Subgroups — C₁₃₁×#33716 #45719600

Order 33536.

826 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#508×C₁₃₁	C₂	1	4	yes	maximal, complemented
2	#3147×C₁₃₁	C₂	1	2	yes	maximal, complemented
3	#2251×C₁₃₁	C₂	1	4	yes	maximal, complemented
4	#2250×C₁₃₁	C₂	1	4	yes	maximal, complemented
5	C₂×#346×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#135×C₁₃₁	—	2	2	no	—
7	C₂×#135×C₁₃₁	C₂²	1	4	yes	—
8	#532×C₁₃₁	—	2	2	no	—
9	#508×C₁₃₁	—	2	16	no	—
10	C₂×Q₃₂×C₁₃₁	C₂²	1	4	yes	—
11	C₂×SD₃₂×C₁₃₁	C₂²	1	8	yes	—
12	C₂×D₁₆×C₁₃₁	C₂²	1	4	yes	—
13	C₂²×C₁₆×C₁₃₁	C₂²	1	2	yes	—
14	#495×C₁₃₁	—	2	2	no	—
15	#495×C₁₃₁	C₂²	1	4	yes	—
16	#494×C₁₃₁	C₂²	1	2	yes	—
17	#493×C₁₃₁	C₂²	1	2	yes	—
18	#492×C₁₃₁	—	2	2	no	—
19	C₂×C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
20	#346×C₁₃₁	C₂²	1	4	yes	—
21	C₂×#41×C₁₃₁	—	2	4	no	—
22	#135×C₁₃₁	—	2	16	no	—
23	#135×C₁₃₁	—	4	8	no	—
24	C₂×Q₁₆×C₁₃₁	—	2	2	no	—
25	C₂×Q₁₆×C₁₃₁	C₂³	1	4	yes	—
26	C₂×SD₁₆×C₁₃₁	—	2	4	no	—
27	C₂×SD₁₆×C₁₃₁	—	2	4	no	—
28	C₂×D₈×C₁₃₁	—	2	2	no	—
29	C₂×D₈×C₁₃₁	C₂³	1	4	yes	—
30	C₂²×C₈×C₁₃₁	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
32	C₄⋊Q₈×C₁₃₁	—	2	2	no	—
33	#127×C₁₃₁	—	2	2	no	—
34	#124×C₁₃₁	—	2	4	no	—
35	#121×C₁₃₁	—	4	4	no	—
36	C₄×D₄×C₁₃₁	—	4	4	no	—
37	C₂×C₄²×C₁₃₁	D₄	1	1	yes	characteristic
38	Q₃₂×C₁₃₁	—	2	8	no	—
39	SD₃₂×C₁₃₁	—	2	16	no	—
40	D₁₆×C₁₃₁	—	2	8	no	—
41	C₂×C₁₆×C₁₃₁	D₄	1	8	yes	complemented
42	C₂×C₁₆×C₁₃₁	C₂³	1	4	yes	—
43	C₄×C₈×C₁₃₁	D₄	1	2	yes	—
44	C₄×C₈×C₁₃₁	C₂³	1	2	yes	—
45	#41×C₁₃₁	—	4	8	no	—
46	#41×C₁₃₁	—	4	8	no	—
47	C₂×Q₈×C₁₃₁	—	2	4	no	—
48	C₂×D₄×C₁₃₁	—	2	4	no	—
49	C₂×D₄×C₁₃₁	—	4	4	no	—
50	C₂×D₄×C₁₃₁	—	4	4	no	—
51	C₂²×C₄×C₁₃₁	—	4	4	no	—
52	C₂²×C₄×C₁₃₁	D₈	1	2	yes	—
53	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
54	Q₁₆×C₁₃₁	—	2	8	no	—
55	Q₁₆×C₁₃₁	—	4	4	no	—
56	SD₁₆×C₁₃₁	—	4	8	no	—
57	SD₁₆×C₁₃₁	—	4	8	no	—
58	D₈×C₁₃₁	—	2	8	no	—
59	D₈×C₁₃₁	—	4	4	no	—
60	C₂×C₈×C₁₃₁	D₈	1	4	yes	—
61	C₂×C₈×C₁₃₁	C₂×D₄	1	1	yes	characteristic
62	C₂×C₈×C₁₃₁	C₂×D₄	1	4	yes	—
63	C₂×C₈×C₁₃₁	C₂×D₄	1	2	yes	—
64	C₂×C₈×C₁₃₁	C₂⁴	1	1	yes	characteristic
65	C₄⋊C₄×C₁₃₁	—	4	4	no	—
66	C₂²⋊C₄×C₁₃₁	—	4	8	no	—
67	C₄²×C₁₃₁	D₈	1	2	yes	—
68	C₄²×C₁₃₁	C₂×D₄	1	2	yes	—
69	C₁₆×C₁₃₁	C₂×D₄	1	8	yes	—
70	C₂³×C₁₃₁	—	4	4	no	—
71	C₂³×C₁₃₁	C₂×D₈	1	1	yes	characteristic
72	Q₈×C₁₃₁	—	4	4	no	—
73	Q₈×C₁₃₁	—	4	4	no	—
74	D₄×C₁₃₁	—	4	4	no	—
75	D₄×C₁₃₁	—	4	4	no	—
76	D₄×C₁₃₁	—	8	8	no	—
77	D₄×C₁₃₁	—	8	8	no	—
78	C₂×C₄×C₁₃₁	—	4	2	no	—
79	C₂×C₄×C₁₃₁	—	4	4	no	—
80	C₂×C₄×C₁₃₁	—	8	8	no	—
81	C₂×C₄×C₁₃₁	#127	1	2	yes	—
82	C₂×C₄×C₁₃₁	C₂×D₈	1	4	yes	—
83	C₂×C₄×C₁₃₁	C₂×D₈	1	2	yes	—
84	C₂×C₄×C₁₃₁	C₂×D₈	1	1	yes	characteristic
85	C₂×C₄×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
86	C₈×C₁₃₁	#127	1	2	yes	—
87	C₈×C₁₃₁	C₂×D₈	1	4	yes	—
88	C₈×C₁₃₁	C₂²×D₄	1	2	yes	—
89	C₂²×C₁₃₁	—	2	2	no	—
90	C₂²×C₁₃₁	—	8	4	no	—
91	C₂²×C₁₃₁	—	8	8	no	—
92	C₂²×C₁₃₁	#493	1	2	yes	—
93	C₂²×C₁₃₁	C₂²×D₈	1	1	yes	characteristic
94	C₄×C₁₃₁	—	2	4	no	—
95	C₄×C₁₃₁	—	8	4	no	—
96	C₄×C₁₃₁	#493	1	2	yes	—
97	C₄×C₁₃₁	C₂×#127	1	1	yes	characteristic
98	C₄×C₁₃₁	C₂²×D₈	1	1	yes	characteristic
99	C₂×C₁₃₁	—	2	2	no	—
100	C₂×C₁₃₁	—	16	4	no	—
101	C₂×C₁₃₁	C₂×#493	1	1	yes	characteristic
102	C₂×C₁₃₁	#3414	1	2	yes	—
103	#33716	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
104	C₁₃₁	#33716	1	1	yes	characteristic, direct product factor, Sylow
105	C₂×#508	C₂×C₁₃₁	1	4	yes	complemented
106	#3147	C₂×C₁₃₁	1	2	yes	complemented
107	#2251	C₂×C₁₃₁	1	4	yes	complemented
108	#2250	C₂×C₁₃₁	1	4	yes	complemented
109	C₂×#346	C₂×C₁₃₁	1	1	yes	characteristic, complemented
110	C₂×#135	—	2	2	no	—
111	C₂×#135	C₂²×C₁₃₁	1	4	yes	—
112	#532	—	2	2	no	—
113	#508	—	2	16	no	—
114	C₂×Q₃₂	C₂²×C₁₃₁	1	4	yes	—
115	C₂×SD₃₂	C₂²×C₁₃₁	1	8	yes	—
116	C₂×D₁₆	C₂²×C₁₃₁	1	4	yes	—
117	C₂²×C₁₆	C₂²×C₁₃₁	1	2	yes	—
118	#495	—	2	2	no	—
119	#495	C₂²×C₁₃₁	1	4	yes	—
120	#494	C₂²×C₁₃₁	1	2	yes	—
121	#493	C₂²×C₁₃₁	1	2	yes	—
122	#492	—	2	2	no	—
123	C₂×C₄×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
124	#346	C₂²×C₁₃₁	1	4	yes	—
125	C₂×#41	—	2	4	no	—
126	#135	—	2	16	no	—
127	#135	—	4	8	no	—
128	C₂×Q₁₆	—	2	2	no	—
129	C₂×Q₁₆	C₂³×C₁₃₁	1	4	yes	—
130	C₂×SD₁₆	—	2	4	no	—
131	C₂×SD₁₆	—	2	4	no	—
132	C₂×D₈	—	2	2	no	—
133	C₂×D₈	C₂³×C₁₃₁	1	4	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	#127	—	2	2	no	—
138	#124	—	2	4	no	—
139	#121	—	4	4	no	—
140	C₄×D₄	—	4	4	no	—
141	C₂×C₄²	D₄×C₁₃₁	1	1	yes	characteristic
142	Q₃₂	—	2	8	no	—
143	SD₃₂	—	2	16	no	—
144	D₁₆	—	2	8	no	—
145	C₂×C₁₆	D₄×C₁₃₁	1	8	yes	complemented
146	C₂×C₁₆	C₂³×C₁₃₁	1	4	yes	—
147	C₄×C₈	D₄×C₁₃₁	1	2	yes	—
148	C₄×C₈	C₂³×C₁₃₁	1	2	yes	—
149	#41	—	4	8	no	—
150	#41	—	4	8	no	—
151	C₂×Q₈	—	2	4	no	—
152	C₂×D₄	—	2	4	no	—
153	C₂×D₄	—	4	4	no	—
154	C₂×D₄	—	4	4	no	—
155	C₂²×C₄	—	4	4	no	—
156	C₂²×C₄	D₈×C₁₃₁	1	2	yes	—
157	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
158	Q₁₆	—	2	8	no	—
159	Q₁₆	—	4	4	no	—
160	SD₁₆	—	4	8	no	—
161	SD₁₆	—	4	8	no	—
162	D₈	—	2	8	no	—
163	D₈	—	4	4	no	—
164	C₂×C₈	D₈×C₁₃₁	1	4	yes	—
165	C₂×C₈	C₂×D₄×C₁₃₁	1	1	yes	characteristic
166	C₂×C₈	C₂×D₄×C₁₃₁	1	4	yes	—
167	C₂×C₈	C₂×D₄×C₁₃₁	1	2	yes	—
168	C₂×C₈	C₂⁴×C₁₃₁	1	1	yes	characteristic
169	C₄⋊C₄	—	4	4	no	—
170	C₂²⋊C₄	—	4	8	no	—
171	C₄²	D₈×C₁₃₁	1	2	yes	—
172	C₄²	C₂×D₄×C₁₃₁	1	2	yes	—
173	C₁₆	C₂×D₄×C₁₃₁	1	8	yes	—
174	C₂³	—	4	4	no	—
175	C₂³	C₂×D₈×C₁₃₁	1	1	yes	characteristic
176	Q₈	—	4	4	no	—
177	Q₈	—	4	4	no	—
178	D₄	—	4	4	no	—
179	D₄	—	4	4	no	—
180	D₄	—	8	8	no	—
181	D₄	—	8	8	no	—
182	C₂×C₄	—	4	2	no	—
183	C₂×C₄	—	4	4	no	—
184	C₂×C₄	—	8	8	no	—
185	C₂×C₄	#127×C₁₃₁	1	2	yes	—
186	C₂×C₄	C₂×D₈×C₁₃₁	1	4	yes	—
187	C₂×C₄	C₂×D₈×C₁₃₁	1	1	yes	characteristic
188	C₂×C₄	C₂×D₈×C₁₃₁	1	2	yes	—
189	C₂×C₄	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
190	C₈	#127×C₁₃₁	1	2	yes	—
191	C₈	C₂×D₈×C₁₃₁	1	4	yes	—
192	C₈	C₂²×D₄×C₁₃₁	1	2	yes	—
193	C₂²	—	2	2	no	—
194	C₂²	—	8	4	no	—
195	C₂²	—	8	8	no	—
196	C₂²	#493×C₁₃₁	1	2	yes	—
197	C₂²	C₂²×D₈×C₁₃₁	1	1	yes	characteristic
198	C₄	—	2	4	no	—
199	C₄	—	8	4	no	—
200	C₄	#493×C₁₃₁	1	2	yes	—
201	C₄	C₂×#127×C₁₃₁	1	1	yes	characteristic
202	C₄	C₂²×D₈×C₁₃₁	1	1	yes	characteristic
203	C₂	—	2	2	no	—
204	C₂	—	16	4	no	—
205	C₂	C₂×#493×C₁₃₁	1	1	yes	characteristic
206	C₂	#3414×C₁₃₁	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₁₃₁	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃₁
socle	C₂²×C₁₃₁	C₂²×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×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
1. C₂×#508×C₁₃₁	C₂	semidirect
2. #3147×C₁₃₁	C₂	semidirect
3. #2251×C₁₃₁	C₂	semidirect
4. #2250×C₁₃₁	C₂	semidirect
5. C₂×#346×C₁₃₁	C₂	semidirect
41. C₂×C₁₆×C₁₃₁	D₄	semidirect
103. #33716	C₁₃₁	direct (N × H)
104. C₁₃₁	#33716	direct (N × H)
105. C₂×#508	C₂×C₁₃₁	semidirect
106. #3147	C₂×C₁₃₁	semidirect
107. #2251	C₂×C₁₃₁	semidirect
108. #2250	C₂×C₁₃₁	semidirect
109. C₂×#346	C₂×C₁₃₁	semidirect
145. C₂×C₁₆	D₄×C₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#508×C₁₃₁	2
#3147×C₁₃₁	2
#2251×C₁₃₁	2
#2250×C₁₃₁	2
C₂×#346×C₁₃₁	2
#33716	131

Subgroups — C131×#33716 #45719600

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#33716 #45719600