Subgroups — C₁₂₇×#13124 #44576797

Order 32512.

332 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₂₇×#2151	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#350×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	C₁₂₇×#2113	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×C₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₂₇×#1764	C₂	1	1	yes	maximal, characteristic
6	C₁₂₇×#1379	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	C₂×M₃₂×C₁₂₇	C₂²	1	1	yes	characteristic
10	C₂×M₃₂×C₁₂₇	C₂²	1	1	yes	characteristic
11	C₂²×C₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
12	C₂²×C₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
13	C₂×#104×C₁₂₇	C₄	1	1	yes	characteristic
14	#416×C₁₂₇	C₄	1	1	yes	characteristic
15	#405×C₁₂₇	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
17	C₄⋊C₁₆×C₁₂₇	—	2	2	no	—
18	#350×C₁₂₇	—	2	2	no	—
19	C₂²⋊C₁₆×C₁₂₇	—	2	2	no	—
20	C₄×C₁₆×C₁₂₇	—	2	2	no	—
21	#341×C₁₂₇	C₄	1	1	yes	characteristic
22	#339×C₁₂₇	C₄	1	1	yes	characteristic
23	C₂×#41×C₁₂₇	C₂×C₄	1	1	yes	characteristic
24	#131×C₁₂₇	—	2	1	no	—
25	#131×C₁₂₇	—	2	1	no	—
26	#131×C₁₂₇	—	2	1	no	—
27	#131×C₁₂₇	—	2	1	no	—
28	C₂×M₁₆×C₁₂₇	—	2	1	no	—
29	C₂×M₁₆×C₁₂₇	C₂×C₄	1	1	yes	characteristic
30	C₂×M₁₆×C₁₂₇	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₁₂₇	—	2	1	no	—
32	C₂²×C₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
34	#117×C₁₂₇	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
36	M₃₂×C₁₂₇	—	2	1	no	—
37	M₃₂×C₁₂₇	—	2	1	no	—
38	M₃₂×C₁₂₇	—	2	1	no	—
39	M₃₂×C₁₂₇	—	2	1	no	—
40	C₂×C₁₆×C₁₂₇	—	2	1	no	—
41	C₂×C₁₆×C₁₂₇	—	2	1	no	—
42	C₂×C₁₆×C₁₂₇	—	2	1	no	—
43	C₂×C₁₆×C₁₂₇	—	2	1	no	—
44	C₂×C₁₆×C₁₂₇	—	2	1	no	—
45	C₂×C₁₆×C₁₂₇	—	2	1	no	—
46	C₂×C₁₆×C₁₂₇	D₄	1	2	yes	—
47	C₂×C₁₆×C₁₂₇	D₄	1	2	yes	—
48	#104×C₁₂₇	—	2	2	no	—
49	#103×C₁₂₇	C₈	1	2	yes	—
50	#102×C₁₂₇	C₈	1	2	yes	—
51	#97×C₁₂₇	—	2	1	no	—
52	C₄×C₈×C₁₂₇	—	2	1	no	—
53	C₄×C₈×C₁₂₇	—	2	1	no	—
54	C₄×C₈×C₁₂₇	—	2	1	no	—
55	#41×C₁₂₇	—	2	1	no	—
56	#41×C₁₂₇	—	2	1	no	—
57	#41×C₁₂₇	—	2	1	no	—
58	#41×C₁₂₇	—	2	1	no	—
59	C₂×Q₈×C₁₂₇	C₂×C₈	1	1	yes	characteristic
60	C₂×D₄×C₁₂₇	—	2	1	no	—
61	C₂×D₄×C₁₂₇	C₂×C₈	1	1	yes	characteristic
62	C₂²×C₄×C₁₂₇	—	2	1	no	—
63	C₂²×C₄×C₁₂₇	—	2	1	no	—
64	C₂²×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
65	M₁₆×C₁₂₇	—	2	1	no	—
66	M₁₆×C₁₂₇	—	2	1	no	—
67	M₁₆×C₁₂₇	—	2	1	no	—
68	M₁₆×C₁₂₇	—	2	1	no	—
69	M₁₆×C₁₂₇	—	4	1	no	—
70	M₁₆×C₁₂₇	—	4	1	no	—
71	C₂×C₈×C₁₂₇	—	2	1	no	—
72	C₂×C₈×C₁₂₇	—	2	1	no	—
73	C₂×C₈×C₁₂₇	—	2	1	no	—
74	C₂×C₈×C₁₂₇	—	2	1	no	—
75	C₂×C₈×C₁₂₇	—	4	1	no	—
76	C₂×C₈×C₁₂₇	—	4	1	no	—
77	C₂×C₈×C₁₂₇	C₂×C₈	1	2	yes	—
78	C₂×C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
79	C₂×C₈×C₁₂₇	C₂×D₄	1	1	yes	characteristic
80	C₂×C₈×C₁₂₇	#41	1	1	yes	characteristic
81	C₂×C₈×C₁₂₇	#41	1	1	yes	characteristic
82	C₂×C₈×C₁₂₇	#41	1	1	yes	characteristic
83	C₂×C₈×C₁₂₇	#41	1	1	yes	characteristic
84	C₄⋊C₄×C₁₂₇	C₂×C₈	1	2	yes	—
85	C₂²⋊C₄×C₁₂₇	—	2	1	no	—
86	C₄²×C₁₂₇	—	2	1	no	—
87	C₄²×C₁₂₇	—	2	1	no	—
88	C₄²×C₁₂₇	M₁₆	1	2	yes	—
89	C₁₆×C₁₂₇	—	2	2	no	—
90	C₁₆×C₁₂₇	—	2	2	no	—
91	C₁₆×C₁₂₇	—	4	1	no	—
92	C₁₆×C₁₂₇	—	4	1	no	—
93	C₂³×C₁₂₇	—	2	1	no	—
94	C₂³×C₁₂₇	#131	1	1	yes	characteristic
95	Q₈×C₁₂₇	—	2	1	no	—
96	Q₈×C₁₂₇	—	2	1	no	—
97	D₄×C₁₂₇	—	2	1	no	—
98	D₄×C₁₂₇	—	2	1	no	—
99	D₄×C₁₂₇	—	4	1	no	—
100	D₄×C₁₂₇	—	4	1	no	—
101	C₂×C₄×C₁₂₇	—	2	4	no	—
102	C₂×C₄×C₁₂₇	—	2	2	no	—
103	C₂×C₄×C₁₂₇	—	2	2	no	—
104	C₂×C₄×C₁₂₇	—	2	1	no	—
105	C₂×C₄×C₁₂₇	—	4	1	no	—
106	C₂×C₄×C₁₂₇	—	4	1	no	—
107	C₂×C₄×C₁₂₇	#117	1	1	yes	characteristic
108	C₂×C₄×C₁₂₇	C₄×D₄	1	1	yes	characteristic
109	C₂×C₄×C₁₂₇	C₄×D₄	1	1	yes	characteristic
110	C₂×C₄×C₁₂₇	C₂²×C₈	1	1	yes	characteristic
111	C₂×C₄×C₁₂₇	C₂×M₁₆	1	1	yes	characteristic
112	C₂×C₄×C₁₂₇	#131	1	1	yes	characteristic
113	C₈×C₁₂₇	—	2	2	no	—
114	C₈×C₁₂₇	—	4	1	no	—
115	C₈×C₁₂₇	—	4	1	no	—
116	C₈×C₁₂₇	#121	1	1	yes	characteristic
117	C₈×C₁₂₇	#123	1	1	yes	characteristic
118	C₈×C₁₂₇	#124	1	1	yes	characteristic
119	C₈×C₁₂₇	#126	1	1	yes	characteristic
120	C₂²×C₁₂₇	—	2	2	no	—
121	C₂²×C₁₂₇	—	4	2	no	—
122	C₂²×C₁₂₇	—	4	1	no	—
123	C₂²×C₁₂₇	#433	1	1	yes	characteristic
124	C₂²×C₁₂₇	C₈×D₄	1	1	yes	characteristic
125	C₂²×C₁₂₇	#435	1	1	yes	characteristic
126	C₄×C₁₂₇	—	2	4	no	—
127	C₄×C₁₂₇	—	4	2	no	—
128	C₄×C₁₂₇	—	4	1	no	—
129	C₄×C₁₂₇	#388	1	1	yes	characteristic
130	C₄×C₁₂₇	#431	1	1	yes	characteristic
131	C₄×C₁₂₇	C₈×D₄	1	1	yes	characteristic
132	C₄×C₁₂₇	#436	1	1	yes	characteristic
133	#13124	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
134	C₂×C₁₂₇	—	2	1	no	—
135	C₂×C₁₂₇	—	2	1	no	—
136	C₂×C₁₂₇	—	8	1	no	—
137	C₂×C₁₂₇	#1923	1	1	yes	characteristic
138	C₂×C₁₂₇	#2170	1	2	yes	—
139	#2151	C₂×C₁₂₇	1	1	yes	characteristic, complemented
140	C₂×#350	C₂×C₁₂₇	1	1	yes	characteristic
141	#2113	C₂×C₁₂₇	1	1	yes	characteristic
142	C₂×C₄×C₁₆	C₂×C₁₂₇	1	1	yes	characteristic, complemented
143	#1764	C₂×C₁₂₇	1	1	yes	characteristic
144	#1379	C₂×C₁₂₇	1	1	yes	characteristic, complemented
145	#1376	C₂×C₁₂₇	1	1	yes	characteristic, complemented
146	C₁₂₇	#13124	1	1	yes	characteristic, direct product factor, Sylow
147	C₂×#131	C₂²×C₁₂₇	1	1	yes	characteristic
148	C₂×M₃₂	C₂²×C₁₂₇	1	1	yes	characteristic
149	C₂×M₃₂	C₂²×C₁₂₇	1	1	yes	characteristic
150	C₂²×C₁₆	C₂²×C₁₂₇	1	1	yes	characteristic
151	C₂²×C₁₆	C₂²×C₁₂₇	1	1	yes	characteristic
152	C₂×#104	C₄×C₁₂₇	1	1	yes	characteristic
153	#416	C₄×C₁₂₇	1	1	yes	characteristic
154	#405	C₂²×C₁₂₇	1	1	yes	characteristic
155	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
156	C₄⋊C₁₆	—	2	2	no	—
157	#350	—	2	2	no	—
158	C₂²⋊C₁₆	—	2	2	no	—
159	C₄×C₁₆	—	2	2	no	—
160	#341	C₄×C₁₂₇	1	1	yes	characteristic
161	#339	C₄×C₁₂₇	1	1	yes	characteristic
162	C₂×#41	C₂×C₄×C₁₂₇	1	1	yes	characteristic
163	#131	—	2	1	no	—
164	#131	—	2	1	no	—
165	#131	—	2	1	no	—
166	#131	—	2	1	no	—
167	C₂×M₁₆	—	2	1	no	—
168	C₂×M₁₆	C₂×C₄×C₁₂₇	1	1	yes	characteristic
169	C₂×M₁₆	C₂×C₄×C₁₂₇	1	1	yes	characteristic
170	C₂²×C₈	—	2	1	no	—
171	C₂²×C₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
172	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
173	#117	C₂×C₄×C₁₂₇	1	1	yes	characteristic
174	C₂×C₄²	C₂×C₄×C₁₂₇	1	1	yes	characteristic
175	M₃₂	—	2	1	no	—
176	M₃₂	—	2	1	no	—
177	M₃₂	—	2	1	no	—
178	M₃₂	—	2	1	no	—
179	C₂×C₁₆	—	2	1	no	—
180	C₂×C₁₆	—	2	1	no	—
181	C₂×C₁₆	—	2	1	no	—
182	C₂×C₁₆	—	2	1	no	—
183	C₂×C₁₆	—	2	1	no	—
184	C₂×C₁₆	—	2	1	no	—
185	C₂×C₁₆	D₄×C₁₂₇	1	2	yes	—
186	C₂×C₁₆	D₄×C₁₂₇	1	2	yes	—
187	#104	—	2	2	no	—
188	#103	C₈×C₁₂₇	1	2	yes	—
189	#102	C₈×C₁₂₇	1	2	yes	—
190	#97	—	2	1	no	—
191	C₄×C₈	—	2	1	no	—
192	C₄×C₈	—	2	1	no	—
193	C₄×C₈	—	2	1	no	—
194	#41	—	2	1	no	—
195	#41	—	2	1	no	—
196	#41	—	2	1	no	—
197	#41	—	2	1	no	—
198	C₂×Q₈	C₂×C₈×C₁₂₇	1	1	yes	characteristic
199	C₂×D₄	—	2	1	no	—
200	C₂×D₄	C₂×C₈×C₁₂₇	1	1	yes	characteristic
201	C₂²×C₄	—	2	1	no	—
202	C₂²×C₄	—	2	1	no	—
203	C₂²×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
204	M₁₆	—	2	1	no	—
205	M₁₆	—	2	1	no	—
206	M₁₆	—	2	1	no	—
207	M₁₆	—	2	1	no	—
208	M₁₆	—	4	1	no	—
209	M₁₆	—	4	1	no	—
210	C₂×C₈	—	2	1	no	—
211	C₂×C₈	—	2	1	no	—
212	C₂×C₈	—	2	1	no	—
213	C₂×C₈	—	2	1	no	—
214	C₂×C₈	—	4	1	no	—
215	C₂×C₈	—	4	1	no	—
216	C₂×C₈	C₂×C₈×C₁₂₇	1	2	yes	—
217	C₂×C₈	C₂×D₄×C₁₂₇	1	1	yes	characteristic
218	C₂×C₈	C₂×D₄×C₁₂₇	1	1	yes	characteristic
219	C₂×C₈	#41×C₁₂₇	1	1	yes	characteristic
220	C₂×C₈	#41×C₁₂₇	1	1	yes	characteristic
221	C₂×C₈	#41×C₁₂₇	1	1	yes	characteristic
222	C₂×C₈	#41×C₁₂₇	1	1	yes	characteristic
223	C₄⋊C₄	C₂×C₈×C₁₂₇	1	2	yes	—
224	C₂²⋊C₄	—	2	1	no	—
225	C₄²	—	2	1	no	—
226	C₄²	—	2	1	no	—
227	C₄²	M₁₆×C₁₂₇	1	2	yes	—
228	C₁₆	—	2	2	no	—
229	C₁₆	—	2	2	no	—
230	C₁₆	—	4	1	no	—
231	C₁₆	—	4	1	no	—
232	C₂³	—	2	1	no	—
233	C₂³	#131×C₁₂₇	1	1	yes	characteristic
234	Q₈	—	2	1	no	—
235	Q₈	—	2	1	no	—
236	D₄	—	2	1	no	—
237	D₄	—	2	1	no	—
238	D₄	—	4	1	no	—
239	D₄	—	4	1	no	—
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	4	1	no	—
245	C₂×C₄	—	4	1	no	—
246	C₂×C₄	#117×C₁₂₇	1	1	yes	characteristic
247	C₂×C₄	C₄×D₄×C₁₂₇	1	1	yes	characteristic
248	C₂×C₄	C₄×D₄×C₁₂₇	1	1	yes	characteristic
249	C₂×C₄	C₂²×C₈×C₁₂₇	1	1	yes	characteristic
250	C₂×C₄	C₂×M₁₆×C₁₂₇	1	1	yes	characteristic
251	C₂×C₄	#131×C₁₂₇	1	1	yes	characteristic
252	C₈	—	2	2	no	—
253	C₈	—	4	1	no	—
254	C₈	—	4	1	no	—
255	C₈	#121×C₁₂₇	1	1	yes	characteristic
256	C₈	#123×C₁₂₇	1	1	yes	characteristic
257	C₈	#124×C₁₂₇	1	1	yes	characteristic
258	C₈	#126×C₁₂₇	1	1	yes	characteristic
259	C₂²	—	2	2	no	—
260	C₂²	—	4	2	no	—
261	C₂²	—	4	1	no	—
262	C₂²	#433×C₁₂₇	1	1	yes	characteristic
263	C₂²	C₈×D₄×C₁₂₇	1	1	yes	characteristic
264	C₂²	#435×C₁₂₇	1	1	yes	characteristic
265	C₄	—	2	4	no	—
266	C₄	—	4	2	no	—
267	C₄	—	4	1	no	—
268	C₄	#388×C₁₂₇	1	1	yes	characteristic
269	C₄	#431×C₁₂₇	1	1	yes	characteristic
270	C₄	C₈×D₄×C₁₂₇	1	1	yes	characteristic
271	C₄	#436×C₁₂₇	1	1	yes	characteristic
272	C₂	—	2	1	no	—
273	C₂	—	2	1	no	—
274	C₂	—	8	1	no	—
275	C₂	C₁₂₇×#1923	1	1	yes	characteristic
276	C₂	C₁₂₇×#2170	1	2	yes	—

	subgroup	G/H
center	C₂×C₈×C₁₂₇	C₂×D₄
commutator	C₂×C₄	C₂²×C₈×C₁₂₇
Frattini, cosocle	C₂²×C₈	C₂³×C₁₂₇
socle	C₂²×C₁₂₇	C₈×D₄
prime radical	C₂×#41×C₁₂₇	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₁₂₇ ◁ C₂²×C₁₆×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₂₇×#2151	C₂	semidirect
4. C₂×C₄×C₁₆×C₁₂₇	C₂	semidirect
6. C₁₂₇×#1379	C₂	semidirect
7. C₁₂₇×#1376	C₂	semidirect
133. #13124	C₁₂₇	direct (N × H)
139. #2151	C₂×C₁₂₇	semidirect
142. C₂×C₄×C₁₆	C₂×C₁₂₇	semidirect
144. #1379	C₂×C₁₂₇	semidirect
145. #1376	C₂×C₁₂₇	semidirect
146. C₁₂₇	#13124	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×#2151	2
C₂×#350×C₁₂₇	2
C₁₂₇×#2113	2
C₂×C₄×C₁₆×C₁₂₇	2
C₁₂₇×#1764	2
C₁₂₇×#1379	2
C₁₂₇×#1376	2
#13124	127

Subgroups — C127×#13124 #44576797

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#13124 #44576797