Subgroups — C₃₁×#20429 #21477188

Order 7936.

1150 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#423	C₂	1	2	yes	maximal
2	C₂×C₄×M₁₆×C₃₁	C₂	1	2	yes	maximal
3	C₂×C₃₁×#376	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1768	C₂	1	8	yes	maximal, complemented
5	C₃₁×#1728	C₂	1	2	yes	maximal
6	C₂²×M₁₆×C₃₁	C₂²	1	4	yes	—
7	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
8	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
9	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
10	C₃₁×#423	C₄	1	16	yes	complemented
11	C₂×C₃₁×C₄⋊C₈	C₂²	1	8	yes	—
12	C₄×M₁₆×C₃₁	—	2	8	no	—
13	C₂×C₃₁×#97	C₂²	1	4	yes	—
14	C₂×C₄×C₈×C₃₁	C₂²	1	4	yes	—
15	C₃₁×#376	C₄	1	4	yes	—
16	C₃₁×#376	C₂²	1	4	yes	—
17	C₂×C₃₁×#95	C₄	1	4	yes	—
18	C₃₁×#336	C₂²	1	8	yes	—
19	C₂³×C₄×C₃₁	C₂×C₄	1	2	yes	—
20	C₂³×C₄×C₃₁	C₂×C₄	1	2	yes	—
21	C₂³×C₄×C₃₁	C₂×C₄	1	2	yes	—
22	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
23	C₂×M₁₆×C₃₁	—	2	8	no	—
24	C₂×M₁₆×C₃₁	—	2	8	no	—
25	C₂×M₁₆×C₃₁	C₂×C₄	1	16	yes	—
26	C₂²×C₈×C₃₁	C₂³	1	8	yes	—
27	C₂×C₄²×C₃₁	—	2	4	no	—
28	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₃₁	C₂×C₄	1	8	yes	—
31	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₃₁	D₄	1	4	yes	—
33	C₂×C₄²×C₃₁	Q₈	1	4	yes	—
34	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
35	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
36	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
37	C₃₁×C₄⋊C₈	C₂×C₄	1	32	yes	complemented
38	C₃₁×#97	—	2	8	no	—
39	C₄×C₈×C₃₁	—	2	8	no	—
40	C₃₁×#95	C₂×C₄	1	16	yes	—
41	C₂⁴×C₃₁	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₃₁	—	2	4	no	—
43	C₂²×C₄×C₃₁	—	4	4	no	—
44	C₂²×C₄×C₃₁	—	4	4	no	—
45	C₂²×C₄×C₃₁	—	4	4	no	—
46	C₂²×C₄×C₃₁	C₄²	1	8	yes	—
47	C₂²×C₄×C₃₁	C₄⋊C₄	1	8	yes	—
48	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
49	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
50	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
51	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
53	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
54	C₂²×C₄×C₃₁	C₂×Q₈	1	2	yes	—
55	C₂²×C₄×C₃₁	#41	1	4	yes	—
56	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
57	M₁₆×C₃₁	—	2	32	no	—
58	C₂×C₈×C₃₁	—	2	8	no	—
59	C₂×C₈×C₃₁	—	2	8	no	—
60	C₂×C₈×C₃₁	C₂²×C₄	1	16	yes	—
61	C₄²×C₃₁	—	2	4	no	—
62	C₄²×C₃₁	—	2	4	no	—
63	C₄²×C₃₁	C₄²	1	8	yes	—
64	C₄²×C₃₁	C₄⋊C₄	1	8	yes	—
65	C₄²×C₃₁	C₂²×C₄	1	8	yes	—
66	C₄²×C₃₁	C₂×D₄	1	4	yes	—
67	C₄²×C₃₁	C₂×Q₈	1	4	yes	—
68	#20429	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
69	C₂³×C₃₁	—	2	4	no	—
70	C₂³×C₃₁	C₂×C₄²	1	2	yes	—
71	C₂³×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
72	C₂³×C₃₁	#117	1	2	yes	—
73	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
74	C₂×C₄×C₃₁	—	2	8	no	—
75	C₂×C₄×C₃₁	—	2	8	no	—
76	C₂×C₄×C₃₁	—	2	4	no	—
77	C₂×C₄×C₃₁	—	2	8	no	—
78	C₂×C₄×C₃₁	—	4	8	no	—
79	C₂×C₄×C₃₁	—	4	8	no	—
80	C₂×C₄×C₃₁	—	4	8	no	—
81	C₂×C₄×C₃₁	C₂×C₄²	1	8	yes	—
82	C₂×C₄×C₃₁	C₂×C₄²	1	2	yes	—
83	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	8	yes	—
84	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
85	C₂×C₄×C₃₁	#117	1	2	yes	—
86	C₂×C₄×C₃₁	C₄×D₄	1	8	yes	—
87	C₂×C₄×C₃₁	C₄×Q₈	1	8	yes	—
88	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
89	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
90	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
91	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
92	C₈×C₃₁	—	2	16	no	—
93	C₂×#423	C₂×C₃₁	1	2	yes	—
94	C₂×C₄×M₁₆	C₂×C₃₁	1	2	yes	—
95	C₂×#376	C₂×C₃₁	1	1	yes	characteristic
96	#1768	C₂×C₃₁	1	8	yes	complemented
97	#1728	C₂×C₃₁	1	2	yes	—
98	C₂²×C₃₁	—	2	4	no	—
99	C₂²×C₃₁	—	2	4	no	—
100	C₂²×C₃₁	—	2	4	no	—
101	C₂²×C₃₁	C₄×C₄⋊C₄	1	4	yes	—
102	C₂²×C₃₁	C₂²×C₄²	1	1	yes	characteristic
103	C₂²×C₃₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
104	C₂²×C₃₁	C₂×#117	1	1	yes	characteristic
105	C₂²×C₃₁	#568	1	4	yes	—
106	C₄×C₃₁	—	2	8	no	—
107	C₄×C₃₁	—	2	8	no	—
108	C₄×C₃₁	—	4	4	no	—
109	C₄×C₃₁	C₄×C₄⋊C₄	1	4	yes	—
110	C₄×C₃₁	C₂×C₄×D₄	1	2	yes	—
111	C₄×C₃₁	C₂×C₄×Q₈	1	2	yes	—
112	C₂²×M₁₆	C₂²×C₃₁	1	4	yes	—
113	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
114	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
115	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
116	#423	C₄×C₃₁	1	16	yes	complemented
117	C₂×C₄⋊C₈	C₂²×C₃₁	1	8	yes	—
118	C₄×M₁₆	—	2	8	no	—
119	C₂×#97	C₂²×C₃₁	1	4	yes	—
120	C₂×C₄×C₈	C₂²×C₃₁	1	4	yes	—
121	#376	C₄×C₃₁	1	4	yes	—
122	#376	C₂²×C₃₁	1	4	yes	—
123	C₂×#95	C₄×C₃₁	1	4	yes	—
124	#336	C₂²×C₃₁	1	8	yes	—
125	C₂×C₃₁	—	2	4	no	—
126	C₂×C₃₁	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
127	C₂×C₃₁	#2873	1	2	yes	—
128	C₂×C₃₁	#2905	1	2	yes	—
129	C₂×C₃₁	#2921	1	2	yes	—
130	C₂³×C₄	C₂×C₄×C₃₁	1	2	yes	—
131	C₂³×C₄	C₂×C₄×C₃₁	1	2	yes	—
132	C₂³×C₄	C₂×C₄×C₃₁	1	2	yes	—
133	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
134	C₂×M₁₆	—	2	8	no	—
135	C₂×M₁₆	—	2	8	no	—
136	C₂×M₁₆	C₂×C₄×C₃₁	1	16	yes	—
137	C₂²×C₈	C₂³×C₃₁	1	8	yes	—
138	C₂×C₄²	—	2	4	no	—
139	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
140	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
141	C₂×C₄²	C₂×C₄×C₃₁	1	8	yes	—
142	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
143	C₂×C₄²	D₄×C₃₁	1	4	yes	—
144	C₂×C₄²	Q₈×C₃₁	1	4	yes	—
145	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
146	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
147	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
148	C₄⋊C₈	C₂×C₄×C₃₁	1	32	yes	complemented
149	#97	—	2	8	no	—
150	C₄×C₈	—	2	8	no	—
151	#95	C₂×C₄×C₃₁	1	16	yes	—
152	C₃₁	#20429	1	1	yes	characteristic, direct product factor, Sylow
153	C₂⁴	C₂²×C₄×C₃₁	1	1	yes	characteristic
154	C₂²×C₄	—	2	4	no	—
155	C₂²×C₄	—	4	4	no	—
156	C₂²×C₄	—	4	4	no	—
157	C₂²×C₄	—	4	4	no	—
158	C₂²×C₄	C₄²×C₃₁	1	8	yes	—
159	C₂²×C₄	C₄⋊C₄×C₃₁	1	8	yes	—
160	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
161	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
162	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
163	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
164	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
165	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
166	C₂²×C₄	C₂×Q₈×C₃₁	1	2	yes	—
167	C₂²×C₄	#41×C₃₁	1	4	yes	—
168	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
169	M₁₆	—	2	32	no	—
170	C₂×C₈	—	2	8	no	—
171	C₂×C₈	—	2	8	no	—
172	C₂×C₈	C₂²×C₄×C₃₁	1	16	yes	—
173	C₄²	—	2	4	no	—
174	C₄²	—	2	4	no	—
175	C₄²	C₄²×C₃₁	1	8	yes	—
176	C₄²	C₄⋊C₄×C₃₁	1	8	yes	—
177	C₄²	C₂²×C₄×C₃₁	1	8	yes	—
178	C₄²	C₂×D₄×C₃₁	1	4	yes	—
179	C₄²	C₂×Q₈×C₃₁	1	4	yes	—
180	C₂³	—	2	4	no	—
181	C₂³	C₂×C₄²×C₃₁	1	2	yes	—
182	C₂³	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
183	C₂³	C₃₁×#117	1	2	yes	—
184	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
185	C₂×C₄	—	2	8	no	—
186	C₂×C₄	—	2	8	no	—
187	C₂×C₄	—	2	4	no	—
188	C₂×C₄	—	2	8	no	—
189	C₂×C₄	—	4	8	no	—
190	C₂×C₄	—	4	8	no	—
191	C₂×C₄	—	4	8	no	—
192	C₂×C₄	C₂×C₄²×C₃₁	1	8	yes	—
193	C₂×C₄	C₂×C₄²×C₃₁	1	2	yes	—
194	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	8	yes	—
195	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
196	C₂×C₄	C₃₁×#117	1	2	yes	—
197	C₂×C₄	C₄×D₄×C₃₁	1	8	yes	—
198	C₂×C₄	C₄×Q₈×C₃₁	1	8	yes	—
199	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
200	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
201	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
202	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
203	C₈	—	2	16	no	—
204	C₂²	—	2	4	no	—
205	C₂²	—	2	4	no	—
206	C₂²	—	2	4	no	—
207	C₂²	C₄×C₄⋊C₄×C₃₁	1	4	yes	—
208	C₂²	C₂²×C₄²×C₃₁	1	1	yes	characteristic
209	C₂²	C₂²×C₄⋊C₄×C₃₁	1	1	yes	characteristic
210	C₂²	C₂×C₃₁×#117	1	1	yes	characteristic
211	C₂²	C₃₁×#568	1	4	yes	—
212	C₄	—	2	8	no	—
213	C₄	—	2	8	no	—
214	C₄	—	4	4	no	—
215	C₄	C₄×C₄⋊C₄×C₃₁	1	4	yes	—
216	C₄	C₂×C₄×D₄×C₃₁	1	2	yes	—
217	C₄	C₂×C₄×Q₈×C₃₁	1	2	yes	—
218	C₂	—	2	4	no	—
219	C₂	C₂×C₄×C₄⋊C₄×C₃₁	1	1	yes	characteristic
220	C₂	C₃₁×#2873	1	2	yes	—
221	C₂	C₃₁×#2905	1	2	yes	—
222	C₂	C₃₁×#2921	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
4. C₃₁×#1768	C₂	semidirect
10. C₃₁×#423	C₄	semidirect
37. C₃₁×C₄⋊C₈	C₂×C₄	semidirect
68. #20429	C₃₁	direct (N × H)
96. #1768	C₂×C₃₁	semidirect
116. #423	C₄×C₃₁	semidirect
148. C₄⋊C₈	C₂×C₄×C₃₁	semidirect
152. C₃₁	#20429	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×#423	2
C₂×C₄×M₁₆×C₃₁	2
C₂×C₃₁×#376	2
C₃₁×#1768	2
C₃₁×#1728	2
#20429	31

Subgroups — C31×#20429 #21477188

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#20429 #21477188