Subgroups — C₃₁×#21735 #21478494

Order 7936.

662 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3162	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2960	C₂	1	2	yes	maximal
3	C₃₁×#1995	C₂	1	2	yes	maximal
4	C₃₁×#1946	C₂	1	4	yes	maximal
5	C₃₁×#1849	C₂	1	1	yes	maximal, characteristic
6	C₃₁×#1772	C₂	1	4	yes	maximal
7	C₃₁×#1726	C₂	1	1	yes	maximal, characteristic
8	C₃₁×#533	C₂²	1	2	yes	—
9	C₃₁×#501	C₄	1	8	yes	complemented
10	C₃₁×#446	—	2	4	no	—
11	C₈×Q₈×C₃₁	—	2	4	no	—
12	C₂×C₃₁×#108	C₂²	1	4	yes	—
13	C₃₁×#427	C₂²	1	4	yes	—
14	C₃₁×#424	C₂²	1	4	yes	—
15	C₃₁×#423	C₂²	1	2	yes	—
16	C₃₁×#405	C₂²	1	4	yes	—
17	C₂×C₃₁×#97	C₂²	1	2	yes	—
18	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
21	C₃₁×#376	C₂²	1	1	yes	characteristic
22	C₃₁×#339	C₂²	1	8	yes	—
23	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
24	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
25	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
26	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
27	C₃₁×C₄⋊Q₈	C₂×C₄	1	4	yes	—
28	C₃₁×#125	C₂×C₄	1	4	yes	—
29	C₃₁×C₂²⋊Q₈	—	2	4	no	—
30	C₄×Q₈×C₃₁	—	2	4	no	—
31	C₃₁×#117	C₂³	1	4	yes	—
32	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
33	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
34	C₃₁×#108	—	2	8	no	—
35	C₃₁×#107	C₂×C₄	1	8	yes	—
36	C₃₁×#106	C₂×C₄	1	8	yes	—
37	C₃₁×C₄⋊C₈	—	2	8	no	—
38	C₃₁×C₄⋊C₈	—	2	4	no	—
39	C₃₁×#97	—	2	4	no	—
40	C₃₁×#97	—	2	4	no	—
41	C₃₁×#97	C₂×C₄	1	4	yes	—
42	C₄×C₈×C₃₁	—	2	4	no	—
43	C₄×C₈×C₃₁	D₄	1	2	yes	—
44	C₄×C₈×C₃₁	D₄	1	2	yes	—
45	C₄×C₈×C₃₁	Q₈	1	8	yes	complemented
46	C₃₁×#95	—	2	2	no	—
47	C₂×Q₈×C₃₁	—	2	4	no	—
48	C₂²×C₄×C₃₁	—	2	2	no	—
49	C₂²×C₄×C₃₁	#41	1	2	yes	—
50	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
51	M₁₆×C₃₁	—	2	8	no	—
52	C₂×C₈×C₃₁	—	2	2	no	—
53	C₂×C₈×C₃₁	—	2	4	no	—
54	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
55	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
56	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
57	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
58	C₂×C₈×C₃₁	C₂×Q₈	1	4	yes	—
59	C₂×C₈×C₃₁	C₂×Q₈	1	4	yes	—
60	C₂×C₈×C₃₁	#41	1	4	yes	—
61	C₄⋊C₄×C₃₁	—	2	8	no	—
62	C₄⋊C₄×C₃₁	—	2	4	no	—
63	C₄⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
64	C₂²⋊C₄×C₃₁	—	2	4	no	—
65	C₄²×C₃₁	—	2	4	no	—
66	C₄²×C₃₁	—	2	2	no	—
67	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
68	C₄²×C₃₁	C₂×D₄	1	2	yes	—
69	C₄²×C₃₁	C₂×Q₈	1	4	yes	—
70	#21735	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
71	C₂³×C₃₁	C₂×#41	1	1	yes	characteristic
72	Q₈×C₃₁	—	4	8	no	—
73	C₂×C₄×C₃₁	—	2	8	no	—
74	C₂×C₄×C₃₁	—	2	4	no	—
75	C₂×C₄×C₃₁	—	4	2	no	—
76	C₂×C₄×C₃₁	—	4	4	no	—
77	C₂×C₄×C₃₁	C₄×D₄	1	4	yes	—
78	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
79	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
80	C₂×C₄×C₃₁	C₂²×Q₈	1	2	yes	—
81	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
82	C₈×C₃₁	—	2	2	no	—
83	C₈×C₃₁	—	4	4	no	—
84	C₈×C₃₁	C₄×Q₈	1	8	yes	—
85	C₈×C₃₁	C₄⋊Q₈	1	2	yes	—
86	C₈×C₃₁	C₄⋊Q₈	1	2	yes	—
87	#3162	C₂×C₃₁	1	1	yes	characteristic
88	#2960	C₂×C₃₁	1	2	yes	—
89	#1995	C₂×C₃₁	1	2	yes	—
90	#1946	C₂×C₃₁	1	4	yes	—
91	#1849	C₂×C₃₁	1	1	yes	characteristic
92	#1772	C₂×C₃₁	1	4	yes	—
93	#1726	C₂×C₃₁	1	1	yes	characteristic
94	C₂²×C₃₁	—	2	2	no	—
95	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
96	C₂²×C₃₁	#533	1	2	yes	—
97	C₄×C₃₁	—	2	4	no	—
98	C₄×C₃₁	—	4	4	no	—
99	C₄×C₃₁	—	4	8	no	—
100	C₄×C₃₁	C₂×C₄×Q₈	1	2	yes	—
101	C₄×C₃₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
102	C₄×C₃₁	#532	1	1	yes	characteristic
103	#533	C₂²×C₃₁	1	2	yes	—
104	#501	C₄×C₃₁	1	8	yes	complemented
105	#446	—	2	4	no	—
106	C₈×Q₈	—	2	4	no	—
107	C₂×#108	C₂²×C₃₁	1	4	yes	—
108	#427	C₂²×C₃₁	1	4	yes	—
109	#424	C₂²×C₃₁	1	4	yes	—
110	#423	C₂²×C₃₁	1	2	yes	—
111	#405	C₂²×C₃₁	1	4	yes	—
112	C₂×#97	C₂²×C₃₁	1	2	yes	—
113	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
114	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
115	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
116	#376	C₂²×C₃₁	1	1	yes	characteristic
117	#339	C₂²×C₃₁	1	8	yes	—
118	C₂×C₃₁	—	2	2	no	—
119	C₂×C₃₁	C₄×C₄⋊Q₈	1	1	yes	characteristic
120	C₂×C₃₁	#2955	1	2	yes	—
121	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
122	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
123	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
124	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
125	C₄⋊Q₈	C₂×C₄×C₃₁	1	4	yes	—
126	#125	C₂×C₄×C₃₁	1	4	yes	—
127	C₂²⋊Q₈	—	2	4	no	—
128	C₄×Q₈	—	2	4	no	—
129	#117	C₂³×C₃₁	1	4	yes	—
130	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
131	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
132	#108	—	2	8	no	—
133	#107	C₂×C₄×C₃₁	1	8	yes	—
134	#106	C₂×C₄×C₃₁	1	8	yes	—
135	C₄⋊C₈	—	2	8	no	—
136	C₄⋊C₈	—	2	4	no	—
137	#97	—	2	4	no	—
138	#97	—	2	4	no	—
139	#97	C₂×C₄×C₃₁	1	4	yes	—
140	C₄×C₈	—	2	4	no	—
141	C₄×C₈	D₄×C₃₁	1	2	yes	—
142	C₄×C₈	D₄×C₃₁	1	2	yes	—
143	C₄×C₈	Q₈×C₃₁	1	8	yes	complemented
144	#95	—	2	2	no	—
145	C₃₁	#21735	1	1	yes	characteristic, direct product factor, Sylow
146	C₂×Q₈	—	2	4	no	—
147	C₂²×C₄	—	2	2	no	—
148	C₂²×C₄	#41×C₃₁	1	2	yes	—
149	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
150	M₁₆	—	2	8	no	—
151	C₂×C₈	—	2	2	no	—
152	C₂×C₈	—	2	4	no	—
153	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
154	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
155	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
156	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
157	C₂×C₈	C₂×Q₈×C₃₁	1	4	yes	—
158	C₂×C₈	C₂×Q₈×C₃₁	1	4	yes	—
159	C₂×C₈	#41×C₃₁	1	4	yes	—
160	C₄⋊C₄	—	2	8	no	—
161	C₄⋊C₄	—	2	4	no	—
162	C₄⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
163	C₂²⋊C₄	—	2	4	no	—
164	C₄²	—	2	4	no	—
165	C₄²	—	2	2	no	—
166	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
167	C₄²	C₂×D₄×C₃₁	1	2	yes	—
168	C₄²	C₂×Q₈×C₃₁	1	4	yes	—
169	C₂³	C₂×#41×C₃₁	1	1	yes	characteristic
170	Q₈	—	4	8	no	—
171	C₂×C₄	—	2	8	no	—
172	C₂×C₄	—	2	4	no	—
173	C₂×C₄	—	4	2	no	—
174	C₂×C₄	—	4	4	no	—
175	C₂×C₄	C₄×D₄×C₃₁	1	4	yes	—
176	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
177	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
178	C₂×C₄	C₂²×Q₈×C₃₁	1	2	yes	—
179	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
180	C₈	—	2	2	no	—
181	C₈	—	4	4	no	—
182	C₈	C₄×Q₈×C₃₁	1	8	yes	—
183	C₈	C₃₁×C₄⋊Q₈	1	2	yes	—
184	C₈	C₃₁×C₄⋊Q₈	1	2	yes	—
185	C₂²	—	2	2	no	—
186	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
187	C₂²	C₃₁×#533	1	2	yes	—
188	C₄	—	2	4	no	—
189	C₄	—	4	4	no	—
190	C₄	—	4	8	no	—
191	C₄	C₂×C₄×Q₈×C₃₁	1	2	yes	—
192	C₄	C₂×C₃₁×C₄⋊Q₈	1	1	yes	characteristic
193	C₄	C₃₁×#532	1	1	yes	characteristic
194	C₂	—	2	2	no	—
195	C₂	C₄×C₃₁×C₄⋊Q₈	1	1	yes	characteristic
196	C₂	C₃₁×#2955	1	2	yes	—

	subgroup	G/H
center	C₂×C₈×C₃₁	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₂×C₄×D₄
prime radical	C₂³×C₃₁	C₂×#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₃₁ ◁ C₂×C₄×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
9. C₃₁×#501	C₄	semidirect
45. C₄×C₈×C₃₁	Q₈	semidirect
70. #21735	C₃₁	direct (N × H)
104. #501	C₄×C₃₁	semidirect
143. C₄×C₈	Q₈×C₃₁	semidirect
145. C₃₁	#21735	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃₁×#3162	2
C₃₁×#2960	2
C₃₁×#1995	2
C₃₁×#1946	2
C₃₁×#1849	2
C₃₁×#1772	2
C₃₁×#1726	2
#21735	31

Subgroups — C31×#21735 #21478494

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#21735 #21478494