Subgroups — C₃₁×#32808 #21489567

Order 7936.

732 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×#2862	C₂	1	2	yes	maximal
2	C₃₁×#2056	C₂	1	4	yes	maximal
3	C₃₁×#2024	C₂	1	4	yes	maximal
4	C₃₁×#2016	C₂	1	4	yes	maximal
5	C₃₁×#1831	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₁₆×C₃₁	C₂²	1	4	yes	—
7	C₂×C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
8	C₂×C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
9	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
10	C₂×C₃₁×#103	C₂²	1	8	yes	—
11	C₃₁×#391	—	2	2	no	—
12	C₃₁×#391	—	2	2	no	—
13	C₃₁×#391	C₂²	1	4	yes	—
14	C₃₁×#391	C₂²	1	4	yes	—
15	C₃₁×#382	—	2	2	no	—
16	C₃₁×#382	—	2	2	no	—
17	C₄³×C₃₁	C₂²	1	1	yes	characteristic
18	C₃₁×#340	C₂²	1	4	yes	—
19	C₃₁×#336	C₂²	1	4	yes	—
20	C₂²×Q₈×C₃₁	—	2	2	no	—
21	C₂²×Q₈×C₃₁	C₂³	1	4	yes	—
22	C₂×Q₁₆×C₃₁	—	4	8	no	—
23	C₂×Q₁₆×C₃₁	—	4	4	no	—
24	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
25	C₃₁×C₄⋊Q₈	—	4	4	no	—
26	C₃₁×C₄⋊Q₈	—	4	4	no	—
27	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
28	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
29	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
30	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
31	C₂×C₄²×C₃₁	D₄	1	2	yes	—
32	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
33	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
35	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
36	C₃₁×C₄⋊C₈	—	4	2	no	—
37	C₃₁×#103	—	4	8	no	—
38	C₃₁×#95	—	2	4	no	—
39	C₃₁×#95	—	2	4	no	—
40	C₃₁×#95	—	2	4	no	—
41	C₃₁×#95	—	2	4	no	—
42	C₂×Q₈×C₃₁	—	2	8	no	—
43	C₂×Q₈×C₃₁	—	2	8	no	—
44	C₂×Q₈×C₃₁	—	2	4	no	—
45	C₂×Q₈×C₃₁	—	2	4	no	—
46	C₂×Q₈×C₃₁	—	4	4	no	—
47	C₂×Q₈×C₃₁	—	4	4	no	—
48	C₂×Q₈×C₃₁	—	4	4	no	—
49	C₂²×C₄×C₃₁	—	2	4	no	—
50	C₂²×C₄×C₃₁	—	2	4	no	—
51	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
54	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
55	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
56	Q₁₆×C₃₁	—	4	16	no	—
57	C₂×C₈×C₃₁	—	2	8	no	—
58	C₂×C₈×C₃₁	—	2	4	no	—
59	C₄⋊C₄×C₃₁	—	2	8	no	—
60	C₄⋊C₄×C₃₁	—	4	4	no	—
61	C₄⋊C₄×C₃₁	—	4	4	no	—
62	C₄⋊C₄×C₃₁	—	4	4	no	—
63	C₄²×C₃₁	—	2	4	no	—
64	C₄²×C₃₁	—	2	1	no	—
65	C₄²×C₃₁	—	2	4	no	—
66	C₄²×C₃₁	—	2	1	no	—
67	C₄²×C₃₁	—	4	1	no	—
68	C₄²×C₃₁	Q₁₆	1	4	yes	complemented
69	#32808	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
70	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
71	Q₈×C₃₁	—	4	8	no	—
72	Q₈×C₃₁	—	4	8	no	—
73	Q₈×C₃₁	—	8	8	no	—
74	C₂×C₄×C₃₁	—	2	2	no	—
75	C₂×C₄×C₃₁	—	2	2	no	—
76	C₂×C₄×C₃₁	—	2	1	no	—
77	C₂×C₄×C₃₁	—	2	2	no	—
78	C₂×C₄×C₃₁	—	2	1	no	—
79	C₂×C₄×C₃₁	—	2	2	no	—
80	C₂×C₄×C₃₁	—	2	1	no	—
81	C₂×C₄×C₃₁	—	2	2	no	—
82	C₂×C₄×C₃₁	—	4	4	no	—
83	C₂×C₄×C₃₁	—	4	4	no	—
84	C₂×C₄×C₃₁	—	4	4	no	—
85	C₂×C₄×C₃₁	—	4	4	no	—
86	C₂×C₄×C₃₁	—	4	4	no	—
87	C₂×C₄×C₃₁	—	4	4	no	—
88	C₂×C₄×C₃₁	C₂×Q₁₆	1	4	yes	—
89	C₂×C₄×C₃₁	C₂×Q₁₆	1	2	yes	—
90	C₂×C₄×C₃₁	#137	1	4	yes	—
91	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
92	C₂×C₄×C₃₁	H(5,𝔽₂)	1	4	yes	—
93	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
94	C₈×C₃₁	—	4	4	no	—
95	#2862	C₂×C₃₁	1	2	yes	—
96	#2056	C₂×C₃₁	1	4	yes	—
97	#2024	C₂×C₃₁	1	4	yes	—
98	#2016	C₂×C₃₁	1	4	yes	—
99	#1831	C₂×C₃₁	1	1	yes	characteristic
100	C₂²×C₃₁	#535	1	2	yes	—
101	C₂²×C₃₁	#537	1	1	yes	characteristic
102	C₂²×C₃₁	C₂²×Q₁₆	1	1	yes	characteristic
103	C₂²×C₃₁	C₂×#137	1	2	yes	—
104	C₂²×C₃₁	#575	1	1	yes	characteristic
105	C₄×C₃₁	—	2	4	no	—
106	C₄×C₃₁	—	2	4	no	—
107	C₄×C₃₁	—	2	2	no	—
108	C₄×C₃₁	—	4	1	no	—
109	C₄×C₃₁	—	8	4	no	—
110	C₄×C₃₁	—	8	4	no	—
111	C₄×C₃₁	H(3,𝔽₄)	1	2	yes	—
112	C₄×C₃₁	#562	1	2	yes	—
113	C₂²×Q₁₆	C₂²×C₃₁	1	4	yes	—
114	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
115	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
116	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
117	C₂×#103	C₂²×C₃₁	1	8	yes	—
118	#391	—	2	2	no	—
119	#391	—	2	2	no	—
120	#391	C₂²×C₃₁	1	4	yes	—
121	#391	C₂²×C₃₁	1	4	yes	—
122	#382	—	2	2	no	—
123	#382	—	2	2	no	—
124	C₄³	C₂²×C₃₁	1	1	yes	characteristic
125	#340	C₂²×C₃₁	1	4	yes	—
126	#336	C₂²×C₃₁	1	4	yes	—
127	C₂×C₃₁	#2819	1	1	yes	characteristic
128	C₂×C₃₁	#3209	1	2	yes	—
129	C₂×C₃₁	#3215	1	2	yes	—
130	C₂×C₃₁	#3249	1	1	yes	characteristic
131	C₂×C₃₁	#3255	1	1	yes	characteristic
132	C₂²×Q₈	—	2	2	no	—
133	C₂²×Q₈	C₂³×C₃₁	1	4	yes	—
134	C₂×Q₁₆	—	4	8	no	—
135	C₂×Q₁₆	—	4	4	no	—
136	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
137	C₄⋊Q₈	—	4	4	no	—
138	C₄⋊Q₈	—	4	4	no	—
139	C₂×C₄⋊C₄	—	2	2	no	—
140	C₂×C₄⋊C₄	—	2	2	no	—
141	C₂×C₄⋊C₄	—	2	2	no	—
142	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
143	C₂×C₄²	D₄×C₃₁	1	2	yes	—
144	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
145	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
146	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
147	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
148	C₄⋊C₈	—	4	2	no	—
149	#103	—	4	8	no	—
150	#95	—	2	4	no	—
151	#95	—	2	4	no	—
152	#95	—	2	4	no	—
153	#95	—	2	4	no	—
154	C₃₁	#32808	1	1	yes	characteristic, direct product factor, Sylow
155	C₂×Q₈	—	2	8	no	—
156	C₂×Q₈	—	2	8	no	—
157	C₂×Q₈	—	2	4	no	—
158	C₂×Q₈	—	2	4	no	—
159	C₂×Q₈	—	4	4	no	—
160	C₂×Q₈	—	4	4	no	—
161	C₂×Q₈	—	4	4	no	—
162	C₂²×C₄	—	2	4	no	—
163	C₂²×C₄	—	2	4	no	—
164	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
165	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
166	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
167	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
168	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
169	Q₁₆	—	4	16	no	—
170	C₂×C₈	—	2	8	no	—
171	C₂×C₈	—	2	4	no	—
172	C₄⋊C₄	—	2	8	no	—
173	C₄⋊C₄	—	4	4	no	—
174	C₄⋊C₄	—	4	4	no	—
175	C₄⋊C₄	—	4	4	no	—
176	C₄²	—	2	4	no	—
177	C₄²	—	2	1	no	—
178	C₄²	—	2	4	no	—
179	C₄²	—	2	1	no	—
180	C₄²	—	4	1	no	—
181	C₄²	Q₁₆×C₃₁	1	4	yes	complemented
182	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
183	Q₈	—	4	8	no	—
184	Q₈	—	4	8	no	—
185	Q₈	—	8	8	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	4	4	no	—
195	C₂×C₄	—	4	4	no	—
196	C₂×C₄	—	4	4	no	—
197	C₂×C₄	—	4	4	no	—
198	C₂×C₄	—	4	4	no	—
199	C₂×C₄	—	4	4	no	—
200	C₂×C₄	C₂×Q₁₆×C₃₁	1	4	yes	—
201	C₂×C₄	C₂×Q₁₆×C₃₁	1	2	yes	—
202	C₂×C₄	C₃₁×#137	1	4	yes	—
203	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
204	C₂×C₄	C₃₁×H(5,𝔽₂)	1	4	yes	—
205	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
206	C₈	—	4	4	no	—
207	C₂²	C₃₁×#535	1	2	yes	—
208	C₂²	C₃₁×#537	1	1	yes	characteristic
209	C₂²	C₂²×Q₁₆×C₃₁	1	1	yes	characteristic
210	C₂²	C₂×C₃₁×#137	1	2	yes	—
211	C₂²	C₃₁×#575	1	1	yes	characteristic
212	C₄	—	2	4	no	—
213	C₄	—	2	4	no	—
214	C₄	—	2	2	no	—
215	C₄	—	4	1	no	—
216	C₄	—	8	4	no	—
217	C₄	—	8	4	no	—
218	C₄	C₃₁×H(3,𝔽₄)	1	2	yes	—
219	C₄	C₃₁×#562	1	2	yes	—
220	C₂	C₃₁×#2819	1	1	yes	characteristic
221	C₂	C₃₁×#3209	1	2	yes	—
222	C₂	C₃₁×#3215	1	2	yes	—
223	C₂	C₃₁×#3249	1	1	yes	characteristic
224	C₂	C₃₁×#3255	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₄³×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
68. C₄²×C₃₁	Q₁₆	semidirect
69. #32808	C₃₁	direct (N × H)
154. C₃₁	#32808	direct (N × H)
181. C₄²	Q₁₆×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#2862	2
C₃₁×#2056	2
C₃₁×#2024	2
C₃₁×#2016	2
C₃₁×#1831	2
#32808	31

Subgroups — C31×#32808 #21489567

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#32808 #21489567