Subgroups — C₁₃₁×#36592 #45722476

Order 33536.

4122 conjugacy classes of subgroups.

The subgroup lattice has 178 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#550×C₁₃₁	C₂	1	6	yes	maximal, complemented
2	#2867×C₁₃₁	C₂	1	3	yes	maximal, complemented
3	#2541×C₁₃₁	C₂	1	18	yes	maximal, complemented
4	C₄×#127×C₁₃₁	C₂	1	3	yes	maximal, complemented
5	C₄²×Q₈×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#41×C₁₃₁	C₂²	1	4	yes	—
7	#550×C₁₃₁	—	2	48	no	—
8	C₂×#127×C₁₃₁	C₂²	1	18	yes	—
9	C₂×#127×C₁₃₁	C₂²	1	3	yes	—
10	C₂×#127×C₁₃₁	C₂²	1	18	yes	—
11	C₂×#121×C₁₃₁	C₂²	1	36	yes	—
12	C₂×C₄×Q₈×C₁₃₁	C₂²	1	3	yes	—
13	C₂×C₄×D₄×C₁₃₁	C₂²	1	18	yes	—
14	#392×C₁₃₁	C₂²	1	24	yes	—
15	#390×C₁₃₁	C₂²	1	18	yes	—
16	#384×C₁₃₁	C₂²	1	1	yes	characteristic
17	C₄×C₄⋊C₄×C₁₃₁	C₂²	1	9	yes	—
18	C₄³×C₁₃₁	C₂²	1	3	yes	—
19	C₂×#41×C₁₃₁	—	2	48	no	—
20	C₂²×Q₈×C₁₃₁	C₂³	1	1	yes	characteristic
21	C₂²×D₄×C₁₃₁	C₂³	1	18	yes	—
22	C₂²×D₄×C₁₃₁	C₂³	1	12	yes	—
23	C₂²×D₄×C₁₃₁	C₂³	1	36	yes	—
24	C₂³×C₄×C₁₃₁	C₂³	1	12	yes	—
25	#127×C₁₃₁	—	2	72	no	—
26	#127×C₁₃₁	—	2	12	no	—
27	#127×C₁₃₁	—	4	36	no	—
28	#121×C₁₃₁	—	2	144	no	—
29	C₄×Q₈×C₁₃₁	D₄	1	24	yes	complemented
30	C₄×D₄×C₁₃₁	—	2	72	no	—
31	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	6	yes	—
32	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	9	yes	—
33	C₂×C₂²⋊C₄×C₁₃₁	C₂³	1	36	yes	—
34	C₂×C₄²×C₁₃₁	C₂³	1	9	yes	—
35	C₂×C₄²×C₁₃₁	C₂³	1	9	yes	—
36	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
37	#95×C₁₃₁	C₂³	1	6	yes	—
38	C₂⁴×C₁₃₁	C₂⁴	1	12	yes	—
39	#41×C₁₃₁	—	4	64	no	—
40	C₂×Q₈×C₁₃₁	C₂×D₄	1	12	yes	—
41	C₂×D₄×C₁₃₁	—	2	36	no	—
42	C₂×D₄×C₁₃₁	—	2	72	no	—
43	C₂×D₄×C₁₃₁	—	2	72	no	—
44	C₂×D₄×C₁₃₁	—	2	36	no	—
45	C₂×D₄×C₁₃₁	—	2	36	no	—
46	C₂×D₄×C₁₃₁	—	4	72	no	—
47	C₂²×C₄×C₁₃₁	—	2	72	no	—
48	C₂²×C₄×C₁₃₁	C₂⁴	1	4	yes	—
49	C₂²×C₄×C₁₃₁	C₂⁴	1	9	yes	—
50	C₂²×C₄×C₁₃₁	C₂⁴	1	3	yes	—
51	C₂²×C₄×C₁₃₁	C₂⁴	1	3	yes	—
52	C₄⋊C₄×C₁₃₁	—	2	12	no	—
53	C₄⋊C₄×C₁₃₁	C₂×D₄	1	36	yes	—
54	C₂²⋊C₄×C₁₃₁	—	2	72	no	—
55	C₄²×C₁₃₁	—	2	18	no	—
56	C₄²×C₁₃₁	C₂×D₄	1	36	yes	—
57	C₄²×C₁₃₁	#41	1	4	yes	complemented
58	C₂³×C₁₃₁	—	2	36	no	—
59	C₂³×C₁₃₁	—	2	12	no	—
60	C₂³×C₁₃₁	—	2	36	no	—
61	C₂³×C₁₃₁	C₂⁵	1	1	yes	characteristic
62	Q₈×C₁₃₁	#127	1	16	yes	complemented
63	D₄×C₁₃₁	—	4	48	no	—
64	D₄×C₁₃₁	—	4	72	no	—
65	D₄×C₁₃₁	—	8	72	no	—
66	C₂×C₄×C₁₃₁	—	2	18	no	—
67	C₂×C₄×C₁₃₁	—	2	12	no	—
68	C₂×C₄×C₁₃₁	—	4	48	no	—
69	C₂×C₄×C₁₃₁	C₂²×D₄	1	18	yes	—
70	C₂×C₄×C₁₃₁	C₂²×D₄	1	18	yes	—
71	C₂×C₄×C₁₃₁	C₂²×D₄	1	6	yes	—
72	C₂×C₄×C₁₃₁	C₂×#41	1	6	yes	—
73	C₂×C₄×C₁₃₁	H(5,𝔽₂)	1	6	yes	—
74	C₂²×C₁₃₁	—	4	12	no	—
75	C₂²×C₁₃₁	—	4	36	no	—
76	C₂²×C₁₃₁	—	4	36	no	—
77	C₂²×C₁₃₁	C₂³×D₄	1	3	yes	—
78	C₂²×C₁₃₁	C₂²×#41	1	1	yes	characteristic
79	C₂²×C₁₃₁	C₂×H(5,𝔽₂)	1	3	yes	—
80	C₄×C₁₃₁	—	2	18	no	—
81	C₄×C₁₃₁	—	4	4	no	—
82	C₄×C₁₃₁	C₂×#127	1	12	yes	—
83	C₄×C₁₃₁	#550	1	12	yes	complemented
84	C₂×C₁₃₁	—	8	12	no	—
85	C₂×C₁₃₁	C₂²×#127	1	1	yes	characteristic
86	C₂×C₁₃₁	C₂×#550	1	3	yes	—
87	C₂×C₁₃₁	#3498	1	3	yes	—
88	#36592	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
89	C₁₃₁	#36592	1	1	yes	characteristic, direct product factor, Sylow
90	C₂×#550	C₂×C₁₃₁	1	6	yes	complemented
91	#2867	C₂×C₁₃₁	1	3	yes	complemented
92	#2541	C₂×C₁₃₁	1	18	yes	complemented
93	C₄×#127	C₂×C₁₃₁	1	3	yes	complemented
94	C₄²×Q₈	C₂×C₁₃₁	1	1	yes	characteristic, complemented
95	C₂²×#41	C₂²×C₁₃₁	1	4	yes	—
96	#550	—	2	48	no	—
97	C₂×#127	C₂²×C₁₃₁	1	3	yes	—
98	C₂×#127	C₂²×C₁₃₁	1	18	yes	—
99	C₂×#127	C₂²×C₁₃₁	1	18	yes	—
100	C₂×#121	C₂²×C₁₃₁	1	36	yes	—
101	C₂×C₄×Q₈	C₂²×C₁₃₁	1	3	yes	—
102	C₂×C₄×D₄	C₂²×C₁₃₁	1	18	yes	—
103	#392	C₂²×C₁₃₁	1	24	yes	—
104	#390	C₂²×C₁₃₁	1	18	yes	—
105	#384	C₂²×C₁₃₁	1	1	yes	characteristic
106	C₄×C₄⋊C₄	C₂²×C₁₃₁	1	9	yes	—
107	C₄³	C₂²×C₁₃₁	1	3	yes	—
108	C₂×#41	—	2	48	no	—
109	C₂²×Q₈	C₂³×C₁₃₁	1	1	yes	characteristic
110	C₂²×D₄	C₂³×C₁₃₁	1	18	yes	—
111	C₂²×D₄	C₂³×C₁₃₁	1	12	yes	—
112	C₂²×D₄	C₂³×C₁₃₁	1	36	yes	—
113	C₂³×C₄	C₂³×C₁₃₁	1	12	yes	—
114	#127	—	2	72	no	—
115	#127	—	2	12	no	—
116	#127	—	4	36	no	—
117	#121	—	2	144	no	—
118	C₄×Q₈	D₄×C₁₃₁	1	24	yes	complemented
119	C₄×D₄	—	2	72	no	—
120	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	6	yes	—
121	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	9	yes	—
122	C₂×C₂²⋊C₄	C₂³×C₁₃₁	1	36	yes	—
123	C₂×C₄²	C₂³×C₁₃₁	1	9	yes	—
124	C₂×C₄²	C₂³×C₁₃₁	1	9	yes	—
125	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
126	#95	C₂³×C₁₃₁	1	6	yes	—
127	C₂⁴	C₂⁴×C₁₃₁	1	12	yes	—
128	#41	—	4	64	no	—
129	C₂×Q₈	C₂×D₄×C₁₃₁	1	12	yes	—
130	C₂×D₄	—	2	36	no	—
131	C₂×D₄	—	2	72	no	—
132	C₂×D₄	—	2	72	no	—
133	C₂×D₄	—	2	36	no	—
134	C₂×D₄	—	2	36	no	—
135	C₂×D₄	—	4	72	no	—
136	C₂²×C₄	—	2	72	no	—
137	C₂²×C₄	C₂⁴×C₁₃₁	1	4	yes	—
138	C₂²×C₄	C₂⁴×C₁₃₁	1	9	yes	—
139	C₂²×C₄	C₂⁴×C₁₃₁	1	3	yes	—
140	C₂²×C₄	C₂⁴×C₁₃₁	1	3	yes	—
141	C₄⋊C₄	—	2	12	no	—
142	C₄⋊C₄	C₂×D₄×C₁₃₁	1	36	yes	—
143	C₂²⋊C₄	—	2	72	no	—
144	C₄²	—	2	18	no	—
145	C₄²	C₂×D₄×C₁₃₁	1	36	yes	—
146	C₄²	#41×C₁₃₁	1	4	yes	complemented
147	C₂³	—	2	36	no	—
148	C₂³	—	2	12	no	—
149	C₂³	—	2	36	no	—
150	C₂³	C₂⁵×C₁₃₁	1	1	yes	characteristic
151	Q₈	#127×C₁₃₁	1	16	yes	complemented
152	D₄	—	4	48	no	—
153	D₄	—	4	72	no	—
154	D₄	—	8	72	no	—
155	C₂×C₄	—	2	18	no	—
156	C₂×C₄	—	2	12	no	—
157	C₂×C₄	—	4	48	no	—
158	C₂×C₄	C₂²×D₄×C₁₃₁	1	18	yes	—
159	C₂×C₄	C₂²×D₄×C₁₃₁	1	18	yes	—
160	C₂×C₄	C₂²×D₄×C₁₃₁	1	6	yes	—
161	C₂×C₄	C₂×#41×C₁₃₁	1	6	yes	—
162	C₂×C₄	H(5,𝔽₂)×C₁₃₁	1	6	yes	—
163	C₂²	—	4	36	no	—
164	C₂²	—	4	12	no	—
165	C₂²	—	4	36	no	—
166	C₂²	C₂³×D₄×C₁₃₁	1	3	yes	—
167	C₂²	C₂²×#41×C₁₃₁	1	1	yes	characteristic
168	C₂²	C₂×H(5,𝔽₂)×C₁₃₁	1	3	yes	—
169	C₄	—	2	18	no	—
170	C₄	—	4	4	no	—
171	C₄	C₂×#127×C₁₃₁	1	12	yes	—
172	C₄	#550×C₁₃₁	1	12	yes	complemented
173	C₂	—	8	12	no	—
174	C₂	C₂²×#127×C₁₃₁	1	1	yes	characteristic
175	C₂	C₂×#550×C₁₃₁	1	3	yes	—
176	C₂	#3498×C₁₃₁	1	3	yes	—

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

Complements

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

N	H ≅ G/N	type
1. C₂×#550×C₁₃₁	C₂	semidirect
2. #2867×C₁₃₁	C₂	semidirect
3. #2541×C₁₃₁	C₂	semidirect
4. C₄×#127×C₁₃₁	C₂	semidirect
5. C₄²×Q₈×C₁₃₁	C₂	semidirect
29. C₄×Q₈×C₁₃₁	D₄	semidirect
57. C₄²×C₁₃₁	#41	semidirect
62. Q₈×C₁₃₁	#127	semidirect
83. C₄×C₁₃₁	#550	semidirect
88. #36592	C₁₃₁	direct (N × H)
89. C₁₃₁	#36592	direct (N × H)
90. C₂×#550	C₂×C₁₃₁	semidirect
91. #2867	C₂×C₁₃₁	semidirect
92. #2541	C₂×C₁₃₁	semidirect
93. C₄×#127	C₂×C₁₃₁	semidirect
94. C₄²×Q₈	C₂×C₁₃₁	semidirect
118. C₄×Q₈	D₄×C₁₃₁	semidirect
146. C₄²	#41×C₁₃₁	semidirect
151. Q₈	#127×C₁₃₁	semidirect
172. C₄	#550×C₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#550×C₁₃₁	2
#2867×C₁₃₁	2
#2541×C₁₃₁	2
C₄×#127×C₁₃₁	2
C₄²×Q₈×C₁₃₁	2
#36592	131

Subgroups — C131×#36592 #45722476

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#36592 #45722476