Subgroups — C₁₃₁×#8278 #45694162

Order 33536.

394 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#1767×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊C₈×C₁₃₁	C₂	1	1	yes	maximal, characteristic
3	#1744×C₁₃₁	C₂	1	1	yes	maximal, characteristic
4	C₂×#335×C₁₃₁	C₂	1	2	yes	maximal
5	#1277×C₁₃₁	C₂	1	2	yes	maximal
6	C₂×C₄⋊C₈×C₁₃₁	C₂²	1	2	yes	—
7	C₂×C₄⋊C₈×C₁₃₁	C₂²	1	2	yes	—
8	C₂×#97×C₁₃₁	—	2	1	no	—
9	C₂×C₄×C₈×C₁₃₁	—	2	1	no	—
10	C₂×C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
12	C₄³×C₁₃₁	C₂²	1	1	yes	characteristic
13	#336×C₁₃₁	—	2	1	no	—
14	#336×C₁₃₁	—	2	1	no	—
15	#336×C₁₃₁	C₄	1	4	yes	—
16	#335×C₁₃₁	C₄	1	8	yes	complemented
17	C₂²×C₈×C₁₃₁	—	2	2	no	—
18	C₂²×C₈×C₁₃₁	—	2	2	no	—
19	C₂²×C₈×C₁₃₁	C₂×C₄	1	2	yes	—
20	C₂²×C₈×C₁₃₁	C₂×C₄	1	2	yes	—
21	C₂×C₄²×C₁₃₁	C₂×C₄	1	2	yes	—
22	C₂×C₄²×C₁₃₁	D₄	1	1	yes	characteristic
23	C₂×C₄²×C₁₃₁	D₄	1	1	yes	characteristic
24	C₂×C₄²×C₁₃₁	Q₈	1	2	yes	—
25	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
26	C₄⋊C₈×C₁₃₁	—	2	2	no	—
27	C₄⋊C₈×C₁₃₁	—	2	2	no	—
28	C₄⋊C₈×C₁₃₁	C₂×C₄	1	4	yes	—
29	C₄⋊C₈×C₁₃₁	C₂×C₄	1	4	yes	—
30	#97×C₁₃₁	—	4	2	no	—
31	C₄×C₈×C₁₃₁	—	2	2	no	—
32	C₄×C₈×C₁₃₁	—	4	2	no	—
33	C₄×C₈×C₁₃₁	C₂×C₄	1	4	yes	—
34	C₂²×C₄×C₁₃₁	C₄⋊C₄	1	2	yes	—
35	C₂²×C₄×C₁₃₁	C₄⋊C₄	1	2	yes	—
36	C₂²×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
38	C₂²×C₄×C₁₃₁	C₂×Q₈	1	1	yes	characteristic
39	C₂×C₈×C₁₃₁	—	2	4	no	—
40	C₂×C₈×C₁₃₁	—	2	2	no	—
41	C₂×C₈×C₁₃₁	—	2	4	no	—
42	C₂×C₈×C₁₃₁	—	2	4	no	—
43	C₂×C₈×C₁₃₁	—	2	4	no	—
44	C₂×C₈×C₁₃₁	—	4	2	no	—
45	C₂×C₈×C₁₃₁	—	4	2	no	—
46	C₂×C₈×C₁₃₁	—	4	2	no	—
47	C₂×C₈×C₁₃₁	—	4	2	no	—
48	C₂×C₈×C₁₃₁	C₄²	1	4	yes	—
49	C₄²×C₁₃₁	—	2	4	no	—
50	C₄²×C₁₃₁	—	2	2	no	—
51	C₄²×C₁₃₁	—	4	2	no	—
52	C₄²×C₁₃₁	D₈	1	2	yes	—
53	C₄²×C₁₃₁	Q₁₆	1	2	yes	—
54	C₄²×C₁₃₁	C₂²×C₄	1	2	yes	—
55	C₄²×C₁₃₁	#41	1	2	yes	—
56	C₂³×C₁₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₄×C₁₃₁	—	2	2	no	—
58	C₂×C₄×C₁₃₁	—	2	2	no	—
59	C₂×C₄×C₁₃₁	—	2	2	no	—
60	C₂×C₄×C₁₃₁	—	2	2	no	—
61	C₂×C₄×C₁₃₁	—	2	2	no	—
62	C₂×C₄×C₁₃₁	—	2	2	no	—
63	C₂×C₄×C₁₃₁	#107	1	4	yes	—
64	C₂×C₄×C₁₃₁	C₂×C₄²	1	1	yes	characteristic
65	C₂×C₄×C₁₃₁	#117	1	1	yes	characteristic
66	C₂×C₄×C₁₃₁	C₄×D₄	1	2	yes	—
67	C₂×C₄×C₁₃₁	C₄×Q₈	1	2	yes	—
68	C₂×C₄×C₁₃₁	#131	1	4	yes	—
69	C₂×C₄×C₁₃₁	C₂×D₈	1	1	yes	characteristic
70	C₂×C₄×C₁₃₁	C₂×Q₁₆	1	1	yes	characteristic
71	C₂×C₄×C₁₃₁	#135	1	2	yes	—
72	C₈×C₁₃₁	—	2	4	no	—
73	C₈×C₁₃₁	—	4	2	no	—
74	C₈×C₁₃₁	—	4	4	no	—
75	C₈×C₁₃₁	—	4	4	no	—
76	C₂²×C₁₃₁	C₄×C₄⋊C₄	1	1	yes	characteristic
77	C₂²×C₁₃₁	#424	1	2	yes	—
78	C₂²×C₁₃₁	C₂×#107	1	1	yes	characteristic
79	C₂²×C₁₃₁	#427	1	1	yes	characteristic
80	C₂²×C₁₃₁	#430	1	2	yes	—
81	C₄×C₁₃₁	—	2	4	no	—
82	C₄×C₁₃₁	—	2	2	no	—
83	C₄×C₁₃₁	—	4	2	no	—
84	C₄×C₁₃₁	#405	1	2	yes	—
85	C₄×C₁₃₁	#433	1	2	yes	—
86	C₄×C₁₃₁	C₄×D₈	1	2	yes	—
87	C₄×C₁₃₁	C₄×Q₁₆	1	2	yes	—
88	C₂×C₁₃₁	#1571	1	2	yes	—
89	C₂×C₁₃₁	#1572	1	2	yes	—
90	C₂×C₁₃₁	#1768	1	1	yes	characteristic
91	C₂×C₁₃₁	C₄×#107	1	1	yes	characteristic
92	C₂×C₁₃₁	#1778	1	1	yes	characteristic
93	#8278	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
94	C₁₃₁	#8278	1	1	yes	characteristic, direct product factor, Sylow
95	#1767	C₂×C₁₃₁	1	1	yes	characteristic
96	C₄×C₄⋊C₈	C₂×C₁₃₁	1	1	yes	characteristic
97	#1744	C₂×C₁₃₁	1	1	yes	characteristic
98	C₂×#335	C₂×C₁₃₁	1	2	yes	—
99	#1277	C₂×C₁₃₁	1	2	yes	—
100	C₂×C₄⋊C₈	C₂²×C₁₃₁	1	2	yes	—
101	C₂×C₄⋊C₈	C₂²×C₁₃₁	1	2	yes	—
102	C₂×#97	—	2	1	no	—
103	C₂×C₄×C₈	—	2	1	no	—
104	C₂×C₄×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
105	C₂×C₄×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
106	C₄³	C₂²×C₁₃₁	1	1	yes	characteristic
107	#336	—	2	1	no	—
108	#336	—	2	1	no	—
109	#336	C₄×C₁₃₁	1	4	yes	—
110	#335	C₄×C₁₃₁	1	8	yes	complemented
111	C₂²×C₈	—	2	2	no	—
112	C₂²×C₈	—	2	2	no	—
113	C₂²×C₈	C₂×C₄×C₁₃₁	1	2	yes	—
114	C₂²×C₈	C₂×C₄×C₁₃₁	1	2	yes	—
115	C₂×C₄²	C₂×C₄×C₁₃₁	1	2	yes	—
116	C₂×C₄²	D₄×C₁₃₁	1	1	yes	characteristic
117	C₂×C₄²	D₄×C₁₃₁	1	1	yes	characteristic
118	C₂×C₄²	Q₈×C₁₃₁	1	2	yes	—
119	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
120	C₄⋊C₈	—	2	2	no	—
121	C₄⋊C₈	—	2	2	no	—
122	C₄⋊C₈	C₂×C₄×C₁₃₁	1	4	yes	—
123	C₄⋊C₈	C₂×C₄×C₁₃₁	1	4	yes	—
124	#97	—	4	2	no	—
125	C₄×C₈	—	2	2	no	—
126	C₄×C₈	—	4	2	no	—
127	C₄×C₈	C₂×C₄×C₁₃₁	1	4	yes	—
128	C₂²×C₄	C₄⋊C₄×C₁₃₁	1	2	yes	—
129	C₂²×C₄	C₄⋊C₄×C₁₃₁	1	2	yes	—
130	C₂²×C₄	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
131	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
132	C₂²×C₄	C₂×Q₈×C₁₃₁	1	1	yes	characteristic
133	C₂×C₈	—	2	4	no	—
134	C₂×C₈	—	2	2	no	—
135	C₂×C₈	—	2	4	no	—
136	C₂×C₈	—	2	4	no	—
137	C₂×C₈	—	2	4	no	—
138	C₂×C₈	—	4	2	no	—
139	C₂×C₈	—	4	2	no	—
140	C₂×C₈	—	4	2	no	—
141	C₂×C₈	—	4	2	no	—
142	C₂×C₈	C₄²×C₁₃₁	1	4	yes	—
143	C₄²	—	2	4	no	—
144	C₄²	—	2	2	no	—
145	C₄²	—	4	2	no	—
146	C₄²	D₈×C₁₃₁	1	2	yes	—
147	C₄²	Q₁₆×C₁₃₁	1	2	yes	—
148	C₄²	C₂²×C₄×C₁₃₁	1	2	yes	—
149	C₄²	#41×C₁₃₁	1	2	yes	—
150	C₂³	C₂×C₄⋊C₄×C₁₃₁	1	1	yes	characteristic
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	#107×C₁₃₁	1	4	yes	—
158	C₂×C₄	C₂×C₄²×C₁₃₁	1	1	yes	characteristic
159	C₂×C₄	#117×C₁₃₁	1	1	yes	characteristic
160	C₂×C₄	C₄×D₄×C₁₃₁	1	2	yes	—
161	C₂×C₄	C₄×Q₈×C₁₃₁	1	2	yes	—
162	C₂×C₄	#131×C₁₃₁	1	4	yes	—
163	C₂×C₄	C₂×D₈×C₁₃₁	1	1	yes	characteristic
164	C₂×C₄	C₂×Q₁₆×C₁₃₁	1	1	yes	characteristic
165	C₂×C₄	#135×C₁₃₁	1	2	yes	—
166	C₈	—	2	4	no	—
167	C₈	—	4	2	no	—
168	C₈	—	4	4	no	—
169	C₈	—	4	4	no	—
170	C₂²	C₄×C₄⋊C₄×C₁₃₁	1	1	yes	characteristic
171	C₂²	#424×C₁₃₁	1	2	yes	—
172	C₂²	C₂×#107×C₁₃₁	1	1	yes	characteristic
173	C₂²	#427×C₁₃₁	1	1	yes	characteristic
174	C₂²	#430×C₁₃₁	1	2	yes	—
175	C₄	—	2	2	no	—
176	C₄	—	2	4	no	—
177	C₄	—	4	2	no	—
178	C₄	#405×C₁₃₁	1	2	yes	—
179	C₄	#433×C₁₃₁	1	2	yes	—
180	C₄	C₄×D₈×C₁₃₁	1	2	yes	—
181	C₄	C₄×Q₁₆×C₁₃₁	1	2	yes	—
182	C₂	#1571×C₁₃₁	1	2	yes	—
183	C₂	#1572×C₁₃₁	1	2	yes	—
184	C₂	#1768×C₁₃₁	1	1	yes	characteristic
185	C₂	C₄×#107×C₁₃₁	1	1	yes	characteristic
186	C₂	#1778×C₁₃₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₁₃₁	C₂×D₄
commutator	C₂×C₄	C₂×C₄²×C₁₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₁₃₁
socle, prime radical	C₂³×C₁₃₁	C₂×C₄⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₃₁ ◁ C₄³×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
16. #335×C₁₃₁	C₄	semidirect
93. #8278	C₁₃₁	direct (N × H)
94. C₁₃₁	#8278	direct (N × H)
110. #335	C₄×C₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
#1767×C₁₃₁	2
C₄×C₄⋊C₈×C₁₃₁	2
#1744×C₁₃₁	2
C₂×#335×C₁₃₁	2
#1277×C₁₃₁	2
#8278	131

Subgroups — C131×#8278 #45694162

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#8278 #45694162