Subgroups — C₁₂₇×#12618 #44576291

Order 32512.

382 conjugacy classes of subgroups.

The subgroup lattice has 286 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×#2117	C₂	1	1	yes	maximal, characteristic
3	C₁₂₇×#2110	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#347×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₂₇×#1765	C₂	1	1	yes	maximal, characteristic
6	C₁₂₇×#1378	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₂₇×#1377	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	C₄	1	4	yes	complemented
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×#97×C₁₂₇	C₂²	1	1	yes	characteristic
20	#350×C₁₂₇	C₄	1	4	yes	complemented
21	#347×C₁₂₇	—	2	2	no	—
22	#347×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	C₂×C₄	1	2	yes	—
29	#131×C₁₂₇	C₂×C₄	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	1	no	—
39	M₃₂×C₁₂₇	—	2	1	no	—
40	M₃₂×C₁₂₇	—	2	1	no	—
41	M₃₂×C₁₂₇	—	2	1	no	—
42	M₃₂×C₁₂₇	—	2	4	no	—
43	M₃₂×C₁₂₇	C₂×C₄	1	2	yes	—
44	M₃₂×C₁₂₇	C₂×C₄	1	2	yes	—
45	M₃₂×C₁₂₇	D₄	1	2	yes	—
46	M₃₂×C₁₂₇	D₄	1	2	yes	—
47	C₂×C₁₆×C₁₂₇	—	2	1	no	—
48	C₂×C₁₆×C₁₂₇	—	2	1	no	—
49	C₂×C₁₆×C₁₂₇	—	2	4	no	—
50	C₂×C₁₆×C₁₂₇	C₂×C₄	1	2	yes	—
51	C₂×C₁₆×C₁₂₇	C₂×C₄	1	2	yes	—
52	#104×C₁₂₇	—	2	2	no	—
53	#103×C₁₂₇	—	2	1	no	—
54	#102×C₁₂₇	—	2	1	no	—
55	#97×C₁₂₇	—	2	1	no	—
56	#97×C₁₂₇	—	2	1	no	—
57	#97×C₁₂₇	—	2	1	no	—
58	C₄×C₈×C₁₂₇	—	2	1	no	—
59	#41×C₁₂₇	—	2	1	no	—
60	#41×C₁₂₇	—	2	1	no	—
61	#41×C₁₂₇	C₄²	1	2	yes	—
62	#41×C₁₂₇	M₁₆	1	2	yes	—
63	C₂×Q₈×C₁₂₇	M₁₆	1	1	yes	characteristic
64	C₂×D₄×C₁₂₇	—	2	1	no	—
65	C₂×D₄×C₁₂₇	M₁₆	1	1	yes	characteristic
66	C₂²×C₄×C₁₂₇	—	2	1	no	—
67	C₂²×C₄×C₁₂₇	—	2	1	no	—
68	C₂²×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
69	M₁₆×C₁₂₇	—	2	1	no	—
70	M₁₆×C₁₂₇	—	2	2	no	—
71	M₁₆×C₁₂₇	—	2	2	no	—
72	M₁₆×C₁₂₇	—	2	1	no	—
73	M₁₆×C₁₂₇	—	2	1	no	—
74	M₁₆×C₁₂₇	C₄²	1	2	yes	—
75	C₂×C₈×C₁₂₇	—	2	1	no	—
76	C₂×C₈×C₁₂₇	—	2	2	no	—
77	C₂×C₈×C₁₂₇	—	2	2	no	—
78	C₂×C₈×C₁₂₇	—	2	1	no	—
79	C₂×C₈×C₁₂₇	—	2	1	no	—
80	C₂×C₈×C₁₂₇	—	2	1	no	—
81	C₂×C₈×C₁₂₇	—	2	1	no	—
82	C₂×C₈×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₈×C₁₂₇	C₂²×C₄	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	#41	1	1	yes	characteristic
87	C₂×C₈×C₁₂₇	#41	1	1	yes	characteristic
88	C₄⋊C₄×C₁₂₇	—	2	1	no	—
89	C₂²⋊C₄×C₁₂₇	—	2	1	no	—
90	C₄²×C₁₂₇	—	2	1	no	—
91	C₄²×C₁₂₇	—	2	1	no	—
92	C₄²×C₁₂₇	—	2	1	no	—
93	C₁₆×C₁₂₇	—	2	2	no	—
94	C₁₆×C₁₂₇	—	2	2	no	—
95	C₁₆×C₁₂₇	—	2	2	no	—
96	C₁₆×C₁₂₇	C₂²⋊C₄	1	4	yes	—
97	C₂³×C₁₂₇	—	2	1	no	—
98	C₂³×C₁₂₇	#131	1	1	yes	characteristic
99	Q₈×C₁₂₇	—	2	1	no	—
100	Q₈×C₁₂₇	#97	1	2	yes	—
101	D₄×C₁₂₇	—	2	2	no	—
102	D₄×C₁₂₇	—	2	2	no	—
103	D₄×C₁₂₇	—	2	1	no	—
104	D₄×C₁₂₇	#97	1	2	yes	—
105	C₂×C₄×C₁₂₇	—	2	1	no	—
106	C₂×C₄×C₁₂₇	—	2	2	no	—
107	C₂×C₄×C₁₂₇	—	2	2	no	—
108	C₂×C₄×C₁₂₇	—	4	1	no	—
109	C₂×C₄×C₁₂₇	—	4	1	no	—
110	C₂×C₄×C₁₂₇	—	8	1	no	—
111	C₂×C₄×C₁₂₇	C₂×C₄²	1	1	yes	characteristic
112	C₂×C₄×C₁₂₇	C₄×D₄	1	1	yes	characteristic
113	C₂×C₄×C₁₂₇	C₄×D₄	1	1	yes	characteristic
114	C₂×C₄×C₁₂₇	C₂×M₁₆	1	1	yes	characteristic
115	C₂×C₄×C₁₂₇	C₂×M₁₆	1	1	yes	characteristic
116	C₂×C₄×C₁₂₇	#131	1	1	yes	characteristic
117	C₈×C₁₂₇	—	2	2	no	—
118	C₈×C₁₂₇	—	4	1	no	—
119	C₈×C₁₂₇	—	4	1	no	—
120	C₈×C₁₂₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₈×C₁₂₇	#117	1	1	yes	characteristic
122	C₈×C₁₂₇	C₄×D₄	1	1	yes	characteristic
123	C₈×C₁₂₇	C₄×D₄	1	1	yes	characteristic
124	C₂²×C₁₂₇	—	2	2	no	—
125	C₂²×C₁₂₇	—	4	1	no	—
126	C₂²×C₁₂₇	—	8	1	no	—
127	C₂²×C₁₂₇	#405	1	1	yes	characteristic
128	C₂²×C₁₂₇	#435	1	1	yes	characteristic
129	C₂²×C₁₂₇	#435	1	1	yes	characteristic
130	C₄×C₁₂₇	—	2	2	no	—
131	C₄×C₁₂₇	—	8	1	no	—
132	C₄×C₁₂₇	—	8	1	no	—
133	C₄×C₁₂₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
134	C₄×C₁₂₇	C₂×#97	1	1	yes	characteristic
135	C₄×C₁₂₇	#435	1	1	yes	characteristic
136	C₄×C₁₂₇	#435	1	1	yes	characteristic
137	#12618	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
138	C₂×C₁₂₇	—	2	1	no	—
139	C₂×C₁₂₇	—	2	1	no	—
140	C₂×C₁₂₇	—	2	1	no	—
141	C₂×C₁₂₇	—	4	2	no	—
142	C₂×C₁₂₇	#1752	1	1	yes	characteristic
143	#3407	C₂×C₁₂₇	1	1	yes	characteristic
144	#2117	C₂×C₁₂₇	1	1	yes	characteristic
145	#2110	C₂×C₁₂₇	1	1	yes	characteristic, complemented
146	C₂×#347	C₂×C₁₂₇	1	1	yes	characteristic, complemented
147	#1765	C₂×C₁₂₇	1	1	yes	characteristic
148	#1378	C₂×C₁₂₇	1	1	yes	characteristic, complemented
149	#1377	C₂×C₁₂₇	1	1	yes	characteristic, complemented
150	C₁₂₇	#12618	1	1	yes	characteristic, direct product factor, Sylow
151	C₂×#131	C₂²×C₁₂₇	1	1	yes	characteristic
152	#504	—	2	2	no	—
153	#504	C₄×C₁₂₇	1	4	yes	complemented
154	C₂×M₃₂	—	2	2	no	—
155	C₂×M₃₂	C₂²×C₁₂₇	1	1	yes	characteristic
156	C₂×M₃₂	C₂²×C₁₂₇	1	1	yes	characteristic
157	C₂×M₃₂	C₂²×C₁₂₇	1	1	yes	characteristic
158	C₂×M₃₂	C₂²×C₁₂₇	1	1	yes	characteristic
159	C₂×#104	C₄×C₁₂₇	1	1	yes	characteristic
160	#416	C₄×C₁₂₇	1	1	yes	characteristic
161	#405	C₂²×C₁₂₇	1	1	yes	characteristic
162	C₂×#97	C₂²×C₁₂₇	1	1	yes	characteristic
163	#350	C₄×C₁₂₇	1	4	yes	complemented
164	#347	—	2	2	no	—
165	#347	—	2	2	no	—
166	#341	C₄×C₁₂₇	1	1	yes	characteristic
167	#339	C₄×C₁₂₇	1	1	yes	characteristic
168	C₂×#41	C₂×C₄×C₁₂₇	1	1	yes	characteristic
169	#131	—	2	1	no	—
170	#131	—	2	1	no	—
171	#131	C₂×C₄×C₁₂₇	1	2	yes	—
172	#131	C₂×C₄×C₁₂₇	1	2	yes	—
173	C₂×M₁₆	—	2	1	no	—
174	C₂×M₁₆	C₂×C₄×C₁₂₇	1	1	yes	characteristic
175	C₂×M₁₆	C₂×C₄×C₁₂₇	1	1	yes	characteristic
176	C₂²×C₈	—	2	1	no	—
177	C₂²×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
178	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
179	#117	C₂×C₄×C₁₂₇	1	1	yes	characteristic
180	C₂×C₄²	C₂×C₄×C₁₂₇	1	1	yes	characteristic
181	M₃₂	—	2	1	no	—
182	M₃₂	—	2	1	no	—
183	M₃₂	—	2	1	no	—
184	M₃₂	—	2	1	no	—
185	M₃₂	—	2	4	no	—
186	M₃₂	C₂×C₄×C₁₂₇	1	2	yes	—
187	M₃₂	C₂×C₄×C₁₂₇	1	2	yes	—
188	M₃₂	D₄×C₁₂₇	1	2	yes	—
189	M₃₂	D₄×C₁₂₇	1	2	yes	—
190	C₂×C₁₆	—	2	1	no	—
191	C₂×C₁₆	—	2	1	no	—
192	C₂×C₁₆	—	2	4	no	—
193	C₂×C₁₆	C₂×C₄×C₁₂₇	1	2	yes	—
194	C₂×C₁₆	C₂×C₄×C₁₂₇	1	2	yes	—
195	#104	—	2	2	no	—
196	#103	—	2	1	no	—
197	#102	—	2	1	no	—
198	#97	—	2	1	no	—
199	#97	—	2	1	no	—
200	#97	—	2	1	no	—
201	C₄×C₈	—	2	1	no	—
202	#41	—	2	1	no	—
203	#41	—	2	1	no	—
204	#41	C₄²×C₁₂₇	1	2	yes	—
205	#41	M₁₆×C₁₂₇	1	2	yes	—
206	C₂×Q₈	M₁₆×C₁₂₇	1	1	yes	characteristic
207	C₂×D₄	—	2	1	no	—
208	C₂×D₄	M₁₆×C₁₂₇	1	1	yes	characteristic
209	C₂²×C₄	—	2	1	no	—
210	C₂²×C₄	—	2	1	no	—
211	C₂²×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
212	M₁₆	—	2	1	no	—
213	M₁₆	—	2	2	no	—
214	M₁₆	—	2	2	no	—
215	M₁₆	—	2	1	no	—
216	M₁₆	—	2	1	no	—
217	M₁₆	C₄²×C₁₂₇	1	2	yes	—
218	C₂×C₈	—	2	1	no	—
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	—	2	1	no	—
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	2	1	no	—
225	C₂×C₈	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
226	C₂×C₈	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
227	C₂×C₈	C₂×D₄×C₁₂₇	1	1	yes	characteristic
228	C₂×C₈	C₂×D₄×C₁₂₇	1	1	yes	characteristic
229	C₂×C₈	#41×C₁₂₇	1	1	yes	characteristic
230	C₂×C₈	#41×C₁₂₇	1	1	yes	characteristic
231	C₄⋊C₄	—	2	1	no	—
232	C₂²⋊C₄	—	2	1	no	—
233	C₄²	—	2	1	no	—
234	C₄²	—	2	1	no	—
235	C₄²	—	2	1	no	—
236	C₁₆	—	2	2	no	—
237	C₁₆	—	2	2	no	—
238	C₁₆	—	2	2	no	—
239	C₁₆	C₂²⋊C₄×C₁₂₇	1	4	yes	—
240	C₂³	—	2	1	no	—
241	C₂³	#131×C₁₂₇	1	1	yes	characteristic
242	Q₈	—	2	1	no	—
243	Q₈	#97×C₁₂₇	1	2	yes	—
244	D₄	—	2	2	no	—
245	D₄	—	2	2	no	—
246	D₄	—	2	1	no	—
247	D₄	#97×C₁₂₇	1	2	yes	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	4	1	no	—
252	C₂×C₄	—	4	1	no	—
253	C₂×C₄	—	8	1	no	—
254	C₂×C₄	C₂×C₄²×C₁₂₇	1	1	yes	characteristic
255	C₂×C₄	C₄×D₄×C₁₂₇	1	1	yes	characteristic
256	C₂×C₄	C₄×D₄×C₁₂₇	1	1	yes	characteristic
257	C₂×C₄	C₂×M₁₆×C₁₂₇	1	1	yes	characteristic
258	C₂×C₄	C₂×M₁₆×C₁₂₇	1	1	yes	characteristic
259	C₂×C₄	#131×C₁₂₇	1	1	yes	characteristic
260	C₈	—	2	2	no	—
261	C₈	—	4	1	no	—
262	C₈	—	4	1	no	—
263	C₈	C₂×C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
264	C₈	#117×C₁₂₇	1	1	yes	characteristic
265	C₈	C₄×D₄×C₁₂₇	1	1	yes	characteristic
266	C₈	C₄×D₄×C₁₂₇	1	1	yes	characteristic
267	C₂²	—	2	2	no	—
268	C₂²	—	4	1	no	—
269	C₂²	—	8	1	no	—
270	C₂²	#405×C₁₂₇	1	1	yes	characteristic
271	C₂²	#435×C₁₂₇	1	1	yes	characteristic
272	C₂²	#435×C₁₂₇	1	1	yes	characteristic
273	C₄	—	2	2	no	—
274	C₄	—	8	1	no	—
275	C₄	—	8	1	no	—
276	C₄	C₄×C₂²⋊C₄×C₁₂₇	1	1	yes	characteristic
277	C₄	C₂×#97×C₁₂₇	1	1	yes	characteristic
278	C₄	#435×C₁₂₇	1	1	yes	characteristic
279	C₄	#435×C₁₂₇	1	1	yes	characteristic
280	C₂	—	2	1	no	—
281	C₂	—	2	1	no	—
282	C₂	—	2	1	no	—
283	C₂	—	4	2	no	—
284	C₂	C₁₂₇×#1752	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₁₂₇	C₄×C₂²⋊C₄
commutator	C₂×C₄	C₂×C₄²×C₁₂₇
Frattini, cosocle	C₂²×C₈	C₂³×C₁₂₇
socle	C₂×C₁₂₇	#1752
prime radical	C₂×#41×C₁₂₇	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₂₇ ◁ C₂×M₃₂×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₁₂₇×#2110	C₂	semidirect
4. C₂×#347×C₁₂₇	C₂	semidirect
6. C₁₂₇×#1378	C₂	semidirect
7. C₁₂₇×#1377	C₂	semidirect
10. #504×C₁₂₇	C₄	semidirect
20. #350×C₁₂₇	C₄	semidirect
137. #12618	C₁₂₇	direct (N × H)
145. #2110	C₂×C₁₂₇	semidirect
146. C₂×#347	C₂×C₁₂₇	semidirect
148. #1378	C₂×C₁₂₇	semidirect
149. #1377	C₂×C₁₂₇	semidirect
150. C₁₂₇	#12618	direct (N × H)
153. #504	C₄×C₁₂₇	semidirect
163. #350	C₄×C₁₂₇	semidirect

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×#2117	2
C₁₂₇×#2110	2
C₂×#347×C₁₂₇	2
C₁₂₇×#1765	2
C₁₂₇×#1378	2
C₁₂₇×#1377	2
#12618	127

Subgroups — C127×#12618 #44576291

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#12618 #44576291