Subgroups — C₃₁×#22739 #21479498

Order 7936.

734 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	C₂×C₈×Q₈×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2355	C₂	1	2	yes	maximal
3	C₃₁×#1949	C₂	1	4	yes	maximal
4	C₃₁×#1947	C₂	1	2	yes	maximal
5	C₃₁×#1835	C₂	1	2	yes	maximal
6	C₃₁×#1834	C₂	1	1	yes	maximal, characteristic
7	C₄×C₃₁×#107	C₂	1	2	yes	maximal
8	C₄×C₃₁×#106	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
10	C₈×Q₈×C₃₁	C₄	1	8	yes	complemented
11	C₂×C₃₁×#107	C₂²	1	4	yes	—
12	C₂×C₃₁×#106	C₂²	1	2	yes	—
13	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
17	C₃₁×#391	—	2	2	no	—
18	C₃₁×#387	—	2	4	no	—
19	C₃₁×#384	C₂²	1	2	yes	—
20	C₃₁×#382	C₂²	1	4	yes	—
21	C₄×C₄⋊C₄×C₃₁	—	2	2	no	—
22	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
23	C₄×C₄⋊C₄×C₃₁	C₂²	1	4	yes	—
24	C₃₁×#340	C₂²	1	2	yes	—
25	C₃₁×#340	C₂²	1	2	yes	—
26	C₃₁×#340	C₂²	1	1	yes	characteristic
27	C₃₁×#340	C₂²	1	4	yes	—
28	C₃₁×#340	C₂²	1	1	yes	characteristic
29	C₂²×Q₈×C₃₁	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
31	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
32	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
33	C₄×Q₈×C₃₁	C₂×C₄	1	4	yes	—
34	C₄×Q₈×C₃₁	Q₈	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	1	yes	characteristic
38	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
39	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
40	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
41	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
42	C₂×C₄²×C₃₁	—	2	2	no	—
43	C₂×C₄²×C₃₁	—	2	4	no	—
44	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
45	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
46	C₃₁×#107	—	2	8	no	—
47	C₃₁×#106	—	2	4	no	—
48	C₃₁×C₄⋊C₈	C₂×C₄	1	4	yes	—
49	C₃₁×C₄⋊C₈	C₂×C₄	1	8	yes	—
50	C₄×C₈×C₃₁	C₂×C₄	1	4	yes	—
51	C₄×C₈×C₃₁	C₂×C₄	1	8	yes	—
52	C₃₁×#95	—	2	4	no	—
53	C₃₁×#95	—	2	2	no	—
54	C₃₁×#95	—	2	4	no	—
55	C₃₁×#95	—	2	4	no	—
56	C₂×Q₈×C₃₁	—	2	4	no	—
57	C₂×Q₈×C₃₁	C₄⋊C₄	1	4	yes	—
58	C₂²×C₄×C₃₁	—	2	2	no	—
59	C₂²×C₄×C₃₁	—	2	2	no	—
60	C₂²×C₄×C₃₁	—	2	4	no	—
61	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
62	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
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	C₂²×C₄	1	2	yes	—
70	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
71	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
72	C₄⋊C₄×C₃₁	—	2	2	no	—
73	C₄⋊C₄×C₃₁	—	2	2	no	—
74	C₄⋊C₄×C₃₁	—	2	4	no	—
75	C₄⋊C₄×C₃₁	—	2	2	no	—
76	C₄⋊C₄×C₃₁	—	2	4	no	—
77	C₄⋊C₄×C₃₁	—	4	4	no	—
78	C₄⋊C₄×C₃₁	—	8	2	no	—
79	C₄⋊C₄×C₃₁	C₄⋊C₄	1	8	yes	—
80	C₄⋊C₄×C₃₁	C₄⋊C₄	1	4	yes	—
81	C₄⋊C₄×C₃₁	#41	1	4	yes	—
82	C₄²×C₃₁	—	4	4	no	—
83	C₄²×C₃₁	—	8	4	no	—
84	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
85	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
86	C₄²×C₃₁	C₂×Q₈	1	4	yes	—
87	C₄²×C₃₁	C₂×Q₈	1	2	yes	—
88	#22739	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
89	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
90	Q₈×C₃₁	—	2	8	no	—
91	C₂×C₄×C₃₁	—	2	4	no	—
92	C₂×C₄×C₃₁	—	2	2	no	—
93	C₂×C₄×C₃₁	—	2	1	no	—
94	C₂×C₄×C₃₁	—	2	2	no	—
95	C₂×C₄×C₃₁	—	2	1	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	—	4	4	no	—
100	C₂×C₄×C₃₁	—	4	2	no	—
101	C₂×C₄×C₃₁	—	4	2	no	—
102	C₂×C₄×C₃₁	—	4	4	no	—
103	C₂×C₄×C₃₁	—	4	4	no	—
104	C₂×C₄×C₃₁	—	4	2	no	—
105	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	4	yes	—
106	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
107	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	4	yes	—
108	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
109	C₂×C₄×C₃₁	#135	1	4	yes	—
110	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
111	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
112	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
113	C₂×C₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
114	C₈×C₃₁	—	2	2	no	—
115	C₈×C₃₁	—	2	4	no	—
116	C₈×C₃₁	—	2	2	no	—
117	C₂×C₈×Q₈	C₂×C₃₁	1	1	yes	characteristic
118	#2355	C₂×C₃₁	1	2	yes	—
119	#1949	C₂×C₃₁	1	4	yes	—
120	#1947	C₂×C₃₁	1	2	yes	—
121	#1835	C₂×C₃₁	1	2	yes	—
122	#1834	C₂×C₃₁	1	1	yes	characteristic
123	C₄×#107	C₂×C₃₁	1	2	yes	—
124	C₄×#106	C₂×C₃₁	1	1	yes	characteristic
125	C₂²×C₃₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
126	C₂²×C₃₁	#548	1	2	yes	—
127	C₂²×C₃₁	C₂×#135	1	2	yes	—
128	C₂²×C₃₁	#576	1	1	yes	characteristic
129	C₂²×C₃₁	#578	1	1	yes	characteristic
130	C₄×C₃₁	—	2	4	no	—
131	C₄×C₃₁	—	2	4	no	—
132	C₄×C₃₁	—	2	2	no	—
133	C₄×C₃₁	—	4	4	no	—
134	C₄×C₃₁	—	8	4	no	—
135	C₄×C₃₁	—	8	2	no	—
136	C₄×C₃₁	#427	1	4	yes	—
137	C₄×C₃₁	C₄×#41	1	1	yes	characteristic
138	C₄×C₃₁	#519	1	1	yes	characteristic
139	C₄×C₃₁	#557	1	2	yes	—
140	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
141	C₈×Q₈	C₄×C₃₁	1	8	yes	complemented
142	C₂×#107	C₂²×C₃₁	1	4	yes	—
143	C₂×#106	C₂²×C₃₁	1	2	yes	—
144	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
145	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
146	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
147	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
148	#391	—	2	2	no	—
149	#387	—	2	4	no	—
150	#384	C₂²×C₃₁	1	2	yes	—
151	#382	C₂²×C₃₁	1	4	yes	—
152	C₄×C₄⋊C₄	—	2	2	no	—
153	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
154	C₄×C₄⋊C₄	C₂²×C₃₁	1	4	yes	—
155	#340	C₂²×C₃₁	1	2	yes	—
156	#340	C₂²×C₃₁	1	2	yes	—
157	#340	C₂²×C₃₁	1	1	yes	characteristic
158	#340	C₂²×C₃₁	1	4	yes	—
159	#340	C₂²×C₃₁	1	1	yes	characteristic
160	C₂×C₃₁	#2351	1	1	yes	characteristic
161	C₂×C₃₁	C₂×#427	1	1	yes	characteristic
162	C₂×C₃₁	#2913	1	1	yes	characteristic
163	C₂×C₃₁	#3361	1	2	yes	—
164	C₂×C₃₁	#3364	1	2	yes	—
165	C₂²×Q₈	D₄×C₃₁	1	1	yes	characteristic
166	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
167	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
168	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
169	C₄×Q₈	C₂×C₄×C₃₁	1	4	yes	—
170	C₄×Q₈	Q₈×C₃₁	1	4	yes	—
171	C₂×C₄⋊C₄	—	2	2	no	—
172	C₂×C₄⋊C₄	—	2	2	no	—
173	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
174	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
175	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
176	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
177	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
178	C₂×C₄²	—	2	2	no	—
179	C₂×C₄²	—	2	4	no	—
180	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
181	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
182	#107	—	2	8	no	—
183	#106	—	2	4	no	—
184	C₄⋊C₈	C₂×C₄×C₃₁	1	4	yes	—
185	C₄⋊C₈	C₂×C₄×C₃₁	1	8	yes	—
186	C₄×C₈	C₂×C₄×C₃₁	1	4	yes	—
187	C₄×C₈	C₂×C₄×C₃₁	1	8	yes	—
188	#95	—	2	4	no	—
189	#95	—	2	2	no	—
190	#95	—	2	4	no	—
191	#95	—	2	4	no	—
192	C₃₁	#22739	1	1	yes	characteristic, direct product factor, Sylow
193	C₂×Q₈	—	2	4	no	—
194	C₂×Q₈	C₄⋊C₄×C₃₁	1	4	yes	—
195	C₂²×C₄	—	2	2	no	—
196	C₂²×C₄	—	2	2	no	—
197	C₂²×C₄	—	2	4	no	—
198	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
199	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
200	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
201	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
202	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	—	2	4	no	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
207	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
208	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
209	C₄⋊C₄	—	2	2	no	—
210	C₄⋊C₄	—	2	2	no	—
211	C₄⋊C₄	—	2	4	no	—
212	C₄⋊C₄	—	2	2	no	—
213	C₄⋊C₄	—	2	4	no	—
214	C₄⋊C₄	—	4	4	no	—
215	C₄⋊C₄	—	8	2	no	—
216	C₄⋊C₄	C₄⋊C₄×C₃₁	1	8	yes	—
217	C₄⋊C₄	C₄⋊C₄×C₃₁	1	4	yes	—
218	C₄⋊C₄	#41×C₃₁	1	4	yes	—
219	C₄²	—	4	4	no	—
220	C₄²	—	8	4	no	—
221	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
222	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
223	C₄²	C₂×Q₈×C₃₁	1	4	yes	—
224	C₄²	C₂×Q₈×C₃₁	1	2	yes	—
225	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
226	Q₈	—	2	8	no	—
227	C₂×C₄	—	2	4	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	4	no	—
233	C₂×C₄	—	4	2	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	4	4	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	—	4	4	no	—
239	C₂×C₄	—	4	4	no	—
240	C₂×C₄	—	4	2	no	—
241	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	4	yes	—
242	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
243	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	4	yes	—
244	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
245	C₂×C₄	C₃₁×#135	1	4	yes	—
246	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
247	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
248	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
249	C₂×C₄	C₃₁×2₋¹⁺⁴	1	2	yes	—
250	C₈	—	2	2	no	—
251	C₈	—	2	4	no	—
252	C₈	—	2	2	no	—
253	C₂²	C₂²×C₄⋊C₄×C₃₁	1	1	yes	characteristic
254	C₂²	C₃₁×#548	1	2	yes	—
255	C₂²	C₂×C₃₁×#135	1	2	yes	—
256	C₂²	C₃₁×#576	1	1	yes	characteristic
257	C₂²	C₃₁×#578	1	1	yes	characteristic
258	C₄	—	2	4	no	—
259	C₄	—	2	4	no	—
260	C₄	—	2	2	no	—
261	C₄	—	4	4	no	—
262	C₄	—	8	4	no	—
263	C₄	—	8	2	no	—
264	C₄	C₃₁×#427	1	4	yes	—
265	C₄	C₄×#41×C₃₁	1	1	yes	characteristic
266	C₄	C₃₁×#519	1	1	yes	characteristic
267	C₄	C₃₁×#557	1	2	yes	—
268	C₂	C₃₁×#2351	1	1	yes	characteristic
269	C₂	C₂×C₃₁×#427	1	1	yes	characteristic
270	C₂	C₃₁×#2913	1	1	yes	characteristic
271	C₂	C₃₁×#3361	1	2	yes	—
272	C₂	C₃₁×#3364	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂×C₄×Q₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
10. C₈×Q₈×C₃₁	C₄	semidirect
88. #22739	C₃₁	direct (N × H)
141. C₈×Q₈	C₄×C₃₁	semidirect
192. C₃₁	#22739	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×C₈×Q₈×C₃₁	2
C₃₁×#2355	2
C₃₁×#1949	2
C₃₁×#1947	2
C₃₁×#1835	2
C₃₁×#1834	2
C₄×C₃₁×#107	2
C₄×C₃₁×#106	2
#22739	31

Subgroups — C31×#22739 #21479498

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22739 #21479498