Subgroups — C₃₁×#33265 #21490024

Order 7936.

732 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2863	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#2860	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#1968	C₂	1	3	yes	maximal
4	C₃₁×#1941	C₂	1	3	yes	maximal, complemented
5	C₃₁×#1934	C₂	1	3	yes	maximal
6	C₃₁×#1933	C₂	1	3	yes	maximal, complemented
7	C₄²×C₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×SD₁₆×C₃₁	C₂²	1	1	yes	characteristic
9	C₂×C₃₁×#126	—	2	1	no	—
10	C₂×C₃₁×#106	C₂²	1	3	yes	—
11	C₂×C₃₁×#103	C₂²	1	3	yes	—
12	C₂×C₃₁×#102	C₂²	1	3	yes	—
13	C₂×C₄×C₈×C₃₁	C₂²	1	3	yes	—
14	C₂×C₄×C₈×C₃₁	C₂²	1	3	yes	—
15	C₃₁×#391	C₂²	1	3	yes	—
16	C₃₁×#390	C₂²	1	3	yes	—
17	C₃₁×#389	C₂²	1	3	yes	—
18	C₃₁×#389	C₂²	1	3	yes	—
19	C₃₁×#388	—	2	3	no	—
20	C₃₁×#387	—	2	3	no	—
21	C₃₁×#383	—	2	1	no	—
22	C₄³×C₃₁	C₂²	1	1	yes	characteristic
23	C₃₁×#340	C₂²	1	6	yes	—
24	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
25	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
26	C₂×SD₁₆×C₃₁	—	2	6	no	—
27	C₂²×C₈×C₃₁	C₂³	1	3	yes	—
28	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
29	C₃₁×#126	—	4	4	no	—
30	C₂×C₄⋊C₄×C₃₁	—	2	3	no	—
31	C₂×C₄⋊C₄×C₃₁	—	2	3	no	—
32	C₂×C₄⋊C₄×C₃₁	C₂³	1	3	yes	—
33	C₂×C₄⋊C₄×C₃₁	C₂³	1	3	yes	—
34	C₂×C₂²⋊C₄×C₃₁	—	2	3	no	—
35	C₂×C₂²⋊C₄×C₃₁	—	2	3	no	—
36	C₂×C₄²×C₃₁	D₄	1	3	yes	—
37	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₃₁	C₂³	1	3	yes	—
39	C₃₁×#106	—	2	6	no	—
40	C₃₁×#103	—	2	6	no	—
41	C₃₁×#102	—	2	6	no	—
42	C₄×C₈×C₃₁	—	2	6	no	—
43	C₄×C₈×C₃₁	—	2	6	no	—
44	C₃₁×#95	—	2	3	no	—
45	C₃₁×#95	—	2	3	no	—
46	C₃₁×#95	—	2	3	no	—
47	C₃₁×#95	—	2	3	no	—
48	C₂⁴×C₃₁	—	2	1	no	—
49	C₂×Q₈×C₃₁	—	2	3	no	—
50	C₂×Q₈×C₃₁	—	2	3	no	—
51	C₂×D₄×C₃₁	—	2	3	no	—
52	C₂×D₄×C₃₁	—	2	3	no	—
53	C₂²×C₄×C₃₁	—	2	1	no	—
54	C₂²×C₄×C₃₁	—	2	3	no	—
55	C₂²×C₄×C₃₁	—	2	3	no	—
56	C₂²×C₄×C₃₁	C₂×D₄	1	3	yes	—
57	C₂²×C₄×C₃₁	C₂×D₄	1	3	yes	—
58	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
59	SD₁₆×C₃₁	—	4	4	no	—
60	C₂×C₈×C₃₁	—	2	3	no	—
61	C₂×C₈×C₃₁	—	2	3	no	—
62	C₂×C₈×C₃₁	#41	1	6	yes	—
63	C₂×C₈×C₃₁	#41	1	6	yes	—
64	C₄⋊C₄×C₃₁	—	2	3	no	—
65	C₄⋊C₄×C₃₁	—	2	3	no	—
66	C₄⋊C₄×C₃₁	—	2	3	no	—
67	C₄⋊C₄×C₃₁	—	2	3	no	—
68	C₄⋊C₄×C₃₁	—	4	6	no	—
69	C₄⋊C₄×C₃₁	—	4	6	no	—
70	C₂²⋊C₄×C₃₁	—	4	6	no	—
71	C₂²⋊C₄×C₃₁	—	4	6	no	—
72	C₄²×C₃₁	—	2	6	no	—
73	C₄²×C₃₁	—	2	3	no	—
74	C₄²×C₃₁	—	2	3	no	—
75	C₄²×C₃₁	SD₁₆	1	4	yes	complemented
76	#33265	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
77	C₂³×C₃₁	—	4	3	no	—
78	C₂³×C₃₁	—	4	1	no	—
79	C₂³×C₃₁	—	4	3	no	—
80	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
81	Q₈×C₃₁	—	4	1	no	—
82	Q₈×C₃₁	—	4	3	no	—
83	D₄×C₃₁	—	4	1	no	—
84	D₄×C₃₁	—	4	3	no	—
85	C₂×C₄×C₃₁	—	2	3	no	—
86	C₂×C₄×C₃₁	—	2	3	no	—
87	C₂×C₄×C₃₁	—	2	3	no	—
88	C₂×C₄×C₃₁	—	2	3	no	—
89	C₂×C₄×C₃₁	—	4	3	no	—
90	C₂×C₄×C₃₁	—	4	3	no	—
91	C₂×C₄×C₃₁	—	4	3	no	—
92	C₂×C₄×C₃₁	—	4	3	no	—
93	C₂×C₄×C₃₁	—	4	3	no	—
94	C₂×C₄×C₃₁	—	4	6	no	—
95	C₂×C₄×C₃₁	C₂×SD₁₆	1	6	yes	—
96	C₂×C₄×C₃₁	#135	1	6	yes	—
97	C₂×C₄×C₃₁	C₂×#41	1	3	yes	—
98	C₂×C₄×C₃₁	C₂×#41	1	3	yes	—
99	C₈×C₃₁	—	2	6	no	—
100	C₈×C₃₁	#126	1	4	yes	complemented
101	#2863	C₂×C₃₁	1	1	yes	characteristic, complemented
102	#2860	C₂×C₃₁	1	1	yes	characteristic
103	#1968	C₂×C₃₁	1	3	yes	—
104	#1941	C₂×C₃₁	1	3	yes	complemented
105	#1934	C₂×C₃₁	1	3	yes	—
106	#1933	C₂×C₃₁	1	3	yes	complemented
107	C₄²×C₈	C₂×C₃₁	1	1	yes	characteristic, complemented
108	C₂²×C₃₁	—	8	1	no	—
109	C₂²×C₃₁	—	8	3	no	—
110	C₂²×C₃₁	—	8	3	no	—
111	C₂²×C₃₁	#532	1	3	yes	—
112	C₂²×C₃₁	C₂²×SD₁₆	1	1	yes	characteristic
113	C₂²×C₃₁	C₂×#135	1	3	yes	—
114	C₄×C₃₁	—	2	6	no	—
115	C₄×C₃₁	—	2	6	no	—
116	C₄×C₃₁	—	8	1	no	—
117	C₄×C₃₁	—	8	3	no	—
118	C₄×C₃₁	—	8	3	no	—
119	C₄×C₃₁	C₂×#126	1	1	yes	characteristic
120	C₄×C₃₁	#529	1	3	yes	—
121	C₂²×SD₁₆	C₂²×C₃₁	1	1	yes	characteristic
122	C₂×#126	—	2	1	no	—
123	C₂×#106	C₂²×C₃₁	1	3	yes	—
124	C₂×#103	C₂²×C₃₁	1	3	yes	—
125	C₂×#102	C₂²×C₃₁	1	3	yes	—
126	C₂×C₄×C₈	C₂²×C₃₁	1	3	yes	—
127	C₂×C₄×C₈	C₂²×C₃₁	1	3	yes	—
128	#391	C₂²×C₃₁	1	3	yes	—
129	#390	C₂²×C₃₁	1	3	yes	—
130	#389	C₂²×C₃₁	1	3	yes	—
131	#389	C₂²×C₃₁	1	3	yes	—
132	#388	—	2	3	no	—
133	#387	—	2	3	no	—
134	#383	—	2	1	no	—
135	C₄³	C₂²×C₃₁	1	1	yes	characteristic
136	#340	C₂²×C₃₁	1	6	yes	—
137	C₂×C₃₁	—	16	1	no	—
138	C₂×C₃₁	#2860	1	1	yes	characteristic
139	C₂×C₃₁	#3167	1	3	yes	—
140	C₂×C₃₁	#3168	1	3	yes	—
141	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
142	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
143	C₂×SD₁₆	—	2	6	no	—
144	C₂²×C₈	C₂³×C₃₁	1	3	yes	—
145	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
146	#126	—	4	4	no	—
147	C₂×C₄⋊C₄	—	2	3	no	—
148	C₂×C₄⋊C₄	—	2	3	no	—
149	C₂×C₄⋊C₄	C₂³×C₃₁	1	3	yes	—
150	C₂×C₄⋊C₄	C₂³×C₃₁	1	3	yes	—
151	C₂×C₂²⋊C₄	—	2	3	no	—
152	C₂×C₂²⋊C₄	—	2	3	no	—
153	C₂×C₄²	D₄×C₃₁	1	3	yes	—
154	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
155	C₂×C₄²	C₂³×C₃₁	1	3	yes	—
156	#106	—	2	6	no	—
157	#103	—	2	6	no	—
158	#102	—	2	6	no	—
159	C₄×C₈	—	2	6	no	—
160	C₄×C₈	—	2	6	no	—
161	#95	—	2	3	no	—
162	#95	—	2	3	no	—
163	#95	—	2	3	no	—
164	#95	—	2	3	no	—
165	C₃₁	#33265	1	1	yes	characteristic, direct product factor, Sylow
166	C₂⁴	—	2	1	no	—
167	C₂×Q₈	—	2	3	no	—
168	C₂×Q₈	—	2	3	no	—
169	C₂×D₄	—	2	3	no	—
170	C₂×D₄	—	2	3	no	—
171	C₂²×C₄	—	2	1	no	—
172	C₂²×C₄	—	2	3	no	—
173	C₂²×C₄	—	2	3	no	—
174	C₂²×C₄	C₂×D₄×C₃₁	1	3	yes	—
175	C₂²×C₄	C₂×D₄×C₃₁	1	3	yes	—
176	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
177	SD₁₆	—	4	4	no	—
178	C₂×C₈	—	2	3	no	—
179	C₂×C₈	—	2	3	no	—
180	C₂×C₈	#41×C₃₁	1	6	yes	—
181	C₂×C₈	#41×C₃₁	1	6	yes	—
182	C₄⋊C₄	—	2	3	no	—
183	C₄⋊C₄	—	2	3	no	—
184	C₄⋊C₄	—	2	3	no	—
185	C₄⋊C₄	—	2	3	no	—
186	C₄⋊C₄	—	4	6	no	—
187	C₄⋊C₄	—	4	6	no	—
188	C₂²⋊C₄	—	4	6	no	—
189	C₂²⋊C₄	—	4	6	no	—
190	C₄²	—	2	6	no	—
191	C₄²	—	2	3	no	—
192	C₄²	—	2	3	no	—
193	C₄²	SD₁₆×C₃₁	1	4	yes	complemented
194	C₂³	—	4	3	no	—
195	C₂³	—	4	1	no	—
196	C₂³	—	4	3	no	—
197	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
198	Q₈	—	4	1	no	—
199	Q₈	—	4	3	no	—
200	D₄	—	4	1	no	—
201	D₄	—	4	3	no	—
202	C₂×C₄	—	2	3	no	—
203	C₂×C₄	—	2	3	no	—
204	C₂×C₄	—	2	3	no	—
205	C₂×C₄	—	2	3	no	—
206	C₂×C₄	—	4	3	no	—
207	C₂×C₄	—	4	3	no	—
208	C₂×C₄	—	4	3	no	—
209	C₂×C₄	—	4	3	no	—
210	C₂×C₄	—	4	3	no	—
211	C₂×C₄	—	4	6	no	—
212	C₂×C₄	C₂×SD₁₆×C₃₁	1	6	yes	—
213	C₂×C₄	C₃₁×#135	1	6	yes	—
214	C₂×C₄	C₂×#41×C₃₁	1	3	yes	—
215	C₂×C₄	C₂×#41×C₃₁	1	3	yes	—
216	C₈	—	2	6	no	—
217	C₈	C₃₁×#126	1	4	yes	complemented
218	C₂²	—	8	1	no	—
219	C₂²	—	8	3	no	—
220	C₂²	—	8	3	no	—
221	C₂²	C₃₁×#532	1	3	yes	—
222	C₂²	C₂²×SD₁₆×C₃₁	1	1	yes	characteristic
223	C₂²	C₂×C₃₁×#135	1	3	yes	—
224	C₄	—	2	6	no	—
225	C₄	—	2	6	no	—
226	C₄	—	8	1	no	—
227	C₄	—	8	3	no	—
228	C₄	—	8	3	no	—
229	C₄	C₂×C₃₁×#126	1	1	yes	characteristic
230	C₄	C₃₁×#529	1	3	yes	—
231	C₂	—	16	1	no	—
232	C₂	C₃₁×#2860	1	1	yes	characteristic
233	C₂	C₃₁×#3167	1	3	yes	—
234	C₂	C₃₁×#3168	1	3	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂²×D₄×C₃₁	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₄³×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#2863	C₂	semidirect
4. C₃₁×#1941	C₂	semidirect
6. C₃₁×#1933	C₂	semidirect
7. C₄²×C₈×C₃₁	C₂	semidirect
75. C₄²×C₃₁	SD₁₆	semidirect
76. #33265	C₃₁	direct (N × H)
100. C₈×C₃₁	#126	semidirect
101. #2863	C₂×C₃₁	semidirect
104. #1941	C₂×C₃₁	semidirect
106. #1933	C₂×C₃₁	semidirect
107. C₄²×C₈	C₂×C₃₁	semidirect
165. C₃₁	#33265	direct (N × H)
193. C₄²	SD₁₆×C₃₁	semidirect
217. C₈	C₃₁×#126	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#2863	2
C₃₁×#2860	2
C₃₁×#1968	2
C₃₁×#1941	2
C₃₁×#1934	2
C₃₁×#1933	2
C₄²×C₈×C₃₁	2
#33265	31

Subgroups — C31×#33265 #21490024

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33265 #21490024