Subgroups — C₁₂₇×#9601 #44573274

Order 32512.

404 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×#1842	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊C₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	C₄×C₄⋊C₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#335×C₁₂₇	C₂	1	2	yes	maximal
5	C₁₂₇×#1277	C₂	1	2	yes	maximal
6	C₂×C₄⋊C₈×C₁₂₇	—	2	1	no	—
7	C₂×C₄⋊C₈×C₁₂₇	—	2	1	no	—
8	C₂×C₄⋊C₈×C₁₂₇	C₄	1	1	yes	characteristic
9	C₂×C₄⋊C₈×C₁₂₇	C₄	1	1	yes	characteristic
10	C₂×C₄⋊C₈×C₁₂₇	C₂²	1	2	yes	—
11	C₂×C₄⋊C₈×C₁₂₇	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊C₈×C₁₂₇	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₁₂₇	—	2	1	no	—
14	C₂×C₄×C₈×C₁₂₇	C₂²	1	2	yes	—
15	C₄³×C₁₂₇	C₂²	1	1	yes	characteristic
16	#336×C₁₂₇	—	2	1	no	—
17	#336×C₁₂₇	C₄	1	2	yes	—
18	#335×C₁₂₇	—	2	4	no	—
19	C₂²×C₈×C₁₂₇	—	2	1	no	—
20	C₂²×C₈×C₁₂₇	—	2	1	no	—
21	C₂²×C₈×C₁₂₇	—	2	2	no	—
22	C₂²×C₈×C₁₂₇	C₂×C₄	1	2	yes	—
23	C₂²×C₈×C₁₂₇	C₂×C₄	1	2	yes	—
24	C₂×C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²×C₁₂₇	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₁₂₇	D₄	1	1	yes	characteristic
27	C₂×C₄²×C₁₂₇	D₄	1	1	yes	characteristic
28	C₂×C₄²×C₁₂₇	Q₈	1	1	yes	characteristic
29	C₂×C₄²×C₁₂₇	Q₈	1	1	yes	characteristic
30	C₂×C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
31	C₄⋊C₈×C₁₂₇	—	2	2	no	—
32	C₄⋊C₈×C₁₂₇	—	2	2	no	—
33	C₄⋊C₈×C₁₂₇	—	2	1	no	—
34	C₄⋊C₈×C₁₂₇	—	2	1	no	—
35	C₄⋊C₈×C₁₂₇	—	2	2	no	—
36	C₄⋊C₈×C₁₂₇	—	4	2	no	—
37	C₄⋊C₈×C₁₂₇	—	4	2	no	—
38	C₄⋊C₈×C₁₂₇	C₈	1	4	yes	complemented
39	C₄⋊C₈×C₁₂₇	D₄	1	2	yes	—
40	C₄⋊C₈×C₁₂₇	Q₈	1	2	yes	—
41	C₄×C₈×C₁₂₇	—	2	2	no	—
42	C₄×C₈×C₁₂₇	—	2	2	no	—
43	C₄×C₈×C₁₂₇	—	4	2	no	—
44	C₂²×C₄×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
45	C₂²×C₄×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
46	C₂²×C₄×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
47	C₂²×C₄×C₁₂₇	C₄⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₈×C₁₂₇	—	2	2	no	—
52	C₂×C₈×C₁₂₇	—	2	2	no	—
53	C₂×C₈×C₁₂₇	—	2	2	no	—
54	C₂×C₈×C₁₂₇	—	2	2	no	—
55	C₂×C₈×C₁₂₇	—	2	2	no	—
56	C₂×C₈×C₁₂₇	—	2	2	no	—
57	C₂×C₈×C₁₂₇	—	2	2	no	—
58	C₂×C₈×C₁₂₇	—	4	1	no	—
59	C₂×C₈×C₁₂₇	—	4	1	no	—
60	C₂×C₈×C₁₂₇	—	4	1	no	—
61	C₂×C₈×C₁₂₇	—	4	1	no	—
62	C₂×C₈×C₁₂₇	—	4	2	no	—
63	C₂×C₈×C₁₂₇	—	4	2	no	—
64	C₂×C₈×C₁₂₇	—	4	2	no	—
65	C₂×C₈×C₁₂₇	C₂×C₈	1	4	yes	—
66	C₄²×C₁₂₇	—	2	2	no	—
67	C₄²×C₁₂₇	—	2	1	no	—
68	C₄²×C₁₂₇	—	2	1	no	—
69	C₄²×C₁₂₇	—	2	1	no	—
70	C₄²×C₁₂₇	—	2	1	no	—
71	C₄²×C₁₂₇	—	4	1	no	—
72	C₄²×C₁₂₇	C₂×C₈	1	2	yes	—
73	C₄²×C₁₂₇	M₁₆	1	2	yes	—
74	C₄²×C₁₂₇	D₈	1	2	yes	—
75	C₄²×C₁₂₇	Q₁₆	1	2	yes	—
76	C₄²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
77	C₄²×C₁₂₇	C₂×Q₈	1	1	yes	characteristic
78	C₄²×C₁₂₇	#41	1	1	yes	characteristic
79	C₄²×C₁₂₇	#41	1	1	yes	characteristic
80	C₂³×C₁₂₇	C₂×C₄⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₁₂₇	—	2	1	no	—
82	C₂×C₄×C₁₂₇	—	2	1	no	—
83	C₂×C₄×C₁₂₇	—	2	1	no	—
84	C₂×C₄×C₁₂₇	—	2	1	no	—
85	C₂×C₄×C₁₂₇	—	2	1	no	—
86	C₂×C₄×C₁₂₇	—	2	1	no	—
87	C₂×C₄×C₁₂₇	—	2	1	no	—
88	C₂×C₄×C₁₂₇	—	2	1	no	—
89	C₂×C₄×C₁₂₇	—	2	1	no	—
90	C₂×C₄×C₁₂₇	C₄⋊C₈	1	2	yes	—
91	C₂×C₄×C₁₂₇	C₄⋊C₈	1	2	yes	—
92	C₂×C₄×C₁₂₇	#107	1	2	yes	—
93	C₂×C₄×C₁₂₇	#107	1	2	yes	—
94	C₂×C₄×C₁₂₇	#108	1	2	yes	—
95	C₂×C₄×C₁₂₇	C₄×D₄	1	1	yes	characteristic
96	C₂×C₄×C₁₂₇	C₄×Q₈	1	1	yes	characteristic
97	C₂×C₄×C₁₂₇	#121	1	1	yes	characteristic
98	C₂×C₄×C₁₂₇	C₂²⋊Q₈	1	1	yes	characteristic
99	C₂×C₄×C₁₂₇	#125	1	1	yes	characteristic
100	C₂×C₄×C₁₂₇	C₄⋊Q₈	1	1	yes	characteristic
101	C₂×C₄×C₁₂₇	C₂²×C₈	1	1	yes	characteristic
102	C₂×C₄×C₁₂₇	C₂×M₁₆	1	1	yes	characteristic
103	C₂×C₄×C₁₂₇	#131	1	1	yes	characteristic
104	C₂×C₄×C₁₂₇	#131	1	1	yes	characteristic
105	C₂×C₄×C₁₂₇	C₂×D₈	1	1	yes	characteristic
106	C₂×C₄×C₁₂₇	C₂×Q₁₆	1	1	yes	characteristic
107	C₂×C₄×C₁₂₇	AGL(1,ℤ/8)	1	1	yes	characteristic
108	C₂×C₄×C₁₂₇	#137	1	1	yes	characteristic
109	C₈×C₁₂₇	—	2	4	no	—
110	C₈×C₁₂₇	—	4	2	no	—
111	C₈×C₁₂₇	—	4	2	no	—
112	C₈×C₁₂₇	—	4	2	no	—
113	C₈×C₁₂₇	—	8	2	no	—
114	C₂²×C₁₂₇	#389	1	1	yes	characteristic
115	C₂²×C₁₂₇	C₂×C₄⋊C₈	1	1	yes	characteristic
116	C₂²×C₁₂₇	#424	1	1	yes	characteristic
117	C₂²×C₁₂₇	C₂×#107	1	1	yes	characteristic
118	C₂²×C₁₂₇	#428	1	1	yes	characteristic
119	C₂²×C₁₂₇	C₂×#108	1	1	yes	characteristic
120	C₂²×C₁₂₇	#430	1	1	yes	characteristic
121	C₄×C₁₂₇	—	2	2	no	—
122	C₄×C₁₂₇	—	2	2	no	—
123	C₄×C₁₂₇	—	2	1	no	—
124	C₄×C₁₂₇	—	2	1	no	—
125	C₄×C₁₂₇	—	4	1	no	—
126	C₄×C₁₂₇	#335	1	4	yes	complemented
127	C₄×C₁₂₇	C₈×D₄	1	1	yes	characteristic
128	C₄×C₁₂₇	#436	1	1	yes	characteristic
129	C₄×C₁₂₇	C₈×Q₈	1	1	yes	characteristic
130	C₄×C₁₂₇	#446	1	1	yes	characteristic
131	C₄×C₁₂₇	#459	1	1	yes	characteristic
132	C₄×C₁₂₇	#462	1	1	yes	characteristic
133	C₄×C₁₂₇	#474	1	1	yes	characteristic
134	C₄×C₁₂₇	#477	1	1	yes	characteristic
135	#9601	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
136	C₂×C₁₂₇	C₂×#335	1	1	yes	characteristic
137	C₂×C₁₂₇	#1565	1	1	yes	characteristic
138	C₂×C₁₂₇	#1571	1	1	yes	characteristic
139	C₂×C₁₂₇	#1574	1	1	yes	characteristic
140	C₂×C₁₂₇	#1939	1	1	yes	characteristic
141	C₂×C₁₂₇	#1947	1	1	yes	characteristic
142	C₂×C₁₂₇	#1950	1	1	yes	characteristic
143	#1842	C₂×C₁₂₇	1	1	yes	characteristic
144	C₄×C₄⋊C₈	C₂×C₁₂₇	1	1	yes	characteristic
145	C₄×C₄⋊C₈	C₂×C₁₂₇	1	1	yes	characteristic
146	C₂×#335	C₂×C₁₂₇	1	2	yes	—
147	#1277	C₂×C₁₂₇	1	2	yes	—
148	C₁₂₇	#9601	1	1	yes	characteristic, direct product factor, Sylow
149	C₂×C₄⋊C₈	—	2	1	no	—
150	C₂×C₄⋊C₈	—	2	1	no	—
151	C₂×C₄⋊C₈	C₄×C₁₂₇	1	1	yes	characteristic
152	C₂×C₄⋊C₈	C₄×C₁₂₇	1	1	yes	characteristic
153	C₂×C₄⋊C₈	C₂²×C₁₂₇	1	2	yes	—
154	C₂×C₄⋊C₈	C₂²×C₁₂₇	1	1	yes	characteristic
155	C₂×C₄⋊C₈	C₂²×C₁₂₇	1	1	yes	characteristic
156	C₂×C₄×C₈	—	2	1	no	—
157	C₂×C₄×C₈	C₂²×C₁₂₇	1	2	yes	—
158	C₄³	C₂²×C₁₂₇	1	1	yes	characteristic
159	#336	—	2	1	no	—
160	#336	C₄×C₁₂₇	1	2	yes	—
161	#335	—	2	4	no	—
162	C₂²×C₈	—	2	1	no	—
163	C₂²×C₈	—	2	1	no	—
164	C₂²×C₈	—	2	2	no	—
165	C₂²×C₈	C₂×C₄×C₁₂₇	1	2	yes	—
166	C₂²×C₈	C₂×C₄×C₁₂₇	1	2	yes	—
167	C₂×C₄²	C₂×C₄×C₁₂₇	1	1	yes	characteristic
168	C₂×C₄²	C₂×C₄×C₁₂₇	1	1	yes	characteristic
169	C₂×C₄²	D₄×C₁₂₇	1	1	yes	characteristic
170	C₂×C₄²	D₄×C₁₂₇	1	1	yes	characteristic
171	C₂×C₄²	Q₈×C₁₂₇	1	1	yes	characteristic
172	C₂×C₄²	Q₈×C₁₂₇	1	1	yes	characteristic
173	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
174	C₄⋊C₈	—	2	2	no	—
175	C₄⋊C₈	—	2	2	no	—
176	C₄⋊C₈	—	2	1	no	—
177	C₄⋊C₈	—	2	1	no	—
178	C₄⋊C₈	—	2	2	no	—
179	C₄⋊C₈	—	4	2	no	—
180	C₄⋊C₈	—	4	2	no	—
181	C₄⋊C₈	C₈×C₁₂₇	1	4	yes	complemented
182	C₄⋊C₈	D₄×C₁₂₇	1	2	yes	—
183	C₄⋊C₈	Q₈×C₁₂₇	1	2	yes	—
184	C₄×C₈	—	2	2	no	—
185	C₄×C₈	—	2	2	no	—
186	C₄×C₈	—	4	2	no	—
187	C₂²×C₄	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
188	C₂²×C₄	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
189	C₂²×C₄	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
190	C₂²×C₄	C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
191	C₂²×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
192	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
193	C₂²×C₄	C₂×Q₈×C₁₂₇	1	1	yes	characteristic
194	C₂×C₈	—	2	2	no	—
195	C₂×C₈	—	2	2	no	—
196	C₂×C₈	—	2	2	no	—
197	C₂×C₈	—	2	2	no	—
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	—	4	1	no	—
202	C₂×C₈	—	4	1	no	—
203	C₂×C₈	—	4	1	no	—
204	C₂×C₈	—	4	1	no	—
205	C₂×C₈	—	4	2	no	—
206	C₂×C₈	—	4	2	no	—
207	C₂×C₈	—	4	2	no	—
208	C₂×C₈	C₂×C₈×C₁₂₇	1	4	yes	—
209	C₄²	—	2	2	no	—
210	C₄²	—	2	1	no	—
211	C₄²	—	2	1	no	—
212	C₄²	—	2	1	no	—
213	C₄²	—	2	1	no	—
214	C₄²	—	4	1	no	—
215	C₄²	C₂×C₈×C₁₂₇	1	2	yes	—
216	C₄²	M₁₆×C₁₂₇	1	2	yes	—
217	C₄²	D₈×C₁₂₇	1	2	yes	—
218	C₄²	Q₁₆×C₁₂₇	1	2	yes	—
219	C₄²	C₂×D₄×C₁₂₇	1	1	yes	characteristic
220	C₄²	C₂×Q₈×C₁₂₇	1	1	yes	characteristic
221	C₄²	#41×C₁₂₇	1	1	yes	characteristic
222	C₄²	#41×C₁₂₇	1	1	yes	characteristic
223	C₂³	C₂×C₄⋊C₄×C₁₂₇	1	1	yes	characteristic
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	C₄⋊C₈×C₁₂₇	1	2	yes	—
234	C₂×C₄	C₄⋊C₈×C₁₂₇	1	2	yes	—
235	C₂×C₄	#107×C₁₂₇	1	2	yes	—
236	C₂×C₄	#107×C₁₂₇	1	2	yes	—
237	C₂×C₄	#108×C₁₂₇	1	2	yes	—
238	C₂×C₄	C₄×D₄×C₁₂₇	1	1	yes	characteristic
239	C₂×C₄	C₄×Q₈×C₁₂₇	1	1	yes	characteristic
240	C₂×C₄	#121×C₁₂₇	1	1	yes	characteristic
241	C₂×C₄	C₂²⋊Q₈×C₁₂₇	1	1	yes	characteristic
242	C₂×C₄	#125×C₁₂₇	1	1	yes	characteristic
243	C₂×C₄	C₄⋊Q₈×C₁₂₇	1	1	yes	characteristic
244	C₂×C₄	C₂²×C₈×C₁₂₇	1	1	yes	characteristic
245	C₂×C₄	C₂×M₁₆×C₁₂₇	1	1	yes	characteristic
246	C₂×C₄	#131×C₁₂₇	1	1	yes	characteristic
247	C₂×C₄	#131×C₁₂₇	1	1	yes	characteristic
248	C₂×C₄	C₂×D₈×C₁₂₇	1	1	yes	characteristic
249	C₂×C₄	C₂×Q₁₆×C₁₂₇	1	1	yes	characteristic
250	C₂×C₄	AGL(1,ℤ/8)×C₁₂₇	1	1	yes	characteristic
251	C₂×C₄	#137×C₁₂₇	1	1	yes	characteristic
252	C₈	—	2	4	no	—
253	C₈	—	4	2	no	—
254	C₈	—	4	2	no	—
255	C₈	—	4	2	no	—
256	C₈	—	8	2	no	—
257	C₂²	#389×C₁₂₇	1	1	yes	characteristic
258	C₂²	C₂×C₄⋊C₈×C₁₂₇	1	1	yes	characteristic
259	C₂²	#424×C₁₂₇	1	1	yes	characteristic
260	C₂²	C₂×#107×C₁₂₇	1	1	yes	characteristic
261	C₂²	#428×C₁₂₇	1	1	yes	characteristic
262	C₂²	C₂×#108×C₁₂₇	1	1	yes	characteristic
263	C₂²	#430×C₁₂₇	1	1	yes	characteristic
264	C₄	—	2	2	no	—
265	C₄	—	2	2	no	—
266	C₄	—	2	1	no	—
267	C₄	—	2	1	no	—
268	C₄	—	4	1	no	—
269	C₄	#335×C₁₂₇	1	4	yes	complemented
270	C₄	C₈×D₄×C₁₂₇	1	1	yes	characteristic
271	C₄	#436×C₁₂₇	1	1	yes	characteristic
272	C₄	C₈×Q₈×C₁₂₇	1	1	yes	characteristic
273	C₄	#446×C₁₂₇	1	1	yes	characteristic
274	C₄	#459×C₁₂₇	1	1	yes	characteristic
275	C₄	#462×C₁₂₇	1	1	yes	characteristic
276	C₄	#474×C₁₂₇	1	1	yes	characteristic
277	C₄	#477×C₁₂₇	1	1	yes	characteristic
278	C₂	C₂×#335×C₁₂₇	1	1	yes	characteristic
279	C₂	C₁₂₇×#1565	1	1	yes	characteristic
280	C₂	C₁₂₇×#1571	1	1	yes	characteristic
281	C₂	C₁₂₇×#1574	1	1	yes	characteristic
282	C₂	C₁₂₇×#1939	1	1	yes	characteristic
283	C₂	C₁₂₇×#1947	1	1	yes	characteristic
284	C₂	C₁₂₇×#1950	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₁₂₇	C₂×D₄
commutator	C₂×C₄	C₂²×C₈×C₁₂₇
Frattini, cosocle	C₂×C₄²	C₂³×C₁₂₇
socle, prime radical	C₂³×C₁₂₇	C₂×C₄⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₂₇ ◁ C₄³×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
38. C₄⋊C₈×C₁₂₇	C₈	semidirect
126. C₄×C₁₂₇	#335	semidirect
135. #9601	C₁₂₇	direct (N × H)
148. C₁₂₇	#9601	direct (N × H)
181. C₄⋊C₈	C₈×C₁₂₇	semidirect
269. C₄	#335×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₁₂₇×#1842	2
C₄×C₄⋊C₈×C₁₂₇	2
C₂×#335×C₁₂₇	2
C₁₂₇×#1277	2
#9601	127

Subgroups — C127×#9601 #44573274

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#9601 #44573274