Subgroups — C₃₁×#22254 #21479013

Order 7936.

674 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#500	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2331	C₂	1	2	yes	maximal
3	C₃₁×#1993	C₂	1	2	yes	maximal
4	C₃₁×#1921	C₂	1	4	yes	maximal
5	C₃₁×#1918	C₂	1	4	yes	maximal
6	C₃₁×#1835	C₂	1	1	yes	maximal, characteristic
7	C₃₁×#1746	C₂	1	1	yes	maximal, characteristic
8	C₂×C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
9	C₃₁×#500	C₄	1	8	yes	complemented
10	C₂×C₃₁×#107	C₂²	1	2	yes	—
11	C₂×C₃₁×#107	C₂²	1	2	yes	—
12	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
14	C₃₁×#391	—	2	4	no	—
15	C₃₁×#389	C₂²	1	4	yes	—
16	C₃₁×#387	—	2	4	no	—
17	C₃₁×#382	C₂²	1	2	yes	—
18	C₄×C₄⋊C₄×C₃₁	C₂²	1	4	yes	—
19	C₃₁×#376	C₂²	1	1	yes	characteristic
20	C₃₁×#340	C₂²	1	2	yes	—
21	C₃₁×#340	C₂²	1	2	yes	—
22	C₃₁×#340	C₂²	1	8	yes	—
23	C₃₁×#336	C₂²	1	2	yes	—
24	C₃₁×#336	C₂²	1	2	yes	—
25	C₂²×Q₈×C₃₁	—	2	2	no	—
26	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
27	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
28	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
29	C₃₁×C₄⋊Q₈	—	2	4	no	—
30	C₃₁×C₄⋊Q₈	C₂×C₄	1	8	yes	—
31	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
32	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
33	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
34	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
35	C₂×C₄²×C₃₁	—	2	4	no	—
36	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
37	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
38	C₃₁×#107	C₂×C₄	1	8	yes	—
39	C₃₁×#107	C₂×C₄	1	8	yes	—
40	C₃₁×#97	—	2	2	no	—
41	C₄×C₈×C₃₁	C₂×C₄	1	4	yes	—
42	C₃₁×#95	—	2	4	no	—
43	C₃₁×#95	—	2	2	no	—
44	C₃₁×#95	—	2	2	no	—
45	C₃₁×#95	—	2	4	no	—
46	C₃₁×#95	D₄	1	2	yes	—
47	C₃₁×#95	D₄	1	2	yes	—
48	C₂×Q₈×C₃₁	—	2	8	no	—
49	C₂×Q₈×C₃₁	—	8	4	no	—
50	C₂²×C₄×C₃₁	—	2	4	no	—
51	C₂²×C₄×C₃₁	—	2	4	no	—
52	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
53	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
56	C₂×C₈×C₃₁	—	2	2	no	—
57	C₂×C₈×C₃₁	—	2	2	no	—
58	C₂×C₈×C₃₁	—	2	2	no	—
59	C₂×C₈×C₃₁	—	2	2	no	—
60	C₂×C₈×C₃₁	—	2	2	no	—
61	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
62	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
63	C₄⋊C₄×C₃₁	—	2	8	no	—
64	C₄⋊C₄×C₃₁	—	2	4	no	—
65	C₄⋊C₄×C₃₁	—	2	4	no	—
66	C₄⋊C₄×C₃₁	—	2	4	no	—
67	C₄⋊C₄×C₃₁	—	8	4	no	—
68	C₄⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
69	C₄²×C₃₁	—	2	2	no	—
70	C₄²×C₃₁	—	2	2	no	—
71	C₄²×C₃₁	—	8	4	no	—
72	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
73	C₄²×C₃₁	#41	1	2	yes	—
74	#22254	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
75	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
76	Q₈×C₃₁	—	8	8	no	—
77	C₂×C₄×C₃₁	—	2	2	no	—
78	C₂×C₄×C₃₁	—	2	2	no	—
79	C₂×C₄×C₃₁	—	2	8	no	—
80	C₂×C₄×C₃₁	—	2	4	no	—
81	C₂×C₄×C₃₁	—	2	4	no	—
82	C₂×C₄×C₃₁	—	2	4	no	—
83	C₂×C₄×C₃₁	—	4	4	no	—
84	C₂×C₄×C₃₁	—	4	4	no	—
85	C₂×C₄×C₃₁	—	4	4	no	—
86	C₂×C₄×C₃₁	—	4	4	no	—
87	C₂×C₄×C₃₁	—	4	4	no	—
88	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
89	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
90	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
91	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
92	C₂×C₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
93	C₂×C₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
94	C₈×C₃₁	—	2	2	no	—
95	C₈×C₃₁	—	2	2	no	—
96	C₈×C₃₁	—	4	2	no	—
97	C₂×#500	C₂×C₃₁	1	1	yes	characteristic
98	#2331	C₂×C₃₁	1	2	yes	—
99	#1993	C₂×C₃₁	1	2	yes	—
100	#1921	C₂×C₃₁	1	4	yes	—
101	#1918	C₂×C₃₁	1	4	yes	—
102	#1835	C₂×C₃₁	1	1	yes	characteristic
103	#1746	C₂×C₃₁	1	1	yes	characteristic
104	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
105	C₂²×C₃₁	#536	1	2	yes	—
106	C₂²×C₃₁	#576	1	2	yes	—
107	C₂²×C₃₁	#578	1	2	yes	—
108	C₄×C₃₁	—	2	2	no	—
109	C₄×C₃₁	—	2	2	no	—
110	C₄×C₃₁	—	4	8	no	—
111	C₄×C₃₁	—	4	4	no	—
112	C₄×C₃₁	—	8	4	no	—
113	C₄×C₃₁	#519	1	1	yes	characteristic
114	C₄×C₃₁	#519	1	1	yes	characteristic
115	C₄×C₃₁	#541	1	2	yes	—
116	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
117	#500	C₄×C₃₁	1	8	yes	complemented
118	C₂×#107	C₂²×C₃₁	1	2	yes	—
119	C₂×#107	C₂²×C₃₁	1	2	yes	—
120	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
121	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
122	#391	—	2	4	no	—
123	#389	C₂²×C₃₁	1	4	yes	—
124	#387	—	2	4	no	—
125	#382	C₂²×C₃₁	1	2	yes	—
126	C₄×C₄⋊C₄	C₂²×C₃₁	1	4	yes	—
127	#376	C₂²×C₃₁	1	1	yes	characteristic
128	#340	C₂²×C₃₁	1	2	yes	—
129	#340	C₂²×C₃₁	1	2	yes	—
130	#340	C₂²×C₃₁	1	8	yes	—
131	#336	C₂²×C₃₁	1	2	yes	—
132	#336	C₂²×C₃₁	1	2	yes	—
133	C₂×C₃₁	#2327	1	1	yes	characteristic
134	C₂×C₃₁	#2948	1	1	yes	characteristic
135	C₂×C₃₁	#2950	1	1	yes	characteristic
136	C₂×C₃₁	#3243	1	4	yes	—
137	C₂²×Q₈	—	2	2	no	—
138	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
139	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
140	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
141	C₄⋊Q₈	—	2	4	no	—
142	C₄⋊Q₈	C₂×C₄×C₃₁	1	8	yes	—
143	C₂×C₄⋊C₄	—	2	2	no	—
144	C₂×C₄⋊C₄	—	2	4	no	—
145	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
146	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
147	C₂×C₄²	—	2	4	no	—
148	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
149	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
150	#107	C₂×C₄×C₃₁	1	8	yes	—
151	#107	C₂×C₄×C₃₁	1	8	yes	—
152	#97	—	2	2	no	—
153	C₄×C₈	C₂×C₄×C₃₁	1	4	yes	—
154	#95	—	2	4	no	—
155	#95	—	2	2	no	—
156	#95	—	2	2	no	—
157	#95	—	2	4	no	—
158	#95	D₄×C₃₁	1	2	yes	—
159	#95	D₄×C₃₁	1	2	yes	—
160	C₃₁	#22254	1	1	yes	characteristic, direct product factor, Sylow
161	C₂×Q₈	—	2	8	no	—
162	C₂×Q₈	—	8	4	no	—
163	C₂²×C₄	—	2	4	no	—
164	C₂²×C₄	—	2	4	no	—
165	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
166	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
167	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
168	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	—	2	2	no	—
171	C₂×C₈	—	2	2	no	—
172	C₂×C₈	—	2	2	no	—
173	C₂×C₈	—	2	2	no	—
174	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
175	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
176	C₄⋊C₄	—	2	8	no	—
177	C₄⋊C₄	—	2	4	no	—
178	C₄⋊C₄	—	2	4	no	—
179	C₄⋊C₄	—	2	4	no	—
180	C₄⋊C₄	—	8	4	no	—
181	C₄⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
182	C₄²	—	2	2	no	—
183	C₄²	—	2	2	no	—
184	C₄²	—	8	4	no	—
185	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
186	C₄²	#41×C₃₁	1	2	yes	—
187	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
188	Q₈	—	8	8	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	8	no	—
192	C₂×C₄	—	2	4	no	—
193	C₂×C₄	—	2	4	no	—
194	C₂×C₄	—	2	4	no	—
195	C₂×C₄	—	4	4	no	—
196	C₂×C₄	—	4	4	no	—
197	C₂×C₄	—	4	4	no	—
198	C₂×C₄	—	4	4	no	—
199	C₂×C₄	—	4	4	no	—
200	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
201	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
202	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
203	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
204	C₂×C₄	C₃₁×2₋¹⁺⁴	1	2	yes	—
205	C₂×C₄	C₃₁×2₋¹⁺⁴	1	2	yes	—
206	C₈	—	2	2	no	—
207	C₈	—	2	2	no	—
208	C₈	—	4	2	no	—
209	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
210	C₂²	C₃₁×#536	1	2	yes	—
211	C₂²	C₃₁×#576	1	2	yes	—
212	C₂²	C₃₁×#578	1	2	yes	—
213	C₄	—	2	2	no	—
214	C₄	—	2	2	no	—
215	C₄	—	4	8	no	—
216	C₄	—	4	4	no	—
217	C₄	—	8	4	no	—
218	C₄	C₃₁×#519	1	1	yes	characteristic
219	C₄	C₃₁×#519	1	1	yes	characteristic
220	C₄	C₃₁×#541	1	2	yes	—
221	C₂	C₃₁×#2327	1	1	yes	characteristic
222	C₂	C₃₁×#2948	1	1	yes	characteristic
223	C₂	C₃₁×#2950	1	1	yes	characteristic
224	C₂	C₃₁×#3243	1	4	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₃₁×#376 ◁ G

Complements

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

N	H ≅ G/N	type
9. C₃₁×#500	C₄	semidirect
74. #22254	C₃₁	direct (N × H)
117. #500	C₄×C₃₁	semidirect
160. C₃₁	#22254	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#500	2
C₃₁×#2331	2
C₃₁×#1993	2
C₃₁×#1921	2
C₃₁×#1918	2
C₃₁×#1835	2
C₃₁×#1746	2
#22254	31

Subgroups — C31×#22254 #21479013

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22254 #21479013