Subgroups — C₂×C₁₂₇×#1706 #44583266

Order 32512.

1046 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

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

	subgroup	G/H
center, socle	C₂³×C₁₂₇	#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₁₂₇
prime radical	C₂×#493×C₁₂₇	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₂₇ ◁ C₂×C₄²×C₁₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#500×C₁₂₇	C₂	semidirect
3. C₂×#492×C₁₂₇	C₂	semidirect
4. C₂×#486×C₁₂₇	C₂	semidirect
5. C₁₂₇×#1706	C₂	direct (N × H)
6. C₂×C₈²×C₁₂₇	C₂	semidirect
12. #500×C₁₂₇	C₂²	semidirect
16. #492×C₁₂₇	C₂²	semidirect
18. #486×C₁₂₇	C₂²	semidirect
24. C₈²×C₁₂₇	C₂²	semidirect
75. C₂×C₈×C₁₂₇	D₈	semidirect
76. C₂×C₈×C₁₂₇	SD₁₆	semidirect
113. C₈×C₁₂₇	C₂×D₈	semidirect
114. C₈×C₁₂₇	C₂×SD₁₆	semidirect
130. C₂×#1706	C₁₂₇	direct (N × H)
132. C₂×C₁₂₇	#1706	direct (N × H)
136. C₂×#500	C₂×C₁₂₇	semidirect
138. C₂×#492	C₂×C₁₂₇	semidirect
139. C₂×#486	C₂×C₁₂₇	semidirect
140. #1706	C₂×C₁₂₇	direct (N × H)
141. C₂×C₈²	C₂×C₁₂₇	semidirect
142. C₁₂₇	C₂×#1706	direct (N × H)
148. #500	C₂²×C₁₂₇	semidirect
152. #492	C₂²×C₁₂₇	semidirect
154. #486	C₂²×C₁₂₇	semidirect
160. C₈²	C₂²×C₁₂₇	semidirect
211. C₂×C₈	D₈×C₁₂₇	semidirect
212. C₂×C₈	SD₁₆×C₁₂₇	semidirect
249. C₈	C₂×D₈×C₁₂₇	semidirect
250. C₈	C₂×SD₁₆×C₁₂₇	semidirect
267. C₂	C₁₂₇×#1706	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#500×C₁₂₇	2
C₂×#493×C₁₂₇	2
C₂×#492×C₁₂₇	2
C₂×#486×C₁₂₇	2
C₁₂₇×#1706	2
C₂×C₈²×C₁₂₇	2
C₂×#1706	127

Subgroups — C2×C127×#1706 #44583266

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₂₇×#1706 #44583266