Subgroups — C₃₁×#13202 #21469961

Order 7936.

614 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#493	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#357	C₂	1	4	yes	maximal, complemented
3	C₂×C₄×C₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1845	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₈×C₃₁	—	2	1	no	—
6	C₂²×D₈×C₃₁	C₂²	1	2	yes	—
7	C₂×C₃₁×#127	—	2	1	no	—
8	C₂²×C₁₆×C₃₁	C₂²	1	2	yes	—
9	C₃₁×#493	—	2	2	no	—
10	C₃₁×#493	C₄	1	4	yes	complemented
11	C₂×C₃₁×#107	—	2	1	no	—
12	C₂×C₃₁×#107	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
14	C₃₁×#384	—	2	1	no	—
15	C₃₁×#357	—	2	8	no	—
16	C₄×C₁₆×C₃₁	—	2	2	no	—
17	C₂²×D₄×C₃₁	—	2	2	no	—
18	C₂²×D₄×C₃₁	—	4	2	no	—
19	C₂×D₈×C₃₁	—	2	4	no	—
20	C₂×D₈×C₃₁	—	2	4	no	—
21	C₂×D₈×C₃₁	—	2	2	no	—
22	C₂×D₈×C₃₁	—	2	4	no	—
23	C₂×D₈×C₃₁	—	4	2	no	—
24	C₂×D₈×C₃₁	—	4	2	no	—
25	C₂×D₈×C₃₁	C₂×C₄	1	4	yes	—
26	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
28	C₃₁×#127	—	2	4	no	—
29	C₃₁×#127	—	4	2	no	—
30	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
31	C₂×C₄⋊C₄×C₃₁	—	4	2	no	—
32	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
33	C₂×C₁₆×C₃₁	—	2	2	no	—
34	C₂×C₁₆×C₃₁	—	2	2	no	—
35	C₂×C₁₆×C₃₁	D₄	1	4	yes	complemented
36	C₃₁×#107	—	2	4	no	—
37	C₃₁×#107	—	4	2	no	—
38	C₄×C₈×C₃₁	C₂×C₄	1	2	yes	—
39	C₄×C₈×C₃₁	D₄	1	2	yes	—
40	C₂⁴×C₃₁	—	4	2	no	—
41	C₂×D₄×C₃₁	—	2	4	no	—
42	C₂×D₄×C₃₁	—	4	8	no	—
43	C₂×D₄×C₃₁	—	4	2	no	—
44	C₂×D₄×C₃₁	—	4	2	no	—
45	C₂×D₄×C₃₁	—	4	2	no	—
46	C₂×D₄×C₃₁	—	4	2	no	—
47	C₂×D₄×C₃₁	—	4	2	no	—
48	C₂×D₄×C₃₁	—	8	4	no	—
49	C₂×D₄×C₃₁	—	8	4	no	—
50	C₂²×C₄×C₃₁	—	4	2	no	—
51	C₂²×C₄×C₃₁	D₈	1	2	yes	—
52	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
53	D₈×C₃₁	—	2	8	no	—
54	D₈×C₃₁	—	4	8	no	—
55	D₈×C₃₁	—	4	4	no	—
56	C₂×C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
57	C₂×C₈×C₃₁	D₈	1	2	yes	—
58	C₂×C₈×C₃₁	SD₁₆	1	2	yes	—
59	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
61	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
62	C₂×C₈×C₃₁	#41	1	2	yes	—
63	C₄⋊C₄×C₃₁	—	4	2	no	—
64	C₄⋊C₄×C₃₁	—	4	2	no	—
65	C₄⋊C₄×C₃₁	—	8	4	no	—
66	C₄²×C₃₁	C₂²⋊C₄	1	2	yes	—
67	C₄²×C₃₁	SD₁₆	1	2	yes	—
68	C₁₆×C₃₁	—	2	4	no	—
69	#13202	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
70	C₂³×C₃₁	—	4	4	no	—
71	C₂³×C₃₁	—	8	2	no	—
72	C₂³×C₃₁	—	8	2	no	—
73	C₂³×C₃₁	—	8	4	no	—
74	C₂³×C₃₁	—	8	4	no	—
75	C₂³×C₃₁	C₂×D₈	1	1	yes	characteristic
76	D₄×C₃₁	—	4	4	no	—
77	D₄×C₃₁	—	4	4	no	—
78	D₄×C₃₁	—	8	16	no	—
79	D₄×C₃₁	—	8	2	no	—
80	D₄×C₃₁	—	8	2	no	—
81	C₂×C₄×C₃₁	—	8	2	no	—
82	C₂×C₄×C₃₁	—	8	4	no	—
83	C₂×C₄×C₃₁	#102	1	4	yes	—
84	C₂×C₄×C₃₁	D₁₆	1	4	yes	—
85	C₂×C₄×C₃₁	SD₃₂	1	4	yes	—
86	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₄×C₃₁	#124	1	1	yes	characteristic
88	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
89	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
90	C₂×C₄×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
91	C₂×C₄×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
92	C₈×C₃₁	#102	1	4	yes	—
93	C₈×C₃₁	C₄×D₄	1	2	yes	—
94	C₈×C₃₁	#121	1	2	yes	—
95	C₂×#493	C₂×C₃₁	1	1	yes	characteristic
96	C₂×#357	C₂×C₃₁	1	4	yes	complemented
97	C₂×C₄×C₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
98	#1845	C₂×C₃₁	1	1	yes	characteristic, complemented
99	C₂²×C₃₁	—	8	8	no	—
100	C₂²×C₃₁	—	8	4	no	—
101	C₂²×C₃₁	—	16	4	no	—
102	C₂²×C₃₁	—	16	2	no	—
103	C₂²×C₃₁	—	16	2	no	—
104	C₂²×C₃₁	C₂×#102	1	1	yes	characteristic
105	C₂²×C₃₁	#486	1	1	yes	characteristic
106	C₂²×C₃₁	#493	1	1	yes	characteristic
107	C₂²×C₃₁	C₂×D₁₆	1	2	yes	—
108	C₂²×C₃₁	C₂×SD₃₂	1	2	yes	—
109	C₄×C₃₁	—	16	2	no	—
110	C₄×C₃₁	#357	1	8	yes	complemented
111	C₄×C₃₁	#390	1	1	yes	characteristic
112	C₄×C₃₁	C₂×#102	1	1	yes	characteristic
113	C₄×C₃₁	#486	1	1	yes	characteristic
114	C₄×C₃₁	#492	1	1	yes	characteristic
115	C₂²×D₈	—	2	1	no	—
116	C₂²×D₈	C₂²×C₃₁	1	2	yes	—
117	C₂×#127	—	2	1	no	—
118	C₂²×C₁₆	C₂²×C₃₁	1	2	yes	—
119	#493	—	2	2	no	—
120	#493	C₄×C₃₁	1	4	yes	complemented
121	C₂×#107	—	2	1	no	—
122	C₂×#107	C₂²×C₃₁	1	2	yes	—
123	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
124	#384	—	2	1	no	—
125	#357	—	2	8	no	—
126	C₄×C₁₆	—	2	2	no	—
127	C₂×C₃₁	—	16	4	no	—
128	C₂×C₃₁	#1957	1	1	yes	characteristic
129	C₂×C₃₁	C₂×#357	1	2	yes	—
130	C₂×C₃₁	#2240	1	2	yes	—
131	C₂×C₃₁	#2246	1	1	yes	characteristic
132	C₂×C₃₁	#2248	1	1	yes	characteristic
133	C₂²×D₄	—	2	2	no	—
134	C₂²×D₄	—	4	2	no	—
135	C₂×D₈	—	2	4	no	—
136	C₂×D₈	—	2	4	no	—
137	C₂×D₈	—	2	2	no	—
138	C₂×D₈	—	2	4	no	—
139	C₂×D₈	—	4	2	no	—
140	C₂×D₈	—	4	2	no	—
141	C₂×D₈	C₂×C₄×C₃₁	1	4	yes	—
142	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
143	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
144	#127	—	2	4	no	—
145	#127	—	4	2	no	—
146	C₂×C₄⋊C₄	—	2	2	no	—
147	C₂×C₄⋊C₄	—	4	2	no	—
148	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
149	C₂×C₁₆	—	2	2	no	—
150	C₂×C₁₆	—	2	2	no	—
151	C₂×C₁₆	D₄×C₃₁	1	4	yes	complemented
152	#107	—	2	4	no	—
153	#107	—	4	2	no	—
154	C₄×C₈	C₂×C₄×C₃₁	1	2	yes	—
155	C₄×C₈	D₄×C₃₁	1	2	yes	—
156	C₃₁	#13202	1	1	yes	characteristic, direct product factor, Sylow
157	C₂⁴	—	4	2	no	—
158	C₂×D₄	—	2	4	no	—
159	C₂×D₄	—	4	8	no	—
160	C₂×D₄	—	4	2	no	—
161	C₂×D₄	—	4	2	no	—
162	C₂×D₄	—	4	2	no	—
163	C₂×D₄	—	4	2	no	—
164	C₂×D₄	—	4	2	no	—
165	C₂×D₄	—	8	4	no	—
166	C₂×D₄	—	8	4	no	—
167	C₂²×C₄	—	4	2	no	—
168	C₂²×C₄	D₈×C₃₁	1	2	yes	—
169	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
170	D₈	—	2	8	no	—
171	D₈	—	4	8	no	—
172	D₈	—	4	4	no	—
173	C₂×C₈	C₂²⋊C₄×C₃₁	1	2	yes	—
174	C₂×C₈	D₈×C₃₁	1	2	yes	—
175	C₂×C₈	SD₁₆×C₃₁	1	2	yes	—
176	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
177	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
178	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
179	C₂×C₈	#41×C₃₁	1	2	yes	—
180	C₄⋊C₄	—	4	2	no	—
181	C₄⋊C₄	—	4	2	no	—
182	C₄⋊C₄	—	8	4	no	—
183	C₄²	C₂²⋊C₄×C₃₁	1	2	yes	—
184	C₄²	SD₁₆×C₃₁	1	2	yes	—
185	C₁₆	—	2	4	no	—
186	C₂³	—	4	4	no	—
187	C₂³	—	8	2	no	—
188	C₂³	—	8	2	no	—
189	C₂³	—	8	4	no	—
190	C₂³	—	8	4	no	—
191	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
192	D₄	—	4	4	no	—
193	D₄	—	4	4	no	—
194	D₄	—	8	2	no	—
195	D₄	—	8	2	no	—
196	D₄	—	8	16	no	—
197	C₂×C₄	—	8	2	no	—
198	C₂×C₄	—	8	4	no	—
199	C₂×C₄	C₃₁×#102	1	4	yes	—
200	C₂×C₄	C₃₁×D₁₆	1	4	yes	—
201	C₂×C₄	C₃₁×SD₃₂	1	4	yes	—
202	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
203	C₂×C₄	C₃₁×#124	1	1	yes	characteristic
204	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
205	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
206	C₂×C₄	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
207	C₂×C₄	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
208	C₈	C₃₁×#102	1	4	yes	—
209	C₈	C₄×D₄×C₃₁	1	2	yes	—
210	C₈	C₃₁×#121	1	2	yes	—
211	C₂²	—	8	8	no	—
212	C₂²	—	8	4	no	—
213	C₂²	—	16	4	no	—
214	C₂²	—	16	2	no	—
215	C₂²	—	16	2	no	—
216	C₂²	C₂×C₃₁×#102	1	1	yes	characteristic
217	C₂²	C₃₁×#486	1	1	yes	characteristic
218	C₂²	C₃₁×#493	1	1	yes	characteristic
219	C₂²	C₂×C₃₁×D₁₆	1	2	yes	—
220	C₂²	C₂×C₃₁×SD₃₂	1	2	yes	—
221	C₄	—	16	2	no	—
222	C₄	C₃₁×#357	1	8	yes	complemented
223	C₄	C₃₁×#390	1	1	yes	characteristic
224	C₄	C₂×C₃₁×#102	1	1	yes	characteristic
225	C₄	C₃₁×#486	1	1	yes	characteristic
226	C₄	C₃₁×#492	1	1	yes	characteristic
227	C₂	—	16	4	no	—
228	C₂	C₃₁×#1957	1	1	yes	characteristic
229	C₂	C₂×C₃₁×#357	1	2	yes	—
230	C₂	C₃₁×#2240	1	2	yes	—
231	C₂	C₃₁×#2246	1	1	yes	characteristic
232	C₂	C₃₁×#2248	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₃₁
Frattini, cosocle	C₂²×C₈	C₂³×C₃₁
prime radical	C₂×C₃₁×#493	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ C₂×C₄×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃₁×#357	C₂	semidirect
3. C₂×C₄×C₁₆×C₃₁	C₂	semidirect
4. C₃₁×#1845	C₂	semidirect
10. C₃₁×#493	C₄	semidirect
35. C₂×C₁₆×C₃₁	D₄	semidirect
69. #13202	C₃₁	direct (N × H)
96. C₂×#357	C₂×C₃₁	semidirect
97. C₂×C₄×C₁₆	C₂×C₃₁	semidirect
98. #1845	C₂×C₃₁	semidirect
110. C₄×C₃₁	#357	semidirect
120. #493	C₄×C₃₁	semidirect
151. C₂×C₁₆	D₄×C₃₁	semidirect
156. C₃₁	#13202	direct (N × H)
222. C₄	C₃₁×#357	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#493	2
C₂×C₃₁×#357	2
C₂×C₄×C₁₆×C₃₁	2
C₃₁×#1845	2
#13202	31

Subgroups — C31×#13202 #21469961

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#13202 #21469961