Subgroups — C₃₁×#20010 #21476769

Order 7936.

566 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3395	C₂	1	2	yes	maximal
2	C₃₁×#3383	C₂	1	2	yes	maximal
3	C₃₁×#1713	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1674	C₂	1	2	yes	maximal
5	C₃₁×#1670	C₂	1	2	yes	maximal
6	C₃₁×#1648	C₂	1	4	yes	maximal
7	C₈×Q₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic
8	C₈×Q₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic
9	C₃₁×#557	C₂²	1	4	yes	—
10	C₃₁×#500	C₂²	1	2	yes	—
11	C₃₁×#500	C₂²	1	2	yes	—
12	C₃₁×#499	—	2	2	no	—
13	C₃₁×#494	—	2	2	no	—
14	C₃₁×#494	C₂²	1	2	yes	—
15	C₃₁×#479	—	2	4	no	—
16	C₃₁×#475	C₂²	1	4	yes	—
17	C₃₁×#462	—	2	4	no	—
18	C₃₁×#462	C₂²	1	4	yes	—
19	C₈×Q₈×C₃₁	C₂²	1	2	yes	—
20	C₈×Q₈×C₃₁	C₂²	1	2	yes	—
21	C₄×Q₁₆×C₃₁	—	2	2	no	—
22	C₄×Q₁₆×C₃₁	—	2	2	no	—
23	C₄×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
24	C₄×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
25	C₃₁×#335	C₂²	1	1	yes	characteristic
26	C₃₁×#335	C₂²	1	1	yes	characteristic
27	C₃₁×#332	C₂²	1	4	yes	—
28	C₃₁×Q₈⋊C₈	C₂²	1	2	yes	—
29	C₃₁×Q₈⋊C₈	C₂²	1	2	yes	—
30	C₈²×C₃₁	C₂²	1	1	yes	characteristic
31	C₂×Q₁₆×C₃₁	—	2	4	no	—
32	C₂×Q₁₆×C₃₁	—	2	2	no	—
33	C₂×Q₁₆×C₃₁	—	2	2	no	—
34	C₂×Q₁₆×C₃₁	D₄	1	1	yes	characteristic
35	C₂×Q₁₆×C₃₁	D₄	1	1	yes	characteristic
36	C₃₁×C₄⋊Q₈	—	2	4	no	—
37	C₃₁×C₄⋊Q₈	C₂³	1	4	yes	—
38	C₃₁×#125	—	2	4	no	—
39	C₃₁×#125	—	4	4	no	—
40	C₄×Q₈×C₃₁	—	2	4	no	—
41	C₄×Q₈×C₃₁	—	2	4	no	—
42	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
43	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
44	C₃₁×#107	—	2	2	no	—
45	C₃₁×#107	—	2	2	no	—
46	C₃₁×#107	—	4	2	no	—
47	C₃₁×#107	D₄	1	1	yes	characteristic
48	C₃₁×#107	D₄	1	1	yes	characteristic
49	C₃₁×#106	—	2	4	no	—
50	C₃₁×C₄⋊C₈	—	2	2	no	—
51	C₃₁×C₄⋊C₈	—	2	2	no	—
52	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
53	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
54	C₃₁×#103	—	2	4	no	—
55	C₃₁×#103	—	2	4	no	—
56	C₃₁×#103	D₄	1	2	yes	—
57	C₃₁×#103	D₄	1	2	yes	—
58	C₄×C₈×C₃₁	—	2	2	no	—
59	C₄×C₈×C₃₁	—	2	2	no	—
60	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
61	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
62	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
63	C₂×Q₈×C₃₁	—	2	4	no	—
64	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
65	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
66	Q₁₆×C₃₁	—	2	2	no	—
67	Q₁₆×C₃₁	—	4	4	no	—
68	Q₁₆×C₃₁	—	8	2	no	—
69	Q₁₆×C₃₁	—	8	4	no	—
70	Q₁₆×C₃₁	Q₁₆	1	4	yes	complemented
71	C₂×C₈×C₃₁	—	2	1	no	—
72	C₂×C₈×C₃₁	—	2	2	no	—
73	C₂×C₈×C₃₁	—	2	2	no	—
74	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
75	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
78	C₄⋊C₄×C₃₁	—	2	2	no	—
79	C₄⋊C₄×C₃₁	—	2	2	no	—
80	C₄⋊C₄×C₃₁	—	2	4	no	—
81	C₄⋊C₄×C₃₁	—	2	4	no	—
82	C₄⋊C₄×C₃₁	—	4	4	no	—
83	C₄⋊C₄×C₃₁	—	4	4	no	—
84	C₄⋊C₄×C₃₁	—	4	4	no	—
85	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
86	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
87	C₄²×C₃₁	—	2	2	no	—
88	C₄²×C₃₁	—	2	2	no	—
89	C₄²×C₃₁	—	4	4	no	—
90	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
91	#20010	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
92	Q₈×C₃₁	—	2	2	no	—
93	Q₈×C₃₁	—	2	2	no	—
94	Q₈×C₃₁	—	2	2	no	—
95	Q₈×C₃₁	—	4	4	no	—
96	Q₈×C₃₁	—	4	4	no	—
97	Q₈×C₃₁	C₂×Q₁₆	1	4	yes	—
98	C₂×C₄×C₃₁	—	2	2	no	—
99	C₂×C₄×C₃₁	—	2	2	no	—
100	C₂×C₄×C₃₁	—	2	2	no	—
101	C₂×C₄×C₃₁	—	2	2	no	—
102	C₂×C₄×C₃₁	—	4	4	no	—
103	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
104	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
105	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
106	C₈×C₃₁	—	2	2	no	—
107	C₈×C₃₁	—	2	1	no	—
108	C₈×C₃₁	—	2	1	no	—
109	C₈×C₃₁	—	4	2	no	—
110	C₈×C₃₁	—	4	2	no	—
111	C₈×C₃₁	C₂×Q₁₆	1	2	yes	—
112	C₈×C₃₁	#135	1	2	yes	—
113	#3395	C₂×C₃₁	1	2	yes	—
114	#3383	C₂×C₃₁	1	2	yes	—
115	#1713	C₂×C₃₁	1	1	yes	characteristic
116	#1674	C₂×C₃₁	1	2	yes	—
117	#1670	C₂×C₃₁	1	2	yes	—
118	#1648	C₂×C₃₁	1	4	yes	—
119	C₈×Q₁₆	C₂×C₃₁	1	1	yes	characteristic
120	C₈×Q₁₆	C₂×C₃₁	1	1	yes	characteristic
121	C₂²×C₃₁	D₄²	1	1	yes	characteristic
122	C₄×C₃₁	—	2	4	no	—
123	C₄×C₃₁	—	2	1	no	—
124	C₄×C₃₁	—	4	2	no	—
125	C₄×C₃₁	—	4	2	no	—
126	C₄×C₃₁	—	4	2	no	—
127	C₄×C₃₁	—	8	4	no	—
128	C₄×C₃₁	C₂²×Q₁₆	1	1	yes	characteristic
129	C₄×C₃₁	C₂×#135	1	1	yes	characteristic
130	C₄×C₃₁	#576	1	1	yes	characteristic
131	C₄×C₃₁	#578	1	1	yes	characteristic
132	#557	C₂²×C₃₁	1	4	yes	—
133	#500	C₂²×C₃₁	1	2	yes	—
134	#500	C₂²×C₃₁	1	2	yes	—
135	#499	—	2	2	no	—
136	#494	—	2	2	no	—
137	#494	C₂²×C₃₁	1	2	yes	—
138	#479	—	2	4	no	—
139	#475	C₂²×C₃₁	1	4	yes	—
140	#462	—	2	4	no	—
141	#462	C₂²×C₃₁	1	4	yes	—
142	C₈×Q₈	C₂²×C₃₁	1	2	yes	—
143	C₈×Q₈	C₂²×C₃₁	1	2	yes	—
144	C₄×Q₁₆	—	2	2	no	—
145	C₄×Q₁₆	—	2	2	no	—
146	C₄×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
147	C₄×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
148	#335	C₂²×C₃₁	1	1	yes	characteristic
149	#335	C₂²×C₃₁	1	1	yes	characteristic
150	#332	C₂²×C₃₁	1	4	yes	—
151	Q₈⋊C₈	C₂²×C₃₁	1	2	yes	—
152	Q₈⋊C₈	C₂²×C₃₁	1	2	yes	—
153	C₈²	C₂²×C₃₁	1	1	yes	characteristic
154	C₂×C₃₁	D₄×Q₁₆	1	1	yes	characteristic
155	C₂×C₃₁	#3287	1	1	yes	characteristic
156	C₂×C₃₁	#3293	1	1	yes	characteristic
157	C₂×Q₁₆	—	2	4	no	—
158	C₂×Q₁₆	—	2	2	no	—
159	C₂×Q₁₆	—	2	2	no	—
160	C₂×Q₁₆	D₄×C₃₁	1	1	yes	characteristic
161	C₂×Q₁₆	D₄×C₃₁	1	1	yes	characteristic
162	C₄⋊Q₈	—	2	4	no	—
163	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	—
164	#125	—	2	4	no	—
165	#125	—	4	4	no	—
166	C₄×Q₈	—	2	4	no	—
167	C₄×Q₈	—	2	4	no	—
168	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
169	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
170	#107	—	2	2	no	—
171	#107	—	2	2	no	—
172	#107	—	4	2	no	—
173	#107	D₄×C₃₁	1	1	yes	characteristic
174	#107	D₄×C₃₁	1	1	yes	characteristic
175	#106	—	2	4	no	—
176	C₄⋊C₈	—	2	2	no	—
177	C₄⋊C₈	—	2	2	no	—
178	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
179	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
180	#103	—	2	4	no	—
181	#103	—	2	4	no	—
182	#103	D₄×C₃₁	1	2	yes	—
183	#103	D₄×C₃₁	1	2	yes	—
184	C₄×C₈	—	2	2	no	—
185	C₄×C₈	—	2	2	no	—
186	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
187	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
188	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
189	C₃₁	#20010	1	1	yes	characteristic, direct product factor, Sylow
190	C₂×Q₈	—	2	4	no	—
191	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
192	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
193	Q₁₆	—	2	2	no	—
194	Q₁₆	—	4	4	no	—
195	Q₁₆	—	8	2	no	—
196	Q₁₆	—	8	4	no	—
197	Q₁₆	Q₁₆×C₃₁	1	4	yes	complemented
198	C₂×C₈	—	2	1	no	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
202	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
203	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
204	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
205	C₄⋊C₄	—	2	2	no	—
206	C₄⋊C₄	—	2	2	no	—
207	C₄⋊C₄	—	2	4	no	—
208	C₄⋊C₄	—	2	4	no	—
209	C₄⋊C₄	—	4	4	no	—
210	C₄⋊C₄	—	4	4	no	—
211	C₄⋊C₄	—	4	4	no	—
212	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
213	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
214	C₄²	—	2	2	no	—
215	C₄²	—	2	2	no	—
216	C₄²	—	4	4	no	—
217	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
218	Q₈	—	2	2	no	—
219	Q₈	—	2	2	no	—
220	Q₈	—	2	2	no	—
221	Q₈	—	4	4	no	—
222	Q₈	—	4	4	no	—
223	Q₈	C₂×Q₁₆×C₃₁	1	4	yes	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	4	4	no	—
229	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
230	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
231	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
232	C₈	—	2	2	no	—
233	C₈	—	2	1	no	—
234	C₈	—	2	1	no	—
235	C₈	—	4	2	no	—
236	C₈	—	4	2	no	—
237	C₈	C₂×Q₁₆×C₃₁	1	2	yes	—
238	C₈	C₃₁×#135	1	2	yes	—
239	C₂²	D₄²×C₃₁	1	1	yes	characteristic
240	C₄	—	2	4	no	—
241	C₄	—	2	1	no	—
242	C₄	—	4	2	no	—
243	C₄	—	4	2	no	—
244	C₄	—	4	2	no	—
245	C₄	—	8	4	no	—
246	C₄	C₂²×Q₁₆×C₃₁	1	1	yes	characteristic
247	C₄	C₂×C₃₁×#135	1	1	yes	characteristic
248	C₄	C₃₁×#576	1	1	yes	characteristic
249	C₄	C₃₁×#578	1	1	yes	characteristic
250	C₂	D₄×Q₁₆×C₃₁	1	1	yes	characteristic
251	C₂	C₃₁×#3287	1	1	yes	characteristic
252	C₂	C₃₁×#3293	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂²×C₃₁	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
70. Q₁₆×C₃₁	Q₁₆	semidirect
91. #20010	C₃₁	direct (N × H)
189. C₃₁	#20010	direct (N × H)
197. Q₁₆	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×#3395	2
C₃₁×#3383	2
C₃₁×#1713	2
C₃₁×#1674	2
C₃₁×#1670	2
C₃₁×#1648	2
C₈×Q₁₆×C₃₁	2
#20010	31

Subgroups — C31×#20010 #21476769

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#20010 #21476769