Subgroups — C₃₁×#13241 #21470000

Order 7936.

426 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#508	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#368	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#359	C₂	1	2	yes	maximal, complemented
4	C₂×C₄×C₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1974	C₂	1	2	yes	maximal, complemented
6	C₂×C₃₁×#135	C₂²	1	2	yes	—
7	C₃₁×#508	—	2	2	no	—
8	C₃₁×#508	—	2	2	no	—
9	C₂×C₃₁×Q₃₂	C₄	1	1	yes	characteristic, complemented
10	C₂×C₃₁×SD₃₂	C₄	1	2	yes	complemented
11	C₂×C₃₁×D₁₆	C₄	1	1	yes	characteristic, complemented
12	C₂²×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
13	C₂²×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
14	C₂×C₃₁×#108	C₂²	1	2	yes	—
15	C₂×C₃₁×#104	—	2	2	no	—
16	C₃₁×#408	—	2	2	no	—
17	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
18	C₃₁×#368	—	2	2	no	—
19	C₃₁×#359	—	2	4	no	—
20	C₄×C₁₆×C₃₁	—	2	2	no	—
21	C₂×#41×C₃₁	—	2	2	no	—
22	C₃₁×#135	—	2	2	no	—
23	C₃₁×#135	—	2	2	no	—
24	C₃₁×#135	—	2	2	no	—
25	C₃₁×#135	—	2	2	no	—
26	C₂×Q₁₆×C₃₁	C₂×C₄	1	2	yes	—
27	C₂×SD₁₆×C₃₁	—	2	2	no	—
28	C₂×D₈×C₃₁	C₂×C₄	1	2	yes	—
29	C₂×M₁₆×C₃₁	—	2	2	no	—
30	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
32	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
33	C₃₁×Q₃₂	—	2	2	no	—
34	C₃₁×SD₃₂	—	2	4	no	—
35	C₃₁×D₁₆	—	2	2	no	—
36	C₂×C₁₆×C₃₁	—	2	1	no	—
37	C₂×C₁₆×C₃₁	—	2	1	no	—
38	C₂×C₁₆×C₃₁	C₂×C₄	1	1	yes	characteristic
39	C₂×C₁₆×C₃₁	C₂×C₄	1	1	yes	characteristic
40	C₂×C₁₆×C₃₁	D₄	1	2	yes	—
41	C₂×C₁₆×C₃₁	D₄	1	2	yes	—
42	C₂×C₁₆×C₃₁	D₄	1	2	yes	—
43	C₃₁×#108	—	2	2	no	—
44	C₃₁×#108	—	2	2	no	—
45	C₃₁×#104	—	4	4	no	—
46	C₃₁×C₂²⋊C₈	—	4	4	no	—
47	C₄×C₈×C₃₁	—	2	1	no	—
48	C₄×C₈×C₃₁	—	2	1	no	—
49	#41×C₃₁	—	4	2	no	—
50	#41×C₃₁	—	4	2	no	—
51	#41×C₃₁	—	4	2	no	—
52	#41×C₃₁	—	4	2	no	—
53	C₂×Q₈×C₃₁	—	2	2	no	—
54	C₂×D₄×C₃₁	—	2	2	no	—
55	C₂×D₄×C₃₁	—	4	2	no	—
56	C₂²×C₄×C₃₁	—	2	1	no	—
57	C₂²×C₄×C₃₁	—	4	2	no	—
58	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
59	Q₁₆×C₃₁	—	2	2	no	—
60	Q₁₆×C₃₁	—	2	2	no	—
61	SD₁₆×C₃₁	—	4	2	no	—
62	SD₁₆×C₃₁	—	4	2	no	—
63	D₈×C₃₁	—	2	2	no	—
64	D₈×C₃₁	—	2	2	no	—
65	M₁₆×C₃₁	—	4	2	no	—
66	M₁₆×C₃₁	—	4	2	no	—
67	C₂×C₈×C₃₁	—	2	1	no	—
68	C₂×C₈×C₃₁	—	2	1	no	—
69	C₂×C₈×C₃₁	—	4	2	no	—
70	C₂×C₈×C₃₁	D₈	1	2	yes	—
71	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
73	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
75	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
76	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
77	C₄²×C₃₁	—	2	1	no	—
78	C₄²×C₃₁	D₈	1	2	yes	—
79	C₁₆×C₃₁	—	2	2	no	—
80	C₁₆×C₃₁	C₂²⋊C₄	1	2	yes	—
81	C₁₆×C₃₁	C₂²⋊C₄	1	2	yes	—
82	#13241	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
83	C₂³×C₃₁	—	4	2	no	—
84	C₂³×C₃₁	#135	1	1	yes	characteristic
85	Q₈×C₃₁	—	4	2	no	—
86	Q₈×C₃₁	—	4	2	no	—
87	D₄×C₃₁	—	4	2	no	—
88	D₄×C₃₁	—	4	2	no	—
89	D₄×C₃₁	—	8	2	no	—
90	D₄×C₃₁	—	8	2	no	—
91	C₂×C₄×C₃₁	—	2	4	no	—
92	C₂×C₄×C₃₁	—	2	2	no	—
93	C₂×C₄×C₃₁	—	4	2	no	—
94	C₂×C₄×C₃₁	—	8	2	no	—
95	C₂×C₄×C₃₁	—	8	2	no	—
96	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
97	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
98	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
99	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
100	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
101	C₂×C₄×C₃₁	#135	1	1	yes	characteristic
102	C₈×C₃₁	—	2	2	no	—
103	C₈×C₃₁	—	8	2	no	—
104	C₈×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₈×C₃₁	C₄×D₄	1	1	yes	characteristic
106	C₈×C₃₁	#121	1	1	yes	characteristic
107	C₈×C₃₁	#124	1	1	yes	characteristic
108	C₂×#508	C₂×C₃₁	1	1	yes	characteristic
109	C₂×#368	C₂×C₃₁	1	1	yes	characteristic, complemented
110	C₂×#359	C₂×C₃₁	1	2	yes	complemented
111	C₂×C₄×C₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
112	#1974	C₂×C₃₁	1	2	yes	complemented
113	C₂²×C₃₁	—	2	2	no	—
114	C₂²×C₃₁	—	8	4	no	—
115	C₂²×C₃₁	—	8	2	no	—
116	C₂²×C₃₁	C₄×D₈	1	1	yes	characteristic
117	C₂²×C₃₁	#466	1	1	yes	characteristic
118	C₂²×C₃₁	#495	1	1	yes	characteristic
119	C₄×C₃₁	—	2	4	no	—
120	C₄×C₃₁	—	8	2	no	—
121	C₄×C₃₁	#390	1	1	yes	characteristic
122	C₄×C₃₁	C₄×D₈	1	1	yes	characteristic
123	C₄×C₃₁	#466	1	1	yes	characteristic
124	C₄×C₃₁	#493	1	1	yes	characteristic
125	C₂×#135	C₂²×C₃₁	1	2	yes	—
126	#508	—	2	2	no	—
127	#508	—	2	2	no	—
128	C₂×Q₃₂	C₄×C₃₁	1	1	yes	characteristic, complemented
129	C₂×SD₃₂	C₄×C₃₁	1	2	yes	complemented
130	C₂×D₁₆	C₄×C₃₁	1	1	yes	characteristic, complemented
131	C₂²×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
132	C₂²×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
133	C₂×#108	C₂²×C₃₁	1	2	yes	—
134	C₂×#104	—	2	2	no	—
135	#408	—	2	2	no	—
136	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
137	#368	—	2	2	no	—
138	#359	—	2	4	no	—
139	C₄×C₁₆	—	2	2	no	—
140	C₂×C₃₁	—	2	1	no	—
141	C₂×C₃₁	—	2	1	no	—
142	C₂×C₃₁	—	16	2	no	—
143	C₂×C₃₁	#1970	1	1	yes	characteristic
144	C₂×C₃₁	#2178	1	2	yes	—
145	C₂×#41	—	2	2	no	—
146	#135	—	2	2	no	—
147	#135	—	2	2	no	—
148	#135	—	2	2	no	—
149	#135	—	2	2	no	—
150	C₂×Q₁₆	C₂×C₄×C₃₁	1	2	yes	—
151	C₂×SD₁₆	—	2	2	no	—
152	C₂×D₈	C₂×C₄×C₃₁	1	2	yes	—
153	C₂×M₁₆	—	2	2	no	—
154	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
155	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
156	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
157	Q₃₂	—	2	2	no	—
158	SD₃₂	—	2	4	no	—
159	D₁₆	—	2	2	no	—
160	C₂×C₁₆	—	2	1	no	—
161	C₂×C₁₆	—	2	1	no	—
162	C₂×C₁₆	C₂×C₄×C₃₁	1	1	yes	characteristic
163	C₂×C₁₆	C₂×C₄×C₃₁	1	1	yes	characteristic
164	C₂×C₁₆	D₄×C₃₁	1	2	yes	—
165	C₂×C₁₆	D₄×C₃₁	1	2	yes	—
166	C₂×C₁₆	D₄×C₃₁	1	2	yes	—
167	#108	—	2	2	no	—
168	#108	—	2	2	no	—
169	#104	—	4	4	no	—
170	C₂²⋊C₈	—	4	4	no	—
171	C₄×C₈	—	2	1	no	—
172	C₄×C₈	—	2	1	no	—
173	C₃₁	#13241	1	1	yes	characteristic, direct product factor, Sylow
174	#41	—	4	2	no	—
175	#41	—	4	2	no	—
176	#41	—	4	2	no	—
177	#41	—	4	2	no	—
178	C₂×Q₈	—	2	2	no	—
179	C₂×D₄	—	2	2	no	—
180	C₂×D₄	—	4	2	no	—
181	C₂²×C₄	—	2	1	no	—
182	C₂²×C₄	—	4	2	no	—
183	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
184	Q₁₆	—	2	2	no	—
185	Q₁₆	—	2	2	no	—
186	SD₁₆	—	4	2	no	—
187	SD₁₆	—	4	2	no	—
188	D₈	—	2	2	no	—
189	D₈	—	2	2	no	—
190	M₁₆	—	4	2	no	—
191	M₁₆	—	4	2	no	—
192	C₂×C₈	—	2	1	no	—
193	C₂×C₈	—	2	1	no	—
194	C₂×C₈	—	4	2	no	—
195	C₂×C₈	D₈×C₃₁	1	2	yes	—
196	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
197	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
198	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
199	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
200	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
201	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
202	C₄²	—	2	1	no	—
203	C₄²	D₈×C₃₁	1	2	yes	—
204	C₁₆	—	2	2	no	—
205	C₁₆	C₂²⋊C₄×C₃₁	1	2	yes	—
206	C₁₆	C₂²⋊C₄×C₃₁	1	2	yes	—
207	C₂³	—	4	2	no	—
208	C₂³	C₃₁×#135	1	1	yes	characteristic
209	Q₈	—	4	2	no	—
210	Q₈	—	4	2	no	—
211	D₄	—	4	2	no	—
212	D₄	—	4	2	no	—
213	D₄	—	8	2	no	—
214	D₄	—	8	2	no	—
215	C₂×C₄	—	2	4	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	4	2	no	—
218	C₂×C₄	—	8	2	no	—
219	C₂×C₄	—	8	2	no	—
220	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
221	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
222	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
223	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
224	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
225	C₂×C₄	C₃₁×#135	1	1	yes	characteristic
226	C₈	—	2	2	no	—
227	C₈	—	8	2	no	—
228	C₈	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
229	C₈	C₄×D₄×C₃₁	1	1	yes	characteristic
230	C₈	C₃₁×#121	1	1	yes	characteristic
231	C₈	C₃₁×#124	1	1	yes	characteristic
232	C₂²	—	2	2	no	—
233	C₂²	—	8	4	no	—
234	C₂²	—	8	2	no	—
235	C₂²	C₄×D₈×C₃₁	1	1	yes	characteristic
236	C₂²	C₃₁×#466	1	1	yes	characteristic
237	C₂²	C₃₁×#495	1	1	yes	characteristic
238	C₄	—	2	4	no	—
239	C₄	—	8	2	no	—
240	C₄	C₃₁×#390	1	1	yes	characteristic
241	C₄	C₄×D₈×C₃₁	1	1	yes	characteristic
242	C₄	C₃₁×#466	1	1	yes	characteristic
243	C₄	C₃₁×#493	1	1	yes	characteristic
244	C₂	—	2	1	no	—
245	C₂	—	2	1	no	—
246	C₂	—	16	2	no	—
247	C₂	C₃₁×#1970	1	1	yes	characteristic
248	C₂	C₃₁×#2178	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₃₁
Frattini, cosocle	C₂²×C₈	C₂³×C₃₁
socle	C₂²×C₃₁	C₄×D₈
prime radical	C₂×C₃₁×#508	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂²×C₈×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×#368	C₂	semidirect
3. C₂×C₃₁×#359	C₂	semidirect
4. C₂×C₄×C₁₆×C₃₁	C₂	semidirect
5. C₃₁×#1974	C₂	semidirect
9. C₂×C₃₁×Q₃₂	C₄	semidirect
10. C₂×C₃₁×SD₃₂	C₄	semidirect
11. C₂×C₃₁×D₁₆	C₄	semidirect
82. #13241	C₃₁	direct (N × H)
109. C₂×#368	C₂×C₃₁	semidirect
110. C₂×#359	C₂×C₃₁	semidirect
111. C₂×C₄×C₁₆	C₂×C₃₁	semidirect
112. #1974	C₂×C₃₁	semidirect
128. C₂×Q₃₂	C₄×C₃₁	semidirect
129. C₂×SD₃₂	C₄×C₃₁	semidirect
130. C₂×D₁₆	C₄×C₃₁	semidirect
173. C₃₁	#13241	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×#508	2
C₂×C₃₁×#368	2
C₂×C₃₁×#359	2
C₂×C₄×C₁₆×C₃₁	2
C₃₁×#1974	2
#13241	31

Subgroups — C31×#13241 #21470000

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#13241 #21470000