Subgroups — C₃₁×#13062 #21469821

Order 7936.

506 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃₁×M₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#467	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₁×#358	C₂	1	2	yes	maximal, complemented
4	C₃₁×#1808	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#1380	C₂	1	2	yes	maximal, complemented
6	C₂²×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
7	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
8	C₂×C₃₁×C₂²⋊Q₈	—	2	1	no	—
9	C₂×C₃₁×M₃₂	—	2	2	no	—
10	C₂×C₃₁×M₃₂	—	2	4	no	—
11	C₂²×C₁₆×C₃₁	C₂²	1	2	yes	—
12	C₃₁×#467	—	2	2	no	—
13	C₃₁×#467	C₄	1	4	yes	complemented
14	C₂×C₃₁×#107	C₂²	1	2	yes	—
15	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×#103	—	2	1	no	—
17	C₃₁×#380	—	2	1	no	—
18	C₃₁×#358	—	2	4	no	—
19	C₃₁×#340	—	2	1	no	—
20	C₂²×Q₈×C₃₁	—	2	1	no	—
21	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
22	C₂×Q₁₆×C₃₁	—	2	2	no	—
23	C₂×Q₁₆×C₃₁	—	2	1	no	—
24	C₂×Q₁₆×C₃₁	—	2	2	no	—
25	C₂×Q₁₆×C₃₁	C₂×C₄	1	2	yes	—
26	C₂²×C₈×C₃₁	—	2	2	no	—
27	C₂²×C₈×C₃₁	—	2	2	no	—
28	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
29	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₃₁	D₄	1	2	yes	—
31	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
32	C₃₁×C₂²⋊Q₈	—	2	4	no	—
33	C₃₁×C₂²⋊Q₈	—	4	2	no	—
34	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
35	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
36	C₂×C₄⋊C₄×C₃₁	—	4	1	no	—
37	C₂×C₂²⋊C₄×C₃₁	—	4	1	no	—
38	C₂×C₂²⋊C₄×C₃₁	—	4	1	no	—
39	C₃₁×M₃₂	—	2	8	no	—
40	C₂×C₁₆×C₃₁	—	2	2	no	—
41	C₂×C₁₆×C₃₁	—	2	2	no	—
42	C₂×C₁₆×C₃₁	D₄	1	4	yes	—
43	C₃₁×#107	—	2	1	no	—
44	C₃₁×#107	—	2	2	no	—
45	C₃₁×#107	—	2	2	no	—
46	C₃₁×#107	C₂×C₄	1	2	yes	—
47	C₃₁×#103	—	2	4	no	—
48	C₃₁×#95	—	4	1	no	—
49	C₂⁴×C₃₁	D₈	1	1	yes	characteristic
50	C₂×Q₈×C₃₁	—	2	2	no	—
51	C₂×Q₈×C₃₁	—	4	1	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	—	2	2	no	—
56	C₂²×C₄×C₃₁	—	4	1	no	—
57	C₂²×C₄×C₃₁	—	4	1	no	—
58	C₂²×C₄×C₃₁	—	4	2	no	—
59	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
60	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
61	C₂²×C₄×C₃₁	SD₁₆	1	2	yes	—
62	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
63	Q₁₆×C₃₁	—	2	4	no	—
64	Q₁₆×C₃₁	—	4	1	no	—
65	Q₁₆×C₃₁	—	4	1	no	—
66	C₂×C₈×C₃₁	—	2	2	no	—
67	C₂×C₈×C₃₁	—	2	4	no	—
68	C₂×C₈×C₃₁	—	2	2	no	—
69	C₂×C₈×C₃₁	—	2	2	no	—
70	C₂×C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
71	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
73	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈×C₃₁	#41	1	2	yes	—
75	C₄⋊C₄×C₃₁	—	2	2	no	—
76	C₄⋊C₄×C₃₁	—	4	4	no	—
77	C₄⋊C₄×C₃₁	—	4	1	no	—
78	C₄⋊C₄×C₃₁	—	4	2	no	—
79	C₄⋊C₄×C₃₁	—	4	2	no	—
80	C₂²⋊C₄×C₃₁	—	4	4	no	—
81	C₂²⋊C₄×C₃₁	—	8	2	no	—
82	C₁₆×C₃₁	—	2	4	no	—
83	#13062	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
84	C₂³×C₃₁	—	2	2	no	—
85	C₂³×C₃₁	—	2	1	no	—
86	C₂³×C₃₁	—	2	1	no	—
87	C₂³×C₃₁	—	2	2	no	—
88	C₂³×C₃₁	#102	1	2	yes	—
89	C₂³×C₃₁	C₂×D₈	1	1	yes	characteristic
90	Q₈×C₃₁	—	4	2	no	—
91	Q₈×C₃₁	—	4	2	no	—
92	Q₈×C₃₁	—	8	2	no	—
93	C₂×C₄×C₃₁	—	2	2	no	—
94	C₂×C₄×C₃₁	—	2	2	no	—
95	C₂×C₄×C₃₁	—	2	2	no	—
96	C₂×C₄×C₃₁	—	2	4	no	—
97	C₂×C₄×C₃₁	—	4	2	no	—
98	C₂×C₄×C₃₁	—	4	2	no	—
99	C₂×C₄×C₃₁	—	8	2	no	—
100	C₂×C₄×C₃₁	—	8	1	no	—
101	C₂×C₄×C₃₁	—	8	1	no	—
102	C₂×C₄×C₃₁	—	8	2	no	—
103	C₂×C₄×C₃₁	—	8	2	no	—
104	C₂×C₄×C₃₁	—	8	2	no	—
105	C₂×C₄×C₃₁	#102	1	2	yes	—
106	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₃₁	#121	1	2	yes	—
108	C₂×C₄×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
109	C₂×C₄×C₃₁	#135	1	2	yes	—
110	C₈×C₃₁	—	2	2	no	—
111	C₈×C₃₁	C₄×D₄	1	2	yes	—
112	C₈×C₃₁	#120	1	1	yes	characteristic
113	C₈×C₃₁	#123	1	1	yes	characteristic
114	C₂²×M₃₂	C₂×C₃₁	1	1	yes	characteristic
115	C₂×#467	C₂×C₃₁	1	1	yes	characteristic
116	C₂×#358	C₂×C₃₁	1	2	yes	complemented
117	#1808	C₂×C₃₁	1	1	yes	characteristic
118	#1380	C₂×C₃₁	1	2	yes	complemented
119	C₂²×C₃₁	—	2	2	no	—
120	C₂²×C₃₁	—	2	2	no	—
121	C₂²×C₃₁	—	2	2	no	—
122	C₂²×C₃₁	—	4	2	no	—
123	C₂²×C₃₁	—	4	2	no	—
124	C₂²×C₃₁	C₂×#102	1	1	yes	characteristic
125	C₂²×C₃₁	#466	1	2	yes	—
126	C₂²×C₃₁	#509	1	1	yes	characteristic
127	C₂²×C₃₁	#510	1	1	yes	characteristic
128	C₂²×C₃₁	#510	1	2	yes	—
129	C₄×C₃₁	—	2	2	no	—
130	C₄×C₃₁	—	8	2	no	—
131	C₄×C₃₁	—	8	2	no	—
132	C₄×C₃₁	—	16	2	no	—
133	C₄×C₃₁	#386	1	1	yes	characteristic
134	C₄×C₃₁	#416	1	1	yes	characteristic
135	C₄×C₃₁	#465	1	2	yes	—
136	C₂²×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
137	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
138	C₂×C₂²⋊Q₈	—	2	1	no	—
139	C₂×M₃₂	—	2	2	no	—
140	C₂×M₃₂	—	2	4	no	—
141	C₂²×C₁₆	C₂²×C₃₁	1	2	yes	—
142	#467	—	2	2	no	—
143	#467	C₄×C₃₁	1	4	yes	complemented
144	C₂×#107	C₂²×C₃₁	1	2	yes	—
145	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
146	C₂×#103	—	2	1	no	—
147	#380	—	2	1	no	—
148	#358	—	2	4	no	—
149	#340	—	2	1	no	—
150	C₂×C₃₁	—	4	2	no	—
151	C₂×C₃₁	#1893	1	1	yes	characteristic
152	C₂×C₃₁	#2139	1	1	yes	characteristic
153	C₂×C₃₁	#2141	1	1	yes	characteristic
154	C₂×C₃₁	#2219	1	2	yes	—
155	C₂×C₃₁	#2220	1	2	yes	—
156	C₂²×Q₈	—	2	1	no	—
157	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
158	C₂×Q₁₆	—	2	2	no	—
159	C₂×Q₁₆	—	2	1	no	—
160	C₂×Q₁₆	—	2	2	no	—
161	C₂×Q₁₆	C₂×C₄×C₃₁	1	2	yes	—
162	C₂²×C₈	—	2	2	no	—
163	C₂²×C₈	—	2	2	no	—
164	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
165	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
166	C₂²×C₈	D₄×C₃₁	1	2	yes	—
167	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
168	C₂²⋊Q₈	—	2	4	no	—
169	C₂²⋊Q₈	—	4	2	no	—
170	C₂×C₄⋊C₄	—	2	1	no	—
171	C₂×C₄⋊C₄	—	2	2	no	—
172	C₂×C₄⋊C₄	—	4	1	no	—
173	C₂×C₂²⋊C₄	—	4	1	no	—
174	C₂×C₂²⋊C₄	—	4	1	no	—
175	M₃₂	—	2	8	no	—
176	C₂×C₁₆	—	2	2	no	—
177	C₂×C₁₆	—	2	2	no	—
178	C₂×C₁₆	D₄×C₃₁	1	4	yes	—
179	#107	—	2	1	no	—
180	#107	—	2	2	no	—
181	#107	—	2	2	no	—
182	#107	C₂×C₄×C₃₁	1	2	yes	—
183	#103	—	2	4	no	—
184	#95	—	4	1	no	—
185	C₃₁	#13062	1	1	yes	characteristic, direct product factor, Sylow
186	C₂⁴	D₈×C₃₁	1	1	yes	characteristic
187	C₂×Q₈	—	2	2	no	—
188	C₂×Q₈	—	4	1	no	—
189	C₂×Q₈	—	4	2	no	—
190	C₂×Q₈	—	4	2	no	—
191	C₂²×C₄	—	2	2	no	—
192	C₂²×C₄	—	2	2	no	—
193	C₂²×C₄	—	4	1	no	—
194	C₂²×C₄	—	4	1	no	—
195	C₂²×C₄	—	4	2	no	—
196	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
197	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
198	C₂²×C₄	SD₁₆×C₃₁	1	2	yes	—
199	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
200	Q₁₆	—	2	4	no	—
201	Q₁₆	—	4	1	no	—
202	Q₁₆	—	4	1	no	—
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	—	2	4	no	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	C₂²⋊C₄×C₃₁	1	2	yes	—
208	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
209	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
210	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
211	C₂×C₈	#41×C₃₁	1	2	yes	—
212	C₄⋊C₄	—	2	2	no	—
213	C₄⋊C₄	—	4	4	no	—
214	C₄⋊C₄	—	4	1	no	—
215	C₄⋊C₄	—	4	2	no	—
216	C₄⋊C₄	—	4	2	no	—
217	C₂²⋊C₄	—	4	4	no	—
218	C₂²⋊C₄	—	8	2	no	—
219	C₁₆	—	2	4	no	—
220	C₂³	—	2	2	no	—
221	C₂³	—	2	1	no	—
222	C₂³	—	2	1	no	—
223	C₂³	—	2	2	no	—
224	C₂³	C₃₁×#102	1	2	yes	—
225	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
226	Q₈	—	4	2	no	—
227	Q₈	—	4	2	no	—
228	Q₈	—	8	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	4	2	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	8	2	no	—
236	C₂×C₄	—	8	1	no	—
237	C₂×C₄	—	8	1	no	—
238	C₂×C₄	—	8	2	no	—
239	C₂×C₄	—	8	2	no	—
240	C₂×C₄	—	8	2	no	—
241	C₂×C₄	C₃₁×#102	1	2	yes	—
242	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
243	C₂×C₄	C₃₁×#121	1	2	yes	—
244	C₂×C₄	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
245	C₂×C₄	C₃₁×#135	1	2	yes	—
246	C₈	—	2	2	no	—
247	C₈	C₄×D₄×C₃₁	1	2	yes	—
248	C₈	C₃₁×#120	1	1	yes	characteristic
249	C₈	C₃₁×#123	1	1	yes	characteristic
250	C₂²	—	2	2	no	—
251	C₂²	—	2	2	no	—
252	C₂²	—	2	2	no	—
253	C₂²	—	4	2	no	—
254	C₂²	—	4	2	no	—
255	C₂²	C₂×C₃₁×#102	1	1	yes	characteristic
256	C₂²	C₃₁×#466	1	2	yes	—
257	C₂²	C₃₁×#509	1	1	yes	characteristic
258	C₂²	C₃₁×#510	1	1	yes	characteristic
259	C₂²	C₃₁×#510	1	2	yes	—
260	C₄	—	2	2	no	—
261	C₄	—	8	2	no	—
262	C₄	—	8	2	no	—
263	C₄	—	16	2	no	—
264	C₄	C₃₁×#386	1	1	yes	characteristic
265	C₄	C₃₁×#416	1	1	yes	characteristic
266	C₄	C₃₁×#465	1	2	yes	—
267	C₂	—	4	2	no	—
268	C₂	C₃₁×#1893	1	1	yes	characteristic
269	C₂	C₃₁×#2139	1	1	yes	characteristic
270	C₂	C₃₁×#2141	1	1	yes	characteristic
271	C₂	C₃₁×#2219	1	2	yes	—
272	C₂	C₃₁×#2220	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₃₁
Frattini, cosocle	C₂²×C₈	C₂³×C₃₁
prime radical	C₂⁴×C₃₁	D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂³×C₄×C₃₁ ◁ C₂³×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×C₃₁×#358	C₂	semidirect
5. C₃₁×#1380	C₂	semidirect
13. C₃₁×#467	C₄	semidirect
83. #13062	C₃₁	direct (N × H)
116. C₂×#358	C₂×C₃₁	semidirect
118. #1380	C₂×C₃₁	semidirect
143. #467	C₄×C₃₁	semidirect
185. C₃₁	#13062	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃₁×M₃₂	2
C₂×C₃₁×#467	2
C₂×C₃₁×#358	2
C₃₁×#1808	2
C₃₁×#1380	2
#13062	31

Subgroups — C31×#13062 #21469821

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#13062 #21469821