Subgroups — C₃₁×#17325 #21474084

Order 7936.

772 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	1	yes	maximal, characteristic
2	C₂×C₃₁×#431	C₂	1	2	yes	maximal
3	C₃₁×#1453	C₂	1	4	yes	maximal, complemented
4	C₃₁×#1452	C₂	1	4	yes	maximal, complemented
5	C₃₁×#1450	C₂	1	4	yes	maximal, complemented
6	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
7	C₃₁×#432	C₄	1	4	yes	—
8	C₃₁×#431	C₄	1	4	yes	—
9	C₃₁×#431	C₄	1	4	yes	—
10	C₃₁×#431	C₂²	1	2	yes	—
11	C₃₁×#431	C₂²	1	2	yes	—
12	C₃₁×#431	C₂²	1	4	yes	—
13	C₃₁×#431	C₂²	1	4	yes	—
14	C₃₁×#423	C₄	1	2	yes	—
15	C₂×C₃₁×C₄⋊C₈	C₄	1	4	yes	—
16	C₃₁×#407	C₄	1	2	yes	—
17	C₂×C₃₁×C₂²⋊C₈	C₄	1	4	yes	—
18	C₄×M₁₆×C₃₁	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₃₁	C₂²	1	4	yes	—
20	C₃₁×#322	C₂²	1	8	yes	complemented
21	C₃₁×#322	C₂²	1	4	yes	complemented
22	C₈²×C₃₁	C₂²	1	4	yes	complemented
23	C₂³×C₄×C₃₁	C₂×C₄	1	2	yes	—
24	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
25	C₂×M₁₆×C₃₁	C₂×C₄	1	4	yes	—
26	C₂²×C₈×C₃₁	C₂×C₄	1	8	yes	—
27	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
33	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
34	C₃₁×C₄⋊C₈	C₂×C₄	1	4	yes	—
35	C₃₁×C₄⋊C₈	C₂×C₄	1	4	yes	—
36	C₃₁×C₄⋊C₈	C₂×C₄	1	8	yes	—
37	C₃₁×C₄⋊C₈	C₂×C₄	1	8	yes	—
38	C₃₁×C₂²⋊C₈	C₂×C₄	1	4	yes	—
39	C₃₁×C₂²⋊C₈	C₂×C₄	1	4	yes	—
40	C₃₁×C₂²⋊C₈	C₂×C₄	1	8	yes	—
41	C₃₁×C₂²⋊C₈	C₂×C₄	1	8	yes	—
42	C₃₁×#97	C₂×C₄	1	4	yes	—
43	C₄×C₈×C₃₁	C₂×C₄	1	8	yes	—
44	C₄×C₈×C₃₁	C₂³	1	4	yes	—
45	C₄×C₈×C₃₁	C₂³	1	4	yes	—
46	C₄×C₈×C₃₁	C₂³	1	4	yes	—
47	C₂⁴×C₃₁	C₄²	1	1	yes	characteristic
48	C₂²×C₄×C₃₁	—	2	8	no	—
49	C₂²×C₄×C₃₁	—	4	2	no	—
50	C₂²×C₄×C₃₁	C₄²	1	2	yes	—
51	C₂²×C₄×C₃₁	C₄²	1	2	yes	—
52	C₂²×C₄×C₃₁	C₄²	1	2	yes	—
53	C₂²×C₄×C₃₁	C₄²	1	1	yes	characteristic
54	C₂²×C₄×C₃₁	C₄²	1	2	yes	—
55	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
56	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
57	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
58	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
60	M₁₆×C₃₁	—	4	4	no	—
61	C₂×C₈×C₃₁	—	2	16	no	—
62	C₂×C₈×C₃₁	C₂²×C₄	1	8	yes	—
63	C₂×C₈×C₃₁	C₂²×C₄	1	8	yes	—
64	C₂×C₈×C₃₁	C₂²×C₄	1	8	yes	—
65	C₄²×C₃₁	C₄²	1	1	yes	characteristic
66	C₄²×C₃₁	C₄²	1	2	yes	—
67	C₄²×C₃₁	C₄²	1	2	yes	—
68	C₄²×C₃₁	C₄²	1	1	yes	characteristic
69	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
70	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
71	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
72	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
73	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
74	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
75	#17325	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
76	C₂³×C₃₁	—	2	4	no	—
77	C₂³×C₃₁	—	4	1	no	—
78	C₂³×C₃₁	C₂×C₄²	1	1	yes	characteristic
79	C₂³×C₃₁	C₂×C₄²	1	2	yes	—
80	C₂×C₄×C₃₁	—	2	4	no	—
81	C₂×C₄×C₃₁	—	2	4	no	—
82	C₂×C₄×C₃₁	—	2	4	no	—
83	C₂×C₄×C₃₁	—	2	2	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	—	4	2	no	—
88	C₂×C₄×C₃₁	—	4	1	no	—
89	C₂×C₄×C₃₁	C₂×C₄²	1	2	yes	—
90	C₂×C₄×C₃₁	C₂×C₄²	1	2	yes	—
91	C₂×C₄×C₃₁	C₂×C₄²	1	2	yes	—
92	C₂×C₄×C₃₁	C₂×C₄²	1	1	yes	characteristic
93	C₂×C₄×C₃₁	C₂×C₄²	1	2	yes	—
94	C₂×C₄×C₃₁	#131	1	8	yes	—
95	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
96	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
97	C₈×C₃₁	—	2	8	no	—
98	C₈×C₃₁	—	2	8	no	—
99	C₈×C₃₁	—	2	8	no	—
100	#2920	C₂×C₃₁	1	1	yes	characteristic
101	C₂×#431	C₂×C₃₁	1	2	yes	—
102	#1453	C₂×C₃₁	1	4	yes	complemented
103	#1452	C₂×C₃₁	1	4	yes	complemented
104	#1450	C₂×C₃₁	1	4	yes	complemented
105	C₂²×C₃₁	—	2	2	no	—
106	C₂²×C₃₁	—	2	2	no	—
107	C₂²×C₃₁	—	2	2	no	—
108	C₂²×C₃₁	—	2	1	no	—
109	C₂²×C₃₁	—	4	4	no	—
110	C₂²×C₃₁	#405	1	4	yes	—
111	C₂²×C₃₁	C₂²×C₄²	1	1	yes	characteristic
112	C₄×C₃₁	—	2	2	no	—
113	C₄×C₃₁	—	2	2	no	—
114	C₄×C₃₁	—	2	1	no	—
115	C₄×C₃₁	—	2	2	no	—
116	C₄×C₃₁	#405	1	4	yes	—
117	C₄×C₃₁	C₂×#131	1	4	yes	—
118	C₄×C₃₁	#568	1	2	yes	—
119	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
120	#432	C₄×C₃₁	1	4	yes	—
121	#431	C₄×C₃₁	1	4	yes	—
122	#431	C₄×C₃₁	1	4	yes	—
123	#431	C₂²×C₃₁	1	2	yes	—
124	#431	C₂²×C₃₁	1	2	yes	—
125	#431	C₂²×C₃₁	1	4	yes	—
126	#431	C₂²×C₃₁	1	4	yes	—
127	#423	C₄×C₃₁	1	2	yes	—
128	C₂×C₄⋊C₈	C₄×C₃₁	1	4	yes	—
129	#407	C₄×C₃₁	1	2	yes	—
130	C₂×C₂²⋊C₈	C₄×C₃₁	1	4	yes	—
131	C₄×M₁₆	C₂²×C₃₁	1	2	yes	—
132	C₂×C₄×C₈	C₂²×C₃₁	1	4	yes	—
133	#322	C₂²×C₃₁	1	8	yes	complemented
134	#322	C₂²×C₃₁	1	4	yes	complemented
135	C₈²	C₂²×C₃₁	1	4	yes	complemented
136	C₂×C₃₁	—	2	4	no	—
137	C₂×C₃₁	—	4	1	no	—
138	C₂×C₃₁	C₂×#405	1	2	yes	—
139	C₂×C₃₁	#2873	1	1	yes	characteristic
140	C₂³×C₄	C₂×C₄×C₃₁	1	2	yes	—
141	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
142	C₂×M₁₆	C₂×C₄×C₃₁	1	4	yes	—
143	C₂²×C₈	C₂×C₄×C₃₁	1	8	yes	—
144	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
145	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
146	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
147	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
148	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
149	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
150	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
151	C₄⋊C₈	C₂×C₄×C₃₁	1	4	yes	—
152	C₄⋊C₈	C₂×C₄×C₃₁	1	4	yes	—
153	C₄⋊C₈	C₂×C₄×C₃₁	1	8	yes	—
154	C₄⋊C₈	C₂×C₄×C₃₁	1	8	yes	—
155	C₂²⋊C₈	C₂×C₄×C₃₁	1	4	yes	—
156	C₂²⋊C₈	C₂×C₄×C₃₁	1	4	yes	—
157	C₂²⋊C₈	C₂×C₄×C₃₁	1	8	yes	—
158	C₂²⋊C₈	C₂×C₄×C₃₁	1	8	yes	—
159	#97	C₂×C₄×C₃₁	1	4	yes	—
160	C₄×C₈	C₂×C₄×C₃₁	1	8	yes	—
161	C₄×C₈	C₂³×C₃₁	1	4	yes	—
162	C₄×C₈	C₂³×C₃₁	1	4	yes	—
163	C₄×C₈	C₂³×C₃₁	1	4	yes	—
164	C₃₁	#17325	1	1	yes	characteristic, direct product factor, Sylow
165	C₂⁴	C₄²×C₃₁	1	1	yes	characteristic
166	C₂²×C₄	—	2	8	no	—
167	C₂²×C₄	—	4	2	no	—
168	C₂²×C₄	C₄²×C₃₁	1	2	yes	—
169	C₂²×C₄	C₄²×C₃₁	1	2	yes	—
170	C₂²×C₄	C₄²×C₃₁	1	2	yes	—
171	C₂²×C₄	C₄²×C₃₁	1	1	yes	characteristic
172	C₂²×C₄	C₄²×C₃₁	1	2	yes	—
173	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
174	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
175	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
176	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
177	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
178	M₁₆	—	4	4	no	—
179	C₂×C₈	—	2	16	no	—
180	C₂×C₈	C₂²×C₄×C₃₁	1	8	yes	—
181	C₂×C₈	C₂²×C₄×C₃₁	1	8	yes	—
182	C₂×C₈	C₂²×C₄×C₃₁	1	8	yes	—
183	C₄²	C₄²×C₃₁	1	1	yes	characteristic
184	C₄²	C₄²×C₃₁	1	2	yes	—
185	C₄²	C₄²×C₃₁	1	2	yes	—
186	C₄²	C₄²×C₃₁	1	1	yes	characteristic
187	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
188	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
189	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
190	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
191	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
192	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
193	C₂³	—	2	4	no	—
194	C₂³	—	4	1	no	—
195	C₂³	C₂×C₄²×C₃₁	1	1	yes	characteristic
196	C₂³	C₂×C₄²×C₃₁	1	2	yes	—
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	2	4	no	—
199	C₂×C₄	—	2	4	no	—
200	C₂×C₄	—	2	4	no	—
201	C₂×C₄	—	2	4	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	4	no	—
204	C₂×C₄	—	4	2	no	—
205	C₂×C₄	—	4	1	no	—
206	C₂×C₄	C₂×C₄²×C₃₁	1	2	yes	—
207	C₂×C₄	C₂×C₄²×C₃₁	1	2	yes	—
208	C₂×C₄	C₂×C₄²×C₃₁	1	2	yes	—
209	C₂×C₄	C₂×C₄²×C₃₁	1	1	yes	characteristic
210	C₂×C₄	C₂×C₄²×C₃₁	1	2	yes	—
211	C₂×C₄	C₃₁×#131	1	8	yes	—
212	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
213	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
214	C₈	—	2	8	no	—
215	C₈	—	2	8	no	—
216	C₈	—	2	8	no	—
217	C₂²	—	2	2	no	—
218	C₂²	—	2	2	no	—
219	C₂²	—	2	2	no	—
220	C₂²	—	2	1	no	—
221	C₂²	—	4	4	no	—
222	C₂²	C₃₁×#405	1	4	yes	—
223	C₂²	C₂²×C₄²×C₃₁	1	1	yes	characteristic
224	C₄	—	2	2	no	—
225	C₄	—	2	2	no	—
226	C₄	—	2	1	no	—
227	C₄	—	2	2	no	—
228	C₄	C₃₁×#405	1	4	yes	—
229	C₄	C₂×C₃₁×#131	1	4	yes	—
230	C₄	C₃₁×#568	1	2	yes	—
231	C₂	—	2	4	no	—
232	C₂	—	4	1	no	—
233	C₂	C₂×C₃₁×#405	1	2	yes	—
234	C₂	C₃₁×#2873	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
3. C₃₁×#1453	C₂	semidirect
4. C₃₁×#1452	C₂	semidirect
5. C₃₁×#1450	C₂	semidirect
20. C₃₁×#322	C₂²	semidirect
21. C₃₁×#322	C₂²	semidirect
22. C₈²×C₃₁	C₂²	semidirect
75. #17325	C₃₁	direct (N × H)
102. #1453	C₂×C₃₁	semidirect
103. #1452	C₂×C₃₁	semidirect
104. #1450	C₂×C₃₁	semidirect
133. #322	C₂²×C₃₁	semidirect
134. #322	C₂²×C₃₁	semidirect
135. C₈²	C₂²×C₃₁	semidirect
164. C₃₁	#17325	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃₁×#2920	2
C₂×C₃₁×#431	2
C₃₁×#1453	2
C₃₁×#1452	2
C₃₁×#1450	2
#17325	31

Subgroups — C31×#17325 #21474084

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#17325 #21474084