Subgroups — #2930×C₁₃₁ #30663113

Order 16768.

492 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#131×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×M₁₆×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#525×C₁₃₁	C₂	1	1	yes	maximal, characteristic
4	#436×C₁₃₁	C₂	1	4	yes	maximal, complemented
5	#435×C₁₃₁	C₂	1	4	yes	maximal, complemented
6	#424×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
7	#408×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
8	#407×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
9	#405×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#41×C₁₃₁	C₂²	1	1	yes	characteristic
11	C₂³×C₄×C₁₃₁	C₂²	1	1	yes	characteristic
12	#131×C₁₃₁	—	2	4	no	—
13	C₂×M₁₆×C₁₃₁	—	2	4	no	—
14	C₂×M₁₆×C₁₃₁	C₂²	1	1	yes	characteristic
15	C₂×M₁₆×C₁₃₁	C₂²	1	1	yes	characteristic
16	C₂×M₁₆×C₁₃₁	C₂²	1	2	yes	—
17	C₂×M₁₆×C₁₃₁	C₂²	1	2	yes	—
18	C₂×M₁₆×C₁₃₁	C₂²	1	2	yes	complemented
19	C₂²×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
20	C₂²×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
21	C₂²×C₈×C₁₃₁	C₂²	1	2	yes	complemented
22	#123×C₁₃₁	C₄	1	2	yes	—
23	C₂²⋊Q₈×C₁₃₁	C₄	1	2	yes	—
24	#121×C₁₃₁	C₄	1	2	yes	—
25	#120×C₁₃₁	C₄	1	2	yes	—
26	C₄×D₄×C₁₃₁	C₂²	1	4	yes	—
27	#117×C₁₃₁	C₂²	1	1	yes	characteristic
28	C₄⋊C₈×C₁₃₁	C₂²	1	4	yes	complemented
29	C₂²⋊C₈×C₁₃₁	C₂²	1	4	yes	—
30	C₂²⋊C₈×C₁₃₁	C₂²	1	4	yes	complemented
31	#97×C₁₃₁	C₂²	1	2	yes	complemented
32	C₄×C₈×C₁₃₁	C₂²	1	2	yes	complemented
33	C₂⁴×C₁₃₁	C₂×C₄	1	1	yes	characteristic
34	#41×C₁₃₁	—	2	2	no	—
35	#41×C₁₃₁	—	2	2	no	—
36	C₂×Q₈×C₁₃₁	C₂×C₄	1	1	yes	characteristic
37	C₂×D₄×C₁₃₁	C₂×C₄	1	2	yes	—
38	C₂×D₄×C₁₃₁	C₂×C₄	1	1	yes	characteristic
39	C₂×D₄×C₁₃₁	C₂×C₄	1	4	yes	—
40	C₂²×C₄×C₁₃₁	—	2	2	no	—
41	C₂²×C₄×C₁₃₁	—	2	2	no	—
42	C₂²×C₄×C₁₃₁	C₂×C₄	1	2	yes	—
43	C₂²×C₄×C₁₃₁	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₁₃₁	C₂³	1	2	yes	—
45	C₂²×C₄×C₁₃₁	C₂³	1	2	yes	—
46	C₂²×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
47	M₁₆×C₁₃₁	—	2	2	no	—
48	M₁₆×C₁₃₁	—	2	2	no	—
49	M₁₆×C₁₃₁	—	2	4	no	—
50	M₁₆×C₁₃₁	—	2	4	no	—
51	M₁₆×C₁₃₁	—	2	4	no	—
52	C₂×C₈×C₁₃₁	—	2	2	no	—
53	C₂×C₈×C₁₃₁	—	2	4	no	—
54	C₂×C₈×C₁₃₁	D₄	1	4	yes	complemented
55	C₂×C₈×C₁₃₁	C₂³	1	2	yes	—
56	C₂×C₈×C₁₃₁	C₂³	1	2	yes	—
57	C₂×C₈×C₁₃₁	C₂³	1	2	yes	—
58	C₂×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
59	C₂×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
60	C₄⋊C₄×C₁₃₁	C₂×C₄	1	2	yes	—
61	C₄⋊C₄×C₁₃₁	C₂×C₄	1	4	yes	—
62	C₂²⋊C₄×C₁₃₁	C₂×C₄	1	2	yes	—
63	C₂²⋊C₄×C₁₃₁	C₂×C₄	1	4	yes	—
64	C₂²⋊C₄×C₁₃₁	C₂×C₄	1	4	yes	—
65	C₄²×C₁₃₁	C₂³	1	2	yes	—
66	C₂³×C₁₃₁	—	2	2	no	—
67	C₂³×C₁₃₁	—	4	2	no	—
68	C₂³×C₁₃₁	C₂²×C₄	1	2	yes	—
69	C₂³×C₁₃₁	C₂²×C₄	1	2	yes	—
70	C₂³×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
71	Q₈×C₁₃₁	—	2	2	no	—
72	D₄×C₁₃₁	—	2	4	no	—
73	D₄×C₁₃₁	—	2	2	no	—
74	D₄×C₁₃₁	—	4	4	no	—
75	C₂×C₄×C₁₃₁	—	2	2	no	—
76	C₂×C₄×C₁₃₁	—	2	4	no	—
77	C₂×C₄×C₁₃₁	—	2	2	no	—
78	C₂×C₄×C₁₃₁	—	2	2	no	—
79	C₂×C₄×C₁₃₁	—	2	2	no	—
80	C₂×C₄×C₁₃₁	—	2	2	no	—
81	C₂×C₄×C₁₃₁	C₂²×C₄	1	4	yes	—
82	C₂×C₄×C₁₃₁	C₂²×C₄	1	2	yes	—
83	C₂×C₄×C₁₃₁	C₂²×C₄	1	2	yes	—
84	C₂×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₄×C₁₃₁	C₂×D₄	1	2	yes	—
86	C₂×C₄×C₁₃₁	#41	1	2	yes	—
87	C₂×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
88	C₈×C₁₃₁	—	2	2	no	—
89	C₈×C₁₃₁	—	2	2	no	—
90	C₈×C₁₃₁	—	2	2	no	—
91	C₈×C₁₃₁	C₂×D₄	1	2	yes	—
92	C₈×C₁₃₁	C₂×D₄	1	2	yes	—
93	C₂²×C₁₃₁	—	2	2	no	—
94	C₂²×C₁₃₁	—	2	2	no	—
95	C₂²×C₁₃₁	—	2	2	no	—
96	C₂²×C₁₃₁	—	2	2	no	—
97	C₂²×C₁₃₁	—	2	4	no	—
98	C₂²×C₁₃₁	—	2	2	no	—
99	C₂²×C₁₃₁	—	4	4	no	—
100	C₂²×C₁₃₁	C₄×D₄	1	2	yes	—
101	C₂²×C₁₃₁	C₂³×C₄	1	1	yes	characteristic
102	C₄×C₁₃₁	—	2	2	no	—
103	C₄×C₁₃₁	—	2	2	no	—
104	C₄×C₁₃₁	—	2	4	no	—
105	C₄×C₁₃₁	C₄×D₄	1	2	yes	—
106	C₄×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
107	C₄×C₁₃₁	C₂×#41	1	1	yes	characteristic
108	C₂×C₁₃₁	—	2	2	no	—
109	C₂×C₁₃₁	—	4	2	no	—
110	C₂×C₁₃₁	—	4	2	no	—
111	C₂×C₁₃₁	C₂×C₄×D₄	1	1	yes	characteristic
112	C₂×C₁₃₁	#568	1	2	yes	—
113	C₁₃₁	#2930	1	1	yes	characteristic, direct product factor, Sylow
114	#2930	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
115	C₂×#131	C₂×C₁₃₁	1	1	yes	characteristic, complemented
116	C₂²×M₁₆	C₂×C₁₃₁	1	1	yes	characteristic, complemented
117	#525	C₂×C₁₃₁	1	1	yes	characteristic
118	#436	C₂×C₁₃₁	1	4	yes	complemented
119	#435	C₂×C₁₃₁	1	4	yes	complemented
120	#424	C₂×C₁₃₁	1	1	yes	characteristic, complemented
121	#408	C₂×C₁₃₁	1	1	yes	characteristic, complemented
122	#407	C₂×C₁₃₁	1	1	yes	characteristic, complemented
123	#405	C₂×C₁₃₁	1	1	yes	characteristic, complemented
124	C₂×#41	C₂²×C₁₃₁	1	1	yes	characteristic
125	C₂³×C₄	C₂²×C₁₃₁	1	1	yes	characteristic
126	#131	—	2	4	no	—
127	C₂×M₁₆	—	2	4	no	—
128	C₂×M₁₆	C₂²×C₁₃₁	1	2	yes	—
129	C₂×M₁₆	C₂²×C₁₃₁	1	2	yes	—
130	C₂×M₁₆	C₂²×C₁₃₁	1	1	yes	characteristic
131	C₂×M₁₆	C₂²×C₁₃₁	1	1	yes	characteristic
132	C₂×M₁₆	C₂²×C₁₃₁	1	2	yes	complemented
133	C₂²×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
134	C₂²×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
135	C₂²×C₈	C₂²×C₁₃₁	1	2	yes	complemented
136	#123	C₄×C₁₃₁	1	2	yes	—
137	C₂²⋊Q₈	C₄×C₁₃₁	1	2	yes	—
138	#121	C₄×C₁₃₁	1	2	yes	—
139	#120	C₄×C₁₃₁	1	2	yes	—
140	C₄×D₄	C₂²×C₁₃₁	1	4	yes	—
141	#117	C₂²×C₁₃₁	1	1	yes	characteristic
142	C₄⋊C₈	C₂²×C₁₃₁	1	4	yes	complemented
143	C₂²⋊C₈	C₂²×C₁₃₁	1	4	yes	—
144	C₂²⋊C₈	C₂²×C₁₃₁	1	4	yes	complemented
145	#97	C₂²×C₁₃₁	1	2	yes	complemented
146	C₄×C₈	C₂²×C₁₃₁	1	2	yes	complemented
147	C₂⁴	C₂×C₄×C₁₃₁	1	1	yes	characteristic
148	#41	—	2	2	no	—
149	#41	—	2	2	no	—
150	C₂×Q₈	C₂×C₄×C₁₃₁	1	1	yes	characteristic
151	C₂×D₄	C₂×C₄×C₁₃₁	1	2	yes	—
152	C₂×D₄	C₂×C₄×C₁₃₁	1	1	yes	characteristic
153	C₂×D₄	C₂×C₄×C₁₃₁	1	4	yes	—
154	C₂²×C₄	—	2	2	no	—
155	C₂²×C₄	—	2	2	no	—
156	C₂²×C₄	C₂×C₄×C₁₃₁	1	2	yes	—
157	C₂²×C₄	C₂×C₄×C₁₃₁	1	1	yes	characteristic
158	C₂²×C₄	C₂³×C₁₃₁	1	2	yes	—
159	C₂²×C₄	C₂³×C₁₃₁	1	2	yes	—
160	C₂²×C₄	C₂³×C₁₃₁	1	1	yes	characteristic
161	M₁₆	—	2	2	no	—
162	M₁₆	—	2	2	no	—
163	M₁₆	—	2	4	no	—
164	M₁₆	—	2	4	no	—
165	M₁₆	—	2	4	no	—
166	C₂×C₈	—	2	2	no	—
167	C₂×C₈	—	2	4	no	—
168	C₂×C₈	D₄×C₁₃₁	1	4	yes	complemented
169	C₂×C₈	C₂³×C₁₃₁	1	2	yes	—
170	C₂×C₈	C₂³×C₁₃₁	1	2	yes	—
171	C₂×C₈	C₂³×C₁₃₁	1	2	yes	—
172	C₂×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
173	C₂×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
174	C₄⋊C₄	C₂×C₄×C₁₃₁	1	2	yes	—
175	C₄⋊C₄	C₂×C₄×C₁₃₁	1	4	yes	—
176	C₂²⋊C₄	C₂×C₄×C₁₃₁	1	2	yes	—
177	C₂²⋊C₄	C₂×C₄×C₁₃₁	1	4	yes	—
178	C₂²⋊C₄	C₂×C₄×C₁₃₁	1	4	yes	—
179	C₄²	C₂³×C₁₃₁	1	2	yes	—
180	C₂³	—	2	2	no	—
181	C₂³	—	4	2	no	—
182	C₂³	C₂²×C₄×C₁₃₁	1	2	yes	—
183	C₂³	C₂²×C₄×C₁₃₁	1	2	yes	—
184	C₂³	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
185	Q₈	—	2	2	no	—
186	D₄	—	2	4	no	—
187	D₄	—	2	2	no	—
188	D₄	—	4	4	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	4	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	C₂²×C₄×C₁₃₁	1	4	yes	—
196	C₂×C₄	C₂²×C₄×C₁₃₁	1	2	yes	—
197	C₂×C₄	C₂²×C₄×C₁₃₁	1	2	yes	—
198	C₂×C₄	C₂²×C₄×C₁₃₁	1	1	yes	characteristic
199	C₂×C₄	C₂×D₄×C₁₃₁	1	2	yes	—
200	C₂×C₄	#41×C₁₃₁	1	2	yes	—
201	C₂×C₄	C₂⁴×C₁₃₁	1	1	yes	characteristic
202	C₈	—	2	2	no	—
203	C₈	—	2	2	no	—
204	C₈	—	2	2	no	—
205	C₈	C₂×D₄×C₁₃₁	1	2	yes	—
206	C₈	C₂×D₄×C₁₃₁	1	2	yes	—
207	C₂²	—	2	2	no	—
208	C₂²	—	2	2	no	—
209	C₂²	—	2	2	no	—
210	C₂²	—	2	2	no	—
211	C₂²	—	2	4	no	—
212	C₂²	—	2	2	no	—
213	C₂²	—	4	4	no	—
214	C₂²	C₄×D₄×C₁₃₁	1	2	yes	—
215	C₂²	C₂³×C₄×C₁₃₁	1	1	yes	characteristic
216	C₄	—	2	2	no	—
217	C₄	—	2	2	no	—
218	C₄	—	2	4	no	—
219	C₄	C₄×D₄×C₁₃₁	1	2	yes	—
220	C₄	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
221	C₄	C₂×#41×C₁₃₁	1	1	yes	characteristic
222	C₂	—	2	2	no	—
223	C₂	—	4	2	no	—
224	C₂	—	4	2	no	—
225	C₂	C₂×C₄×D₄×C₁₃₁	1	1	yes	characteristic
226	C₂	#568×C₁₃₁	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₁₃₁	C₂⁴
commutator	C₂²	C₂³×C₄×C₁₃₁
Frattini, cosocle	C₂×C₄	C₂⁴×C₁₃₁
socle	C₂²×C₁₃₁	C₂³×C₄
prime radical	#525×C₁₃₁	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#131×C₁₃₁	C₂	semidirect
2. C₂²×M₁₆×C₁₃₁	C₂	semidirect
4. #436×C₁₃₁	C₂	semidirect
5. #435×C₁₃₁	C₂	semidirect
6. #424×C₁₃₁	C₂	semidirect
7. #408×C₁₃₁	C₂	semidirect
8. #407×C₁₃₁	C₂	semidirect
9. #405×C₁₃₁	C₂	semidirect
18. C₂×M₁₆×C₁₃₁	C₂²	semidirect
21. C₂²×C₈×C₁₃₁	C₂²	semidirect
28. C₄⋊C₈×C₁₃₁	C₂²	semidirect
30. C₂²⋊C₈×C₁₃₁	C₂²	semidirect
31. #97×C₁₃₁	C₂²	semidirect
32. C₄×C₈×C₁₃₁	C₂²	semidirect
54. C₂×C₈×C₁₃₁	D₄	semidirect
113. C₁₃₁	#2930	direct (N × H)
114. #2930	C₁₃₁	direct (N × H)
115. C₂×#131	C₂×C₁₃₁	semidirect
116. C₂²×M₁₆	C₂×C₁₃₁	semidirect
118. #436	C₂×C₁₃₁	semidirect
119. #435	C₂×C₁₃₁	semidirect
120. #424	C₂×C₁₃₁	semidirect
121. #408	C₂×C₁₃₁	semidirect
122. #407	C₂×C₁₃₁	semidirect
123. #405	C₂×C₁₃₁	semidirect
132. C₂×M₁₆	C₂²×C₁₃₁	semidirect
135. C₂²×C₈	C₂²×C₁₃₁	semidirect
142. C₄⋊C₈	C₂²×C₁₃₁	semidirect
144. C₂²⋊C₈	C₂²×C₁₃₁	semidirect
145. #97	C₂²×C₁₃₁	semidirect
146. C₄×C₈	C₂²×C₁₃₁	semidirect
168. C₂×C₈	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×#131×C₁₃₁	2
C₂²×M₁₆×C₁₃₁	2
#525×C₁₃₁	2
#436×C₁₃₁	2
#435×C₁₃₁	2
#424×C₁₃₁	2
#408×C₁₃₁	2
#407×C₁₃₁	2
#405×C₁₃₁	2
#2930	131

Subgroups — #2930×C131 #30663113

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2930×C₁₃₁ #30663113