Subgroups — C₁₃₁×#33730 #45719614

Order 33536.

474 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3164×C₁₃₁	C₂	1	2	yes	maximal
2	#2255×C₁₃₁	C₂	1	4	yes	maximal, complemented
3	#2254×C₁₃₁	C₂	1	2	yes	maximal, complemented
4	#2253×C₁₃₁	C₂	1	2	yes	maximal, complemented
5	#2159×C₁₃₁	C₂	1	2	yes	maximal
6	#2158×C₁₃₁	C₂	1	1	yes	maximal, characteristic
7	C₂×#367×C₁₃₁	C₂	1	1	yes	maximal, characteristic
8	#2108×C₁₃₁	C₂	1	1	yes	maximal, characteristic
9	#544×C₁₃₁	—	2	2	no	—
10	C₂×M₃₂×C₁₃₁	C₂²	1	1	yes	characteristic
11	C₂²×C₁₆×C₁₃₁	C₂²	1	1	yes	characteristic
12	#501×C₁₃₁	—	2	4	no	—
13	#500×C₁₃₁	C₂²	1	4	yes	—
14	#499×C₁₃₁	C₂²	1	4	yes	—
15	#428×C₁₃₁	—	2	2	no	—
16	#427×C₁₃₁	C₂²	1	2	yes	—
17	C₂×#107×C₁₃₁	C₂²	1	2	yes	—
18	#405×C₁₃₁	C₂²	1	1	yes	characteristic
19	#367×C₁₃₁	C₂²	1	1	yes	characteristic
20	#367×C₁₃₁	C₂²	1	2	yes	—
21	#367×C₁₃₁	C₂²	1	1	yes	characteristic
22	#367×C₁₃₁	C₂²	1	4	yes	—
23	#367×C₁₃₁	C₂²	1	2	yes	—
24	#366×C₁₃₁	C₂²	1	4	yes	—
25	#366×C₁₃₁	C₂²	1	2	yes	—
26	#346×C₁₃₁	C₂²	1	2	yes	—
27	C₄×C₁₆×C₁₃₁	C₂²	1	2	yes	—
28	C₂×M₁₆×C₁₃₁	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
30	C₄⋊Q₈×C₁₃₁	—	2	4	no	—
31	C₄⋊Q₈×C₁₃₁	—	4	2	no	—
32	#125×C₁₃₁	—	2	4	no	—
33	#125×C₁₃₁	—	4	2	no	—
34	C₂²⋊Q₈×C₁₃₁	—	4	4	no	—
35	#117×C₁₃₁	—	2	2	no	—
36	#117×C₁₃₁	D₄	1	1	yes	characteristic
37	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
38	M₃₂×C₁₃₁	Q₈	1	4	yes	complemented
39	C₂×C₁₆×C₁₃₁	Q₈	1	4	yes	complemented
40	C₂×C₁₆×C₁₃₁	C₂³	1	1	yes	characteristic
41	C₂×C₁₆×C₁₃₁	C₂³	1	1	yes	characteristic
42	C₂×C₁₆×C₁₃₁	C₂³	1	2	yes	—
43	#107×C₁₃₁	—	2	2	no	—
44	#107×C₁₃₁	—	2	4	no	—
45	#107×C₁₃₁	C₂³	1	4	yes	—
46	#107×C₁₃₁	C₂³	1	2	yes	—
47	#107×C₁₃₁	C₂³	1	2	yes	—
48	#106×C₁₃₁	—	2	2	no	—
49	#106×C₁₃₁	—	2	4	no	—
50	#97×C₁₃₁	D₄	1	2	yes	—
51	C₄×C₈×C₁₃₁	C₂³	1	2	yes	—
52	C₂×Q₈×C₁₃₁	—	4	4	no	—
53	C₂²×C₄×C₁₃₁	—	4	2	no	—
54	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
55	M₁₆×C₁₃₁	Q₁₆	1	4	yes	—
56	C₂×C₈×C₁₃₁	C₂×D₄	1	1	yes	characteristic
57	C₂×C₈×C₁₃₁	C₂×D₄	1	2	yes	—
58	C₂×C₈×C₁₃₁	C₂×Q₈	1	2	yes	—
59	C₂×C₈×C₁₃₁	C₂×Q₈	1	2	yes	—
60	C₂×C₈×C₁₃₁	C₂⁴	1	1	yes	characteristic
61	C₄⋊C₄×C₁₃₁	—	2	2	no	—
62	C₄⋊C₄×C₁₃₁	—	2	4	no	—
63	C₄⋊C₄×C₁₃₁	—	2	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	4	no	—
68	C₄⋊C₄×C₁₃₁	D₈	1	2	yes	—
69	C₂²⋊C₄×C₁₃₁	—	4	2	no	—
70	C₂²⋊C₄×C₁₃₁	D₈	1	2	yes	—
71	C₄²×C₁₃₁	—	4	2	no	—
72	C₄²×C₁₃₁	C₂×D₄	1	2	yes	—
73	C₁₆×C₁₃₁	C₂×Q₈	1	2	yes	—
74	C₁₆×C₁₃₁	C₂×Q₈	1	2	yes	—
75	C₁₆×C₁₃₁	C₂×Q₈	1	4	yes	—
76	C₂³×C₁₃₁	C₂×D₈	1	1	yes	characteristic
77	Q₈×C₁₃₁	—	16	4	no	—
78	C₂×C₄×C₁₃₁	—	4	2	no	—
79	C₂×C₄×C₁₃₁	—	4	2	no	—
80	C₂×C₄×C₁₃₁	—	4	4	no	—
81	C₂×C₄×C₁₃₁	—	16	2	no	—
82	C₂×C₄×C₁₃₁	C₄⋊Q₈	1	2	yes	—
83	C₂×C₄×C₁₃₁	C₂×D₈	1	4	yes	—
84	C₂×C₄×C₁₃₁	C₂×D₈	1	1	yes	characteristic
85	C₂×C₄×C₁₃₁	C₂×Q₁₆	1	2	yes	—
86	C₂×C₄×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
87	C₈×C₁₃₁	C₄⋊Q₈	1	2	yes	—
88	C₈×C₁₃₁	C₂×Q₁₆	1	4	yes	—
89	C₈×C₁₃₁	C₂²×Q₈	1	1	yes	characteristic
90	C₈×C₁₃₁	C₂²×Q₈	1	1	yes	characteristic
91	C₂²×C₁₃₁	—	2	2	no	—
92	C₂²×C₁₃₁	#500	1	2	yes	—
93	C₂²×C₁₃₁	C₂²×D₈	1	1	yes	characteristic
94	C₄×C₁₃₁	—	2	4	no	—
95	C₄×C₁₃₁	—	8	2	no	—
96	C₄×C₁₃₁	—	8	4	no	—
97	C₄×C₁₃₁	—	8	2	no	—
98	C₄×C₁₃₁	#500	1	2	yes	—
99	C₄×C₁₃₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
100	C₄×C₁₃₁	C₂²×Q₁₆	1	1	yes	characteristic
101	C₂×C₁₃₁	—	2	2	no	—
102	C₂×C₁₃₁	C₂×#500	1	1	yes	characteristic
103	C₂×C₁₃₁	#3416	1	2	yes	—
104	#33730	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
105	C₁₃₁	#33730	1	1	yes	characteristic, direct product factor, Sylow
106	#3164	C₂×C₁₃₁	1	2	yes	—
107	#2255	C₂×C₁₃₁	1	4	yes	complemented
108	#2254	C₂×C₁₃₁	1	2	yes	complemented
109	#2253	C₂×C₁₃₁	1	2	yes	complemented
110	#2159	C₂×C₁₃₁	1	2	yes	—
111	#2158	C₂×C₁₃₁	1	1	yes	characteristic
112	C₂×#367	C₂×C₁₃₁	1	1	yes	characteristic
113	#2108	C₂×C₁₃₁	1	1	yes	characteristic
114	#544	—	2	2	no	—
115	C₂×M₃₂	C₂²×C₁₃₁	1	1	yes	characteristic
116	C₂²×C₁₆	C₂²×C₁₃₁	1	1	yes	characteristic
117	#501	—	2	4	no	—
118	#500	C₂²×C₁₃₁	1	4	yes	—
119	#499	C₂²×C₁₃₁	1	4	yes	—
120	#428	—	2	2	no	—
121	#427	C₂²×C₁₃₁	1	2	yes	—
122	C₂×#107	C₂²×C₁₃₁	1	2	yes	—
123	#405	C₂²×C₁₃₁	1	1	yes	characteristic
124	#367	C₂²×C₁₃₁	1	2	yes	—
125	#367	C₂²×C₁₃₁	1	2	yes	—
126	#367	C₂²×C₁₃₁	1	1	yes	characteristic
127	#367	C₂²×C₁₃₁	1	1	yes	characteristic
128	#367	C₂²×C₁₃₁	1	4	yes	—
129	#366	C₂²×C₁₃₁	1	2	yes	—
130	#366	C₂²×C₁₃₁	1	4	yes	—
131	#346	C₂²×C₁₃₁	1	2	yes	—
132	C₄×C₁₆	C₂²×C₁₃₁	1	2	yes	—
133	C₂×M₁₆	D₄×C₁₃₁	1	1	yes	characteristic
134	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
135	C₄⋊Q₈	—	2	4	no	—
136	C₄⋊Q₈	—	4	2	no	—
137	#125	—	2	4	no	—
138	#125	—	4	2	no	—
139	C₂²⋊Q₈	—	4	4	no	—
140	#117	—	2	2	no	—
141	#117	D₄×C₁₃₁	1	1	yes	characteristic
142	C₂×C₄⋊C₄	—	2	2	no	—
143	M₃₂	Q₈×C₁₃₁	1	4	yes	complemented
144	C₂×C₁₆	Q₈×C₁₃₁	1	4	yes	complemented
145	C₂×C₁₆	C₂³×C₁₃₁	1	1	yes	characteristic
146	C₂×C₁₆	C₂³×C₁₃₁	1	2	yes	—
147	C₂×C₁₆	C₂³×C₁₃₁	1	1	yes	characteristic
148	#107	—	2	2	no	—
149	#107	—	2	4	no	—
150	#107	C₂³×C₁₃₁	1	2	yes	—
151	#107	C₂³×C₁₃₁	1	4	yes	—
152	#107	C₂³×C₁₃₁	1	2	yes	—
153	#106	—	2	2	no	—
154	#106	—	2	4	no	—
155	#97	D₄×C₁₃₁	1	2	yes	—
156	C₄×C₈	C₂³×C₁₃₁	1	2	yes	—
157	C₂×Q₈	—	4	4	no	—
158	C₂²×C₄	—	4	2	no	—
159	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
160	M₁₆	Q₁₆×C₁₃₁	1	4	yes	—
161	C₂×C₈	C₂×D₄×C₁₃₁	1	1	yes	characteristic
162	C₂×C₈	C₂×D₄×C₁₃₁	1	2	yes	—
163	C₂×C₈	C₂×Q₈×C₁₃₁	1	2	yes	—
164	C₂×C₈	C₂×Q₈×C₁₃₁	1	2	yes	—
165	C₂×C₈	C₂⁴×C₁₃₁	1	1	yes	characteristic
166	C₄⋊C₄	—	2	2	no	—
167	C₄⋊C₄	—	2	4	no	—
168	C₄⋊C₄	—	2	2	no	—
169	C₄⋊C₄	—	4	2	no	—
170	C₄⋊C₄	—	4	4	no	—
171	C₄⋊C₄	—	4	4	no	—
172	C₄⋊C₄	—	4	4	no	—
173	C₄⋊C₄	D₈×C₁₃₁	1	2	yes	—
174	C₂²⋊C₄	—	4	2	no	—
175	C₂²⋊C₄	D₈×C₁₃₁	1	2	yes	—
176	C₄²	—	4	2	no	—
177	C₄²	C₂×D₄×C₁₃₁	1	2	yes	—
178	C₁₆	C₂×Q₈×C₁₃₁	1	2	yes	—
179	C₁₆	C₂×Q₈×C₁₃₁	1	2	yes	—
180	C₁₆	C₂×Q₈×C₁₃₁	1	4	yes	—
181	C₂³	C₂×D₈×C₁₃₁	1	1	yes	characteristic
182	Q₈	—	16	4	no	—
183	C₂×C₄	—	4	2	no	—
184	C₂×C₄	—	4	2	no	—
185	C₂×C₄	—	4	4	no	—
186	C₂×C₄	—	16	2	no	—
187	C₂×C₄	C₄⋊Q₈×C₁₃₁	1	2	yes	—
188	C₂×C₄	C₂×D₈×C₁₃₁	1	4	yes	—
189	C₂×C₄	C₂×D₈×C₁₃₁	1	1	yes	characteristic
190	C₂×C₄	C₂×Q₁₆×C₁₃₁	1	2	yes	—
191	C₂×C₄	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
192	C₈	C₄⋊Q₈×C₁₃₁	1	2	yes	—
193	C₈	C₂×Q₁₆×C₁₃₁	1	4	yes	—
194	C₈	C₂²×Q₈×C₁₃₁	1	1	yes	characteristic
195	C₈	C₂²×Q₈×C₁₃₁	1	1	yes	characteristic
196	C₂²	—	2	2	no	—
197	C₂²	#500×C₁₃₁	1	2	yes	—
198	C₂²	C₂²×D₈×C₁₃₁	1	1	yes	characteristic
199	C₄	—	2	4	no	—
200	C₄	—	8	4	no	—
201	C₄	—	8	2	no	—
202	C₄	—	8	2	no	—
203	C₄	#500×C₁₃₁	1	2	yes	—
204	C₄	C₂×C₄⋊Q₈×C₁₃₁	1	1	yes	characteristic
205	C₄	C₂²×Q₁₆×C₁₃₁	1	1	yes	characteristic
206	C₂	—	2	2	no	—
207	C₂	C₂×#500×C₁₃₁	1	1	yes	characteristic
208	C₂	#3416×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 ◁ #117×C₁₃₁ ◁ #405×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. #2255×C₁₃₁	C₂	semidirect
3. #2254×C₁₃₁	C₂	semidirect
4. #2253×C₁₃₁	C₂	semidirect
38. M₃₂×C₁₃₁	Q₈	semidirect
39. C₂×C₁₆×C₁₃₁	Q₈	semidirect
104. #33730	C₁₃₁	direct (N × H)
105. C₁₃₁	#33730	direct (N × H)
107. #2255	C₂×C₁₃₁	semidirect
108. #2254	C₂×C₁₃₁	semidirect
109. #2253	C₂×C₁₃₁	semidirect
143. M₃₂	Q₈×C₁₃₁	semidirect
144. C₂×C₁₆	Q₈×C₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
#3164×C₁₃₁	2
#2255×C₁₃₁	2
#2254×C₁₃₁	2
#2253×C₁₃₁	2
#2159×C₁₃₁	2
#2158×C₁₃₁	2
C₂×#367×C₁₃₁	2
#2108×C₁₃₁	2
#33730	131

Subgroups — C131×#33730 #45719614

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#33730 #45719614