Subgroups — C₁₃₁×#33732 #45719616

Order 33536.

482 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3160×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	C₂×#499×C₁₃₁	C₂	1	1	yes	maximal, characteristic
3	#2255×C₁₃₁	C₂	1	4	yes	maximal, complemented
4	#2253×C₁₃₁	C₂	1	4	yes	maximal, complemented
5	#2159×C₁₃₁	C₂	1	4	yes	maximal
6	C₄×M₃₂×C₁₃₁	C₂	1	1	yes	maximal, characteristic
7	#533×C₁₃₁	—	2	1	no	—
8	C₂×#125×C₁₃₁	—	2	1	no	—
9	C₂×M₃₂×C₁₃₁	C₂²	1	2	yes	—
10	#500×C₁₃₁	C₂²	1	2	yes	—
11	#499×C₁₃₁	—	2	1	no	—
12	#499×C₁₃₁	—	2	1	no	—
13	#499×C₁₃₁	—	2	1	no	—
14	#499×C₁₃₁	C₂²	1	2	yes	—
15	#499×C₁₃₁	C₂²	1	2	yes	—
16	#499×C₁₃₁	C₂²	1	2	yes	—
17	#498×C₁₃₁	—	2	1	no	—
18	#427×C₁₃₁	—	2	1	no	—
19	#427×C₁₃₁	C₂²	1	2	yes	—
20	C₂×#107×C₁₃₁	C₂²	1	2	yes	—
21	C₂×#106×C₁₃₁	—	2	1	no	—
22	C₂×C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
23	#367×C₁₃₁	C₂²	1	8	yes	—
24	#366×C₁₃₁	C₂²	1	8	yes	—
25	#346×C₁₃₁	C₂²	1	2	yes	—
26	C₄×C₁₆×C₁₃₁	C₂²	1	2	yes	—
27	C₂²×C₈×C₁₃₁	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
29	C₄⋊Q₈×C₁₃₁	—	2	2	no	—
30	#125×C₁₃₁	—	2	2	no	—
31	#125×C₁₃₁	—	2	2	no	—
32	#125×C₁₃₁	—	2	2	no	—
33	#125×C₁₃₁	—	4	2	no	—
34	C₂²⋊Q₈×C₁₃₁	—	4	2	no	—
35	C₄×Q₈×C₁₃₁	—	4	2	no	—
36	#117×C₁₃₁	—	2	2	no	—
37	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
38	C₂×C₄⋊C₄×C₁₃₁	—	4	2	no	—
39	C₂×C₄²×C₁₃₁	D₄	1	1	yes	characteristic
40	M₃₂×C₁₃₁	Q₈	1	8	yes	complemented
41	C₂×C₁₆×C₁₃₁	C₂³	1	4	yes	—
42	#107×C₁₃₁	—	2	2	no	—
43	#107×C₁₃₁	—	2	2	no	—
44	#107×C₁₃₁	C₂³	1	4	yes	—
45	#107×C₁₃₁	C₂³	1	4	yes	—
46	#106×C₁₃₁	—	2	2	no	—
47	#106×C₁₃₁	—	2	2	no	—
48	#106×C₁₃₁	—	2	2	no	—
49	#106×C₁₃₁	—	2	2	no	—
50	C₄×C₈×C₁₃₁	D₄	1	1	yes	characteristic
51	C₄×C₈×C₁₃₁	D₄	1	1	yes	characteristic
52	C₄×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
53	C₄×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
54	C₂×Q₈×C₁₃₁	—	4	2	no	—
55	C₂²×C₄×C₁₃₁	—	4	2	no	—
56	C₂²×C₄×C₁₃₁	D₈	1	2	yes	—
57	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
58	C₂×C₈×C₁₃₁	Q₁₆	1	4	yes	—
59	C₂×C₈×C₁₃₁	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₁₃₁	C₂×D₄	1	1	yes	characteristic
61	C₂×C₈×C₁₃₁	C₂×D₄	1	1	yes	characteristic
62	C₂×C₈×C₁₃₁	C₂×Q₈	1	4	yes	—
63	C₂×C₈×C₁₃₁	C₂⁴	1	1	yes	characteristic
64	C₄⋊C₄×C₁₃₁	—	2	4	no	—
65	C₄⋊C₄×C₁₃₁	—	2	4	no	—
66	C₄⋊C₄×C₁₃₁	—	4	2	no	—
67	C₄⋊C₄×C₁₃₁	—	4	4	no	—
68	C₄⋊C₄×C₁₃₁	—	4	4	no	—
69	C₄⋊C₄×C₁₃₁	—	4	4	no	—
70	C₄⋊C₄×C₁₃₁	—	4	2	no	—
71	C₂²⋊C₄×C₁₃₁	—	4	2	no	—
72	C₄²×C₁₃₁	—	4	2	no	—
73	C₄²×C₁₃₁	D₈	1	2	yes	—
74	C₄²×C₁₃₁	C₂×D₄	1	1	yes	characteristic
75	C₄²×C₁₃₁	C₂×D₄	1	1	yes	characteristic
76	C₁₆×C₁₃₁	C₂×Q₈	1	8	yes	—
77	C₂³×C₁₃₁	C₂×D₈	1	1	yes	characteristic
78	Q₈×C₁₃₁	—	8	4	no	—
79	C₂×C₄×C₁₃₁	—	4	2	no	—
80	C₂×C₄×C₁₃₁	—	4	4	no	—
81	C₂×C₄×C₁₃₁	—	4	4	no	—
82	C₂×C₄×C₁₃₁	—	8	4	no	—
83	C₂×C₄×C₁₃₁	C₄⋊Q₈	1	2	yes	—
84	C₂×C₄×C₁₃₁	C₂×D₈	1	2	yes	—
85	C₂×C₄×C₁₃₁	C₂×D₈	1	2	yes	—
86	C₂×C₄×C₁₃₁	C₂×D₈	1	1	yes	characteristic
87	C₂×C₄×C₁₃₁	C₂×Q₁₆	1	2	yes	—
88	C₂×C₄×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
89	C₈×C₁₃₁	C₄⋊Q₈	1	2	yes	—
90	C₈×C₁₃₁	C₂×Q₁₆	1	2	yes	—
91	C₈×C₁₃₁	C₂×Q₁₆	1	2	yes	—
92	C₈×C₁₃₁	C₂²×Q₈	1	2	yes	—
93	C₂²×C₁₃₁	—	2	2	no	—
94	C₂²×C₁₃₁	#500	1	2	yes	—
95	C₂²×C₁₃₁	C₂²×D₈	1	1	yes	characteristic
96	C₄×C₁₃₁	—	2	2	no	—
97	C₄×C₁₃₁	—	2	2	no	—
98	C₄×C₁₃₁	—	8	4	no	—
99	C₄×C₁₃₁	—	8	4	no	—
100	C₄×C₁₃₁	#500	1	2	yes	—
101	C₄×C₁₃₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
102	C₄×C₁₃₁	C₂²×Q₁₆	1	1	yes	characteristic
103	C₂×C₁₃₁	—	2	2	no	—
104	C₂×C₁₃₁	C₂×#500	1	1	yes	characteristic
105	C₂×C₁₃₁	#3414	1	2	yes	—
106	#33732	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
107	C₁₃₁	#33732	1	1	yes	characteristic, direct product factor, Sylow
108	#3160	C₂×C₁₃₁	1	1	yes	characteristic
109	C₂×#499	C₂×C₁₃₁	1	1	yes	characteristic
110	#2255	C₂×C₁₃₁	1	4	yes	complemented
111	#2253	C₂×C₁₃₁	1	4	yes	complemented
112	#2159	C₂×C₁₃₁	1	4	yes	—
113	C₄×M₃₂	C₂×C₁₃₁	1	1	yes	characteristic
114	#533	—	2	1	no	—
115	C₂×#125	—	2	1	no	—
116	C₂×M₃₂	C₂²×C₁₃₁	1	2	yes	—
117	#500	C₂²×C₁₃₁	1	2	yes	—
118	#499	—	2	1	no	—
119	#499	—	2	1	no	—
120	#499	—	2	1	no	—
121	#499	C₂²×C₁₃₁	1	2	yes	—
122	#499	C₂²×C₁₃₁	1	2	yes	—
123	#499	C₂²×C₁₃₁	1	2	yes	—
124	#498	—	2	1	no	—
125	#427	—	2	1	no	—
126	#427	C₂²×C₁₃₁	1	2	yes	—
127	C₂×#107	C₂²×C₁₃₁	1	2	yes	—
128	C₂×#106	—	2	1	no	—
129	C₂×C₄×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
130	#367	C₂²×C₁₃₁	1	8	yes	—
131	#366	C₂²×C₁₃₁	1	8	yes	—
132	#346	C₂²×C₁₃₁	1	2	yes	—
133	C₄×C₁₆	C₂²×C₁₃₁	1	2	yes	—
134	C₂²×C₈	D₄×C₁₃₁	1	1	yes	characteristic
135	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
136	C₄⋊Q₈	—	2	2	no	—
137	#125	—	2	2	no	—
138	#125	—	2	2	no	—
139	#125	—	2	2	no	—
140	#125	—	4	2	no	—
141	C₂²⋊Q₈	—	4	2	no	—
142	C₄×Q₈	—	4	2	no	—
143	#117	—	2	2	no	—
144	C₂×C₄⋊C₄	—	2	2	no	—
145	C₂×C₄⋊C₄	—	4	2	no	—
146	C₂×C₄²	D₄×C₁₃₁	1	1	yes	characteristic
147	M₃₂	Q₈×C₁₃₁	1	8	yes	complemented
148	C₂×C₁₆	C₂³×C₁₃₁	1	4	yes	—
149	#107	—	2	2	no	—
150	#107	—	2	2	no	—
151	#107	C₂³×C₁₃₁	1	4	yes	—
152	#107	C₂³×C₁₃₁	1	4	yes	—
153	#106	—	2	2	no	—
154	#106	—	2	2	no	—
155	#106	—	2	2	no	—
156	#106	—	2	2	no	—
157	C₄×C₈	D₄×C₁₃₁	1	1	yes	characteristic
158	C₄×C₈	D₄×C₁₃₁	1	1	yes	characteristic
159	C₄×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
160	C₄×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
161	C₂×Q₈	—	4	2	no	—
162	C₂²×C₄	—	4	2	no	—
163	C₂²×C₄	D₈×C₁₃₁	1	2	yes	—
164	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
165	C₂×C₈	Q₁₆×C₁₃₁	1	4	yes	—
166	C₂×C₈	C₂×D₄×C₁₃₁	1	1	yes	characteristic
167	C₂×C₈	C₂×D₄×C₁₃₁	1	1	yes	characteristic
168	C₂×C₈	C₂×D₄×C₁₃₁	1	1	yes	characteristic
169	C₂×C₈	C₂×Q₈×C₁₃₁	1	4	yes	—
170	C₂×C₈	C₂⁴×C₁₃₁	1	1	yes	characteristic
171	C₄⋊C₄	—	2	4	no	—
172	C₄⋊C₄	—	2	4	no	—
173	C₄⋊C₄	—	4	2	no	—
174	C₄⋊C₄	—	4	4	no	—
175	C₄⋊C₄	—	4	4	no	—
176	C₄⋊C₄	—	4	4	no	—
177	C₄⋊C₄	—	4	2	no	—
178	C₂²⋊C₄	—	4	2	no	—
179	C₄²	—	4	2	no	—
180	C₄²	D₈×C₁₃₁	1	2	yes	—
181	C₄²	C₂×D₄×C₁₃₁	1	1	yes	characteristic
182	C₄²	C₂×D₄×C₁₃₁	1	1	yes	characteristic
183	C₁₆	C₂×Q₈×C₁₃₁	1	8	yes	—
184	C₂³	C₂×D₈×C₁₃₁	1	1	yes	characteristic
185	Q₈	—	8	4	no	—
186	C₂×C₄	—	4	2	no	—
187	C₂×C₄	—	4	4	no	—
188	C₂×C₄	—	4	4	no	—
189	C₂×C₄	—	8	4	no	—
190	C₂×C₄	C₄⋊Q₈×C₁₃₁	1	2	yes	—
191	C₂×C₄	C₂×D₈×C₁₃₁	1	2	yes	—
192	C₂×C₄	C₂×D₈×C₁₃₁	1	2	yes	—
193	C₂×C₄	C₂×D₈×C₁₃₁	1	1	yes	characteristic
194	C₂×C₄	C₂×Q₁₆×C₁₃₁	1	2	yes	—
195	C₂×C₄	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
196	C₈	C₄⋊Q₈×C₁₃₁	1	2	yes	—
197	C₈	C₂×Q₁₆×C₁₃₁	1	2	yes	—
198	C₈	C₂×Q₁₆×C₁₃₁	1	2	yes	—
199	C₈	C₂²×Q₈×C₁₃₁	1	2	yes	—
200	C₂²	—	2	2	no	—
201	C₂²	#500×C₁₃₁	1	2	yes	—
202	C₂²	C₂²×D₈×C₁₃₁	1	1	yes	characteristic
203	C₄	—	2	2	no	—
204	C₄	—	2	2	no	—
205	C₄	—	8	4	no	—
206	C₄	—	8	4	no	—
207	C₄	#500×C₁₃₁	1	2	yes	—
208	C₄	C₂×C₄⋊Q₈×C₁₃₁	1	1	yes	characteristic
209	C₄	C₂²×Q₁₆×C₁₃₁	1	1	yes	characteristic
210	C₂	—	2	2	no	—
211	C₂	C₂×#500×C₁₃₁	1	1	yes	characteristic
212	C₂	#3414×C₁₃₁	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₃₁	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃₁
prime radical	C₂³×C₁₃₁	C₂×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×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. #2255×C₁₃₁	C₂	semidirect
4. #2253×C₁₃₁	C₂	semidirect
40. M₃₂×C₁₃₁	Q₈	semidirect
106. #33732	C₁₃₁	direct (N × H)
107. C₁₃₁	#33732	direct (N × H)
110. #2255	C₂×C₁₃₁	semidirect
111. #2253	C₂×C₁₃₁	semidirect
147. M₃₂	Q₈×C₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
#3160×C₁₃₁	2
C₂×#499×C₁₃₁	2
#2255×C₁₃₁	2
#2253×C₁₃₁	2
#2159×C₁₃₁	2
C₄×M₃₂×C₁₃₁	2
#33732	131

Subgroups — C131×#33732 #45719616

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#33732 #45719616