Subgroups — #2591×C₁₃₁ #30662774

Order 16768.

596 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#121×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊C₄×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#400×C₁₃₁	C₂	1	2	yes	maximal, complemented
4	#394×C₁₃₁	C₂	1	2	yes	maximal, complemented
5	#388×C₁₃₁	C₂	1	4	yes	maximal, complemented
6	#386×C₁₃₁	C₂	1	2	yes	maximal, complemented
7	#386×C₁₃₁	C₂	1	2	yes	maximal, complemented
8	C₄×C₂²⋊C₄×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₄×C₁₃₁	C₂²	1	2	yes	—
10	C₂²×D₄×C₁₃₁	C₂²	1	1	yes	characteristic
11	C₂³×C₄×C₁₃₁	C₂²	1	1	yes	characteristic
12	C₂³×C₄×C₁₃₁	C₂²	1	1	yes	characteristic
13	C₂³×C₄×C₁₃₁	C₂²	1	1	yes	characteristic
14	#121×C₁₃₁	—	2	4	no	—
15	C₂×C₄⋊C₄×C₁₃₁	—	2	4	no	—
16	C₂×C₄⋊C₄×C₁₃₁	C₂²	1	2	yes	—
17	C₂×C₄⋊C₄×C₁₃₁	C₂²	1	2	yes	—
18	C₂×C₄⋊C₄×C₁₃₁	C₂²	1	1	yes	characteristic
19	C₂×C₂²⋊C₄×C₁₃₁	C₂²	1	2	yes	—
20	C₂×C₂²⋊C₄×C₁₃₁	C₂²	1	4	yes	—
21	C₂×C₂²⋊C₄×C₁₃₁	C₂²	1	2	yes	—
22	C₂×C₂²⋊C₄×C₁₃₁	C₂²	1	4	yes	—
23	C₂×C₄²×C₁₃₁	C₂²	1	2	yes	—
24	#95×C₁₃₁	C₂²	1	4	yes	—
25	#95×C₁₃₁	C₂²	1	4	yes	—
26	#95×C₁₃₁	C₂²	1	2	yes	—
27	C₂⁴×C₁₃₁	C₂³	1	2	yes	—
28	C₂⁴×C₁₃₁	C₂³	1	1	yes	characteristic
29	C₂×D₄×C₁₃₁	—	2	4	no	—
30	C₂×D₄×C₁₃₁	—	2	8	no	—
31	C₂×D₄×C₁₃₁	—	2	2	no	—
32	C₂×D₄×C₁₃₁	—	4	2	no	—
33	C₂²×C₄×C₁₃₁	—	2	2	no	—
34	C₂²×C₄×C₁₃₁	—	2	4	no	—
35	C₂²×C₄×C₁₃₁	—	2	2	no	—
36	C₂²×C₄×C₁₃₁	—	4	2	no	—
37	C₂²×C₄×C₁₃₁	—	4	2	no	—
38	C₂²×C₄×C₁₃₁	D₄	1	4	yes	—
39	C₂²×C₄×C₁₃₁	C₂³	1	2	yes	—
40	C₂²×C₄×C₁₃₁	C₂³	1	4	yes	—
41	C₂²×C₄×C₁₃₁	C₂³	1	2	yes	—
42	C₂²×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
43	C₂²×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
44	C₂²×C₄×C₁₃₁	C₂³	1	2	yes	—
45	C₄⋊C₄×C₁₃₁	—	2	4	no	—
46	C₄⋊C₄×C₁₃₁	—	2	4	no	—
47	C₄⋊C₄×C₁₃₁	—	2	2	no	—
48	C₂²⋊C₄×C₁₃₁	—	2	4	no	—
49	C₂²⋊C₄×C₁₃₁	—	2	4	no	—
50	C₂²⋊C₄×C₁₃₁	—	2	8	no	—
51	C₂²⋊C₄×C₁₃₁	—	4	4	no	—
52	C₄²×C₁₃₁	—	4	2	no	—
53	C₂³×C₁₃₁	—	2	2	no	—
54	C₂³×C₁₃₁	—	2	2	no	—
55	C₂³×C₁₃₁	—	2	2	no	—
56	C₂³×C₁₃₁	—	2	4	no	—
57	C₂³×C₁₃₁	—	2	4	no	—
58	C₂³×C₁₃₁	—	2	4	no	—
59	C₂³×C₁₃₁	C₂×D₄	1	2	yes	—
60	C₂³×C₁₃₁	#41	1	4	yes	—
61	C₂³×C₁₃₁	C₂⁴	1	1	yes	characteristic
62	D₄×C₁₃₁	—	4	4	no	—
63	D₄×C₁₃₁	—	4	8	no	—
64	C₂×C₄×C₁₃₁	—	2	8	no	—
65	C₂×C₄×C₁₃₁	—	2	2	no	—
66	C₂×C₄×C₁₃₁	—	2	2	no	—
67	C₂×C₄×C₁₃₁	—	2	2	no	—
68	C₂×C₄×C₁₃₁	—	2	2	no	—
69	C₂×C₄×C₁₃₁	—	2	2	no	—
70	C₂×C₄×C₁₃₁	—	2	2	no	—
71	C₂×C₄×C₁₃₁	—	2	4	no	—
72	C₂×C₄×C₁₃₁	—	2	4	no	—
73	C₂×C₄×C₁₃₁	—	2	4	no	—
74	C₂×C₄×C₁₃₁	—	2	4	no	—
75	C₂×C₄×C₁₃₁	—	2	4	no	—
76	C₂×C₄×C₁₃₁	—	4	4	no	—
77	C₂×C₄×C₁₃₁	—	4	4	no	—
78	C₂×C₄×C₁₃₁	C₂×D₄	1	2	yes	—
79	C₂×C₄×C₁₃₁	C₂×D₄	1	2	yes	—
80	C₂×C₄×C₁₃₁	#41	1	4	yes	—
81	C₂²×C₁₃₁	—	2	2	no	—
82	C₂²×C₁₃₁	—	2	2	no	—
83	C₂²×C₁₃₁	—	2	8	no	—
84	C₂²×C₁₃₁	—	4	2	no	—
85	C₂²×C₁₃₁	—	4	2	no	—
86	C₂²×C₁₃₁	—	4	2	no	—
87	C₂²×C₁₃₁	—	4	8	no	—
88	C₂²×C₁₃₁	#124	1	4	yes	—
89	C₂²×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
90	C₂²×C₁₃₁	C₂×#41	1	2	yes	—
91	C₂²×C₁₃₁	C₂×#41	1	2	yes	—
92	C₂²×C₁₃₁	H(5,𝔽₂)	1	2	yes	—
93	C₄×C₁₃₁	—	2	4	no	—
94	C₄×C₁₃₁	—	2	2	no	—
95	C₄×C₁₃₁	—	2	2	no	—
96	C₄×C₁₃₁	—	4	2	no	—
97	C₄×C₁₃₁	—	4	2	no	—
98	C₄×C₁₃₁	—	4	4	no	—
99	C₂×C₁₃₁	—	2	4	no	—
100	C₂×C₁₃₁	—	8	2	no	—
101	C₂×C₁₃₁	#525	1	1	yes	characteristic
102	C₂×C₁₃₁	C₂×#124	1	1	yes	characteristic
103	C₂×C₁₃₁	#534	1	1	yes	characteristic
104	C₂×C₁₃₁	#553	1	4	yes	—
105	C₁₃₁	#2591	1	1	yes	characteristic, direct product factor, Sylow
106	#2591	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
107	C₂×#121	C₂×C₁₃₁	1	1	yes	characteristic
108	C₂²×C₄⋊C₄	C₂×C₁₃₁	1	1	yes	characteristic, complemented
109	#400	C₂×C₁₃₁	1	2	yes	complemented
110	#394	C₂×C₁₃₁	1	2	yes	complemented
111	#388	C₂×C₁₃₁	1	4	yes	complemented
112	#386	C₂×C₁₃₁	1	2	yes	complemented
113	#386	C₂×C₁₃₁	1	2	yes	complemented
114	C₄×C₂²⋊C₄	C₂×C₁₃₁	1	1	yes	characteristic, complemented
115	C₂²×D₄	C₂²×C₁₃₁	1	2	yes	—
116	C₂²×D₄	C₂²×C₁₃₁	1	1	yes	characteristic
117	C₂³×C₄	C₂²×C₁₃₁	1	1	yes	characteristic
118	C₂³×C₄	C₂²×C₁₃₁	1	1	yes	characteristic
119	C₂³×C₄	C₂²×C₁₃₁	1	1	yes	characteristic
120	#121	—	2	4	no	—
121	C₂×C₄⋊C₄	—	2	4	no	—
122	C₂×C₄⋊C₄	C₂²×C₁₃₁	1	2	yes	—
123	C₂×C₄⋊C₄	C₂²×C₁₃₁	1	2	yes	—
124	C₂×C₄⋊C₄	C₂²×C₁₃₁	1	1	yes	characteristic
125	C₂×C₂²⋊C₄	C₂²×C₁₃₁	1	2	yes	—
126	C₂×C₂²⋊C₄	C₂²×C₁₃₁	1	2	yes	—
127	C₂×C₂²⋊C₄	C₂²×C₁₃₁	1	4	yes	—
128	C₂×C₂²⋊C₄	C₂²×C₁₃₁	1	4	yes	—
129	C₂×C₄²	C₂²×C₁₃₁	1	2	yes	—
130	#95	C₂²×C₁₃₁	1	2	yes	—
131	#95	C₂²×C₁₃₁	1	4	yes	—
132	#95	C₂²×C₁₃₁	1	4	yes	—
133	C₂⁴	C₂³×C₁₃₁	1	2	yes	—
134	C₂⁴	C₂³×C₁₃₁	1	1	yes	characteristic
135	C₂×D₄	—	2	4	no	—
136	C₂×D₄	—	2	8	no	—
137	C₂×D₄	—	2	2	no	—
138	C₂×D₄	—	4	2	no	—
139	C₂²×C₄	—	2	2	no	—
140	C₂²×C₄	—	2	4	no	—
141	C₂²×C₄	—	2	2	no	—
142	C₂²×C₄	—	4	2	no	—
143	C₂²×C₄	—	4	2	no	—
144	C₂²×C₄	D₄×C₁₃₁	1	4	yes	—
145	C₂²×C₄	C₂³×C₁₃₁	1	2	yes	—
146	C₂²×C₄	C₂³×C₁₃₁	1	4	yes	—
147	C₂²×C₄	C₂³×C₁₃₁	1	2	yes	—
148	C₂²×C₄	C₂³×C₁₃₁	1	1	yes	characteristic
149	C₂²×C₄	C₂³×C₁₃₁	1	1	yes	characteristic
150	C₂²×C₄	C₂³×C₁₃₁	1	2	yes	—
151	C₄⋊C₄	—	2	4	no	—
152	C₄⋊C₄	—	2	4	no	—
153	C₄⋊C₄	—	2	2	no	—
154	C₂²⋊C₄	—	2	4	no	—
155	C₂²⋊C₄	—	2	4	no	—
156	C₂²⋊C₄	—	2	8	no	—
157	C₂²⋊C₄	—	4	4	no	—
158	C₄²	—	4	2	no	—
159	C₂³	—	2	2	no	—
160	C₂³	—	2	2	no	—
161	C₂³	—	2	2	no	—
162	C₂³	—	2	4	no	—
163	C₂³	—	2	4	no	—
164	C₂³	—	2	4	no	—
165	C₂³	C₂×D₄×C₁₃₁	1	2	yes	—
166	C₂³	#41×C₁₃₁	1	4	yes	—
167	C₂³	C₂⁴×C₁₃₁	1	1	yes	characteristic
168	D₄	—	4	4	no	—
169	D₄	—	4	8	no	—
170	C₂×C₄	—	2	8	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	4	no	—
178	C₂×C₄	—	2	4	no	—
179	C₂×C₄	—	2	4	no	—
180	C₂×C₄	—	2	4	no	—
181	C₂×C₄	—	2	4	no	—
182	C₂×C₄	—	4	4	no	—
183	C₂×C₄	—	4	4	no	—
184	C₂×C₄	C₂×D₄×C₁₃₁	1	2	yes	—
185	C₂×C₄	C₂×D₄×C₁₃₁	1	2	yes	—
186	C₂×C₄	#41×C₁₃₁	1	4	yes	—
187	C₂²	—	2	2	no	—
188	C₂²	—	2	2	no	—
189	C₂²	—	2	8	no	—
190	C₂²	—	4	2	no	—
191	C₂²	—	4	2	no	—
192	C₂²	—	4	2	no	—
193	C₂²	—	4	8	no	—
194	C₂²	#124×C₁₃₁	1	4	yes	—
195	C₂²	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
196	C₂²	C₂×#41×C₁₃₁	1	2	yes	—
197	C₂²	C₂×#41×C₁₃₁	1	2	yes	—
198	C₂²	H(5,𝔽₂)×C₁₃₁	1	2	yes	—
199	C₄	—	2	4	no	—
200	C₄	—	2	2	no	—
201	C₄	—	2	2	no	—
202	C₄	—	4	2	no	—
203	C₄	—	4	2	no	—
204	C₄	—	4	4	no	—
205	C₂	—	2	4	no	—
206	C₂	—	8	2	no	—
207	C₂	#525×C₁₃₁	1	1	yes	characteristic
208	C₂	C₂×#124×C₁₃₁	1	1	yes	characteristic
209	C₂	#534×C₁₃₁	1	1	yes	characteristic
210	C₂	#553×C₁₃₁	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₃₁	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₁₃₁
prime radical	C₂×#121×C₁₃₁	C₂
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
2. C₂²×C₄⋊C₄×C₁₃₁	C₂	semidirect
3. #400×C₁₃₁	C₂	semidirect
4. #394×C₁₃₁	C₂	semidirect
5. #388×C₁₃₁	C₂	semidirect
6. #386×C₁₃₁	C₂	semidirect
7. #386×C₁₃₁	C₂	semidirect
8. C₄×C₂²⋊C₄×C₁₃₁	C₂	semidirect
105. C₁₃₁	#2591	direct (N × H)
106. #2591	C₁₃₁	direct (N × H)
108. C₂²×C₄⋊C₄	C₂×C₁₃₁	semidirect
109. #400	C₂×C₁₃₁	semidirect
110. #394	C₂×C₁₃₁	semidirect
111. #388	C₂×C₁₃₁	semidirect
112. #386	C₂×C₁₃₁	semidirect
113. #386	C₂×C₁₃₁	semidirect
114. C₄×C₂²⋊C₄	C₂×C₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#121×C₁₃₁	2
C₂²×C₄⋊C₄×C₁₃₁	2
#400×C₁₃₁	2
#394×C₁₃₁	2
#388×C₁₃₁	2
#386×C₁₃₁	2
C₄×C₂²⋊C₄×C₁₃₁	2
#2591	131

Subgroups — #2591×C131 #30662774

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2591×C₁₃₁ #30662774