Subgroups — C₂×C₁₂₇×#1715 #44583279

Order 32512.

1030 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#498×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×#1715	C₂	1	8	yes	maximal, direct product factor
6	C₂×#322×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	1	yes	characteristic
11	C₂×#127×C₁₂₇	C₂²	1	1	yes	characteristic
12	#498×C₁₂₇	—	2	2	no	—
13	#498×C₁₂₇	C₂²	1	4	yes	complemented
14	#493×C₁₂₇	—	2	2	no	—
15	#493×C₁₂₇	C₂²	1	4	yes	—
16	#492×C₁₂₇	—	2	4	no	—
17	#492×C₁₂₇	C₂²	1	8	yes	complemented
18	#486×C₁₂₇	—	2	4	no	—
19	#486×C₁₂₇	C₂²	1	8	yes	complemented
20	C₂×#106×C₁₂₇	—	2	2	no	—
21	C₂×#102×C₁₂₇	—	2	4	no	—
22	C₂×C₄×C₈×C₁₂₇	C₂²	1	2	yes	—
23	C₂×C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
24	#322×C₁₂₇	C₂²	1	4	yes	complemented
25	C₂²×Q₈×C₁₂₇	—	2	2	no	—
26	C₂²×D₄×C₁₂₇	—	2	1	no	—
27	C₂²×D₄×C₁₂₇	—	2	1	no	—
28	C₂²×D₄×C₁₂₇	—	2	2	no	—
29	C₂²×D₄×C₁₂₇	—	2	2	no	—
30	C₂×SD₁₆×C₁₂₇	—	2	16	no	—
31	C₂×SD₁₆×C₁₂₇	—	4	16	no	—
32	C₂×D₈×C₁₂₇	—	2	8	no	—
33	C₂×D₈×C₁₂₇	—	4	8	no	—
34	C₂²×C₈×C₁₂₇	D₄	1	4	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	1	no	—
43	#127×C₁₂₇	—	2	1	no	—
44	#127×C₁₂₇	C₂³	1	2	yes	—
45	#127×C₁₂₇	C₂³	1	2	yes	—
46	C₂×C₄⋊C₄×C₁₂₇	—	2	2	no	—
47	C₂×C₄⋊C₄×C₁₂₇	—	2	2	no	—
48	C₂×C₄²×C₁₂₇	C₂³	1	1	yes	characteristic
49	#106×C₁₂₇	—	2	8	no	—
50	#102×C₁₂₇	—	2	16	no	—
51	C₄×C₈×C₁₂₇	D₄	1	2	yes	—
52	C₄×C₈×C₁₂₇	D₄	1	4	yes	—
53	C₄×C₈×C₁₂₇	C₂³	1	2	yes	—
54	C₄×C₈×C₁₂₇	C₂³	1	4	yes	—
55	C₂⁴×C₁₂₇	—	4	1	no	—
56	C₂⁴×C₁₂₇	—	4	1	no	—
57	C₂×Q₈×C₁₂₇	—	2	4	no	—
58	C₂×Q₈×C₁₂₇	—	4	2	no	—
59	C₂×Q₈×C₁₂₇	—	4	8	no	—
60	C₂×D₄×C₁₂₇	—	2	4	no	—
61	C₂×D₄×C₁₂₇	—	2	2	no	—
62	C₂×D₄×C₁₂₇	—	2	2	no	—
63	C₂×D₄×C₁₂₇	—	2	4	no	—
64	C₂×D₄×C₁₂₇	—	4	1	no	—
65	C₂×D₄×C₁₂₇	—	4	1	no	—
66	C₂×D₄×C₁₂₇	—	4	2	no	—
67	C₂×D₄×C₁₂₇	—	4	2	no	—
68	C₂×D₄×C₁₂₇	—	4	4	no	—
69	C₂×D₄×C₁₂₇	—	4	4	no	—
70	C₂×D₄×C₁₂₇	—	4	8	no	—
71	C₂×D₄×C₁₂₇	—	4	8	no	—
72	C₂²×C₄×C₁₂₇	—	4	2	no	—
73	C₂²×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₁₂₇	C₂×D₄	1	2	yes	—
75	SD₁₆×C₁₂₇	—	4	32	no	—
76	D₈×C₁₂₇	—	4	16	no	—
77	C₂×C₈×C₁₂₇	—	2	8	no	—
78	C₂×C₈×C₁₂₇	SD₁₆	1	8	yes	complemented
79	C₂×C₈×C₁₂₇	C₂×D₄	1	8	yes	—
80	C₂×C₈×C₁₂₇	C₂×D₄	1	4	yes	—
81	C₄⋊C₄×C₁₂₇	—	2	4	no	—
82	C₄⋊C₄×C₁₂₇	—	2	4	no	—
83	C₄⋊C₄×C₁₂₇	—	4	2	no	—
84	C₄⋊C₄×C₁₂₇	—	4	2	no	—
85	C₄²×C₁₂₇	C₂×D₄	1	2	yes	—
86	C₄²×C₁₂₇	C₂×D₄	1	1	yes	characteristic
87	C₄²×C₁₂₇	C₂⁴	1	1	yes	characteristic
88	C₂³×C₁₂₇	—	4	2	no	—
89	C₂³×C₁₂₇	—	4	2	no	—
90	C₂³×C₁₂₇	—	8	2	no	—
91	C₂³×C₁₂₇	—	8	2	no	—
92	C₂³×C₁₂₇	—	8	4	no	—
93	C₂³×C₁₂₇	—	8	4	no	—
94	C₂³×C₁₂₇	#127	1	1	yes	characteristic
95	Q₈×C₁₂₇	—	4	8	no	—
96	Q₈×C₁₂₇	—	8	4	no	—
97	D₄×C₁₂₇	—	4	4	no	—
98	D₄×C₁₂₇	—	4	4	no	—
99	D₄×C₁₂₇	—	4	8	no	—
100	D₄×C₁₂₇	—	4	8	no	—
101	D₄×C₁₂₇	—	8	2	no	—
102	D₄×C₁₂₇	—	8	2	no	—
103	D₄×C₁₂₇	—	8	4	no	—
104	D₄×C₁₂₇	—	8	4	no	—
105	C₂×C₄×C₁₂₇	—	4	4	no	—
106	C₂×C₄×C₁₂₇	—	8	4	no	—
107	C₂×C₄×C₁₂₇	#127	1	1	yes	characteristic
108	C₂×C₄×C₁₂₇	#127	1	2	yes	—
109	C₂×C₄×C₁₂₇	C₂×SD₁₆	1	4	yes	—
110	C₂×C₄×C₁₂₇	AGL(1,ℤ/8)	1	8	yes	—
111	C₂×C₄×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
112	C₂×C₄×C₁₂₇	C₂²×D₄	1	2	yes	—
113	C₈×C₁₂₇	—	2	8	no	—
114	C₈×C₁₂₇	C₂×SD₁₆	1	8	yes	complemented
115	C₂²×C₁₂₇	—	8	2	no	—
116	C₂²×C₁₂₇	—	8	2	no	—
117	C₂²×C₁₂₇	—	8	4	no	—
118	C₂²×C₁₂₇	—	8	4	no	—
119	C₂²×C₁₂₇	—	16	4	no	—
120	C₂²×C₁₂₇	—	16	4	no	—
121	C₂²×C₁₂₇	#492	1	2	yes	—
122	C₂²×C₁₂₇	#496	1	4	yes	—
123	C₂²×C₁₂₇	C₂×#127	1	1	yes	characteristic
124	C₄×C₁₂₇	—	8	4	no	—
125	C₄×C₁₂₇	#492	1	2	yes	—
126	C₄×C₁₂₇	#496	1	4	yes	—
127	C₄×C₁₂₇	C₂²×SD₁₆	1	2	yes	—
128	C₄×C₁₂₇	C₂×AGL(1,ℤ/8)	1	4	yes	—
129	C₂×#1715	C₁₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
130	C₂×C₁₂₇	—	16	2	no	—
131	C₂×C₁₂₇	—	16	2	no	—
132	C₂×C₁₂₇	#1715	1	4	yes	direct product factor
133	C₂×C₁₂₇	C₂×#492	1	1	yes	characteristic
134	C₂×C₁₂₇	C₂×#496	1	2	yes	—
135	C₂×#498	C₂×C₁₂₇	1	1	yes	characteristic, complemented
136	C₂×#493	C₂×C₁₂₇	1	1	yes	characteristic
137	C₂×#492	C₂×C₁₂₇	1	2	yes	complemented
138	C₂×#486	C₂×C₁₂₇	1	2	yes	complemented
139	#1715	C₂×C₁₂₇	1	8	yes	direct product factor
140	C₂×#322	C₂×C₁₂₇	1	1	yes	characteristic, complemented
141	C₁₂₇	C₂×#1715	1	1	yes	characteristic, direct product factor, Sylow
142	C₂²×SD₁₆	—	2	4	no	—
143	C₂²×D₈	—	2	2	no	—
144	C₂×C₄⋊Q₈	C₂²×C₁₂₇	1	2	yes	—
145	C₂×#127	C₂²×C₁₂₇	1	1	yes	characteristic
146	C₂×#127	C₂²×C₁₂₇	1	1	yes	characteristic
147	#498	—	2	2	no	—
148	#498	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×#106	—	2	2	no	—
156	C₂×#102	—	2	4	no	—
157	C₂×C₄×C₈	C₂²×C₁₂₇	1	2	yes	—
158	C₂×C₄×C₈	C₂²×C₁₂₇	1	1	yes	characteristic
159	#322	C₂²×C₁₂₇	1	4	yes	complemented
160	C₂²×Q₈	—	2	2	no	—
161	C₂²×D₄	—	2	1	no	—
162	C₂²×D₄	—	2	1	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	4	yes	—
170	C₂²×C₈	D₄×C₁₂₇	1	2	yes	—
171	C₄⋊Q₈	—	2	2	no	—
172	C₄⋊Q₈	—	2	2	no	—
173	C₄⋊Q₈	—	2	2	no	—
174	C₄⋊Q₈	C₂³×C₁₂₇	1	4	yes	—
175	#127	—	2	2	no	—
176	#127	—	2	2	no	—
177	#127	—	2	1	no	—
178	#127	—	2	1	no	—
179	#127	C₂³×C₁₂₇	1	2	yes	—
180	#127	C₂³×C₁₂₇	1	2	yes	—
181	C₂×C₄⋊C₄	—	2	2	no	—
182	C₂×C₄⋊C₄	—	2	2	no	—
183	C₂×C₄²	C₂³×C₁₂₇	1	1	yes	characteristic
184	#106	—	2	8	no	—
185	#102	—	2	16	no	—
186	C₄×C₈	D₄×C₁₂₇	1	4	yes	—
187	C₄×C₈	D₄×C₁₂₇	1	2	yes	—
188	C₄×C₈	C₂³×C₁₂₇	1	4	yes	—
189	C₄×C₈	C₂³×C₁₂₇	1	2	yes	—
190	C₂⁴	—	4	1	no	—
191	C₂⁴	—	4	1	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	2	no	—
197	C₂×D₄	—	2	2	no	—
198	C₂×D₄	—	2	4	no	—
199	C₂×D₄	—	4	2	no	—
200	C₂×D₄	—	4	2	no	—
201	C₂×D₄	—	4	1	no	—
202	C₂×D₄	—	4	1	no	—
203	C₂×D₄	—	4	4	no	—
204	C₂×D₄	—	4	4	no	—
205	C₂×D₄	—	4	8	no	—
206	C₂×D₄	—	4	8	no	—
207	C₂²×C₄	—	4	2	no	—
208	C₂²×C₄	C₂×D₄×C₁₂₇	1	1	yes	characteristic
209	C₂²×C₄	C₂×D₄×C₁₂₇	1	2	yes	—
210	SD₁₆	—	4	32	no	—
211	D₈	—	4	16	no	—
212	C₂×C₈	—	2	8	no	—
213	C₂×C₈	SD₁₆×C₁₂₇	1	8	yes	complemented
214	C₂×C₈	C₂×D₄×C₁₂₇	1	8	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	2	yes	—
222	C₄²	C₂⁴×C₁₂₇	1	1	yes	characteristic
223	C₂³	—	4	2	no	—
224	C₂³	—	4	2	no	—
225	C₂³	—	8	4	no	—
226	C₂³	—	8	4	no	—
227	C₂³	—	8	2	no	—
228	C₂³	—	8	2	no	—
229	C₂³	#127×C₁₂₇	1	1	yes	characteristic
230	Q₈	—	4	8	no	—
231	Q₈	—	8	4	no	—
232	D₄	—	4	4	no	—
233	D₄	—	4	4	no	—
234	D₄	—	4	8	no	—
235	D₄	—	4	8	no	—
236	D₄	—	8	4	no	—
237	D₄	—	8	4	no	—
238	D₄	—	8	2	no	—
239	D₄	—	8	2	no	—
240	C₂×C₄	—	4	4	no	—
241	C₂×C₄	—	8	4	no	—
242	C₂×C₄	#127×C₁₂₇	1	2	yes	—
243	C₂×C₄	#127×C₁₂₇	1	1	yes	characteristic
244	C₂×C₄	C₂×SD₁₆×C₁₂₇	1	4	yes	—
245	C₂×C₄	AGL(1,ℤ/8)×C₁₂₇	1	8	yes	—
246	C₂×C₄	C₂²×D₄×C₁₂₇	1	1	yes	characteristic
247	C₂×C₄	C₂²×D₄×C₁₂₇	1	2	yes	—
248	C₈	—	2	8	no	—
249	C₈	C₂×SD₁₆×C₁₂₇	1	8	yes	complemented
250	C₂²	—	8	2	no	—
251	C₂²	—	8	2	no	—
252	C₂²	—	8	4	no	—
253	C₂²	—	8	4	no	—
254	C₂²	—	16	4	no	—
255	C₂²	—	16	4	no	—
256	C₂²	#492×C₁₂₇	1	2	yes	—
257	C₂²	#496×C₁₂₇	1	4	yes	—
258	C₂²	C₂×#127×C₁₂₇	1	1	yes	characteristic
259	C₄	—	8	4	no	—
260	C₄	#492×C₁₂₇	1	2	yes	—
261	C₄	#496×C₁₂₇	1	4	yes	—
262	C₄	C₂²×SD₁₆×C₁₂₇	1	2	yes	—
263	C₄	C₂×AGL(1,ℤ/8)×C₁₂₇	1	4	yes	—
264	C₂	—	16	2	no	—
265	C₂	—	16	2	no	—
266	C₂	C₁₂₇×#1715	1	4	yes	direct product factor
267	C₂	C₂×#492×C₁₂₇	1	1	yes	characteristic
268	C₂	C₂×#496×C₁₂₇	1	2	yes	—

	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×#498×C₁₂₇	C₂	semidirect
3. C₂×#492×C₁₂₇	C₂	semidirect
4. C₂×#486×C₁₂₇	C₂	semidirect
5. C₁₂₇×#1715	C₂	direct (N × H)
6. C₂×#322×C₁₂₇	C₂	semidirect
13. #498×C₁₂₇	C₂²	semidirect
17. #492×C₁₂₇	C₂²	semidirect
19. #486×C₁₂₇	C₂²	semidirect
24. #322×C₁₂₇	C₂²	semidirect
78. C₂×C₈×C₁₂₇	SD₁₆	semidirect
114. C₈×C₁₂₇	C₂×SD₁₆	semidirect
129. C₂×#1715	C₁₂₇	direct (N × H)
132. C₂×C₁₂₇	#1715	direct (N × H)
135. C₂×#498	C₂×C₁₂₇	semidirect
137. C₂×#492	C₂×C₁₂₇	semidirect
138. C₂×#486	C₂×C₁₂₇	semidirect
139. #1715	C₂×C₁₂₇	direct (N × H)
140. C₂×#322	C₂×C₁₂₇	semidirect
141. C₁₂₇	C₂×#1715	direct (N × H)
148. #498	C₂²×C₁₂₇	semidirect
152. #492	C₂²×C₁₂₇	semidirect
154. #486	C₂²×C₁₂₇	semidirect
159. #322	C₂²×C₁₂₇	semidirect
213. C₂×C₈	SD₁₆×C₁₂₇	semidirect
249. C₈	C₂×SD₁₆×C₁₂₇	semidirect
266. C₂	C₁₂₇×#1715	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×#498×C₁₂₇	2
C₂×#493×C₁₂₇	2
C₂×#492×C₁₂₇	2
C₂×#486×C₁₂₇	2
C₁₂₇×#1715	2
C₂×#322×C₁₂₇	2
C₂×#1715	127

Subgroups — C2×C127×#1715 #44583279

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₂₇×#1715 #44583279