Subgroups — C₃₁×#17789 #21474548

Order 7936.

734 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2988	C₂	1	2	yes	maximal
2	C₄×C₃₁×#131	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#1565	C₂	1	6	yes	maximal, complemented
4	C₃₁×#1564	C₂	1	3	yes	maximal, complemented
5	C₂×C₃₁×#335	C₂	1	3	yes	maximal, complemented
6	C₂×C₃₁×#131	C₄	1	1	yes	characteristic
7	C₂×C₃₁×#131	C₄	1	1	yes	characteristic
8	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
9	C₈×Q₈×C₃₁	—	2	2	no	—
10	C₈×D₄×C₃₁	—	2	6	no	—
11	C₃₁×#431	C₂²	1	6	yes	—
12	C₂×C₃₁×C₄⋊C₈	C₂²	1	6	yes	—
13	C₃₁×#405	C₄	1	3	yes	—
14	C₃₁×#405	C₄	1	3	yes	—
15	C₄×M₁₆×C₃₁	C₂²	1	3	yes	—
16	C₂×C₄×C₈×C₃₁	C₂²	1	3	yes	—
17	C₃₁×#335	C₂²	1	3	yes	—
18	C₃₁×#335	C₂²	1	6	yes	—
19	C₃₁×#335	C₂²	1	1	yes	characteristic
20	C₃₁×#334	C₂²	1	6	yes	—
21	C₂×#41×C₃₁	C₂×C₄	1	1	yes	characteristic
22	C₃₁×#131	—	2	4	no	—
23	C₃₁×#131	C₈	1	8	yes	complemented
24	C₂×M₁₆×C₃₁	C₂×C₄	1	3	yes	—
25	C₂×M₁₆×C₃₁	C₂×C₄	1	3	yes	—
26	C₂²×C₈×C₃₁	—	2	6	no	—
27	C₂²×C₈×C₃₁	C₂×C₄	1	3	yes	—
28	C₂²×C₈×C₃₁	C₂×C₄	1	3	yes	—
29	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
30	C₄×Q₈×C₃₁	Q₈	1	1	yes	characteristic
31	C₄×D₄×C₃₁	D₄	1	3	yes	—
32	C₄×D₄×C₃₁	Q₈	1	3	yes	—
33	C₃₁×#117	C₂×C₄	1	3	yes	—
34	C₂×C₄²×C₃₁	C₂³	1	3	yes	—
35	C₃₁×C₄⋊C₈	—	2	6	no	—
36	C₃₁×C₄⋊C₈	C₂³	1	6	yes	—
37	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
38	C₃₁×C₂²⋊C₈	—	2	6	no	—
39	C₃₁×#97	C₂×C₄	1	6	yes	—
40	C₄×C₈×C₃₁	—	2	6	no	—
41	C₄×C₈×C₃₁	C₂×C₄	1	3	yes	—
42	C₄×C₈×C₃₁	C₂×C₄	1	3	yes	—
43	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
44	C₄×C₈×C₃₁	C₂³	1	3	yes	—
45	#41×C₃₁	C₂×C₈	1	4	yes	—
46	#41×C₃₁	M₁₆	1	4	yes	—
47	C₂×Q₈×C₃₁	C₄⋊C₄	1	1	yes	characteristic
48	C₂×D₄×C₃₁	C₄⋊C₄	1	3	yes	—
49	C₂²×C₄×C₃₁	C₂²×C₄	1	3	yes	—
50	C₂²×C₄×C₃₁	C₂×D₄	1	3	yes	—
51	C₂²×C₄×C₃₁	C₂×Q₈	1	3	yes	—
52	M₁₆×C₃₁	—	2	6	no	—
53	M₁₆×C₃₁	C₂×C₈	1	12	yes	complemented
54	C₂×C₈×C₃₁	—	2	6	no	—
55	C₂×C₈×C₃₁	—	2	2	no	—
56	C₂×C₈×C₃₁	—	2	6	no	—
57	C₂×C₈×C₃₁	—	8	6	no	—
58	C₂×C₈×C₃₁	C₂×C₈	1	12	yes	complemented
59	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₈×C₃₁	C₂²×C₄	1	3	yes	—
61	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₈×C₃₁	C₂²×C₄	1	3	yes	—
63	C₄⋊C₄×C₃₁	C₄⋊C₄	1	3	yes	—
64	C₄⋊C₄×C₃₁	C₄⋊C₄	1	3	yes	—
65	C₂²⋊C₄×C₃₁	C₄⋊C₄	1	6	yes	—
66	C₄²×C₃₁	C₂²×C₄	1	3	yes	—
67	C₄²×C₃₁	C₂×D₄	1	3	yes	—
68	C₄²×C₃₁	C₂×Q₈	1	3	yes	—
69	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
70	#17789	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
71	C₂³×C₃₁	C₂×C₄⋊C₄	1	3	yes	—
72	Q₈×C₃₁	C₄⋊C₈	1	4	yes	—
73	D₄×C₃₁	C₄⋊C₈	1	12	yes	—
74	C₂×C₄×C₃₁	—	2	3	no	—
75	C₂×C₄×C₃₁	—	2	3	no	—
76	C₂×C₄×C₃₁	—	4	3	no	—
77	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	3	yes	—
78	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	3	yes	—
79	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	3	yes	—
80	C₂×C₄×C₃₁	C₂²×C₈	1	6	yes	—
81	C₂×C₄×C₃₁	C₂×M₁₆	1	6	yes	—
82	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
83	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
84	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
85	C₈×C₃₁	—	2	1	no	—
86	C₈×C₃₁	—	2	3	no	—
87	C₈×C₃₁	—	4	6	no	—
88	C₈×C₃₁	—	4	2	no	—
89	C₈×C₃₁	C₂²×C₈	1	2	yes	—
90	C₈×C₃₁	C₂²×C₈	1	6	yes	—
91	#2988	C₂×C₃₁	1	2	yes	—
92	C₄×#131	C₂×C₃₁	1	1	yes	characteristic
93	#1565	C₂×C₃₁	1	6	yes	complemented
94	#1564	C₂×C₃₁	1	3	yes	complemented
95	C₂×#335	C₂×C₃₁	1	3	yes	complemented
96	C₂²×C₃₁	—	2	3	no	—
97	C₂²×C₃₁	—	2	3	no	—
98	C₂²×C₃₁	C₂×C₄⋊C₈	1	6	yes	—
99	C₂²×C₃₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
100	C₄×C₃₁	—	2	3	no	—
101	C₄×C₃₁	—	2	1	no	—
102	C₄×C₃₁	—	2	3	no	—
103	C₄×C₃₁	C₂×C₄⋊C₈	1	6	yes	—
104	C₄×C₃₁	C₂³×C₈	1	1	yes	characteristic
105	C₄×C₃₁	C₂²×M₁₆	1	1	yes	characteristic
106	C₄×C₃₁	#576	1	1	yes	characteristic
107	C₄×C₃₁	#578	1	1	yes	characteristic
108	C₂×#131	C₄×C₃₁	1	1	yes	characteristic
109	C₂×#131	C₄×C₃₁	1	1	yes	characteristic
110	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
111	C₈×Q₈	—	2	2	no	—
112	C₈×D₄	—	2	6	no	—
113	#431	C₂²×C₃₁	1	6	yes	—
114	C₂×C₄⋊C₈	C₂²×C₃₁	1	6	yes	—
115	#405	C₄×C₃₁	1	3	yes	—
116	#405	C₄×C₃₁	1	3	yes	—
117	C₄×M₁₆	C₂²×C₃₁	1	3	yes	—
118	C₂×C₄×C₈	C₂²×C₃₁	1	3	yes	—
119	#335	C₂²×C₃₁	1	3	yes	—
120	#335	C₂²×C₃₁	1	6	yes	—
121	#335	C₂²×C₃₁	1	1	yes	characteristic
122	#334	C₂²×C₃₁	1	6	yes	—
123	C₂×C₃₁	—	2	6	no	—
124	C₂×C₃₁	C₂²×C₄⋊C₈	1	1	yes	characteristic
125	C₂×C₃₁	#2913	1	1	yes	characteristic
126	C₂×C₃₁	#2916	1	1	yes	characteristic
127	C₂×#41	C₂×C₄×C₃₁	1	1	yes	characteristic
128	#131	—	2	4	no	—
129	#131	C₈×C₃₁	1	8	yes	complemented
130	C₂×M₁₆	C₂×C₄×C₃₁	1	3	yes	—
131	C₂×M₁₆	C₂×C₄×C₃₁	1	3	yes	—
132	C₂²×C₈	—	2	6	no	—
133	C₂²×C₈	C₂×C₄×C₃₁	1	3	yes	—
134	C₂²×C₈	C₂×C₄×C₃₁	1	3	yes	—
135	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
136	C₄×Q₈	Q₈×C₃₁	1	1	yes	characteristic
137	C₄×D₄	D₄×C₃₁	1	3	yes	—
138	C₄×D₄	Q₈×C₃₁	1	3	yes	—
139	#117	C₂×C₄×C₃₁	1	3	yes	—
140	C₂×C₄²	C₂³×C₃₁	1	3	yes	—
141	C₄⋊C₈	—	2	6	no	—
142	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
143	C₄⋊C₈	C₂³×C₃₁	1	6	yes	—
144	C₂²⋊C₈	—	2	6	no	—
145	#97	C₂×C₄×C₃₁	1	6	yes	—
146	C₄×C₈	—	2	6	no	—
147	C₄×C₈	C₂×C₄×C₃₁	1	3	yes	—
148	C₄×C₈	C₂×C₄×C₃₁	1	3	yes	—
149	C₄×C₈	C₂³×C₃₁	1	3	yes	—
150	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
151	C₃₁	#17789	1	1	yes	characteristic, direct product factor, Sylow
152	#41	C₂×C₈×C₃₁	1	4	yes	—
153	#41	M₁₆×C₃₁	1	4	yes	—
154	C₂×Q₈	C₄⋊C₄×C₃₁	1	1	yes	characteristic
155	C₂×D₄	C₄⋊C₄×C₃₁	1	3	yes	—
156	C₂²×C₄	C₂²×C₄×C₃₁	1	3	yes	—
157	C₂²×C₄	C₂×D₄×C₃₁	1	3	yes	—
158	C₂²×C₄	C₂×Q₈×C₃₁	1	3	yes	—
159	M₁₆	—	2	6	no	—
160	M₁₆	C₂×C₈×C₃₁	1	12	yes	complemented
161	C₂×C₈	—	2	6	no	—
162	C₂×C₈	—	2	2	no	—
163	C₂×C₈	—	2	6	no	—
164	C₂×C₈	—	8	6	no	—
165	C₂×C₈	C₂×C₈×C₃₁	1	12	yes	complemented
166	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
167	C₂×C₈	C₂²×C₄×C₃₁	1	3	yes	—
168	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
169	C₂×C₈	C₂²×C₄×C₃₁	1	3	yes	—
170	C₄⋊C₄	C₄⋊C₄×C₃₁	1	3	yes	—
171	C₄⋊C₄	C₄⋊C₄×C₃₁	1	3	yes	—
172	C₂²⋊C₄	C₄⋊C₄×C₃₁	1	6	yes	—
173	C₄²	C₂²×C₄×C₃₁	1	3	yes	—
174	C₄²	C₂×D₄×C₃₁	1	3	yes	—
175	C₄²	C₂×Q₈×C₃₁	1	3	yes	—
176	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
177	C₂³	C₂×C₄⋊C₄×C₃₁	1	3	yes	—
178	Q₈	C₃₁×C₄⋊C₈	1	4	yes	—
179	D₄	C₃₁×C₄⋊C₈	1	12	yes	—
180	C₂×C₄	—	2	3	no	—
181	C₂×C₄	—	2	3	no	—
182	C₂×C₄	—	4	3	no	—
183	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	3	yes	—
184	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	3	yes	—
185	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	3	yes	—
186	C₂×C₄	C₂²×C₈×C₃₁	1	6	yes	—
187	C₂×C₄	C₂×M₁₆×C₃₁	1	6	yes	—
188	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
189	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
190	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
191	C₈	—	2	1	no	—
192	C₈	—	2	3	no	—
193	C₈	—	4	6	no	—
194	C₈	—	4	2	no	—
195	C₈	C₂²×C₈×C₃₁	1	2	yes	—
196	C₈	C₂²×C₈×C₃₁	1	6	yes	—
197	C₂²	—	2	3	no	—
198	C₂²	—	2	3	no	—
199	C₂²	C₂×C₃₁×C₄⋊C₈	1	6	yes	—
200	C₂²	C₂²×C₄⋊C₄×C₃₁	1	1	yes	characteristic
201	C₄	—	2	3	no	—
202	C₄	—	2	1	no	—
203	C₄	—	2	3	no	—
204	C₄	C₂×C₃₁×C₄⋊C₈	1	6	yes	—
205	C₄	C₂³×C₈×C₃₁	1	1	yes	characteristic
206	C₄	C₂²×M₁₆×C₃₁	1	1	yes	characteristic
207	C₄	C₃₁×#576	1	1	yes	characteristic
208	C₄	C₃₁×#578	1	1	yes	characteristic
209	C₂	—	2	6	no	—
210	C₂	C₂²×C₃₁×C₄⋊C₈	1	1	yes	characteristic
211	C₂	C₃₁×#2913	1	1	yes	characteristic
212	C₂	C₃₁×#2916	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂²×D₄
commutator	C₄	C₂³×C₈×C₃₁
Frattini, cosocle	C₄²	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₂²×C₄⋊C₄
prime radical	C₂×#41×C₃₁	C₂×C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₄×#41×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃₁×#1565	C₂	semidirect
4. C₃₁×#1564	C₂	semidirect
5. C₂×C₃₁×#335	C₂	semidirect
23. C₃₁×#131	C₈	semidirect
53. M₁₆×C₃₁	C₂×C₈	semidirect
58. C₂×C₈×C₃₁	C₂×C₈	semidirect
70. #17789	C₃₁	direct (N × H)
93. #1565	C₂×C₃₁	semidirect
94. #1564	C₂×C₃₁	semidirect
95. C₂×#335	C₂×C₃₁	semidirect
129. #131	C₈×C₃₁	semidirect
151. C₃₁	#17789	direct (N × H)
160. M₁₆	C₂×C₈×C₃₁	semidirect
165. C₂×C₈	C₂×C₈×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#2988	2
C₄×C₃₁×#131	2
C₃₁×#1565	2
C₃₁×#1564	2
C₂×C₃₁×#335	2
#17789	31

Subgroups — C31×#17789 #21474548

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#17789 #21474548