Subgroups — C₁₂₇×#18055 #44581728

Order 32512.

660 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₂₇×#2990	C₂	1	2	yes	maximal, complemented
2	C₄×#135×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	C₁₂₇×#1591	C₂	1	2	yes	maximal, complemented
4	C₁₂₇×#1585	C₂	1	4	yes	maximal, complemented
5	C₈×D₈×C₁₂₇	C₂	1	2	yes	maximal, complemented
6	C₁₂₇×#1567	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₂₇×#1489	C₂	1	2	yes	maximal, complemented
8	C₈×M₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#135×C₁₂₇	C₄	1	1	yes	characteristic
10	C₄×#41×C₁₂₇	C₂²	1	2	yes	—
11	#446×C₁₂₇	C₂²	1	4	yes	—
12	C₄×Q₁₆×C₁₂₇	C₄	1	1	yes	characteristic
13	C₄×Q₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
14	C₄×SD₁₆×C₁₂₇	C₄	1	2	yes	—
15	C₄×SD₁₆×C₁₂₇	C₂²	1	2	yes	—
16	C₄×D₈×C₁₂₇	C₄	1	1	yes	characteristic
17	C₄×D₈×C₁₂₇	C₂²	1	1	yes	characteristic
18	#435×C₁₂₇	—	2	4	no	—
19	C₈×D₄×C₁₂₇	C₂²	1	4	yes	—
20	#432×C₁₂₇	—	2	2	no	—
21	#431×C₁₂₇	C₂²	1	1	yes	characteristic
22	#427×C₁₂₇	C₄	1	1	yes	characteristic
23	#423×C₁₂₇	C₂²	1	2	yes	—
24	C₂×C₄⋊C₈×C₁₂₇	—	2	2	no	—
25	#416×C₁₂₇	C₄	1	2	yes	—
26	C₄×M₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
28	#335×C₁₂₇	C₂²	1	2	yes	—
29	#335×C₁₂₇	C₂²	1	2	yes	—
30	Q₈⋊C₈×C₁₂₇	C₂²	1	4	yes	—
31	D₄⋊C₈×C₁₂₇	C₂²	1	4	yes	—
32	#322×C₁₂₇	C₂²	1	2	yes	—
33	C₈²×C₁₂₇	C₂²	1	2	yes	—
34	C₂×#41×C₁₂₇	C₂×C₄	1	2	yes	—
35	#135×C₁₂₇	—	2	4	no	—
36	C₂×Q₁₆×C₁₂₇	C₂×C₄	1	1	yes	characteristic
37	C₂×SD₁₆×C₁₂₇	C₂×C₄	1	2	yes	—
38	C₂×D₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
39	C₂×M₁₆×C₁₂₇	—	2	1	no	—
40	C₂×M₁₆×C₁₂₇	—	2	2	no	—
41	C₂²×C₈×C₁₂₇	—	2	4	no	—
42	C₂²×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
44	C₄×Q₈×C₁₂₇	C₂×C₄	1	2	yes	—
45	C₄×Q₈×C₁₂₇	C₂³	1	2	yes	—
46	C₄×D₄×C₁₂₇	—	2	2	no	—
47	C₄×D₄×C₁₂₇	—	2	2	no	—
48	C₄×D₄×C₁₂₇	C₂×C₄	1	2	yes	—
49	C₄×D₄×C₁₂₇	C₂³	1	2	yes	—
50	#117×C₁₂₇	—	2	2	no	—
51	#117×C₁₂₇	C₂×C₄	1	2	yes	—
52	C₂×C₄²×C₁₂₇	—	2	2	no	—
53	C₂×C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
54	#107×C₁₂₇	C₂×C₄	1	1	yes	characteristic
55	#107×C₁₂₇	C₂×C₄	1	1	yes	characteristic
56	#106×C₁₂₇	C₂×C₄	1	2	yes	—
57	C₄⋊C₈×C₁₂₇	—	2	4	no	—
58	C₄⋊C₈×C₁₂₇	—	2	2	no	—
59	C₄⋊C₈×C₁₂₇	D₄	1	1	yes	characteristic
60	C₄⋊C₈×C₁₂₇	D₄	1	1	yes	characteristic
61	C₄⋊C₈×C₁₂₇	C₂³	1	4	yes	—
62	#103×C₁₂₇	C₂×C₄	1	2	yes	—
63	#103×C₁₂₇	C₂×C₄	1	2	yes	—
64	#102×C₁₂₇	C₂×C₄	1	2	yes	—
65	#102×C₁₂₇	C₂×C₄	1	2	yes	—
66	C₂²⋊C₈×C₁₂₇	—	2	4	no	—
67	C₂²⋊C₈×C₁₂₇	D₄	1	2	yes	—
68	#97×C₁₂₇	—	2	4	no	—
69	#97×C₁₂₇	—	2	1	no	—
70	C₄×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
71	C₄×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
72	C₄×C₈×C₁₂₇	C₂³	1	2	yes	—
73	C₄×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
74	C₄×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
75	C₄×C₈×C₁₂₇	C₂³	1	2	yes	—
76	#41×C₁₂₇	—	2	4	no	—
77	#41×C₁₂₇	—	4	2	no	—
78	C₂×Q₈×C₁₂₇	C₂²×C₄	1	2	yes	—
79	C₂×D₄×C₁₂₇	—	2	2	no	—
80	C₂×D₄×C₁₂₇	C₂²×C₄	1	2	yes	—
81	C₂²×C₄×C₁₂₇	—	2	2	no	—
82	C₂²×C₄×C₁₂₇	—	2	2	no	—
83	C₂²×C₄×C₁₂₇	—	2	2	no	—
84	C₂²×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
85	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₁₂₇	#41	1	1	yes	characteristic
87	Q₁₆×C₁₂₇	—	2	2	no	—
88	SD₁₆×C₁₂₇	—	2	4	no	—
89	D₈×C₁₂₇	M₁₆	1	4	yes	complemented
90	M₁₆×C₁₂₇	—	2	4	no	—
91	M₁₆×C₁₂₇	—	8	2	no	—
92	C₂×C₈×C₁₂₇	—	2	2	no	—
93	C₂×C₈×C₁₂₇	—	2	2	no	—
94	C₂×C₈×C₁₂₇	—	2	4	no	—
95	C₂×C₈×C₁₂₇	—	4	1	no	—
96	C₂×C₈×C₁₂₇	—	8	4	no	—
97	C₂×C₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₈×C₁₂₇	C₂×D₄	1	2	yes	—
102	C₂×C₈×C₁₂₇	C₂×D₄	1	2	yes	—
103	C₄⋊C₄×C₁₂₇	—	2	2	no	—
104	C₄⋊C₄×C₁₂₇	—	2	2	no	—
105	C₄⋊C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
106	C₄⋊C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
107	C₂²⋊C₄×C₁₂₇	—	2	2	no	—
108	C₂²⋊C₄×C₁₂₇	—	2	2	no	—
109	C₂²⋊C₄×C₁₂₇	—	2	2	no	—
110	C₄²×C₁₂₇	—	2	2	no	—
111	C₄²×C₁₂₇	—	2	2	no	—
112	C₄²×C₁₂₇	—	2	2	no	—
113	C₄²×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
114	C₄²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
115	C₄²×C₁₂₇	#41	1	1	yes	characteristic
116	C₄²×C₁₂₇	C₂⁴	1	1	yes	characteristic
117	C₂³×C₁₂₇	—	2	2	no	—
118	C₂³×C₁₂₇	C₄×D₄	1	1	yes	characteristic
119	Q₈×C₁₂₇	—	2	2	no	—
120	Q₈×C₁₂₇	—	2	2	no	—
121	D₄×C₁₂₇	—	2	2	no	—
122	D₄×C₁₂₇	—	4	4	no	—
123	D₄×C₁₂₇	C₂×M₁₆	1	4	yes	—
124	C₂×C₄×C₁₂₇	—	2	1	no	—
125	C₂×C₄×C₁₂₇	—	2	1	no	—
126	C₂×C₄×C₁₂₇	—	2	1	no	—
127	C₂×C₄×C₁₂₇	—	2	1	no	—
128	C₂×C₄×C₁₂₇	—	2	2	no	—
129	C₂×C₄×C₁₂₇	—	2	2	no	—
130	C₂×C₄×C₁₂₇	—	2	2	no	—
131	C₂×C₄×C₁₂₇	—	2	2	no	—
132	C₂×C₄×C₁₂₇	—	4	4	no	—
133	C₂×C₄×C₁₂₇	—	4	2	no	—
134	C₂×C₄×C₁₂₇	—	4	2	no	—
135	C₂×C₄×C₁₂₇	—	4	4	no	—
136	C₂×C₄×C₁₂₇	C₄×D₄	1	1	yes	characteristic
137	C₂×C₄×C₁₂₇	C₄×D₄	1	1	yes	characteristic
138	C₂×C₄×C₁₂₇	C₄×D₄	1	1	yes	characteristic
139	C₂×C₄×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
140	C₂×C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
141	C₂×C₄×C₁₂₇	C₂×#41	1	1	yes	characteristic
142	C₈×C₁₂₇	—	2	4	no	—
143	C₈×C₁₂₇	—	2	1	no	—
144	C₈×C₁₂₇	—	2	1	no	—
145	C₈×C₁₂₇	—	2	1	no	—
146	C₈×C₁₂₇	—	2	2	no	—
147	C₈×C₁₂₇	—	2	2	no	—
148	C₈×C₁₂₇	—	4	4	no	—
149	C₈×C₁₂₇	C₂×M₁₆	1	2	yes	—
150	C₂²×C₁₂₇	—	2	1	no	—
151	C₂²×C₁₂₇	—	2	4	no	—
152	C₂²×C₁₂₇	—	2	1	no	—
153	C₂²×C₁₂₇	—	2	1	no	—
154	C₂²×C₁₂₇	—	4	2	no	—
155	C₂²×C₁₂₇	—	4	2	no	—
156	C₂²×C₁₂₇	C₂×C₄×D₄	1	1	yes	characteristic
157	C₄×C₁₂₇	—	2	1	no	—
158	C₄×C₁₂₇	—	2	1	no	—
159	C₄×C₁₂₇	—	2	1	no	—
160	C₄×C₁₂₇	—	4	2	no	—
161	C₄×C₁₂₇	—	4	2	no	—
162	C₄×C₁₂₇	—	4	2	no	—
163	C₄×C₁₂₇	#443	1	2	yes	—
164	C₄×C₁₂₇	C₂²×M₁₆	1	1	yes	characteristic
165	C₄×C₁₂₇	#568	1	1	yes	characteristic
166	C₄×C₁₂₇	#576	1	1	yes	characteristic
167	C₄×C₁₂₇	#578	1	1	yes	characteristic
168	#18055	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
169	C₂×C₁₂₇	—	4	1	no	—
170	C₂×C₁₂₇	—	4	4	no	—
171	C₂×C₁₂₇	D₄×M₁₆	1	1	yes	characteristic
172	C₂×C₁₂₇	#2951	1	1	yes	characteristic
173	C₂×C₁₂₇	C₂×#443	1	1	yes	characteristic
174	#2990	C₂×C₁₂₇	1	2	yes	complemented
175	C₄×#135	C₂×C₁₂₇	1	1	yes	characteristic
176	#1591	C₂×C₁₂₇	1	2	yes	complemented
177	#1585	C₂×C₁₂₇	1	4	yes	complemented
178	C₈×D₈	C₂×C₁₂₇	1	2	yes	complemented
179	#1567	C₂×C₁₂₇	1	1	yes	characteristic, complemented
180	#1489	C₂×C₁₂₇	1	2	yes	complemented
181	C₈×M₁₆	C₂×C₁₂₇	1	1	yes	characteristic, complemented
182	C₁₂₇	#18055	1	1	yes	characteristic, direct product factor, Sylow
183	C₂×#135	C₄×C₁₂₇	1	1	yes	characteristic
184	C₄×#41	C₂²×C₁₂₇	1	2	yes	—
185	#446	C₂²×C₁₂₇	1	4	yes	—
186	C₄×Q₁₆	C₄×C₁₂₇	1	1	yes	characteristic
187	C₄×Q₁₆	C₂²×C₁₂₇	1	1	yes	characteristic
188	C₄×SD₁₆	C₄×C₁₂₇	1	2	yes	—
189	C₄×SD₁₆	C₂²×C₁₂₇	1	2	yes	—
190	C₄×D₈	C₄×C₁₂₇	1	1	yes	characteristic
191	C₄×D₈	C₂²×C₁₂₇	1	1	yes	characteristic
192	#435	—	2	4	no	—
193	C₈×D₄	C₂²×C₁₂₇	1	4	yes	—
194	#432	—	2	2	no	—
195	#431	C₂²×C₁₂₇	1	1	yes	characteristic
196	#427	C₄×C₁₂₇	1	1	yes	characteristic
197	#423	C₂²×C₁₂₇	1	2	yes	—
198	C₂×C₄⋊C₈	—	2	2	no	—
199	#416	C₄×C₁₂₇	1	2	yes	—
200	C₄×M₁₆	C₂²×C₁₂₇	1	1	yes	characteristic
201	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
202	#335	C₂²×C₁₂₇	1	2	yes	—
203	#335	C₂²×C₁₂₇	1	2	yes	—
204	Q₈⋊C₈	C₂²×C₁₂₇	1	4	yes	—
205	D₄⋊C₈	C₂²×C₁₂₇	1	4	yes	—
206	#322	C₂²×C₁₂₇	1	2	yes	—
207	C₈²	C₂²×C₁₂₇	1	2	yes	—
208	C₂×#41	C₂×C₄×C₁₂₇	1	2	yes	—
209	#135	—	2	4	no	—
210	C₂×Q₁₆	C₂×C₄×C₁₂₇	1	1	yes	characteristic
211	C₂×SD₁₆	C₂×C₄×C₁₂₇	1	2	yes	—
212	C₂×D₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
213	C₂×M₁₆	—	2	1	no	—
214	C₂×M₁₆	—	2	2	no	—
215	C₂²×C₈	—	2	4	no	—
216	C₂²×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
217	C₂²×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
218	C₄×Q₈	C₂×C₄×C₁₂₇	1	2	yes	—
219	C₄×Q₈	C₂³×C₁₂₇	1	2	yes	—
220	C₄×D₄	—	2	2	no	—
221	C₄×D₄	—	2	2	no	—
222	C₄×D₄	C₂×C₄×C₁₂₇	1	2	yes	—
223	C₄×D₄	C₂³×C₁₂₇	1	2	yes	—
224	#117	—	2	2	no	—
225	#117	C₂×C₄×C₁₂₇	1	2	yes	—
226	C₂×C₄²	—	2	2	no	—
227	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
228	#107	C₂×C₄×C₁₂₇	1	1	yes	characteristic
229	#107	C₂×C₄×C₁₂₇	1	1	yes	characteristic
230	#106	C₂×C₄×C₁₂₇	1	2	yes	—
231	C₄⋊C₈	—	2	4	no	—
232	C₄⋊C₈	—	2	2	no	—
233	C₄⋊C₈	D₄×C₁₂₇	1	1	yes	characteristic
234	C₄⋊C₈	D₄×C₁₂₇	1	1	yes	characteristic
235	C₄⋊C₈	C₂³×C₁₂₇	1	4	yes	—
236	#103	C₂×C₄×C₁₂₇	1	2	yes	—
237	#103	C₂×C₄×C₁₂₇	1	2	yes	—
238	#102	C₂×C₄×C₁₂₇	1	2	yes	—
239	#102	C₂×C₄×C₁₂₇	1	2	yes	—
240	C₂²⋊C₈	—	2	4	no	—
241	C₂²⋊C₈	D₄×C₁₂₇	1	2	yes	—
242	#97	—	2	4	no	—
243	#97	—	2	1	no	—
244	C₄×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
245	C₄×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
246	C₄×C₈	C₂³×C₁₂₇	1	2	yes	—
247	C₄×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
248	C₄×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
249	C₄×C₈	C₂³×C₁₂₇	1	2	yes	—
250	#41	—	2	4	no	—
251	#41	—	4	2	no	—
252	C₂×Q₈	C₂²×C₄×C₁₂₇	1	2	yes	—
253	C₂×D₄	—	2	2	no	—
254	C₂×D₄	C₂²×C₄×C₁₂₇	1	2	yes	—
255	C₂²×C₄	—	2	2	no	—
256	C₂²×C₄	—	2	2	no	—
257	C₂²×C₄	—	2	2	no	—
258	C₂²×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
259	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
260	C₂²×C₄	#41×C₁₂₇	1	1	yes	characteristic
261	Q₁₆	—	2	2	no	—
262	SD₁₆	—	2	4	no	—
263	D₈	M₁₆×C₁₂₇	1	4	yes	complemented
264	M₁₆	—	2	4	no	—
265	M₁₆	—	8	2	no	—
266	C₂×C₈	—	2	2	no	—
267	C₂×C₈	—	2	2	no	—
268	C₂×C₈	—	2	4	no	—
269	C₂×C₈	—	4	1	no	—
270	C₂×C₈	—	8	4	no	—
271	C₂×C₈	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
272	C₂×C₈	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
273	C₂×C₈	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
274	C₂×C₈	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
275	C₂×C₈	C₂×D₄×C₁₂₇	1	2	yes	—
276	C₂×C₈	C₂×D₄×C₁₂₇	1	2	yes	—
277	C₄⋊C₄	—	2	2	no	—
278	C₄⋊C₄	—	2	2	no	—
279	C₄⋊C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
280	C₄⋊C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
281	C₂²⋊C₄	—	2	2	no	—
282	C₂²⋊C₄	—	2	2	no	—
283	C₂²⋊C₄	—	2	2	no	—
284	C₄²	—	2	2	no	—
285	C₄²	—	2	2	no	—
286	C₄²	—	2	2	no	—
287	C₄²	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
288	C₄²	C₂×D₄×C₁₂₇	1	1	yes	characteristic
289	C₄²	#41×C₁₂₇	1	1	yes	characteristic
290	C₄²	C₂⁴×C₁₂₇	1	1	yes	characteristic
291	C₂³	—	2	2	no	—
292	C₂³	C₄×D₄×C₁₂₇	1	1	yes	characteristic
293	Q₈	—	2	2	no	—
294	Q₈	—	2	2	no	—
295	D₄	—	2	2	no	—
296	D₄	—	4	4	no	—
297	D₄	C₂×M₁₆×C₁₂₇	1	4	yes	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	4	4	no	—
307	C₂×C₄	—	4	2	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	—	4	4	no	—
310	C₂×C₄	C₄×D₄×C₁₂₇	1	1	yes	characteristic
311	C₂×C₄	C₄×D₄×C₁₂₇	1	1	yes	characteristic
312	C₂×C₄	C₄×D₄×C₁₂₇	1	1	yes	characteristic
313	C₂×C₄	C₂³×C₄×C₁₂₇	1	1	yes	characteristic
314	C₂×C₄	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
315	C₂×C₄	C₂×#41×C₁₂₇	1	1	yes	characteristic
316	C₈	—	2	4	no	—
317	C₈	—	2	1	no	—
318	C₈	—	2	1	no	—
319	C₈	—	2	1	no	—
320	C₈	—	2	2	no	—
321	C₈	—	2	2	no	—
322	C₈	—	4	4	no	—
323	C₈	C₂×M₁₆×C₁₂₇	1	2	yes	—
324	C₂²	—	2	1	no	—
325	C₂²	—	2	4	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	2	1	no	—
328	C₂²	—	4	2	no	—
329	C₂²	—	4	2	no	—
330	C₂²	C₂×C₄×D₄×C₁₂₇	1	1	yes	characteristic
331	C₄	—	2	1	no	—
332	C₄	—	2	1	no	—
333	C₄	—	2	1	no	—
334	C₄	—	4	2	no	—
335	C₄	—	4	2	no	—
336	C₄	—	4	2	no	—
337	C₄	#443×C₁₂₇	1	2	yes	—
338	C₄	C₂²×M₁₆×C₁₂₇	1	1	yes	characteristic
339	C₄	#568×C₁₂₇	1	1	yes	characteristic
340	C₄	#576×C₁₂₇	1	1	yes	characteristic
341	C₄	#578×C₁₂₇	1	1	yes	characteristic
342	C₂	—	4	1	no	—
343	C₂	—	4	4	no	—
344	C₂	D₄×M₁₆×C₁₂₇	1	1	yes	characteristic
345	C₂	C₁₂₇×#2951	1	1	yes	characteristic
346	C₂	C₂×#443×C₁₂₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₂₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₁₂₇
Frattini, cosocle	C₄²	C₂⁴×C₁₂₇
socle	C₂²×C₁₂₇	C₂×C₄×D₄
prime radical	C₂×#135×C₁₂₇	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₂₇ ◁ #431×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₂₇×#2990	C₂	semidirect
3. C₁₂₇×#1591	C₂	semidirect
4. C₁₂₇×#1585	C₂	semidirect
5. C₈×D₈×C₁₂₇	C₂	semidirect
6. C₁₂₇×#1567	C₂	semidirect
7. C₁₂₇×#1489	C₂	semidirect
8. C₈×M₁₆×C₁₂₇	C₂	semidirect
89. D₈×C₁₂₇	M₁₆	semidirect
168. #18055	C₁₂₇	direct (N × H)
174. #2990	C₂×C₁₂₇	semidirect
176. #1591	C₂×C₁₂₇	semidirect
177. #1585	C₂×C₁₂₇	semidirect
178. C₈×D₈	C₂×C₁₂₇	semidirect
179. #1567	C₂×C₁₂₇	semidirect
180. #1489	C₂×C₁₂₇	semidirect
181. C₈×M₁₆	C₂×C₁₂₇	semidirect
182. C₁₂₇	#18055	direct (N × H)
263. D₈	M₁₆×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₁₂₇×#2990	2
C₄×#135×C₁₂₇	2
C₁₂₇×#1591	2
C₁₂₇×#1585	2
C₈×D₈×C₁₂₇	2
C₁₂₇×#1567	2
C₁₂₇×#1489	2
C₈×M₁₆×C₁₂₇	2
#18055	127

Subgroups — C127×#18055 #44581728

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#18055 #44581728