Subgroups — C₁₃₁×#33734 #45719618

Order 33536.

514 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3170×C₁₃₁	C₂	1	2	yes	maximal
2	#2252×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#2247×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
4	#2241×C₁₃₁	C₂	1	2	yes	maximal, complemented
5	#2216×C₁₃₁	C₂	1	4	yes	maximal
6	C₄×Q₃₂×C₁₃₁	C₂	1	4	yes	maximal, complemented
7	C₂×C₄×C₁₆×C₁₃₁	C₂	1	1	yes	maximal, characteristic
8	#533×C₁₃₁	—	2	2	no	—
9	C₂×Q₃₂×C₁₃₁	C₂²	1	4	yes	—
10	C₂²×C₁₆×C₁₃₁	C₂²	1	2	yes	—
11	#500×C₁₃₁	C₂²	1	2	yes	—
12	#494×C₁₃₁	C₂²	1	2	yes	—
13	#487×C₁₃₁	—	2	2	no	—
14	#467×C₁₃₁	—	2	2	no	—
15	#467×C₁₃₁	C₂²	1	4	yes	—
16	C₄×Q₁₆×C₁₃₁	—	2	2	no	—
17	C₄×Q₁₆×C₁₃₁	C₂²	1	4	yes	—
18	C₂×C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
19	#366×C₁₃₁	C₂²	1	4	yes	—
20	#358×C₁₃₁	C₂²	1	8	yes	—
21	C₄×C₁₆×C₁₃₁	C₂²	1	1	yes	characteristic
22	C₄×C₁₆×C₁₃₁	C₂²	1	1	yes	characteristic
23	C₄×C₁₆×C₁₃₁	C₂²	1	2	yes	—
24	C₂×Q₁₆×C₁₃₁	—	2	2	no	—
25	C₂×Q₁₆×C₁₃₁	C₂³	1	4	yes	—
26	C₂²×C₈×C₁₃₁	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
28	C₄⋊Q₈×C₁₃₁	—	2	2	no	—
29	C₄⋊Q₈×C₁₃₁	—	2	2	no	—
30	#125×C₁₃₁	—	4	2	no	—
31	C₂²⋊Q₈×C₁₃₁	—	2	4	no	—
32	C₂²⋊Q₈×C₁₃₁	—	4	2	no	—
33	C₄×Q₈×C₁₃₁	—	2	4	no	—
34	C₄×Q₈×C₁₃₁	—	4	2	no	—
35	#117×C₁₃₁	—	4	2	no	—
36	C₂×C₄²×C₁₃₁	D₄	1	1	yes	characteristic
37	Q₃₂×C₁₃₁	—	2	8	no	—
38	C₂×C₁₆×C₁₃₁	—	2	4	no	—
39	C₂×C₁₆×C₁₃₁	C₂³	1	2	yes	—
40	C₂×C₁₆×C₁₃₁	C₂³	1	2	yes	—
41	#107×C₁₃₁	—	2	2	no	—
42	#107×C₁₃₁	C₂³	1	4	yes	—
43	#103×C₁₃₁	—	2	4	no	—
44	#103×C₁₃₁	—	2	4	no	—
45	C₄×C₈×C₁₃₁	D₄	1	1	yes	characteristic
46	C₄×C₈×C₁₃₁	D₄	1	1	yes	characteristic
47	C₄×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
48	C₄×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
49	C₂×Q₈×C₁₃₁	—	2	4	no	—
50	C₂×Q₈×C₁₃₁	—	4	2	no	—
51	C₂²×C₄×C₁₃₁	D₈	1	1	yes	characteristic
52	C₂²×C₄×C₁₃₁	D₈	1	1	yes	characteristic
53	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
54	Q₁₆×C₁₃₁	—	2	8	no	—
55	Q₁₆×C₁₃₁	—	4	4	no	—
56	C₂×C₈×C₁₃₁	—	2	2	no	—
57	C₂×C₈×C₁₃₁	—	2	2	no	—
58	C₂×C₈×C₁₃₁	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈×C₁₃₁	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₁₃₁	C₂×D₄	1	1	yes	characteristic
61	C₂×C₈×C₁₃₁	C₂⁴	1	1	yes	characteristic
62	C₄⋊C₄×C₁₃₁	—	2	4	no	—
63	C₄⋊C₄×C₁₃₁	—	4	2	no	—
64	C₄⋊C₄×C₁₃₁	—	4	2	no	—
65	C₄⋊C₄×C₁₃₁	—	4	4	no	—
66	C₄⋊C₄×C₁₃₁	—	4	4	no	—
67	C₄⋊C₄×C₁₃₁	—	4	2	no	—
68	C₂²⋊C₄×C₁₃₁	—	4	4	no	—
69	C₄²×C₁₃₁	—	4	4	no	—
70	C₄²×C₁₃₁	D₈	1	1	yes	characteristic
71	C₄²×C₁₃₁	D₈	1	1	yes	characteristic
72	C₄²×C₁₃₁	C₂×D₄	1	1	yes	characteristic
73	C₄²×C₁₃₁	C₂×D₄	1	1	yes	characteristic
74	C₁₆×C₁₃₁	—	2	2	no	—
75	C₁₆×C₁₃₁	#41	1	4	yes	—
76	C₂³×C₁₃₁	C₂×D₈	1	1	yes	characteristic
77	Q₈×C₁₃₁	—	4	4	no	—
78	Q₈×C₁₃₁	—	4	4	no	—
79	Q₈×C₁₃₁	—	8	4	no	—
80	C₂×C₄×C₁₃₁	—	2	1	no	—
81	C₂×C₄×C₁₃₁	—	2	1	no	—
82	C₂×C₄×C₁₃₁	—	2	2	no	—
83	C₂×C₄×C₁₃₁	—	4	4	no	—
84	C₂×C₄×C₁₃₁	—	4	4	no	—
85	C₂×C₄×C₁₃₁	Q₃₂	1	4	yes	complemented
86	C₂×C₄×C₁₃₁	C₂×D₈	1	1	yes	characteristic
87	C₂×C₄×C₁₃₁	C₂×D₈	1	1	yes	characteristic
88	C₂×C₄×C₁₃₁	C₂×D₈	1	1	yes	characteristic
89	C₂×C₄×C₁₃₁	C₂×D₈	1	1	yes	characteristic
90	C₂×C₄×C₁₃₁	C₂×D₈	1	1	yes	characteristic
91	C₂×C₄×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
92	C₈×C₁₃₁	—	2	1	no	—
93	C₈×C₁₃₁	—	2	1	no	—
94	C₈×C₁₃₁	#135	1	2	yes	—
95	C₈×C₁₃₁	C₂×#41	1	2	yes	—
96	C₂²×C₁₃₁	—	2	1	no	—
97	C₂²×C₁₃₁	—	2	1	no	—
98	C₂²×C₁₃₁	C₂×Q₃₂	1	2	yes	—
99	C₂²×C₁₃₁	C₂²×D₈	1	1	yes	characteristic
100	C₄×C₁₃₁	—	2	1	no	—
101	C₄×C₁₃₁	—	2	1	no	—
102	C₄×C₁₃₁	—	8	4	no	—
103	C₄×C₁₃₁	—	8	4	no	—
104	C₄×C₁₃₁	C₂×Q₃₂	1	2	yes	—
105	C₄×C₁₃₁	C₂×Q₃₂	1	2	yes	—
106	C₄×C₁₃₁	#508	1	2	yes	—
107	C₄×C₁₃₁	#532	1	1	yes	characteristic
108	C₄×C₁₃₁	C₂×#135	1	1	yes	characteristic
109	C₂×C₁₃₁	—	2	2	no	—
110	C₂×C₁₃₁	#3169	1	1	yes	characteristic
111	C₂×C₁₃₁	C₂²×Q₃₂	1	1	yes	characteristic
112	C₂×C₁₃₁	C₂×#508	1	1	yes	characteristic
113	#33734	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
114	C₁₃₁	#33734	1	1	yes	characteristic, direct product factor, Sylow
115	#3170	C₂×C₁₃₁	1	2	yes	—
116	#2252	C₂×C₁₃₁	1	1	yes	characteristic, complemented
117	#2247	C₂×C₁₃₁	1	1	yes	characteristic, complemented
118	#2241	C₂×C₁₃₁	1	2	yes	complemented
119	#2216	C₂×C₁₃₁	1	4	yes	—
120	C₄×Q₃₂	C₂×C₁₃₁	1	4	yes	complemented
121	C₂×C₄×C₁₆	C₂×C₁₃₁	1	1	yes	characteristic
122	#533	—	2	2	no	—
123	C₂×Q₃₂	C₂²×C₁₃₁	1	4	yes	—
124	C₂²×C₁₆	C₂²×C₁₃₁	1	2	yes	—
125	#500	C₂²×C₁₃₁	1	2	yes	—
126	#494	C₂²×C₁₃₁	1	2	yes	—
127	#487	—	2	2	no	—
128	#467	—	2	2	no	—
129	#467	C₂²×C₁₃₁	1	4	yes	—
130	C₄×Q₁₆	—	2	2	no	—
131	C₄×Q₁₆	C₂²×C₁₃₁	1	4	yes	—
132	C₂×C₄×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
133	#366	C₂²×C₁₃₁	1	4	yes	—
134	#358	C₂²×C₁₃₁	1	8	yes	—
135	C₄×C₁₆	C₂²×C₁₃₁	1	2	yes	—
136	C₄×C₁₆	C₂²×C₁₃₁	1	1	yes	characteristic
137	C₄×C₁₆	C₂²×C₁₃₁	1	1	yes	characteristic
138	C₂×Q₁₆	—	2	2	no	—
139	C₂×Q₁₆	C₂³×C₁₃₁	1	4	yes	—
140	C₂²×C₈	D₄×C₁₃₁	1	1	yes	characteristic
141	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
142	C₄⋊Q₈	—	2	2	no	—
143	C₄⋊Q₈	—	2	2	no	—
144	#125	—	4	2	no	—
145	C₂²⋊Q₈	—	2	4	no	—
146	C₂²⋊Q₈	—	4	2	no	—
147	C₄×Q₈	—	2	4	no	—
148	C₄×Q₈	—	4	2	no	—
149	#117	—	4	2	no	—
150	C₂×C₄²	D₄×C₁₃₁	1	1	yes	characteristic
151	Q₃₂	—	2	8	no	—
152	C₂×C₁₆	—	2	4	no	—
153	C₂×C₁₆	C₂³×C₁₃₁	1	2	yes	—
154	C₂×C₁₆	C₂³×C₁₃₁	1	2	yes	—
155	#107	—	2	2	no	—
156	#107	C₂³×C₁₃₁	1	4	yes	—
157	#103	—	2	4	no	—
158	#103	—	2	4	no	—
159	C₄×C₈	D₄×C₁₃₁	1	1	yes	characteristic
160	C₄×C₈	D₄×C₁₃₁	1	1	yes	characteristic
161	C₄×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
162	C₄×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
163	C₂×Q₈	—	2	4	no	—
164	C₂×Q₈	—	4	2	no	—
165	C₂²×C₄	D₈×C₁₃₁	1	1	yes	characteristic
166	C₂²×C₄	D₈×C₁₃₁	1	1	yes	characteristic
167	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
168	Q₁₆	—	2	8	no	—
169	Q₁₆	—	4	4	no	—
170	C₂×C₈	—	2	2	no	—
171	C₂×C₈	—	2	2	no	—
172	C₂×C₈	C₂×D₄×C₁₃₁	1	1	yes	characteristic
173	C₂×C₈	C₂×D₄×C₁₃₁	1	1	yes	characteristic
174	C₂×C₈	C₂×D₄×C₁₃₁	1	1	yes	characteristic
175	C₂×C₈	C₂⁴×C₁₃₁	1	1	yes	characteristic
176	C₄⋊C₄	—	2	4	no	—
177	C₄⋊C₄	—	4	2	no	—
178	C₄⋊C₄	—	4	2	no	—
179	C₄⋊C₄	—	4	4	no	—
180	C₄⋊C₄	—	4	4	no	—
181	C₄⋊C₄	—	4	2	no	—
182	C₂²⋊C₄	—	4	4	no	—
183	C₄²	—	4	4	no	—
184	C₄²	D₈×C₁₃₁	1	1	yes	characteristic
185	C₄²	D₈×C₁₃₁	1	1	yes	characteristic
186	C₄²	C₂×D₄×C₁₃₁	1	1	yes	characteristic
187	C₄²	C₂×D₄×C₁₃₁	1	1	yes	characteristic
188	C₁₆	—	2	2	no	—
189	C₁₆	#41×C₁₃₁	1	4	yes	—
190	C₂³	C₂×D₈×C₁₃₁	1	1	yes	characteristic
191	Q₈	—	4	4	no	—
192	Q₈	—	4	4	no	—
193	Q₈	—	8	4	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	—	4	4	no	—
198	C₂×C₄	—	4	4	no	—
199	C₂×C₄	Q₃₂×C₁₃₁	1	4	yes	complemented
200	C₂×C₄	C₂×D₈×C₁₃₁	1	1	yes	characteristic
201	C₂×C₄	C₂×D₈×C₁₃₁	1	1	yes	characteristic
202	C₂×C₄	C₂×D₈×C₁₃₁	1	1	yes	characteristic
203	C₂×C₄	C₂×D₈×C₁₃₁	1	1	yes	characteristic
204	C₂×C₄	C₂×D₈×C₁₃₁	1	1	yes	characteristic
205	C₂×C₄	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
206	C₈	—	2	1	no	—
207	C₈	—	2	1	no	—
208	C₈	#135×C₁₃₁	1	2	yes	—
209	C₈	C₂×#41×C₁₃₁	1	2	yes	—
210	C₂²	—	2	1	no	—
211	C₂²	—	2	1	no	—
212	C₂²	C₂×Q₃₂×C₁₃₁	1	2	yes	—
213	C₂²	C₂²×D₈×C₁₃₁	1	1	yes	characteristic
214	C₄	—	2	1	no	—
215	C₄	—	2	1	no	—
216	C₄	—	8	4	no	—
217	C₄	—	8	4	no	—
218	C₄	C₂×Q₃₂×C₁₃₁	1	2	yes	—
219	C₄	C₂×Q₃₂×C₁₃₁	1	2	yes	—
220	C₄	#508×C₁₃₁	1	2	yes	—
221	C₄	#532×C₁₃₁	1	1	yes	characteristic
222	C₄	C₂×#135×C₁₃₁	1	1	yes	characteristic
223	C₂	—	2	2	no	—
224	C₂	#3169×C₁₃₁	1	1	yes	characteristic
225	C₂	C₂²×Q₃₂×C₁₃₁	1	1	yes	characteristic
226	C₂	C₂×#508×C₁₃₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₃₁	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃₁
socle	C₂²×C₁₃₁	C₂²×D₈
prime radical	C₂³×C₁₃₁	C₂×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
2. #2252×C₁₃₁	C₂	semidirect
3. #2247×C₁₃₁	C₂	semidirect
4. #2241×C₁₃₁	C₂	semidirect
6. C₄×Q₃₂×C₁₃₁	C₂	semidirect
85. C₂×C₄×C₁₃₁	Q₃₂	semidirect
113. #33734	C₁₃₁	direct (N × H)
114. C₁₃₁	#33734	direct (N × H)
116. #2252	C₂×C₁₃₁	semidirect
117. #2247	C₂×C₁₃₁	semidirect
118. #2241	C₂×C₁₃₁	semidirect
120. C₄×Q₃₂	C₂×C₁₃₁	semidirect
199. C₂×C₄	Q₃₂×C₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
#3170×C₁₃₁	2
#2252×C₁₃₁	2
#2247×C₁₃₁	2
#2241×C₁₃₁	2
#2216×C₁₃₁	2
C₄×Q₃₂×C₁₃₁	2
C₂×C₄×C₁₆×C₁₃₁	2
#33734	131

Subgroups — C131×#33734 #45719618

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#33734 #45719618