Subgroups — C₁₂₇×#21698 #44585371

Order 32512.

1014 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#494×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#441×C₁₂₇	C₂	1	2	yes	maximal
3	C₂×C₄×Q₁₆×C₁₂₇	C₂	1	2	yes	maximal
4	C₁₂₇×#2327	C₂	1	2	yes	maximal
5	C₁₂₇×#1971	C₂	1	2	yes	maximal
6	C₁₂₇×#1969	C₂	1	2	yes	maximal
7	C₁₂₇×#1960	C₂	1	2	yes	maximal
8	C₁₂₇×#1846	C₂	1	1	yes	maximal, characteristic
9	C₁₂₇×#1726	C₂	1	1	yes	maximal, characteristic
10	C₂²×Q₁₆×C₁₂₇	C₂²	1	2	yes	—
11	C₂²×Q₁₆×C₁₂₇	C₂²	1	2	yes	—
12	C₂×C₄⋊Q₈×C₁₂₇	C₂²	1	2	yes	—
13	C₂×C₄×Q₈×C₁₂₇	C₂²	1	4	yes	—
14	#494×C₁₂₇	C₄	1	8	yes	complemented
15	#441×C₁₂₇	—	2	8	no	—
16	C₄×Q₁₆×C₁₂₇	—	2	8	no	—
17	C₂×#107×C₁₂₇	C₂²	1	2	yes	—
18	C₂×#106×C₁₂₇	C₂²	1	2	yes	—
19	C₂×#103×C₁₂₇	C₂²	1	4	yes	—
20	C₂×#103×C₁₂₇	C₂²	1	4	yes	—
21	C₂×#97×C₁₂₇	C₂²	1	1	yes	characteristic
22	C₂×#97×C₁₂₇	C₂²	1	1	yes	characteristic
23	C₂×#97×C₁₂₇	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
27	#391×C₁₂₇	C₂²	1	4	yes	—
28	#387×C₁₂₇	—	2	8	no	—
29	#382×C₁₂₇	C₂²	1	2	yes	—
30	#376×C₁₂₇	C₂²	1	1	yes	characteristic
31	C₂²×Q₈×C₁₂₇	C₂³	1	4	yes	—
32	C₂×Q₁₆×C₁₂₇	—	2	4	no	—
33	C₂×Q₁₆×C₁₂₇	—	2	4	no	—
34	C₂×Q₁₆×C₁₂₇	—	2	4	no	—
35	C₂×Q₁₆×C₁₂₇	—	2	4	no	—
36	C₂×Q₁₆×C₁₂₇	C₂×C₄	1	8	yes	—
37	C₂×Q₁₆×C₁₂₇	C₂×C₄	1	8	yes	—
38	C₂²×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
39	C₂²×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
40	C₂²×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
41	C₂²×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
42	C₄⋊Q₈×C₁₂₇	—	2	4	no	—
43	C₄⋊Q₈×C₁₂₇	C₂×C₄	1	8	yes	—
44	C₄×Q₈×C₁₂₇	—	2	8	no	—
45	C₄×Q₈×C₁₂₇	—	2	8	no	—
46	C₂×C₄⋊C₄×C₁₂₇	—	2	4	no	—
47	C₂×C₄⋊C₄×C₁₂₇	—	2	4	no	—
48	C₂×C₄⋊C₄×C₁₂₇	C₂³	1	4	yes	—
49	C₂×C₄²×C₁₂₇	—	2	4	no	—
50	C₂×C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
51	C₂×C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
52	C₂×C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
53	#107×C₁₂₇	—	2	4	no	—
54	#106×C₁₂₇	—	2	4	no	—
55	#103×C₁₂₇	—	2	8	no	—
56	#103×C₁₂₇	—	2	8	no	—
57	#97×C₁₂₇	—	2	2	no	—
58	#97×C₁₂₇	—	2	2	no	—
59	#97×C₁₂₇	D₄	1	4	yes	—
60	C₄×C₈×C₁₂₇	—	2	2	no	—
61	C₄×C₈×C₁₂₇	C₂×C₄	1	4	yes	—
62	C₄×C₈×C₁₂₇	D₄	1	4	yes	—
63	#95×C₁₂₇	—	2	4	no	—
64	#95×C₁₂₇	—	2	8	no	—
65	#95×C₁₂₇	D₄	1	4	yes	—
66	C₂×Q₈×C₁₂₇	—	2	4	no	—
67	C₂×Q₈×C₁₂₇	—	2	4	no	—
68	C₂×Q₈×C₁₂₇	—	2	4	no	—
69	C₂×Q₈×C₁₂₇	—	2	4	no	—
70	C₂×Q₈×C₁₂₇	—	2	4	no	—
71	C₂×Q₈×C₁₂₇	C₂²×C₄	1	8	yes	—
72	C₂²×C₄×C₁₂₇	—	2	4	no	—
73	C₂²×C₄×C₁₂₇	—	2	4	no	—
74	C₂²×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
75	C₂²×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
76	C₂²×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
77	C₂²×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
78	Q₁₆×C₁₂₇	—	2	16	no	—
79	Q₁₆×C₁₂₇	—	2	16	no	—
80	C₂×C₈×C₁₂₇	—	2	1	no	—
81	C₂×C₈×C₁₂₇	—	2	1	no	—
82	C₂×C₈×C₁₂₇	—	2	1	no	—
83	C₂×C₈×C₁₂₇	—	2	1	no	—
84	C₂×C₈×C₁₂₇	C₂²×C₄	1	2	yes	—
85	C₂×C₈×C₁₂₇	C₂²×C₄	1	2	yes	—
86	C₂×C₈×C₁₂₇	C₂×D₄	1	2	yes	—
87	C₂×C₈×C₁₂₇	C₂×D₄	1	2	yes	—
88	C₂×C₈×C₁₂₇	C₂×D₄	1	2	yes	—
89	C₂×C₈×C₁₂₇	C₂×D₄	1	2	yes	—
90	C₂×C₈×C₁₂₇	#41	1	2	yes	—
91	C₂×C₈×C₁₂₇	#41	1	2	yes	—
92	C₄⋊C₄×C₁₂₇	—	2	16	no	—
93	C₄⋊C₄×C₁₂₇	—	2	4	no	—
94	C₄⋊C₄×C₁₂₇	—	2	4	no	—
95	C₄⋊C₄×C₁₂₇	—	4	8	no	—
96	C₄²×C₁₂₇	—	2	1	no	—
97	C₄²×C₁₂₇	—	2	1	no	—
98	C₄²×C₁₂₇	—	4	8	no	—
99	C₄²×C₁₂₇	C₂²×C₄	1	2	yes	—
100	C₄²×C₁₂₇	C₂×D₄	1	2	yes	—
101	C₄²×C₁₂₇	C₂×D₄	1	2	yes	—
102	C₄²×C₁₂₇	#41	1	2	yes	—
103	C₂³×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
104	Q₈×C₁₂₇	—	2	8	no	—
105	Q₈×C₁₂₇	—	2	8	no	—
106	Q₈×C₁₂₇	—	4	4	no	—
107	Q₈×C₁₂₇	—	4	4	no	—
108	C₂×C₄×C₁₂₇	—	2	4	no	—
109	C₂×C₄×C₁₂₇	—	2	4	no	—
110	C₂×C₄×C₁₂₇	—	2	8	no	—
111	C₂×C₄×C₁₂₇	—	2	2	no	—
112	C₂×C₄×C₁₂₇	—	2	4	no	—
113	C₂×C₄×C₁₂₇	—	4	4	no	—
114	C₂×C₄×C₁₂₇	—	4	4	no	—
115	C₂×C₄×C₁₂₇	—	4	4	no	—
116	C₂×C₄×C₁₂₇	—	4	8	no	—
117	C₂×C₄×C₁₂₇	C₄×D₄	1	4	yes	—
118	C₂×C₄×C₁₂₇	#127	1	4	yes	—
119	C₂×C₄×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
120	C₂×C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
121	C₂×C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
122	C₂×C₄×C₁₂₇	C₂×#41	1	1	yes	characteristic
123	C₂×C₄×C₁₂₇	C₂×#41	1	1	yes	characteristic
124	C₂×C₄×C₁₂₇	C₂×#41	1	1	yes	characteristic
125	C₈×C₁₂₇	—	2	2	no	—
126	C₈×C₁₂₇	—	2	2	no	—
127	C₈×C₁₂₇	C₄×D₄	1	4	yes	—
128	C₈×C₁₂₇	C₄×D₄	1	4	yes	—
129	C₂²×C₁₂₇	C₂×C₄×D₄	1	1	yes	characteristic
130	C₂²×C₁₂₇	C₂×#127	1	1	yes	characteristic
131	C₂²×C₁₂₇	#532	1	1	yes	characteristic
132	C₂²×C₁₂₇	#578	1	4	yes	—
133	C₄×C₁₂₇	—	2	4	no	—
134	C₄×C₁₂₇	—	2	4	no	—
135	C₄×C₁₂₇	—	4	8	no	—
136	C₄×C₁₂₇	—	4	2	no	—
137	C₄×C₁₂₇	—	8	4	no	—
138	C₄×C₁₂₇	C₂×C₄×D₄	1	1	yes	characteristic
139	C₄×C₁₂₇	C₂×C₄×D₄	1	1	yes	characteristic
140	C₄×C₁₂₇	#532	1	1	yes	characteristic
141	C₄×C₁₂₇	#532	1	1	yes	characteristic
142	#21698	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
143	C₂×C₁₂₇	C₄×#127	1	1	yes	characteristic
144	C₂×C₁₂₇	#2950	1	2	yes	—
145	C₂×C₁₂₇	#3156	1	2	yes	—
146	C₂×C₁₂₇	#3177	1	2	yes	—
147	C₂×#494	C₂×C₁₂₇	1	1	yes	characteristic
148	C₂×#441	C₂×C₁₂₇	1	2	yes	—
149	C₂×C₄×Q₁₆	C₂×C₁₂₇	1	2	yes	—
150	#2327	C₂×C₁₂₇	1	2	yes	—
151	#1971	C₂×C₁₂₇	1	2	yes	—
152	#1969	C₂×C₁₂₇	1	2	yes	—
153	#1960	C₂×C₁₂₇	1	2	yes	—
154	#1846	C₂×C₁₂₇	1	1	yes	characteristic
155	#1726	C₂×C₁₂₇	1	1	yes	characteristic
156	C₁₂₇	#21698	1	1	yes	characteristic, direct product factor, Sylow
157	C₂²×Q₁₆	C₂²×C₁₂₇	1	2	yes	—
158	C₂²×Q₁₆	C₂²×C₁₂₇	1	2	yes	—
159	C₂×C₄⋊Q₈	C₂²×C₁₂₇	1	2	yes	—
160	C₂×C₄×Q₈	C₂²×C₁₂₇	1	4	yes	—
161	#494	C₄×C₁₂₇	1	8	yes	complemented
162	#441	—	2	8	no	—
163	C₄×Q₁₆	—	2	8	no	—
164	C₂×#107	C₂²×C₁₂₇	1	2	yes	—
165	C₂×#106	C₂²×C₁₂₇	1	2	yes	—
166	C₂×#103	C₂²×C₁₂₇	1	4	yes	—
167	C₂×#103	C₂²×C₁₂₇	1	4	yes	—
168	C₂×#97	C₂²×C₁₂₇	1	1	yes	characteristic
169	C₂×#97	C₂²×C₁₂₇	1	1	yes	characteristic
170	C₂×#97	C₂²×C₁₂₇	1	1	yes	characteristic
171	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
172	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
173	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
174	#391	C₂²×C₁₂₇	1	4	yes	—
175	#387	—	2	8	no	—
176	#382	C₂²×C₁₂₇	1	2	yes	—
177	#376	C₂²×C₁₂₇	1	1	yes	characteristic
178	C₂²×Q₈	C₂³×C₁₂₇	1	4	yes	—
179	C₂×Q₁₆	—	2	4	no	—
180	C₂×Q₁₆	—	2	4	no	—
181	C₂×Q₁₆	—	2	4	no	—
182	C₂×Q₁₆	—	2	4	no	—
183	C₂×Q₁₆	C₂×C₄×C₁₂₇	1	8	yes	—
184	C₂×Q₁₆	C₂×C₄×C₁₂₇	1	8	yes	—
185	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
186	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
187	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
188	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
189	C₄⋊Q₈	—	2	4	no	—
190	C₄⋊Q₈	C₂×C₄×C₁₂₇	1	8	yes	—
191	C₄×Q₈	—	2	8	no	—
192	C₄×Q₈	—	2	8	no	—
193	C₂×C₄⋊C₄	—	2	4	no	—
194	C₂×C₄⋊C₄	—	2	4	no	—
195	C₂×C₄⋊C₄	C₂³×C₁₂₇	1	4	yes	—
196	C₂×C₄²	—	2	4	no	—
197	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
198	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
199	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
200	#107	—	2	4	no	—
201	#106	—	2	4	no	—
202	#103	—	2	8	no	—
203	#103	—	2	8	no	—
204	#97	—	2	2	no	—
205	#97	—	2	2	no	—
206	#97	D₄×C₁₂₇	1	4	yes	—
207	C₄×C₈	—	2	2	no	—
208	C₄×C₈	C₂×C₄×C₁₂₇	1	4	yes	—
209	C₄×C₈	D₄×C₁₂₇	1	4	yes	—
210	#95	—	2	4	no	—
211	#95	—	2	8	no	—
212	#95	D₄×C₁₂₇	1	4	yes	—
213	C₂×Q₈	—	2	4	no	—
214	C₂×Q₈	—	2	4	no	—
215	C₂×Q₈	—	2	4	no	—
216	C₂×Q₈	—	2	4	no	—
217	C₂×Q₈	—	2	4	no	—
218	C₂×Q₈	C₂²×C₄×C₁₂₇	1	8	yes	—
219	C₂²×C₄	—	2	4	no	—
220	C₂²×C₄	—	2	4	no	—
221	C₂²×C₄	C₂×D₄×C₁₂₇	1	2	yes	—
222	C₂²×C₄	C₂×D₄×C₁₂₇	1	2	yes	—
223	C₂²×C₄	C₂×D₄×C₁₂₇	1	2	yes	—
224	C₂²×C₄	C₂⁴×C₁₂₇	1	1	yes	characteristic
225	Q₁₆	—	2	16	no	—
226	Q₁₆	—	2	16	no	—
227	C₂×C₈	—	2	1	no	—
228	C₂×C₈	—	2	1	no	—
229	C₂×C₈	—	2	1	no	—
230	C₂×C₈	—	2	1	no	—
231	C₂×C₈	C₂²×C₄×C₁₂₇	1	2	yes	—
232	C₂×C₈	C₂²×C₄×C₁₂₇	1	2	yes	—
233	C₂×C₈	C₂×D₄×C₁₂₇	1	2	yes	—
234	C₂×C₈	C₂×D₄×C₁₂₇	1	2	yes	—
235	C₂×C₈	C₂×D₄×C₁₂₇	1	2	yes	—
236	C₂×C₈	C₂×D₄×C₁₂₇	1	2	yes	—
237	C₂×C₈	#41×C₁₂₇	1	2	yes	—
238	C₂×C₈	#41×C₁₂₇	1	2	yes	—
239	C₄⋊C₄	—	2	16	no	—
240	C₄⋊C₄	—	2	4	no	—
241	C₄⋊C₄	—	2	4	no	—
242	C₄⋊C₄	—	4	8	no	—
243	C₄²	—	2	1	no	—
244	C₄²	—	2	1	no	—
245	C₄²	—	4	8	no	—
246	C₄²	C₂²×C₄×C₁₂₇	1	2	yes	—
247	C₄²	C₂×D₄×C₁₂₇	1	2	yes	—
248	C₄²	C₂×D₄×C₁₂₇	1	2	yes	—
249	C₄²	#41×C₁₂₇	1	2	yes	—
250	C₂³	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
251	Q₈	—	2	8	no	—
252	Q₈	—	2	8	no	—
253	Q₈	—	4	4	no	—
254	Q₈	—	4	4	no	—
255	C₂×C₄	—	2	4	no	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	2	8	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	4	no	—
260	C₂×C₄	—	4	4	no	—
261	C₂×C₄	—	4	4	no	—
262	C₂×C₄	—	4	4	no	—
263	C₂×C₄	—	4	8	no	—
264	C₂×C₄	C₄×D₄×C₁₂₇	1	4	yes	—
265	C₂×C₄	#127×C₁₂₇	1	4	yes	—
266	C₂×C₄	C₂³×C₄×C₁₂₇	1	1	yes	characteristic
267	C₂×C₄	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
268	C₂×C₄	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
269	C₂×C₄	C₂×#41×C₁₂₇	1	1	yes	characteristic
270	C₂×C₄	C₂×#41×C₁₂₇	1	1	yes	characteristic
271	C₂×C₄	C₂×#41×C₁₂₇	1	1	yes	characteristic
272	C₈	—	2	2	no	—
273	C₈	—	2	2	no	—
274	C₈	C₄×D₄×C₁₂₇	1	4	yes	—
275	C₈	C₄×D₄×C₁₂₇	1	4	yes	—
276	C₂²	C₂×C₄×D₄×C₁₂₇	1	1	yes	characteristic
277	C₂²	C₂×#127×C₁₂₇	1	1	yes	characteristic
278	C₂²	#532×C₁₂₇	1	1	yes	characteristic
279	C₂²	#578×C₁₂₇	1	4	yes	—
280	C₄	—	2	4	no	—
281	C₄	—	2	4	no	—
282	C₄	—	4	2	no	—
283	C₄	—	4	8	no	—
284	C₄	—	8	4	no	—
285	C₄	C₂×C₄×D₄×C₁₂₇	1	1	yes	characteristic
286	C₄	C₂×C₄×D₄×C₁₂₇	1	1	yes	characteristic
287	C₄	#532×C₁₂₇	1	1	yes	characteristic
288	C₄	#532×C₁₂₇	1	1	yes	characteristic
289	C₂	C₄×#127×C₁₂₇	1	1	yes	characteristic
290	C₂	C₁₂₇×#2950	1	2	yes	—
291	C₂	C₁₂₇×#3156	1	2	yes	—
292	C₂	C₁₂₇×#3177	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₂₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₁₂₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₂₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₂₇ ◁ #376×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
14. #494×C₁₂₇	C₄	semidirect
142. #21698	C₁₂₇	direct (N × H)
156. C₁₂₇	#21698	direct (N × H)
161. #494	C₄×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#494×C₁₂₇	2
C₂×#441×C₁₂₇	2
C₂×C₄×Q₁₆×C₁₂₇	2
C₁₂₇×#2327	2
C₁₂₇×#1971	2
C₁₂₇×#1969	2
C₁₂₇×#1960	2
C₁₂₇×#1846	2
C₁₂₇×#1726	2
#21698	127

Subgroups — C127×#21698 #44585371

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#21698 #44585371