Subgroups — C₁₂₇×#21718 #44585391

Order 32512.

774 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#446×C₁₂₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×Q₈×C₁₂₇	C₂	1	1	yes	maximal, characteristic
3	C₁₂₇×#2399	C₂	1	1	yes	maximal, characteristic
4	C₁₂₇×#1985	C₂	1	1	yes	maximal, characteristic
5	C₁₂₇×#1985	C₂	1	1	yes	maximal, characteristic
6	C₁₂₇×#1939	C₂	1	4	yes	maximal
7	C₁₂₇×#1843	C₂	1	1	yes	maximal, characteristic
8	C₁₂₇×#1771	C₂	1	4	yes	maximal
9	C₁₂₇×#1726	C₂	1	1	yes	maximal, characteristic
10	C₂×C₄×Q₈×C₁₂₇	C₂²	1	1	yes	characteristic
11	C₂×C₄×Q₈×C₁₂₇	C₂²	1	1	yes	characteristic
12	#446×C₁₂₇	—	2	4	no	—
13	C₈×Q₈×C₁₂₇	—	2	4	no	—
14	C₂×C₄⋊C₈×C₁₂₇	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₈×C₁₂₇	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₈×C₁₂₇	C₂²	1	4	yes	—
17	C₂×C₄⋊C₈×C₁₂₇	C₂²	1	2	yes	—
18	C₂×C₄⋊C₈×C₁₂₇	C₂²	1	2	yes	—
19	C₂×#97×C₁₂₇	C₂²	1	2	yes	—
20	C₂×#97×C₁₂₇	C₂²	1	1	yes	characteristic
21	C₂×#97×C₁₂₇	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₁₂₇	C₂²	1	2	yes	—
23	C₂×C₄×C₈×C₁₂₇	C₂²	1	2	yes	—
24	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
25	#398×C₁₂₇	C₄	1	2	yes	—
26	#398×C₁₂₇	C₄	1	2	yes	—
27	#395×C₁₂₇	C₄	1	2	yes	—
28	#395×C₁₂₇	C₄	1	2	yes	—
29	C₄×C₄⋊C₄×C₁₂₇	C₂²	1	4	yes	—
30	#376×C₁₂₇	C₂²	1	1	yes	characteristic
31	#336×C₁₂₇	C₂²	1	4	yes	—
32	#336×C₁₂₇	C₂²	1	4	yes	—
33	C₂²×Q₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
34	C₂²×Q₈×C₁₂₇	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₈×C₁₂₇	C₂³	1	2	yes	—
36	C₂²×C₈×C₁₂₇	C₂³	1	2	yes	—
37	C₂²×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
38	C₂²×C₈×C₁₂₇	C₂³	1	2	yes	—
39	C₂²×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
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	—	2	2	no	—
44	C₂×C₄⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
45	C₂×C₄⋊C₄×C₁₂₇	C₂×C₄	1	2	yes	—
46	C₂×C₄⋊C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄⋊C₄×C₁₂₇	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄⋊C₄×C₁₂₇	C₂×C₄	1	4	yes	—
49	C₂×C₄⋊C₄×C₁₂₇	C₂×C₄	1	4	yes	—
50	C₂×C₄²×C₁₂₇	C₂³	1	2	yes	—
51	C₂×C₄²×C₁₂₇	C₂³	1	2	yes	—
52	C₂×C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
53	C₂×C₄²×C₁₂₇	C₂³	1	2	yes	—
54	C₄⋊C₈×C₁₂₇	—	2	4	no	—
55	C₄⋊C₈×C₁₂₇	—	2	4	no	—
56	C₄⋊C₈×C₁₂₇	—	2	2	no	—
57	C₄⋊C₈×C₁₂₇	—	2	2	no	—
58	C₄⋊C₈×C₁₂₇	—	2	8	no	—
59	#97×C₁₂₇	—	2	4	no	—
60	#97×C₁₂₇	—	2	2	no	—
61	#97×C₁₂₇	Q₈	1	8	yes	complemented
62	C₄×C₈×C₁₂₇	—	2	4	no	—
63	C₄×C₈×C₁₂₇	—	2	4	no	—
64	C₄×C₈×C₁₂₇	D₄	1	4	yes	—
65	#95×C₁₂₇	C₂×C₄	1	2	yes	—
66	#95×C₁₂₇	C₂×C₄	1	2	yes	—
67	#95×C₁₂₇	C₂×C₄	1	4	yes	—
68	#95×C₁₂₇	C₂×C₄	1	4	yes	—
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	2	no	—
73	C₂²×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
74	C₂²×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
75	C₂²×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
76	C₂²×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₂₇	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₁₂₇	C₂²×C₄	1	4	yes	—
79	C₂²×C₄×C₁₂₇	C₂²×C₄	1	2	yes	—
80	C₂²×C₄×C₁₂₇	C₂⁴	1	1	yes	characteristic
81	C₂×C₈×C₁₂₇	—	2	2	no	—
82	C₂×C₈×C₁₂₇	—	2	2	no	—
83	C₂×C₈×C₁₂₇	—	2	2	no	—
84	C₂×C₈×C₁₂₇	—	2	2	no	—
85	C₂×C₈×C₁₂₇	—	2	2	no	—
86	C₂×C₈×C₁₂₇	—	2	4	no	—
87	C₂×C₈×C₁₂₇	—	2	4	no	—
88	C₂×C₈×C₁₂₇	C₂×D₄	1	2	yes	—
89	C₂×C₈×C₁₂₇	C₂×D₄	1	2	yes	—
90	C₂×C₈×C₁₂₇	C₂×Q₈	1	4	yes	—
91	C₂×C₈×C₁₂₇	C₂×Q₈	1	4	yes	—
92	C₄⋊C₄×C₁₂₇	—	2	4	no	—
93	C₄⋊C₄×C₁₂₇	—	2	2	no	—
94	C₄⋊C₄×C₁₂₇	—	2	2	no	—
95	C₄⋊C₄×C₁₂₇	—	2	4	no	—
96	C₄⋊C₄×C₁₂₇	—	2	8	no	—
97	C₄⋊C₄×C₁₂₇	—	4	4	no	—
98	C₄²×C₁₂₇	—	2	2	no	—
99	C₄²×C₁₂₇	—	2	2	no	—
100	C₄²×C₁₂₇	—	2	2	no	—
101	C₄²×C₁₂₇	—	2	2	no	—
102	C₄²×C₁₂₇	C₂×D₄	1	2	yes	—
103	C₄²×C₁₂₇	C₂×Q₈	1	4	yes	—
104	C₄²×C₁₂₇	#41	1	2	yes	—
105	C₄²×C₁₂₇	#41	1	4	yes	—
106	C₂³×C₁₂₇	C₂³×C₄	1	1	yes	characteristic
107	Q₈×C₁₂₇	—	2	8	no	—
108	Q₈×C₁₂₇	—	4	4	no	—
109	C₂×C₄×C₁₂₇	—	2	2	no	—
110	C₂×C₄×C₁₂₇	—	2	4	no	—
111	C₂×C₄×C₁₂₇	—	2	2	no	—
112	C₂×C₄×C₁₂₇	—	2	2	no	—
113	C₂×C₄×C₁₂₇	—	2	4	no	—
114	C₂×C₄×C₁₂₇	—	2	2	no	—
115	C₂×C₄×C₁₂₇	—	2	2	no	—
116	C₂×C₄×C₁₂₇	—	2	2	no	—
117	C₂×C₄×C₁₂₇	—	2	2	no	—
118	C₂×C₄×C₁₂₇	—	2	4	no	—
119	C₂×C₄×C₁₂₇	—	2	2	no	—
120	C₂×C₄×C₁₂₇	—	2	4	no	—
121	C₂×C₄×C₁₂₇	—	2	2	no	—
122	C₂×C₄×C₁₂₇	C₄×D₄	1	2	yes	—
123	C₂×C₄×C₁₂₇	C₄×D₄	1	2	yes	—
124	C₂×C₄×C₁₂₇	C₄×Q₈	1	8	yes	—
125	C₂×C₄×C₁₂₇	#131	1	4	yes	—
126	C₂×C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
127	C₂×C₄×C₁₂₇	C₂²×Q₈	1	2	yes	—
128	C₂×C₄×C₁₂₇	C₂×#41	1	2	yes	—
129	C₂×C₄×C₁₂₇	C₂×#41	1	1	yes	characteristic
130	C₈×C₁₂₇	—	2	2	no	—
131	C₈×C₁₂₇	—	2	4	no	—
132	C₈×C₁₂₇	—	4	2	no	—
133	C₈×C₁₂₇	—	4	2	no	—
134	C₈×C₁₂₇	C₄⋊Q₈	1	4	yes	—
135	C₂²×C₁₂₇	C₂×C₄×D₄	1	1	yes	characteristic
136	C₂²×C₁₂₇	C₂×C₄×Q₈	1	2	yes	—
137	C₂²×C₁₂₇	C₂×#131	1	2	yes	—
138	C₂²×C₁₂₇	#568	1	2	yes	—
139	C₄×C₁₂₇	—	2	2	no	—
140	C₄×C₁₂₇	—	2	4	no	—
141	C₄×C₁₂₇	—	2	2	no	—
142	C₄×C₁₂₇	—	2	8	no	—
143	C₄×C₁₂₇	—	4	2	no	—
144	C₄×C₁₂₇	—	4	2	no	—
145	C₄×C₁₂₇	—	4	2	no	—
146	C₄×C₁₂₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
147	C₄×C₁₂₇	#532	1	1	yes	characteristic
148	C₄×C₁₂₇	#533	1	2	yes	—
149	#21718	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
150	C₂×C₁₂₇	C₄×C₄⋊Q₈	1	1	yes	characteristic
151	C₂×C₁₂₇	#2929	1	1	yes	characteristic
152	C₂×C₁₂₇	#2930	1	1	yes	characteristic
153	C₂×C₁₂₇	#2962	1	4	yes	—
154	C₂×#446	C₂×C₁₂₇	1	1	yes	characteristic
155	C₂×C₈×Q₈	C₂×C₁₂₇	1	1	yes	characteristic
156	#2399	C₂×C₁₂₇	1	1	yes	characteristic
157	#1985	C₂×C₁₂₇	1	1	yes	characteristic
158	#1985	C₂×C₁₂₇	1	1	yes	characteristic
159	#1939	C₂×C₁₂₇	1	4	yes	—
160	#1843	C₂×C₁₂₇	1	1	yes	characteristic
161	#1771	C₂×C₁₂₇	1	4	yes	—
162	#1726	C₂×C₁₂₇	1	1	yes	characteristic
163	C₁₂₇	#21718	1	1	yes	characteristic, direct product factor, Sylow
164	C₂×C₄×Q₈	C₂²×C₁₂₇	1	1	yes	characteristic
165	C₂×C₄×Q₈	C₂²×C₁₂₇	1	1	yes	characteristic
166	#446	—	2	4	no	—
167	C₈×Q₈	—	2	4	no	—
168	C₂×C₄⋊C₈	C₂²×C₁₂₇	1	1	yes	characteristic
169	C₂×C₄⋊C₈	C₂²×C₁₂₇	1	1	yes	characteristic
170	C₂×C₄⋊C₈	C₂²×C₁₂₇	1	4	yes	—
171	C₂×C₄⋊C₈	C₂²×C₁₂₇	1	2	yes	—
172	C₂×C₄⋊C₈	C₂²×C₁₂₇	1	2	yes	—
173	C₂×#97	C₂²×C₁₂₇	1	2	yes	—
174	C₂×#97	C₂²×C₁₂₇	1	1	yes	characteristic
175	C₂×#97	C₂²×C₁₂₇	1	2	yes	—
176	C₂×C₄×C₈	C₂²×C₁₂₇	1	2	yes	—
177	C₂×C₄×C₈	C₂²×C₁₂₇	1	2	yes	—
178	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
179	#398	C₄×C₁₂₇	1	2	yes	—
180	#398	C₄×C₁₂₇	1	2	yes	—
181	#395	C₄×C₁₂₇	1	2	yes	—
182	#395	C₄×C₁₂₇	1	2	yes	—
183	C₄×C₄⋊C₄	C₂²×C₁₂₇	1	4	yes	—
184	#376	C₂²×C₁₂₇	1	1	yes	characteristic
185	#336	C₂²×C₁₂₇	1	4	yes	—
186	#336	C₂²×C₁₂₇	1	4	yes	—
187	C₂²×Q₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
188	C₂²×Q₈	C₂×C₄×C₁₂₇	1	1	yes	characteristic
189	C₂²×C₈	C₂³×C₁₂₇	1	2	yes	—
190	C₂²×C₈	C₂³×C₁₂₇	1	2	yes	—
191	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
192	C₂²×C₈	C₂³×C₁₂₇	1	2	yes	—
193	C₂²×C₈	C₂³×C₁₂₇	1	1	yes	characteristic
194	C₄×Q₈	—	2	2	no	—
195	C₄×Q₈	—	2	2	no	—
196	C₄×Q₈	—	2	2	no	—
197	C₄×Q₈	—	2	2	no	—
198	C₂×C₄⋊C₄	C₂×C₄×C₁₂₇	1	2	yes	—
199	C₂×C₄⋊C₄	C₂×C₄×C₁₂₇	1	2	yes	—
200	C₂×C₄⋊C₄	C₂×C₄×C₁₂₇	1	1	yes	characteristic
201	C₂×C₄⋊C₄	C₂×C₄×C₁₂₇	1	1	yes	characteristic
202	C₂×C₄⋊C₄	C₂×C₄×C₁₂₇	1	4	yes	—
203	C₂×C₄⋊C₄	C₂×C₄×C₁₂₇	1	4	yes	—
204	C₂×C₄²	C₂³×C₁₂₇	1	2	yes	—
205	C₂×C₄²	C₂³×C₁₂₇	1	2	yes	—
206	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
207	C₂×C₄²	C₂³×C₁₂₇	1	2	yes	—
208	C₄⋊C₈	—	2	4	no	—
209	C₄⋊C₈	—	2	4	no	—
210	C₄⋊C₈	—	2	2	no	—
211	C₄⋊C₈	—	2	2	no	—
212	C₄⋊C₈	—	2	8	no	—
213	#97	—	2	4	no	—
214	#97	—	2	2	no	—
215	#97	Q₈×C₁₂₇	1	8	yes	complemented
216	C₄×C₈	—	2	4	no	—
217	C₄×C₈	—	2	4	no	—
218	C₄×C₈	D₄×C₁₂₇	1	4	yes	—
219	#95	C₂×C₄×C₁₂₇	1	2	yes	—
220	#95	C₂×C₄×C₁₂₇	1	2	yes	—
221	#95	C₂×C₄×C₁₂₇	1	4	yes	—
222	#95	C₂×C₄×C₁₂₇	1	4	yes	—
223	C₂×Q₈	—	2	2	no	—
224	C₂×Q₈	—	2	2	no	—
225	C₂×Q₈	—	2	4	no	—
226	C₂×Q₈	—	4	2	no	—
227	C₂²×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
228	C₂²×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
229	C₂²×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
230	C₂²×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
231	C₂²×C₄	C₂²×C₄×C₁₂₇	1	1	yes	characteristic
232	C₂²×C₄	C₂²×C₄×C₁₂₇	1	4	yes	—
233	C₂²×C₄	C₂²×C₄×C₁₂₇	1	2	yes	—
234	C₂²×C₄	C₂⁴×C₁₂₇	1	1	yes	characteristic
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	2	4	no	—
241	C₂×C₈	—	2	4	no	—
242	C₂×C₈	C₂×D₄×C₁₂₇	1	2	yes	—
243	C₂×C₈	C₂×D₄×C₁₂₇	1	2	yes	—
244	C₂×C₈	C₂×Q₈×C₁₂₇	1	4	yes	—
245	C₂×C₈	C₂×Q₈×C₁₂₇	1	4	yes	—
246	C₄⋊C₄	—	2	4	no	—
247	C₄⋊C₄	—	2	2	no	—
248	C₄⋊C₄	—	2	2	no	—
249	C₄⋊C₄	—	2	4	no	—
250	C₄⋊C₄	—	2	8	no	—
251	C₄⋊C₄	—	4	4	no	—
252	C₄²	—	2	2	no	—
253	C₄²	—	2	2	no	—
254	C₄²	—	2	2	no	—
255	C₄²	—	2	2	no	—
256	C₄²	C₂×D₄×C₁₂₇	1	2	yes	—
257	C₄²	C₂×Q₈×C₁₂₇	1	4	yes	—
258	C₄²	#41×C₁₂₇	1	2	yes	—
259	C₄²	#41×C₁₂₇	1	4	yes	—
260	C₂³	C₂³×C₄×C₁₂₇	1	1	yes	characteristic
261	Q₈	—	2	8	no	—
262	Q₈	—	4	4	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	4	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	C₄×D₄×C₁₂₇	1	2	yes	—
277	C₂×C₄	C₄×D₄×C₁₂₇	1	2	yes	—
278	C₂×C₄	C₄×Q₈×C₁₂₇	1	8	yes	—
279	C₂×C₄	#131×C₁₂₇	1	4	yes	—
280	C₂×C₄	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
281	C₂×C₄	C₂²×Q₈×C₁₂₇	1	2	yes	—
282	C₂×C₄	C₂×#41×C₁₂₇	1	2	yes	—
283	C₂×C₄	C₂×#41×C₁₂₇	1	1	yes	characteristic
284	C₈	—	2	2	no	—
285	C₈	—	2	4	no	—
286	C₈	—	4	2	no	—
287	C₈	—	4	2	no	—
288	C₈	C₄⋊Q₈×C₁₂₇	1	4	yes	—
289	C₂²	C₂×C₄×D₄×C₁₂₇	1	1	yes	characteristic
290	C₂²	C₂×C₄×Q₈×C₁₂₇	1	2	yes	—
291	C₂²	C₂×#131×C₁₂₇	1	2	yes	—
292	C₂²	#568×C₁₂₇	1	2	yes	—
293	C₄	—	2	2	no	—
294	C₄	—	2	4	no	—
295	C₄	—	2	2	no	—
296	C₄	—	2	8	no	—
297	C₄	—	4	2	no	—
298	C₄	—	4	2	no	—
299	C₄	—	4	2	no	—
300	C₄	C₂×C₄⋊Q₈×C₁₂₇	1	1	yes	characteristic
301	C₄	#532×C₁₂₇	1	1	yes	characteristic
302	C₄	#533×C₁₂₇	1	2	yes	—
303	C₂	C₄×C₄⋊Q₈×C₁₂₇	1	1	yes	characteristic
304	C₂	C₁₂₇×#2929	1	1	yes	characteristic
305	C₂	C₁₂₇×#2930	1	1	yes	characteristic
306	C₂	C₁₂₇×#2962	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
61. #97×C₁₂₇	Q₈	semidirect
149. #21718	C₁₂₇	direct (N × H)
163. C₁₂₇	#21718	direct (N × H)
215. #97	Q₈×C₁₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#446×C₁₂₇	2
C₂×C₈×Q₈×C₁₂₇	2
C₁₂₇×#2399	2
C₁₂₇×#1985	2
C₁₂₇×#1939	2
C₁₂₇×#1843	2
C₁₂₇×#1771	2
C₁₂₇×#1726	2
#21718	127

Subgroups — C127×#21718 #44585391

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₂₇×#21718 #44585391