Subgroups — C₄×#496×C₁₂₇ #44585372

Order 32512.

1262 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#496×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×SD₁₆×C₁₂₇	C₂	1	2	yes	maximal, complemented
3	C₂×C₄×D₈×C₁₂₇	C₂	1	2	yes	maximal, complemented
4	C₄×#127×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×#124×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₂₇×#1973	C₂	1	2	yes	maximal, complemented
7	C₁₂₇×#1972	C₂	1	2	yes	maximal, complemented
8	C₁₂₇×#1961	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₂₇×#1959	C₂	1	1	yes	maximal, characteristic, complemented
10	C₁₂₇×#1847	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#97×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×SD₁₆×C₁₂₇	C₂²	1	2	yes	—
13	C₂²×D₈×C₁₂₇	C₂²	1	2	yes	—
14	C₂×#127×C₁₂₇	C₂²	1	1	yes	characteristic
15	C₂×#124×C₁₂₇	C₂²	1	1	yes	characteristic
16	C₂×C₄×Q₈×C₁₂₇	C₂²	1	1	yes	characteristic
17	C₂×C₄×D₄×C₁₂₇	—	2	2	no	—
18	C₂×C₄×D₄×C₁₂₇	C₂²	1	1	yes	characteristic
19	C₂×C₄×D₄×C₁₂₇	C₂²	1	2	yes	—
20	#496×C₁₂₇	C₄	1	8	yes	direct product factor
21	C₄×SD₁₆×C₁₂₇	—	2	8	no	—
22	C₄×D₈×C₁₂₇	—	2	8	no	—
23	C₂×#107×C₁₂₇	C₂²	1	2	yes	—
24	C₂×#106×C₁₂₇	C₂²	1	2	yes	—
25	C₂×#103×C₁₂₇	C₂²	1	2	yes	—
26	C₂×#102×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	2	yes	—
31	C₂×#97×C₁₂₇	C₂²	1	1	yes	characteristic
32	C₂×C₄×C₈×C₁₂₇	C₂²	1	2	yes	—
33	#391×C₁₂₇	C₂²	1	1	yes	characteristic
34	#390×C₁₂₇	—	2	2	no	—
35	#390×C₁₂₇	C₂²	1	1	yes	characteristic
36	#390×C₁₂₇	C₂²	1	2	yes	—
37	#388×C₁₂₇	—	2	2	no	—
38	#384×C₁₂₇	C₂²	1	1	yes	characteristic
39	#382×C₁₂₇	C₂²	1	1	yes	characteristic
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×D₄×C₁₂₇	—	2	2	no	—
44	C₂²×D₄×C₁₂₇	C₂³	1	1	yes	characteristic
45	C₂²×D₄×C₁₂₇	C₂³	1	2	yes	—
46	C₂³×C₄×C₁₂₇	—	2	2	no	—
47	C₂³×C₄×C₁₂₇	—	2	1	no	—
48	C₂×SD₁₆×C₁₂₇	—	2	4	no	—
49	C₂×SD₁₆×C₁₂₇	—	2	4	no	—
50	C₂×SD₁₆×C₁₂₇	C₂×C₄	1	8	yes	complemented
51	C₂×D₈×C₁₂₇	—	2	4	no	—
52	C₂×D₈×C₁₂₇	—	2	4	no	—
53	C₂×D₈×C₁₂₇	C₂×C₄	1	8	yes	complemented
54	C₂²×C₈×C₁₂₇	C₂³	1	2	yes	—
55	C₂²×C₈×C₁₂₇	C₂³	1	2	yes	—
56	#127×C₁₂₇	—	2	2	no	—
57	#127×C₁₂₇	C₂×C₄	1	4	yes	complemented
58	#124×C₁₂₇	—	2	2	no	—
59	#124×C₁₂₇	C₂×C₄	1	4	yes	complemented
60	C₄×Q₈×C₁₂₇	—	2	4	no	—
61	C₄×D₄×C₁₂₇	—	2	4	no	—
62	C₄×D₄×C₁₂₇	—	2	4	no	—
63	C₄×D₄×C₁₂₇	—	2	4	no	—
64	C₄×D₄×C₁₂₇	—	4	8	no	—
65	C₂×C₄⋊C₄×C₁₂₇	—	2	2	no	—
66	C₂×C₄⋊C₄×C₁₂₇	—	2	1	no	—
67	C₂×C₄⋊C₄×C₁₂₇	C₂³	1	1	yes	characteristic
68	C₂×C₄⋊C₄×C₁₂₇	C₂³	1	2	yes	—
69	C₂×C₄⋊C₄×C₁₂₇	C₂³	1	1	yes	characteristic
70	C₂×C₂²⋊C₄×C₁₂₇	—	2	2	no	—
71	C₂×C₂²⋊C₄×C₁₂₇	—	2	4	no	—
72	C₂×C₂²⋊C₄×C₁₂₇	—	2	1	no	—
73	C₂×C₂²⋊C₄×C₁₂₇	—	2	2	no	—
74	C₂×C₂²⋊C₄×C₁₂₇	—	2	2	no	—
75	C₂×C₄²×C₁₂₇	—	2	1	no	—
76	C₂×C₄²×C₁₂₇	D₄	1	2	yes	—
77	C₂×C₄²×C₁₂₇	D₄	1	2	yes	—
78	C₂×C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
79	C₂×C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
80	C₂×C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
81	#107×C₁₂₇	—	2	4	no	—
82	#106×C₁₂₇	—	2	4	no	—
83	#103×C₁₂₇	—	2	4	no	—
84	#102×C₁₂₇	—	2	4	no	—
85	#102×C₁₂₇	—	2	4	no	—
86	#102×C₁₂₇	—	2	4	no	—
87	#97×C₁₂₇	—	2	4	no	—
88	#97×C₁₂₇	—	2	2	no	—
89	#97×C₁₂₇	C₂×C₄	1	4	yes	complemented
90	C₄×C₈×C₁₂₇	D₄	1	8	yes	complemented
91	#95×C₁₂₇	—	2	2	no	—
92	#95×C₁₂₇	—	2	2	no	—
93	C₂⁴×C₁₂₇	—	2	1	no	—
94	C₂⁴×C₁₂₇	—	2	2	no	—
95	C₂×Q₈×C₁₂₇	—	2	1	no	—
96	C₂×Q₈×C₁₂₇	—	2	2	no	—
97	C₂×Q₈×C₁₂₇	—	2	2	no	—
98	C₂×Q₈×C₁₂₇	C₂²×C₄	1	2	yes	—
99	C₂×D₄×C₁₂₇	—	2	8	no	—
100	C₂×D₄×C₁₂₇	—	2	2	no	—
101	C₂×D₄×C₁₂₇	—	2	2	no	—
102	C₂×D₄×C₁₂₇	—	2	2	no	—
103	C₂×D₄×C₁₂₇	—	2	2	no	—
104	C₂×D₄×C₁₂₇	—	2	2	no	—
105	C₂×D₄×C₁₂₇	—	2	2	no	—
106	C₂×D₄×C₁₂₇	—	2	1	no	—
107	C₂×D₄×C₁₂₇	—	2	2	no	—
108	C₂×D₄×C₁₂₇	—	4	4	no	—
109	C₂×D₄×C₁₂₇	—	4	4	no	—
110	C₂×D₄×C₁₂₇	C₂²×C₄	1	2	yes	—
111	C₂×D₄×C₁₂₇	C₂²×C₄	1	4	yes	—
112	C₂²×C₄×C₁₂₇	—	2	1	no	—
113	C₂²×C₄×C₁₂₇	—	2	1	no	—
114	C₂²×C₄×C₁₂₇	—	2	1	no	—
115	C₂²×C₄×C₁₂₇	—	2	2	no	—
116	C₂²×C₄×C₁₂₇	—	4	2	no	—
117	C₂²×C₄×C₁₂₇	—	4	4	no	—
118	C₂²×C₄×C₁₂₇	—	4	8	no	—
119	C₂²×C₄×C₁₂₇	—	4	4	no	—
120	C₂²×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
121	C₂²×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
122	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
123	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
124	C₂²×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
125	SD₁₆×C₁₂₇	—	2	16	no	—
126	D₈×C₁₂₇	—	2	16	no	—
127	C₂×C₈×C₁₂₇	—	2	2	no	—
128	C₂×C₈×C₁₂₇	—	2	2	no	—
129	C₂×C₈×C₁₂₇	C₂²×C₄	1	4	yes	—
130	C₂×C₈×C₁₂₇	C₂×D₄	1	4	yes	—
131	C₂×C₈×C₁₂₇	C₂×D₄	1	4	yes	—
132	C₂×C₈×C₁₂₇	#41	1	4	yes	—
133	C₄⋊C₄×C₁₂₇	—	2	2	no	—
134	C₄⋊C₄×C₁₂₇	—	2	2	no	—
135	C₄⋊C₄×C₁₂₇	—	2	2	no	—
136	C₄⋊C₄×C₁₂₇	—	2	2	no	—
137	C₄⋊C₄×C₁₂₇	—	4	4	no	—
138	C₄⋊C₄×C₁₂₇	—	4	2	no	—
139	C₂²⋊C₄×C₁₂₇	—	2	8	no	—
140	C₂²⋊C₄×C₁₂₇	—	4	4	no	—
141	C₂²⋊C₄×C₁₂₇	—	4	8	no	—
142	C₂²⋊C₄×C₁₂₇	—	4	2	no	—
143	C₂²⋊C₄×C₁₂₇	—	4	4	no	—
144	C₄²×C₁₂₇	—	2	4	no	—
145	C₄²×C₁₂₇	—	2	2	no	—
146	C₄²×C₁₂₇	—	4	2	no	—
147	C₄²×C₁₂₇	—	4	2	no	—
148	C₄²×C₁₂₇	C₂²×C₄	1	2	yes	—
149	C₄²×C₁₂₇	C₂×D₄	1	4	yes	—
150	C₄²×C₁₂₇	#41	1	2	yes	—
151	C₂³×C₁₂₇	—	2	2	no	—
152	C₂³×C₁₂₇	—	2	4	no	—
153	C₂³×C₁₂₇	—	4	1	no	—
154	C₂³×C₁₂₇	—	4	2	no	—
155	C₂³×C₁₂₇	—	4	1	no	—
156	C₂³×C₁₂₇	—	4	2	no	—
157	C₂³×C₁₂₇	—	4	4	no	—
158	C₂³×C₁₂₇	—	4	2	no	—
159	C₂³×C₁₂₇	—	4	2	no	—
160	C₂³×C₁₂₇	—	4	4	no	—
161	C₂³×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
162	Q₈×C₁₂₇	—	2	4	no	—
163	Q₈×C₁₂₇	—	4	2	no	—
164	D₄×C₁₂₇	—	2	4	no	—
165	D₄×C₁₂₇	—	2	4	no	—
166	D₄×C₁₂₇	—	2	4	no	—
167	D₄×C₁₂₇	—	4	16	no	—
168	D₄×C₁₂₇	—	4	2	no	—
169	D₄×C₁₂₇	—	4	2	no	—
170	D₄×C₁₂₇	—	4	2	no	—
171	C₂×C₄×C₁₂₇	—	2	2	no	—
172	C₂×C₄×C₁₂₇	—	2	2	no	—
173	C₂×C₄×C₁₂₇	—	2	2	no	—
174	C₂×C₄×C₁₂₇	—	2	2	no	—
175	C₂×C₄×C₁₂₇	—	2	2	no	—
176	C₂×C₄×C₁₂₇	—	2	2	no	—
177	C₂×C₄×C₁₂₇	—	2	2	no	—
178	C₂×C₄×C₁₂₇	—	4	1	no	—
179	C₂×C₄×C₁₂₇	—	4	1	no	—
180	C₂×C₄×C₁₂₇	—	4	1	no	—
181	C₂×C₄×C₁₂₇	—	4	1	no	—
182	C₂×C₄×C₁₂₇	—	4	2	no	—
183	C₂×C₄×C₁₂₇	—	4	4	no	—
184	C₂×C₄×C₁₂₇	—	4	2	no	—
185	C₂×C₄×C₁₂₇	—	4	2	no	—
186	C₂×C₄×C₁₂₇	—	8	8	no	—
187	C₂×C₄×C₁₂₇	—	8	4	no	—
188	C₂×C₄×C₁₂₇	C₄×D₄	1	4	yes	—
189	C₂×C₄×C₁₂₇	#127	1	2	yes	—
190	C₂×C₄×C₁₂₇	#127	1	2	yes	—
191	C₂×C₄×C₁₂₇	AGL(1,ℤ/8)	1	4	yes	—
192	C₂×C₄×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
193	C₂×C₄×C₁₂₇	C₂²×D₄	1	2	yes	—
194	C₂×C₄×C₁₂₇	C₂×#41	1	2	yes	—
195	C₂×C₄×C₁₂₇	C₂×#41	1	1	yes	characteristic
196	C₈×C₁₂₇	—	2	4	no	—
197	C₈×C₁₂₇	C₄×D₄	1	8	yes	complemented
198	C₂²×C₁₂₇	—	4	2	no	—
199	C₂²×C₁₂₇	—	4	4	no	—
200	C₂²×C₁₂₇	—	4	8	no	—
201	C₂²×C₁₂₇	—	4	4	no	—
202	C₂²×C₁₂₇	—	8	2	no	—
203	C₂²×C₁₂₇	—	8	1	no	—
204	C₂²×C₁₂₇	—	8	1	no	—
205	C₂²×C₁₂₇	—	8	2	no	—
206	C₂²×C₁₂₇	—	8	4	no	—
207	C₂²×C₁₂₇	—	8	2	no	—
208	C₂²×C₁₂₇	C₂×C₄×D₄	1	1	yes	characteristic
209	C₂²×C₁₂₇	C₂×#127	1	1	yes	characteristic
210	C₂²×C₁₂₇	#532	1	1	yes	characteristic
211	C₂²×C₁₂₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
212	C₂²×C₁₂₇	#575	1	2	yes	—
213	C₄×C₁₂₇	—	2	2	no	—
214	C₄×C₁₂₇	—	2	4	no	—
215	C₄×C₁₂₇	—	4	2	no	—
216	C₄×C₁₂₇	—	4	2	no	—
217	C₄×C₁₂₇	—	8	2	no	—
218	C₄×C₁₂₇	—	8	1	no	—
219	C₄×C₁₂₇	—	8	1	no	—
220	C₄×C₁₂₇	#496	1	4	yes	direct product factor
221	C₄×C₁₂₇	C₂×C₄×D₄	1	2	yes	—
222	C₄×C₁₂₇	#532	1	2	yes	—
223	C₄×#496	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
224	C₂×C₁₂₇	—	8	4	no	—
225	C₂×C₁₂₇	—	8	2	no	—
226	C₂×C₁₂₇	C₄×#127	1	1	yes	characteristic
227	C₂×C₁₂₇	C₄×AGL(1,ℤ/8)	1	2	yes	—
228	C₂×C₁₂₇	C₂×#496	1	1	yes	characteristic
229	C₂×C₁₂₇	#3150	1	1	yes	characteristic
230	C₂×C₁₂₇	#3171	1	2	yes	—
231	C₂×#496	C₂×C₁₂₇	1	1	yes	characteristic
232	C₂×C₄×SD₁₆	C₂×C₁₂₇	1	2	yes	complemented
233	C₂×C₄×D₈	C₂×C₁₂₇	1	2	yes	complemented
234	C₄×#127	C₂×C₁₂₇	1	1	yes	characteristic, complemented
235	C₄×#124	C₂×C₁₂₇	1	1	yes	characteristic, complemented
236	#1973	C₂×C₁₂₇	1	2	yes	complemented
237	#1972	C₂×C₁₂₇	1	2	yes	complemented
238	#1961	C₂×C₁₂₇	1	1	yes	characteristic, complemented
239	#1959	C₂×C₁₂₇	1	1	yes	characteristic, complemented
240	#1847	C₂×C₁₂₇	1	1	yes	characteristic, complemented
241	C₄×#97	C₂×C₁₂₇	1	1	yes	characteristic, complemented
242	C₁₂₇	C₄×#496	1	1	yes	characteristic, direct product factor, Sylow
243	C₂²×SD₁₆	C₂²×C₁₂₇	1	2	yes	—
244	C₂²×D₈	C₂²×C₁₂₇	1	2	yes	—
245	C₂×#127	C₂²×C₁₂₇	1	1	yes	characteristic
246	C₂×#124	C₂²×C₁₂₇	1	1	yes	characteristic
247	C₂×C₄×Q₈	C₂²×C₁₂₇	1	1	yes	characteristic
248	C₂×C₄×D₄	—	2	2	no	—
249	C₂×C₄×D₄	C₂²×C₁₂₇	1	2	yes	—
250	C₂×C₄×D₄	C₂²×C₁₂₇	1	1	yes	characteristic
251	#496	C₄×C₁₂₇	1	8	yes	direct product factor
252	C₄×SD₁₆	—	2	8	no	—
253	C₄×D₈	—	2	8	no	—
254	C₂×#107	C₂²×C₁₂₇	1	2	yes	—
255	C₂×#106	C₂²×C₁₂₇	1	2	yes	—
256	C₂×#103	C₂²×C₁₂₇	1	2	yes	—
257	C₂×#102	C₂²×C₁₂₇	1	2	yes	—
258	C₂×#102	C₂²×C₁₂₇	1	2	yes	—
259	C₂×#102	C₂²×C₁₂₇	1	2	yes	—
260	C₂×#97	C₂²×C₁₂₇	1	1	yes	characteristic
261	C₂×#97	C₂²×C₁₂₇	1	2	yes	—
262	C₂×#97	C₂²×C₁₂₇	1	1	yes	characteristic
263	C₂×C₄×C₈	C₂²×C₁₂₇	1	2	yes	—
264	#391	C₂²×C₁₂₇	1	1	yes	characteristic
265	#390	—	2	2	no	—
266	#390	C₂²×C₁₂₇	1	2	yes	—
267	#390	C₂²×C₁₂₇	1	1	yes	characteristic
268	#388	—	2	2	no	—
269	#384	C₂²×C₁₂₇	1	1	yes	characteristic
270	#382	C₂²×C₁₂₇	1	1	yes	characteristic
271	C₄×C₂²⋊C₄	—	2	2	no	—
272	C₄³	C₂²×C₁₂₇	1	1	yes	characteristic
273	C₂²×Q₈	C₂³×C₁₂₇	1	1	yes	characteristic
274	C₂²×D₄	—	2	2	no	—
275	C₂²×D₄	C₂³×C₁₂₇	1	1	yes	characteristic
276	C₂²×D₄	C₂³×C₁₂₇	1	2	yes	—
277	C₂³×C₄	—	2	2	no	—
278	C₂³×C₄	—	2	1	no	—
279	C₂×SD₁₆	—	2	4	no	—
280	C₂×SD₁₆	—	2	4	no	—
281	C₂×SD₁₆	C₂×C₄×C₁₂₇	1	8	yes	complemented
282	C₂×D₈	—	2	4	no	—
283	C₂×D₈	—	2	4	no	—
284	C₂×D₈	C₂×C₄×C₁₂₇	1	8	yes	complemented
285	C₂²×C₈	C₂³×C₁₂₇	1	2	yes	—
286	C₂²×C₈	C₂³×C₁₂₇	1	2	yes	—
287	#127	—	2	2	no	—
288	#127	C₂×C₄×C₁₂₇	1	4	yes	complemented
289	#124	—	2	2	no	—
290	#124	C₂×C₄×C₁₂₇	1	4	yes	complemented
291	C₄×Q₈	—	2	4	no	—
292	C₄×D₄	—	2	4	no	—
293	C₄×D₄	—	2	4	no	—
294	C₄×D₄	—	2	4	no	—
295	C₄×D₄	—	4	8	no	—
296	C₂×C₄⋊C₄	—	2	2	no	—
297	C₂×C₄⋊C₄	—	2	1	no	—
298	C₂×C₄⋊C₄	C₂³×C₁₂₇	1	1	yes	characteristic
299	C₂×C₄⋊C₄	C₂³×C₁₂₇	1	2	yes	—
300	C₂×C₄⋊C₄	C₂³×C₁₂₇	1	1	yes	characteristic
301	C₂×C₂²⋊C₄	—	2	2	no	—
302	C₂×C₂²⋊C₄	—	2	4	no	—
303	C₂×C₂²⋊C₄	—	2	1	no	—
304	C₂×C₂²⋊C₄	—	2	2	no	—
305	C₂×C₂²⋊C₄	—	2	2	no	—
306	C₂×C₄²	—	2	1	no	—
307	C₂×C₄²	D₄×C₁₂₇	1	2	yes	—
308	C₂×C₄²	D₄×C₁₂₇	1	2	yes	—
309	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
310	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
311	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
312	#107	—	2	4	no	—
313	#106	—	2	4	no	—
314	#103	—	2	4	no	—
315	#102	—	2	4	no	—
316	#102	—	2	4	no	—
317	#102	—	2	4	no	—
318	#97	—	2	4	no	—
319	#97	—	2	2	no	—
320	#97	C₂×C₄×C₁₂₇	1	4	yes	complemented
321	C₄×C₈	D₄×C₁₂₇	1	8	yes	complemented
322	#95	—	2	2	no	—
323	#95	—	2	2	no	—
324	C₂⁴	—	2	1	no	—
325	C₂⁴	—	2	2	no	—
326	C₂×Q₈	—	2	1	no	—
327	C₂×Q₈	—	2	2	no	—
328	C₂×Q₈	—	2	2	no	—
329	C₂×Q₈	C₂²×C₄×C₁₂₇	1	2	yes	—
330	C₂×D₄	—	2	8	no	—
331	C₂×D₄	—	2	2	no	—
332	C₂×D₄	—	2	2	no	—
333	C₂×D₄	—	2	2	no	—
334	C₂×D₄	—	2	2	no	—
335	C₂×D₄	—	2	2	no	—
336	C₂×D₄	—	2	2	no	—
337	C₂×D₄	—	2	1	no	—
338	C₂×D₄	—	2	2	no	—
339	C₂×D₄	—	4	4	no	—
340	C₂×D₄	—	4	4	no	—
341	C₂×D₄	C₂²×C₄×C₁₂₇	1	2	yes	—
342	C₂×D₄	C₂²×C₄×C₁₂₇	1	4	yes	—
343	C₂²×C₄	—	2	1	no	—
344	C₂²×C₄	—	2	1	no	—
345	C₂²×C₄	—	2	1	no	—
346	C₂²×C₄	—	2	2	no	—
347	C₂²×C₄	—	4	2	no	—
348	C₂²×C₄	—	4	4	no	—
349	C₂²×C₄	—	4	8	no	—
350	C₂²×C₄	—	4	4	no	—
351	C₂²×C₄	C₂×D₄×C₁₂₇	1	2	yes	—
352	C₂²×C₄	C₂×D₄×C₁₂₇	1	2	yes	—
353	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
354	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
355	C₂²×C₄	C₂⁴×C₁₂₇	1	1	yes	characteristic
356	SD₁₆	—	2	16	no	—
357	D₈	—	2	16	no	—
358	C₂×C₈	—	2	2	no	—
359	C₂×C₈	—	2	2	no	—
360	C₂×C₈	C₂²×C₄×C₁₂₇	1	4	yes	—
361	C₂×C₈	C₂×D₄×C₁₂₇	1	4	yes	—
362	C₂×C₈	C₂×D₄×C₁₂₇	1	4	yes	—
363	C₂×C₈	#41×C₁₂₇	1	4	yes	—
364	C₄⋊C₄	—	2	2	no	—
365	C₄⋊C₄	—	2	2	no	—
366	C₄⋊C₄	—	2	2	no	—
367	C₄⋊C₄	—	2	2	no	—
368	C₄⋊C₄	—	4	4	no	—
369	C₄⋊C₄	—	4	2	no	—
370	C₂²⋊C₄	—	2	8	no	—
371	C₂²⋊C₄	—	4	4	no	—
372	C₂²⋊C₄	—	4	8	no	—
373	C₂²⋊C₄	—	4	2	no	—
374	C₂²⋊C₄	—	4	4	no	—
375	C₄²	—	2	4	no	—
376	C₄²	—	2	2	no	—
377	C₄²	—	4	2	no	—
378	C₄²	—	4	2	no	—
379	C₄²	C₂²×C₄×C₁₂₇	1	2	yes	—
380	C₄²	C₂×D₄×C₁₂₇	1	4	yes	—
381	C₄²	#41×C₁₂₇	1	2	yes	—
382	C₂³	—	2	2	no	—
383	C₂³	—	2	4	no	—
384	C₂³	—	4	1	no	—
385	C₂³	—	4	2	no	—
386	C₂³	—	4	1	no	—
387	C₂³	—	4	2	no	—
388	C₂³	—	4	4	no	—
389	C₂³	—	4	2	no	—
390	C₂³	—	4	2	no	—
391	C₂³	—	4	4	no	—
392	C₂³	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
393	Q₈	—	2	4	no	—
394	Q₈	—	4	2	no	—
395	D₄	—	2	4	no	—
396	D₄	—	2	4	no	—
397	D₄	—	2	4	no	—
398	D₄	—	4	16	no	—
399	D₄	—	4	2	no	—
400	D₄	—	4	2	no	—
401	D₄	—	4	2	no	—
402	C₂×C₄	—	2	2	no	—
403	C₂×C₄	—	2	2	no	—
404	C₂×C₄	—	2	2	no	—
405	C₂×C₄	—	2	2	no	—
406	C₂×C₄	—	2	2	no	—
407	C₂×C₄	—	2	2	no	—
408	C₂×C₄	—	2	2	no	—
409	C₂×C₄	—	4	1	no	—
410	C₂×C₄	—	4	1	no	—
411	C₂×C₄	—	4	1	no	—
412	C₂×C₄	—	4	1	no	—
413	C₂×C₄	—	4	2	no	—
414	C₂×C₄	—	4	4	no	—
415	C₂×C₄	—	4	2	no	—
416	C₂×C₄	—	4	2	no	—
417	C₂×C₄	—	8	8	no	—
418	C₂×C₄	—	8	4	no	—
419	C₂×C₄	C₄×D₄×C₁₂₇	1	4	yes	—
420	C₂×C₄	#127×C₁₂₇	1	2	yes	—
421	C₂×C₄	#127×C₁₂₇	1	2	yes	—
422	C₂×C₄	AGL(1,ℤ/8)×C₁₂₇	1	4	yes	—
423	C₂×C₄	C₂³×C₄×C₁₂₇	1	1	yes	characteristic
424	C₂×C₄	C₂²×D₄×C₁₂₇	1	2	yes	—
425	C₂×C₄	C₂×#41×C₁₂₇	1	2	yes	—
426	C₂×C₄	C₂×#41×C₁₂₇	1	1	yes	characteristic
427	C₈	—	2	4	no	—
428	C₈	C₄×D₄×C₁₂₇	1	8	yes	complemented
429	C₂²	—	4	8	no	—
430	C₂²	—	4	4	no	—
431	C₂²	—	4	2	no	—
432	C₂²	—	4	4	no	—
433	C₂²	—	8	2	no	—
434	C₂²	—	8	1	no	—
435	C₂²	—	8	1	no	—
436	C₂²	—	8	2	no	—
437	C₂²	—	8	4	no	—
438	C₂²	—	8	2	no	—
439	C₂²	C₂×C₄×D₄×C₁₂₇	1	1	yes	characteristic
440	C₂²	C₂×#127×C₁₂₇	1	1	yes	characteristic
441	C₂²	#532×C₁₂₇	1	1	yes	characteristic
442	C₂²	C₂×AGL(1,ℤ/8)×C₁₂₇	1	2	yes	—
443	C₂²	#575×C₁₂₇	1	2	yes	—
444	C₄	—	2	2	no	—
445	C₄	—	2	4	no	—
446	C₄	—	4	2	no	—
447	C₄	—	4	2	no	—
448	C₄	—	8	2	no	—
449	C₄	—	8	1	no	—
450	C₄	—	8	1	no	—
451	C₄	#496×C₁₂₇	1	4	yes	direct product factor
452	C₄	C₂×C₄×D₄×C₁₂₇	1	2	yes	—
453	C₄	#532×C₁₂₇	1	2	yes	—
454	C₂	—	8	4	no	—
455	C₂	—	8	2	no	—
456	C₂	C₄×#127×C₁₂₇	1	1	yes	characteristic
457	C₂	C₄×AGL(1,ℤ/8)×C₁₂₇	1	2	yes	—
458	C₂	C₂×#496×C₁₂₇	1	1	yes	characteristic
459	C₂	C₁₂₇×#3150	1	1	yes	characteristic
460	C₂	C₁₂₇×#3171	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₁₂₇	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₁₂₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₂₇
socle	C₂³×C₁₂₇	C₂²×D₄
prime radical	C₂×#496×C₁₂₇	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₂₇ ◁ C₄³×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×SD₁₆×C₁₂₇	C₂	semidirect
3. C₂×C₄×D₈×C₁₂₇	C₂	semidirect
4. C₄×#127×C₁₂₇	C₂	semidirect
5. C₄×#124×C₁₂₇	C₂	semidirect
6. C₁₂₇×#1973	C₂	semidirect
7. C₁₂₇×#1972	C₂	semidirect
8. C₁₂₇×#1961	C₂	semidirect
9. C₁₂₇×#1959	C₂	semidirect
10. C₁₂₇×#1847	C₂	semidirect
11. C₄×#97×C₁₂₇	C₂	semidirect
20. #496×C₁₂₇	C₄	direct (N × H)
50. C₂×SD₁₆×C₁₂₇	C₂×C₄	semidirect
53. C₂×D₈×C₁₂₇	C₂×C₄	semidirect
57. #127×C₁₂₇	C₂×C₄	semidirect
59. #124×C₁₂₇	C₂×C₄	semidirect
89. #97×C₁₂₇	C₂×C₄	semidirect
90. C₄×C₈×C₁₂₇	D₄	semidirect
197. C₈×C₁₂₇	C₄×D₄	semidirect
220. C₄×C₁₂₇	#496	direct (N × H)
223. C₄×#496	C₁₂₇	direct (N × H)
232. C₂×C₄×SD₁₆	C₂×C₁₂₇	semidirect
233. C₂×C₄×D₈	C₂×C₁₂₇	semidirect
234. C₄×#127	C₂×C₁₂₇	semidirect
235. C₄×#124	C₂×C₁₂₇	semidirect
236. #1973	C₂×C₁₂₇	semidirect
237. #1972	C₂×C₁₂₇	semidirect
238. #1961	C₂×C₁₂₇	semidirect
239. #1959	C₂×C₁₂₇	semidirect
240. #1847	C₂×C₁₂₇	semidirect
241. C₄×#97	C₂×C₁₂₇	semidirect
242. C₁₂₇	C₄×#496	direct (N × H)
251. #496	C₄×C₁₂₇	direct (N × H)
281. C₂×SD₁₆	C₂×C₄×C₁₂₇	semidirect
284. C₂×D₈	C₂×C₄×C₁₂₇	semidirect
288. #127	C₂×C₄×C₁₂₇	semidirect
290. #124	C₂×C₄×C₁₂₇	semidirect
320. #97	C₂×C₄×C₁₂₇	semidirect
321. C₄×C₈	D₄×C₁₂₇	semidirect
428. C₈	C₄×D₄×C₁₂₇	semidirect
451. C₄	#496×C₁₂₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#496×C₁₂₇	2
C₂×C₄×SD₁₆×C₁₂₇	2
C₂×C₄×D₈×C₁₂₇	2
C₄×#127×C₁₂₇	2
C₄×#124×C₁₂₇	2
C₁₂₇×#1973	2
C₁₂₇×#1972	2
C₁₂₇×#1961	2
C₁₂₇×#1959	2
C₁₂₇×#1847	2
C₄×#97×C₁₂₇	2
C₄×#496	127

Subgroups — C4×#496×C127 #44585372

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#496×C₁₂₇ #44585372