Subgroups — C₁₂₇×#12918 #44576591

Order 32512.

376 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₂₇×#3407	C₂	1	1	yes	maximal, characteristic
2	C₁₂₇×#2153	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₂₇×#2150	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₂₇×#2117	C₂	1	1	yes	maximal, characteristic
5	C₁₂₇×#1764	C₂	1	1	yes	maximal, characteristic
6	C₁₂₇×#1378	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₂₇×#1376	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#131×C₁₂₇	C₂²	1	1	yes	characteristic
9	#504×C₁₂₇	—	2	2	no	—
10	#504×C₁₂₇	—	2	2	no	—
11	C₂×M₃₂×C₁₂₇	—	2	2	no	—
12	C₂×M₃₂×C₁₂₇	C₂²	1	1	yes	characteristic
13	C₂×M₃₂×C₁₂₇	C₂²	1	1	yes	characteristic
14	C₂×M₃₂×C₁₂₇	C₂²	1	1	yes	characteristic
15	C₂×M₃₂×C₁₂₇	C₂²	1	1	yes	characteristic
16	C₂×#104×C₁₂₇	C₄	1	1	yes	characteristic
17	#416×C₁₂₇	C₄	1	1	yes	characteristic
18	#405×C₁₂₇	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
20	#364×C₁₂₇	—	2	2	no	—
21	C₄⋊C₁₆×C₁₂₇	—	2	2	no	—
22	#350×C₁₂₇	—	2	2	no	—
23	#341×C₁₂₇	C₄	1	1	yes	characteristic
24	#339×C₁₂₇	C₄	1	1	yes	characteristic
25	C₂×#41×C₁₂₇	C₂×C₄	1	1	yes	characteristic
26	#131×C₁₂₇	—	2	1	no	—
27	#131×C₁₂₇	—	2	1	no	—
28	#131×C₁₂₇	D₄	1	2	yes	—
29	#131×C₁₂₇	Q₈	1	2	yes	—
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	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
36	#117×C₁₂₇	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
38	M₃₂×C₁₂₇	—	2	4	no	—
39	M₃₂×C₁₂₇	—	2	1	no	—
40	M₃₂×C₁₂₇	—	2	1	no	—
41	M₃₂×C₁₂₇	—	2	1	no	—
42	M₃₂×C₁₂₇	—	2	1	no	—
43	M₃₂×C₁₂₇	—	2	1	no	—
44	M₃₂×C₁₂₇	—	2	1	no	—
45	M₃₂×C₁₂₇	—	2	1	no	—
46	M₃₂×C₁₂₇	D₄	1	2	yes	—
47	C₂×C₁₆×C₁₂₇	—	2	4	no	—
48	C₂×C₁₆×C₁₂₇	—	2	1	no	—
49	C₂×C₁₆×C₁₂₇	—	2	1	no	—
50	C₂×C₁₆×C₁₂₇	—	2	1	no	—
51	C₂×C₁₆×C₁₂₇	D₄	1	2	yes	—
52	#104×C₁₂₇	C₈	1	2	yes	—
53	#104×C₁₂₇	C₈	1	2	yes	—
54	#103×C₁₂₇	—	2	1	no	—
55	#102×C₁₂₇	—	2	1	no	—
56	#97×C₁₂₇	—	2	1	no	—
57	C₄×C₈×C₁₂₇	—	2	1	no	—
58	C₄×C₈×C₁₂₇	—	2	1	no	—
59	C₄×C₈×C₁₂₇	—	2	1	no	—
60	#41×C₁₂₇	—	2	1	no	—
61	#41×C₁₂₇	—	2	1	no	—
62	#41×C₁₂₇	C₄⋊C₄	1	2	yes	—
63	#41×C₁₂₇	C₂×C₈	1	2	yes	—
64	C₂×Q₈×C₁₂₇	M₁₆	1	1	yes	characteristic
65	C₂×D₄×C₁₂₇	—	2	1	no	—
66	C₂×D₄×C₁₂₇	M₁₆	1	1	yes	characteristic
67	C₂²×C₄×C₁₂₇	—	2	1	no	—
68	C₂²×C₄×C₁₂₇	—	2	1	no	—
69	C₂²×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
70	M₁₆×C₁₂₇	—	2	1	no	—
71	M₁₆×C₁₂₇	—	2	1	no	—
72	M₁₆×C₁₂₇	—	2	2	no	—
73	M₁₆×C₁₂₇	—	2	2	no	—
74	M₁₆×C₁₂₇	C₄⋊C₄	1	2	yes	—
75	M₁₆×C₁₂₇	C₂×C₈	1	2	yes	—
76	C₂×C₈×C₁₂₇	—	2	1	no	—
77	C₂×C₈×C₁₂₇	—	2	1	no	—
78	C₂×C₈×C₁₂₇	—	2	2	no	—
79	C₂×C₈×C₁₂₇	—	2	2	no	—
80	C₂×C₈×C₁₂₇	—	2	1	no	—
81	C₂×C₈×C₁₂₇	—	2	1	no	—
82	C₂×C₈×C₁₂₇	—	2	1	no	—
83	C₂×C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
84	C₂×C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
85	C₂×C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
87	C₂×C₈×C₁₂₇	#41	1	1	yes	characteristic
88	C₂×C₈×C₁₂₇	#41	1	1	yes	characteristic
89	C₄⋊C₄×C₁₂₇	—	2	1	no	—
90	C₂²⋊C₄×C₁₂₇	—	2	1	no	—
91	C₄²×C₁₂₇	—	2	1	no	—
92	C₄²×C₁₂₇	—	2	1	no	—
93	C₄²×C₁₂₇	C₂×C₈	1	1	yes	characteristic
94	C₄²×C₁₂₇	C₂×C₈	1	1	yes	characteristic
95	C₁₆×C₁₂₇	—	2	2	no	—
96	C₁₆×C₁₂₇	—	2	2	no	—
97	C₁₆×C₁₂₇	—	4	1	no	—
98	C₁₆×C₁₂₇	—	4	1	no	—
99	C₂³×C₁₂₇	—	2	1	no	—
100	C₂³×C₁₂₇	#131	1	1	yes	characteristic
101	Q₈×C₁₂₇	—	2	1	no	—
102	Q₈×C₁₂₇	C₄⋊C₈	1	2	yes	—
103	D₄×C₁₂₇	—	2	2	no	—
104	D₄×C₁₂₇	—	2	2	no	—
105	D₄×C₁₂₇	—	2	1	no	—
106	D₄×C₁₂₇	C₄⋊C₈	1	2	yes	—
107	C₂×C₄×C₁₂₇	—	2	1	no	—
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	1	no	—
112	C₂×C₄×C₁₂₇	—	4	2	no	—
113	C₂×C₄×C₁₂₇	—	4	1	no	—
114	C₂×C₄×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₁₂₇	C₄×D₄	1	1	yes	characteristic
116	C₂×C₄×C₁₂₇	C₄×D₄	1	1	yes	characteristic
117	C₂×C₄×C₁₂₇	C₂²×C₈	1	1	yes	characteristic
118	C₂×C₄×C₁₂₇	C₂×M₁₆	1	1	yes	characteristic
119	C₂×C₄×C₁₂₇	#131	1	1	yes	characteristic
120	C₈×C₁₂₇	—	2	2	no	—
121	C₈×C₁₂₇	—	2	2	no	—
122	C₈×C₁₂₇	—	4	1	no	—
123	C₈×C₁₂₇	#120	1	1	yes	characteristic
124	C₈×C₁₂₇	C₂²⋊Q₈	1	1	yes	characteristic
125	C₈×C₁₂₇	C₂²⋊Q₈	1	1	yes	characteristic
126	C₈×C₁₂₇	#123	1	1	yes	characteristic
127	C₂²×C₁₂₇	—	2	2	no	—
128	C₂²×C₁₂₇	—	4	1	no	—
129	C₂²×C₁₂₇	—	8	1	no	—
130	C₂²×C₁₂₇	#424	1	1	yes	characteristic
131	C₂²×C₁₂₇	C₈×D₄	1	1	yes	characteristic
132	C₂²×C₁₂₇	#435	1	1	yes	characteristic
133	C₄×C₁₂₇	—	2	2	no	—
134	C₄×C₁₂₇	—	2	2	no	—
135	C₄×C₁₂₇	—	4	1	no	—
136	C₄×C₁₂₇	—	8	1	no	—
137	C₄×C₁₂₇	#385	1	1	yes	characteristic
138	C₄×C₁₂₇	C₂×C₄⋊C₈	1	1	yes	characteristic
139	C₄×C₁₂₇	C₈×D₄	1	1	yes	characteristic
140	C₄×C₁₂₇	#435	1	1	yes	characteristic
141	#12918	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
142	C₂×C₁₂₇	—	2	1	no	—
143	C₂×C₁₂₇	—	2	1	no	—
144	C₂×C₁₂₇	—	2	1	no	—
145	C₂×C₁₂₇	—	4	2	no	—
146	C₂×C₁₂₇	#1850	1	1	yes	characteristic
147	#3407	C₂×C₁₂₇	1	1	yes	characteristic
148	#2153	C₂×C₁₂₇	1	1	yes	characteristic, complemented
149	#2150	C₂×C₁₂₇	1	1	yes	characteristic, complemented
150	#2117	C₂×C₁₂₇	1	1	yes	characteristic
151	#1764	C₂×C₁₂₇	1	1	yes	characteristic
152	#1378	C₂×C₁₂₇	1	1	yes	characteristic, complemented
153	#1376	C₂×C₁₂₇	1	1	yes	characteristic, complemented
154	C₁₂₇	#12918	1	1	yes	characteristic, direct product factor, Sylow
155	C₂×#131	C₂²×C₁₂₇	1	1	yes	characteristic
156	#504	—	2	2	no	—
157	#504	—	2	2	no	—
158	C₂×M₃₂	—	2	2	no	—
159	C₂×M₃₂	C₂²×C₁₂₇	1	1	yes	characteristic
160	C₂×M₃₂	C₂²×C₁₂₇	1	1	yes	characteristic
161	C₂×M₃₂	C₂²×C₁₂₇	1	1	yes	characteristic
162	C₂×M₃₂	C₂²×C₁₂₇	1	1	yes	characteristic
163	C₂×#104	C₄×C₁₂₇	1	1	yes	characteristic
164	#416	C₄×C₁₂₇	1	1	yes	characteristic
165	#405	C₂²×C₁₂₇	1	1	yes	characteristic
166	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
167	#364	—	2	2	no	—
168	C₄⋊C₁₆	—	2	2	no	—
169	#350	—	2	2	no	—
170	#341	C₄×C₁₂₇	1	1	yes	characteristic
171	#339	C₄×C₁₂₇	1	1	yes	characteristic
172	C₂×#41	C₂×C₄×C₁₂₇	1	1	yes	characteristic
173	#131	—	2	1	no	—
174	#131	—	2	1	no	—
175	#131	D₄×C₁₂₇	1	2	yes	—
176	#131	Q₈×C₁₂₇	1	2	yes	—
177	C₂×M₁₆	—	2	1	no	—
178	C₂×M₁₆	C₂×C₄×C₁₂₇	1	1	yes	characteristic
179	C₂×M₁₆	C₂×C₄×C₁₂₇	1	1	yes	characteristic
180	C₂²×C₈	—	2	1	no	—
181	C₂²×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
182	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
183	#117	C₂×C₄×C₁₂₇	1	1	yes	characteristic
184	C₂×C₄²	C₂×C₄×C₁₂₇	1	1	yes	characteristic
185	M₃₂	—	2	4	no	—
186	M₃₂	—	2	1	no	—
187	M₃₂	—	2	1	no	—
188	M₃₂	—	2	1	no	—
189	M₃₂	—	2	1	no	—
190	M₃₂	—	2	1	no	—
191	M₃₂	—	2	1	no	—
192	M₃₂	—	2	1	no	—
193	M₃₂	D₄×C₁₂₇	1	2	yes	—
194	C₂×C₁₆	—	2	4	no	—
195	C₂×C₁₆	—	2	1	no	—
196	C₂×C₁₆	—	2	1	no	—
197	C₂×C₁₆	—	2	1	no	—
198	C₂×C₁₆	D₄×C₁₂₇	1	2	yes	—
199	#104	C₈×C₁₂₇	1	2	yes	—
200	#104	C₈×C₁₂₇	1	2	yes	—
201	#103	—	2	1	no	—
202	#102	—	2	1	no	—
203	#97	—	2	1	no	—
204	C₄×C₈	—	2	1	no	—
205	C₄×C₈	—	2	1	no	—
206	C₄×C₈	—	2	1	no	—
207	#41	—	2	1	no	—
208	#41	—	2	1	no	—
209	#41	C₄⋊C₄×C₁₂₇	1	2	yes	—
210	#41	C₂×C₈×C₁₂₇	1	2	yes	—
211	C₂×Q₈	M₁₆×C₁₂₇	1	1	yes	characteristic
212	C₂×D₄	—	2	1	no	—
213	C₂×D₄	M₁₆×C₁₂₇	1	1	yes	characteristic
214	C₂²×C₄	—	2	1	no	—
215	C₂²×C₄	—	2	1	no	—
216	C₂²×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
217	M₁₆	—	2	1	no	—
218	M₁₆	—	2	1	no	—
219	M₁₆	—	2	2	no	—
220	M₁₆	—	2	2	no	—
221	M₁₆	C₄⋊C₄×C₁₂₇	1	2	yes	—
222	M₁₆	C₂×C₈×C₁₂₇	1	2	yes	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	2	1	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	1	no	—
228	C₂×C₈	—	2	1	no	—
229	C₂×C₈	—	2	1	no	—
230	C₂×C₈	C₂×D₄×C₁₂₇	1	1	yes	characteristic
231	C₂×C₈	C₂×D₄×C₁₂₇	1	1	yes	characteristic
232	C₂×C₈	C₂×D₄×C₁₂₇	1	1	yes	characteristic
233	C₂×C₈	C₂×Q₈×C₁₂₇	1	1	yes	characteristic
234	C₂×C₈	#41×C₁₂₇	1	1	yes	characteristic
235	C₂×C₈	#41×C₁₂₇	1	1	yes	characteristic
236	C₄⋊C₄	—	2	1	no	—
237	C₂²⋊C₄	—	2	1	no	—
238	C₄²	—	2	1	no	—
239	C₄²	—	2	1	no	—
240	C₄²	C₂×C₈×C₁₂₇	1	1	yes	characteristic
241	C₄²	C₂×C₈×C₁₂₇	1	1	yes	characteristic
242	C₁₆	—	2	2	no	—
243	C₁₆	—	2	2	no	—
244	C₁₆	—	4	1	no	—
245	C₁₆	—	4	1	no	—
246	C₂³	—	2	1	no	—
247	C₂³	#131×C₁₂₇	1	1	yes	characteristic
248	Q₈	—	2	1	no	—
249	Q₈	C₄⋊C₈×C₁₂₇	1	2	yes	—
250	D₄	—	2	2	no	—
251	D₄	—	2	2	no	—
252	D₄	—	2	1	no	—
253	D₄	C₄⋊C₈×C₁₂₇	1	2	yes	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	—	4	1	no	—
261	C₂×C₄	C₂×C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
262	C₂×C₄	C₄×D₄×C₁₂₇	1	1	yes	characteristic
263	C₂×C₄	C₄×D₄×C₁₂₇	1	1	yes	characteristic
264	C₂×C₄	C₂²×C₈×C₁₂₇	1	1	yes	characteristic
265	C₂×C₄	C₂×M₁₆×C₁₂₇	1	1	yes	characteristic
266	C₂×C₄	#131×C₁₂₇	1	1	yes	characteristic
267	C₈	—	2	2	no	—
268	C₈	—	2	2	no	—
269	C₈	—	4	1	no	—
270	C₈	#120×C₁₂₇	1	1	yes	characteristic
271	C₈	C₂²⋊Q₈×C₁₂₇	1	1	yes	characteristic
272	C₈	C₂²⋊Q₈×C₁₂₇	1	1	yes	characteristic
273	C₈	#123×C₁₂₇	1	1	yes	characteristic
274	C₂²	—	2	2	no	—
275	C₂²	—	4	1	no	—
276	C₂²	—	8	1	no	—
277	C₂²	#424×C₁₂₇	1	1	yes	characteristic
278	C₂²	C₈×D₄×C₁₂₇	1	1	yes	characteristic
279	C₂²	#435×C₁₂₇	1	1	yes	characteristic
280	C₄	—	2	2	no	—
281	C₄	—	2	2	no	—
282	C₄	—	4	1	no	—
283	C₄	—	8	1	no	—
284	C₄	#385×C₁₂₇	1	1	yes	characteristic
285	C₄	C₂×C₄⋊C₈×C₁₂₇	1	1	yes	characteristic
286	C₄	C₈×D₄×C₁₂₇	1	1	yes	characteristic
287	C₄	#435×C₁₂₇	1	1	yes	characteristic
288	C₂	—	2	1	no	—
289	C₂	—	2	1	no	—
290	C₂	—	2	1	no	—
291	C₂	—	4	2	no	—
292	C₂	C₁₂₇×#1850	1	1	yes	characteristic

	subgroup	G/H
center	C₈×C₁₂₇	#120
commutator	C₂×C₄	C₂²×C₈×C₁₂₇
Frattini, cosocle	C₂²×C₈	C₂³×C₁₂₇
socle	C₂×C₁₂₇	#1850
prime radical	C₂×#41×C₁₂₇	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₁₂₇ ◁ C₂²×C₈×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₂₇×#2153	C₂	semidirect
3. C₁₂₇×#2150	C₂	semidirect
6. C₁₂₇×#1378	C₂	semidirect
7. C₁₂₇×#1376	C₂	semidirect
141. #12918	C₁₂₇	direct (N × H)
148. #2153	C₂×C₁₂₇	semidirect
149. #2150	C₂×C₁₂₇	semidirect
152. #1378	C₂×C₁₂₇	semidirect
153. #1376	C₂×C₁₂₇	semidirect
154. C₁₂₇	#12918	direct (N × H)

Maximal subgroups

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

subgroup	index
C₁₂₇×#3407	2
C₁₂₇×#2153	2
C₁₂₇×#2150	2
C₁₂₇×#2117	2
C₁₂₇×#1764	2
C₁₂₇×#1378	2
C₁₂₇×#1376	2
#12918	127

Subgroups — C127×#12918 #44576591

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#12918 #44576591