Subgroups — C₃₁×#9733 #21466492

Order 7936.

518 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#494	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1988	C₂	1	2	yes	maximal
3	C₈×C₄⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃₁×#335	C₂	1	1	yes	maximal, characteristic
5	C₂×C₃₁×#332	C₂	1	2	yes	maximal
6	C₂²×Q₁₆×C₃₁	—	2	1	no	—
7	C₂²×Q₁₆×C₃₁	—	2	1	no	—
8	C₂×C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
9	C₃₁×#494	—	2	2	no	—
10	C₃₁×#494	C₄	1	4	yes	complemented
11	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
12	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₃₁	—	2	1	no	—
14	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
15	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
16	C₃₁×#340	—	2	4	no	—
17	C₃₁×#336	—	2	1	no	—
18	C₃₁×#335	—	2	2	no	—
19	C₃₁×#332	—	2	4	no	—
20	C₂²×Q₈×C₃₁	—	2	2	no	—
21	C₂×Q₁₆×C₃₁	—	2	4	no	—
22	C₂×Q₁₆×C₃₁	—	2	4	no	—
23	C₂×Q₁₆×C₃₁	—	4	2	no	—
24	C₂×Q₁₆×C₃₁	—	4	2	no	—
25	C₂×Q₁₆×C₃₁	—	4	2	no	—
26	C₂×Q₁₆×C₃₁	—	4	2	no	—
27	C₂²×C₈×C₃₁	—	2	1	no	—
28	C₂²×C₈×C₃₁	—	2	2	no	—
29	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
31	C₃₁×C₄⋊Q₈	—	2	2	no	—
32	C₃₁×C₄⋊Q₈	—	2	2	no	—
33	C₃₁×C₄⋊Q₈	—	2	2	no	—
34	C₃₁×C₄⋊Q₈	C₂×C₄	1	4	yes	—
35	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
36	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
37	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
38	C₂×C₄²×C₃₁	—	2	1	no	—
39	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
40	C₃₁×C₄⋊C₈	—	2	2	no	—
41	C₃₁×C₄⋊C₈	—	2	2	no	—
42	C₃₁×C₄⋊C₈	—	2	1	no	—
43	C₃₁×C₄⋊C₈	D₄	1	2	yes	—
44	C₄×C₈×C₃₁	—	4	2	no	—
45	C₄×C₈×C₃₁	C₂×C₄	1	2	yes	—
46	C₄×C₈×C₃₁	D₄	1	2	yes	—
47	C₃₁×#95	—	2	1	no	—
48	C₂×Q₈×C₃₁	—	2	4	no	—
49	C₂×Q₈×C₃₁	—	4	2	no	—
50	C₂×Q₈×C₃₁	—	4	2	no	—
51	C₂×Q₈×C₃₁	—	4	2	no	—
52	C₂×Q₈×C₃₁	—	4	2	no	—
53	C₂×Q₈×C₃₁	—	4	2	no	—
54	C₂²×C₄×C₃₁	—	2	2	no	—
55	C₂²×C₄×C₃₁	—	4	2	no	—
56	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
59	Q₁₆×C₃₁	—	4	8	no	—
60	Q₁₆×C₃₁	—	4	8	no	—
61	C₂×C₈×C₃₁	—	2	1	no	—
62	C₂×C₈×C₃₁	—	2	1	no	—
63	C₂×C₈×C₃₁	—	2	2	no	—
64	C₂×C₈×C₃₁	—	4	1	no	—
65	C₂×C₈×C₃₁	—	4	1	no	—
66	C₂×C₈×C₃₁	—	4	2	no	—
67	C₂×C₈×C₃₁	—	4	4	no	—
68	C₂×C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
69	C₂×C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
70	C₂×C₈×C₃₁	D₈	1	2	yes	—
71	C₂×C₈×C₃₁	SD₁₆	1	2	yes	—
72	C₄⋊C₄×C₃₁	—	2	2	no	—
73	C₄⋊C₄×C₃₁	—	2	4	no	—
74	C₄⋊C₄×C₃₁	—	2	4	no	—
75	C₄⋊C₄×C₃₁	—	4	2	no	—
76	C₄⋊C₄×C₃₁	—	4	2	no	—
77	C₄⋊C₄×C₃₁	Q₁₆	1	4	yes	complemented
78	C₄²×C₃₁	—	4	1	no	—
79	C₄²×C₃₁	—	4	1	no	—
80	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
81	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
82	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
83	C₄²×C₃₁	#41	1	1	yes	characteristic
84	#9733	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
85	C₂³×C₃₁	#121	1	1	yes	characteristic
86	Q₈×C₃₁	—	4	4	no	—
87	Q₈×C₃₁	—	4	4	no	—
88	Q₈×C₃₁	—	8	2	no	—
89	Q₈×C₃₁	—	8	2	no	—
90	C₂×C₄×C₃₁	—	2	2	no	—
91	C₂×C₄×C₃₁	—	2	4	no	—
92	C₂×C₄×C₃₁	—	4	2	no	—
93	C₂×C₄×C₃₁	—	4	2	no	—
94	C₂×C₄×C₃₁	—	4	4	no	—
95	C₂×C₄×C₃₁	—	8	2	no	—
96	C₂×C₄×C₃₁	—	8	2	no	—
97	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
99	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
100	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
101	C₂×C₄×C₃₁	#124	1	1	yes	characteristic
102	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
103	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
104	C₂×C₄×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
105	C₂×C₄×C₃₁	C₂×Q₁₆	1	2	yes	—
106	C₂×C₄×C₃₁	#135	1	2	yes	—
107	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
108	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
109	C₈×C₃₁	—	2	2	no	—
110	C₈×C₃₁	—	4	2	no	—
111	C₈×C₃₁	—	8	2	no	—
112	C₈×C₃₁	#102	1	4	yes	—
113	C₂×#494	C₂×C₃₁	1	1	yes	characteristic
114	#1988	C₂×C₃₁	1	2	yes	—
115	C₈×C₄⋊C₄	C₂×C₃₁	1	1	yes	characteristic
116	C₂×#335	C₂×C₃₁	1	1	yes	characteristic
117	C₂×#332	C₂×C₃₁	1	2	yes	—
118	C₂²×C₃₁	#390	1	1	yes	characteristic
119	C₂²×C₃₁	#459	1	1	yes	characteristic
120	C₂²×C₃₁	#460	1	1	yes	characteristic
121	C₂²×C₃₁	#467	1	2	yes	—
122	C₂²×C₃₁	#473	1	2	yes	—
123	C₄×C₃₁	—	2	1	no	—
124	C₄×C₃₁	—	2	1	no	—
125	C₄×C₃₁	—	4	4	no	—
126	C₄×C₃₁	—	8	4	no	—
127	C₄×C₃₁	C₂×#102	1	1	yes	characteristic
128	C₄×C₃₁	#418	1	1	yes	characteristic
129	C₄×C₃₁	C₄×Q₁₆	1	2	yes	—
130	C₄×C₃₁	#459	1	1	yes	characteristic
131	C₄×C₃₁	#460	1	1	yes	characteristic
132	C₄×C₃₁	#494	1	1	yes	characteristic
133	C₄×C₃₁	#495	1	1	yes	characteristic
134	C₂²×Q₁₆	—	2	1	no	—
135	C₂²×Q₁₆	—	2	1	no	—
136	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
137	#494	—	2	2	no	—
138	#494	C₄×C₃₁	1	4	yes	complemented
139	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
140	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
141	C₂×C₄×C₈	—	2	1	no	—
142	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
143	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
144	#340	—	2	4	no	—
145	#336	—	2	1	no	—
146	#335	—	2	2	no	—
147	#332	—	2	4	no	—
148	C₂×C₃₁	#1669	1	2	yes	—
149	C₂×C₃₁	#1670	1	2	yes	—
150	C₂×C₃₁	#1969	1	1	yes	characteristic
151	C₂×C₃₁	#1979	1	1	yes	characteristic
152	C₂×C₃₁	#1982	1	1	yes	characteristic
153	C₂²×Q₈	—	2	2	no	—
154	C₂×Q₁₆	—	2	4	no	—
155	C₂×Q₁₆	—	2	4	no	—
156	C₂×Q₁₆	—	4	2	no	—
157	C₂×Q₁₆	—	4	2	no	—
158	C₂×Q₁₆	—	4	2	no	—
159	C₂×Q₁₆	—	4	2	no	—
160	C₂²×C₈	—	2	1	no	—
161	C₂²×C₈	—	2	2	no	—
162	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
163	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
164	C₄⋊Q₈	—	2	2	no	—
165	C₄⋊Q₈	—	2	2	no	—
166	C₄⋊Q₈	—	2	2	no	—
167	C₄⋊Q₈	C₂×C₄×C₃₁	1	4	yes	—
168	C₂×C₄⋊C₄	—	2	2	no	—
169	C₂×C₄⋊C₄	—	2	2	no	—
170	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
171	C₂×C₄²	—	2	1	no	—
172	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
173	C₄⋊C₈	—	2	2	no	—
174	C₄⋊C₈	—	2	2	no	—
175	C₄⋊C₈	—	2	1	no	—
176	C₄⋊C₈	D₄×C₃₁	1	2	yes	—
177	C₄×C₈	—	4	2	no	—
178	C₄×C₈	C₂×C₄×C₃₁	1	2	yes	—
179	C₄×C₈	D₄×C₃₁	1	2	yes	—
180	#95	—	2	1	no	—
181	C₃₁	#9733	1	1	yes	characteristic, direct product factor, Sylow
182	C₂×Q₈	—	2	4	no	—
183	C₂×Q₈	—	4	2	no	—
184	C₂×Q₈	—	4	2	no	—
185	C₂×Q₈	—	4	2	no	—
186	C₂×Q₈	—	4	2	no	—
187	C₂×Q₈	—	4	2	no	—
188	C₂²×C₄	—	2	2	no	—
189	C₂²×C₄	—	4	2	no	—
190	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
191	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
192	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
193	Q₁₆	—	4	8	no	—
194	Q₁₆	—	4	8	no	—
195	C₂×C₈	—	2	1	no	—
196	C₂×C₈	—	2	1	no	—
197	C₂×C₈	—	2	2	no	—
198	C₂×C₈	—	4	1	no	—
199	C₂×C₈	—	4	1	no	—
200	C₂×C₈	—	4	2	no	—
201	C₂×C₈	—	4	4	no	—
202	C₂×C₈	C₂²⋊C₄×C₃₁	1	2	yes	—
203	C₂×C₈	C₂²⋊C₄×C₃₁	1	2	yes	—
204	C₂×C₈	D₈×C₃₁	1	2	yes	—
205	C₂×C₈	SD₁₆×C₃₁	1	2	yes	—
206	C₄⋊C₄	—	2	2	no	—
207	C₄⋊C₄	—	2	4	no	—
208	C₄⋊C₄	—	2	4	no	—
209	C₄⋊C₄	—	4	2	no	—
210	C₄⋊C₄	—	4	2	no	—
211	C₄⋊C₄	Q₁₆×C₃₁	1	4	yes	complemented
212	C₄²	—	4	1	no	—
213	C₄²	—	4	1	no	—
214	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
215	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
216	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
217	C₄²	#41×C₃₁	1	1	yes	characteristic
218	C₂³	C₃₁×#121	1	1	yes	characteristic
219	Q₈	—	4	4	no	—
220	Q₈	—	4	4	no	—
221	Q₈	—	8	2	no	—
222	Q₈	—	8	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	4	no	—
225	C₂×C₄	—	4	2	no	—
226	C₂×C₄	—	4	2	no	—
227	C₂×C₄	—	4	4	no	—
228	C₂×C₄	—	8	2	no	—
229	C₂×C₄	—	8	2	no	—
230	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
231	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
232	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
233	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
234	C₂×C₄	C₃₁×#124	1	1	yes	characteristic
235	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
236	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
237	C₂×C₄	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
238	C₂×C₄	C₂×Q₁₆×C₃₁	1	2	yes	—
239	C₂×C₄	C₃₁×#135	1	2	yes	—
240	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
241	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
242	C₈	—	2	2	no	—
243	C₈	—	4	2	no	—
244	C₈	—	8	2	no	—
245	C₈	C₃₁×#102	1	4	yes	—
246	C₂²	C₃₁×#390	1	1	yes	characteristic
247	C₂²	C₃₁×#459	1	1	yes	characteristic
248	C₂²	C₃₁×#460	1	1	yes	characteristic
249	C₂²	C₃₁×#467	1	2	yes	—
250	C₂²	C₃₁×#473	1	2	yes	—
251	C₄	—	2	1	no	—
252	C₄	—	2	1	no	—
253	C₄	—	4	4	no	—
254	C₄	—	8	4	no	—
255	C₄	C₂×C₃₁×#102	1	1	yes	characteristic
256	C₄	C₃₁×#418	1	1	yes	characteristic
257	C₄	C₄×Q₁₆×C₃₁	1	2	yes	—
258	C₄	C₃₁×#459	1	1	yes	characteristic
259	C₄	C₃₁×#460	1	1	yes	characteristic
260	C₄	C₃₁×#494	1	1	yes	characteristic
261	C₄	C₃₁×#495	1	1	yes	characteristic
262	C₂	C₃₁×#1669	1	2	yes	—
263	C₂	C₃₁×#1670	1	2	yes	—
264	C₂	C₃₁×#1969	1	1	yes	characteristic
265	C₂	C₃₁×#1979	1	1	yes	characteristic
266	C₂	C₃₁×#1982	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₃₁	#121
commutator	C₄²	C₂²×C₄×C₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
10. C₃₁×#494	C₄	semidirect
77. C₄⋊C₄×C₃₁	Q₁₆	semidirect
84. #9733	C₃₁	direct (N × H)
138. #494	C₄×C₃₁	semidirect
181. C₃₁	#9733	direct (N × H)
211. C₄⋊C₄	Q₁₆×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#494	2
C₃₁×#1988	2
C₈×C₄⋊C₄×C₃₁	2
C₂×C₃₁×#335	2
C₂×C₃₁×#332	2
#9733	31

Subgroups — C31×#9733 #21466492

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#9733 #21466492