Subgroups — C₁₂₇×#19632 #44583305

Order 32512.

750 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×#3230	C₂	1	2	yes	maximal
2	C₁₂₇×#3170	C₂	1	1	yes	maximal, characteristic
3	C₁₂₇×#3165	C₂	1	2	yes	maximal
4	C₂×#494×C₁₂₇	C₂	1	1	yes	maximal, characteristic
5	C₁₂₇×#1723	C₂	1	4	yes	maximal, complemented
6	C₁₂₇×#1713	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	C₂²×Q₁₆×C₁₂₇	—	2	2	no	—
10	#533×C₁₂₇	C₂²	1	2	yes	—
11	C₂×C₄⋊Q₈×C₁₂₇	C₂²	1	2	yes	—
12	#501×C₁₂₇	—	2	4	no	—
13	#500×C₁₂₇	C₂²	1	4	yes	—
14	#500×C₁₂₇	C₂²	1	1	yes	characteristic
15	#500×C₁₂₇	C₂²	1	4	yes	—
16	#498×C₁₂₇	C₂²	1	4	yes	—
17	#494×C₁₂₇	—	2	2	no	—
18	#494×C₁₂₇	C₂²	1	2	yes	—
19	#494×C₁₂₇	C₂²	1	2	yes	—
20	#494×C₁₂₇	C₂²	1	1	yes	characteristic
21	#487×C₁₂₇	C₂²	1	2	yes	—
22	#487×C₁₂₇	C₂²	1	4	yes	—
23	#484×C₁₂₇	—	2	4	no	—
24	#477×C₁₂₇	—	2	4	no	—
25	#467×C₁₂₇	—	2	2	no	—
26	C₄×Q₁₆×C₁₂₇	—	2	2	no	—
27	#431×C₁₂₇	C₂²	1	1	yes	characteristic
28	#428×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	4	no	—
35	C₂×Q₁₆×C₁₂₇	—	2	4	no	—
36	C₂×Q₁₆×C₁₂₇	—	2	2	no	—
37	C₂×Q₁₆×C₁₂₇	—	4	8	no	—
38	C₂×M₁₆×C₁₂₇	D₄	1	2	yes	—
39	C₂²×C₈×C₁₂₇	D₄	1	2	yes	—
40	C₄⋊Q₈×C₁₂₇	—	2	2	no	—
41	C₄⋊Q₈×C₁₂₇	—	2	2	no	—
42	C₄⋊Q₈×C₁₂₇	—	2	2	no	—
43	C₄⋊Q₈×C₁₂₇	C₂³	1	2	yes	—
44	C₄⋊Q₈×C₁₂₇	C₂³	1	4	yes	—
45	C₄⋊Q₈×C₁₂₇	C₂³	1	2	yes	—
46	#125×C₁₂₇	—	2	2	no	—
47	C₂²⋊Q₈×C₁₂₇	—	2	2	no	—
48	C₂²⋊Q₈×C₁₂₇	—	2	2	no	—
49	C₄×Q₈×C₁₂₇	—	2	2	no	—
50	C₄×Q₈×C₁₂₇	—	2	2	no	—
51	#117×C₁₂₇	—	2	2	no	—
52	C₂×C₄⋊C₄×C₁₂₇	—	2	2	no	—
53	C₂×C₄⋊C₄×C₁₂₇	—	2	2	no	—
54	C₂×C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
55	#107×C₁₂₇	—	2	8	no	—
56	#107×C₁₂₇	—	2	2	no	—
57	#107×C₁₂₇	—	2	8	no	—
58	#106×C₁₂₇	—	2	8	no	—
59	C₄⋊C₈×C₁₂₇	D₄	1	2	yes	—
60	#103×C₁₂₇	—	2	4	no	—
61	#103×C₁₂₇	—	2	8	no	—
62	C₂²⋊C₈×C₁₂₇	D₄	1	2	yes	—
63	#97×C₁₂₇	D₄	1	2	yes	—
64	C₄×C₈×C₁₂₇	D₄	1	2	yes	—
65	C₄×C₈×C₁₂₇	C₂³	1	2	yes	—
66	C₄×C₈×C₁₂₇	C₂³	1	2	yes	—
67	C₄×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
68	C₄×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
69	C₂×Q₈×C₁₂₇	—	2	2	no	—
70	C₂×Q₈×C₁₂₇	—	2	2	no	—
71	C₂×Q₈×C₁₂₇	—	2	4	no	—
72	C₂×Q₈×C₁₂₇	—	4	8	no	—
73	C₂×Q₈×C₁₂₇	—	4	2	no	—
74	C₂²×C₄×C₁₂₇	—	4	2	no	—
75	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
78	Q₁₆×C₁₂₇	—	4	8	no	—
79	Q₁₆×C₁₂₇	—	4	4	no	—
80	Q₁₆×C₁₂₇	—	4	8	no	—
81	M₁₆×C₁₂₇	Q₁₆	1	8	yes	complemented
82	C₂×C₈×C₁₂₇	—	2	4	no	—
83	C₂×C₈×C₁₂₇	C₂×D₄	1	2	yes	—
84	C₂×C₈×C₁₂₇	C₂×D₄	1	4	yes	—
85	C₂×C₈×C₁₂₇	C₂×D₄	1	4	yes	—
86	C₂×C₈×C₁₂₇	C₂×D₄	1	2	yes	—
87	C₄⋊C₄×C₁₂₇	—	2	4	no	—
88	C₄⋊C₄×C₁₂₇	—	2	2	no	—
89	C₄⋊C₄×C₁₂₇	—	2	2	no	—
90	C₄⋊C₄×C₁₂₇	—	2	4	no	—
91	C₄⋊C₄×C₁₂₇	—	2	2	no	—
92	C₄⋊C₄×C₁₂₇	—	2	2	no	—
93	C₄⋊C₄×C₁₂₇	—	4	2	no	—
94	C₄⋊C₄×C₁₂₇	—	4	2	no	—
95	C₄⋊C₄×C₁₂₇	—	4	2	no	—
96	C₄⋊C₄×C₁₂₇	—	4	2	no	—
97	C₂²⋊C₄×C₁₂₇	—	4	2	no	—
98	C₄²×C₁₂₇	—	4	2	no	—
99	C₄²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
100	C₄²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
101	C₄²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
102	C₄²×C₁₂₇	C₂⁴	1	1	yes	characteristic
103	C₂³×C₁₂₇	#127	1	1	yes	characteristic
104	Q₈×C₁₂₇	—	4	8	no	—
105	Q₈×C₁₂₇	—	4	4	no	—
106	Q₈×C₁₂₇	—	4	4	no	—
107	Q₈×C₁₂₇	—	8	4	no	—
108	C₂×C₄×C₁₂₇	—	2	2	no	—
109	C₂×C₄×C₁₂₇	—	2	2	no	—
110	C₂×C₄×C₁₂₇	—	4	2	no	—
111	C₂×C₄×C₁₂₇	—	4	2	no	—
112	C₂×C₄×C₁₂₇	—	4	4	no	—
113	C₂×C₄×C₁₂₇	—	8	4	no	—
114	C₂×C₄×C₁₂₇	#127	1	1	yes	characteristic
115	C₂×C₄×C₁₂₇	#127	1	1	yes	characteristic
116	C₂×C₄×C₁₂₇	#127	1	1	yes	characteristic
117	C₂×C₄×C₁₂₇	C₂×Q₁₆	1	4	yes	—
118	C₂×C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
119	C₂×C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
120	C₂×C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
121	C₈×C₁₂₇	—	2	2	no	—
122	C₈×C₁₂₇	—	2	4	no	—
123	C₈×C₁₂₇	C₂×Q₁₆	1	8	yes	complemented
124	C₈×C₁₂₇	AGL(1,ℤ/8)	1	4	yes	—
125	C₂²×C₁₂₇	—	2	1	no	—
126	C₂²×C₁₂₇	—	2	1	no	—
127	C₂²×C₁₂₇	#494	1	2	yes	—
128	C₂²×C₁₂₇	C₂×#127	1	1	yes	characteristic
129	C₄×C₁₂₇	—	2	1	no	—
130	C₄×C₁₂₇	—	2	1	no	—
131	C₄×C₁₂₇	—	8	4	no	—
132	C₄×C₁₂₇	—	8	2	no	—
133	C₄×C₁₂₇	—	8	2	no	—
134	C₄×C₁₂₇	#494	1	2	yes	—
135	C₄×C₁₂₇	C₂²×Q₁₆	1	2	yes	—
136	C₄×C₁₂₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
137	C₄×C₁₂₇	#578	1	2	yes	—
138	#19632	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
139	C₂×C₁₂₇	—	2	2	no	—
140	C₂×C₁₂₇	C₂×#494	1	1	yes	characteristic
141	C₂×C₁₂₇	#3151	1	1	yes	characteristic
142	C₂×C₁₂₇	#3156	1	1	yes	characteristic
143	#3230	C₂×C₁₂₇	1	2	yes	—
144	#3170	C₂×C₁₂₇	1	1	yes	characteristic
145	#3165	C₂×C₁₂₇	1	2	yes	—
146	C₂×#494	C₂×C₁₂₇	1	1	yes	characteristic
147	#1723	C₂×C₁₂₇	1	4	yes	complemented
148	#1713	C₂×C₁₂₇	1	2	yes	complemented
149	#1707	C₂×C₁₂₇	1	2	yes	complemented
150	C₈×M₁₆	C₂×C₁₂₇	1	1	yes	characteristic
151	C₁₂₇	#19632	1	1	yes	characteristic, direct product factor, Sylow
152	C₂²×Q₁₆	—	2	2	no	—
153	#533	C₂²×C₁₂₇	1	2	yes	—
154	C₂×C₄⋊Q₈	C₂²×C₁₂₇	1	2	yes	—
155	#501	—	2	4	no	—
156	#500	C₂²×C₁₂₇	1	4	yes	—
157	#500	C₂²×C₁₂₇	1	1	yes	characteristic
158	#500	C₂²×C₁₂₇	1	4	yes	—
159	#498	C₂²×C₁₂₇	1	4	yes	—
160	#494	—	2	2	no	—
161	#494	C₂²×C₁₂₇	1	2	yes	—
162	#494	C₂²×C₁₂₇	1	2	yes	—
163	#494	C₂²×C₁₂₇	1	1	yes	characteristic
164	#487	C₂²×C₁₂₇	1	2	yes	—
165	#487	C₂²×C₁₂₇	1	4	yes	—
166	#484	—	2	4	no	—
167	#477	—	2	4	no	—
168	#467	—	2	2	no	—
169	C₄×Q₁₆	—	2	2	no	—
170	#431	C₂²×C₁₂₇	1	1	yes	characteristic
171	#428	—	2	4	no	—
172	C₄×M₁₆	C₂²×C₁₂₇	1	1	yes	characteristic
173	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
174	#322	C₂²×C₁₂₇	1	2	yes	—
175	C₈²	C₂²×C₁₂₇	1	2	yes	—
176	C₂²×Q₈	—	2	2	no	—
177	C₂×Q₁₆	—	2	4	no	—
178	C₂×Q₁₆	—	2	4	no	—
179	C₂×Q₁₆	—	2	2	no	—
180	C₂×Q₁₆	—	4	8	no	—
181	C₂×M₁₆	D₄×C₁₂₇	1	2	yes	—
182	C₂²×C₈	D₄×C₁₂₇	1	2	yes	—
183	C₄⋊Q₈	—	2	2	no	—
184	C₄⋊Q₈	—	2	2	no	—
185	C₄⋊Q₈	—	2	2	no	—
186	C₄⋊Q₈	C₂³×C₁₂₇	1	4	yes	—
187	C₄⋊Q₈	C₂³×C₁₂₇	1	2	yes	—
188	C₄⋊Q₈	C₂³×C₁₂₇	1	2	yes	—
189	#125	—	2	2	no	—
190	C₂²⋊Q₈	—	2	2	no	—
191	C₂²⋊Q₈	—	2	2	no	—
192	C₄×Q₈	—	2	2	no	—
193	C₄×Q₈	—	2	2	no	—
194	#117	—	2	2	no	—
195	C₂×C₄⋊C₄	—	2	2	no	—
196	C₂×C₄⋊C₄	—	2	2	no	—
197	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
198	#107	—	2	8	no	—
199	#107	—	2	2	no	—
200	#107	—	2	8	no	—
201	#106	—	2	8	no	—
202	C₄⋊C₈	D₄×C₁₂₇	1	2	yes	—
203	#103	—	2	4	no	—
204	#103	—	2	8	no	—
205	C₂²⋊C₈	D₄×C₁₂₇	1	2	yes	—
206	#97	D₄×C₁₂₇	1	2	yes	—
207	C₄×C₈	D₄×C₁₂₇	1	2	yes	—
208	C₄×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
209	C₄×C₈	C₂³×C₁₂₇	1	2	yes	—
210	C₄×C₈	C₂³×C₁₂₇	1	2	yes	—
211	C₄×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
212	C₂×Q₈	—	2	2	no	—
213	C₂×Q₈	—	2	2	no	—
214	C₂×Q₈	—	2	4	no	—
215	C₂×Q₈	—	4	8	no	—
216	C₂×Q₈	—	4	2	no	—
217	C₂²×C₄	—	4	2	no	—
218	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
219	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
220	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
221	Q₁₆	—	4	8	no	—
222	Q₁₆	—	4	4	no	—
223	Q₁₆	—	4	8	no	—
224	M₁₆	Q₁₆×C₁₂₇	1	8	yes	complemented
225	C₂×C₈	—	2	4	no	—
226	C₂×C₈	C₂×D₄×C₁₂₇	1	2	yes	—
227	C₂×C₈	C₂×D₄×C₁₂₇	1	4	yes	—
228	C₂×C₈	C₂×D₄×C₁₂₇	1	4	yes	—
229	C₂×C₈	C₂×D₄×C₁₂₇	1	2	yes	—
230	C₄⋊C₄	—	2	4	no	—
231	C₄⋊C₄	—	2	2	no	—
232	C₄⋊C₄	—	2	2	no	—
233	C₄⋊C₄	—	2	4	no	—
234	C₄⋊C₄	—	2	2	no	—
235	C₄⋊C₄	—	2	2	no	—
236	C₄⋊C₄	—	4	2	no	—
237	C₄⋊C₄	—	4	2	no	—
238	C₄⋊C₄	—	4	2	no	—
239	C₄⋊C₄	—	4	2	no	—
240	C₂²⋊C₄	—	4	2	no	—
241	C₄²	—	4	2	no	—
242	C₄²	C₂×D₄×C₁₂₇	1	1	yes	characteristic
243	C₄²	C₂×D₄×C₁₂₇	1	1	yes	characteristic
244	C₄²	C₂×D₄×C₁₂₇	1	1	yes	characteristic
245	C₄²	C₂⁴×C₁₂₇	1	1	yes	characteristic
246	C₂³	#127×C₁₂₇	1	1	yes	characteristic
247	Q₈	—	4	8	no	—
248	Q₈	—	4	4	no	—
249	Q₈	—	4	4	no	—
250	Q₈	—	8	4	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	—	4	2	no	—
255	C₂×C₄	—	4	4	no	—
256	C₂×C₄	—	8	4	no	—
257	C₂×C₄	#127×C₁₂₇	1	1	yes	characteristic
258	C₂×C₄	#127×C₁₂₇	1	1	yes	characteristic
259	C₂×C₄	#127×C₁₂₇	1	1	yes	characteristic
260	C₂×C₄	C₂×Q₁₆×C₁₂₇	1	4	yes	—
261	C₂×C₄	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
262	C₂×C₄	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
263	C₂×C₄	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
264	C₈	—	2	2	no	—
265	C₈	—	2	4	no	—
266	C₈	C₂×Q₁₆×C₁₂₇	1	8	yes	complemented
267	C₈	AGL(1,ℤ/8)×C₁₂₇	1	4	yes	—
268	C₂²	—	2	1	no	—
269	C₂²	—	2	1	no	—
270	C₂²	#494×C₁₂₇	1	2	yes	—
271	C₂²	C₂×#127×C₁₂₇	1	1	yes	characteristic
272	C₄	—	2	1	no	—
273	C₄	—	2	1	no	—
274	C₄	—	8	4	no	—
275	C₄	—	8	2	no	—
276	C₄	—	8	2	no	—
277	C₄	#494×C₁₂₇	1	2	yes	—
278	C₄	C₂²×Q₁₆×C₁₂₇	1	2	yes	—
279	C₄	C₂×AGL(1,ℤ/8)×C₁₂₇	1	2	yes	—
280	C₄	#578×C₁₂₇	1	2	yes	—
281	C₂	—	2	2	no	—
282	C₂	C₂×#494×C₁₂₇	1	1	yes	characteristic
283	C₂	C₁₂₇×#3151	1	1	yes	characteristic
284	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×#1723	C₂	semidirect
6. C₁₂₇×#1713	C₂	semidirect
7. C₁₂₇×#1707	C₂	semidirect
81. M₁₆×C₁₂₇	Q₁₆	semidirect
123. C₈×C₁₂₇	C₂×Q₁₆	semidirect
138. #19632	C₁₂₇	direct (N × H)
147. #1723	C₂×C₁₂₇	semidirect
148. #1713	C₂×C₁₂₇	semidirect
149. #1707	C₂×C₁₂₇	semidirect
151. C₁₂₇	#19632	direct (N × H)
224. M₁₆	Q₁₆×C₁₂₇	semidirect
266. C₈	C₂×Q₁₆×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₁₂₇×#3230	2
C₁₂₇×#3170	2
C₁₂₇×#3165	2
C₂×#494×C₁₂₇	2
C₁₂₇×#1723	2
C₁₂₇×#1713	2
C₁₂₇×#1707	2
C₈×M₁₆×C₁₂₇	2
#19632	127

Subgroups — C127×#19632 #44583305

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#19632 #44583305