Subgroups — C₃₁×#8158 #21464917

Order 7936.

400 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#433	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1458	C₂	1	2	yes	maximal
3	C₃₁×#1452	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1450	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1281	C₂	1	2	yes	maximal, complemented
6	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
7	C₃₁×#433	—	2	1	no	—
8	C₃₁×#433	—	2	1	no	—
9	C₃₁×#433	—	2	2	no	—
10	C₃₁×#431	C₂²	1	2	yes	—
11	C₃₁×#431	C₂²	1	2	yes	—
12	C₂×C₃₁×C₄⋊C₈	C₄	1	2	yes	—
13	C₂×C₃₁×C₂²⋊C₈	C₄	1	2	yes	—
14	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
16	C₃₁×#324	C₄	1	4	yes	—
17	C₃₁×#323	C₄	1	4	yes	—
18	C₃₁×#322	—	2	1	no	—
19	C₃₁×#322	—	2	2	no	—
20	C₈²×C₃₁	—	2	1	no	—
21	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
23	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
24	C₃₁×#117	—	2	1	no	—
25	C₃₁×#117	—	2	1	no	—
26	C₃₁×#117	—	2	2	no	—
27	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
28	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
31	C₃₁×C₄⋊C₈	—	2	2	no	—
32	C₃₁×C₄⋊C₈	—	2	2	no	—
33	C₃₁×C₄⋊C₈	—	2	2	no	—
34	C₃₁×C₄⋊C₈	—	2	2	no	—
35	C₃₁×C₂²⋊C₈	—	2	2	no	—
36	C₃₁×C₂²⋊C₈	—	2	2	no	—
37	C₃₁×C₂²⋊C₈	C₂×C₄	1	4	yes	—
38	C₃₁×C₂²⋊C₈	C₂×C₄	1	4	yes	—
39	C₃₁×#97	—	2	1	no	—
40	C₃₁×#97	—	2	1	no	—
41	C₄×C₈×C₃₁	—	2	2	no	—
42	C₄×C₈×C₃₁	—	2	2	no	—
43	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
44	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
45	C₄×C₈×C₃₁	D₄	1	2	yes	—
46	C₂⁴×C₃₁	C₄²	1	1	yes	characteristic
47	C₂²×C₄×C₃₁	—	2	1	no	—
48	C₂²×C₄×C₃₁	—	2	1	no	—
49	C₂²×C₄×C₃₁	—	2	4	no	—
50	C₂²×C₄×C₃₁	C₄²	1	1	yes	characteristic
51	C₂²×C₄×C₃₁	C₄²	1	2	yes	—
52	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
53	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₈×C₃₁	—	2	4	no	—
55	C₂×C₈×C₃₁	—	2	2	no	—
56	C₂×C₈×C₃₁	—	2	4	no	—
57	C₂×C₈×C₃₁	—	2	4	no	—
58	C₂×C₈×C₃₁	—	2	4	no	—
59	C₂×C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
60	C₂×C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
61	C₄⋊C₄×C₃₁	—	2	2	no	—
62	C₄⋊C₄×C₃₁	—	2	2	no	—
63	C₂²⋊C₄×C₃₁	—	2	2	no	—
64	C₂²⋊C₄×C₃₁	—	2	2	no	—
65	C₄²×C₃₁	—	2	1	no	—
66	C₄²×C₃₁	—	2	1	no	—
67	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
68	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
69	C₄²×C₃₁	#41	1	2	yes	—
70	#8158	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
71	C₂³×C₃₁	—	2	1	no	—
72	C₂³×C₃₁	—	2	2	no	—
73	C₂³×C₃₁	—	4	2	no	—
74	C₂³×C₃₁	C₂×C₄²	1	1	yes	characteristic
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	—	4	2	no	—
79	C₂×C₄×C₃₁	—	4	2	no	—
80	C₂×C₄×C₃₁	—	4	2	no	—
81	C₂×C₄×C₃₁	—	4	2	no	—
82	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₃₁	#117	1	1	yes	characteristic
84	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
85	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
86	C₂×C₄×C₃₁	#131	1	4	yes	—
87	C₈×C₃₁	—	2	2	no	—
88	C₈×C₃₁	—	2	2	no	—
89	C₈×C₃₁	—	4	4	no	—
90	C₈×C₃₁	—	4	4	no	—
91	C₈×C₃₁	—	4	2	no	—
92	C₂×#433	C₂×C₃₁	1	1	yes	characteristic
93	#1458	C₂×C₃₁	1	2	yes	—
94	#1452	C₂×C₃₁	1	1	yes	characteristic, complemented
95	#1450	C₂×C₃₁	1	1	yes	characteristic, complemented
96	#1281	C₂×C₃₁	1	2	yes	complemented
97	C₂²×C₃₁	—	2	2	no	—
98	C₂²×C₃₁	—	4	1	no	—
99	C₂²×C₃₁	—	4	2	no	—
100	C₂²×C₃₁	—	8	2	no	—
101	C₂²×C₃₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
102	C₂²×C₃₁	#405	1	2	yes	—
103	C₄×C₃₁	—	2	2	no	—
104	C₄×C₃₁	—	2	2	no	—
105	C₄×C₃₁	—	4	1	no	—
106	C₄×C₃₁	—	4	2	no	—
107	C₄×C₃₁	#408	1	2	yes	—
108	C₄×C₃₁	#433	1	2	yes	—
109	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
110	#433	—	2	1	no	—
111	#433	—	2	1	no	—
112	#433	—	2	2	no	—
113	#431	C₂²×C₃₁	1	2	yes	—
114	#431	C₂²×C₃₁	1	2	yes	—
115	C₂×C₄⋊C₈	C₄×C₃₁	1	2	yes	—
116	C₂×C₂²⋊C₈	C₄×C₃₁	1	2	yes	—
117	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
118	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
119	#324	C₄×C₃₁	1	4	yes	—
120	#323	C₄×C₃₁	1	4	yes	—
121	#322	—	2	1	no	—
122	#322	—	2	2	no	—
123	C₈²	—	2	1	no	—
124	C₂×C₃₁	—	2	2	no	—
125	C₂×C₃₁	—	8	1	no	—
126	C₂×C₃₁	#1750	1	1	yes	characteristic
127	C₂×C₃₁	#1757	1	2	yes	—
128	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
129	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
130	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
131	#117	—	2	1	no	—
132	#117	—	2	1	no	—
133	#117	—	2	2	no	—
134	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
135	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
136	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
137	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
138	C₄⋊C₈	—	2	2	no	—
139	C₄⋊C₈	—	2	2	no	—
140	C₄⋊C₈	—	2	2	no	—
141	C₄⋊C₈	—	2	2	no	—
142	C₂²⋊C₈	—	2	2	no	—
143	C₂²⋊C₈	—	2	2	no	—
144	C₂²⋊C₈	C₂×C₄×C₃₁	1	4	yes	—
145	C₂²⋊C₈	C₂×C₄×C₃₁	1	4	yes	—
146	#97	—	2	1	no	—
147	#97	—	2	1	no	—
148	C₄×C₈	—	2	2	no	—
149	C₄×C₈	—	2	2	no	—
150	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
151	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
152	C₄×C₈	D₄×C₃₁	1	2	yes	—
153	C₃₁	#8158	1	1	yes	characteristic, direct product factor, Sylow
154	C₂⁴	C₄²×C₃₁	1	1	yes	characteristic
155	C₂²×C₄	—	2	1	no	—
156	C₂²×C₄	—	2	1	no	—
157	C₂²×C₄	—	2	4	no	—
158	C₂²×C₄	C₄²×C₃₁	1	1	yes	characteristic
159	C₂²×C₄	C₄²×C₃₁	1	2	yes	—
160	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
161	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
162	C₂×C₈	—	2	4	no	—
163	C₂×C₈	—	2	2	no	—
164	C₂×C₈	—	2	4	no	—
165	C₂×C₈	—	2	4	no	—
166	C₂×C₈	—	2	4	no	—
167	C₂×C₈	C₂²⋊C₄×C₃₁	1	2	yes	—
168	C₂×C₈	C₂²⋊C₄×C₃₁	1	2	yes	—
169	C₄⋊C₄	—	2	2	no	—
170	C₄⋊C₄	—	2	2	no	—
171	C₂²⋊C₄	—	2	2	no	—
172	C₂²⋊C₄	—	2	2	no	—
173	C₄²	—	2	1	no	—
174	C₄²	—	2	1	no	—
175	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
176	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
177	C₄²	#41×C₃₁	1	2	yes	—
178	C₂³	—	2	1	no	—
179	C₂³	—	2	2	no	—
180	C₂³	—	4	2	no	—
181	C₂³	C₂×C₄²×C₃₁	1	1	yes	characteristic
182	C₂×C₄	—	2	2	no	—
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	4	2	no	—
186	C₂×C₄	—	4	2	no	—
187	C₂×C₄	—	4	2	no	—
188	C₂×C₄	—	4	2	no	—
189	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
190	C₂×C₄	C₃₁×#117	1	1	yes	characteristic
191	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
192	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
193	C₂×C₄	C₃₁×#131	1	4	yes	—
194	C₈	—	2	2	no	—
195	C₈	—	2	2	no	—
196	C₈	—	4	4	no	—
197	C₈	—	4	4	no	—
198	C₈	—	4	2	no	—
199	C₂²	—	2	2	no	—
200	C₂²	—	4	1	no	—
201	C₂²	—	4	2	no	—
202	C₂²	—	8	2	no	—
203	C₂²	C₄×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
204	C₂²	C₃₁×#405	1	2	yes	—
205	C₄	—	2	2	no	—
206	C₄	—	2	2	no	—
207	C₄	—	4	1	no	—
208	C₄	—	4	2	no	—
209	C₄	C₃₁×#408	1	2	yes	—
210	C₄	C₃₁×#433	1	2	yes	—
211	C₂	—	2	2	no	—
212	C₂	—	8	1	no	—
213	C₂	C₃₁×#1750	1	1	yes	characteristic
214	C₂	C₃₁×#1757	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₃₁×#1452	C₂	semidirect
4. C₃₁×#1450	C₂	semidirect
5. C₃₁×#1281	C₂	semidirect
70. #8158	C₃₁	direct (N × H)
94. #1452	C₂×C₃₁	semidirect
95. #1450	C₂×C₃₁	semidirect
96. #1281	C₂×C₃₁	semidirect
153. C₃₁	#8158	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#433	2
C₃₁×#1458	2
C₃₁×#1452	2
C₃₁×#1450	2
C₃₁×#1281	2
#8158	31

Subgroups — C31×#8158 #21464917

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#8158 #21464917