Subgroups — C₁₂₇×#19636 #44583309

Order 32512.

686 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₂₇×#3229	C₂	1	2	yes	maximal
2	C₁₂₇×#3170	C₂	1	1	yes	maximal, characteristic
3	C₂×#500×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₁₂₇×#3139	C₂	1	2	yes	maximal
5	C₁₂₇×#1718	C₂	1	4	yes	maximal, complemented
6	C₁₂₇×#1710	C₂	1	2	yes	maximal, complemented
7	C₁₂₇×#1707	C₂	1	2	yes	maximal, complemented
8	C₈×M₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic
9	#533×C₁₂₇	C₂²	1	2	yes	—
10	C₂×C₄⋊Q₈×C₁₂₇	C₂²	1	2	yes	—
11	#500×C₁₂₇	—	2	2	no	—
12	#500×C₁₂₇	C₂²	1	2	yes	—
13	#500×C₁₂₇	C₂²	1	2	yes	—
14	#500×C₁₂₇	C₂²	1	1	yes	characteristic
15	#498×C₁₂₇	C₂²	1	4	yes	—
16	#494×C₁₂₇	C₂²	1	1	yes	characteristic
17	#491×C₁₂₇	—	2	4	no	—
18	#487×C₁₂₇	C₂²	1	4	yes	—
19	#487×C₁₂₇	C₂²	1	4	yes	—
20	#487×C₁₂₇	C₂²	1	2	yes	—
21	#487×C₁₂₇	C₂²	1	4	yes	—
22	#483×C₁₂₇	—	2	4	no	—
23	#475×C₁₂₇	—	2	4	no	—
24	#467×C₁₂₇	—	2	2	no	—
25	C₄×Q₁₆×C₁₂₇	—	2	2	no	—
26	#431×C₁₂₇	C₂²	1	1	yes	characteristic
27	C₂×#107×C₁₂₇	—	2	2	no	—
28	#419×C₁₂₇	—	2	4	no	—
29	C₄×M₁₆×C₁₂₇	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
31	#322×C₁₂₇	C₂²	1	2	yes	—
32	C₈²×C₁₂₇	C₂²	1	2	yes	—
33	C₂²×Q₈×C₁₂₇	—	2	2	no	—
34	C₂×Q₁₆×C₁₂₇	—	2	2	no	—
35	C₂×M₁₆×C₁₂₇	D₄	1	2	yes	—
36	C₂²×C₈×C₁₂₇	D₄	1	2	yes	—
37	C₄⋊Q₈×C₁₂₇	—	2	2	no	—
38	C₄⋊Q₈×C₁₂₇	—	2	2	no	—
39	C₄⋊Q₈×C₁₂₇	—	2	2	no	—
40	C₄⋊Q₈×C₁₂₇	C₂³	1	2	yes	—
41	C₄⋊Q₈×C₁₂₇	C₂³	1	4	yes	—
42	C₄⋊Q₈×C₁₂₇	C₂³	1	2	yes	—
43	#125×C₁₂₇	—	2	2	no	—
44	C₂²⋊Q₈×C₁₂₇	—	2	2	no	—
45	C₂²⋊Q₈×C₁₂₇	—	2	2	no	—
46	C₄×Q₈×C₁₂₇	—	2	2	no	—
47	C₄×Q₈×C₁₂₇	—	2	2	no	—
48	#117×C₁₂₇	—	2	2	no	—
49	C₂×C₄⋊C₄×C₁₂₇	—	2	2	no	—
50	C₂×C₄⋊C₄×C₁₂₇	—	2	2	no	—
51	C₂×C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
52	#107×C₁₂₇	—	2	4	no	—
53	#107×C₁₂₇	—	2	4	no	—
54	#107×C₁₂₇	—	2	2	no	—
55	#106×C₁₂₇	—	2	8	no	—
56	C₄⋊C₈×C₁₂₇	D₄	1	2	yes	—
57	#103×C₁₂₇	—	2	8	no	—
58	#103×C₁₂₇	—	2	8	no	—
59	#103×C₁₂₇	—	2	4	no	—
60	#103×C₁₂₇	—	2	8	no	—
61	C₂²⋊C₈×C₁₂₇	D₄	1	2	yes	—
62	#97×C₁₂₇	D₄	1	2	yes	—
63	C₄×C₈×C₁₂₇	D₄	1	2	yes	—
64	C₄×C₈×C₁₂₇	C₂³	1	2	yes	—
65	C₄×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
66	C₄×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
67	C₄×C₈×C₁₂₇	C₂³	1	2	yes	—
68	C₂×Q₈×C₁₂₇	—	2	2	no	—
69	C₂×Q₈×C₁₂₇	—	2	4	no	—
70	C₂×Q₈×C₁₂₇	—	2	2	no	—
71	C₂×Q₈×C₁₂₇	—	4	2	no	—
72	C₂×Q₈×C₁₂₇	—	4	8	no	—
73	C₂²×C₄×C₁₂₇	—	4	2	no	—
74	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
77	Q₁₆×C₁₂₇	—	4	4	no	—
78	M₁₆×C₁₂₇	—	2	4	no	—
79	C₂×C₈×C₁₂₇	—	2	4	no	—
80	C₂×C₈×C₁₂₇	C₂×D₄	1	2	yes	—
81	C₂×C₈×C₁₂₇	C₂×D₄	1	4	yes	—
82	C₂×C₈×C₁₂₇	C₂×D₄	1	4	yes	—
83	C₂×C₈×C₁₂₇	C₂×D₄	1	2	yes	—
84	C₄⋊C₄×C₁₂₇	—	2	2	no	—
85	C₄⋊C₄×C₁₂₇	—	2	2	no	—
86	C₄⋊C₄×C₁₂₇	—	2	4	no	—
87	C₄⋊C₄×C₁₂₇	—	2	2	no	—
88	C₄⋊C₄×C₁₂₇	—	2	2	no	—
89	C₄⋊C₄×C₁₂₇	—	2	4	no	—
90	C₄⋊C₄×C₁₂₇	—	4	2	no	—
91	C₄⋊C₄×C₁₂₇	—	4	2	no	—
92	C₄⋊C₄×C₁₂₇	—	4	2	no	—
93	C₄⋊C₄×C₁₂₇	—	4	2	no	—
94	C₂²⋊C₄×C₁₂₇	—	4	2	no	—
95	C₄²×C₁₂₇	—	4	2	no	—
96	C₄²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
97	C₄²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
98	C₄²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
99	C₄²×C₁₂₇	C₂⁴	1	1	yes	characteristic
100	C₂³×C₁₂₇	#127	1	1	yes	characteristic
101	Q₈×C₁₂₇	—	4	8	no	—
102	Q₈×C₁₂₇	—	4	4	no	—
103	Q₈×C₁₂₇	—	4	4	no	—
104	Q₈×C₁₂₇	—	8	4	no	—
105	C₂×C₄×C₁₂₇	—	2	2	no	—
106	C₂×C₄×C₁₂₇	—	2	2	no	—
107	C₂×C₄×C₁₂₇	—	4	2	no	—
108	C₂×C₄×C₁₂₇	—	4	2	no	—
109	C₂×C₄×C₁₂₇	—	4	4	no	—
110	C₂×C₄×C₁₂₇	—	8	4	no	—
111	C₂×C₄×C₁₂₇	#127	1	1	yes	characteristic
112	C₂×C₄×C₁₂₇	#127	1	1	yes	characteristic
113	C₂×C₄×C₁₂₇	#127	1	1	yes	characteristic
114	C₂×C₄×C₁₂₇	#135	1	4	yes	—
115	C₂×C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
116	C₂×C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
117	C₂×C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
118	C₈×C₁₂₇	—	2	4	no	—
119	C₈×C₁₂₇	—	2	2	no	—
120	C₈×C₁₂₇	—	2	4	no	—
121	C₈×C₁₂₇	#137	1	4	yes	—
122	C₂²×C₁₂₇	—	2	1	no	—
123	C₂²×C₁₂₇	—	2	1	no	—
124	C₂²×C₁₂₇	#495	1	2	yes	—
125	C₂²×C₁₂₇	C₂×#127	1	1	yes	characteristic
126	C₄×C₁₂₇	—	2	1	no	—
127	C₄×C₁₂₇	—	2	1	no	—
128	C₄×C₁₂₇	—	8	4	no	—
129	C₄×C₁₂₇	—	8	2	no	—
130	C₄×C₁₂₇	—	8	2	no	—
131	C₄×C₁₂₇	#495	1	2	yes	—
132	C₄×C₁₂₇	C₂×#135	1	2	yes	—
133	C₄×C₁₂₇	C₂×#137	1	2	yes	—
134	C₄×C₁₂₇	#578	1	2	yes	—
135	#19636	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
136	C₂×C₁₂₇	—	2	2	no	—
137	C₂×C₁₂₇	C₂×#495	1	1	yes	characteristic
138	C₂×C₁₂₇	#3152	1	1	yes	characteristic
139	C₂×C₁₂₇	#3156	1	1	yes	characteristic
140	#3229	C₂×C₁₂₇	1	2	yes	—
141	#3170	C₂×C₁₂₇	1	1	yes	characteristic
142	C₂×#500	C₂×C₁₂₇	1	1	yes	characteristic
143	#3139	C₂×C₁₂₇	1	2	yes	—
144	#1718	C₂×C₁₂₇	1	4	yes	complemented
145	#1710	C₂×C₁₂₇	1	2	yes	complemented
146	#1707	C₂×C₁₂₇	1	2	yes	complemented
147	C₈×M₁₆	C₂×C₁₂₇	1	1	yes	characteristic
148	C₁₂₇	#19636	1	1	yes	characteristic, direct product factor, Sylow
149	#533	C₂²×C₁₂₇	1	2	yes	—
150	C₂×C₄⋊Q₈	C₂²×C₁₂₇	1	2	yes	—
151	#500	—	2	2	no	—
152	#500	C₂²×C₁₂₇	1	2	yes	—
153	#500	C₂²×C₁₂₇	1	2	yes	—
154	#500	C₂²×C₁₂₇	1	1	yes	characteristic
155	#498	C₂²×C₁₂₇	1	4	yes	—
156	#494	C₂²×C₁₂₇	1	1	yes	characteristic
157	#491	—	2	4	no	—
158	#487	C₂²×C₁₂₇	1	4	yes	—
159	#487	C₂²×C₁₂₇	1	4	yes	—
160	#487	C₂²×C₁₂₇	1	2	yes	—
161	#487	C₂²×C₁₂₇	1	4	yes	—
162	#483	—	2	4	no	—
163	#475	—	2	4	no	—
164	#467	—	2	2	no	—
165	C₄×Q₁₆	—	2	2	no	—
166	#431	C₂²×C₁₂₇	1	1	yes	characteristic
167	C₂×#107	—	2	2	no	—
168	#419	—	2	4	no	—
169	C₄×M₁₆	C₂²×C₁₂₇	1	1	yes	characteristic
170	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
171	#322	C₂²×C₁₂₇	1	2	yes	—
172	C₈²	C₂²×C₁₂₇	1	2	yes	—
173	C₂²×Q₈	—	2	2	no	—
174	C₂×Q₁₆	—	2	2	no	—
175	C₂×M₁₆	D₄×C₁₂₇	1	2	yes	—
176	C₂²×C₈	D₄×C₁₂₇	1	2	yes	—
177	C₄⋊Q₈	—	2	2	no	—
178	C₄⋊Q₈	—	2	2	no	—
179	C₄⋊Q₈	—	2	2	no	—
180	C₄⋊Q₈	C₂³×C₁₂₇	1	4	yes	—
181	C₄⋊Q₈	C₂³×C₁₂₇	1	2	yes	—
182	C₄⋊Q₈	C₂³×C₁₂₇	1	2	yes	—
183	#125	—	2	2	no	—
184	C₂²⋊Q₈	—	2	2	no	—
185	C₂²⋊Q₈	—	2	2	no	—
186	C₄×Q₈	—	2	2	no	—
187	C₄×Q₈	—	2	2	no	—
188	#117	—	2	2	no	—
189	C₂×C₄⋊C₄	—	2	2	no	—
190	C₂×C₄⋊C₄	—	2	2	no	—
191	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
192	#107	—	2	4	no	—
193	#107	—	2	4	no	—
194	#107	—	2	2	no	—
195	#106	—	2	8	no	—
196	C₄⋊C₈	D₄×C₁₂₇	1	2	yes	—
197	#103	—	2	8	no	—
198	#103	—	2	8	no	—
199	#103	—	2	4	no	—
200	#103	—	2	8	no	—
201	C₂²⋊C₈	D₄×C₁₂₇	1	2	yes	—
202	#97	D₄×C₁₂₇	1	2	yes	—
203	C₄×C₈	D₄×C₁₂₇	1	2	yes	—
204	C₄×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
205	C₄×C₈	C₂³×C₁₂₇	1	2	yes	—
206	C₄×C₈	C₂³×C₁₂₇	1	2	yes	—
207	C₄×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
208	C₂×Q₈	—	2	2	no	—
209	C₂×Q₈	—	2	4	no	—
210	C₂×Q₈	—	2	2	no	—
211	C₂×Q₈	—	4	2	no	—
212	C₂×Q₈	—	4	8	no	—
213	C₂²×C₄	—	4	2	no	—
214	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
215	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
216	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
217	Q₁₆	—	4	4	no	—
218	M₁₆	—	2	4	no	—
219	C₂×C₈	—	2	4	no	—
220	C₂×C₈	C₂×D₄×C₁₂₇	1	2	yes	—
221	C₂×C₈	C₂×D₄×C₁₂₇	1	4	yes	—
222	C₂×C₈	C₂×D₄×C₁₂₇	1	4	yes	—
223	C₂×C₈	C₂×D₄×C₁₂₇	1	2	yes	—
224	C₄⋊C₄	—	2	2	no	—
225	C₄⋊C₄	—	2	2	no	—
226	C₄⋊C₄	—	2	4	no	—
227	C₄⋊C₄	—	2	2	no	—
228	C₄⋊C₄	—	2	2	no	—
229	C₄⋊C₄	—	2	4	no	—
230	C₄⋊C₄	—	4	2	no	—
231	C₄⋊C₄	—	4	2	no	—
232	C₄⋊C₄	—	4	2	no	—
233	C₄⋊C₄	—	4	2	no	—
234	C₂²⋊C₄	—	4	2	no	—
235	C₄²	—	4	2	no	—
236	C₄²	C₂×D₄×C₁₂₇	1	1	yes	characteristic
237	C₄²	C₂×D₄×C₁₂₇	1	1	yes	characteristic
238	C₄²	C₂×D₄×C₁₂₇	1	1	yes	characteristic
239	C₄²	C₂⁴×C₁₂₇	1	1	yes	characteristic
240	C₂³	#127×C₁₂₇	1	1	yes	characteristic
241	Q₈	—	4	8	no	—
242	Q₈	—	4	4	no	—
243	Q₈	—	4	4	no	—
244	Q₈	—	8	4	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	—	4	2	no	—
249	C₂×C₄	—	4	4	no	—
250	C₂×C₄	—	8	4	no	—
251	C₂×C₄	#127×C₁₂₇	1	1	yes	characteristic
252	C₂×C₄	#127×C₁₂₇	1	1	yes	characteristic
253	C₂×C₄	#127×C₁₂₇	1	1	yes	characteristic
254	C₂×C₄	#135×C₁₂₇	1	4	yes	—
255	C₂×C₄	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
256	C₂×C₄	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
257	C₂×C₄	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
258	C₈	—	2	4	no	—
259	C₈	—	2	2	no	—
260	C₈	—	2	4	no	—
261	C₈	#137×C₁₂₇	1	4	yes	—
262	C₂²	—	2	1	no	—
263	C₂²	—	2	1	no	—
264	C₂²	#495×C₁₂₇	1	2	yes	—
265	C₂²	C₂×#127×C₁₂₇	1	1	yes	characteristic
266	C₄	—	2	1	no	—
267	C₄	—	2	1	no	—
268	C₄	—	8	4	no	—
269	C₄	—	8	2	no	—
270	C₄	—	8	2	no	—
271	C₄	#495×C₁₂₇	1	2	yes	—
272	C₄	C₂×#135×C₁₂₇	1	2	yes	—
273	C₄	C₂×#137×C₁₂₇	1	2	yes	—
274	C₄	#578×C₁₂₇	1	2	yes	—
275	C₂	—	2	2	no	—
276	C₂	C₂×#495×C₁₂₇	1	1	yes	characteristic
277	C₂	C₁₂₇×#3152	1	1	yes	characteristic
278	C₂	C₁₂₇×#3156	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₂₇	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₁₂₇
prime radical	C₂³×C₁₂₇	#127
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₂₇ ◁ C₂×C₄²×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
5. C₁₂₇×#1718	C₂	semidirect
6. C₁₂₇×#1710	C₂	semidirect
7. C₁₂₇×#1707	C₂	semidirect
135. #19636	C₁₂₇	direct (N × H)
144. #1718	C₂×C₁₂₇	semidirect
145. #1710	C₂×C₁₂₇	semidirect
146. #1707	C₂×C₁₂₇	semidirect
148. C₁₂₇	#19636	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×#3229	2
C₁₂₇×#3170	2
C₂×#500×C₁₂₇	2
C₁₂₇×#3139	2
C₁₂₇×#1718	2
C₁₂₇×#1710	2
C₁₂₇×#1707	2
C₈×M₁₆×C₁₂₇	2
#19636	127

Subgroups — C127×#19636 #44583309

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#19636 #44583309