Subgroups — C₃₁×#33260 #21490019

Order 7936.

606 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2647	C₂	1	2	yes	maximal
2	C₃₁×#1944	C₂	1	2	yes	maximal
3	C₃₁×#1943	C₂	1	2	yes	maximal
4	C₃₁×#1941	C₂	1	2	yes	maximal
5	C₃₁×#1941	C₂	1	2	yes	maximal
6	C₃₁×#1846	C₂	1	1	yes	maximal, characteristic
7	C₃₁×#1844	C₂	1	1	yes	maximal, characteristic
8	C₃₁×#1836	C₂	1	2	yes	maximal
9	C₃₁×#1726	C₂	1	1	yes	maximal, characteristic
10	C₂×C₃₁×#107	C₂²	1	2	yes	—
11	C₂×C₃₁×#106	C₂²	1	2	yes	—
12	C₂×C₃₁×#106	C₂²	1	2	yes	—
13	C₂×C₃₁×#106	C₂²	1	2	yes	—
14	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
20	C₃₁×#400	—	2	4	no	—
21	C₃₁×#398	—	2	4	no	—
22	C₃₁×#389	C₂²	1	4	yes	—
23	C₃₁×#389	C₂²	1	4	yes	—
24	C₃₁×#384	C₂²	1	2	yes	—
25	C₃₁×#382	C₂²	1	2	yes	—
26	C₃₁×#376	C₂²	1	1	yes	characteristic
27	C₃₁×#340	C₂²	1	4	yes	—
28	C₃₁×#340	C₂²	1	4	yes	—
29	C₂²×C₈×C₃₁	C₂³	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	1	yes	characteristic
33	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
34	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
35	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
36	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
37	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
38	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
39	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
40	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
41	C₃₁×#107	—	2	4	no	—
42	C₃₁×#106	—	2	4	no	—
43	C₃₁×#106	—	2	4	no	—
44	C₃₁×#106	—	2	4	no	—
45	C₃₁×#97	—	2	2	no	—
46	C₃₁×#97	—	2	2	no	—
47	C₃₁×#97	Q₈	1	4	yes	—
48	C₄×C₈×C₃₁	—	2	2	no	—
49	C₄×C₈×C₃₁	—	2	2	no	—
50	C₄×C₈×C₃₁	Q₈	1	4	yes	—
51	C₃₁×#95	—	2	4	no	—
52	C₃₁×#95	—	2	4	no	—
53	C₃₁×#95	—	2	4	no	—
54	C₃₁×#95	D₄	1	4	yes	—
55	C₂²×C₄×C₃₁	—	2	4	no	—
56	C₂²×C₄×C₃₁	—	2	4	no	—
57	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
58	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
59	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
60	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
61	C₂×C₈×C₃₁	—	2	1	no	—
62	C₂×C₈×C₃₁	—	2	1	no	—
63	C₂×C₈×C₃₁	—	2	1	no	—
64	C₂×C₈×C₃₁	—	2	1	no	—
65	C₂×C₈×C₃₁	C₂×Q₈	1	2	yes	—
66	C₂×C₈×C₃₁	C₂×Q₈	1	2	yes	—
67	C₂×C₈×C₃₁	C₂×Q₈	1	2	yes	—
68	C₂×C₈×C₃₁	C₂×Q₈	1	2	yes	—
69	C₂×C₈×C₃₁	#41	1	2	yes	—
70	C₂×C₈×C₃₁	#41	1	2	yes	—
71	C₂×C₈×C₃₁	#41	1	2	yes	—
72	C₂×C₈×C₃₁	#41	1	2	yes	—
73	C₄⋊C₄×C₃₁	—	2	4	no	—
74	C₄⋊C₄×C₃₁	—	2	4	no	—
75	C₄⋊C₄×C₃₁	—	2	4	no	—
76	C₄⋊C₄×C₃₁	—	2	4	no	—
77	C₄⋊C₄×C₃₁	—	8	4	no	—
78	C₄⋊C₄×C₃₁	—	8	4	no	—
79	C₄⋊C₄×C₃₁	—	8	4	no	—
80	C₄²×C₃₁	—	2	1	no	—
81	C₄²×C₃₁	—	2	1	no	—
82	C₄²×C₃₁	—	2	1	no	—
83	C₄²×C₃₁	—	2	1	no	—
84	C₄²×C₃₁	C₂×Q₈	1	2	yes	—
85	C₄²×C₃₁	C₂×Q₈	1	2	yes	—
86	#33260	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
87	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
88	C₂×C₄×C₃₁	—	2	4	no	—
89	C₂×C₄×C₃₁	—	2	2	no	—
90	C₂×C₄×C₃₁	—	2	2	no	—
91	C₂×C₄×C₃₁	—	2	4	no	—
92	C₂×C₄×C₃₁	—	2	4	no	—
93	C₂×C₄×C₃₁	—	4	4	no	—
94	C₂×C₄×C₃₁	—	4	4	no	—
95	C₂×C₄×C₃₁	—	4	4	no	—
96	C₂×C₄×C₃₁	—	4	4	no	—
97	C₂×C₄×C₃₁	—	4	4	no	—
98	C₂×C₄×C₃₁	—	4	4	no	—
99	C₂×C₄×C₃₁	C₄⋊Q₈	1	4	yes	—
100	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
101	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
102	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
103	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
104	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
105	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
106	C₈×C₃₁	—	2	2	no	—
107	C₈×C₃₁	—	2	2	no	—
108	C₈×C₃₁	#125	1	4	yes	—
109	C₈×C₃₁	#125	1	4	yes	—
110	#2647	C₂×C₃₁	1	2	yes	—
111	#1944	C₂×C₃₁	1	2	yes	—
112	#1943	C₂×C₃₁	1	2	yes	—
113	#1941	C₂×C₃₁	1	2	yes	—
114	#1941	C₂×C₃₁	1	2	yes	—
115	#1846	C₂×C₃₁	1	1	yes	characteristic
116	#1844	C₂×C₃₁	1	1	yes	characteristic
117	#1836	C₂×C₃₁	1	2	yes	—
118	#1726	C₂×C₃₁	1	1	yes	characteristic
119	C₂²×C₃₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
120	C₂²×C₃₁	#532	1	1	yes	characteristic
121	C₂²×C₃₁	#532	1	1	yes	characteristic
122	C₂²×C₃₁	#577	1	4	yes	—
123	C₄×C₃₁	—	2	4	no	—
124	C₄×C₃₁	—	4	2	no	—
125	C₄×C₃₁	—	4	2	no	—
126	C₄×C₃₁	—	8	4	no	—
127	C₄×C₃₁	—	8	4	no	—
128	C₄×C₃₁	C₂×#125	1	1	yes	characteristic
129	C₄×C₃₁	C₂×#125	1	1	yes	characteristic
130	C₄×C₃₁	#533	1	1	yes	characteristic
131	C₄×C₃₁	#533	1	1	yes	characteristic
132	C₂×#107	C₂²×C₃₁	1	2	yes	—
133	C₂×#106	C₂²×C₃₁	1	2	yes	—
134	C₂×#106	C₂²×C₃₁	1	2	yes	—
135	C₂×#106	C₂²×C₃₁	1	2	yes	—
136	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
137	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
138	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
139	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
140	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
141	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
142	#400	—	2	4	no	—
143	#398	—	2	4	no	—
144	#389	C₂²×C₃₁	1	4	yes	—
145	#389	C₂²×C₃₁	1	4	yes	—
146	#384	C₂²×C₃₁	1	2	yes	—
147	#382	C₂²×C₃₁	1	2	yes	—
148	#376	C₂²×C₃₁	1	1	yes	characteristic
149	#340	C₂²×C₃₁	1	4	yes	—
150	#340	C₂²×C₃₁	1	4	yes	—
151	C₂×C₃₁	#2859	1	1	yes	characteristic
152	C₂×C₃₁	#3163	1	2	yes	—
153	C₂×C₃₁	#3173	1	2	yes	—
154	C₂×C₃₁	#3174	1	2	yes	—
155	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
156	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
157	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
158	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
159	C₂×C₄⋊C₄	—	2	4	no	—
160	C₂×C₄⋊C₄	—	2	4	no	—
161	C₂×C₄⋊C₄	—	2	4	no	—
162	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
163	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
164	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
165	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
166	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
167	#107	—	2	4	no	—
168	#106	—	2	4	no	—
169	#106	—	2	4	no	—
170	#106	—	2	4	no	—
171	#97	—	2	2	no	—
172	#97	—	2	2	no	—
173	#97	Q₈×C₃₁	1	4	yes	—
174	C₄×C₈	—	2	2	no	—
175	C₄×C₈	—	2	2	no	—
176	C₄×C₈	Q₈×C₃₁	1	4	yes	—
177	#95	—	2	4	no	—
178	#95	—	2	4	no	—
179	#95	—	2	4	no	—
180	#95	D₄×C₃₁	1	4	yes	—
181	C₃₁	#33260	1	1	yes	characteristic, direct product factor, Sylow
182	C₂²×C₄	—	2	4	no	—
183	C₂²×C₄	—	2	4	no	—
184	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
185	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
186	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
187	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
188	C₂×C₈	—	2	1	no	—
189	C₂×C₈	—	2	1	no	—
190	C₂×C₈	—	2	1	no	—
191	C₂×C₈	—	2	1	no	—
192	C₂×C₈	C₂×Q₈×C₃₁	1	2	yes	—
193	C₂×C₈	C₂×Q₈×C₃₁	1	2	yes	—
194	C₂×C₈	C₂×Q₈×C₃₁	1	2	yes	—
195	C₂×C₈	C₂×Q₈×C₃₁	1	2	yes	—
196	C₂×C₈	#41×C₃₁	1	2	yes	—
197	C₂×C₈	#41×C₃₁	1	2	yes	—
198	C₂×C₈	#41×C₃₁	1	2	yes	—
199	C₂×C₈	#41×C₃₁	1	2	yes	—
200	C₄⋊C₄	—	2	4	no	—
201	C₄⋊C₄	—	2	4	no	—
202	C₄⋊C₄	—	2	4	no	—
203	C₄⋊C₄	—	2	4	no	—
204	C₄⋊C₄	—	8	4	no	—
205	C₄⋊C₄	—	8	4	no	—
206	C₄⋊C₄	—	8	4	no	—
207	C₄²	—	2	1	no	—
208	C₄²	—	2	1	no	—
209	C₄²	—	2	1	no	—
210	C₄²	—	2	1	no	—
211	C₄²	C₂×Q₈×C₃₁	1	2	yes	—
212	C₄²	C₂×Q₈×C₃₁	1	2	yes	—
213	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
214	C₂×C₄	—	2	4	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	4	no	—
218	C₂×C₄	—	2	4	no	—
219	C₂×C₄	—	4	4	no	—
220	C₂×C₄	—	4	4	no	—
221	C₂×C₄	—	4	4	no	—
222	C₂×C₄	—	4	4	no	—
223	C₂×C₄	—	4	4	no	—
224	C₂×C₄	—	4	4	no	—
225	C₂×C₄	C₃₁×C₄⋊Q₈	1	4	yes	—
226	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
227	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
228	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
229	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
230	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
231	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
232	C₈	—	2	2	no	—
233	C₈	—	2	2	no	—
234	C₈	C₃₁×#125	1	4	yes	—
235	C₈	C₃₁×#125	1	4	yes	—
236	C₂²	C₂×C₃₁×C₄⋊Q₈	1	1	yes	characteristic
237	C₂²	C₃₁×#532	1	1	yes	characteristic
238	C₂²	C₃₁×#532	1	1	yes	characteristic
239	C₂²	C₃₁×#577	1	4	yes	—
240	C₄	—	2	4	no	—
241	C₄	—	4	2	no	—
242	C₄	—	4	2	no	—
243	C₄	—	8	4	no	—
244	C₄	—	8	4	no	—
245	C₄	C₂×C₃₁×#125	1	1	yes	characteristic
246	C₄	C₂×C₃₁×#125	1	1	yes	characteristic
247	C₄	C₃₁×#533	1	1	yes	characteristic
248	C₄	C₃₁×#533	1	1	yes	characteristic
249	C₂	C₃₁×#2859	1	1	yes	characteristic
250	C₂	C₃₁×#3163	1	2	yes	—
251	C₂	C₃₁×#3173	1	2	yes	—
252	C₂	C₃₁×#3174	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₃₁×#376 ◁ G

Complements

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

N	H ≅ G/N	type
86. #33260	C₃₁	direct (N × H)
181. C₃₁	#33260	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃₁×#2647	2
C₃₁×#1944	2
C₃₁×#1943	2
C₃₁×#1941	2
C₃₁×#1846	2
C₃₁×#1844	2
C₃₁×#1836	2
C₃₁×#1726	2
#33260	31

Subgroups — C31×#33260 #21490019

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33260 #21490019