Subgroups — C₃₁×#33451 #21490210

Order 7936.

682 conjugacy classes of subgroups.

The subgroup lattice has 252 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	C₄×C₃₁×#135	C₂	1	2	yes	maximal, complemented
3	C₃₁×#2177	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2176	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2175	C₂	1	4	yes	maximal, complemented
6	C₃₁×#2158	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#2138	C₂	1	2	yes	maximal, complemented
8	C₂×C₃₁×#346	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₃₁×#135	C₂²	1	2	yes	—
10	C₄×#41×C₃₁	—	2	2	no	—
11	C₃₁×#508	C₄	1	8	yes	complemented
12	C₂×C₃₁×Q₃₂	C₂²	1	1	yes	characteristic
13	C₂×C₃₁×SD₃₂	C₂²	1	2	yes	—
14	C₂×C₃₁×D₁₆	C₂²	1	1	yes	characteristic
15	C₂²×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
16	C₂²×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
17	C₄×Q₁₆×C₃₁	C₂²	1	4	yes	—
18	C₄×SD₁₆×C₃₁	—	2	4	no	—
19	C₄×D₈×C₃₁	C₂²	1	4	yes	—
20	C₃₁×#427	C₂²	1	2	yes	—
21	C₃₁×#416	—	2	2	no	—
22	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
23	C₃₁×#367	C₂²	1	2	yes	—
24	C₃₁×#366	C₂²	1	2	yes	—
25	C₃₁×#358	C₂²	1	4	yes	—
26	C₃₁×#357	C₂²	1	4	yes	—
27	C₃₁×#346	C₂²	1	2	yes	—
28	C₃₁×#346	C₂²	1	2	yes	—
29	C₂×#41×C₃₁	—	2	2	no	—
30	C₃₁×#135	—	2	4	no	—
31	C₃₁×#135	C₂×C₄	1	8	yes	complemented
32	C₂×Q₁₆×C₃₁	C₂³	1	2	yes	—
33	C₂×SD₁₆×C₃₁	—	2	2	no	—
34	C₂×D₈×C₃₁	C₂³	1	2	yes	—
35	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
37	C₄×Q₈×C₃₁	—	2	4	no	—
38	C₄×D₄×C₃₁	—	2	4	no	—
39	C₄×D₄×C₃₁	—	4	4	no	—
40	C₃₁×#117	—	2	2	no	—
41	C₃₁×#117	—	4	2	no	—
42	C₂×C₄²×C₃₁	—	4	2	no	—
43	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
44	C₃₁×Q₃₂	C₂×C₄	1	4	yes	complemented
45	C₃₁×SD₃₂	C₂×C₄	1	8	yes	complemented
46	C₃₁×D₁₆	C₂×C₄	1	4	yes	complemented
47	C₂×C₁₆×C₃₁	—	2	2	no	—
48	C₂×C₁₆×C₃₁	C₂×C₄	1	4	yes	complemented
49	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
50	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
51	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
52	C₃₁×#107	C₂³	1	4	yes	—
53	C₃₁×#106	—	2	2	no	—
54	C₃₁×#103	—	2	4	no	—
55	C₃₁×#102	—	2	4	no	—
56	C₄×C₈×C₃₁	D₄	1	2	yes	—
57	C₄×C₈×C₃₁	C₂³	1	2	yes	—
58	#41×C₃₁	—	2	8	no	—
59	#41×C₃₁	—	4	4	no	—
60	C₂×Q₈×C₃₁	—	2	2	no	—
61	C₂×D₄×C₃₁	—	2	2	no	—
62	C₂×D₄×C₃₁	—	4	2	no	—
63	C₂²×C₄×C₃₁	—	4	2	no	—
64	C₂²×C₄×C₃₁	—	4	4	no	—
65	C₂²×C₄×C₃₁	D₈	1	2	yes	—
66	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
67	Q₁₆×C₃₁	—	2	2	no	—
68	Q₁₆×C₃₁	C₂²×C₄	1	4	yes	—
69	SD₁₆×C₃₁	—	2	8	no	—
70	D₈×C₃₁	—	2	2	no	—
71	D₈×C₃₁	C₂²×C₄	1	4	yes	—
72	C₂×C₈×C₃₁	—	2	2	no	—
73	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
74	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
75	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₃₁	#41	1	2	yes	—
77	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
78	C₄⋊C₄×C₃₁	—	2	4	no	—
79	C₄⋊C₄×C₃₁	—	4	4	no	—
80	C₂²⋊C₄×C₃₁	—	4	2	no	—
81	C₂²⋊C₄×C₃₁	—	4	4	no	—
82	C₄²×C₃₁	—	4	4	no	—
83	C₄²×C₃₁	—	4	2	no	—
84	C₄²×C₃₁	D₈	1	2	yes	—
85	C₄²×C₃₁	C₂×D₄	1	2	yes	—
86	C₁₆×C₃₁	—	2	2	no	—
87	C₁₆×C₃₁	C₂²×C₄	1	2	yes	—
88	C₁₆×C₃₁	C₂²×C₄	1	2	yes	—
89	#33451	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
90	C₂³×C₃₁	—	4	2	no	—
91	C₂³×C₃₁	C₂×D₈	1	1	yes	characteristic
92	Q₈×C₃₁	—	2	4	no	—
93	Q₈×C₃₁	—	4	2	no	—
94	D₄×C₃₁	—	2	4	no	—
95	D₄×C₃₁	—	4	8	no	—
96	D₄×C₃₁	—	4	2	no	—
97	C₂×C₄×C₃₁	—	4	2	no	—
98	C₂×C₄×C₃₁	—	4	2	no	—
99	C₂×C₄×C₃₁	—	4	8	no	—
100	C₂×C₄×C₃₁	—	4	4	no	—
101	C₂×C₄×C₃₁	—	8	8	no	—
102	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
103	C₂×C₄×C₃₁	C₂×D₈	1	4	yes	—
104	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
105	C₂×C₄×C₃₁	#135	1	2	yes	—
106	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
107	C₈×C₃₁	—	2	2	no	—
108	C₈×C₃₁	C₄×D₄	1	2	yes	—
109	C₈×C₃₁	C₂³×C₄	1	1	yes	characteristic
110	C₈×C₃₁	C₂×#41	1	1	yes	characteristic
111	C₂×#508	C₂×C₃₁	1	1	yes	characteristic
112	C₄×#135	C₂×C₃₁	1	2	yes	complemented
113	#2177	C₂×C₃₁	1	2	yes	complemented
114	#2176	C₂×C₃₁	1	2	yes	complemented
115	#2175	C₂×C₃₁	1	4	yes	complemented
116	#2158	C₂×C₃₁	1	1	yes	characteristic, complemented
117	#2138	C₂×C₃₁	1	2	yes	complemented
118	C₂×#346	C₂×C₃₁	1	1	yes	characteristic, complemented
119	C₂²×C₃₁	—	2	2	no	—
120	C₂²×C₃₁	—	4	4	no	—
121	C₂²×C₃₁	—	8	4	no	—
122	C₂²×C₃₁	C₄×D₈	1	2	yes	—
123	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
124	C₄×C₃₁	—	2	4	no	—
125	C₄×C₃₁	—	4	4	no	—
126	C₄×C₃₁	—	8	4	no	—
127	C₄×C₃₁	C₄×D₈	1	2	yes	—
128	C₄×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
129	C₄×C₃₁	C₂×#135	1	1	yes	characteristic
130	C₂×#135	C₂²×C₃₁	1	2	yes	—
131	C₄×#41	—	2	2	no	—
132	#508	C₄×C₃₁	1	8	yes	complemented
133	C₂×Q₃₂	C₂²×C₃₁	1	1	yes	characteristic
134	C₂×SD₃₂	C₂²×C₃₁	1	2	yes	—
135	C₂×D₁₆	C₂²×C₃₁	1	1	yes	characteristic
136	C₂²×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
137	C₂²×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
138	C₄×Q₁₆	C₂²×C₃₁	1	4	yes	—
139	C₄×SD₁₆	—	2	4	no	—
140	C₄×D₈	C₂²×C₃₁	1	4	yes	—
141	#427	C₂²×C₃₁	1	2	yes	—
142	#416	—	2	2	no	—
143	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
144	#367	C₂²×C₃₁	1	2	yes	—
145	#366	C₂²×C₃₁	1	2	yes	—
146	#358	C₂²×C₃₁	1	4	yes	—
147	#357	C₂²×C₃₁	1	4	yes	—
148	#346	C₂²×C₃₁	1	2	yes	—
149	#346	C₂²×C₃₁	1	2	yes	—
150	C₂×C₃₁	—	2	2	no	—
151	C₂×C₃₁	—	8	4	no	—
152	C₂×C₃₁	C₂×C₄×D₈	1	1	yes	characteristic
153	C₂×C₃₁	#3414	1	2	yes	—
154	C₂×#41	—	2	2	no	—
155	#135	—	2	4	no	—
156	#135	C₂×C₄×C₃₁	1	8	yes	complemented
157	C₂×Q₁₆	C₂³×C₃₁	1	2	yes	—
158	C₂×SD₁₆	—	2	2	no	—
159	C₂×D₈	C₂³×C₃₁	1	2	yes	—
160	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
161	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
162	C₄×Q₈	—	2	4	no	—
163	C₄×D₄	—	2	4	no	—
164	C₄×D₄	—	4	4	no	—
165	#117	—	2	2	no	—
166	#117	—	4	2	no	—
167	C₂×C₄²	—	4	2	no	—
168	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
169	Q₃₂	C₂×C₄×C₃₁	1	4	yes	complemented
170	SD₃₂	C₂×C₄×C₃₁	1	8	yes	complemented
171	D₁₆	C₂×C₄×C₃₁	1	4	yes	complemented
172	C₂×C₁₆	—	2	2	no	—
173	C₂×C₁₆	C₂×C₄×C₃₁	1	4	yes	complemented
174	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
175	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
176	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
177	#107	C₂³×C₃₁	1	4	yes	—
178	#106	—	2	2	no	—
179	#103	—	2	4	no	—
180	#102	—	2	4	no	—
181	C₄×C₈	D₄×C₃₁	1	2	yes	—
182	C₄×C₈	C₂³×C₃₁	1	2	yes	—
183	C₃₁	#33451	1	1	yes	characteristic, direct product factor, Sylow
184	#41	—	2	8	no	—
185	#41	—	4	4	no	—
186	C₂×Q₈	—	2	2	no	—
187	C₂×D₄	—	2	2	no	—
188	C₂×D₄	—	4	2	no	—
189	C₂²×C₄	—	4	2	no	—
190	C₂²×C₄	—	4	4	no	—
191	C₂²×C₄	D₈×C₃₁	1	2	yes	—
192	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
193	Q₁₆	—	2	2	no	—
194	Q₁₆	C₂²×C₄×C₃₁	1	4	yes	—
195	SD₁₆	—	2	8	no	—
196	D₈	—	2	2	no	—
197	D₈	C₂²×C₄×C₃₁	1	4	yes	—
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
200	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
201	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
202	C₂×C₈	#41×C₃₁	1	2	yes	—
203	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
204	C₄⋊C₄	—	2	4	no	—
205	C₄⋊C₄	—	4	4	no	—
206	C₂²⋊C₄	—	4	2	no	—
207	C₂²⋊C₄	—	4	4	no	—
208	C₄²	—	4	2	no	—
209	C₄²	—	4	4	no	—
210	C₄²	D₈×C₃₁	1	2	yes	—
211	C₄²	C₂×D₄×C₃₁	1	2	yes	—
212	C₁₆	—	2	2	no	—
213	C₁₆	C₂²×C₄×C₃₁	1	2	yes	—
214	C₁₆	C₂²×C₄×C₃₁	1	2	yes	—
215	C₂³	—	4	2	no	—
216	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
217	Q₈	—	2	4	no	—
218	Q₈	—	4	2	no	—
219	D₄	—	2	4	no	—
220	D₄	—	4	8	no	—
221	D₄	—	4	2	no	—
222	C₂×C₄	—	4	2	no	—
223	C₂×C₄	—	4	2	no	—
224	C₂×C₄	—	4	8	no	—
225	C₂×C₄	—	4	4	no	—
226	C₂×C₄	—	8	8	no	—
227	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
228	C₂×C₄	C₂×D₈×C₃₁	1	4	yes	—
229	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
230	C₂×C₄	C₃₁×#135	1	2	yes	—
231	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
232	C₈	—	2	2	no	—
233	C₈	C₄×D₄×C₃₁	1	2	yes	—
234	C₈	C₂³×C₄×C₃₁	1	1	yes	characteristic
235	C₈	C₂×#41×C₃₁	1	1	yes	characteristic
236	C₂²	—	2	2	no	—
237	C₂²	—	4	4	no	—
238	C₂²	—	8	4	no	—
239	C₂²	C₄×D₈×C₃₁	1	2	yes	—
240	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
241	C₄	—	2	4	no	—
242	C₄	—	4	4	no	—
243	C₄	—	8	4	no	—
244	C₄	C₄×D₈×C₃₁	1	2	yes	—
245	C₄	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
246	C₄	C₂×C₃₁×#135	1	1	yes	characteristic
247	C₂	—	2	2	no	—
248	C₂	—	8	4	no	—
249	C₂	C₂×C₄×D₈×C₃₁	1	1	yes	characteristic
250	C₂	C₃₁×#3414	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂×D₈
commutator	C₈	C₂³×C₄×C₃₁
Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₂²×D₈
prime radical	C₂×C₃₁×#508	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×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. C₄×C₃₁×#135	C₂	semidirect
3. C₃₁×#2177	C₂	semidirect
4. C₃₁×#2176	C₂	semidirect
5. C₃₁×#2175	C₂	semidirect
6. C₃₁×#2158	C₂	semidirect
7. C₃₁×#2138	C₂	semidirect
8. C₂×C₃₁×#346	C₂	semidirect
11. C₃₁×#508	C₄	semidirect
31. C₃₁×#135	C₂×C₄	semidirect
44. C₃₁×Q₃₂	C₂×C₄	semidirect
45. C₃₁×SD₃₂	C₂×C₄	semidirect
46. C₃₁×D₁₆	C₂×C₄	semidirect
48. C₂×C₁₆×C₃₁	C₂×C₄	semidirect
89. #33451	C₃₁	direct (N × H)
112. C₄×#135	C₂×C₃₁	semidirect
113. #2177	C₂×C₃₁	semidirect
114. #2176	C₂×C₃₁	semidirect
115. #2175	C₂×C₃₁	semidirect
116. #2158	C₂×C₃₁	semidirect
117. #2138	C₂×C₃₁	semidirect
118. C₂×#346	C₂×C₃₁	semidirect
132. #508	C₄×C₃₁	semidirect
156. #135	C₂×C₄×C₃₁	semidirect
169. Q₃₂	C₂×C₄×C₃₁	semidirect
170. SD₃₂	C₂×C₄×C₃₁	semidirect
171. D₁₆	C₂×C₄×C₃₁	semidirect
173. C₂×C₁₆	C₂×C₄×C₃₁	semidirect
183. C₃₁	#33451	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
C₄×C₃₁×#135	2
C₃₁×#2177	2
C₃₁×#2176	2
C₃₁×#2175	2
C₃₁×#2158	2
C₃₁×#2138	2
C₂×C₃₁×#346	2
#33451	31

Subgroups — C31×#33451 #21490210

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33451 #21490210