Subgroups — C₁₂₇×#19623 #44583296

Order 32512.

848 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₂₇×#3208	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₂₇×#3206	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₂₇×#3167	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₂₇×#3165	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₂₇×#3151	C₂	1	1	yes	maximal, characteristic
6	C₂×#486×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₂₇×#1715	C₂	1	2	yes	maximal, complemented
8	C₁₂₇×#1714	C₂	1	2	yes	maximal, complemented
9	C₁₂₇×#1708	C₂	1	2	yes	maximal, complemented
10	C₁₂₇×#1706	C₂	1	2	yes	maximal, complemented
11	C₈×M₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×AGL(1,ℤ/8)×C₁₂₇	—	2	2	no	—
13	#533×C₁₂₇	C₂²	1	1	yes	characteristic
14	#532×C₁₂₇	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊Q₈×C₁₂₇	C₂²	1	1	yes	characteristic
16	C₂×#127×C₁₂₇	C₂²	1	1	yes	characteristic
17	#501×C₁₂₇	—	2	2	no	—
18	#500×C₁₂₇	C₂²	1	2	yes	complemented
19	#498×C₁₂₇	C₂²	1	2	yes	complemented
20	#498×C₁₂₇	C₂²	1	1	yes	characteristic, complemented
21	#496×C₁₂₇	—	2	2	no	—
22	#493×C₁₂₇	C₂²	1	2	yes	—
23	#492×C₁₂₇	C₂²	1	2	yes	—
24	#492×C₁₂₇	C₂²	1	2	yes	—
25	#492×C₁₂₇	C₂²	1	1	yes	characteristic, complemented
26	#492×C₁₂₇	C₂²	1	2	yes	complemented
27	#487×C₁₂₇	C₂²	1	1	yes	characteristic, complemented
28	#487×C₁₂₇	C₂²	1	2	yes	complemented
29	#486×C₁₂₇	—	2	2	no	—
30	#486×C₁₂₇	C₂²	1	2	yes	—
31	#486×C₁₂₇	C₂²	1	2	yes	—
32	#486×C₁₂₇	C₂²	1	2	yes	—
33	#486×C₁₂₇	C₂²	1	1	yes	characteristic, complemented
34	#465×C₁₂₇	—	2	2	no	—
35	#461×C₁₂₇	—	2	2	no	—
36	#460×C₁₂₇	—	2	2	no	—
37	#450×C₁₂₇	—	2	2	no	—
38	#448×C₁₂₇	—	2	2	no	—
39	C₄×SD₁₆×C₁₂₇	—	2	2	no	—
40	#431×C₁₂₇	C₂²	1	1	yes	characteristic
41	#428×C₁₂₇	—	2	2	no	—
42	C₂×#102×C₁₂₇	—	2	2	no	—
43	C₄×M₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
44	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
45	#322×C₁₂₇	C₂²	1	2	yes	complemented
46	C₈²×C₁₂₇	C₂²	1	2	yes	complemented
47	C₂×#41×C₁₂₇	—	2	1	no	—
48	C₂²×Q₈×C₁₂₇	—	2	1	no	—
49	C₂²×D₄×C₁₂₇	—	2	1	no	—
50	C₂²×D₄×C₁₂₇	—	2	1	no	—
51	C₂²×D₄×C₁₂₇	—	2	1	no	—
52	AGL(1,ℤ/8)×C₁₂₇	—	4	8	no	—
53	C₂×SD₁₆×C₁₂₇	—	2	4	no	—
54	C₂×SD₁₆×C₁₂₇	—	2	2	no	—
55	C₂×SD₁₆×C₁₂₇	—	2	4	no	—
56	C₂×SD₁₆×C₁₂₇	—	2	4	no	—
57	C₂×D₈×C₁₂₇	—	2	4	no	—
58	C₂×M₁₆×C₁₂₇	D₄	1	2	yes	—
59	C₂²×C₈×C₁₂₇	D₄	1	2	yes	—
60	C₄⋊Q₈×C₁₂₇	—	2	1	no	—
61	C₄⋊Q₈×C₁₂₇	—	2	1	no	—
62	C₄⋊Q₈×C₁₂₇	—	2	1	no	—
63	C₄⋊Q₈×C₁₂₇	C₂³	1	1	yes	characteristic
64	C₄⋊Q₈×C₁₂₇	C₂³	1	1	yes	characteristic
65	C₄⋊Q₈×C₁₂₇	C₂³	1	2	yes	—
66	C₄⋊Q₈×C₁₂₇	C₂³	1	1	yes	characteristic
67	#127×C₁₂₇	—	2	1	no	—
68	#127×C₁₂₇	—	2	1	no	—
69	#127×C₁₂₇	—	2	1	no	—
70	#127×C₁₂₇	C₂³	1	2	yes	—
71	#127×C₁₂₇	C₂³	1	1	yes	characteristic
72	#125×C₁₂₇	—	2	1	no	—
73	#124×C₁₂₇	—	2	1	no	—
74	C₂²⋊Q₈×C₁₂₇	—	2	1	no	—
75	C₂²⋊Q₈×C₁₂₇	—	2	1	no	—
76	#121×C₁₂₇	—	2	1	no	—
77	#121×C₁₂₇	—	2	1	no	—
78	C₄×Q₈×C₁₂₇	—	2	1	no	—
79	C₄×Q₈×C₁₂₇	—	2	1	no	—
80	C₄×D₄×C₁₂₇	—	2	1	no	—
81	C₄×D₄×C₁₂₇	—	2	1	no	—
82	#117×C₁₂₇	—	2	1	no	—
83	C₂×C₄⋊C₄×C₁₂₇	—	2	1	no	—
84	C₂×C₄⋊C₄×C₁₂₇	—	2	1	no	—
85	C₂×C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
86	#107×C₁₂₇	—	2	4	no	—
87	#106×C₁₂₇	—	2	4	no	—
88	#106×C₁₂₇	—	2	2	no	—
89	C₄⋊C₈×C₁₂₇	D₄	1	2	yes	—
90	#103×C₁₂₇	—	2	2	no	—
91	#103×C₁₂₇	—	2	4	no	—
92	#102×C₁₂₇	—	2	4	no	—
93	#102×C₁₂₇	—	2	4	no	—
94	#102×C₁₂₇	—	2	2	no	—
95	#102×C₁₂₇	—	2	4	no	—
96	C₂²⋊C₈×C₁₂₇	D₄	1	2	yes	—
97	#97×C₁₂₇	D₄	1	2	yes	—
98	C₄×C₈×C₁₂₇	D₄	1	2	yes	—
99	C₄×C₈×C₁₂₇	C₂³	1	2	yes	—
100	C₄×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
101	C₄×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
102	C₄×C₈×C₁₂₇	C₂³	1	2	yes	—
103	C₂⁴×C₁₂₇	—	4	1	no	—
104	#41×C₁₂₇	—	4	4	no	—
105	C₂×Q₈×C₁₂₇	—	2	1	no	—
106	C₂×Q₈×C₁₂₇	—	2	2	no	—
107	C₂×Q₈×C₁₂₇	—	2	1	no	—
108	C₂×Q₈×C₁₂₇	—	2	1	no	—
109	C₂×Q₈×C₁₂₇	—	4	1	no	—
110	C₂×Q₈×C₁₂₇	—	8	2	no	—
111	C₂×D₄×C₁₂₇	—	2	2	no	—
112	C₂×D₄×C₁₂₇	—	2	1	no	—
113	C₂×D₄×C₁₂₇	—	2	2	no	—
114	C₂×D₄×C₁₂₇	—	2	1	no	—
115	C₂×D₄×C₁₂₇	—	2	1	no	—
116	C₂×D₄×C₁₂₇	—	2	2	no	—
117	C₂×D₄×C₁₂₇	—	4	4	no	—
118	C₂×D₄×C₁₂₇	—	4	1	no	—
119	C₂×D₄×C₁₂₇	—	4	1	no	—
120	C₂×D₄×C₁₂₇	—	4	1	no	—
121	C₂×D₄×C₁₂₇	—	4	1	no	—
122	C₂×D₄×C₁₂₇	—	4	4	no	—
123	C₂×D₄×C₁₂₇	—	8	2	no	—
124	C₂²×C₄×C₁₂₇	—	4	1	no	—
125	C₂²×C₄×C₁₂₇	—	4	1	no	—
126	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
127	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
128	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
129	SD₁₆×C₁₂₇	—	4	4	no	—
130	SD₁₆×C₁₂₇	—	4	8	no	—
131	SD₁₆×C₁₂₇	—	8	4	no	—
132	SD₁₆×C₁₂₇	—	8	4	no	—
133	D₈×C₁₂₇	—	4	8	no	—
134	M₁₆×C₁₂₇	SD₁₆	1	8	yes	complemented
135	C₂×C₈×C₁₂₇	—	4	2	no	—
136	C₂×C₈×C₁₂₇	C₂×D₄	1	2	yes	—
137	C₂×C₈×C₁₂₇	C₂×D₄	1	2	yes	—
138	C₂×C₈×C₁₂₇	C₂×D₄	1	4	yes	—
139	C₂×C₈×C₁₂₇	C₂×D₄	1	4	yes	—
140	C₄⋊C₄×C₁₂₇	—	2	2	no	—
141	C₄⋊C₄×C₁₂₇	—	2	1	no	—
142	C₄⋊C₄×C₁₂₇	—	2	1	no	—
143	C₄⋊C₄×C₁₂₇	—	2	1	no	—
144	C₄⋊C₄×C₁₂₇	—	2	1	no	—
145	C₄⋊C₄×C₁₂₇	—	2	1	no	—
146	C₄⋊C₄×C₁₂₇	—	2	2	no	—
147	C₄⋊C₄×C₁₂₇	—	2	1	no	—
148	C₄⋊C₄×C₁₂₇	—	4	1	no	—
149	C₄⋊C₄×C₁₂₇	—	4	1	no	—
150	C₄⋊C₄×C₁₂₇	—	4	1	no	—
151	C₄⋊C₄×C₁₂₇	—	4	1	no	—
152	C₂²⋊C₄×C₁₂₇	—	4	1	no	—
153	C₂²⋊C₄×C₁₂₇	—	4	1	no	—
154	C₂²⋊C₄×C₁₂₇	—	4	1	no	—
155	C₄²×C₁₂₇	—	4	1	no	—
156	C₄²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
157	C₄²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
158	C₄²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
159	C₄²×C₁₂₇	C₂⁴	1	1	yes	characteristic
160	C₂³×C₁₂₇	—	4	1	no	—
161	C₂³×C₁₂₇	—	4	2	no	—
162	C₂³×C₁₂₇	—	8	2	no	—
163	C₂³×C₁₂₇	—	8	2	no	—
164	C₂³×C₁₂₇	—	8	2	no	—
165	C₂³×C₁₂₇	#127	1	1	yes	characteristic
166	Q₈×C₁₂₇	—	4	2	no	—
167	Q₈×C₁₂₇	—	4	2	no	—
168	Q₈×C₁₂₇	—	4	2	no	—
169	Q₈×C₁₂₇	—	4	4	no	—
170	Q₈×C₁₂₇	—	8	2	no	—
171	D₄×C₁₂₇	—	4	2	no	—
172	D₄×C₁₂₇	—	4	4	no	—
173	D₄×C₁₂₇	—	4	2	no	—
174	D₄×C₁₂₇	—	4	4	no	—
175	D₄×C₁₂₇	—	4	2	no	—
176	D₄×C₁₂₇	—	4	4	no	—
177	D₄×C₁₂₇	—	8	2	no	—
178	D₄×C₁₂₇	—	8	2	no	—
179	D₄×C₁₂₇	—	8	2	no	—
180	D₄×C₁₂₇	—	8	2	no	—
181	C₂×C₄×C₁₂₇	—	2	2	no	—
182	C₂×C₄×C₁₂₇	—	2	2	no	—
183	C₂×C₄×C₁₂₇	—	4	1	no	—
184	C₂×C₄×C₁₂₇	—	4	1	no	—
185	C₂×C₄×C₁₂₇	—	4	1	no	—
186	C₂×C₄×C₁₂₇	—	4	2	no	—
187	C₂×C₄×C₁₂₇	—	8	2	no	—
188	C₂×C₄×C₁₂₇	—	8	2	no	—
189	C₂×C₄×C₁₂₇	#127	1	1	yes	characteristic
190	C₂×C₄×C₁₂₇	#127	1	1	yes	characteristic
191	C₂×C₄×C₁₂₇	#127	1	1	yes	characteristic
192	C₂×C₄×C₁₂₇	C₂×SD₁₆	1	4	yes	—
193	C₂×C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
194	C₂×C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
195	C₂×C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
196	C₈×C₁₂₇	—	2	2	no	—
197	C₈×C₁₂₇	—	2	2	no	—
198	C₈×C₁₂₇	—	2	4	no	—
199	C₈×C₁₂₇	C₂×SD₁₆	1	8	yes	—
200	C₂²×C₁₂₇	—	2	1	no	—
201	C₂²×C₁₂₇	—	2	1	no	—
202	C₂²×C₁₂₇	—	8	1	no	—
203	C₂²×C₁₂₇	—	8	2	no	—
204	C₂²×C₁₂₇	—	8	2	no	—
205	C₂²×C₁₂₇	—	8	1	no	—
206	C₂²×C₁₂₇	—	8	2	no	—
207	C₂²×C₁₂₇	—	8	1	no	—
208	C₂²×C₁₂₇	—	16	4	no	—
209	C₂²×C₁₂₇	#492	1	2	yes	—
210	C₂²×C₁₂₇	C₂×#127	1	1	yes	characteristic
211	C₄×C₁₂₇	—	2	1	no	—
212	C₄×C₁₂₇	—	2	1	no	—
213	C₄×C₁₂₇	—	8	1	no	—
214	C₄×C₁₂₇	—	8	1	no	—
215	C₄×C₁₂₇	—	8	2	no	—
216	C₄×C₁₂₇	—	8	1	no	—
217	C₄×C₁₂₇	#492	1	2	yes	—
218	C₄×C₁₂₇	C₂²×SD₁₆	1	2	yes	—
219	C₄×C₁₂₇	#575	1	2	yes	—
220	C₄×C₁₂₇	#576	1	2	yes	—
221	#19623	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
222	C₂×C₁₂₇	—	2	2	no	—
223	C₂×C₁₂₇	—	16	1	no	—
224	C₂×C₁₂₇	—	16	2	no	—
225	C₂×C₁₂₇	C₂×#492	1	1	yes	characteristic
226	C₂×C₁₂₇	#3153	1	1	yes	characteristic
227	C₂×C₁₂₇	#3155	1	1	yes	characteristic
228	#3208	C₂×C₁₂₇	1	1	yes	characteristic, complemented
229	#3206	C₂×C₁₂₇	1	1	yes	characteristic, complemented
230	#3167	C₂×C₁₂₇	1	1	yes	characteristic, complemented
231	#3165	C₂×C₁₂₇	1	1	yes	characteristic, complemented
232	#3151	C₂×C₁₂₇	1	1	yes	characteristic
233	C₂×#486	C₂×C₁₂₇	1	1	yes	characteristic, complemented
234	#1715	C₂×C₁₂₇	1	2	yes	complemented
235	#1714	C₂×C₁₂₇	1	2	yes	complemented
236	#1708	C₂×C₁₂₇	1	2	yes	complemented
237	#1706	C₂×C₁₂₇	1	2	yes	complemented
238	C₈×M₁₆	C₂×C₁₂₇	1	1	yes	characteristic, complemented
239	C₁₂₇	#19623	1	1	yes	characteristic, direct product factor, Sylow
240	C₂×AGL(1,ℤ/8)	—	2	2	no	—
241	#533	C₂²×C₁₂₇	1	1	yes	characteristic
242	#532	C₂²×C₁₂₇	1	1	yes	characteristic
243	C₂×C₄⋊Q₈	C₂²×C₁₂₇	1	1	yes	characteristic
244	C₂×#127	C₂²×C₁₂₇	1	1	yes	characteristic
245	#501	—	2	2	no	—
246	#500	C₂²×C₁₂₇	1	2	yes	complemented
247	#498	C₂²×C₁₂₇	1	2	yes	complemented
248	#498	C₂²×C₁₂₇	1	1	yes	characteristic, complemented
249	#496	—	2	2	no	—
250	#493	C₂²×C₁₂₇	1	2	yes	—
251	#492	C₂²×C₁₂₇	1	2	yes	—
252	#492	C₂²×C₁₂₇	1	2	yes	—
253	#492	C₂²×C₁₂₇	1	1	yes	characteristic, complemented
254	#492	C₂²×C₁₂₇	1	2	yes	complemented
255	#487	C₂²×C₁₂₇	1	1	yes	characteristic, complemented
256	#487	C₂²×C₁₂₇	1	2	yes	complemented
257	#486	—	2	2	no	—
258	#486	C₂²×C₁₂₇	1	2	yes	—
259	#486	C₂²×C₁₂₇	1	2	yes	—
260	#486	C₂²×C₁₂₇	1	2	yes	—
261	#486	C₂²×C₁₂₇	1	1	yes	characteristic, complemented
262	#465	—	2	2	no	—
263	#461	—	2	2	no	—
264	#460	—	2	2	no	—
265	#450	—	2	2	no	—
266	#448	—	2	2	no	—
267	C₄×SD₁₆	—	2	2	no	—
268	#431	C₂²×C₁₂₇	1	1	yes	characteristic
269	#428	—	2	2	no	—
270	C₂×#102	—	2	2	no	—
271	C₄×M₁₆	C₂²×C₁₂₇	1	1	yes	characteristic
272	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
273	#322	C₂²×C₁₂₇	1	2	yes	complemented
274	C₈²	C₂²×C₁₂₇	1	2	yes	complemented
275	C₂×#41	—	2	1	no	—
276	C₂²×Q₈	—	2	1	no	—
277	C₂²×D₄	—	2	1	no	—
278	C₂²×D₄	—	2	1	no	—
279	C₂²×D₄	—	2	1	no	—
280	AGL(1,ℤ/8)	—	4	8	no	—
281	C₂×SD₁₆	—	2	4	no	—
282	C₂×SD₁₆	—	2	2	no	—
283	C₂×SD₁₆	—	2	4	no	—
284	C₂×SD₁₆	—	2	4	no	—
285	C₂×D₈	—	2	4	no	—
286	C₂×M₁₆	D₄×C₁₂₇	1	2	yes	—
287	C₂²×C₈	D₄×C₁₂₇	1	2	yes	—
288	C₄⋊Q₈	—	2	1	no	—
289	C₄⋊Q₈	—	2	1	no	—
290	C₄⋊Q₈	—	2	1	no	—
291	C₄⋊Q₈	C₂³×C₁₂₇	1	2	yes	—
292	C₄⋊Q₈	C₂³×C₁₂₇	1	1	yes	characteristic
293	C₄⋊Q₈	C₂³×C₁₂₇	1	1	yes	characteristic
294	C₄⋊Q₈	C₂³×C₁₂₇	1	1	yes	characteristic
295	#127	—	2	1	no	—
296	#127	—	2	1	no	—
297	#127	—	2	1	no	—
298	#127	C₂³×C₁₂₇	1	2	yes	—
299	#127	C₂³×C₁₂₇	1	1	yes	characteristic
300	#125	—	2	1	no	—
301	#124	—	2	1	no	—
302	C₂²⋊Q₈	—	2	1	no	—
303	C₂²⋊Q₈	—	2	1	no	—
304	#121	—	2	1	no	—
305	#121	—	2	1	no	—
306	C₄×Q₈	—	2	1	no	—
307	C₄×Q₈	—	2	1	no	—
308	C₄×D₄	—	2	1	no	—
309	C₄×D₄	—	2	1	no	—
310	#117	—	2	1	no	—
311	C₂×C₄⋊C₄	—	2	1	no	—
312	C₂×C₄⋊C₄	—	2	1	no	—
313	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
314	#107	—	2	4	no	—
315	#106	—	2	4	no	—
316	#106	—	2	2	no	—
317	C₄⋊C₈	D₄×C₁₂₇	1	2	yes	—
318	#103	—	2	2	no	—
319	#103	—	2	4	no	—
320	#102	—	2	4	no	—
321	#102	—	2	4	no	—
322	#102	—	2	2	no	—
323	#102	—	2	4	no	—
324	C₂²⋊C₈	D₄×C₁₂₇	1	2	yes	—
325	#97	D₄×C₁₂₇	1	2	yes	—
326	C₄×C₈	D₄×C₁₂₇	1	2	yes	—
327	C₄×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
328	C₄×C₈	C₂³×C₁₂₇	1	2	yes	—
329	C₄×C₈	C₂³×C₁₂₇	1	2	yes	—
330	C₄×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
331	C₂⁴	—	4	1	no	—
332	#41	—	4	4	no	—
333	C₂×Q₈	—	2	1	no	—
334	C₂×Q₈	—	2	2	no	—
335	C₂×Q₈	—	2	1	no	—
336	C₂×Q₈	—	2	1	no	—
337	C₂×Q₈	—	4	1	no	—
338	C₂×Q₈	—	8	2	no	—
339	C₂×D₄	—	2	2	no	—
340	C₂×D₄	—	2	1	no	—
341	C₂×D₄	—	2	2	no	—
342	C₂×D₄	—	2	1	no	—
343	C₂×D₄	—	2	1	no	—
344	C₂×D₄	—	2	2	no	—
345	C₂×D₄	—	4	4	no	—
346	C₂×D₄	—	4	1	no	—
347	C₂×D₄	—	4	1	no	—
348	C₂×D₄	—	4	1	no	—
349	C₂×D₄	—	4	1	no	—
350	C₂×D₄	—	4	4	no	—
351	C₂×D₄	—	8	2	no	—
352	C₂²×C₄	—	4	1	no	—
353	C₂²×C₄	—	4	1	no	—
354	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
355	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
356	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
357	SD₁₆	—	4	4	no	—
358	SD₁₆	—	4	8	no	—
359	SD₁₆	—	8	4	no	—
360	SD₁₆	—	8	4	no	—
361	D₈	—	4	8	no	—
362	M₁₆	SD₁₆×C₁₂₇	1	8	yes	complemented
363	C₂×C₈	—	4	2	no	—
364	C₂×C₈	C₂×D₄×C₁₂₇	1	2	yes	—
365	C₂×C₈	C₂×D₄×C₁₂₇	1	2	yes	—
366	C₂×C₈	C₂×D₄×C₁₂₇	1	4	yes	—
367	C₂×C₈	C₂×D₄×C₁₂₇	1	4	yes	—
368	C₄⋊C₄	—	2	2	no	—
369	C₄⋊C₄	—	2	1	no	—
370	C₄⋊C₄	—	2	1	no	—
371	C₄⋊C₄	—	2	1	no	—
372	C₄⋊C₄	—	2	1	no	—
373	C₄⋊C₄	—	2	1	no	—
374	C₄⋊C₄	—	2	2	no	—
375	C₄⋊C₄	—	2	1	no	—
376	C₄⋊C₄	—	4	1	no	—
377	C₄⋊C₄	—	4	1	no	—
378	C₄⋊C₄	—	4	1	no	—
379	C₄⋊C₄	—	4	1	no	—
380	C₂²⋊C₄	—	4	1	no	—
381	C₂²⋊C₄	—	4	1	no	—
382	C₂²⋊C₄	—	4	1	no	—
383	C₄²	—	4	1	no	—
384	C₄²	C₂×D₄×C₁₂₇	1	1	yes	characteristic
385	C₄²	C₂×D₄×C₁₂₇	1	1	yes	characteristic
386	C₄²	C₂×D₄×C₁₂₇	1	1	yes	characteristic
387	C₄²	C₂⁴×C₁₂₇	1	1	yes	characteristic
388	C₂³	—	4	1	no	—
389	C₂³	—	4	2	no	—
390	C₂³	—	8	2	no	—
391	C₂³	—	8	2	no	—
392	C₂³	—	8	2	no	—
393	C₂³	#127×C₁₂₇	1	1	yes	characteristic
394	Q₈	—	4	2	no	—
395	Q₈	—	4	2	no	—
396	Q₈	—	4	2	no	—
397	Q₈	—	4	4	no	—
398	Q₈	—	8	2	no	—
399	D₄	—	4	2	no	—
400	D₄	—	4	4	no	—
401	D₄	—	4	2	no	—
402	D₄	—	4	4	no	—
403	D₄	—	4	2	no	—
404	D₄	—	4	4	no	—
405	D₄	—	8	2	no	—
406	D₄	—	8	2	no	—
407	D₄	—	8	2	no	—
408	D₄	—	8	2	no	—
409	C₂×C₄	—	2	2	no	—
410	C₂×C₄	—	2	2	no	—
411	C₂×C₄	—	4	1	no	—
412	C₂×C₄	—	4	1	no	—
413	C₂×C₄	—	4	1	no	—
414	C₂×C₄	—	4	2	no	—
415	C₂×C₄	—	8	2	no	—
416	C₂×C₄	—	8	2	no	—
417	C₂×C₄	#127×C₁₂₇	1	1	yes	characteristic
418	C₂×C₄	#127×C₁₂₇	1	1	yes	characteristic
419	C₂×C₄	#127×C₁₂₇	1	1	yes	characteristic
420	C₂×C₄	C₂×SD₁₆×C₁₂₇	1	4	yes	—
421	C₂×C₄	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
422	C₂×C₄	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
423	C₂×C₄	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
424	C₈	—	2	2	no	—
425	C₈	—	2	2	no	—
426	C₈	—	2	4	no	—
427	C₈	C₂×SD₁₆×C₁₂₇	1	8	yes	—
428	C₂²	—	2	1	no	—
429	C₂²	—	2	1	no	—
430	C₂²	—	8	1	no	—
431	C₂²	—	8	2	no	—
432	C₂²	—	8	2	no	—
433	C₂²	—	8	1	no	—
434	C₂²	—	8	2	no	—
435	C₂²	—	8	1	no	—
436	C₂²	—	16	4	no	—
437	C₂²	#492×C₁₂₇	1	2	yes	—
438	C₂²	C₂×#127×C₁₂₇	1	1	yes	characteristic
439	C₄	—	2	1	no	—
440	C₄	—	2	1	no	—
441	C₄	—	8	1	no	—
442	C₄	—	8	1	no	—
443	C₄	—	8	2	no	—
444	C₄	—	8	1	no	—
445	C₄	#492×C₁₂₇	1	2	yes	—
446	C₄	C₂²×SD₁₆×C₁₂₇	1	2	yes	—
447	C₄	#575×C₁₂₇	1	2	yes	—
448	C₄	#576×C₁₂₇	1	2	yes	—
449	C₂	—	2	2	no	—
450	C₂	—	16	2	no	—
451	C₂	—	16	1	no	—
452	C₂	C₂×#492×C₁₂₇	1	1	yes	characteristic
453	C₂	C₁₂₇×#3153	1	1	yes	characteristic
454	C₂	C₁₂₇×#3155	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₂₇	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₁₂₇
prime radical	C₁₂₇×#3151	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₂₇ ◁ C₂×C₄²×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₂₇×#3208	C₂	semidirect
2. C₁₂₇×#3206	C₂	semidirect
3. C₁₂₇×#3167	C₂	semidirect
4. C₁₂₇×#3165	C₂	semidirect
6. C₂×#486×C₁₂₇	C₂	semidirect
7. C₁₂₇×#1715	C₂	semidirect
8. C₁₂₇×#1714	C₂	semidirect
9. C₁₂₇×#1708	C₂	semidirect
10. C₁₂₇×#1706	C₂	semidirect
11. C₈×M₁₆×C₁₂₇	C₂	semidirect
18. #500×C₁₂₇	C₂²	semidirect
19. #498×C₁₂₇	C₂²	semidirect
20. #498×C₁₂₇	C₂²	semidirect
25. #492×C₁₂₇	C₂²	semidirect
26. #492×C₁₂₇	C₂²	semidirect
27. #487×C₁₂₇	C₂²	semidirect
28. #487×C₁₂₇	C₂²	semidirect
33. #486×C₁₂₇	C₂²	semidirect
45. #322×C₁₂₇	C₂²	semidirect
46. C₈²×C₁₂₇	C₂²	semidirect
134. M₁₆×C₁₂₇	SD₁₆	semidirect
221. #19623	C₁₂₇	direct (N × H)
228. #3208	C₂×C₁₂₇	semidirect
229. #3206	C₂×C₁₂₇	semidirect
230. #3167	C₂×C₁₂₇	semidirect
231. #3165	C₂×C₁₂₇	semidirect
233. C₂×#486	C₂×C₁₂₇	semidirect
234. #1715	C₂×C₁₂₇	semidirect
235. #1714	C₂×C₁₂₇	semidirect
236. #1708	C₂×C₁₂₇	semidirect
237. #1706	C₂×C₁₂₇	semidirect
238. C₈×M₁₆	C₂×C₁₂₇	semidirect
239. C₁₂₇	#19623	direct (N × H)
246. #500	C₂²×C₁₂₇	semidirect
247. #498	C₂²×C₁₂₇	semidirect
248. #498	C₂²×C₁₂₇	semidirect
253. #492	C₂²×C₁₂₇	semidirect
254. #492	C₂²×C₁₂₇	semidirect
255. #487	C₂²×C₁₂₇	semidirect
256. #487	C₂²×C₁₂₇	semidirect
261. #486	C₂²×C₁₂₇	semidirect
273. #322	C₂²×C₁₂₇	semidirect
274. C₈²	C₂²×C₁₂₇	semidirect
362. M₁₆	SD₁₆×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₁₂₇×#3208	2
C₁₂₇×#3206	2
C₁₂₇×#3167	2
C₁₂₇×#3165	2
C₁₂₇×#3151	2
C₂×#486×C₁₂₇	2
C₁₂₇×#1715	2
C₁₂₇×#1714	2
C₁₂₇×#1708	2
C₁₂₇×#1706	2
C₈×M₁₆×C₁₂₇	2
#19623	127

Subgroups — C127×#19623 #44583296

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#19623 #44583296