Subgroups — C₄×#357×C₁₃₁ #45698519

Order 33536.

534 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₈×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	C₂×#357×C₁₃₁	C₂	1	2	yes	maximal
3	C₂×C₄×C₁₆×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×#107×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
5	#1380×C₁₃₁	C₂	1	2	yes	maximal, complemented
6	C₂²×D₈×C₁₃₁	C₂²	1	1	yes	characteristic
7	C₂×C₄×D₄×C₁₃₁	—	2	1	no	—
8	C₂²×C₁₆×C₁₃₁	C₂²	1	2	yes	—
9	C₄×D₈×C₁₃₁	—	2	2	no	—
10	C₄×D₈×C₁₃₁	C₄	1	4	yes	complemented
11	C₂×#107×C₁₃₁	C₂²	1	1	yes	characteristic
12	C₂×#107×C₁₃₁	C₂²	1	2	yes	—
13	C₂×#102×C₁₃₁	—	2	1	no	—
14	C₂×C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
15	C₄×C₄⋊C₄×C₁₃₁	—	2	1	no	—
16	#357×C₁₃₁	C₄	1	8	yes	direct product factor
17	C₄×C₁₆×C₁₃₁	—	2	2	no	—
18	#340×C₁₃₁	—	2	1	no	—
19	C₂²×D₄×C₁₃₁	—	2	1	no	—
20	C₂³×C₄×C₁₃₁	—	4	1	no	—
21	C₂×D₈×C₁₃₁	—	2	1	no	—
22	C₂×D₈×C₁₃₁	—	2	2	no	—
23	C₂×D₈×C₁₃₁	C₂×C₄	1	2	yes	—
24	C₂×D₈×C₁₃₁	C₂×C₄	1	4	yes	—
25	C₂²×C₈×C₁₃₁	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
27	C₄×D₄×C₁₃₁	—	2	4	no	—
28	C₄×D₄×C₁₃₁	—	4	2	no	—
29	C₂×C₄⋊C₄×C₁₃₁	—	2	1	no	—
30	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
31	C₂×C₂²⋊C₄×C₁₃₁	—	4	1	no	—
32	C₂×C₄²×C₁₃₁	—	4	1	no	—
33	C₂×C₄²×C₁₃₁	D₄	1	1	yes	characteristic
34	C₂×C₁₆×C₁₃₁	—	2	2	no	—
35	C₂×C₁₆×C₁₃₁	—	2	2	no	—
36	C₂×C₁₆×C₁₃₁	C₂×C₄	1	4	yes	complemented
37	#107×C₁₃₁	—	2	1	no	—
38	#107×C₁₃₁	—	2	2	no	—
39	#107×C₁₃₁	C₂×C₄	1	2	yes	—
40	#107×C₁₃₁	C₂×C₄	1	4	yes	complemented
41	#102×C₁₃₁	—	2	4	no	—
42	C₄×C₈×C₁₃₁	C₂×C₄	1	2	yes	—
43	C₄×C₈×C₁₃₁	D₄	1	2	yes	—
44	#95×C₁₃₁	—	4	1	no	—
45	C₂⁴×C₁₃₁	—	4	1	no	—
46	C₂×D₄×C₁₃₁	—	2	2	no	—
47	C₂×D₄×C₁₃₁	—	2	4	no	—
48	C₂×D₄×C₁₃₁	—	4	1	no	—
49	C₂×D₄×C₁₃₁	—	4	2	no	—
50	C₂²×C₄×C₁₃₁	—	4	2	no	—
51	C₂²×C₄×C₁₃₁	—	4	1	no	—
52	C₂²×C₄×C₁₃₁	—	4	4	no	—
53	C₂²×C₄×C₁₃₁	—	8	2	no	—
54	C₂²×C₄×C₁₃₁	—	8	2	no	—
55	C₂²×C₄×C₁₃₁	D₈	1	1	yes	characteristic
56	C₂²×C₄×C₁₃₁	D₈	1	1	yes	characteristic
57	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
58	D₈×C₁₃₁	—	2	2	no	—
59	D₈×C₁₃₁	—	2	4	no	—
60	D₈×C₁₃₁	C₄²	1	4	yes	complemented
61	C₂×C₈×C₁₃₁	—	2	2	no	—
62	C₂×C₈×C₁₃₁	C₂²⋊C₄	1	2	yes	—
63	C₂×C₈×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₈×C₁₃₁	C₂²×C₄	1	2	yes	—
65	C₂×C₈×C₁₃₁	C₂×D₄	1	1	yes	characteristic
66	C₂×C₈×C₁₃₁	#41	1	2	yes	—
67	C₄⋊C₄×C₁₃₁	—	2	2	no	—
68	C₄⋊C₄×C₁₃₁	—	2	4	no	—
69	C₄⋊C₄×C₁₃₁	—	4	1	no	—
70	C₄⋊C₄×C₁₃₁	—	4	2	no	—
71	C₂²⋊C₄×C₁₃₁	—	4	4	no	—
72	C₄²×C₁₃₁	—	8	2	no	—
73	C₄²×C₁₃₁	C₂²⋊C₄	1	2	yes	—
74	C₄²×C₁₃₁	SD₁₆	1	2	yes	—
75	C₁₆×C₁₃₁	—	2	4	no	—
76	C₂³×C₁₃₁	—	4	2	no	—
77	C₂³×C₁₃₁	—	4	4	no	—
78	C₂³×C₁₃₁	—	8	1	no	—
79	C₂³×C₁₃₁	—	8	1	no	—
80	C₂³×C₁₃₁	—	8	2	no	—
81	C₂³×C₁₃₁	C₂×D₈	1	1	yes	characteristic
82	D₄×C₁₃₁	—	2	4	no	—
83	D₄×C₁₃₁	—	4	2	no	—
84	D₄×C₁₃₁	—	4	4	no	—
85	C₂×C₄×C₁₃₁	—	2	2	no	—
86	C₂×C₄×C₁₃₁	—	4	2	no	—
87	C₂×C₄×C₁₃₁	—	4	4	no	—
88	C₂×C₄×C₁₃₁	—	8	2	no	—
89	C₂×C₄×C₁₃₁	—	8	1	no	—
90	C₂×C₄×C₁₃₁	—	8	1	no	—
91	C₂×C₄×C₁₃₁	—	8	2	no	—
92	C₂×C₄×C₁₃₁	—	8	8	no	—
93	C₂×C₄×C₁₃₁	#102	1	2	yes	—
94	C₂×C₄×C₁₃₁	#102	1	2	yes	—
95	C₂×C₄×C₁₃₁	D₁₆	1	2	yes	—
96	C₂×C₄×C₁₃₁	SD₃₂	1	2	yes	—
97	C₂×C₄×C₁₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₁₃₁	C₄×D₄	1	2	yes	—
99	C₂×C₄×C₁₃₁	C₂×SD₁₆	1	1	yes	characteristic
100	C₂×C₄×C₁₃₁	#135	1	2	yes	—
101	C₈×C₁₃₁	—	2	2	no	—
102	C₈×C₁₃₁	C₂×C₄²	1	1	yes	characteristic
103	C₈×C₁₃₁	#117	1	1	yes	characteristic
104	C₈×C₁₃₁	C₄×D₄	1	2	yes	—
105	C₂²×C₁₃₁	—	4	4	no	—
106	C₂²×C₁₃₁	—	8	2	no	—
107	C₂²×C₁₃₁	—	8	2	no	—
108	C₂²×C₁₃₁	—	8	4	no	—
109	C₂²×C₁₃₁	—	8	4	no	—
110	C₂²×C₁₃₁	C₂×#102	1	1	yes	characteristic
111	C₂²×C₁₃₁	C₄×D₈	1	2	yes	—
112	C₂²×C₁₃₁	C₂×D₁₆	1	1	yes	characteristic
113	C₂²×C₁₃₁	C₂×SD₃₂	1	1	yes	characteristic
114	C₂²×C₁₃₁	#508	1	2	yes	—
115	C₄×C₁₃₁	—	2	2	no	—
116	C₄×C₁₃₁	—	8	2	no	—
117	C₄×C₁₃₁	—	8	4	no	—
118	C₄×C₁₃₁	#357	1	4	yes	direct product factor
119	C₄×C₁₃₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
120	C₄×C₁₃₁	#416	1	1	yes	characteristic
121	C₄×C₁₃₁	C₄×SD₁₆	1	2	yes	—
122	C₂×C₁₃₁	—	8	4	no	—
123	C₂×C₁₃₁	C₄×#102	1	1	yes	characteristic
124	C₂×C₁₃₁	C₂×#357	1	1	yes	characteristic
125	C₂×C₁₃₁	#2138	1	1	yes	characteristic
126	C₂×C₁₃₁	C₄×D₁₆	1	2	yes	—
127	C₂×C₁₃₁	C₄×SD₃₂	1	2	yes	—
128	C₄×#357	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
129	C₁₃₁	C₄×#357	1	1	yes	characteristic, direct product factor, Sylow
130	C₂×C₄×D₈	C₂×C₁₃₁	1	1	yes	characteristic
131	C₂×#357	C₂×C₁₃₁	1	2	yes	—
132	C₂×C₄×C₁₆	C₂×C₁₃₁	1	1	yes	characteristic, complemented
133	C₄×#107	C₂×C₁₃₁	1	1	yes	characteristic, complemented
134	#1380	C₂×C₁₃₁	1	2	yes	complemented
135	C₂²×D₈	C₂²×C₁₃₁	1	1	yes	characteristic
136	C₂×C₄×D₄	—	2	1	no	—
137	C₂²×C₁₆	C₂²×C₁₃₁	1	2	yes	—
138	C₄×D₈	—	2	2	no	—
139	C₄×D₈	C₄×C₁₃₁	1	4	yes	complemented
140	C₂×#107	C₂²×C₁₃₁	1	1	yes	characteristic
141	C₂×#107	C₂²×C₁₃₁	1	2	yes	—
142	C₂×#102	—	2	1	no	—
143	C₂×C₄×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
144	C₄×C₄⋊C₄	—	2	1	no	—
145	#357	C₄×C₁₃₁	1	8	yes	direct product factor
146	C₄×C₁₆	—	2	2	no	—
147	#340	—	2	1	no	—
148	C₂²×D₄	—	2	1	no	—
149	C₂³×C₄	—	4	1	no	—
150	C₂×D₈	—	2	1	no	—
151	C₂×D₈	—	2	2	no	—
152	C₂×D₈	C₂×C₄×C₁₃₁	1	4	yes	—
153	C₂×D₈	C₂×C₄×C₁₃₁	1	2	yes	—
154	C₂²×C₈	D₄×C₁₃₁	1	1	yes	characteristic
155	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
156	C₄×D₄	—	2	4	no	—
157	C₄×D₄	—	4	2	no	—
158	C₂×C₄⋊C₄	—	2	1	no	—
159	C₂×C₄⋊C₄	—	2	2	no	—
160	C₂×C₂²⋊C₄	—	4	1	no	—
161	C₂×C₄²	—	4	1	no	—
162	C₂×C₄²	D₄×C₁₃₁	1	1	yes	characteristic
163	C₂×C₁₆	—	2	2	no	—
164	C₂×C₁₆	—	2	2	no	—
165	C₂×C₁₆	C₂×C₄×C₁₃₁	1	4	yes	complemented
166	#107	—	2	1	no	—
167	#107	—	2	2	no	—
168	#107	C₂×C₄×C₁₃₁	1	2	yes	—
169	#107	C₂×C₄×C₁₃₁	1	4	yes	complemented
170	#102	—	2	4	no	—
171	C₄×C₈	C₂×C₄×C₁₃₁	1	2	yes	—
172	C₄×C₈	D₄×C₁₃₁	1	2	yes	—
173	#95	—	4	1	no	—
174	C₂⁴	—	4	1	no	—
175	C₂×D₄	—	2	2	no	—
176	C₂×D₄	—	2	4	no	—
177	C₂×D₄	—	4	1	no	—
178	C₂×D₄	—	4	2	no	—
179	C₂²×C₄	—	4	2	no	—
180	C₂²×C₄	—	4	1	no	—
181	C₂²×C₄	—	4	4	no	—
182	C₂²×C₄	—	8	2	no	—
183	C₂²×C₄	—	8	2	no	—
184	C₂²×C₄	D₈×C₁₃₁	1	1	yes	characteristic
185	C₂²×C₄	D₈×C₁₃₁	1	1	yes	characteristic
186	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
187	D₈	—	2	2	no	—
188	D₈	—	2	4	no	—
189	D₈	C₄²×C₁₃₁	1	4	yes	complemented
190	C₂×C₈	—	2	2	no	—
191	C₂×C₈	C₂²⋊C₄×C₁₃₁	1	2	yes	—
192	C₂×C₈	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
193	C₂×C₈	C₂²×C₄×C₁₃₁	1	2	yes	—
194	C₂×C₈	C₂×D₄×C₁₃₁	1	1	yes	characteristic
195	C₂×C₈	#41×C₁₃₁	1	2	yes	—
196	C₄⋊C₄	—	2	2	no	—
197	C₄⋊C₄	—	2	4	no	—
198	C₄⋊C₄	—	4	1	no	—
199	C₄⋊C₄	—	4	2	no	—
200	C₂²⋊C₄	—	4	4	no	—
201	C₄²	—	8	2	no	—
202	C₄²	C₂²⋊C₄×C₁₃₁	1	2	yes	—
203	C₄²	SD₁₆×C₁₃₁	1	2	yes	—
204	C₁₆	—	2	4	no	—
205	C₂³	—	4	2	no	—
206	C₂³	—	4	4	no	—
207	C₂³	—	8	1	no	—
208	C₂³	—	8	1	no	—
209	C₂³	—	8	2	no	—
210	C₂³	C₂×D₈×C₁₃₁	1	1	yes	characteristic
211	D₄	—	2	4	no	—
212	D₄	—	4	2	no	—
213	D₄	—	4	4	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	4	2	no	—
216	C₂×C₄	—	4	4	no	—
217	C₂×C₄	—	8	2	no	—
218	C₂×C₄	—	8	1	no	—
219	C₂×C₄	—	8	1	no	—
220	C₂×C₄	—	8	2	no	—
221	C₂×C₄	—	8	8	no	—
222	C₂×C₄	#102×C₁₃₁	1	2	yes	—
223	C₂×C₄	#102×C₁₃₁	1	2	yes	—
224	C₂×C₄	D₁₆×C₁₃₁	1	2	yes	—
225	C₂×C₄	SD₃₂×C₁₃₁	1	2	yes	—
226	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
227	C₂×C₄	C₄×D₄×C₁₃₁	1	2	yes	—
228	C₂×C₄	C₂×SD₁₆×C₁₃₁	1	1	yes	characteristic
229	C₂×C₄	#135×C₁₃₁	1	2	yes	—
230	C₈	—	2	2	no	—
231	C₈	C₂×C₄²×C₁₃₁	1	1	yes	characteristic
232	C₈	#117×C₁₃₁	1	1	yes	characteristic
233	C₈	C₄×D₄×C₁₃₁	1	2	yes	—
234	C₂²	—	4	4	no	—
235	C₂²	—	8	2	no	—
236	C₂²	—	8	2	no	—
237	C₂²	—	8	4	no	—
238	C₂²	—	8	4	no	—
239	C₂²	C₂×#102×C₁₃₁	1	1	yes	characteristic
240	C₂²	C₄×D₈×C₁₃₁	1	2	yes	—
241	C₂²	C₂×D₁₆×C₁₃₁	1	1	yes	characteristic
242	C₂²	C₂×SD₃₂×C₁₃₁	1	1	yes	characteristic
243	C₂²	#508×C₁₃₁	1	2	yes	—
244	C₄	—	2	2	no	—
245	C₄	—	8	2	no	—
246	C₄	—	8	4	no	—
247	C₄	#357×C₁₃₁	1	4	yes	direct product factor
248	C₄	C₄×C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
249	C₄	#416×C₁₃₁	1	1	yes	characteristic
250	C₄	C₄×SD₁₆×C₁₃₁	1	2	yes	—
251	C₂	—	8	4	no	—
252	C₂	C₄×#102×C₁₃₁	1	1	yes	characteristic
253	C₂	C₂×#357×C₁₃₁	1	1	yes	characteristic
254	C₂	#2138×C₁₃₁	1	1	yes	characteristic
255	C₂	C₄×D₁₆×C₁₃₁	1	2	yes	—
256	C₂	C₄×SD₃₂×C₁₃₁	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₁₃₁	D₈
commutator	C₈	C₂×C₄²×C₁₃₁
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃₁
socle	C₂³×C₁₃₁	C₂×D₈
prime radical	C₂²×D₈×C₁₃₁	C₂²
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₃₁ ◁ C₂×C₄²×C₁₃₁ ◁ C₂×C₄×C₈×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×C₄×C₁₆×C₁₃₁	C₂	semidirect
4. C₄×#107×C₁₃₁	C₂	semidirect
5. #1380×C₁₃₁	C₂	semidirect
10. C₄×D₈×C₁₃₁	C₄	semidirect
16. #357×C₁₃₁	C₄	direct (N × H)
36. C₂×C₁₆×C₁₃₁	C₂×C₄	semidirect
40. #107×C₁₃₁	C₂×C₄	semidirect
60. D₈×C₁₃₁	C₄²	semidirect
118. C₄×C₁₃₁	#357	direct (N × H)
128. C₄×#357	C₁₃₁	direct (N × H)
129. C₁₃₁	C₄×#357	direct (N × H)
132. C₂×C₄×C₁₆	C₂×C₁₃₁	semidirect
133. C₄×#107	C₂×C₁₃₁	semidirect
134. #1380	C₂×C₁₃₁	semidirect
139. C₄×D₈	C₄×C₁₃₁	semidirect
145. #357	C₄×C₁₃₁	direct (N × H)
165. C₂×C₁₆	C₂×C₄×C₁₃₁	semidirect
169. #107	C₂×C₄×C₁₃₁	semidirect
189. D₈	C₄²×C₁₃₁	semidirect
247. C₄	#357×C₁₃₁	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×C₄×D₈×C₁₃₁	2
C₂×#357×C₁₃₁	2
C₂×C₄×C₁₆×C₁₃₁	2
C₄×#107×C₁₃₁	2
#1380×C₁₃₁	2
C₄×#357	131

Subgroups — C4×#357×C131 #45698519

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#357×C₁₃₁ #45698519