Subgroups — C₄×C₃₁×#508 #21490206

Order 7936.

690 conjugacy classes of subgroups.

The subgroup lattice has 258 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	C₄×C₃₁×Q₃₂	C₂	1	2	yes	maximal, complemented
4	C₄×C₃₁×SD₃₂	C₂	1	4	yes	maximal, complemented
5	C₄×C₃₁×D₁₆	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2158	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#2138	C₂	1	2	yes	maximal, complemented
8	C₂×C₄×C₁₆×C₃₁	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	direct product factor
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	C₄×C₁₆×C₃₁	C₂²	1	2	yes	—
28	C₄×C₁₆×C₃₁	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	2	no	—
83	C₄²×C₃₁	—	4	4	no	—
84	C₄²×C₃₁	D₈	1	1	yes	characteristic
85	C₄²×C₃₁	D₈	1	1	yes	characteristic
86	C₄²×C₃₁	C₂×D₄	1	2	yes	—
87	C₁₆×C₃₁	—	2	2	no	—
88	C₁₆×C₃₁	C₂²×C₄	1	2	yes	—
89	C₁₆×C₃₁	C₂²×C₄	1	2	yes	—
90	C₄×#508	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
91	C₂³×C₃₁	—	4	2	no	—
92	C₂³×C₃₁	C₂×D₈	1	1	yes	characteristic
93	Q₈×C₃₁	—	2	4	no	—
94	Q₈×C₃₁	—	4	2	no	—
95	D₄×C₃₁	—	2	4	no	—
96	D₄×C₃₁	—	4	8	no	—
97	D₄×C₃₁	—	4	2	no	—
98	C₂×C₄×C₃₁	—	2	4	no	—
99	C₂×C₄×C₃₁	—	4	2	no	—
100	C₂×C₄×C₃₁	—	4	8	no	—
101	C₂×C₄×C₃₁	—	4	4	no	—
102	C₂×C₄×C₃₁	—	8	8	no	—
103	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
104	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
105	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
106	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
107	C₂×C₄×C₃₁	#135	1	2	yes	—
108	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
109	C₈×C₃₁	—	2	2	no	—
110	C₈×C₃₁	C₄×D₄	1	2	yes	—
111	C₈×C₃₁	C₂³×C₄	1	1	yes	characteristic
112	C₈×C₃₁	C₂×#41	1	1	yes	characteristic
113	C₂×#508	C₂×C₃₁	1	1	yes	characteristic
114	C₄×#135	C₂×C₃₁	1	2	yes	complemented
115	C₄×Q₃₂	C₂×C₃₁	1	2	yes	complemented
116	C₄×SD₃₂	C₂×C₃₁	1	4	yes	complemented
117	C₄×D₁₆	C₂×C₃₁	1	2	yes	complemented
118	#2158	C₂×C₃₁	1	1	yes	characteristic, complemented
119	#2138	C₂×C₃₁	1	2	yes	complemented
120	C₂×C₄×C₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
121	C₂²×C₃₁	—	2	2	no	—
122	C₂²×C₃₁	—	4	4	no	—
123	C₂²×C₃₁	—	8	4	no	—
124	C₂²×C₃₁	C₄×D₈	1	2	yes	—
125	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
126	C₄×C₃₁	—	2	2	no	—
127	C₄×C₃₁	—	4	4	no	—
128	C₄×C₃₁	—	8	4	no	—
129	C₄×C₃₁	C₄×D₈	1	2	yes	—
130	C₄×C₃₁	#508	1	4	yes	direct product factor
131	C₄×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
132	C₄×C₃₁	C₂×#135	1	1	yes	characteristic
133	C₂×#135	C₂²×C₃₁	1	2	yes	—
134	C₄×#41	—	2	2	no	—
135	#508	C₄×C₃₁	1	8	yes	direct product factor
136	C₂×Q₃₂	C₂²×C₃₁	1	1	yes	characteristic
137	C₂×SD₃₂	C₂²×C₃₁	1	2	yes	—
138	C₂×D₁₆	C₂²×C₃₁	1	1	yes	characteristic
139	C₂²×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
140	C₂²×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
141	C₄×Q₁₆	C₂²×C₃₁	1	4	yes	—
142	C₄×SD₁₆	—	2	4	no	—
143	C₄×D₈	C₂²×C₃₁	1	4	yes	—
144	#427	C₂²×C₃₁	1	2	yes	—
145	#416	—	2	2	no	—
146	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
147	#367	C₂²×C₃₁	1	2	yes	—
148	#366	C₂²×C₃₁	1	2	yes	—
149	#358	C₂²×C₃₁	1	4	yes	—
150	#357	C₂²×C₃₁	1	4	yes	—
151	C₄×C₁₆	C₂²×C₃₁	1	2	yes	—
152	C₄×C₁₆	C₂²×C₃₁	1	2	yes	—
153	C₂×C₃₁	—	2	2	no	—
154	C₂×C₃₁	—	8	4	no	—
155	C₂×C₃₁	C₂×C₄×D₈	1	1	yes	characteristic
156	C₂×C₃₁	C₂×#508	1	2	yes	—
157	C₂×#41	—	2	2	no	—
158	#135	—	2	4	no	—
159	#135	C₂×C₄×C₃₁	1	8	yes	complemented
160	C₂×Q₁₆	C₂³×C₃₁	1	2	yes	—
161	C₂×SD₁₆	—	2	2	no	—
162	C₂×D₈	C₂³×C₃₁	1	2	yes	—
163	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
164	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
165	C₄×Q₈	—	2	4	no	—
166	C₄×D₄	—	2	4	no	—
167	C₄×D₄	—	4	4	no	—
168	#117	—	2	2	no	—
169	#117	—	4	2	no	—
170	C₂×C₄²	—	4	2	no	—
171	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
172	Q₃₂	C₂×C₄×C₃₁	1	4	yes	complemented
173	SD₃₂	C₂×C₄×C₃₁	1	8	yes	complemented
174	D₁₆	C₂×C₄×C₃₁	1	4	yes	complemented
175	C₂×C₁₆	—	2	2	no	—
176	C₂×C₁₆	C₂×C₄×C₃₁	1	4	yes	complemented
177	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
178	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
179	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
180	#107	C₂³×C₃₁	1	4	yes	—
181	#106	—	2	2	no	—
182	#103	—	2	4	no	—
183	#102	—	2	4	no	—
184	C₄×C₈	D₄×C₃₁	1	2	yes	—
185	C₄×C₈	C₂³×C₃₁	1	2	yes	—
186	C₃₁	C₄×#508	1	1	yes	characteristic, direct product factor, Sylow
187	#41	—	2	8	no	—
188	#41	—	4	4	no	—
189	C₂×Q₈	—	2	2	no	—
190	C₂×D₄	—	2	2	no	—
191	C₂×D₄	—	4	2	no	—
192	C₂²×C₄	—	4	2	no	—
193	C₂²×C₄	—	4	4	no	—
194	C₂²×C₄	D₈×C₃₁	1	2	yes	—
195	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
196	Q₁₆	—	2	2	no	—
197	Q₁₆	C₂²×C₄×C₃₁	1	4	yes	—
198	SD₁₆	—	2	8	no	—
199	D₈	—	2	2	no	—
200	D₈	C₂²×C₄×C₃₁	1	4	yes	—
201	C₂×C₈	—	2	2	no	—
202	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
203	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
204	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
205	C₂×C₈	#41×C₃₁	1	2	yes	—
206	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
207	C₄⋊C₄	—	2	4	no	—
208	C₄⋊C₄	—	4	4	no	—
209	C₂²⋊C₄	—	4	2	no	—
210	C₂²⋊C₄	—	4	4	no	—
211	C₄²	—	4	2	no	—
212	C₄²	—	4	4	no	—
213	C₄²	D₈×C₃₁	1	1	yes	characteristic
214	C₄²	D₈×C₃₁	1	1	yes	characteristic
215	C₄²	C₂×D₄×C₃₁	1	2	yes	—
216	C₁₆	—	2	2	no	—
217	C₁₆	C₂²×C₄×C₃₁	1	2	yes	—
218	C₁₆	C₂²×C₄×C₃₁	1	2	yes	—
219	C₂³	—	4	2	no	—
220	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
221	Q₈	—	2	4	no	—
222	Q₈	—	4	2	no	—
223	D₄	—	2	4	no	—
224	D₄	—	4	8	no	—
225	D₄	—	4	2	no	—
226	C₂×C₄	—	2	4	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	—	4	8	no	—
229	C₂×C₄	—	4	4	no	—
230	C₂×C₄	—	8	8	no	—
231	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
232	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
233	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
234	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
235	C₂×C₄	C₃₁×#135	1	2	yes	—
236	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
237	C₈	—	2	2	no	—
238	C₈	C₄×D₄×C₃₁	1	2	yes	—
239	C₈	C₂³×C₄×C₃₁	1	1	yes	characteristic
240	C₈	C₂×#41×C₃₁	1	1	yes	characteristic
241	C₂²	—	2	2	no	—
242	C₂²	—	4	4	no	—
243	C₂²	—	8	4	no	—
244	C₂²	C₄×D₈×C₃₁	1	2	yes	—
245	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
246	C₄	—	2	2	no	—
247	C₄	—	4	4	no	—
248	C₄	—	8	4	no	—
249	C₄	C₄×D₈×C₃₁	1	2	yes	—
250	C₄	C₃₁×#508	1	4	yes	direct product factor
251	C₄	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
252	C₄	C₂×C₃₁×#135	1	1	yes	characteristic
253	C₂	—	2	2	no	—
254	C₂	—	8	4	no	—
255	C₂	C₂×C₄×D₈×C₃₁	1	1	yes	characteristic
256	C₂	C₂×C₃₁×#508	1	2	yes	—

	subgroup	G/H
center	C₄²×C₃₁	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 ◁ C₄²×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. C₄×C₃₁×Q₃₂	C₂	semidirect
4. C₄×C₃₁×SD₃₂	C₂	semidirect
5. C₄×C₃₁×D₁₆	C₂	semidirect
6. C₃₁×#2158	C₂	semidirect
7. C₃₁×#2138	C₂	semidirect
8. C₂×C₄×C₁₆×C₃₁	C₂	semidirect
11. C₃₁×#508	C₄	direct (N × H)
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
90. C₄×#508	C₃₁	direct (N × H)
114. C₄×#135	C₂×C₃₁	semidirect
115. C₄×Q₃₂	C₂×C₃₁	semidirect
116. C₄×SD₃₂	C₂×C₃₁	semidirect
117. C₄×D₁₆	C₂×C₃₁	semidirect
118. #2158	C₂×C₃₁	semidirect
119. #2138	C₂×C₃₁	semidirect
120. C₂×C₄×C₁₆	C₂×C₃₁	semidirect
130. C₄×C₃₁	#508	direct (N × H)
135. #508	C₄×C₃₁	direct (N × H)
159. #135	C₂×C₄×C₃₁	semidirect
172. Q₃₂	C₂×C₄×C₃₁	semidirect
173. SD₃₂	C₂×C₄×C₃₁	semidirect
174. D₁₆	C₂×C₄×C₃₁	semidirect
176. C₂×C₁₆	C₂×C₄×C₃₁	semidirect
186. C₃₁	C₄×#508	direct (N × H)
250. C₄	C₃₁×#508	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
C₄×C₃₁×Q₃₂	2
C₄×C₃₁×SD₃₂	2
C₄×C₃₁×D₁₆	2
C₃₁×#2158	2
C₃₁×#2138	2
C₂×C₄×C₁₆×C₃₁	2
C₄×#508	31

Subgroups — C4×C31×#508 #21490206

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₃₁×#508 #21490206