Subgroups — C₃₁×#17619 #21474378

Order 7936.

938 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	C₃₁×#3502	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2982	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2351	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1518	C₂	1	6	yes	maximal
5	C₃₁×#1503	C₂	1	6	yes	maximal
6	Q₈²×C₃₁	C₄	1	4	yes	complemented
7	C₃₁×#557	—	2	4	no	—
8	C₃₁×#557	C₄	1	4	yes	complemented
9	C₃₁×#544	—	2	3	no	—
10	C₃₁×#533	C₂²	1	6	yes	—
11	C₃₁×#519	—	2	1	no	—
12	C₂×C₄×Q₈×C₃₁	—	2	1	no	—
13	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
14	C₈×Q₈×C₃₁	—	2	4	no	—
15	C₃₁×#431	C₂²	1	6	yes	—
16	C₃₁×#389	—	2	3	no	—
17	C₃₁×#382	C₂²	1	3	yes	—
18	C₄×C₄⋊C₄×C₃₁	C₂²	1	3	yes	—
19	C₃₁×#328	C₂²	1	4	yes	—
20	C₃₁×#328	C₂²	1	12	yes	—
21	C₂²×Q₈×C₃₁	D₄	1	1	yes	characteristic
22	C₃₁×C₄⋊Q₈	—	2	12	no	—
23	C₃₁×C₄⋊Q₈	C₂×C₄	1	6	yes	—
24	C₃₁×C₄⋊Q₈	C₂×C₄	1	6	yes	—
25	C₃₁×#125	—	2	12	no	—
26	C₃₁×#125	—	2	6	no	—
27	C₃₁×C₂²⋊Q₈	—	2	6	no	—
28	C₃₁×C₂²⋊Q₈	C₂³	1	4	yes	—
29	C₄×Q₈×C₃₁	—	2	4	no	—
30	C₄×Q₈×C₃₁	—	2	4	no	—
31	C₄×Q₈×C₃₁	—	2	6	no	—
32	C₄×Q₈×C₃₁	—	4	4	no	—
33	C₄×Q₈×C₃₁	C₂×C₄	1	12	yes	—
34	C₄×Q₈×C₃₁	C₂×C₄	1	4	yes	—
35	C₄×Q₈×C₃₁	D₄	1	4	yes	—
36	C₃₁×#117	—	2	6	no	—
37	C₂×C₄⋊C₄×C₃₁	—	2	3	no	—
38	C₂×C₄⋊C₄×C₃₁	—	2	3	no	—
39	C₂×C₄⋊C₄×C₃₁	D₄	1	3	yes	—
40	C₂×C₄²×C₃₁	—	2	3	no	—
41	C₂×C₄²×C₃₁	C₂³	1	3	yes	—
42	C₃₁×C₄⋊C₈	—	2	6	no	—
43	C₃₁×C₂²⋊C₈	C₂³	1	4	yes	—
44	C₄×C₈×C₃₁	—	2	6	no	—
45	C₃₁×#95	—	2	3	no	—
46	C₃₁×#95	C₂³	1	1	yes	characteristic
47	C₃₁×#95	C₂³	1	3	yes	—
48	C₂×Q₈×C₃₁	—	2	6	no	—
49	C₂×Q₈×C₃₁	C₂²⋊C₄	1	4	yes	—
50	C₂×Q₈×C₃₁	SD₁₆	1	4	yes	—
51	C₂×Q₈×C₃₁	Q₁₆	1	4	yes	—
52	C₂×Q₈×C₃₁	C₂²×C₄	1	4	yes	—
53	C₂²×C₄×C₃₁	—	2	1	no	—
54	C₂²×C₄×C₃₁	—	2	3	no	—
55	C₂²×C₄×C₃₁	C₂×D₄	1	3	yes	—
56	C₂²×C₄×C₃₁	C₂×D₄	1	3	yes	—
57	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
58	C₂×C₈×C₃₁	—	2	4	no	—
59	C₄⋊C₄×C₃₁	—	2	6	no	—
60	C₄⋊C₄×C₃₁	—	2	6	no	—
61	C₄⋊C₄×C₃₁	—	2	6	no	—
62	C₄⋊C₄×C₃₁	—	2	12	no	—
63	C₄⋊C₄×C₃₁	—	2	4	no	—
64	C₄⋊C₄×C₃₁	—	4	6	no	—
65	C₄⋊C₄×C₃₁	—	4	6	no	—
66	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	12	yes	—
67	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
68	C₂²⋊C₄×C₃₁	—	2	4	no	—
69	C₄²×C₃₁	—	2	12	no	—
70	C₄²×C₃₁	—	4	6	no	—
71	C₄²×C₃₁	C₂²×C₄	1	6	yes	—
72	C₄²×C₃₁	C₂×D₄	1	6	yes	—
73	#17619	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
74	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
75	Q₈×C₃₁	—	4	4	no	—
76	Q₈×C₃₁	—	8	6	no	—
77	Q₈×C₃₁	#103	1	16	yes	—
78	C₂×C₄×C₃₁	—	2	2	no	—
79	C₂×C₄×C₃₁	—	2	4	no	—
80	C₂×C₄×C₃₁	—	2	4	no	—
81	C₂×C₄×C₃₁	—	4	3	no	—
82	C₂×C₄×C₃₁	—	4	1	no	—
83	C₂×C₄×C₃₁	—	4	1	no	—
84	C₂×C₄×C₃₁	—	4	1	no	—
85	C₂×C₄×C₃₁	—	4	3	no	—
86	C₂×C₄×C₃₁	—	4	3	no	—
87	C₂×C₄×C₃₁	—	4	3	no	—
88	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	6	yes	—
89	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	6	yes	—
90	C₂×C₄×C₃₁	C₂×SD₁₆	1	6	yes	—
91	C₂×C₄×C₃₁	C₂×Q₁₆	1	6	yes	—
92	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
93	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
94	C₈×C₃₁	—	4	4	no	—
95	#3502	C₂×C₃₁	1	1	yes	characteristic
96	#2982	C₂×C₃₁	1	1	yes	characteristic
97	#2351	C₂×C₃₁	1	1	yes	characteristic
98	#1518	C₂×C₃₁	1	6	yes	—
99	#1503	C₂×C₃₁	1	6	yes	—
100	C₂²×C₃₁	—	2	1	no	—
101	C₂²×C₃₁	—	2	1	no	—
102	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
103	C₂²×C₃₁	C₂²×SD₁₆	1	1	yes	characteristic
104	C₂²×C₃₁	C₂²×Q₁₆	1	1	yes	characteristic
105	C₄×C₃₁	—	2	1	no	—
106	C₄×C₃₁	—	2	1	no	—
107	C₄×C₃₁	—	2	6	no	—
108	C₄×C₃₁	—	4	2	no	—
109	C₄×C₃₁	—	4	4	no	—
110	C₄×C₃₁	—	4	4	no	—
111	C₄×C₃₁	—	8	3	no	—
112	C₄×C₃₁	C₂×#103	1	12	yes	—
113	Q₈²	C₄×C₃₁	1	4	yes	complemented
114	#557	—	2	4	no	—
115	#557	C₄×C₃₁	1	4	yes	complemented
116	#544	—	2	3	no	—
117	#533	C₂²×C₃₁	1	6	yes	—
118	#519	—	2	1	no	—
119	C₂×C₄×Q₈	—	2	1	no	—
120	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
121	C₈×Q₈	—	2	4	no	—
122	#431	C₂²×C₃₁	1	6	yes	—
123	#389	—	2	3	no	—
124	#382	C₂²×C₃₁	1	3	yes	—
125	C₄×C₄⋊C₄	C₂²×C₃₁	1	3	yes	—
126	#328	C₂²×C₃₁	1	4	yes	—
127	#328	C₂²×C₃₁	1	12	yes	—
128	C₂×C₃₁	—	2	1	no	—
129	C₂×C₃₁	—	2	1	no	—
130	C₂×C₃₁	#2884	1	1	yes	characteristic
131	C₂×C₃₁	C₂²×#103	1	1	yes	characteristic
132	C₂×C₃₁	#2901	1	1	yes	characteristic
133	C₂²×Q₈	D₄×C₃₁	1	1	yes	characteristic
134	C₄⋊Q₈	—	2	12	no	—
135	C₄⋊Q₈	C₂×C₄×C₃₁	1	6	yes	—
136	C₄⋊Q₈	C₂×C₄×C₃₁	1	6	yes	—
137	#125	—	2	12	no	—
138	#125	—	2	6	no	—
139	C₂²⋊Q₈	—	2	6	no	—
140	C₂²⋊Q₈	C₂³×C₃₁	1	4	yes	—
141	C₄×Q₈	—	2	4	no	—
142	C₄×Q₈	—	2	4	no	—
143	C₄×Q₈	—	2	6	no	—
144	C₄×Q₈	—	4	4	no	—
145	C₄×Q₈	C₂×C₄×C₃₁	1	12	yes	—
146	C₄×Q₈	C₂×C₄×C₃₁	1	4	yes	—
147	C₄×Q₈	D₄×C₃₁	1	4	yes	—
148	#117	—	2	6	no	—
149	C₂×C₄⋊C₄	—	2	3	no	—
150	C₂×C₄⋊C₄	—	2	3	no	—
151	C₂×C₄⋊C₄	D₄×C₃₁	1	3	yes	—
152	C₂×C₄²	—	2	3	no	—
153	C₂×C₄²	C₂³×C₃₁	1	3	yes	—
154	C₄⋊C₈	—	2	6	no	—
155	C₂²⋊C₈	C₂³×C₃₁	1	4	yes	—
156	C₄×C₈	—	2	6	no	—
157	#95	—	2	3	no	—
158	#95	C₂³×C₃₁	1	1	yes	characteristic
159	#95	C₂³×C₃₁	1	3	yes	—
160	C₃₁	#17619	1	1	yes	characteristic, direct product factor, Sylow
161	C₂×Q₈	—	2	6	no	—
162	C₂×Q₈	C₂²⋊C₄×C₃₁	1	4	yes	—
163	C₂×Q₈	SD₁₆×C₃₁	1	4	yes	—
164	C₂×Q₈	Q₁₆×C₃₁	1	4	yes	—
165	C₂×Q₈	C₂²×C₄×C₃₁	1	4	yes	—
166	C₂²×C₄	—	2	1	no	—
167	C₂²×C₄	—	2	3	no	—
168	C₂²×C₄	C₂×D₄×C₃₁	1	3	yes	—
169	C₂²×C₄	C₂×D₄×C₃₁	1	3	yes	—
170	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
171	C₂×C₈	—	2	4	no	—
172	C₄⋊C₄	—	2	6	no	—
173	C₄⋊C₄	—	2	6	no	—
174	C₄⋊C₄	—	2	6	no	—
175	C₄⋊C₄	—	2	12	no	—
176	C₄⋊C₄	—	2	4	no	—
177	C₄⋊C₄	—	4	6	no	—
178	C₄⋊C₄	—	4	6	no	—
179	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	12	yes	—
180	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
181	C₂²⋊C₄	—	2	4	no	—
182	C₄²	—	2	12	no	—
183	C₄²	—	4	6	no	—
184	C₄²	C₂²×C₄×C₃₁	1	6	yes	—
185	C₄²	C₂×D₄×C₃₁	1	6	yes	—
186	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
187	Q₈	—	4	4	no	—
188	Q₈	—	8	6	no	—
189	Q₈	C₃₁×#103	1	16	yes	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	4	no	—
192	C₂×C₄	—	2	4	no	—
193	C₂×C₄	—	4	3	no	—
194	C₂×C₄	—	4	1	no	—
195	C₂×C₄	—	4	1	no	—
196	C₂×C₄	—	4	1	no	—
197	C₂×C₄	—	4	3	no	—
198	C₂×C₄	—	4	3	no	—
199	C₂×C₄	—	4	3	no	—
200	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	6	yes	—
201	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	6	yes	—
202	C₂×C₄	C₂×SD₁₆×C₃₁	1	6	yes	—
203	C₂×C₄	C₂×Q₁₆×C₃₁	1	6	yes	—
204	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
205	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
206	C₈	—	4	4	no	—
207	C₂²	—	2	1	no	—
208	C₂²	—	2	1	no	—
209	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
210	C₂²	C₂²×SD₁₆×C₃₁	1	1	yes	characteristic
211	C₂²	C₂²×Q₁₆×C₃₁	1	1	yes	characteristic
212	C₄	—	2	1	no	—
213	C₄	—	2	1	no	—
214	C₄	—	2	6	no	—
215	C₄	—	4	2	no	—
216	C₄	—	4	4	no	—
217	C₄	—	4	4	no	—
218	C₄	—	8	3	no	—
219	C₄	C₂×C₃₁×#103	1	12	yes	—
220	C₂	—	2	1	no	—
221	C₂	—	2	1	no	—
222	C₂	C₃₁×#2884	1	1	yes	characteristic
223	C₂	C₂²×C₃₁×#103	1	1	yes	characteristic
224	C₂	C₃₁×#2901	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂³×C₃₁	C₂²×D₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄×Q₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
6. Q₈²×C₃₁	C₄	semidirect
8. C₃₁×#557	C₄	semidirect
73. #17619	C₃₁	direct (N × H)
113. Q₈²	C₄×C₃₁	semidirect
115. #557	C₄×C₃₁	semidirect
160. C₃₁	#17619	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×#3502	2
C₃₁×#2982	2
C₃₁×#2351	2
C₃₁×#1518	2
C₃₁×#1503	2
#17619	31

Subgroups — C31×#17619 #21474378

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#17619 #21474378