Subgroups — C₃₁×#22253 #21479012

Order 7936.

674 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#498	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2331	C₂	1	2	yes	maximal
3	C₃₁×#1993	C₂	1	2	yes	maximal
4	C₃₁×#1920	C₂	1	4	yes	maximal
5	C₃₁×#1919	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×#498	C₄	1	8	yes	complemented
10	C₂×C₃₁×#106	C₂²	1	4	yes	—
11	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
13	C₃₁×#391	—	2	4	no	—
14	C₃₁×#389	C₂²	1	4	yes	—
15	C₃₁×#387	—	2	4	no	—
16	C₃₁×#382	C₂²	1	2	yes	—
17	C₄×C₄⋊C₄×C₃₁	C₂²	1	4	yes	—
18	C₃₁×#376	C₂²	1	1	yes	characteristic
19	C₃₁×#340	C₂²	1	4	yes	—
20	C₃₁×#340	C₂²	1	4	yes	—
21	C₃₁×#340	C₂²	1	4	yes	—
22	C₃₁×#336	C₂²	1	2	yes	—
23	C₃₁×#336	C₂²	1	2	yes	—
24	C₂²×Q₈×C₃₁	—	2	2	no	—
25	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
26	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
27	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
28	C₃₁×C₄⋊Q₈	—	2	4	no	—
29	C₃₁×C₄⋊Q₈	C₂×C₄	1	8	yes	—
30	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
31	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
32	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
33	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
34	C₂×C₄²×C₃₁	—	2	4	no	—
35	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
36	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
37	C₃₁×#106	C₂×C₄	1	16	yes	—
38	C₃₁×#97	—	2	2	no	—
39	C₄×C₈×C₃₁	C₂×C₄	1	4	yes	—
40	C₃₁×#95	—	2	4	no	—
41	C₃₁×#95	—	2	2	no	—
42	C₃₁×#95	—	2	2	no	—
43	C₃₁×#95	—	2	4	no	—
44	C₃₁×#95	D₄	1	4	yes	—
45	C₂×Q₈×C₃₁	—	2	8	no	—
46	C₂×Q₈×C₃₁	—	8	4	no	—
47	C₂²×C₄×C₃₁	—	2	4	no	—
48	C₂²×C₄×C₃₁	—	2	4	no	—
49	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
50	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
51	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
52	C₂×C₈×C₃₁	—	2	2	no	—
53	C₂×C₈×C₃₁	—	2	2	no	—
54	C₂×C₈×C₃₁	—	2	1	no	—
55	C₂×C₈×C₃₁	—	2	1	no	—
56	C₂×C₈×C₃₁	—	2	2	no	—
57	C₂×C₈×C₃₁	—	2	2	no	—
58	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
59	C₄⋊C₄×C₃₁	—	2	8	no	—
60	C₄⋊C₄×C₃₁	—	2	4	no	—
61	C₄⋊C₄×C₃₁	—	2	4	no	—
62	C₄⋊C₄×C₃₁	—	2	4	no	—
63	C₄⋊C₄×C₃₁	—	8	4	no	—
64	C₄⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
65	C₄²×C₃₁	—	2	2	no	—
66	C₄²×C₃₁	—	2	2	no	—
67	C₄²×C₃₁	—	8	4	no	—
68	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
69	C₄²×C₃₁	#41	1	2	yes	—
70	#22253	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
71	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
72	Q₈×C₃₁	—	8	8	no	—
73	C₂×C₄×C₃₁	—	2	4	no	—
74	C₂×C₄×C₃₁	—	2	8	no	—
75	C₂×C₄×C₃₁	—	2	4	no	—
76	C₂×C₄×C₃₁	—	2	4	no	—
77	C₂×C₄×C₃₁	—	2	4	no	—
78	C₂×C₄×C₃₁	—	4	4	no	—
79	C₂×C₄×C₃₁	—	4	4	no	—
80	C₂×C₄×C₃₁	—	4	4	no	—
81	C₂×C₄×C₃₁	—	4	4	no	—
82	C₂×C₄×C₃₁	—	4	4	no	—
83	C₂×C₄×C₃₁	C₄×D₄	1	4	yes	—
84	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
85	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
86	C₂×C₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
87	C₂×C₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
88	C₈×C₃₁	—	2	4	no	—
89	C₈×C₃₁	—	4	1	no	—
90	C₈×C₃₁	—	4	1	no	—
91	C₂×#498	C₂×C₃₁	1	1	yes	characteristic
92	#2331	C₂×C₃₁	1	2	yes	—
93	#1993	C₂×C₃₁	1	2	yes	—
94	#1920	C₂×C₃₁	1	4	yes	—
95	#1919	C₂×C₃₁	1	4	yes	—
96	#1835	C₂×C₃₁	1	1	yes	characteristic
97	#1746	C₂×C₃₁	1	1	yes	characteristic
98	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
99	C₂²×C₃₁	#536	1	2	yes	—
100	C₂²×C₃₁	#577	1	4	yes	—
101	C₄×C₃₁	—	2	4	no	—
102	C₄×C₃₁	—	4	8	no	—
103	C₄×C₃₁	—	4	4	no	—
104	C₄×C₃₁	—	8	4	no	—
105	C₄×C₃₁	#519	1	2	yes	—
106	C₄×C₃₁	#541	1	1	yes	characteristic
107	C₄×C₃₁	#541	1	1	yes	characteristic
108	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
109	#498	C₄×C₃₁	1	8	yes	complemented
110	C₂×#106	C₂²×C₃₁	1	4	yes	—
111	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
112	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
113	#391	—	2	4	no	—
114	#389	C₂²×C₃₁	1	4	yes	—
115	#387	—	2	4	no	—
116	#382	C₂²×C₃₁	1	2	yes	—
117	C₄×C₄⋊C₄	C₂²×C₃₁	1	4	yes	—
118	#376	C₂²×C₃₁	1	1	yes	characteristic
119	#340	C₂²×C₃₁	1	4	yes	—
120	#340	C₂²×C₃₁	1	4	yes	—
121	#340	C₂²×C₃₁	1	4	yes	—
122	#336	C₂²×C₃₁	1	2	yes	—
123	#336	C₂²×C₃₁	1	2	yes	—
124	C₂×C₃₁	#2327	1	1	yes	characteristic
125	C₂×C₃₁	#2949	1	2	yes	—
126	C₂×C₃₁	#3244	1	4	yes	—
127	C₂²×Q₈	—	2	2	no	—
128	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
129	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
130	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
131	C₄⋊Q₈	—	2	4	no	—
132	C₄⋊Q₈	C₂×C₄×C₃₁	1	8	yes	—
133	C₂×C₄⋊C₄	—	2	2	no	—
134	C₂×C₄⋊C₄	—	2	4	no	—
135	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
136	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
137	C₂×C₄²	—	2	4	no	—
138	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
139	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
140	#106	C₂×C₄×C₃₁	1	16	yes	—
141	#97	—	2	2	no	—
142	C₄×C₈	C₂×C₄×C₃₁	1	4	yes	—
143	#95	—	2	4	no	—
144	#95	—	2	2	no	—
145	#95	—	2	2	no	—
146	#95	—	2	4	no	—
147	#95	D₄×C₃₁	1	4	yes	—
148	C₃₁	#22253	1	1	yes	characteristic, direct product factor, Sylow
149	C₂×Q₈	—	2	8	no	—
150	C₂×Q₈	—	8	4	no	—
151	C₂²×C₄	—	2	4	no	—
152	C₂²×C₄	—	2	4	no	—
153	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
154	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
155	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
156	C₂×C₈	—	2	2	no	—
157	C₂×C₈	—	2	2	no	—
158	C₂×C₈	—	2	1	no	—
159	C₂×C₈	—	2	1	no	—
160	C₂×C₈	—	2	2	no	—
161	C₂×C₈	—	2	2	no	—
162	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
163	C₄⋊C₄	—	2	8	no	—
164	C₄⋊C₄	—	2	4	no	—
165	C₄⋊C₄	—	2	4	no	—
166	C₄⋊C₄	—	2	4	no	—
167	C₄⋊C₄	—	8	4	no	—
168	C₄⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
169	C₄²	—	2	2	no	—
170	C₄²	—	2	2	no	—
171	C₄²	—	8	4	no	—
172	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
173	C₄²	#41×C₃₁	1	2	yes	—
174	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
175	Q₈	—	8	8	no	—
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	—	2	8	no	—
178	C₂×C₄	—	2	4	no	—
179	C₂×C₄	—	2	4	no	—
180	C₂×C₄	—	2	4	no	—
181	C₂×C₄	—	4	4	no	—
182	C₂×C₄	—	4	4	no	—
183	C₂×C₄	—	4	4	no	—
184	C₂×C₄	—	4	4	no	—
185	C₂×C₄	—	4	4	no	—
186	C₂×C₄	C₄×D₄×C₃₁	1	4	yes	—
187	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
188	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
189	C₂×C₄	C₃₁×2₋¹⁺⁴	1	2	yes	—
190	C₂×C₄	C₃₁×2₋¹⁺⁴	1	2	yes	—
191	C₈	—	2	4	no	—
192	C₈	—	4	1	no	—
193	C₈	—	4	1	no	—
194	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
195	C₂²	C₃₁×#536	1	2	yes	—
196	C₂²	C₃₁×#577	1	4	yes	—
197	C₄	—	2	4	no	—
198	C₄	—	4	8	no	—
199	C₄	—	4	4	no	—
200	C₄	—	8	4	no	—
201	C₄	C₃₁×#519	1	2	yes	—
202	C₄	C₃₁×#541	1	1	yes	characteristic
203	C₄	C₃₁×#541	1	1	yes	characteristic
204	C₂	C₃₁×#2327	1	1	yes	characteristic
205	C₂	C₃₁×#2949	1	2	yes	—
206	C₂	C₃₁×#3244	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×#498	C₄	semidirect
70. #22253	C₃₁	direct (N × H)
109. #498	C₄×C₃₁	semidirect
148. C₃₁	#22253	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×#498	2
C₃₁×#2331	2
C₃₁×#1993	2
C₃₁×#1920	2
C₃₁×#1919	2
C₃₁×#1835	2
C₃₁×#1746	2
#22253	31

Subgroups — C31×#22253 #21479012

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22253 #21479012