Subgroups — #3260×C₁₃₁ #30663443

Order 16768.

346 conjugacy classes of subgroups.

The subgroup lattice has 202 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#541×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
2	#540×C₁₃₁	C₂	1	1	yes	maximal, characteristic
3	#479×C₁₃₁	C₂	1	2	yes	maximal, complemented
4	#478×C₁₃₁	C₂	1	2	yes	maximal, complemented
5	#468×C₁₃₁	C₂	1	2	yes	maximal, complemented
6	#465×C₁₃₁	C₂	1	2	yes	maximal, complemented
7	#461×C₁₃₁	C₂	1	2	yes	maximal, complemented
8	#460×C₁₃₁	C₂	1	2	yes	maximal, complemented
9	#424×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×SD₁₆×C₁₃₁	C₂²	1	2	yes	—
11	C₂×SD₁₆×C₁₃₁	C₂²	1	2	yes	—
12	C₂×M₁₆×C₁₃₁	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
14	C₄⋊Q₈×C₁₃₁	C₂²	1	1	yes	characteristic
15	#127×C₁₃₁	C₂²	1	1	yes	characteristic
16	#126×C₁₃₁	—	2	2	no	—
17	#125×C₁₃₁	—	2	2	no	—
18	#125×C₁₃₁	C₂²	1	1	yes	characteristic
19	#125×C₁₃₁	C₂²	1	1	yes	characteristic
20	#123×C₁₃₁	—	2	2	no	—
21	C₂²⋊Q₈×C₁₃₁	C₂²	1	2	yes	—
22	#121×C₁₃₁	—	2	2	no	—
23	#121×C₁₃₁	C₂²	1	2	yes	—
24	C₄×Q₈×C₁₃₁	C₂²	1	2	yes	—
25	C₄×D₄×C₁₃₁	C₂²	1	2	yes	—
26	#117×C₁₃₁	C₂²	1	1	yes	characteristic
27	#107×C₁₃₁	C₂²	1	2	yes	—
28	#106×C₁₃₁	C₂²	1	2	yes	—
29	C₄⋊C₈×C₁₃₁	C₂²	1	2	yes	—
30	C₄⋊C₈×C₁₃₁	C₂²	1	2	yes	—
31	#103×C₁₃₁	C₂²	1	2	yes	—
32	#103×C₁₃₁	C₂²	1	2	yes	—
33	#102×C₁₃₁	C₂²	1	2	yes	—
34	#102×C₁₃₁	C₂²	1	2	yes	—
35	C₂×Q₈×C₁₃₁	C₂³	1	2	yes	—
36	C₂×D₄×C₁₃₁	—	2	2	no	—
37	C₂×D₄×C₁₃₁	—	2	2	no	—
38	C₂×D₄×C₁₃₁	C₂³	1	2	yes	—
39	C₂²×C₄×C₁₃₁	—	2	2	no	—
40	C₂²×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
41	SD₁₆×C₁₃₁	—	4	2	no	—
42	SD₁₆×C₁₃₁	—	4	2	no	—
43	M₁₆×C₁₃₁	—	4	1	no	—
44	C₂×C₈×C₁₃₁	—	4	1	no	—
45	C₂×C₈×C₁₃₁	C₂³	1	2	yes	—
46	C₂×C₈×C₁₃₁	C₂³	1	2	yes	—
47	C₄⋊C₄×C₁₃₁	—	2	2	no	—
48	C₄⋊C₄×C₁₃₁	—	2	2	no	—
49	C₄⋊C₄×C₁₃₁	—	2	2	no	—
50	C₄⋊C₄×C₁₃₁	—	2	2	no	—
51	C₄⋊C₄×C₁₃₁	—	2	2	no	—
52	C₄⋊C₄×C₁₃₁	D₄	1	2	yes	—
53	C₄⋊C₄×C₁₃₁	C₂³	1	2	yes	—
54	C₄⋊C₄×C₁₃₁	C₂³	1	2	yes	—
55	C₂²⋊C₄×C₁₃₁	—	2	2	no	—
56	C₂²⋊C₄×C₁₃₁	—	2	2	no	—
57	C₂²⋊C₄×C₁₃₁	—	2	2	no	—
58	C₂²⋊C₄×C₁₃₁	D₄	1	2	yes	—
59	C₄²×C₁₃₁	—	2	2	no	—
60	C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
61	C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
62	C₂³×C₁₃₁	—	2	2	no	—
63	C₂³×C₁₃₁	C₂×D₄	1	1	yes	characteristic
64	Q₈×C₁₃₁	—	2	2	no	—
65	Q₈×C₁₃₁	—	2	2	no	—
66	D₄×C₁₃₁	—	2	2	no	—
67	D₄×C₁₃₁	—	2	2	no	—
68	D₄×C₁₃₁	—	8	2	no	—
69	D₄×C₁₃₁	—	8	2	no	—
70	C₂×C₄×C₁₃₁	—	2	1	no	—
71	C₂×C₄×C₁₃₁	—	2	1	no	—
72	C₂×C₄×C₁₃₁	—	2	2	no	—
73	C₂×C₄×C₁₃₁	—	2	2	no	—
74	C₂×C₄×C₁₃₁	—	2	2	no	—
75	C₂×C₄×C₁₃₁	—	8	2	no	—
76	C₂×C₄×C₁₃₁	C₂×D₄	1	2	yes	—
77	C₂×C₄×C₁₃₁	C₂×D₄	1	2	yes	—
78	C₂×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
79	C₂×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
80	C₈×C₁₃₁	—	2	2	no	—
81	C₈×C₁₃₁	—	2	2	no	—
82	C₂²×C₁₃₁	—	2	1	no	—
83	C₂²×C₁₃₁	—	2	2	no	—
84	C₂²×C₁₃₁	—	4	2	no	—
85	C₂²×C₁₃₁	—	4	2	no	—
86	C₂²×C₁₃₁	—	4	2	no	—
87	C₂²×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
88	C₄×C₁₃₁	—	2	1	no	—
89	C₄×C₁₃₁	—	2	2	no	—
90	C₄×C₁₃₁	—	2	2	no	—
91	C₄×C₁₃₁	—	4	2	no	—
92	C₄×C₁₃₁	—	4	2	no	—
93	C₄×C₁₃₁	—	4	2	no	—
94	C₄×C₁₃₁	H(5,𝔽₂)	1	1	yes	characteristic
95	C₄×C₁₃₁	2₋¹⁺⁴	1	1	yes	characteristic
96	C₂×C₁₃₁	—	4	1	no	—
97	C₂×C₁₃₁	—	8	2	no	—
98	C₂×C₁₃₁	#537	1	1	yes	characteristic
99	C₂×C₁₃₁	#577	1	2	yes	—
100	C₁₃₁	#3260	1	1	yes	characteristic, direct product factor, Sylow
101	#3260	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
102	#541	C₂×C₁₃₁	1	1	yes	characteristic, complemented
103	#540	C₂×C₁₃₁	1	1	yes	characteristic
104	#479	C₂×C₁₃₁	1	2	yes	complemented
105	#478	C₂×C₁₃₁	1	2	yes	complemented
106	#468	C₂×C₁₃₁	1	2	yes	complemented
107	#465	C₂×C₁₃₁	1	2	yes	complemented
108	#461	C₂×C₁₃₁	1	2	yes	complemented
109	#460	C₂×C₁₃₁	1	2	yes	complemented
110	#424	C₂×C₁₃₁	1	1	yes	characteristic, complemented
111	C₂×SD₁₆	C₂²×C₁₃₁	1	2	yes	—
112	C₂×SD₁₆	C₂²×C₁₃₁	1	2	yes	—
113	C₂×M₁₆	C₂²×C₁₃₁	1	1	yes	characteristic
114	C₂²×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
115	C₄⋊Q₈	C₂²×C₁₃₁	1	1	yes	characteristic
116	#127	C₂²×C₁₃₁	1	1	yes	characteristic
117	#126	—	2	2	no	—
118	#125	—	2	2	no	—
119	#125	C₂²×C₁₃₁	1	1	yes	characteristic
120	#125	C₂²×C₁₃₁	1	1	yes	characteristic
121	#123	—	2	2	no	—
122	C₂²⋊Q₈	C₂²×C₁₃₁	1	2	yes	—
123	#121	—	2	2	no	—
124	#121	C₂²×C₁₃₁	1	2	yes	—
125	C₄×Q₈	C₂²×C₁₃₁	1	2	yes	—
126	C₄×D₄	C₂²×C₁₃₁	1	2	yes	—
127	#117	C₂²×C₁₃₁	1	1	yes	characteristic
128	#107	C₂²×C₁₃₁	1	2	yes	—
129	#106	C₂²×C₁₃₁	1	2	yes	—
130	C₄⋊C₈	C₂²×C₁₃₁	1	2	yes	—
131	C₄⋊C₈	C₂²×C₁₃₁	1	2	yes	—
132	#103	C₂²×C₁₃₁	1	2	yes	—
133	#103	C₂²×C₁₃₁	1	2	yes	—
134	#102	C₂²×C₁₃₁	1	2	yes	—
135	#102	C₂²×C₁₃₁	1	2	yes	—
136	C₂×Q₈	C₂³×C₁₃₁	1	2	yes	—
137	C₂×D₄	—	2	2	no	—
138	C₂×D₄	—	2	2	no	—
139	C₂×D₄	C₂³×C₁₃₁	1	2	yes	—
140	C₂²×C₄	—	2	2	no	—
141	C₂²×C₄	C₂³×C₁₃₁	1	1	yes	characteristic
142	SD₁₆	—	4	2	no	—
143	SD₁₆	—	4	2	no	—
144	M₁₆	—	4	1	no	—
145	C₂×C₈	—	4	1	no	—
146	C₂×C₈	C₂³×C₁₃₁	1	2	yes	—
147	C₂×C₈	C₂³×C₁₃₁	1	2	yes	—
148	C₄⋊C₄	—	2	2	no	—
149	C₄⋊C₄	—	2	2	no	—
150	C₄⋊C₄	—	2	2	no	—
151	C₄⋊C₄	—	2	2	no	—
152	C₄⋊C₄	—	2	2	no	—
153	C₄⋊C₄	D₄×C₁₃₁	1	2	yes	—
154	C₄⋊C₄	C₂³×C₁₃₁	1	2	yes	—
155	C₄⋊C₄	C₂³×C₁₃₁	1	2	yes	—
156	C₂²⋊C₄	—	2	2	no	—
157	C₂²⋊C₄	—	2	2	no	—
158	C₂²⋊C₄	—	2	2	no	—
159	C₂²⋊C₄	D₄×C₁₃₁	1	2	yes	—
160	C₄²	—	2	2	no	—
161	C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
162	C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
163	C₂³	—	2	2	no	—
164	C₂³	C₂×D₄×C₁₃₁	1	1	yes	characteristic
165	Q₈	—	2	2	no	—
166	Q₈	—	2	2	no	—
167	D₄	—	2	2	no	—
168	D₄	—	2	2	no	—
169	D₄	—	8	2	no	—
170	D₄	—	8	2	no	—
171	C₂×C₄	—	2	1	no	—
172	C₂×C₄	—	2	1	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	8	2	no	—
177	C₂×C₄	C₂×D₄×C₁₃₁	1	2	yes	—
178	C₂×C₄	C₂×D₄×C₁₃₁	1	2	yes	—
179	C₂×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
180	C₂×C₄	C₂⁴×C₁₃₁	1	1	yes	characteristic
181	C₈	—	2	2	no	—
182	C₈	—	2	2	no	—
183	C₂²	—	2	1	no	—
184	C₂²	—	2	2	no	—
185	C₂²	—	4	2	no	—
186	C₂²	—	4	2	no	—
187	C₂²	—	4	2	no	—
188	C₂²	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
189	C₄	—	2	1	no	—
190	C₄	—	2	2	no	—
191	C₄	—	2	2	no	—
192	C₄	—	4	2	no	—
193	C₄	—	4	2	no	—
194	C₄	—	4	2	no	—
195	C₄	H(5,𝔽₂)×C₁₃₁	1	1	yes	characteristic
196	C₄	2₋¹⁺⁴×C₁₃₁	1	1	yes	characteristic
197	C₂	—	4	1	no	—
198	C₂	—	8	2	no	—
199	C₂	#537×C₁₃₁	1	1	yes	characteristic
200	C₂	#577×C₁₃₁	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₃₁
prime radical	#540×C₁₃₁	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₁ ◁ #117×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. #541×C₁₃₁	C₂	semidirect
3. #479×C₁₃₁	C₂	semidirect
4. #478×C₁₃₁	C₂	semidirect
5. #468×C₁₃₁	C₂	semidirect
6. #465×C₁₃₁	C₂	semidirect
7. #461×C₁₃₁	C₂	semidirect
8. #460×C₁₃₁	C₂	semidirect
9. #424×C₁₃₁	C₂	semidirect
100. C₁₃₁	#3260	direct (N × H)
101. #3260	C₁₃₁	direct (N × H)
102. #541	C₂×C₁₃₁	semidirect
104. #479	C₂×C₁₃₁	semidirect
105. #478	C₂×C₁₃₁	semidirect
106. #468	C₂×C₁₃₁	semidirect
107. #465	C₂×C₁₃₁	semidirect
108. #461	C₂×C₁₃₁	semidirect
109. #460	C₂×C₁₃₁	semidirect
110. #424	C₂×C₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
#541×C₁₃₁	2
#540×C₁₃₁	2
#479×C₁₃₁	2
#478×C₁₃₁	2
#468×C₁₃₁	2
#465×C₁₃₁	2
#461×C₁₃₁	2
#460×C₁₃₁	2
#424×C₁₃₁	2
#3260	131

Subgroups — #3260×C131 #30663443

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #3260×C₁₃₁ #30663443