Subgroups — C₃₁×#13236 #21469995

Order 7936.

438 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	C₂²×C₃₁×Q₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#366	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₁×#358	C₂	1	2	yes	maximal
4	C₂×C₄×C₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#1969	C₂	1	2	yes	maximal
6	C₂²×Q₁₆×C₃₁	C₂²	1	2	yes	—
7	C₂×C₃₁×Q₃₂	—	2	2	no	—
8	C₂×C₃₁×Q₃₂	—	2	2	no	—
9	C₂×C₃₁×Q₃₂	C₄	1	4	yes	complemented
10	C₂²×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
11	C₂²×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
12	C₂×C₃₁×#107	C₂²	1	2	yes	—
13	C₂×C₃₁×#103	—	2	2	no	—
14	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
15	C₃₁×#391	—	2	2	no	—
16	C₃₁×#366	—	2	2	no	—
17	C₃₁×#358	—	2	4	no	—
18	C₄×C₁₆×C₃₁	—	2	2	no	—
19	C₂²×Q₈×C₃₁	—	2	2	no	—
20	C₂×Q₁₆×C₃₁	—	2	2	no	—
21	C₂×Q₁₆×C₃₁	—	2	2	no	—
22	C₂×Q₁₆×C₃₁	—	2	2	no	—
23	C₂×Q₁₆×C₃₁	—	2	2	no	—
24	C₂×Q₁₆×C₃₁	—	2	2	no	—
25	C₂×Q₁₆×C₃₁	C₂×C₄	1	4	yes	—
26	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
28	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
29	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
30	C₃₁×Q₃₂	—	2	8	no	—
31	C₂×C₁₆×C₃₁	—	2	1	no	—
32	C₂×C₁₆×C₃₁	—	2	1	no	—
33	C₂×C₁₆×C₃₁	C₂×C₄	1	2	yes	—
34	C₂×C₁₆×C₃₁	D₄	1	2	yes	—
35	C₂×C₁₆×C₃₁	D₄	1	2	yes	—
36	C₂×C₁₆×C₃₁	D₄	1	2	yes	—
37	C₃₁×#107	—	2	2	no	—
38	C₃₁×#107	—	2	2	no	—
39	C₃₁×#103	—	4	4	no	—
40	C₄×C₈×C₃₁	—	2	1	no	—
41	C₄×C₈×C₃₁	—	2	1	no	—
42	C₃₁×#95	—	4	4	no	—
43	C₂×Q₈×C₃₁	—	2	4	no	—
44	C₂×Q₈×C₃₁	—	4	2	no	—
45	C₂×Q₈×C₃₁	—	4	2	no	—
46	C₂×Q₈×C₃₁	—	4	2	no	—
47	C₂×Q₈×C₃₁	—	4	2	no	—
48	C₂×Q₈×C₃₁	—	4	2	no	—
49	C₂²×C₄×C₃₁	—	4	2	no	—
50	C₂²×C₄×C₃₁	—	4	2	no	—
51	C₂²×C₄×C₃₁	D₈	1	2	yes	—
52	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
53	Q₁₆×C₃₁	—	2	4	no	—
54	Q₁₆×C₃₁	—	2	4	no	—
55	Q₁₆×C₃₁	—	4	2	no	—
56	Q₁₆×C₃₁	—	4	2	no	—
57	C₂×C₈×C₃₁	—	2	1	no	—
58	C₂×C₈×C₃₁	—	2	1	no	—
59	C₂×C₈×C₃₁	D₈	1	2	yes	—
60	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
62	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
63	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
64	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
65	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
66	C₄⋊C₄×C₃₁	—	4	2	no	—
67	C₄⋊C₄×C₃₁	—	4	2	no	—
68	C₄²×C₃₁	—	2	1	no	—
69	C₄²×C₃₁	—	2	1	no	—
70	C₁₆×C₃₁	—	2	2	no	—
71	C₁₆×C₃₁	C₂²⋊C₄	1	4	yes	—
72	#13236	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
73	C₂³×C₃₁	C₂×D₈	1	1	yes	characteristic
74	Q₈×C₃₁	—	4	4	no	—
75	Q₈×C₃₁	—	4	4	no	—
76	Q₈×C₃₁	—	8	2	no	—
77	Q₈×C₃₁	—	8	2	no	—
78	C₂×C₄×C₃₁	—	2	2	no	—
79	C₂×C₄×C₃₁	—	2	2	no	—
80	C₂×C₄×C₃₁	—	4	4	no	—
81	C₂×C₄×C₃₁	—	8	2	no	—
82	C₂×C₄×C₃₁	—	8	2	no	—
83	C₂×C₄×C₃₁	—	8	2	no	—
84	C₂×C₄×C₃₁	—	8	4	no	—
85	C₂×C₄×C₃₁	Q₃₂	1	4	yes	complemented
86	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
87	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
88	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
89	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
90	C₂×C₄×C₃₁	#135	1	1	yes	characteristic
91	C₂×C₄×C₃₁	#135	1	1	yes	characteristic
92	C₈×C₃₁	—	2	2	no	—
93	C₈×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₈×C₃₁	C₄×D₄	1	1	yes	characteristic
95	C₈×C₃₁	#121	1	1	yes	characteristic
96	C₈×C₃₁	#124	1	1	yes	characteristic
97	C₂²×Q₃₂	C₂×C₃₁	1	1	yes	characteristic
98	C₂×#366	C₂×C₃₁	1	1	yes	characteristic
99	C₂×#358	C₂×C₃₁	1	2	yes	—
100	C₂×C₄×C₁₆	C₂×C₃₁	1	1	yes	characteristic
101	#1969	C₂×C₃₁	1	2	yes	—
102	C₂²×C₃₁	C₄×D₈	1	1	yes	characteristic
103	C₂²×C₃₁	#466	1	1	yes	characteristic
104	C₂²×C₃₁	#493	1	1	yes	characteristic
105	C₂²×C₃₁	C₂×Q₃₂	1	2	yes	—
106	C₂²×C₃₁	#508	1	2	yes	—
107	C₄×C₃₁	—	2	4	no	—
108	C₄×C₃₁	—	8	4	no	—
109	C₄×C₃₁	—	16	2	no	—
110	C₄×C₃₁	#390	1	1	yes	characteristic
111	C₄×C₃₁	C₄×D₈	1	1	yes	characteristic
112	C₄×C₃₁	#466	1	1	yes	characteristic
113	C₄×C₃₁	#495	1	1	yes	characteristic
114	C₂²×Q₁₆	C₂²×C₃₁	1	2	yes	—
115	C₂×Q₃₂	—	2	2	no	—
116	C₂×Q₃₂	—	2	2	no	—
117	C₂×Q₃₂	C₄×C₃₁	1	4	yes	complemented
118	C₂²×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
119	C₂²×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
120	C₂×#107	C₂²×C₃₁	1	2	yes	—
121	C₂×#103	—	2	2	no	—
122	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
123	#391	—	2	2	no	—
124	#366	—	2	2	no	—
125	#358	—	2	4	no	—
126	C₄×C₁₆	—	2	2	no	—
127	C₂×C₃₁	#1970	1	1	yes	characteristic
128	C₂×C₃₁	C₄×Q₃₂	1	2	yes	—
129	C₂×C₃₁	#2216	1	2	yes	—
130	C₂×C₃₁	#2247	1	1	yes	characteristic
131	C₂×C₃₁	#2249	1	1	yes	characteristic
132	C₂²×Q₈	—	2	2	no	—
133	C₂×Q₁₆	—	2	2	no	—
134	C₂×Q₁₆	—	2	2	no	—
135	C₂×Q₁₆	—	2	2	no	—
136	C₂×Q₁₆	—	2	2	no	—
137	C₂×Q₁₆	—	2	2	no	—
138	C₂×Q₁₆	C₂×C₄×C₃₁	1	4	yes	—
139	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
140	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
141	C₂×C₄⋊C₄	—	2	2	no	—
142	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
143	Q₃₂	—	2	8	no	—
144	C₂×C₁₆	—	2	1	no	—
145	C₂×C₁₆	—	2	1	no	—
146	C₂×C₁₆	C₂×C₄×C₃₁	1	2	yes	—
147	C₂×C₁₆	D₄×C₃₁	1	2	yes	—
148	C₂×C₁₆	D₄×C₃₁	1	2	yes	—
149	C₂×C₁₆	D₄×C₃₁	1	2	yes	—
150	#107	—	2	2	no	—
151	#107	—	2	2	no	—
152	#103	—	4	4	no	—
153	C₄×C₈	—	2	1	no	—
154	C₄×C₈	—	2	1	no	—
155	#95	—	4	4	no	—
156	C₃₁	#13236	1	1	yes	characteristic, direct product factor, Sylow
157	C₂×Q₈	—	2	4	no	—
158	C₂×Q₈	—	4	2	no	—
159	C₂×Q₈	—	4	2	no	—
160	C₂×Q₈	—	4	2	no	—
161	C₂×Q₈	—	4	2	no	—
162	C₂×Q₈	—	4	2	no	—
163	C₂²×C₄	—	4	2	no	—
164	C₂²×C₄	—	4	2	no	—
165	C₂²×C₄	D₈×C₃₁	1	2	yes	—
166	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
167	Q₁₆	—	2	4	no	—
168	Q₁₆	—	2	4	no	—
169	Q₁₆	—	4	2	no	—
170	Q₁₆	—	4	2	no	—
171	C₂×C₈	—	2	1	no	—
172	C₂×C₈	—	2	1	no	—
173	C₂×C₈	D₈×C₃₁	1	2	yes	—
174	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
175	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
176	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
177	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
178	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
179	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
180	C₄⋊C₄	—	4	2	no	—
181	C₄⋊C₄	—	4	2	no	—
182	C₄²	—	2	1	no	—
183	C₄²	—	2	1	no	—
184	C₁₆	—	2	2	no	—
185	C₁₆	C₂²⋊C₄×C₃₁	1	4	yes	—
186	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
187	Q₈	—	4	4	no	—
188	Q₈	—	4	4	no	—
189	Q₈	—	8	2	no	—
190	Q₈	—	8	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	4	4	no	—
194	C₂×C₄	—	8	2	no	—
195	C₂×C₄	—	8	2	no	—
196	C₂×C₄	—	8	2	no	—
197	C₂×C₄	—	8	4	no	—
198	C₂×C₄	C₃₁×Q₃₂	1	4	yes	complemented
199	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
200	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
201	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
202	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
203	C₂×C₄	C₃₁×#135	1	1	yes	characteristic
204	C₂×C₄	C₃₁×#135	1	1	yes	characteristic
205	C₈	—	2	2	no	—
206	C₈	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
207	C₈	C₄×D₄×C₃₁	1	1	yes	characteristic
208	C₈	C₃₁×#121	1	1	yes	characteristic
209	C₈	C₃₁×#124	1	1	yes	characteristic
210	C₂²	C₄×D₈×C₃₁	1	1	yes	characteristic
211	C₂²	C₃₁×#466	1	1	yes	characteristic
212	C₂²	C₃₁×#493	1	1	yes	characteristic
213	C₂²	C₂×C₃₁×Q₃₂	1	2	yes	—
214	C₂²	C₃₁×#508	1	2	yes	—
215	C₄	—	2	4	no	—
216	C₄	—	8	4	no	—
217	C₄	—	16	2	no	—
218	C₄	C₃₁×#390	1	1	yes	characteristic
219	C₄	C₄×D₈×C₃₁	1	1	yes	characteristic
220	C₄	C₃₁×#466	1	1	yes	characteristic
221	C₄	C₃₁×#495	1	1	yes	characteristic
222	C₂	C₃₁×#1970	1	1	yes	characteristic
223	C₂	C₄×C₃₁×Q₃₂	1	2	yes	—
224	C₂	C₃₁×#2216	1	2	yes	—
225	C₂	C₃₁×#2247	1	1	yes	characteristic
226	C₂	C₃₁×#2249	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₃₁	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₃₁
Frattini, cosocle	C₂²×C₈	C₂³×C₃₁
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ C₂×C₄×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
9. C₂×C₃₁×Q₃₂	C₄	semidirect
72. #13236	C₃₁	direct (N × H)
85. C₂×C₄×C₃₁	Q₃₂	semidirect
117. C₂×Q₃₂	C₄×C₃₁	semidirect
156. C₃₁	#13236	direct (N × H)
198. C₂×C₄	C₃₁×Q₃₂	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃₁×Q₃₂	2
C₂×C₃₁×#366	2
C₂×C₃₁×#358	2
C₂×C₄×C₁₆×C₃₁	2
C₃₁×#1969	2
#13236	31

Subgroups — C31×#13236 #21469995

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#13236 #21469995