇	—	8	1	no	—
192	C₂²×C₁₂₇	—	8	1	no	—
193	C₂²×C₁₂₇	—	8	2	no	—
194	C₂²×C₁₂₇	C₂×C₄×D₄	1	1	yes	characteristic
195	C₂²×C₁₂₇	C₂×#127	1	1	yes	characteristic
196	C₂²×C₁₂₇	#532	1	1	yes	characteristic
197	C₂²×C₁₂₇	C₂×#137	1	2	yes	—
198	C₂²×C₁₂₇	#575	1	2	yes	—
199	C₄×C₁₂₇	—	2	2	no	—
200	C₄×C₁₂₇	—	2	2	no	—
201	C₄×C₁₂₇	—	2	4	no	—
202	C₄×C₁₂₇	—	2	4	no	—
203	C₄×C₁₂₇	—	4	2	no	—
204	C₄×C₁₂₇	—	4	4	no	—
205	C₄×C₁₂₇	—	8	1	no	—
206	C₄×C₁₂₇	—	8	2	no	—
207	C₄×C₁₂₇	—	8	1	no	—
208	C₄×C₁₂₇	C₂×C₄×D₄	1	2	yes	—
209	C₄×C₁₂₇	#532	1	2	yes	—
210	#21702	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
211	C₂×C₁₂₇	—	8	2	no	—
212	C₂×C₁₂₇	C₄×#127	1	1	yes	characteristic
213	C₂×C₁₂₇	C₄×#137	1	2	yes	—
214	C₂×C₁₂₇	#3152	1	1	yes	characteristic
215	C₂×C₁₂₇	#3153	1	1	yes	characteristic
216	C₂×C₁₂₇	#3181	1	2	yes	—
217	C₂×#497	C₂×C₁₂₇	1	1	yes	characteristic
218	C₂×#441	C₂×C₁₂₇	1	2	yes	complemented
219	C₂×#440	C₂×C₁₂₇	1	2	yes	—
220	C₄×C₄⋊Q₈	C₂×C₁₂₇	1	1	yes	characteristic, complemented
221	C₄×#124	C₂×C₁₂₇	1	1	yes	characteristic
222	#1969	C₂×C₁₂₇	1	2	yes	complemented
223	#1968	C₂×C₁₂₇	1	2	yes	—
224	#1960	C₂×C₁₂₇	1	1	yes	characteristic, complemented
225	#1959	C₂×C₁₂₇	1	1	yes	characteristic
226	#1847	C₂×C₁₂₇	1	1	yes	characteristic, complemented
227	C₄×#97	C₂×C₁₂₇	1	1	yes	characteristic, complemented
228	C₁₂₇	#21702	1	1	yes	characteristic, direct product factor, Sylow
229	C₂²×Q₁₆	C₂²×C₁₂₇	1	2	yes	—
230	C₂²×SD₁₆	C₂²×C₁₂₇	1	2	yes	—
231	C₂×C₄⋊Q₈	C₂²×C₁₂₇	1	1	yes	characteristic
232	C₂×#124	C₂²×C₁₂₇	1	1	yes	characteristic
233	C₂×C₄×Q₈	C₂²×C₁₂₇	1	2	yes	—
234	C₂×C₄×Q₈	C₂²×C₁₂₇	1	1	yes	characteristic
235	C₂×C₄×D₄	C₂²×C₁₂₇	1	1	yes	characteristic
236	#497	C₄×C₁₂₇	1	8	yes	complemented
237	#441	—	2	8	no	—
238	#440	—	2	8	no	—
239	C₂×#107	C₂²×C₁₂₇	1	2	yes	—
240	C₂×#106	C₂²×C₁₂₇	1	2	yes	—
241	C₂×#103	C₂²×C₁₂₇	1	2	yes	—
242	C₂×#103	C₂²×C₁₂₇	1	2	yes	—
243	C₂×#103	C₂²×C₁₂₇	1	2	yes	—
244	C₂×#102	C₂²×C₁₂₇	1	2	yes	—
245	C₂×#97	C₂²×C₁₂₇	1	1	yes	characteristic
246	C₂×#97	C₂²×C₁₂₇	1	1	yes	characteristic
247	C₂×#97	C₂²×C₁₂₇	1	2	yes	—
248	C₂×C₄×C₈	C₂²×C₁₂₇	1	2	yes	—
249	#391	C₂²×C₁₂₇	1	2	yes	—
250	#391	C₂²×C₁₂₇	1	1	yes	characteristic
251	#390	C₂²×C₁₂₇	1	1	yes	characteristic
252	#389	—	2	2	no	—
253	#388	—	2	2	no	—
254	#384	C₂²×C₁₂₇	1	1	yes	characteristic
255	#382	C₂²×C₁₂₇	1	1	yes	characteristic
256	C₄×C₄⋊C₄	—	2	2	no	—
257	C₄×C₂²⋊C₄	—	2	2	no	—
258	C₄³	C₂²×C₁₂₇	1	1	yes	characteristic
259	C₂²×Q₈	C₂³×C₁₂₇	1	1	yes	characteristic
260	C₂²×Q₈	C₂³×C₁₂₇	1	2	yes	—
261	C₂²×D₄	C₂³×C₁₂₇	1	1	yes	characteristic
262	C₂³×C₄	—	2	1	no	—
263	C₂×Q₁₆	—	2	4	no	—
264	C₂×Q₁₆	—	2	4	no	—
265	C₂×Q₁₆	C₂×C₄×C₁₂₇	1	8	yes	complemented
266	C₂×SD₁₆	—	2	4	no	—
267	C₂×SD₁₆	—	2	4	no	—
268	C₂×SD₁₆	C₂×C₄×C₁₂₇	1	8	yes	—
269	C₂²×C₈	C₂³×C₁₂₇	1	2	yes	—
270	C₂²×C₈	C₂³×C₁₂₇	1	2	yes	—
271	C₄⋊Q₈	—	2	2	no	—
272	C₄⋊Q₈	C₂×C₄×C₁₂₇	1	4	yes	complemented
273	#124	—	2	2	no	—
274	#124	C₂×C₄×C₁₂₇	1	4	yes	—
275	C₄×Q₈	—	2	4	no	—
276	C₄×Q₈	—	2	4	no	—
277	C₄×Q₈	—	2	4	no	—
278	C₄×D₄	—	2	4	no	—
279	C₂×C₄⋊C₄	—	2	2	no	—
280	C₂×C₄⋊C₄	—	2	1	no	—
281	C₂×C₄⋊C₄	—	2	2	no	—
282	C₂×C₄⋊C₄	—	2	2	no	—
283	C₂×C₄⋊C₄	C₂³×C₁₂₇	1	1	yes	characteristic
284	C₂×C₄⋊C₄	C₂³×C₁₂₇	1	1	yes	characteristic
285	C₂×C₄⋊C₄	C₂³×C₁₂₇	1	2	yes	—
286	C₂×C₂²⋊C₄	—	2	2	no	—
287	C₂×C₂²⋊C₄	—	2	1	no	—
288	C₂×C₂²⋊C₄	—	2	2	no	—
289	C₂×C₄²	—	2	1	no	—
290	C₂×C₄²	—	2	2	no	—
291	C₂×C₄²	D₄×C₁₂₇	1	2	yes	—
292	C₂×C₄²	D₄×C₁₂₇	1	2	yes	—
293	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
294	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
295	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
296	#107	—	2	4	no	—
297	#106	—	2	4	no	—
298	#103	—	2	4	no	—
299	#103	—	2	4	no	—
300	#103	—	2	4	no	—
301	#102	—	2	4	no	—
302	#97	—	2	2	no	—
303	#97	C₂×C₄×C₁₂₇	1	4	yes	complemented
304	#97	D₄×C₁₂₇	1	8	yes	—
305	C₄×C₈	—	2	4	no	—
306	#95	—	2	2	no	—
307	#95	—	2	2	no	—
308	#95	—	2	4	no	—
309	C₂⁴	—	2	1	no	—
310	C₂×Q₈	—	2	1	no	—
311	C₂×Q₈	—	2	2	no	—
312	C₂×Q₈	—	2	2	no	—
313	C₂×Q₈	—	2	2	no	—
314	C₂×Q₈	—	2	2	no	—
315	C₂×Q₈	—	2	2	no	—
316	C₂×Q₈	—	2	2	no	—
317	C₂×Q₈	—	2	2	no	—
318	C₂×Q₈	C₂²×C₄×C₁₂₇	1	2	yes	—
319	C₂×Q₈	C₂²×C₄×C₁₂₇	1	4	yes	—
320	C₂×D₄	—	2	2	no	—
321	C₂×D₄	—	2	2	no	—
322	C₂×D₄	—	2	1	no	—
323	C₂×D₄	C₂²×C₄×C₁₂₇	1	2	yes	—
324	C₂²×C₄	—	2	1	no	—
325	C₂²×C₄	—	2	2	no	—
326	C₂²×C₄	—	2	1	no	—
327	C₂²×C₄	—	2	1	no	—
328	C₂²×C₄	—	2	2	no	—
329	C₂²×C₄	—	4	4	no	—
330	C₂²×C₄	—	4	2	no	—
331	C₂²×C₄	C₂×D₄×C₁₂₇	1	2	yes	—
332	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
333	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
334	C₂²×C₄	C₂×D₄×C₁₂₇	1	2	yes	—
335	C₂²×C₄	C₂⁴×C₁₂₇	1	1	yes	characteristic
336	Q₁₆	—	2	16	no	—
337	SD₁₆	—	2	16	no	—
338	C₂×C₈	—	2	2	no	—
339	C₂×C₈	—	2	2	no	—
340	C₂×C₈	C₂²×C₄×C₁₂₇	1	4	yes	—
341	C₂×C₈	C₂×D₄×C₁₂₇	1	4	yes	—
342	C₂×C₈	C₂×D₄×C₁₂₇	1	4	yes	—
343	C₂×C₈	#41×C₁₂₇	1	4	yes	—
344	C₄⋊C₄	—	2	8	no	—
345	C₄⋊C₄	—	2	2	no	—
346	C₄⋊C₄	—	2	2	no	—
347	C₄⋊C₄	—	2	2	no	—
348	C₄⋊C₄	—	2	2	no	—
349	C₄⋊C₄	—	4	4	no	—
350	C₄⋊C₄	—	4	2	no	—
351	C₄⋊C₄	—	4	4	no	—
352	C₂²⋊C₄	—	2	8	no	—
353	C₂²⋊C₄	—	4	2	no	—
354	C₂²⋊C₄	—	4	4	no	—
355	C₄²	—	2	4	no	—
356	C₄²	—	2	2	no	—
357	C₄²	—	4	2	no	—
358	C₄²	—	4	4	no	—
359	C₄²	—	4	2	no	—
360	C₄²	C₂²×C₄×C₁₂₇	1	2	yes	—
361	C₄²	C₂×D₄×C₁₂₇	1	4	yes	—
362	C₄²	#41×C₁₂₇	1	2	yes	—
363	C₂³	—	2	2	no	—
364	C₂³	—	4	1	no	—
365	C₂³	—	4	1	no	—
366	C₂³	—	4	2	no	—
367	C₂³	—	4	2	no	—
368	C₂³	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
369	Q₈	—	2	4	no	—
370	Q₈	—	2	4	no	—
371	Q₈	—	2	4	no	—
372	Q₈	—	4	2	no	—
373	Q₈	—	4	2	no	—
374	Q₈	—	4	2	no	—
375	D₄	—	2	4	no	—
376	D₄	—	4	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	—	2	2	no	—
383	C₂×C₄	—	2	2	no	—
384	C₂×C₄	—	2	4	no	—
385	C₂×C₄	—	4	1	no	—
386	C₂×C₄	—	4	2	no	—
387	C₂×C₄	—	4	1	no	—
388	C₂×C₄	—	4	2	no	—
389	C₂×C₄	—	4	1	no	—
390	C₂×C₄	—	4	2	no	—
391	C₂×C₄	—	4	1	no	—
392	C₂×C₄	—	4	2	no	—
393	C₂×C₄	—	4	4	no	—
394	C₂×C₄	—	4	2	no	—
395	C₂×C₄	—	8	4	no	—
396	C₂×C₄	C₄×D₄×C₁₂₇	1	4	yes	—
397	C₂×C₄	#127×C₁₂₇	1	2	yes	—
398	C₂×C₄	#127×C₁₂₇	1	2	yes	—
399	C₂×C₄	#137×C₁₂₇	1	4	yes	—
400	C₂×C₄	C₂³×C₄×C₁₂₇	1	1	yes	characteristic
401	C₂×C₄	C₂²×D₄×C₁₂₇	1	2	yes	—
402	C₂×C₄	C₂×#41×C₁₂₇	1	2	yes	—
403	C₂×C₄	C₂×#41×C₁₂₇	1	1	yes	characteristic
404	C₈	—	2	4	no	—
405	C₈	C₄×D₄×C₁₂₇	1	8	yes	—
406	C₂²	—	4	4	no	—
407	C₂²	—	4	2	no	—
408	C₂²	—	8	1	no	—
409	C₂²	—	8	1	no	—
410	C₂²	—	8	2	no	—
411	C₂²	C₂×C₄×D₄×C₁₂₇	1	1	yes	characteristic
412	C₂²	C₂×#127×C₁₂₇	1	1	yes	characteristic
413	C₂²	#532×C₁₂₇	1	1	yes	characteristic
414	C₂²	C₂×#137×C₁₂₇	1	2	yes	—
415	C₂²	#575×C₁₂₇	1	2	yes	—
416	C₄	—	2	2	no	—
417	C₄	—	2	2	no	—
418	C₄	—	2	4	no	—
419	C₄	—	2	4	no	—
420	C₄	—	4	2	no	—
421	C₄	—	4	4	no	—
422	C₄	—	8	1	no	—
423	C₄	—	8	1	no	—
424	C₄	—	8	2	no	—
425	C₄	C₂×C₄×D₄×C₁₂₇	1	2	yes	—
426	C₄	#532×C₁₂₇	1	2	yes	—
427	C₂	—	8	2	no	—
428	C₂	C₄×#127×C₁₂₇	1	1	yes	characteristic
429	C₂	C₄×#137×C₁₂₇	1	2	yes	—
430	C₂	C₁₂₇×#3152	1	1	yes	characteristic
431	C₂	C₁₂₇×#3153	1	1	yes	characteristic
432	C₂	C₁₂₇×#3181	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×D₄×C₁₂₇	C₂³
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
2. C₂×#441×C₁₂₇	C₂	semidirect
4. C₄×C₄⋊Q₈×C₁₂₇	C₂	semidirect
6. C₁₂₇×#1969	C₂	semidirect
8. C₁₂₇×#1960	C₂	semidirect
10. C₁₂₇×#1847	C₂	semidirect
11. C₄×#97×C₁₂₇	C₂	semidirect
19. #497×C₁₂₇	C₄	semidirect
48. C₂×Q₁₆×C₁₂₇	C₂×C₄	semidirect
55. C₄⋊Q₈×C₁₂₇	C₂×C₄	semidirect
86. #97×C₁₂₇	C₂×C₄	semidirect
210. #21702	C₁₂₇	direct (N × H)
218. C₂×#441	C₂×C₁₂₇	semidirect
220. C₄×C₄⋊Q₈	C₂×C₁₂₇	semidirect
222. #1969	C₂×C₁₂₇	semidirect
224. #1960	C₂×C₁₂₇	semidirect
226. #1847	C₂×C₁₂₇	semidirect
227. C₄×#97	C₂×C₁₂₇	semidirect
228. C₁₂₇	#21702	direct (N × H)
236. #497	C₄×C₁₂₇	semidirect
265. C₂×Q₁₆	C₂×C₄×C₁₂₇	semidirect
272. C₄⋊Q₈	C₂×C₄×C₁₂₇	semidirect
303. #97	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×#497×C₁₂₇	2
C₂×#441×C₁₂₇	2
C₂×#440×C₁₂₇	2
C₄×C₄⋊Q₈×C₁₂₇	2
C₄×#124×C₁₂₇	2
C₁₂₇×#1969	2
C₁₂₇×#1968	2
C₁₂₇×#1960	2
C₁₂₇×#1959	2
C₁₂₇×#1847	2
C₄×#97×C₁₂₇	2
#21702	127

Subgroups — C127×#21702 #44585375

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#21702 #44585375