Subgroups — C₁₂₇×#19621 #44583294

Order 32512.

900 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₂₇×#3227	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₂₇×#3208	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×#492×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×#1706	C₂	1	2	yes	maximal, complemented
10	C₁₂₇×#1705	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	C₂²×SD₁₆×C₁₂₇	—	2	2	no	—
14	#533×C₁₂₇	C₂²	1	1	yes	characteristic
15	#532×C₁₂₇	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Q₈×C₁₂₇	C₂²	1	1	yes	characteristic
17	C₂×#127×C₁₂₇	C₂²	1	1	yes	characteristic
18	#501×C₁₂₇	—	2	2	no	—
19	#500×C₁₂₇	C₂²	1	2	yes	complemented
20	#498×C₁₂₇	C₂²	1	2	yes	complemented
21	#498×C₁₂₇	C₂²	1	1	yes	characteristic, complemented
22	#498×C₁₂₇	C₂²	1	2	yes	complemented
23	#496×C₁₂₇	—	2	2	no	—
24	#493×C₁₂₇	C₂²	1	2	yes	—
25	#492×C₁₂₇	—	2	2	no	—
26	#492×C₁₂₇	C₂²	1	2	yes	—
27	#492×C₁₂₇	C₂²	1	2	yes	—
28	#492×C₁₂₇	C₂²	1	2	yes	—
29	#492×C₁₂₇	C₂²	1	2	yes	—
30	#492×C₁₂₇	C₂²	1	1	yes	characteristic, complemented
31	#487×C₁₂₇	C₂²	1	1	yes	characteristic, complemented
32	#486×C₁₂₇	C₂²	1	2	yes	—
33	#486×C₁₂₇	C₂²	1	1	yes	characteristic, complemented
34	#486×C₁₂₇	C₂²	1	2	yes	complemented
35	#481×C₁₂₇	—	2	2	no	—
36	#476×C₁₂₇	—	2	2	no	—
37	#465×C₁₂₇	—	2	2	no	—
38	#460×C₁₂₇	—	2	2	no	—
39	#450×C₁₂₇	—	2	2	no	—
40	C₄×SD₁₆×C₁₂₇	—	2	2	no	—
41	#431×C₁₂₇	C₂²	1	1	yes	characteristic
42	#428×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	4	no	—
55	C₂×SD₁₆×C₁₂₇	—	2	4	no	—
56	C₂×SD₁₆×C₁₂₇	—	2	2	no	—
57	C₂×SD₁₆×C₁₂₇	—	2	4	no	—
58	C₂×SD₁₆×C₁₂₇	—	4	8	no	—
59	C₂×D₈×C₁₂₇	—	2	4	no	—
60	C₂×M₁₆×C₁₂₇	D₄	1	2	yes	—
61	C₂²×C₈×C₁₂₇	D₄	1	2	yes	—
62	C₄⋊Q₈×C₁₂₇	—	2	1	no	—
63	C₄⋊Q₈×C₁₂₇	—	2	1	no	—
64	C₄⋊Q₈×C₁₂₇	—	2	1	no	—
65	C₄⋊Q₈×C₁₂₇	C₂³	1	1	yes	characteristic
66	C₄⋊Q₈×C₁₂₇	C₂³	1	1	yes	characteristic
67	C₄⋊Q₈×C₁₂₇	C₂³	1	2	yes	—
68	C₄⋊Q₈×C₁₂₇	C₂³	1	1	yes	characteristic
69	#127×C₁₂₇	—	2	1	no	—
70	#127×C₁₂₇	—	2	1	no	—
71	#127×C₁₂₇	—	2	1	no	—
72	#127×C₁₂₇	C₂³	1	2	yes	—
73	#127×C₁₂₇	C₂³	1	1	yes	characteristic
74	#125×C₁₂₇	—	2	1	no	—
75	#124×C₁₂₇	—	2	1	no	—
76	C₂²⋊Q₈×C₁₂₇	—	2	1	no	—
77	C₂²⋊Q₈×C₁₂₇	—	2	1	no	—
78	#121×C₁₂₇	—	2	1	no	—
79	#121×C₁₂₇	—	2	1	no	—
80	C₄×Q₈×C₁₂₇	—	2	1	no	—
81	C₄×Q₈×C₁₂₇	—	2	1	no	—
82	C₄×D₄×C₁₂₇	—	2	1	no	—
83	C₄×D₄×C₁₂₇	—	2	1	no	—
84	#117×C₁₂₇	—	2	1	no	—
85	C₂×C₄⋊C₄×C₁₂₇	—	2	1	no	—
86	C₂×C₄⋊C₄×C₁₂₇	—	2	1	no	—
87	C₂×C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
88	#107×C₁₂₇	—	2	4	no	—
89	#106×C₁₂₇	—	2	4	no	—
90	#106×C₁₂₇	—	2	2	no	—
91	#106×C₁₂₇	—	2	4	no	—
92	C₄⋊C₈×C₁₂₇	D₄	1	2	yes	—
93	#103×C₁₂₇	—	2	2	no	—
94	#102×C₁₂₇	—	2	2	no	—
95	#102×C₁₂₇	—	2	4	no	—
96	#102×C₁₂₇	—	2	4	no	—
97	C₂²⋊C₈×C₁₂₇	D₄	1	2	yes	—
98	#97×C₁₂₇	D₄	1	2	yes	—
99	C₄×C₈×C₁₂₇	D₄	1	2	yes	—
100	C₄×C₈×C₁₂₇	C₂³	1	2	yes	—
101	C₄×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
102	C₄×C₈×C₁₂₇	C₂³	1	2	yes	—
103	C₄×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
104	C₂⁴×C₁₂₇	—	4	1	no	—
105	#41×C₁₂₇	—	4	4	no	—
106	C₂×Q₈×C₁₂₇	—	2	1	no	—
107	C₂×Q₈×C₁₂₇	—	2	1	no	—
108	C₂×Q₈×C₁₂₇	—	2	1	no	—
109	C₂×Q₈×C₁₂₇	—	2	2	no	—
110	C₂×Q₈×C₁₂₇	—	4	4	no	—
111	C₂×Q₈×C₁₂₇	—	4	1	no	—
112	C₂×D₄×C₁₂₇	—	2	1	no	—
113	C₂×D₄×C₁₂₇	—	2	1	no	—
114	C₂×D₄×C₁₂₇	—	2	2	no	—
115	C₂×D₄×C₁₂₇	—	2	1	no	—
116	C₂×D₄×C₁₂₇	—	2	2	no	—
117	C₂×D₄×C₁₂₇	—	2	2	no	—
118	C₂×D₄×C₁₂₇	—	4	4	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	1	no	—
123	C₂×D₄×C₁₂₇	—	4	4	no	—
124	C₂×D₄×C₁₂₇	—	8	2	no	—
125	C₂²×C₄×C₁₂₇	—	4	1	no	—
126	C₂²×C₄×C₁₂₇	—	4	1	no	—
127	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
128	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
129	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
130	SD₁₆×C₁₂₇	—	4	8	no	—
131	SD₁₆×C₁₂₇	—	4	8	no	—
132	SD₁₆×C₁₂₇	—	4	4	no	—
133	SD₁₆×C₁₂₇	—	4	8	no	—
134	SD₁₆×C₁₂₇	—	8	4	no	—
135	D₈×C₁₂₇	—	4	8	no	—
136	M₁₆×C₁₂₇	SD₁₆	1	8	yes	complemented
137	C₂×C₈×C₁₂₇	—	2	4	no	—
138	C₂×C₈×C₁₂₇	C₂×D₄	1	2	yes	—
139	C₂×C₈×C₁₂₇	C₂×D₄	1	4	yes	—
140	C₂×C₈×C₁₂₇	C₂×D₄	1	4	yes	—
141	C₂×C₈×C₁₂₇	C₂×D₄	1	2	yes	—
142	C₄⋊C₄×C₁₂₇	—	2	1	no	—
143	C₄⋊C₄×C₁₂₇	—	2	2	no	—
144	C₄⋊C₄×C₁₂₇	—	2	1	no	—
145	C₄⋊C₄×C₁₂₇	—	2	1	no	—
146	C₄⋊C₄×C₁₂₇	—	2	1	no	—
147	C₄⋊C₄×C₁₂₇	—	2	1	no	—
148	C₄⋊C₄×C₁₂₇	—	2	2	no	—
149	C₄⋊C₄×C₁₂₇	—	2	1	no	—
150	C₄⋊C₄×C₁₂₇	—	4	1	no	—
151	C₄⋊C₄×C₁₂₇	—	4	1	no	—
152	C₄⋊C₄×C₁₂₇	—	4	1	no	—
153	C₄⋊C₄×C₁₂₇	—	4	1	no	—
154	C₂²⋊C₄×C₁₂₇	—	4	1	no	—
155	C₂²⋊C₄×C₁₂₇	—	4	1	no	—
156	C₂²⋊C₄×C₁₂₇	—	4	1	no	—
157	C₄²×C₁₂₇	—	4	1	no	—
158	C₄²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
159	C₄²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
160	C₄²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
161	C₄²×C₁₂₇	C₂⁴	1	1	yes	characteristic
162	C₂³×C₁₂₇	—	4	1	no	—
163	C₂³×C₁₂₇	—	4	2	no	—
164	C₂³×C₁₂₇	—	8	2	no	—
165	C₂³×C₁₂₇	—	8	2	no	—
166	C₂³×C₁₂₇	—	8	2	no	—
167	C₂³×C₁₂₇	#127	1	1	yes	characteristic
168	Q₈×C₁₂₇	—	4	2	no	—
169	Q₈×C₁₂₇	—	4	4	no	—
170	Q₈×C₁₂₇	—	4	2	no	—
171	Q₈×C₁₂₇	—	4	2	no	—
172	Q₈×C₁₂₇	—	8	2	no	—
173	D₄×C₁₂₇	—	4	2	no	—
174	D₄×C₁₂₇	—	4	4	no	—
175	D₄×C₁₂₇	—	4	4	no	—
176	D₄×C₁₂₇	—	4	2	no	—
177	D₄×C₁₂₇	—	4	4	no	—
178	D₄×C₁₂₇	—	4	2	no	—
179	D₄×C₁₂₇	—	8	2	no	—
180	D₄×C₁₂₇	—	8	2	no	—
181	D₄×C₁₂₇	—	8	2	no	—
182	D₄×C₁₂₇	—	8	2	no	—
183	C₂×C₄×C₁₂₇	—	2	2	no	—
184	C₂×C₄×C₁₂₇	—	2	2	no	—
185	C₂×C₄×C₁₂₇	—	4	1	no	—
186	C₂×C₄×C₁₂₇	—	4	1	no	—
187	C₂×C₄×C₁₂₇	—	4	1	no	—
188	C₂×C₄×C₁₂₇	—	4	2	no	—
189	C₂×C₄×C₁₂₇	—	8	2	no	—
190	C₂×C₄×C₁₂₇	—	8	2	no	—
191	C₂×C₄×C₁₂₇	#127	1	1	yes	characteristic
192	C₂×C₄×C₁₂₇	#127	1	1	yes	characteristic
193	C₂×C₄×C₁₂₇	#127	1	1	yes	characteristic
194	C₂×C₄×C₁₂₇	C₂×SD₁₆	1	4	yes	—
195	C₂×C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
196	C₂×C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
197	C₂×C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
198	C₈×C₁₂₇	—	2	2	no	—
199	C₈×C₁₂₇	—	2	4	no	—
200	C₈×C₁₂₇	C₂×SD₁₆	1	8	yes	complemented
201	C₈×C₁₂₇	AGL(1,ℤ/8)	1	4	yes	complemented
202	C₂²×C₁₂₇	—	2	1	no	—
203	C₂²×C₁₂₇	—	2	1	no	—
204	C₂²×C₁₂₇	—	8	1	no	—
205	C₂²×C₁₂₇	—	8	2	no	—
206	C₂²×C₁₂₇	—	8	2	no	—
207	C₂²×C₁₂₇	—	8	1	no	—
208	C₂²×C₁₂₇	—	8	2	no	—
209	C₂²×C₁₂₇	—	8	1	no	—
210	C₂²×C₁₂₇	—	16	4	no	—
211	C₂²×C₁₂₇	#492	1	2	yes	—
212	C₂²×C₁₂₇	C₂×#127	1	1	yes	characteristic
213	C₄×C₁₂₇	—	2	1	no	—
214	C₄×C₁₂₇	—	2	1	no	—
215	C₄×C₁₂₇	—	8	1	no	—
216	C₄×C₁₂₇	—	8	2	no	—
217	C₄×C₁₂₇	—	8	1	no	—
218	C₄×C₁₂₇	—	8	1	no	—
219	C₄×C₁₂₇	#492	1	2	yes	—
220	C₄×C₁₂₇	C₂²×SD₁₆	1	2	yes	—
221	C₄×C₁₂₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
222	C₄×C₁₂₇	#577	1	2	yes	—
223	#19621	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
224	C₂×C₁₂₇	—	2	2	no	—
225	C₂×C₁₂₇	—	16	1	no	—
226	C₂×C₁₂₇	—	16	2	no	—
227	C₂×C₁₂₇	C₂×#492	1	1	yes	characteristic
228	C₂×C₁₂₇	#3151	1	1	yes	characteristic
229	C₂×C₁₂₇	#3154	1	1	yes	characteristic
230	#3227	C₂×C₁₂₇	1	1	yes	characteristic, complemented
231	#3208	C₂×C₁₂₇	1	1	yes	characteristic, complemented
232	#3167	C₂×C₁₂₇	1	1	yes	characteristic, complemented
233	#3165	C₂×C₁₂₇	1	1	yes	characteristic, complemented
234	#3151	C₂×C₁₂₇	1	1	yes	characteristic
235	C₂×#492	C₂×C₁₂₇	1	1	yes	characteristic, complemented
236	#1715	C₂×C₁₂₇	1	2	yes	complemented
237	#1714	C₂×C₁₂₇	1	2	yes	complemented
238	#1706	C₂×C₁₂₇	1	2	yes	complemented
239	#1705	C₂×C₁₂₇	1	2	yes	complemented
240	C₈×M₁₆	C₂×C₁₂₇	1	1	yes	characteristic, complemented
241	C₁₂₇	#19621	1	1	yes	characteristic, direct product factor, Sylow
242	C₂×AGL(1,ℤ/8)	—	2	2	no	—
243	C₂²×SD₁₆	—	2	2	no	—
244	#533	C₂²×C₁₂₇	1	1	yes	characteristic
245	#532	C₂²×C₁₂₇	1	1	yes	characteristic
246	C₂×C₄⋊Q₈	C₂²×C₁₂₇	1	1	yes	characteristic
247	C₂×#127	C₂²×C₁₂₇	1	1	yes	characteristic
248	#501	—	2	2	no	—
249	#500	C₂²×C₁₂₇	1	2	yes	complemented
250	#498	C₂²×C₁₂₇	1	2	yes	complemented
251	#498	C₂²×C₁₂₇	1	1	yes	characteristic, complemented
252	#498	C₂²×C₁₂₇	1	2	yes	complemented
253	#496	—	2	2	no	—
254	#493	C₂²×C₁₂₇	1	2	yes	—
255	#492	—	2	2	no	—
256	#492	C₂²×C₁₂₇	1	2	yes	—
257	#492	C₂²×C₁₂₇	1	2	yes	—
258	#492	C₂²×C₁₂₇	1	2	yes	—
259	#492	C₂²×C₁₂₇	1	2	yes	—
260	#492	C₂²×C₁₂₇	1	1	yes	characteristic, complemented
261	#487	C₂²×C₁₂₇	1	1	yes	characteristic, complemented
262	#486	C₂²×C₁₂₇	1	2	yes	—
263	#486	C₂²×C₁₂₇	1	1	yes	characteristic, complemented
264	#486	C₂²×C₁₂₇	1	2	yes	complemented
265	#481	—	2	2	no	—
266	#476	—	2	2	no	—
267	#465	—	2	2	no	—
268	#460	—	2	2	no	—
269	#450	—	2	2	no	—
270	C₄×SD₁₆	—	2	2	no	—
271	#431	C₂²×C₁₂₇	1	1	yes	characteristic
272	#428	—	2	2	no	—
273	C₄×M₁₆	C₂²×C₁₂₇	1	1	yes	characteristic
274	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
275	#322	C₂²×C₁₂₇	1	2	yes	complemented
276	C₈²	C₂²×C₁₂₇	1	2	yes	complemented
277	C₂×#41	—	2	1	no	—
278	C₂²×Q₈	—	2	1	no	—
279	C₂²×D₄	—	2	1	no	—
280	C₂²×D₄	—	2	1	no	—
281	C₂²×D₄	—	2	1	no	—
282	AGL(1,ℤ/8)	—	4	8	no	—
283	C₂×SD₁₆	—	2	4	no	—
284	C₂×SD₁₆	—	2	4	no	—
285	C₂×SD₁₆	—	2	4	no	—
286	C₂×SD₁₆	—	2	2	no	—
287	C₂×SD₁₆	—	2	4	no	—
288	C₂×SD₁₆	—	4	8	no	—
289	C₂×D₈	—	2	4	no	—
290	C₂×M₁₆	D₄×C₁₂₇	1	2	yes	—
291	C₂²×C₈	D₄×C₁₂₇	1	2	yes	—
292	C₄⋊Q₈	—	2	1	no	—
293	C₄⋊Q₈	—	2	1	no	—
294	C₄⋊Q₈	—	2	1	no	—
295	C₄⋊Q₈	C₂³×C₁₂₇	1	2	yes	—
296	C₄⋊Q₈	C₂³×C₁₂₇	1	1	yes	characteristic
297	C₄⋊Q₈	C₂³×C₁₂₇	1	1	yes	characteristic
298	C₄⋊Q₈	C₂³×C₁₂₇	1	1	yes	characteristic
299	#127	—	2	1	no	—
300	#127	—	2	1	no	—
301	#127	—	2	1	no	—
302	#127	C₂³×C₁₂₇	1	2	yes	—
303	#127	C₂³×C₁₂₇	1	1	yes	characteristic
304	#125	—	2	1	no	—
305	#124	—	2	1	no	—
306	C₂²⋊Q₈	—	2	1	no	—
307	C₂²⋊Q₈	—	2	1	no	—
308	#121	—	2	1	no	—
309	#121	—	2	1	no	—
310	C₄×Q₈	—	2	1	no	—
311	C₄×Q₈	—	2	1	no	—
312	C₄×D₄	—	2	1	no	—
313	C₄×D₄	—	2	1	no	—
314	#117	—	2	1	no	—
315	C₂×C₄⋊C₄	—	2	1	no	—
316	C₂×C₄⋊C₄	—	2	1	no	—
317	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
318	#107	—	2	4	no	—
319	#106	—	2	4	no	—
320	#106	—	2	2	no	—
321	#106	—	2	4	no	—
322	C₄⋊C₈	D₄×C₁₂₇	1	2	yes	—
323	#103	—	2	2	no	—
324	#102	—	2	2	no	—
325	#102	—	2	4	no	—
326	#102	—	2	4	no	—
327	C₂²⋊C₈	D₄×C₁₂₇	1	2	yes	—
328	#97	D₄×C₁₂₇	1	2	yes	—
329	C₄×C₈	D₄×C₁₂₇	1	2	yes	—
330	C₄×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
331	C₄×C₈	C₂³×C₁₂₇	1	2	yes	—
332	C₄×C₈	C₂³×C₁₂₇	1	2	yes	—
333	C₄×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
334	C₂⁴	—	4	1	no	—
335	#41	—	4	4	no	—
336	C₂×Q₈	—	2	1	no	—
337	C₂×Q₈	—	2	1	no	—
338	C₂×Q₈	—	2	1	no	—
339	C₂×Q₈	—	2	2	no	—
340	C₂×Q₈	—	4	4	no	—
341	C₂×Q₈	—	4	1	no	—
342	C₂×D₄	—	2	1	no	—
343	C₂×D₄	—	2	1	no	—
344	C₂×D₄	—	2	2	no	—
345	C₂×D₄	—	2	1	no	—
346	C₂×D₄	—	2	2	no	—
347	C₂×D₄	—	2	2	no	—
348	C₂×D₄	—	4	4	no	—
349	C₂×D₄	—	4	1	no	—
350	C₂×D₄	—	4	1	no	—
351	C₂×D₄	—	4	1	no	—
352	C₂×D₄	—	4	1	no	—
353	C₂×D₄	—	4	4	no	—
354	C₂×D₄	—	8	2	no	—
355	C₂²×C₄	—	4	1	no	—
356	C₂²×C₄	—	4	1	no	—
357	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
358	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
359	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
360	SD₁₆	—	4	8	no	—
361	SD₁₆	—	4	8	no	—
362	SD₁₆	—	4	4	no	—
363	SD₁₆	—	4	8	no	—
364	SD₁₆	—	8	4	no	—
365	D₈	—	4	8	no	—
366	M₁₆	SD₁₆×C₁₂₇	1	8	yes	complemented
367	C₂×C₈	—	2	4	no	—
368	C₂×C₈	C₂×D₄×C₁₂₇	1	2	yes	—
369	C₂×C₈	C₂×D₄×C₁₂₇	1	4	yes	—
370	C₂×C₈	C₂×D₄×C₁₂₇	1	4	yes	—
371	C₂×C₈	C₂×D₄×C₁₂₇	1	2	yes	—
372	C₄⋊C₄	—	2	1	no	—
373	C₄⋊C₄	—	2	2	no	—
374	C₄⋊C₄	—	2	1	no	—
375	C₄⋊C₄	—	2	1	no	—
376	C₄⋊C₄	—	2	1	no	—
377	C₄⋊C₄	—	2	1	no	—
378	C₄⋊C₄	—	2	2	no	—
379	C₄⋊C₄	—	2	1	no	—
380	C₄⋊C₄	—	4	1	no	—
381	C₄⋊C₄	—	4	1	no	—
382	C₄⋊C₄	—	4	1	no	—
383	C₄⋊C₄	—	4	1	no	—
384	C₂²⋊C₄	—	4	1	no	—
385	C₂²⋊C₄	—	4	1	no	—
386	C₂²⋊C₄	—	4	1	no	—
387	C₄²	—	4	1	no	—
388	C₄²	C₂×D₄×C₁₂₇	1	1	yes	characteristic
389	C₄²	C₂×D₄×C₁₂₇	1	1	yes	characteristic
390	C₄²	C₂×D₄×C₁₂₇	1	1	yes	characteristic
391	C₄²	C₂⁴×C₁₂₇	1	1	yes	characteristic
392	C₂³	—	4	1	no	—
393	C₂³	—	4	2	no	—
394	C₂³	—	8	2	no	—
395	C₂³	—	8	2	no	—
396	C₂³	—	8	2	no	—
397	C₂³	#127×C₁₂₇	1	1	yes	characteristic
398	Q₈	—	4	2	no	—
399	Q₈	—	4	4	no	—
400	Q₈	—	4	2	no	—
401	Q₈	—	4	2	no	—
402	Q₈	—	8	2	no	—
403	D₄	—	4	2	no	—
404	D₄	—	4	4	no	—
405	D₄	—	4	4	no	—
406	D₄	—	4	2	no	—
407	D₄	—	4	4	no	—
408	D₄	—	4	2	no	—
409	D₄	—	8	2	no	—
410	D₄	—	8	2	no	—
411	D₄	—	8	2	no	—
412	D₄	—	8	2	no	—
413	C₂×C₄	—	2	2	no	—
414	C₂×C₄	—	2	2	no	—
415	C₂×C₄	—	4	1	no	—
416	C₂×C₄	—	4	1	no	—
417	C₂×C₄	—	4	1	no	—
418	C₂×C₄	—	4	2	no	—
419	C₂×C₄	—	8	2	no	—
420	C₂×C₄	—	8	2	no	—
421	C₂×C₄	#127×C₁₂₇	1	1	yes	characteristic
422	C₂×C₄	#127×C₁₂₇	1	1	yes	characteristic
423	C₂×C₄	#127×C₁₂₇	1	1	yes	characteristic
424	C₂×C₄	C₂×SD₁₆×C₁₂₇	1	4	yes	—
425	C₂×C₄	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
426	C₂×C₄	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
427	C₂×C₄	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
428	C₈	—	2	2	no	—
429	C₈	—	2	4	no	—
430	C₈	C₂×SD₁₆×C₁₂₇	1	8	yes	complemented
431	C₈	AGL(1,ℤ/8)×C₁₂₇	1	4	yes	complemented
432	C₂²	—	2	1	no	—
433	C₂²	—	2	1	no	—
434	C₂²	—	8	1	no	—
435	C₂²	—	8	2	no	—
436	C₂²	—	8	2	no	—
437	C₂²	—	8	1	no	—
438	C₂²	—	8	2	no	—
439	C₂²	—	8	1	no	—
440	C₂²	—	16	4	no	—
441	C₂²	#492×C₁₂₇	1	2	yes	—
442	C₂²	C₂×#127×C₁₂₇	1	1	yes	characteristic
443	C₄	—	2	1	no	—
444	C₄	—	2	1	no	—
445	C₄	—	8	1	no	—
446	C₄	—	8	2	no	—
447	C₄	—	8	1	no	—
448	C₄	—	8	1	no	—
449	C₄	#492×C₁₂₇	1	2	yes	—
450	C₄	C₂²×SD₁₆×C₁₂₇	1	2	yes	—
451	C₄	C₂×AGL(1,ℤ/8)×C₁₂₇	1	2	yes	—
452	C₄	#577×C₁₂₇	1	2	yes	—
453	C₂	—	2	2	no	—
454	C₂	—	16	2	no	—
455	C₂	—	16	1	no	—
456	C₂	C₂×#492×C₁₂₇	1	1	yes	characteristic
457	C₂	C₁₂₇×#3151	1	1	yes	characteristic
458	C₂	C₁₂₇×#3154	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×#3227	C₂	semidirect
2. C₁₂₇×#3208	C₂	semidirect
3. C₁₂₇×#3167	C₂	semidirect
4. C₁₂₇×#3165	C₂	semidirect
6. C₂×#492×C₁₂₇	C₂	semidirect
7. C₁₂₇×#1715	C₂	semidirect
8. C₁₂₇×#1714	C₂	semidirect
9. C₁₂₇×#1706	C₂	semidirect
10. C₁₂₇×#1705	C₂	semidirect
11. C₈×M₁₆×C₁₂₇	C₂	semidirect
19. #500×C₁₂₇	C₂²	semidirect
20. #498×C₁₂₇	C₂²	semidirect
21. #498×C₁₂₇	C₂²	semidirect
22. #498×C₁₂₇	C₂²	semidirect
30. #492×C₁₂₇	C₂²	semidirect
31. #487×C₁₂₇	C₂²	semidirect
33. #486×C₁₂₇	C₂²	semidirect
34. #486×C₁₂₇	C₂²	semidirect
45. #322×C₁₂₇	C₂²	semidirect
46. C₈²×C₁₂₇	C₂²	semidirect
136. M₁₆×C₁₂₇	SD₁₆	semidirect
200. C₈×C₁₂₇	C₂×SD₁₆	semidirect
201. C₈×C₁₂₇	AGL(1,ℤ/8)	semidirect
223. #19621	C₁₂₇	direct (N × H)
230. #3227	C₂×C₁₂₇	semidirect
231. #3208	C₂×C₁₂₇	semidirect
232. #3167	C₂×C₁₂₇	semidirect
233. #3165	C₂×C₁₂₇	semidirect
235. C₂×#492	C₂×C₁₂₇	semidirect
236. #1715	C₂×C₁₂₇	semidirect
237. #1714	C₂×C₁₂₇	semidirect
238. #1706	C₂×C₁₂₇	semidirect
239. #1705	C₂×C₁₂₇	semidirect
240. C₈×M₁₆	C₂×C₁₂₇	semidirect
241. C₁₂₇	#19621	direct (N × H)
249. #500	C₂²×C₁₂₇	semidirect
250. #498	C₂²×C₁₂₇	semidirect
251. #498	C₂²×C₁₂₇	semidirect
252. #498	C₂²×C₁₂₇	semidirect
260. #492	C₂²×C₁₂₇	semidirect
261. #487	C₂²×C₁₂₇	semidirect
263. #486	C₂²×C₁₂₇	semidirect
264. #486	C₂²×C₁₂₇	semidirect
275. #322	C₂²×C₁₂₇	semidirect
276. C₈²	C₂²×C₁₂₇	semidirect
366. M₁₆	SD₁₆×C₁₂₇	semidirect
430. C₈	C₂×SD₁₆×C₁₂₇	semidirect
431. C₈	AGL(1,ℤ/8)×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₁₂₇×#3227	2
C₁₂₇×#3208	2
C₁₂₇×#3167	2
C₁₂₇×#3165	2
C₁₂₇×#3151	2
C₂×#492×C₁₂₇	2
C₁₂₇×#1715	2
C₁₂₇×#1714	2
C₁₂₇×#1706	2
C₁₂₇×#1705	2
C₈×M₁₆×C₁₂₇	2
#19621	127

Subgroups — C127×#19621 #44583294

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#19621 #44583294