Subgroups — C₁₂₇×#12604 #44576277

Order 32512.

482 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#131×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#350×C₁₂₇	C₂	1	2	yes	maximal
3	C₄×M₃₂×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#346×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₂₇×#1379	C₂	1	2	yes	maximal, complemented
6	C₂×#131×C₁₂₇	C₄	1	1	yes	characteristic
7	C₂×#131×C₁₂₇	C₂²	1	1	yes	characteristic
8	C₄×#41×C₁₂₇	C₄	1	1	yes	characteristic
9	C₂×M₃₂×C₁₂₇	C₂²	1	2	yes	—
10	C₂²×C₁₆×C₁₂₇	C₂²	1	2	yes	—
11	#405×C₁₂₇	—	2	1	no	—
12	#405×C₁₂₇	—	2	1	no	—
13	#405×C₁₂₇	C₄	1	1	yes	characteristic
14	#405×C₁₂₇	C₂²	1	1	yes	characteristic
15	C₄×M₁₆×C₁₂₇	—	2	1	no	—
16	C₄×M₁₆×C₁₂₇	C₄	1	1	yes	characteristic
17	C₂×C₄×C₈×C₁₂₇	—	2	1	no	—
18	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
19	#350×C₁₂₇	C₄	1	8	yes	complemented
20	#346×C₁₂₇	—	2	1	no	—
21	#346×C₁₂₇	—	2	1	no	—
22	#346×C₁₂₇	—	2	1	no	—
23	C₄×C₁₆×C₁₂₇	—	2	1	no	—
24	C₂×#41×C₁₂₇	C₂×C₄	1	1	yes	characteristic
25	#131×C₁₂₇	—	2	2	no	—
26	#131×C₁₂₇	—	2	2	no	—
27	#131×C₁₂₇	C₈	1	4	yes	—
28	#131×C₁₂₇	C₂×C₄	1	4	yes	—
29	C₂×M₁₆×C₁₂₇	—	2	1	no	—
30	C₂×M₁₆×C₁₂₇	—	2	1	no	—
31	C₂×M₁₆×C₁₂₇	C₂×C₄	1	1	yes	characteristic
32	C₂×M₁₆×C₁₂₇	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₁₂₇	—	2	1	no	—
34	C₂²×C₈×C₁₂₇	—	2	1	no	—
35	C₂²×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
37	C₄×Q₈×C₁₂₇	C₈	1	2	yes	—
38	C₄×D₄×C₁₂₇	—	2	2	no	—
39	C₄×D₄×C₁₂₇	C₈	1	2	yes	—
40	#117×C₁₂₇	—	2	1	no	—
41	#117×C₁₂₇	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₁₂₇	—	2	1	no	—
43	C₂×C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
44	M₃₂×C₁₂₇	—	2	2	no	—
45	M₃₂×C₁₂₇	C₂×C₄	1	4	yes	complemented
46	C₂×C₁₆×C₁₂₇	—	2	2	no	—
47	C₂×C₁₆×C₁₂₇	—	2	2	no	—
48	C₂×C₁₆×C₁₂₇	—	2	2	no	—
49	C₂×C₁₆×C₁₂₇	C₂×C₄	1	4	yes	complemented
50	#97×C₁₂₇	—	2	1	no	—
51	#97×C₁₂₇	—	2	1	no	—
52	#97×C₁₂₇	—	2	1	no	—
53	C₄×C₈×C₁₂₇	—	2	1	no	—
54	C₄×C₈×C₁₂₇	—	2	1	no	—
55	C₄×C₈×C₁₂₇	—	2	1	no	—
56	C₄×C₈×C₁₂₇	D₄	1	1	yes	characteristic
57	C₄×C₈×C₁₂₇	D₄	1	1	yes	characteristic
58	C₄×C₈×C₁₂₇	D₄	1	1	yes	characteristic
59	C₄×C₈×C₁₂₇	D₄	1	1	yes	characteristic
60	#41×C₁₂₇	—	2	1	no	—
61	#41×C₁₂₇	—	2	1	no	—
62	#41×C₁₂₇	C₄²	1	2	yes	—
63	#41×C₁₂₇	C₂×C₈	1	2	yes	—
64	C₂×Q₈×C₁₂₇	C₂×C₈	1	1	yes	characteristic
65	C₂×D₄×C₁₂₇	—	2	1	no	—
66	C₂×D₄×C₁₂₇	C₂×C₈	1	1	yes	characteristic
67	C₂²×C₄×C₁₂₇	—	2	1	no	—
68	C₂²×C₄×C₁₂₇	—	2	2	no	—
69	C₂²×C₄×C₁₂₇	M₁₆	1	2	yes	—
70	C₂²×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
71	M₁₆×C₁₂₇	—	2	1	no	—
72	M₁₆×C₁₂₇	—	2	1	no	—
73	M₁₆×C₁₂₇	—	2	4	no	—
74	M₁₆×C₁₂₇	—	2	4	no	—
75	M₁₆×C₁₂₇	C₄²	1	2	yes	—
76	M₁₆×C₁₂₇	C₂×C₈	1	2	yes	—
77	C₂×C₈×C₁₂₇	—	2	1	no	—
78	C₂×C₈×C₁₂₇	—	2	1	no	—
79	C₂×C₈×C₁₂₇	—	2	4	no	—
80	C₂×C₈×C₁₂₇	—	2	4	no	—
81	C₂×C₈×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₈×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₈×C₁₂₇	C₂×C₈	1	2	yes	—
84	C₂×C₈×C₁₂₇	M₁₆	1	2	yes	—
85	C₂×C₈×C₁₂₇	C₂²×C₄	1	2	yes	—
86	C₂×C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
87	C₂×C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₁₂₇	#41	1	2	yes	—
89	C₄⋊C₄×C₁₂₇	—	2	2	no	—
90	C₄⋊C₄×C₁₂₇	C₂×C₈	1	2	yes	—
91	C₂²⋊C₄×C₁₂₇	—	2	1	no	—
92	C₂²⋊C₄×C₁₂₇	—	2	2	no	—
93	C₄²×C₁₂₇	—	2	2	no	—
94	C₄²×C₁₂₇	—	2	1	no	—
95	C₄²×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
96	C₄²×C₁₂₇	C₂²⋊C₄	1	1	yes	characteristic
97	C₄²×C₁₂₇	C₂×C₈	1	2	yes	—
98	C₁₆×C₁₂₇	—	2	4	no	—
99	C₁₆×C₁₂₇	—	2	4	no	—
100	C₂³×C₁₂₇	—	2	1	no	—
101	C₂³×C₁₂₇	C₂×M₁₆	1	1	yes	characteristic
102	Q₈×C₁₂₇	—	2	1	no	—
103	Q₈×C₁₂₇	C₄×C₈	1	2	yes	—
104	D₄×C₁₂₇	—	2	2	no	—
105	D₄×C₁₂₇	—	2	2	no	—
106	D₄×C₁₂₇	—	2	1	no	—
107	D₄×C₁₂₇	C₄×C₈	1	2	yes	—
108	C₂×C₄×C₁₂₇	—	2	1	no	—
109	C₂×C₄×C₁₂₇	—	2	2	no	—
110	C₂×C₄×C₁₂₇	—	2	2	no	—
111	C₂×C₄×C₁₂₇	—	2	2	no	—
112	C₂×C₄×C₁₂₇	—	4	2	no	—
113	C₂×C₄×C₁₂₇	—	4	4	no	—
114	C₂×C₄×C₁₂₇	C₂²⋊C₈	1	2	yes	—
115	C₂×C₄×C₁₂₇	C₂²⋊C₈	1	2	yes	—
116	C₂×C₄×C₁₂₇	C₂×C₄²	1	1	yes	characteristic
117	C₂×C₄×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₁₂₇	#117	1	1	yes	characteristic
119	C₂×C₄×C₁₂₇	C₂²×C₈	1	1	yes	characteristic
120	C₂×C₄×C₁₂₇	C₂²×C₈	1	1	yes	characteristic
121	C₂×C₄×C₁₂₇	C₂×M₁₆	1	1	yes	characteristic
122	C₈×C₁₂₇	—	2	2	no	—
123	C₈×C₁₂₇	—	2	2	no	—
124	C₈×C₁₂₇	C₂²⋊C₈	1	2	yes	—
125	C₈×C₁₂₇	C₂²⋊C₈	1	2	yes	—
126	C₈×C₁₂₇	C₄×D₄	1	2	yes	—
127	C₈×C₁₂₇	C₄×D₄	1	2	yes	—
128	C₂²×C₁₂₇	—	2	2	no	—
129	C₂²×C₁₂₇	—	2	2	no	—
130	C₂²×C₁₂₇	—	4	2	no	—
131	C₂²×C₁₂₇	C₄×M₁₆	1	1	yes	characteristic
132	C₂²×C₁₂₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
133	C₂²×C₁₂₇	#431	1	1	yes	characteristic
134	C₄×C₁₂₇	—	2	2	no	—
135	C₄×C₁₂₇	—	2	2	no	—
136	C₄×C₁₂₇	—	2	2	no	—
137	C₄×C₁₂₇	—	4	2	no	—
138	C₄×C₁₂₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
139	C₄×C₁₂₇	C₂×C₄×C₈	1	1	yes	characteristic
140	C₄×C₁₂₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
141	C₄×C₁₂₇	#431	1	1	yes	characteristic
142	#12604	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
143	C₂×C₁₂₇	—	2	1	no	—
144	C₂×C₁₂₇	—	2	1	no	—
145	C₂×C₁₂₇	—	4	2	no	—
146	C₂×C₁₂₇	C₄×C₂²⋊C₈	1	1	yes	characteristic
147	C₂×C₁₂₇	#2117	1	2	yes	—
148	C₄×#131	C₂×C₁₂₇	1	1	yes	characteristic
149	C₂×#350	C₂×C₁₂₇	1	2	yes	—
150	C₄×M₃₂	C₂×C₁₂₇	1	1	yes	characteristic, complemented
151	C₂×#346	C₂×C₁₂₇	1	1	yes	characteristic, complemented
152	#1379	C₂×C₁₂₇	1	2	yes	complemented
153	C₁₂₇	#12604	1	1	yes	characteristic, direct product factor, Sylow
154	C₂×#131	C₄×C₁₂₇	1	1	yes	characteristic
155	C₂×#131	C₂²×C₁₂₇	1	1	yes	characteristic
156	C₄×#41	C₄×C₁₂₇	1	1	yes	characteristic
157	C₂×M₃₂	C₂²×C₁₂₇	1	2	yes	—
158	C₂²×C₁₆	C₂²×C₁₂₇	1	2	yes	—
159	#405	—	2	1	no	—
160	#405	—	2	1	no	—
161	#405	C₄×C₁₂₇	1	1	yes	characteristic
162	#405	C₂²×C₁₂₇	1	1	yes	characteristic
163	C₄×M₁₆	—	2	1	no	—
164	C₄×M₁₆	C₄×C₁₂₇	1	1	yes	characteristic
165	C₂×C₄×C₈	—	2	1	no	—
166	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
167	#350	C₄×C₁₂₇	1	8	yes	complemented
168	#346	—	2	1	no	—
169	#346	—	2	1	no	—
170	#346	—	2	1	no	—
171	C₄×C₁₆	—	2	1	no	—
172	C₂×#41	C₂×C₄×C₁₂₇	1	1	yes	characteristic
173	#131	—	2	2	no	—
174	#131	—	2	2	no	—
175	#131	C₈×C₁₂₇	1	4	yes	—
176	#131	C₂×C₄×C₁₂₇	1	4	yes	—
177	C₂×M₁₆	—	2	1	no	—
178	C₂×M₁₆	—	2	1	no	—
179	C₂×M₁₆	C₂×C₄×C₁₂₇	1	1	yes	characteristic
180	C₂×M₁₆	C₂×C₄×C₁₂₇	1	1	yes	characteristic
181	C₂²×C₈	—	2	1	no	—
182	C₂²×C₈	—	2	1	no	—
183	C₂²×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
184	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
185	C₄×Q₈	C₈×C₁₂₇	1	2	yes	—
186	C₄×D₄	—	2	2	no	—
187	C₄×D₄	C₈×C₁₂₇	1	2	yes	—
188	#117	—	2	1	no	—
189	#117	C₂×C₄×C₁₂₇	1	1	yes	characteristic
190	C₂×C₄²	—	2	1	no	—
191	C₂×C₄²	C₂×C₄×C₁₂₇	1	1	yes	characteristic
192	M₃₂	—	2	2	no	—
193	M₃₂	C₂×C₄×C₁₂₇	1	4	yes	complemented
194	C₂×C₁₆	—	2	2	no	—
195	C₂×C₁₆	—	2	2	no	—
196	C₂×C₁₆	—	2	2	no	—
197	C₂×C₁₆	C₂×C₄×C₁₂₇	1	4	yes	complemented
198	#97	—	2	1	no	—
199	#97	—	2	1	no	—
200	#97	—	2	1	no	—
201	C₄×C₈	—	2	1	no	—
202	C₄×C₈	—	2	1	no	—
203	C₄×C₈	—	2	1	no	—
204	C₄×C₈	D₄×C₁₂₇	1	1	yes	characteristic
205	C₄×C₈	D₄×C₁₂₇	1	1	yes	characteristic
206	C₄×C₈	D₄×C₁₂₇	1	1	yes	characteristic
207	C₄×C₈	D₄×C₁₂₇	1	1	yes	characteristic
208	#41	—	2	1	no	—
209	#41	—	2	1	no	—
210	#41	C₄²×C₁₂₇	1	2	yes	—
211	#41	C₂×C₈×C₁₂₇	1	2	yes	—
212	C₂×Q₈	C₂×C₈×C₁₂₇	1	1	yes	characteristic
213	C₂×D₄	—	2	1	no	—
214	C₂×D₄	C₂×C₈×C₁₂₇	1	1	yes	characteristic
215	C₂²×C₄	—	2	1	no	—
216	C₂²×C₄	—	2	2	no	—
217	C₂²×C₄	M₁₆×C₁₂₇	1	2	yes	—
218	C₂²×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
219	M₁₆	—	2	1	no	—
220	M₁₆	—	2	1	no	—
221	M₁₆	—	2	4	no	—
222	M₁₆	—	2	4	no	—
223	M₁₆	C₄²×C₁₂₇	1	2	yes	—
224	M₁₆	C₂×C₈×C₁₂₇	1	2	yes	—
225	C₂×C₈	—	2	1	no	—
226	C₂×C₈	—	2	1	no	—
227	C₂×C₈	—	2	4	no	—
228	C₂×C₈	—	2	4	no	—
229	C₂×C₈	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
230	C₂×C₈	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
231	C₂×C₈	C₂×C₈×C₁₂₇	1	2	yes	—
232	C₂×C₈	M₁₆×C₁₂₇	1	2	yes	—
233	C₂×C₈	C₂²×C₄×C₁₂₇	1	2	yes	—
234	C₂×C₈	C₂×D₄×C₁₂₇	1	1	yes	characteristic
235	C₂×C₈	C₂×D₄×C₁₂₇	1	1	yes	characteristic
236	C₂×C₈	#41×C₁₂₇	1	2	yes	—
237	C₄⋊C₄	—	2	2	no	—
238	C₄⋊C₄	C₂×C₈×C₁₂₇	1	2	yes	—
239	C₂²⋊C₄	—	2	1	no	—
240	C₂²⋊C₄	—	2	2	no	—
241	C₄²	—	2	2	no	—
242	C₄²	—	2	1	no	—
243	C₄²	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
244	C₄²	C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
245	C₄²	C₂×C₈×C₁₂₇	1	2	yes	—
246	C₁₆	—	2	4	no	—
247	C₁₆	—	2	4	no	—
248	C₂³	—	2	1	no	—
249	C₂³	C₂×M₁₆×C₁₂₇	1	1	yes	characteristic
250	Q₈	—	2	1	no	—
251	Q₈	C₄×C₈×C₁₂₇	1	2	yes	—
252	D₄	—	2	2	no	—
253	D₄	—	2	2	no	—
254	D₄	—	2	1	no	—
255	D₄	C₄×C₈×C₁₂₇	1	2	yes	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	4	4	no	—
262	C₂×C₄	C₂²⋊C₈×C₁₂₇	1	2	yes	—
263	C₂×C₄	C₂²⋊C₈×C₁₂₇	1	2	yes	—
264	C₂×C₄	C₂×C₄²×C₁₂₇	1	1	yes	characteristic
265	C₂×C₄	C₂×C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
266	C₂×C₄	#117×C₁₂₇	1	1	yes	characteristic
267	C₂×C₄	C₂²×C₈×C₁₂₇	1	1	yes	characteristic
268	C₂×C₄	C₂²×C₈×C₁₂₇	1	1	yes	characteristic
269	C₂×C₄	C₂×M₁₆×C₁₂₇	1	1	yes	characteristic
270	C₈	—	2	2	no	—
271	C₈	—	2	2	no	—
272	C₈	C₂²⋊C₈×C₁₂₇	1	2	yes	—
273	C₈	C₂²⋊C₈×C₁₂₇	1	2	yes	—
274	C₈	C₄×D₄×C₁₂₇	1	2	yes	—
275	C₈	C₄×D₄×C₁₂₇	1	2	yes	—
276	C₂²	—	2	2	no	—
277	C₂²	—	2	2	no	—
278	C₂²	—	4	2	no	—
279	C₂²	C₄×M₁₆×C₁₂₇	1	1	yes	characteristic
280	C₂²	C₂×C₂²⋊C₈×C₁₂₇	1	1	yes	characteristic
281	C₂²	#431×C₁₂₇	1	1	yes	characteristic
282	C₄	—	2	2	no	—
283	C₄	—	2	2	no	—
284	C₄	—	2	2	no	—
285	C₄	—	4	2	no	—
286	C₄	C₄×C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
287	C₄	C₂×C₄×C₈×C₁₂₇	1	1	yes	characteristic
288	C₄	C₂×C₂²⋊C₈×C₁₂₇	1	1	yes	characteristic
289	C₄	#431×C₁₂₇	1	1	yes	characteristic
290	C₂	—	2	1	no	—
291	C₂	—	2	1	no	—
292	C₂	—	4	2	no	—
293	C₂	C₄×C₂²⋊C₈×C₁₂₇	1	1	yes	characteristic
294	C₂	C₁₂₇×#2117	1	2	yes	—

	subgroup	G/H
center	C₂×C₈×C₁₂₇	C₂×D₄
commutator	C₄	C₂×C₄×C₈×C₁₂₇
Frattini, cosocle	C₂²×C₈	C₂³×C₁₂₇
socle	C₂²×C₁₂₇	C₂×C₂²⋊C₈
prime radical	C₂×#41×C₁₂₇	C₂×C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₁₂₇ ◁ C₂×C₄×C₈×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₄×M₃₂×C₁₂₇	C₂	semidirect
4. C₂×#346×C₁₂₇	C₂	semidirect
5. C₁₂₇×#1379	C₂	semidirect
19. #350×C₁₂₇	C₄	semidirect
45. M₃₂×C₁₂₇	C₂×C₄	semidirect
49. C₂×C₁₆×C₁₂₇	C₂×C₄	semidirect
142. #12604	C₁₂₇	direct (N × H)
150. C₄×M₃₂	C₂×C₁₂₇	semidirect
151. C₂×#346	C₂×C₁₂₇	semidirect
152. #1379	C₂×C₁₂₇	semidirect
153. C₁₂₇	#12604	direct (N × H)
167. #350	C₄×C₁₂₇	semidirect
193. M₃₂	C₂×C₄×C₁₂₇	semidirect
197. C₂×C₁₆	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
C₄×#131×C₁₂₇	2
C₂×#350×C₁₂₇	2
C₄×M₃₂×C₁₂₇	2
C₂×#346×C₁₂₇	2
C₁₂₇×#1379	2
#12604	127

Subgroups — C127×#12604 #44576277

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#12604 #44576277