Subgroups — C₁₃₁×#33456 #45719340

Order 33536.

718 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₁₆×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	#2948×C₁₃₁	C₂	1	2	yes	maximal, complemented
3	#2177×C₁₃₁	C₂	1	4	yes	maximal, complemented
4	C₄×D₁₆×C₁₃₁	C₂	1	4	yes	maximal, complemented
5	#2158×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#2140×C₁₃₁	C₂	1	2	yes	maximal, complemented
7	#2108×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₈×C₁₃₁	C₂²	1	2	yes	—
9	#518×C₁₃₁	—	2	2	no	—
10	C₂×D₁₆×C₁₃₁	C₄	1	8	yes	complemented
11	C₂×D₁₆×C₁₃₁	C₂²	1	2	yes	—
12	C₂×D₁₆×C₁₃₁	C₂²	1	2	yes	—
13	C₂×M₃₂×C₁₃₁	C₂²	1	1	yes	characteristic
14	C₂²×C₁₆×C₁₃₁	C₂²	1	1	yes	characteristic
15	#442×C₁₃₁	—	2	4	no	—
16	C₄×D₈×C₁₃₁	C₂²	1	8	yes	complemented
17	#427×C₁₃₁	C₂²	1	2	yes	—
18	#418×C₁₃₁	—	2	2	no	—
19	#405×C₁₃₁	C₂²	1	1	yes	characteristic
20	#367×C₁₃₁	C₂²	1	2	yes	complemented
21	#366×C₁₃₁	C₂²	1	2	yes	complemented
22	#357×C₁₃₁	C₂²	1	8	yes	complemented
23	#346×C₁₃₁	C₂²	1	2	yes	complemented
24	C₄×C₁₆×C₁₃₁	C₂²	1	2	yes	complemented
25	C₂²×D₄×C₁₃₁	—	2	2	no	—
26	C₂×D₈×C₁₃₁	—	2	2	no	—
27	C₂×D₈×C₁₃₁	—	2	4	no	—
28	C₂×D₈×C₁₃₁	C₂×C₄	1	8	yes	complemented
29	C₂×D₈×C₁₃₁	C₂³	1	4	yes	—
30	C₂×M₁₆×C₁₃₁	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
32	C₄×D₄×C₁₃₁	—	2	8	no	—
33	C₄×D₄×C₁₃₁	—	4	4	no	—
34	#117×C₁₃₁	—	2	2	no	—
35	#117×C₁₃₁	D₄	1	1	yes	characteristic
36	C₂×C₂²⋊C₄×C₁₃₁	—	4	4	no	—
37	D₁₆×C₁₃₁	C₂×C₄	1	8	yes	complemented
38	D₁₆×C₁₃₁	C₂×C₄	1	8	yes	complemented
39	M₃₂×C₁₃₁	—	2	2	no	—
40	C₂×C₁₆×C₁₃₁	C₂×C₄	1	4	yes	complemented
41	C₂×C₁₆×C₁₃₁	C₂³	1	2	yes	—
42	C₂×C₁₆×C₁₃₁	C₂³	1	1	yes	characteristic
43	C₂×C₁₆×C₁₃₁	C₂³	1	1	yes	characteristic
44	#107×C₁₃₁	C₂³	1	4	yes	—
45	#106×C₁₃₁	—	2	2	no	—
46	#102×C₁₃₁	—	2	8	no	—
47	#97×C₁₃₁	D₄	1	2	yes	—
48	C₄×C₈×C₁₃₁	C₂³	1	2	yes	—
49	C₂⁴×C₁₃₁	—	4	2	no	—
50	C₂×D₄×C₁₃₁	—	2	8	no	—
51	C₂×D₄×C₁₃₁	—	2	4	no	—
52	C₂×D₄×C₁₃₁	—	4	2	no	—
53	C₂×D₄×C₁₃₁	—	4	4	no	—
54	C₂²×C₄×C₁₃₁	—	4	8	no	—
55	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
56	D₈×C₁₃₁	—	2	8	no	—
57	D₈×C₁₃₁	—	2	4	no	—
58	D₈×C₁₃₁	C₂²×C₄	1	8	yes	—
59	M₁₆×C₁₃₁	—	2	2	no	—
60	C₂×C₈×C₁₃₁	C₂²×C₄	1	2	yes	—
61	C₂×C₈×C₁₃₁	C₂×D₄	1	2	yes	—
62	C₂×C₈×C₁₃₁	C₂×D₄	1	1	yes	characteristic
63	C₂×C₈×C₁₃₁	#41	1	2	yes	—
64	C₂×C₈×C₁₃₁	C₂⁴	1	1	yes	characteristic
65	C₄⋊C₄×C₁₃₁	—	2	4	no	—
66	C₄⋊C₄×C₁₃₁	D₈	1	2	yes	—
67	C₂²⋊C₄×C₁₃₁	—	4	2	no	—
68	C₂²⋊C₄×C₁₃₁	—	4	8	no	—
69	C₂²⋊C₄×C₁₃₁	D₈	1	2	yes	—
70	C₄²×C₁₃₁	—	4	2	no	—
71	C₄²×C₁₃₁	C₂×D₄	1	2	yes	—
72	C₁₆×C₁₃₁	—	2	2	no	—
73	C₁₆×C₁₃₁	C₂²×C₄	1	2	yes	—
74	C₁₆×C₁₃₁	C₂²×C₄	1	2	yes	—
75	C₂³×C₁₃₁	—	4	4	no	—
76	C₂³×C₁₃₁	—	4	8	no	—
77	C₂³×C₁₃₁	—	16	4	no	—
78	C₂³×C₁₃₁	C₂×D₈	1	1	yes	characteristic
79	D₄×C₁₃₁	—	2	8	no	—
80	D₄×C₁₃₁	—	4	4	no	—
81	D₄×C₁₃₁	—	4	8	no	—
82	C₂×C₄×C₁₃₁	—	4	4	no	—
83	C₂×C₄×C₁₃₁	—	16	8	no	—
84	C₂×C₄×C₁₃₁	C₄×D₄	1	2	yes	—
85	C₂×C₄×C₁₃₁	C₂×D₈	1	4	yes	—
86	C₂×C₄×C₁₃₁	C₂×D₈	1	1	yes	characteristic
87	C₂×C₄×C₁₃₁	#135	1	2	yes	—
88	C₂×C₄×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
89	C₈×C₁₃₁	—	2	2	no	—
90	C₈×C₁₃₁	C₄×D₄	1	2	yes	—
91	C₈×C₁₃₁	C₂³×C₄	1	1	yes	characteristic
92	C₈×C₁₃₁	C₂×#41	1	1	yes	characteristic
93	C₂²×C₁₃₁	—	2	2	no	—
94	C₂²×C₁₃₁	—	4	8	no	—
95	C₂²×C₁₃₁	—	8	8	no	—
96	C₂²×C₁₃₁	—	16	8	no	—
97	C₂²×C₁₃₁	C₄×D₈	1	2	yes	—
98	C₂²×C₁₃₁	C₂²×D₈	1	1	yes	characteristic
99	C₄×C₁₃₁	—	2	4	no	—
100	C₄×C₁₃₁	—	8	4	no	—
101	C₄×C₁₃₁	C₄×D₈	1	2	yes	—
102	C₄×C₁₃₁	C₂×C₄×D₄	1	1	yes	characteristic
103	C₄×C₁₃₁	C₂×#135	1	1	yes	characteristic
104	C₂×C₁₃₁	—	2	2	no	—
105	C₂×C₁₃₁	—	8	8	no	—
106	C₂×C₁₃₁	C₂×C₄×D₈	1	1	yes	characteristic
107	C₂×C₁₃₁	#3415	1	2	yes	—
108	#33456	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
109	C₁₃₁	#33456	1	1	yes	characteristic, direct product factor, Sylow
110	C₂²×D₁₆	C₂×C₁₃₁	1	1	yes	characteristic
111	#2948	C₂×C₁₃₁	1	2	yes	complemented
112	#2177	C₂×C₁₃₁	1	4	yes	complemented
113	C₄×D₁₆	C₂×C₁₃₁	1	4	yes	complemented
114	#2158	C₂×C₁₃₁	1	1	yes	characteristic, complemented
115	#2140	C₂×C₁₃₁	1	2	yes	complemented
116	#2108	C₂×C₁₃₁	1	1	yes	characteristic, complemented
117	C₂²×D₈	C₂²×C₁₃₁	1	2	yes	—
118	#518	—	2	2	no	—
119	C₂×D₁₆	C₄×C₁₃₁	1	8	yes	complemented
120	C₂×D₁₆	C₂²×C₁₃₁	1	2	yes	—
121	C₂×D₁₆	C₂²×C₁₃₁	1	2	yes	—
122	C₂×M₃₂	C₂²×C₁₃₁	1	1	yes	characteristic
123	C₂²×C₁₆	C₂²×C₁₃₁	1	1	yes	characteristic
124	#442	—	2	4	no	—
125	C₄×D₈	C₂²×C₁₃₁	1	8	yes	complemented
126	#427	C₂²×C₁₃₁	1	2	yes	—
127	#418	—	2	2	no	—
128	#405	C₂²×C₁₃₁	1	1	yes	characteristic
129	#367	C₂²×C₁₃₁	1	2	yes	complemented
130	#366	C₂²×C₁₃₁	1	2	yes	complemented
131	#357	C₂²×C₁₃₁	1	8	yes	complemented
132	#346	C₂²×C₁₃₁	1	2	yes	complemented
133	C₄×C₁₆	C₂²×C₁₃₁	1	2	yes	complemented
134	C₂²×D₄	—	2	2	no	—
135	C₂×D₈	—	2	2	no	—
136	C₂×D₈	—	2	4	no	—
137	C₂×D₈	C₂×C₄×C₁₃₁	1	8	yes	complemented
138	C₂×D₈	C₂³×C₁₃₁	1	4	yes	—
139	C₂×M₁₆	D₄×C₁₃₁	1	1	yes	characteristic
140	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
141	C₄×D₄	—	2	8	no	—
142	C₄×D₄	—	4	4	no	—
143	#117	—	2	2	no	—
144	#117	D₄×C₁₃₁	1	1	yes	characteristic
145	C₂×C₂²⋊C₄	—	4	4	no	—
146	D₁₆	C₂×C₄×C₁₃₁	1	8	yes	complemented
147	D₁₆	C₂×C₄×C₁₃₁	1	8	yes	complemented
148	M₃₂	—	2	2	no	—
149	C₂×C₁₆	C₂×C₄×C₁₃₁	1	4	yes	complemented
150	C₂×C₁₆	C₂³×C₁₃₁	1	2	yes	—
151	C₂×C₁₆	C₂³×C₁₃₁	1	1	yes	characteristic
152	C₂×C₁₆	C₂³×C₁₃₁	1	1	yes	characteristic
153	#107	C₂³×C₁₃₁	1	4	yes	—
154	#106	—	2	2	no	—
155	#102	—	2	8	no	—
156	#97	D₄×C₁₃₁	1	2	yes	—
157	C₄×C₈	C₂³×C₁₃₁	1	2	yes	—
158	C₂⁴	—	4	2	no	—
159	C₂×D₄	—	2	8	no	—
160	C₂×D₄	—	2	4	no	—
161	C₂×D₄	—	4	2	no	—
162	C₂×D₄	—	4	4	no	—
163	C₂²×C₄	—	4	8	no	—
164	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
165	D₈	—	2	8	no	—
166	D₈	—	2	4	no	—
167	D₈	C₂²×C₄×C₁₃₁	1	8	yes	—
168	M₁₆	—	2	2	no	—
169	C₂×C₈	C₂²×C₄×C₁₃₁	1	2	yes	—
170	C₂×C₈	C₂×D₄×C₁₃₁	1	2	yes	—
171	C₂×C₈	C₂×D₄×C₁₃₁	1	1	yes	characteristic
172	C₂×C₈	#41×C₁₃₁	1	2	yes	—
173	C₂×C₈	C₂⁴×C₁₃₁	1	1	yes	characteristic
174	C₄⋊C₄	—	2	4	no	—
175	C₄⋊C₄	D₈×C₁₃₁	1	2	yes	—
176	C₂²⋊C₄	—	4	2	no	—
177	C₂²⋊C₄	—	4	8	no	—
178	C₂²⋊C₄	D₈×C₁₃₁	1	2	yes	—
179	C₄²	—	4	2	no	—
180	C₄²	C₂×D₄×C₁₃₁	1	2	yes	—
181	C₁₆	—	2	2	no	—
182	C₁₆	C₂²×C₄×C₁₃₁	1	2	yes	—
183	C₁₆	C₂²×C₄×C₁₃₁	1	2	yes	—
184	C₂³	—	4	4	no	—
185	C₂³	—	4	8	no	—
186	C₂³	—	16	4	no	—
187	C₂³	C₂×D₈×C₁₃₁	1	1	yes	characteristic
188	D₄	—	2	8	no	—
189	D₄	—	4	8	no	—
190	D₄	—	4	4	no	—
191	C₂×C₄	—	4	4	no	—
192	C₂×C₄	—	16	8	no	—
193	C₂×C₄	C₄×D₄×C₁₃₁	1	2	yes	—
194	C₂×C₄	C₂×D₈×C₁₃₁	1	4	yes	—
195	C₂×C₄	C₂×D₈×C₁₃₁	1	1	yes	characteristic
196	C₂×C₄	#135×C₁₃₁	1	2	yes	—
197	C₂×C₄	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
198	C₈	—	2	2	no	—
199	C₈	C₄×D₄×C₁₃₁	1	2	yes	—
200	C₈	C₂³×C₄×C₁₃₁	1	1	yes	characteristic
201	C₈	C₂×#41×C₁₃₁	1	1	yes	characteristic
202	C₂²	—	2	2	no	—
203	C₂²	—	4	8	no	—
204	C₂²	—	8	8	no	—
205	C₂²	—	16	8	no	—
206	C₂²	C₄×D₈×C₁₃₁	1	2	yes	—
207	C₂²	C₂²×D₈×C₁₃₁	1	1	yes	characteristic
208	C₄	—	2	4	no	—
209	C₄	—	8	4	no	—
210	C₄	C₄×D₈×C₁₃₁	1	2	yes	—
211	C₄	C₂×C₄×D₄×C₁₃₁	1	1	yes	characteristic
212	C₄	C₂×#135×C₁₃₁	1	1	yes	characteristic
213	C₂	—	2	2	no	—
214	C₂	—	8	8	no	—
215	C₂	C₂×C₄×D₈×C₁₃₁	1	1	yes	characteristic
216	C₂	#3415×C₁₃₁	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₃₁	C₂²×D₈
commutator	C₈	C₂³×C₄×C₁₃₁
Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃₁
prime radical	C₂²×D₁₆×C₁₃₁	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₁ ◁ #117×C₁₃₁ ◁ #405×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. #2948×C₁₃₁	C₂	semidirect
3. #2177×C₁₃₁	C₂	semidirect
4. C₄×D₁₆×C₁₃₁	C₂	semidirect
5. #2158×C₁₃₁	C₂	semidirect
6. #2140×C₁₃₁	C₂	semidirect
7. #2108×C₁₃₁	C₂	semidirect
10. C₂×D₁₆×C₁₃₁	C₄	semidirect
16. C₄×D₈×C₁₃₁	C₂²	semidirect
20. #367×C₁₃₁	C₂²	semidirect
21. #366×C₁₃₁	C₂²	semidirect
22. #357×C₁₃₁	C₂²	semidirect
23. #346×C₁₃₁	C₂²	semidirect
24. C₄×C₁₆×C₁₃₁	C₂²	semidirect
28. C₂×D₈×C₁₃₁	C₂×C₄	semidirect
37. D₁₆×C₁₃₁	C₂×C₄	semidirect
38. D₁₆×C₁₃₁	C₂×C₄	semidirect
40. C₂×C₁₆×C₁₃₁	C₂×C₄	semidirect
108. #33456	C₁₃₁	direct (N × H)
109. C₁₃₁	#33456	direct (N × H)
111. #2948	C₂×C₁₃₁	semidirect
112. #2177	C₂×C₁₃₁	semidirect
113. C₄×D₁₆	C₂×C₁₃₁	semidirect
114. #2158	C₂×C₁₃₁	semidirect
115. #2140	C₂×C₁₃₁	semidirect
116. #2108	C₂×C₁₃₁	semidirect
119. C₂×D₁₆	C₄×C₁₃₁	semidirect
125. C₄×D₈	C₂²×C₁₃₁	semidirect
129. #367	C₂²×C₁₃₁	semidirect
130. #366	C₂²×C₁₃₁	semidirect
131. #357	C₂²×C₁₃₁	semidirect
132. #346	C₂²×C₁₃₁	semidirect
133. C₄×C₁₆	C₂²×C₁₃₁	semidirect
137. C₂×D₈	C₂×C₄×C₁₃₁	semidirect
146. D₁₆	C₂×C₄×C₁₃₁	semidirect
147. D₁₆	C₂×C₄×C₁₃₁	semidirect
149. C₂×C₁₆	C₂×C₄×C₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂²×D₁₆×C₁₃₁	2
#2948×C₁₃₁	2
#2177×C₁₃₁	2
C₄×D₁₆×C₁₃₁	2
#2158×C₁₃₁	2
#2140×C₁₃₁	2
#2108×C₁₃₁	2
#33456	131

Subgroups — C131×#33456 #45719340

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#33456 #45719340