Subgroups — C₃₁×#33263 #21490022

Order 7936.

694 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2859	C₂	1	2	yes	maximal
2	C₃₁×#1944	C₂	1	4	yes	maximal
3	C₃₁×#1943	C₂	1	4	yes	maximal
4	C₃₁×#1845	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#1844	C₂	1	1	yes	maximal, characteristic
6	C₃₁×#1834	C₂	1	2	yes	maximal
7	C₄×C₃₁×#97	C₂	1	1	yes	maximal, characteristic
8	C₂×C₃₁×#125	—	2	2	no	—
9	C₂×C₃₁×#107	C₂²	1	4	yes	—
10	C₂×C₃₁×#106	C₂²	1	4	yes	—
11	C₂×C₃₁×#97	C₂²	1	2	yes	—
12	C₂×C₃₁×#97	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
15	C₃₁×#389	—	2	2	no	—
16	C₃₁×#389	—	2	2	no	—
17	C₃₁×#389	C₂²	1	4	yes	—
18	C₃₁×#389	C₂²	1	4	yes	—
19	C₃₁×#384	C₂²	1	2	yes	—
20	C₃₁×#384	C₂²	1	2	yes	—
21	C₃₁×#382	—	2	2	no	—
22	C₄³×C₃₁	C₂²	1	1	yes	characteristic
23	C₃₁×#340	C₂²	1	8	yes	—
24	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
25	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
26	C₃₁×#125	—	4	8	no	—
27	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
28	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
29	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
30	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
31	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
32	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
33	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
34	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
35	C₂×C₄²×C₃₁	D₄	1	2	yes	—
36	C₂×C₄²×C₃₁	D₄	1	2	yes	—
37	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
38	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
39	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
40	C₃₁×#107	—	2	8	no	—
41	C₃₁×#106	—	2	8	no	—
42	C₃₁×#97	—	2	4	no	—
43	C₃₁×#97	—	2	4	no	—
44	C₄×C₈×C₃₁	Q₈	1	4	yes	—
45	C₄×C₈×C₃₁	Q₈	1	4	yes	—
46	C₃₁×#95	—	2	4	no	—
47	C₃₁×#95	—	2	4	no	—
48	C₂²×C₄×C₃₁	—	2	4	no	—
49	C₂²×C₄×C₃₁	—	2	4	no	—
50	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
52	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
54	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
55	C₂×C₈×C₃₁	—	2	2	no	—
56	C₂×C₈×C₃₁	—	2	2	no	—
57	C₂×C₈×C₃₁	C₂×Q₈	1	4	yes	—
58	C₂×C₈×C₃₁	C₂×Q₈	1	4	yes	—
59	C₂×C₈×C₃₁	#41	1	4	yes	—
60	C₂×C₈×C₃₁	#41	1	4	yes	—
61	C₄⋊C₄×C₃₁	—	2	8	no	—
62	C₄⋊C₄×C₃₁	—	2	8	no	—
63	C₄⋊C₄×C₃₁	—	4	8	no	—
64	C₄⋊C₄×C₃₁	—	4	8	no	—
65	C₄⋊C₄×C₃₁	—	4	4	no	—
66	C₄⋊C₄×C₃₁	—	4	4	no	—
67	C₄⋊C₄×C₃₁	—	4	4	no	—
68	C₄⋊C₄×C₃₁	—	4	4	no	—
69	C₄²×C₃₁	—	2	4	no	—
70	C₄²×C₃₁	—	2	4	no	—
71	C₄²×C₃₁	—	2	2	no	—
72	C₄²×C₃₁	—	2	2	no	—
73	C₄²×C₃₁	C₂×Q₈	1	2	yes	—
74	C₄²×C₃₁	C₂×Q₈	1	2	yes	—
75	#33263	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
76	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
77	C₂×C₄×C₃₁	—	2	2	no	—
78	C₂×C₄×C₃₁	—	2	1	no	—
79	C₂×C₄×C₃₁	—	2	1	no	—
80	C₂×C₄×C₃₁	—	2	2	no	—
81	C₂×C₄×C₃₁	—	2	2	no	—
82	C₂×C₄×C₃₁	—	2	2	no	—
83	C₂×C₄×C₃₁	—	2	2	no	—
84	C₂×C₄×C₃₁	—	4	4	no	—
85	C₂×C₄×C₃₁	—	4	4	no	—
86	C₂×C₄×C₃₁	—	4	4	no	—
87	C₂×C₄×C₃₁	—	4	4	no	—
88	C₂×C₄×C₃₁	—	4	4	no	—
89	C₂×C₄×C₃₁	—	4	4	no	—
90	C₂×C₄×C₃₁	C₄⋊Q₈	1	4	yes	—
91	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	4	yes	—
92	C₂×C₄×C₃₁	#137	1	4	yes	—
93	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
94	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
95	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
96	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
97	C₈×C₃₁	—	2	4	no	—
98	C₈×C₃₁	#125	1	8	yes	complemented
99	#2859	C₂×C₃₁	1	2	yes	—
100	#1944	C₂×C₃₁	1	4	yes	—
101	#1943	C₂×C₃₁	1	4	yes	—
102	#1845	C₂×C₃₁	1	1	yes	characteristic
103	#1844	C₂×C₃₁	1	1	yes	characteristic
104	#1834	C₂×C₃₁	1	2	yes	—
105	C₄×#97	C₂×C₃₁	1	1	yes	characteristic
106	C₂²×C₃₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
107	C₂²×C₃₁	#532	1	1	yes	characteristic
108	C₂²×C₃₁	#532	1	1	yes	characteristic
109	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
110	C₂²×C₃₁	C₂×#137	1	2	yes	—
111	C₄×C₃₁	—	4	2	no	—
112	C₄×C₃₁	—	4	1	no	—
113	C₄×C₃₁	—	4	1	no	—
114	C₄×C₃₁	—	8	4	no	—
115	C₄×C₃₁	—	8	4	no	—
116	C₄×C₃₁	#501	1	8	yes	—
117	C₄×C₃₁	C₂×#125	1	2	yes	—
118	C₄×C₃₁	#533	1	2	yes	—
119	C₂×#125	—	2	2	no	—
120	C₂×#107	C₂²×C₃₁	1	4	yes	—
121	C₂×#106	C₂²×C₃₁	1	4	yes	—
122	C₂×#97	C₂²×C₃₁	1	2	yes	—
123	C₂×#97	C₂²×C₃₁	1	2	yes	—
124	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
125	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
126	#389	—	2	2	no	—
127	#389	—	2	2	no	—
128	#389	C₂²×C₃₁	1	4	yes	—
129	#389	C₂²×C₃₁	1	4	yes	—
130	#384	C₂²×C₃₁	1	2	yes	—
131	#384	C₂²×C₃₁	1	2	yes	—
132	#382	—	2	2	no	—
133	C₄³	C₂²×C₃₁	1	1	yes	characteristic
134	#340	C₂²×C₃₁	1	8	yes	—
135	C₂×C₃₁	#2859	1	1	yes	characteristic
136	C₂×C₃₁	C₂×#501	1	2	yes	—
137	C₂×C₃₁	#3182	1	2	yes	—
138	C₂×C₃₁	#3184	1	1	yes	characteristic
139	C₂×C₃₁	#3185	1	1	yes	characteristic
140	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
141	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
142	#125	—	4	8	no	—
143	C₂×C₄⋊C₄	—	2	2	no	—
144	C₂×C₄⋊C₄	—	2	2	no	—
145	C₂×C₄⋊C₄	—	2	4	no	—
146	C₂×C₄⋊C₄	—	2	4	no	—
147	C₂×C₄⋊C₄	—	2	2	no	—
148	C₂×C₄⋊C₄	—	2	2	no	—
149	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
150	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
151	C₂×C₄²	D₄×C₃₁	1	2	yes	—
152	C₂×C₄²	D₄×C₃₁	1	2	yes	—
153	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
154	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
155	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
156	#107	—	2	8	no	—
157	#106	—	2	8	no	—
158	#97	—	2	4	no	—
159	#97	—	2	4	no	—
160	C₄×C₈	Q₈×C₃₁	1	4	yes	—
161	C₄×C₈	Q₈×C₃₁	1	4	yes	—
162	#95	—	2	4	no	—
163	#95	—	2	4	no	—
164	C₃₁	#33263	1	1	yes	characteristic, direct product factor, Sylow
165	C₂²×C₄	—	2	4	no	—
166	C₂²×C₄	—	2	4	no	—
167	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
168	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
169	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
170	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
171	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
172	C₂×C₈	—	2	2	no	—
173	C₂×C₈	—	2	2	no	—
174	C₂×C₈	C₂×Q₈×C₃₁	1	4	yes	—
175	C₂×C₈	C₂×Q₈×C₃₁	1	4	yes	—
176	C₂×C₈	#41×C₃₁	1	4	yes	—
177	C₂×C₈	#41×C₃₁	1	4	yes	—
178	C₄⋊C₄	—	2	8	no	—
179	C₄⋊C₄	—	2	8	no	—
180	C₄⋊C₄	—	4	8	no	—
181	C₄⋊C₄	—	4	8	no	—
182	C₄⋊C₄	—	4	4	no	—
183	C₄⋊C₄	—	4	4	no	—
184	C₄⋊C₄	—	4	4	no	—
185	C₄⋊C₄	—	4	4	no	—
186	C₄²	—	2	4	no	—
187	C₄²	—	2	4	no	—
188	C₄²	—	2	2	no	—
189	C₄²	—	2	2	no	—
190	C₄²	C₂×Q₈×C₃₁	1	2	yes	—
191	C₄²	C₂×Q₈×C₃₁	1	2	yes	—
192	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	4	4	no	—
201	C₂×C₄	—	4	4	no	—
202	C₂×C₄	—	4	4	no	—
203	C₂×C₄	—	4	4	no	—
204	C₂×C₄	—	4	4	no	—
205	C₂×C₄	—	4	4	no	—
206	C₂×C₄	C₃₁×C₄⋊Q₈	1	4	yes	—
207	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	4	yes	—
208	C₂×C₄	C₃₁×#137	1	4	yes	—
209	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
210	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
211	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
212	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
213	C₈	—	2	4	no	—
214	C₈	C₃₁×#125	1	8	yes	complemented
215	C₂²	C₂×C₃₁×C₄⋊Q₈	1	1	yes	characteristic
216	C₂²	C₃₁×#532	1	1	yes	characteristic
217	C₂²	C₃₁×#532	1	1	yes	characteristic
218	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	2	yes	—
219	C₂²	C₂×C₃₁×#137	1	2	yes	—
220	C₄	—	4	2	no	—
221	C₄	—	4	1	no	—
222	C₄	—	4	1	no	—
223	C₄	—	8	4	no	—
224	C₄	—	8	4	no	—
225	C₄	C₃₁×#501	1	8	yes	—
226	C₄	C₂×C₃₁×#125	1	2	yes	—
227	C₄	C₃₁×#533	1	2	yes	—
228	C₂	C₃₁×#2859	1	1	yes	characteristic
229	C₂	C₂×C₃₁×#501	1	2	yes	—
230	C₂	C₃₁×#3182	1	2	yes	—
231	C₂	C₃₁×#3184	1	1	yes	characteristic
232	C₂	C₃₁×#3185	1	1	yes	characteristic

	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 ◁ C₄³×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
75. #33263	C₃₁	direct (N × H)
98. C₈×C₃₁	#125	semidirect
164. C₃₁	#33263	direct (N × H)
214. C₈	C₃₁×#125	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#2859	2
C₃₁×#1944	2
C₃₁×#1943	2
C₃₁×#1845	2
C₃₁×#1844	2
C₃₁×#1834	2
C₄×C₃₁×#97	2
#33263	31

Subgroups — C31×#33263 #21490022

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33263 #21490022