Subgroups — C₃₁×#21734 #21478493

Order 7936.

678 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3160	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2960	C₂	1	2	yes	maximal
3	C₃₁×#1994	C₂	1	2	yes	maximal
4	C₃₁×#1946	C₂	1	4	yes	maximal
5	C₃₁×#1848	C₂	1	1	yes	maximal, characteristic
6	C₃₁×#1773	C₂	1	4	yes	maximal
7	C₄×C₃₁×#97	C₂	1	1	yes	maximal, characteristic
8	C₃₁×#533	C₂²	1	2	yes	—
9	C₃₁×#500	C₄	1	2	yes	complemented
10	C₃₁×#499	C₄	1	4	yes	complemented
11	C₃₁×#498	C₄	1	2	yes	complemented
12	C₃₁×#446	—	2	4	no	—
13	C₈×Q₈×C₃₁	—	2	4	no	—
14	C₂×C₃₁×#108	C₂²	1	4	yes	—
15	C₃₁×#427	C₂²	1	4	yes	—
16	C₃₁×#424	C₂²	1	4	yes	—
17	C₃₁×#423	C₂²	1	2	yes	—
18	C₃₁×#405	C₂²	1	4	yes	—
19	C₂×C₃₁×#97	C₂²	1	2	yes	—
20	C₂×C₃₁×#97	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
23	C₄³×C₃₁	C₂²	1	1	yes	characteristic
24	C₃₁×#339	C₂²	1	8	yes	—
25	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
26	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
27	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
28	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
29	C₃₁×C₄⋊Q₈	C₂×C₄	1	4	yes	—
30	C₃₁×#125	C₂×C₄	1	4	yes	—
31	C₃₁×C₂²⋊Q₈	—	2	4	no	—
32	C₄×Q₈×C₃₁	—	2	4	no	—
33	C₃₁×#117	C₂³	1	4	yes	—
34	C₂×C₄²×C₃₁	—	2	2	no	—
35	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
36	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
37	C₃₁×#108	—	2	8	no	—
38	C₃₁×#107	C₂×C₄	1	8	yes	—
39	C₃₁×#106	C₂×C₄	1	8	yes	—
40	C₃₁×C₄⋊C₈	—	2	8	no	—
41	C₃₁×C₄⋊C₈	—	2	4	no	—
42	C₃₁×#97	—	2	4	no	—
43	C₃₁×#97	—	2	4	no	—
44	C₃₁×#97	Q₈	1	8	yes	complemented
45	C₄×C₈×C₃₁	—	2	4	no	—
46	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
47	C₄×C₈×C₃₁	C₂×C₄	1	2	yes	—
48	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
49	C₄×C₈×C₃₁	D₄	1	4	yes	—
50	C₂×Q₈×C₃₁	—	2	4	no	—
51	C₂²×C₄×C₃₁	—	2	2	no	—
52	C₂²×C₄×C₃₁	#41	1	2	yes	—
53	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
54	M₁₆×C₃₁	—	2	8	no	—
55	C₂×C₈×C₃₁	—	2	2	no	—
56	C₂×C₈×C₃₁	—	2	4	no	—
57	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
58	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
59	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
60	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
61	C₂×C₈×C₃₁	C₂×Q₈	1	4	yes	—
62	C₂×C₈×C₃₁	C₂×Q₈	1	4	yes	—
63	C₂×C₈×C₃₁	#41	1	4	yes	—
64	C₄⋊C₄×C₃₁	—	2	8	no	—
65	C₄⋊C₄×C₃₁	—	2	4	no	—
66	C₄⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
67	C₂²⋊C₄×C₃₁	—	2	4	no	—
68	C₄²×C₃₁	—	2	4	no	—
69	C₄²×C₃₁	—	2	2	no	—
70	C₄²×C₃₁	—	4	4	no	—
71	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
72	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
73	C₄²×C₃₁	C₂×D₄	1	2	yes	—
74	C₄²×C₃₁	C₂×Q₈	1	4	yes	—
75	#21734	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
76	C₂³×C₃₁	C₂×#41	1	1	yes	characteristic
77	Q₈×C₃₁	—	4	8	no	—
78	C₂×C₄×C₃₁	—	2	4	no	—
79	C₂×C₄×C₃₁	—	2	8	no	—
80	C₂×C₄×C₃₁	—	2	4	no	—
81	C₂×C₄×C₃₁	—	4	4	no	—
82	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
83	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
84	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
85	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
86	C₂×C₄×C₃₁	C₂²×Q₈	1	2	yes	—
87	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
88	C₈×C₃₁	—	2	2	no	—
89	C₈×C₃₁	—	4	4	no	—
90	C₈×C₃₁	C₄×Q₈	1	4	yes	—
91	C₈×C₃₁	C₄×Q₈	1	4	yes	—
92	C₈×C₃₁	C₄⋊Q₈	1	4	yes	—
93	#3160	C₂×C₃₁	1	1	yes	characteristic
94	#2960	C₂×C₃₁	1	2	yes	—
95	#1994	C₂×C₃₁	1	2	yes	—
96	#1946	C₂×C₃₁	1	4	yes	—
97	#1848	C₂×C₃₁	1	1	yes	characteristic
98	#1773	C₂×C₃₁	1	4	yes	—
99	C₄×#97	C₂×C₃₁	1	1	yes	characteristic
100	C₂²×C₃₁	—	2	2	no	—
101	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
102	C₂²×C₃₁	#533	1	2	yes	—
103	C₄×C₃₁	—	2	2	no	—
104	C₄×C₃₁	—	4	4	no	—
105	C₄×C₃₁	—	4	8	no	—
106	C₄×C₃₁	#444	1	4	yes	—
107	C₄×C₃₁	C₂×C₄×Q₈	1	2	yes	—
108	C₄×C₃₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
109	C₄×C₃₁	#532	1	1	yes	characteristic
110	#533	C₂²×C₃₁	1	2	yes	—
111	#500	C₄×C₃₁	1	2	yes	complemented
112	#499	C₄×C₃₁	1	4	yes	complemented
113	#498	C₄×C₃₁	1	2	yes	complemented
114	#446	—	2	4	no	—
115	C₈×Q₈	—	2	4	no	—
116	C₂×#108	C₂²×C₃₁	1	4	yes	—
117	#427	C₂²×C₃₁	1	4	yes	—
118	#424	C₂²×C₃₁	1	4	yes	—
119	#423	C₂²×C₃₁	1	2	yes	—
120	#405	C₂²×C₃₁	1	4	yes	—
121	C₂×#97	C₂²×C₃₁	1	2	yes	—
122	C₂×#97	C₂²×C₃₁	1	2	yes	—
123	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
124	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
125	C₄³	C₂²×C₃₁	1	1	yes	characteristic
126	#339	C₂²×C₃₁	1	8	yes	—
127	C₂×C₃₁	—	2	2	no	—
128	C₂×C₃₁	C₄×C₄⋊Q₈	1	1	yes	characteristic
129	C₂×C₃₁	C₂×#444	1	2	yes	—
130	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
131	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
132	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
133	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
134	C₄⋊Q₈	C₂×C₄×C₃₁	1	4	yes	—
135	#125	C₂×C₄×C₃₁	1	4	yes	—
136	C₂²⋊Q₈	—	2	4	no	—
137	C₄×Q₈	—	2	4	no	—
138	#117	C₂³×C₃₁	1	4	yes	—
139	C₂×C₄²	—	2	2	no	—
140	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
141	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
142	#108	—	2	8	no	—
143	#107	C₂×C₄×C₃₁	1	8	yes	—
144	#106	C₂×C₄×C₃₁	1	8	yes	—
145	C₄⋊C₈	—	2	8	no	—
146	C₄⋊C₈	—	2	4	no	—
147	#97	—	2	4	no	—
148	#97	—	2	4	no	—
149	#97	Q₈×C₃₁	1	8	yes	complemented
150	C₄×C₈	—	2	4	no	—
151	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
152	C₄×C₈	C₂×C₄×C₃₁	1	2	yes	—
153	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
154	C₄×C₈	D₄×C₃₁	1	4	yes	—
155	C₃₁	#21734	1	1	yes	characteristic, direct product factor, Sylow
156	C₂×Q₈	—	2	4	no	—
157	C₂²×C₄	—	2	2	no	—
158	C₂²×C₄	#41×C₃₁	1	2	yes	—
159	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
160	M₁₆	—	2	8	no	—
161	C₂×C₈	—	2	2	no	—
162	C₂×C₈	—	2	4	no	—
163	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
164	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
165	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
166	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
167	C₂×C₈	C₂×Q₈×C₃₁	1	4	yes	—
168	C₂×C₈	C₂×Q₈×C₃₁	1	4	yes	—
169	C₂×C₈	#41×C₃₁	1	4	yes	—
170	C₄⋊C₄	—	2	8	no	—
171	C₄⋊C₄	—	2	4	no	—
172	C₄⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
173	C₂²⋊C₄	—	2	4	no	—
174	C₄²	—	2	4	no	—
175	C₄²	—	2	2	no	—
176	C₄²	—	4	4	no	—
177	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
178	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
179	C₄²	C₂×D₄×C₃₁	1	2	yes	—
180	C₄²	C₂×Q₈×C₃₁	1	4	yes	—
181	C₂³	C₂×#41×C₃₁	1	1	yes	characteristic
182	Q₈	—	4	8	no	—
183	C₂×C₄	—	2	4	no	—
184	C₂×C₄	—	2	8	no	—
185	C₂×C₄	—	2	4	no	—
186	C₂×C₄	—	4	4	no	—
187	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
188	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
189	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
190	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
191	C₂×C₄	C₂²×Q₈×C₃₁	1	2	yes	—
192	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
193	C₈	—	2	2	no	—
194	C₈	—	4	4	no	—
195	C₈	C₄×Q₈×C₃₁	1	4	yes	—
196	C₈	C₄×Q₈×C₃₁	1	4	yes	—
197	C₈	C₃₁×C₄⋊Q₈	1	4	yes	—
198	C₂²	—	2	2	no	—
199	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
200	C₂²	C₃₁×#533	1	2	yes	—
201	C₄	—	2	2	no	—
202	C₄	—	4	4	no	—
203	C₄	—	4	8	no	—
204	C₄	C₃₁×#444	1	4	yes	—
205	C₄	C₂×C₄×Q₈×C₃₁	1	2	yes	—
206	C₄	C₂×C₃₁×C₄⋊Q₈	1	1	yes	characteristic
207	C₄	C₃₁×#532	1	1	yes	characteristic
208	C₂	—	2	2	no	—
209	C₂	C₄×C₃₁×C₄⋊Q₈	1	1	yes	characteristic
210	C₂	C₂×C₃₁×#444	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂²×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×C₄×C₃₁ ◁ C₂×C₄×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
9. C₃₁×#500	C₄	semidirect
10. C₃₁×#499	C₄	semidirect
11. C₃₁×#498	C₄	semidirect
44. C₃₁×#97	Q₈	semidirect
75. #21734	C₃₁	direct (N × H)
111. #500	C₄×C₃₁	semidirect
112. #499	C₄×C₃₁	semidirect
113. #498	C₄×C₃₁	semidirect
149. #97	Q₈×C₃₁	semidirect
155. C₃₁	#21734	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×#3160	2
C₃₁×#2960	2
C₃₁×#1994	2
C₃₁×#1946	2
C₃₁×#1848	2
C₃₁×#1773	2
C₄×C₃₁×#97	2
#21734	31

Subgroups — C31×#21734 #21478493

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#21734 #21478493