Subgroups — C₁₂₇×#19635 #44583308

Order 32512.

878 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₂₇×#3208	C₂	1	2	yes	maximal, complemented
2	C₁₂₇×#3170	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#493×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₁₂₇×#3138	C₂	1	2	yes	maximal, complemented
5	C₁₂₇×#1717	C₂	1	4	yes	maximal, complemented
6	C₁₂₇×#1709	C₂	1	2	yes	maximal, complemented
7	C₁₂₇×#1706	C₂	1	2	yes	maximal, complemented
8	C₈×M₁₆×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₈×C₁₂₇	—	2	2	no	—
10	#533×C₁₂₇	C₂²	1	2	yes	—
11	C₂×#127×C₁₂₇	C₂²	1	2	yes	—
12	#500×C₁₂₇	C₂²	1	1	yes	characteristic, complemented
13	#494×C₁₂₇	C₂²	1	1	yes	characteristic, complemented
14	#493×C₁₂₇	—	2	2	no	—
15	#493×C₁₂₇	C₂²	1	2	yes	—
16	#493×C₁₂₇	C₂²	1	2	yes	—
17	#492×C₁₂₇	C₂²	1	4	yes	complemented
18	#490×C₁₂₇	—	2	4	no	—
19	#487×C₁₂₇	C₂²	1	2	yes	complemented
20	#486×C₁₂₇	C₂²	1	4	yes	complemented
21	#486×C₁₂₇	C₂²	1	4	yes	complemented
22	#486×C₁₂₇	C₂²	1	4	yes	complemented
23	#467×C₁₂₇	—	2	2	no	—
24	#460×C₁₂₇	—	2	4	no	—
25	#450×C₁₂₇	—	2	4	no	—
26	C₄×Q₁₆×C₁₂₇	—	2	2	no	—
27	#431×C₁₂₇	C₂²	1	1	yes	characteristic
28	#418×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	complemented
32	C₈²×C₁₂₇	C₂²	1	2	yes	complemented
33	C₂²×D₄×C₁₂₇	—	2	2	no	—
34	C₂²×D₄×C₁₂₇	—	2	2	no	—
35	C₂²×D₄×C₁₂₇	—	2	2	no	—
36	C₂×Q₁₆×C₁₂₇	—	2	2	no	—
37	C₂×SD₁₆×C₁₂₇	—	2	8	no	—
38	C₂×D₈×C₁₂₇	—	2	4	no	—
39	C₂×D₈×C₁₂₇	—	2	4	no	—
40	C₂×D₈×C₁₂₇	—	4	8	no	—
41	C₂×M₁₆×C₁₂₇	D₄	1	2	yes	—
42	C₂²×C₈×C₁₂₇	D₄	1	2	yes	—
43	C₄⋊Q₈×C₁₂₇	C₂³	1	2	yes	—
44	C₄⋊Q₈×C₁₂₇	C₂³	1	2	yes	—
45	#127×C₁₂₇	—	2	2	no	—
46	#127×C₁₂₇	—	2	2	no	—
47	#127×C₁₂₇	—	2	2	no	—
48	#127×C₁₂₇	C₂³	1	4	yes	—
49	#125×C₁₂₇	—	2	2	no	—
50	C₂²⋊Q₈×C₁₂₇	—	2	2	no	—
51	C₂²⋊Q₈×C₁₂₇	—	2	2	no	—
52	C₄×Q₈×C₁₂₇	—	2	2	no	—
53	C₄×Q₈×C₁₂₇	—	2	2	no	—
54	#117×C₁₂₇	—	2	2	no	—
55	C₂×C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
56	#107×C₁₂₇	—	2	2	no	—
57	C₄⋊C₈×C₁₂₇	D₄	1	2	yes	—
58	#103×C₁₂₇	—	2	4	no	—
59	#102×C₁₂₇	—	2	8	no	—
60	#102×C₁₂₇	—	2	8	no	—
61	#102×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	1	yes	characteristic
66	C₄×C₈×C₁₂₇	C₂³	1	2	yes	—
67	C₄×C₈×C₁₂₇	C₂³	1	2	yes	—
68	C₄×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
69	C₂⁴×C₁₂₇	—	4	2	no	—
70	C₂×Q₈×C₁₂₇	—	2	2	no	—
71	C₂×Q₈×C₁₂₇	—	2	2	no	—
72	C₂×D₄×C₁₂₇	—	2	4	no	—
73	C₂×D₄×C₁₂₇	—	2	4	no	—
74	C₂×D₄×C₁₂₇	—	2	4	no	—
75	C₂×D₄×C₁₂₇	—	4	8	no	—
76	C₂×D₄×C₁₂₇	—	4	2	no	—
77	C₂×D₄×C₁₂₇	—	4	2	no	—
78	C₂×D₄×C₁₂₇	—	4	2	no	—
79	C₂×D₄×C₁₂₇	—	4	8	no	—
80	C₂×D₄×C₁₂₇	—	8	4	no	—
81	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
84	Q₁₆×C₁₂₇	—	4	4	no	—
85	SD₁₆×C₁₂₇	—	8	8	no	—
86	D₈×C₁₂₇	—	4	8	no	—
87	D₈×C₁₂₇	—	4	8	no	—
88	M₁₆×C₁₂₇	—	2	4	no	—
89	C₂×C₈×C₁₂₇	—	2	4	no	—
90	C₂×C₈×C₁₂₇	C₂×D₄	1	2	yes	—
91	C₂×C₈×C₁₂₇	C₂×D₄	1	4	yes	—
92	C₂×C₈×C₁₂₇	C₂×D₄	1	4	yes	—
93	C₂×C₈×C₁₂₇	C₂×D₄	1	2	yes	—
94	C₄⋊C₄×C₁₂₇	—	2	2	no	—
95	C₄⋊C₄×C₁₂₇	—	2	2	no	—
96	C₄⋊C₄×C₁₂₇	—	2	2	no	—
97	C₄⋊C₄×C₁₂₇	—	2	2	no	—
98	C₄⋊C₄×C₁₂₇	—	4	2	no	—
99	C₄⋊C₄×C₁₂₇	—	4	2	no	—
100	C₂²⋊C₄×C₁₂₇	—	4	2	no	—
101	C₄²×C₁₂₇	—	4	2	no	—
102	C₄²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
103	C₄²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
104	C₄²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
105	C₄²×C₁₂₇	C₂⁴	1	1	yes	characteristic
106	C₂³×C₁₂₇	—	4	4	no	—
107	C₂³×C₁₂₇	—	8	4	no	—
108	C₂³×C₁₂₇	—	8	4	no	—
109	C₂³×C₁₂₇	—	8	4	no	—
110	C₂³×C₁₂₇	#127	1	1	yes	characteristic
111	Q₈×C₁₂₇	—	4	4	no	—
112	Q₈×C₁₂₇	—	4	4	no	—
113	D₄×C₁₂₇	—	4	8	no	—
114	D₄×C₁₂₇	—	4	8	no	—
115	D₄×C₁₂₇	—	4	8	no	—
116	D₄×C₁₂₇	—	8	4	no	—
117	D₄×C₁₂₇	—	8	4	no	—
118	D₄×C₁₂₇	—	8	4	no	—
119	C₂×C₄×C₁₂₇	—	2	2	no	—
120	C₂×C₄×C₁₂₇	—	2	2	no	—
121	C₂×C₄×C₁₂₇	—	4	2	no	—
122	C₂×C₄×C₁₂₇	—	4	2	no	—
123	C₂×C₄×C₁₂₇	#127	1	1	yes	characteristic
124	C₂×C₄×C₁₂₇	#127	1	1	yes	characteristic
125	C₂×C₄×C₁₂₇	#127	1	1	yes	characteristic
126	C₂×C₄×C₁₂₇	#135	1	4	yes	—
127	C₂×C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
128	C₂×C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
129	C₂×C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
130	C₈×C₁₂₇	—	2	4	no	—
131	C₈×C₁₂₇	—	2	2	no	—
132	C₈×C₁₂₇	—	2	4	no	—
133	C₈×C₁₂₇	AGL(1,ℤ/8)	1	4	yes	—
134	C₂²×C₁₂₇	—	2	1	no	—
135	C₂²×C₁₂₇	—	2	1	no	—
136	C₂²×C₁₂₇	—	8	4	no	—
137	C₂²×C₁₂₇	—	8	4	no	—
138	C₂²×C₁₂₇	—	8	4	no	—
139	C₂²×C₁₂₇	—	16	8	no	—
140	C₂²×C₁₂₇	#495	1	2	yes	—
141	C₂²×C₁₂₇	C₂×#127	1	1	yes	characteristic
142	C₄×C₁₂₇	—	2	1	no	—
143	C₄×C₁₂₇	—	2	1	no	—
144	C₄×C₁₂₇	—	8	2	no	—
145	C₄×C₁₂₇	—	8	2	no	—
146	C₄×C₁₂₇	#495	1	2	yes	—
147	C₄×C₁₂₇	C₂×#135	1	2	yes	—
148	C₄×C₁₂₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
149	C₄×C₁₂₇	#576	1	2	yes	—
150	#19635	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
151	C₂×C₁₂₇	—	2	2	no	—
152	C₂×C₁₂₇	—	16	4	no	—
153	C₂×C₁₂₇	C₂×#495	1	1	yes	characteristic
154	C₂×C₁₂₇	#3151	1	1	yes	characteristic
155	C₂×C₁₂₇	#3155	1	1	yes	characteristic
156	#3208	C₂×C₁₂₇	1	2	yes	complemented
157	#3170	C₂×C₁₂₇	1	1	yes	characteristic, complemented
158	C₂×#493	C₂×C₁₂₇	1	1	yes	characteristic
159	#3138	C₂×C₁₂₇	1	2	yes	complemented
160	#1717	C₂×C₁₂₇	1	4	yes	complemented
161	#1709	C₂×C₁₂₇	1	2	yes	complemented
162	#1706	C₂×C₁₂₇	1	2	yes	complemented
163	C₈×M₁₆	C₂×C₁₂₇	1	1	yes	characteristic, complemented
164	C₁₂₇	#19635	1	1	yes	characteristic, direct product factor, Sylow
165	C₂²×D₈	—	2	2	no	—
166	#533	C₂²×C₁₂₇	1	2	yes	—
167	C₂×#127	C₂²×C₁₂₇	1	2	yes	—
168	#500	C₂²×C₁₂₇	1	1	yes	characteristic, complemented
169	#494	C₂²×C₁₂₇	1	1	yes	characteristic, complemented
170	#493	—	2	2	no	—
171	#493	C₂²×C₁₂₇	1	2	yes	—
172	#493	C₂²×C₁₂₇	1	2	yes	—
173	#492	C₂²×C₁₂₇	1	4	yes	complemented
174	#490	—	2	4	no	—
175	#487	C₂²×C₁₂₇	1	2	yes	complemented
176	#486	C₂²×C₁₂₇	1	4	yes	complemented
177	#486	C₂²×C₁₂₇	1	4	yes	complemented
178	#486	C₂²×C₁₂₇	1	4	yes	complemented
179	#467	—	2	2	no	—
180	#460	—	2	4	no	—
181	#450	—	2	4	no	—
182	C₄×Q₁₆	—	2	2	no	—
183	#431	C₂²×C₁₂₇	1	1	yes	characteristic
184	#418	—	2	4	no	—
185	C₄×M₁₆	C₂²×C₁₂₇	1	1	yes	characteristic
186	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
187	#322	C₂²×C₁₂₇	1	2	yes	complemented
188	C₈²	C₂²×C₁₂₇	1	2	yes	complemented
189	C₂²×D₄	—	2	2	no	—
190	C₂²×D₄	—	2	2	no	—
191	C₂²×D₄	—	2	2	no	—
192	C₂×Q₁₆	—	2	2	no	—
193	C₂×SD₁₆	—	2	8	no	—
194	C₂×D₈	—	2	4	no	—
195	C₂×D₈	—	2	4	no	—
196	C₂×D₈	—	4	8	no	—
197	C₂×M₁₆	D₄×C₁₂₇	1	2	yes	—
198	C₂²×C₈	D₄×C₁₂₇	1	2	yes	—
199	C₄⋊Q₈	C₂³×C₁₂₇	1	2	yes	—
200	C₄⋊Q₈	C₂³×C₁₂₇	1	2	yes	—
201	#127	—	2	2	no	—
202	#127	—	2	2	no	—
203	#127	—	2	2	no	—
204	#127	C₂³×C₁₂₇	1	4	yes	—
205	#125	—	2	2	no	—
206	C₂²⋊Q₈	—	2	2	no	—
207	C₂²⋊Q₈	—	2	2	no	—
208	C₄×Q₈	—	2	2	no	—
209	C₄×Q₈	—	2	2	no	—
210	#117	—	2	2	no	—
211	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
212	#107	—	2	2	no	—
213	C₄⋊C₈	D₄×C₁₂₇	1	2	yes	—
214	#103	—	2	4	no	—
215	#102	—	2	8	no	—
216	#102	—	2	8	no	—
217	#102	—	2	8	no	—
218	C₂²⋊C₈	D₄×C₁₂₇	1	2	yes	—
219	#97	D₄×C₁₂₇	1	2	yes	—
220	C₄×C₈	D₄×C₁₂₇	1	2	yes	—
221	C₄×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
222	C₄×C₈	C₂³×C₁₂₇	1	2	yes	—
223	C₄×C₈	C₂³×C₁₂₇	1	2	yes	—
224	C₄×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
225	C₂⁴	—	4	2	no	—
226	C₂×Q₈	—	2	2	no	—
227	C₂×Q₈	—	2	2	no	—
228	C₂×D₄	—	2	4	no	—
229	C₂×D₄	—	2	4	no	—
230	C₂×D₄	—	2	4	no	—
231	C₂×D₄	—	4	8	no	—
232	C₂×D₄	—	4	2	no	—
233	C₂×D₄	—	4	2	no	—
234	C₂×D₄	—	4	2	no	—
235	C₂×D₄	—	4	8	no	—
236	C₂×D₄	—	8	4	no	—
237	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
238	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
239	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
240	Q₁₆	—	4	4	no	—
241	SD₁₆	—	8	8	no	—
242	D₈	—	4	8	no	—
243	D₈	—	4	8	no	—
244	M₁₆	—	2	4	no	—
245	C₂×C₈	—	2	4	no	—
246	C₂×C₈	C₂×D₄×C₁₂₇	1	2	yes	—
247	C₂×C₈	C₂×D₄×C₁₂₇	1	4	yes	—
248	C₂×C₈	C₂×D₄×C₁₂₇	1	4	yes	—
249	C₂×C₈	C₂×D₄×C₁₂₇	1	2	yes	—
250	C₄⋊C₄	—	2	2	no	—
251	C₄⋊C₄	—	2	2	no	—
252	C₄⋊C₄	—	2	2	no	—
253	C₄⋊C₄	—	2	2	no	—
254	C₄⋊C₄	—	4	2	no	—
255	C₄⋊C₄	—	4	2	no	—
256	C₂²⋊C₄	—	4	2	no	—
257	C₄²	—	4	2	no	—
258	C₄²	C₂×D₄×C₁₂₇	1	1	yes	characteristic
259	C₄²	C₂×D₄×C₁₂₇	1	1	yes	characteristic
260	C₄²	C₂×D₄×C₁₂₇	1	1	yes	characteristic
261	C₄²	C₂⁴×C₁₂₇	1	1	yes	characteristic
262	C₂³	—	4	4	no	—
263	C₂³	—	8	4	no	—
264	C₂³	—	8	4	no	—
265	C₂³	—	8	4	no	—
266	C₂³	#127×C₁₂₇	1	1	yes	characteristic
267	Q₈	—	4	4	no	—
268	Q₈	—	4	4	no	—
269	D₄	—	4	8	no	—
270	D₄	—	4	8	no	—
271	D₄	—	4	8	no	—
272	D₄	—	8	4	no	—
273	D₄	—	8	4	no	—
274	D₄	—	8	4	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	#127×C₁₂₇	1	1	yes	characteristic
280	C₂×C₄	#127×C₁₂₇	1	1	yes	characteristic
281	C₂×C₄	#127×C₁₂₇	1	1	yes	characteristic
282	C₂×C₄	#135×C₁₂₇	1	4	yes	—
283	C₂×C₄	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
284	C₂×C₄	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
285	C₂×C₄	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
286	C₈	—	2	4	no	—
287	C₈	—	2	2	no	—
288	C₈	—	2	4	no	—
289	C₈	AGL(1,ℤ/8)×C₁₂₇	1	4	yes	—
290	C₂²	—	2	1	no	—
291	C₂²	—	2	1	no	—
292	C₂²	—	8	4	no	—
293	C₂²	—	8	4	no	—
294	C₂²	—	8	4	no	—
295	C₂²	—	16	8	no	—
296	C₂²	#495×C₁₂₇	1	2	yes	—
297	C₂²	C₂×#127×C₁₂₇	1	1	yes	characteristic
298	C₄	—	2	1	no	—
299	C₄	—	2	1	no	—
300	C₄	—	8	2	no	—
301	C₄	—	8	2	no	—
302	C₄	#495×C₁₂₇	1	2	yes	—
303	C₄	C₂×#135×C₁₂₇	1	2	yes	—
304	C₄	C₂×AGL(1,ℤ/8)×C₁₂₇	1	2	yes	—
305	C₄	#576×C₁₂₇	1	2	yes	—
306	C₂	—	2	2	no	—
307	C₂	—	16	4	no	—
308	C₂	C₂×#495×C₁₂₇	1	1	yes	characteristic
309	C₂	C₁₂₇×#3151	1	1	yes	characteristic
310	C₂	C₁₂₇×#3155	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₂₇	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₁₂₇
prime radical	C₂×#493×C₁₂₇	C₂
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
1. C₁₂₇×#3208	C₂	semidirect
2. C₁₂₇×#3170	C₂	semidirect
4. C₁₂₇×#3138	C₂	semidirect
5. C₁₂₇×#1717	C₂	semidirect
6. C₁₂₇×#1709	C₂	semidirect
7. C₁₂₇×#1706	C₂	semidirect
8. C₈×M₁₆×C₁₂₇	C₂	semidirect
12. #500×C₁₂₇	C₂²	semidirect
13. #494×C₁₂₇	C₂²	semidirect
17. #492×C₁₂₇	C₂²	semidirect
19. #487×C₁₂₇	C₂²	semidirect
20. #486×C₁₂₇	C₂²	semidirect
21. #486×C₁₂₇	C₂²	semidirect
22. #486×C₁₂₇	C₂²	semidirect
31. #322×C₁₂₇	C₂²	semidirect
32. C₈²×C₁₂₇	C₂²	semidirect
150. #19635	C₁₂₇	direct (N × H)
156. #3208	C₂×C₁₂₇	semidirect
157. #3170	C₂×C₁₂₇	semidirect
159. #3138	C₂×C₁₂₇	semidirect
160. #1717	C₂×C₁₂₇	semidirect
161. #1709	C₂×C₁₂₇	semidirect
162. #1706	C₂×C₁₂₇	semidirect
163. C₈×M₁₆	C₂×C₁₂₇	semidirect
164. C₁₂₇	#19635	direct (N × H)
168. #500	C₂²×C₁₂₇	semidirect
169. #494	C₂²×C₁₂₇	semidirect
173. #492	C₂²×C₁₂₇	semidirect
175. #487	C₂²×C₁₂₇	semidirect
176. #486	C₂²×C₁₂₇	semidirect
177. #486	C₂²×C₁₂₇	semidirect
178. #486	C₂²×C₁₂₇	semidirect
187. #322	C₂²×C₁₂₇	semidirect
188. C₈²	C₂²×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₁₂₇×#3208	2
C₁₂₇×#3170	2
C₂×#493×C₁₂₇	2
C₁₂₇×#3138	2
C₁₂₇×#1717	2
C₁₂₇×#1709	2
C₁₂₇×#1706	2
C₈×M₁₆×C₁₂₇	2
#19635	127

Subgroups — C127×#19635 #44583308

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#19635 #44583308