Subgroups — C₁₃₁×#34641 #45720525

Order 33536.

3694 conjugacy classes of subgroups.

The subgroup lattice has 182 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#532×C₁₃₁	C₂	1	6	yes	maximal, complemented
2	#2658×C₁₃₁	C₂	1	4	yes	maximal, complemented
3	#2657×C₁₃₁	C₂	1	4	yes	maximal, complemented
4	#2656×C₁₃₁	C₂	1	8	yes	maximal, complemented
5	#2397×C₁₃₁	C₂	1	8	yes	maximal, complemented
6	C₂×#376×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×#41×C₁₃₁	C₂²	1	4	yes	—
8	#532×C₁₃₁	—	2	48	no	—
9	C₂×C₄⋊Q₈×C₁₃₁	C₂²	1	6	yes	—
10	C₂×#127×C₁₃₁	C₂²	1	6	yes	—
11	C₂×#124×C₁₃₁	C₂²	1	12	yes	—
12	C₂×#121×C₁₃₁	C₂²	1	24	yes	—
13	C₂×C₄×D₄×C₁₃₁	C₂²	1	24	yes	—
14	C₂²×C₄²×C₁₃₁	C₂²	1	3	yes	—
15	#399×C₁₃₁	C₂²	1	32	yes	—
16	#392×C₁₃₁	C₂²	1	32	yes	—
17	#376×C₁₃₁	C₂²	1	4	yes	—
18	#376×C₁₃₁	C₂²	1	4	yes	—
19	C₂×#95×C₁₃₁	C₂²	1	4	yes	—
20	C₂×#41×C₁₃₁	—	2	48	no	—
21	C₂²×Q₈×C₁₃₁	C₂³	1	4	yes	—
22	C₂²×D₄×C₁₃₁	C₂³	1	8	yes	—
23	C₂²×D₄×C₁₃₁	C₂³	1	4	yes	—
24	C₂²×D₄×C₁₃₁	C₂³	1	24	yes	—
25	C₂³×C₄×C₁₃₁	C₂³	1	8	yes	—
26	C₂³×C₄×C₁₃₁	C₂³	1	6	yes	—
27	C₂³×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
28	C₄⋊Q₈×C₁₃₁	—	2	24	no	—
29	#127×C₁₃₁	—	2	24	no	—
30	#124×C₁₃₁	—	2	48	no	—
31	#121×C₁₃₁	—	2	96	no	—
32	C₄×D₄×C₁₃₁	—	2	96	no	—
33	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	24	yes	—
34	C₂×C₂²⋊C₄×C₁₃₁	C₂³	1	48	yes	—
35	C₂×C₄²×C₁₃₁	D₄	1	24	yes	complemented
36	C₂×C₄²×C₁₃₁	C₂³	1	6	yes	—
37	C₂×C₄²×C₁₃₁	C₂³	1	6	yes	—
38	#95×C₁₃₁	C₂³	1	16	yes	—
39	C₂⁴×C₁₃₁	C₂⁴	1	8	yes	—
40	C₂⁴×C₁₃₁	C₂⁴	1	1	yes	characteristic
41	#41×C₁₃₁	—	4	64	no	—
42	C₂×Q₈×C₁₃₁	—	2	24	no	—
43	C₂×D₄×C₁₃₁	—	2	48	no	—
44	C₂×D₄×C₁₃₁	—	2	48	no	—
45	C₂×D₄×C₁₃₁	—	2	24	no	—
46	C₂×D₄×C₁₃₁	—	4	48	no	—
47	C₂²×C₄×C₁₃₁	—	2	48	no	—
48	C₂²×C₄×C₁₃₁	—	4	12	no	—
49	C₂²×C₄×C₁₃₁	C₂×D₄	1	24	yes	—
50	C₂²×C₄×C₁₃₁	C₂×D₄	1	12	yes	—
51	C₂²×C₄×C₁₃₁	C₂⁴	1	8	yes	—
52	C₂²×C₄×C₁₃₁	C₂⁴	1	12	yes	—
53	C₂²×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
54	C₂²×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
55	C₄⋊C₄×C₁₃₁	—	2	48	no	—
56	C₂²⋊C₄×C₁₃₁	—	2	96	no	—
57	C₄²×C₁₃₁	C₂×D₄	1	24	yes	—
58	C₄²×C₁₃₁	C₂×D₄	1	24	yes	—
59	C₂³×C₁₃₁	—	2	4	no	—
60	C₂³×C₁₃₁	—	2	24	no	—
61	C₂³×C₁₃₁	—	2	32	no	—
62	C₂³×C₁₃₁	C₂²×D₄	1	6	yes	—
63	C₂³×C₁₃₁	C₂⁵	1	1	yes	characteristic
64	Q₈×C₁₃₁	—	4	16	no	—
65	D₄×C₁₃₁	—	4	96	no	—
66	D₄×C₁₃₁	—	4	32	no	—
67	D₄×C₁₃₁	—	4	16	no	—
68	C₂×C₄×C₁₃₁	—	2	24	no	—
69	C₂×C₄×C₁₃₁	—	2	24	no	—
70	C₂×C₄×C₁₃₁	—	4	24	no	—
71	C₂×C₄×C₁₃₁	—	4	32	no	—
72	C₂×C₄×C₁₃₁	#127	1	16	yes	—
73	C₂×C₄×C₁₃₁	C₂²×D₄	1	24	yes	—
74	C₂×C₄×C₁₃₁	C₂²×D₄	1	6	yes	—
75	C₂×C₄×C₁₃₁	C₂²×D₄	1	6	yes	—
76	C₂²×C₁₃₁	—	2	12	no	—
77	C₂²×C₁₃₁	—	4	8	no	—
78	C₂²×C₁₃₁	—	4	48	no	—
79	C₂²×C₁₃₁	C₂×#127	1	4	yes	—
80	C₂²×C₁₃₁	C₂³×D₄	1	3	yes	—
81	C₂²×C₁₃₁	#585	1	4	yes	—
82	C₄×C₁₃₁	—	2	24	no	—
83	C₄×C₁₃₁	—	4	8	no	—
84	C₄×C₁₃₁	C₂×#127	1	4	yes	—
85	C₄×C₁₃₁	C₂×#127	1	4	yes	—
86	C₂×C₁₃₁	—	2	4	no	—
87	C₂×C₁₃₁	—	8	8	no	—
88	C₂×C₁₃₁	C₂²×#127	1	1	yes	characteristic
89	C₂×C₁₃₁	#3449	1	6	yes	—
90	#34641	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
91	C₁₃₁	#34641	1	1	yes	characteristic, direct product factor, Sylow
92	C₂×#532	C₂×C₁₃₁	1	6	yes	complemented
93	#2658	C₂×C₁₃₁	1	4	yes	complemented
94	#2657	C₂×C₁₃₁	1	4	yes	complemented
95	#2656	C₂×C₁₃₁	1	8	yes	complemented
96	#2397	C₂×C₁₃₁	1	8	yes	complemented
97	C₂×#376	C₂×C₁₃₁	1	1	yes	characteristic, complemented
98	C₂²×#41	C₂²×C₁₃₁	1	4	yes	—
99	#532	—	2	48	no	—
100	C₂×C₄⋊Q₈	C₂²×C₁₃₁	1	6	yes	—
101	C₂×#127	C₂²×C₁₃₁	1	6	yes	—
102	C₂×#124	C₂²×C₁₃₁	1	12	yes	—
103	C₂×#121	C₂²×C₁₃₁	1	24	yes	—
104	C₂×C₄×D₄	C₂²×C₁₃₁	1	24	yes	—
105	C₂²×C₄²	C₂²×C₁₃₁	1	3	yes	—
106	#399	C₂²×C₁₃₁	1	32	yes	—
107	#392	C₂²×C₁₃₁	1	32	yes	—
108	#376	C₂²×C₁₃₁	1	4	yes	—
109	#376	C₂²×C₁₃₁	1	4	yes	—
110	C₂×#95	C₂²×C₁₃₁	1	4	yes	—
111	C₂×#41	—	2	48	no	—
112	C₂²×Q₈	C₂³×C₁₃₁	1	4	yes	—
113	C₂²×D₄	C₂³×C₁₃₁	1	8	yes	—
114	C₂²×D₄	C₂³×C₁₃₁	1	4	yes	—
115	C₂²×D₄	C₂³×C₁₃₁	1	24	yes	—
116	C₂³×C₄	C₂³×C₁₃₁	1	8	yes	—
117	C₂³×C₄	C₂³×C₁₃₁	1	6	yes	—
118	C₂³×C₄	C₂³×C₁₃₁	1	1	yes	characteristic
119	C₄⋊Q₈	—	2	24	no	—
120	#127	—	2	24	no	—
121	#124	—	2	48	no	—
122	#121	—	2	96	no	—
123	C₄×D₄	—	2	96	no	—
124	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	24	yes	—
125	C₂×C₂²⋊C₄	C₂³×C₁₃₁	1	48	yes	—
126	C₂×C₄²	D₄×C₁₃₁	1	24	yes	complemented
127	C₂×C₄²	C₂³×C₁₃₁	1	6	yes	—
128	C₂×C₄²	C₂³×C₁₃₁	1	6	yes	—
129	#95	C₂³×C₁₃₁	1	16	yes	—
130	C₂⁴	C₂⁴×C₁₃₁	1	8	yes	—
131	C₂⁴	C₂⁴×C₁₃₁	1	1	yes	characteristic
132	#41	—	4	64	no	—
133	C₂×Q₈	—	2	24	no	—
134	C₂×D₄	—	2	48	no	—
135	C₂×D₄	—	2	48	no	—
136	C₂×D₄	—	2	24	no	—
137	C₂×D₄	—	4	48	no	—
138	C₂²×C₄	—	2	48	no	—
139	C₂²×C₄	—	4	12	no	—
140	C₂²×C₄	C₂×D₄×C₁₃₁	1	24	yes	—
141	C₂²×C₄	C₂×D₄×C₁₃₁	1	12	yes	—
142	C₂²×C₄	C₂⁴×C₁₃₁	1	8	yes	—
143	C₂²×C₄	C₂⁴×C₁₃₁	1	12	yes	—
144	C₂²×C₄	C₂⁴×C₁₃₁	1	1	yes	characteristic
145	C₂²×C₄	C₂⁴×C₁₃₁	1	1	yes	characteristic
146	C₄⋊C₄	—	2	48	no	—
147	C₂²⋊C₄	—	2	96	no	—
148	C₄²	C₂×D₄×C₁₃₁	1	24	yes	—
149	C₄²	C₂×D₄×C₁₃₁	1	24	yes	—
150	C₂³	—	2	4	no	—
151	C₂³	—	2	24	no	—
152	C₂³	—	2	32	no	—
153	C₂³	C₂²×D₄×C₁₃₁	1	6	yes	—
154	C₂³	C₂⁵×C₁₃₁	1	1	yes	characteristic
155	Q₈	—	4	16	no	—
156	D₄	—	4	96	no	—
157	D₄	—	4	32	no	—
158	D₄	—	4	16	no	—
159	C₂×C₄	—	2	24	no	—
160	C₂×C₄	—	2	24	no	—
161	C₂×C₄	—	4	24	no	—
162	C₂×C₄	—	4	32	no	—
163	C₂×C₄	#127×C₁₃₁	1	16	yes	—
164	C₂×C₄	C₂²×D₄×C₁₃₁	1	24	yes	—
165	C₂×C₄	C₂²×D₄×C₁₃₁	1	6	yes	—
166	C₂×C₄	C₂²×D₄×C₁₃₁	1	6	yes	—
167	C₂²	—	2	12	no	—
168	C₂²	—	4	8	no	—
169	C₂²	—	4	48	no	—
170	C₂²	C₂×#127×C₁₃₁	1	4	yes	—
171	C₂²	C₂³×D₄×C₁₃₁	1	3	yes	—
172	C₂²	#585×C₁₃₁	1	4	yes	—
173	C₄	—	2	24	no	—
174	C₄	—	4	8	no	—
175	C₄	C₂×#127×C₁₃₁	1	4	yes	—
176	C₄	C₂×#127×C₁₃₁	1	4	yes	—
177	C₂	—	2	4	no	—
178	C₂	—	8	8	no	—
179	C₂	C₂²×#127×C₁₃₁	1	1	yes	characteristic
180	C₂	#3449×C₁₃₁	1	6	yes	—

	subgroup	G/H
center	C₂²×C₄×C₁₃₁	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₁₃₁
socle	C₂³×C₁₃₁	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#532×C₁₃₁	C₂	semidirect
2. #2658×C₁₃₁	C₂	semidirect
3. #2657×C₁₃₁	C₂	semidirect
4. #2656×C₁₃₁	C₂	semidirect
5. #2397×C₁₃₁	C₂	semidirect
6. C₂×#376×C₁₃₁	C₂	semidirect
35. C₂×C₄²×C₁₃₁	D₄	semidirect
90. #34641	C₁₃₁	direct (N × H)
91. C₁₃₁	#34641	direct (N × H)
92. C₂×#532	C₂×C₁₃₁	semidirect
93. #2658	C₂×C₁₃₁	semidirect
94. #2657	C₂×C₁₃₁	semidirect
95. #2656	C₂×C₁₃₁	semidirect
96. #2397	C₂×C₁₃₁	semidirect
97. C₂×#376	C₂×C₁₃₁	semidirect
126. C₂×C₄²	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×#532×C₁₃₁	2
#2658×C₁₃₁	2
#2657×C₁₃₁	2
#2656×C₁₃₁	2
#2397×C₁₃₁	2
C₂×#376×C₁₃₁	2
#34641	131

Subgroups — C131×#34641 #45720525

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#34641 #45720525