Subgroups — C₃₁×#20826 #21477585

Order 7936.

810 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2920	C₂	1	2	yes	maximal
2	C₂²×C₄×C₈×C₃₁	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#1848	C₂	1	2	yes	maximal, complemented
4	C₃₁×#1837	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1810	C₂	1	4	yes	maximal
6	C₄×C₃₁×#108	C₂	1	4	yes	maximal, complemented
7	C₂³×C₈×C₃₁	C₂²	1	2	yes	—
8	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
9	C₃₁×#432	—	2	4	no	—
10	C₃₁×#431	—	2	4	no	—
11	C₂×C₃₁×#108	C₂²	1	8	yes	—
12	C₃₁×#423	C₂²	1	4	yes	—
13	C₃₁×#407	C₂²	1	4	yes	—
14	C₄×M₁₆×C₃₁	C₂²	1	4	yes	—
15	C₂×C₄×C₈×C₃₁	C₄	1	4	yes	—
16	C₂×C₄×C₈×C₃₁	C₄	1	4	yes	—
17	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
20	C₃₁×#341	C₂²	1	8	yes	—
21	C₂³×C₄×C₃₁	Q₈	1	2	yes	—
22	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
23	C₂×M₁₆×C₃₁	C₂³	1	8	yes	—
24	C₂²×C₈×C₃₁	—	2	8	no	—
25	C₂²×C₈×C₃₁	C₂×C₄	1	8	yes	—
26	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
27	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
28	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₃₁	D₄	1	2	yes	—
31	C₂×C₄²×C₃₁	D₄	1	2	yes	—
32	C₂×C₄²×C₃₁	Q₈	1	2	yes	—
33	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
34	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
35	C₃₁×#108	—	2	16	no	—
36	C₃₁×C₄⋊C₈	—	2	8	no	—
37	C₃₁×C₂²⋊C₈	—	2	8	no	—
38	C₃₁×#97	—	2	4	no	—
39	C₄×C₈×C₃₁	—	2	4	no	—
40	C₄×C₈×C₃₁	C₂×C₄	1	2	yes	—
41	C₄×C₈×C₃₁	C₂×C₄	1	2	yes	—
42	C₄×C₈×C₃₁	C₂×C₄	1	8	yes	—
43	C₄×C₈×C₃₁	C₂×C₄	1	2	yes	—
44	C₄×C₈×C₃₁	C₂×C₄	1	2	yes	—
45	C₂⁴×C₃₁	C₂×Q₈	1	1	yes	characteristic
46	C₂²×C₄×C₃₁	—	2	8	no	—
47	C₂²×C₄×C₃₁	—	2	4	no	—
48	C₂²×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
49	C₂²×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
50	C₂²×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
51	C₂²×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
52	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
53	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
54	C₂²×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
55	C₂²×C₄×C₃₁	C₂×Q₈	1	2	yes	—
56	C₂²×C₄×C₃₁	C₂×Q₈	1	2	yes	—
57	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
58	M₁₆×C₃₁	—	2	16	no	—
59	C₂×C₈×C₃₁	—	2	4	no	—
60	C₂×C₈×C₃₁	—	2	16	no	—
61	C₂×C₈×C₃₁	—	2	8	no	—
62	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
63	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
64	C₂×C₈×C₃₁	#41	1	8	yes	—
65	C₄²×C₃₁	C₄⋊C₄	1	2	yes	—
66	C₄²×C₃₁	C₄⋊C₄	1	2	yes	—
67	C₄²×C₃₁	C₄⋊C₄	1	2	yes	—
68	C₄²×C₃₁	C₄⋊C₄	1	2	yes	—
69	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
70	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
71	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
72	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
73	C₄²×C₃₁	C₂×D₄	1	2	yes	—
74	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
75	#20826	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
76	C₂³×C₃₁	—	2	2	no	—
77	C₂³×C₃₁	—	2	4	no	—
78	C₂³×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
79	C₂³×C₃₁	C₂²×Q₈	1	1	yes	characteristic
80	C₂×C₄×C₃₁	—	2	2	no	—
81	C₂×C₄×C₃₁	—	2	4	no	—
82	C₂×C₄×C₃₁	—	2	4	no	—
83	C₂×C₄×C₃₁	—	2	4	no	—
84	C₂×C₄×C₃₁	—	2	4	no	—
85	C₂×C₄×C₃₁	—	2	4	no	—
86	C₂×C₄×C₃₁	—	2	4	no	—
87	C₂×C₄×C₃₁	—	2	4	no	—
88	C₂×C₄×C₃₁	#108	1	8	yes	complemented
89	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
90	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
91	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
92	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
93	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
94	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
95	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
96	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
97	C₈×C₃₁	—	2	4	no	—
98	C₈×C₃₁	—	4	8	no	—
99	C₈×C₃₁	#117	1	8	yes	—
100	#2920	C₂×C₃₁	1	2	yes	—
101	C₂²×C₄×C₈	C₂×C₃₁	1	1	yes	characteristic
102	#1848	C₂×C₃₁	1	2	yes	complemented
103	#1837	C₂×C₃₁	1	2	yes	complemented
104	#1810	C₂×C₃₁	1	4	yes	—
105	C₄×#108	C₂×C₃₁	1	4	yes	complemented
106	C₂²×C₃₁	—	2	8	no	—
107	C₂²×C₃₁	—	2	2	no	—
108	C₂²×C₃₁	—	2	2	no	—
109	C₂²×C₃₁	—	2	2	no	—
110	C₂²×C₃₁	C₂×#108	1	4	yes	—
111	C₂²×C₃₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
112	C₂²×C₃₁	#533	1	2	yes	—
113	C₄×C₃₁	—	2	2	no	—
114	C₄×C₃₁	—	2	2	no	—
115	C₄×C₃₁	—	2	2	no	—
116	C₄×C₃₁	C₂×#108	1	4	yes	—
117	C₄×C₃₁	C₂×#108	1	4	yes	—
118	C₄×C₃₁	C₂×#117	1	2	yes	—
119	C₄×C₃₁	#532	1	2	yes	—
120	C₂³×C₈	C₂²×C₃₁	1	2	yes	—
121	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
122	#432	—	2	4	no	—
123	#431	—	2	4	no	—
124	C₂×#108	C₂²×C₃₁	1	8	yes	—
125	#423	C₂²×C₃₁	1	4	yes	—
126	#407	C₂²×C₃₁	1	4	yes	—
127	C₄×M₁₆	C₂²×C₃₁	1	4	yes	—
128	C₂×C₄×C₈	C₄×C₃₁	1	4	yes	—
129	C₂×C₄×C₈	C₄×C₃₁	1	4	yes	—
130	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
131	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
132	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
133	#341	C₂²×C₃₁	1	8	yes	—
134	C₂×C₃₁	—	2	2	no	—
135	C₂×C₃₁	—	2	4	no	—
136	C₂×C₃₁	#2285	1	1	yes	characteristic
137	C₂×C₃₁	C₂²×#108	1	2	yes	—
138	C₂³×C₄	Q₈×C₃₁	1	2	yes	—
139	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
140	C₂×M₁₆	C₂³×C₃₁	1	8	yes	—
141	C₂²×C₈	—	2	8	no	—
142	C₂²×C₈	C₂×C₄×C₃₁	1	8	yes	—
143	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
144	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
145	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
146	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
147	C₂×C₄²	D₄×C₃₁	1	2	yes	—
148	C₂×C₄²	D₄×C₃₁	1	2	yes	—
149	C₂×C₄²	Q₈×C₃₁	1	2	yes	—
150	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
151	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
152	#108	—	2	16	no	—
153	C₄⋊C₈	—	2	8	no	—
154	C₂²⋊C₈	—	2	8	no	—
155	#97	—	2	4	no	—
156	C₄×C₈	—	2	4	no	—
157	C₄×C₈	C₂×C₄×C₃₁	1	2	yes	—
158	C₄×C₈	C₂×C₄×C₃₁	1	2	yes	—
159	C₄×C₈	C₂×C₄×C₃₁	1	8	yes	—
160	C₄×C₈	C₂×C₄×C₃₁	1	2	yes	—
161	C₄×C₈	C₂×C₄×C₃₁	1	2	yes	—
162	C₃₁	#20826	1	1	yes	characteristic, direct product factor, Sylow
163	C₂⁴	C₂×Q₈×C₃₁	1	1	yes	characteristic
164	C₂²×C₄	—	2	8	no	—
165	C₂²×C₄	—	2	4	no	—
166	C₂²×C₄	C₄⋊C₄×C₃₁	1	2	yes	—
167	C₂²×C₄	C₄⋊C₄×C₃₁	1	2	yes	—
168	C₂²×C₄	C₄⋊C₄×C₃₁	1	2	yes	—
169	C₂²×C₄	C₄⋊C₄×C₃₁	1	2	yes	—
170	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
171	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
172	C₂²×C₄	C₂×Q₈×C₃₁	1	1	yes	characteristic
173	C₂²×C₄	C₂×Q₈×C₃₁	1	2	yes	—
174	C₂²×C₄	C₂×Q₈×C₃₁	1	2	yes	—
175	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
176	M₁₆	—	2	16	no	—
177	C₂×C₈	—	2	4	no	—
178	C₂×C₈	—	2	16	no	—
179	C₂×C₈	—	2	8	no	—
180	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
181	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
182	C₂×C₈	#41×C₃₁	1	8	yes	—
183	C₄²	C₄⋊C₄×C₃₁	1	2	yes	—
184	C₄²	C₄⋊C₄×C₃₁	1	2	yes	—
185	C₄²	C₄⋊C₄×C₃₁	1	2	yes	—
186	C₄²	C₄⋊C₄×C₃₁	1	2	yes	—
187	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
188	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
189	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
190	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
191	C₄²	C₂×D₄×C₃₁	1	2	yes	—
192	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
193	C₂³	—	2	4	no	—
194	C₂³	—	2	2	no	—
195	C₂³	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
196	C₂³	C₂²×Q₈×C₃₁	1	1	yes	characteristic
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	4	no	—
200	C₂×C₄	—	2	4	no	—
201	C₂×C₄	—	2	4	no	—
202	C₂×C₄	—	2	4	no	—
203	C₂×C₄	—	2	4	no	—
204	C₂×C₄	—	2	4	no	—
205	C₂×C₄	C₃₁×#108	1	8	yes	complemented
206	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
207	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
208	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
209	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
210	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
211	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
212	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
213	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
214	C₈	—	2	4	no	—
215	C₈	—	4	8	no	—
216	C₈	C₃₁×#117	1	8	yes	—
217	C₂²	—	2	8	no	—
218	C₂²	—	2	2	no	—
219	C₂²	—	2	2	no	—
220	C₂²	—	2	2	no	—
221	C₂²	C₂×C₃₁×#108	1	4	yes	—
222	C₂²	C₂²×C₄⋊C₄×C₃₁	1	1	yes	characteristic
223	C₂²	C₃₁×#533	1	2	yes	—
224	C₄	—	2	2	no	—
225	C₄	—	2	2	no	—
226	C₄	—	2	2	no	—
227	C₄	C₂×C₃₁×#108	1	4	yes	—
228	C₄	C₂×C₃₁×#108	1	4	yes	—
229	C₄	C₂×C₃₁×#117	1	2	yes	—
230	C₄	C₃₁×#532	1	2	yes	—
231	C₂	—	2	4	no	—
232	C₂	—	2	2	no	—
233	C₂	C₃₁×#2285	1	1	yes	characteristic
234	C₂	C₂²×C₃₁×#108	1	2	yes	—

	subgroup	G/H
center	C₄²×C₃₁	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₃₁	C₂×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄²×C₃₁ ◁ C₂²×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃₁×#1848	C₂	semidirect
4. C₃₁×#1837	C₂	semidirect
6. C₄×C₃₁×#108	C₂	semidirect
75. #20826	C₃₁	direct (N × H)
88. C₂×C₄×C₃₁	#108	semidirect
102. #1848	C₂×C₃₁	semidirect
103. #1837	C₂×C₃₁	semidirect
105. C₄×#108	C₂×C₃₁	semidirect
162. C₃₁	#20826	direct (N × H)
205. C₂×C₄	C₃₁×#108	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#2920	2
C₂²×C₄×C₈×C₃₁	2
C₃₁×#1848	2
C₃₁×#1837	2
C₃₁×#1810	2
C₄×C₃₁×#108	2
#20826	31

Subgroups — C31×#20826 #21477585

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#20826 #21477585