Subgroups — C₃₁×#33905 #21490664

Order 7936.

484 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3338	C₂	1	2	yes	maximal
2	C₃₁×#2252	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2236	C₂	1	4	yes	maximal, complemented
4	C₃₁×#2227	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2216	C₂	1	2	yes	maximal, complemented
6	C₄×C₃₁×Q₃₂	C₂	1	1	yes	maximal, characteristic, complemented
7	D₄×C₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic
8	C₂×C₃₁×#366	C₂	1	2	yes	maximal, complemented
9	C₃₁×#554	—	2	2	no	—
10	C₂×C₃₁×Q₃₂	C₂²	1	1	yes	characteristic
11	C₂²×C₁₆×C₃₁	C₂²	1	2	yes	—
12	C₃₁×#500	C₂²	1	2	yes	—
13	C₃₁×#484	—	2	4	no	—
14	C₃₁×#477	—	2	2	no	—
15	C₃₁×#467	C₂²	1	4	yes	—
16	C₄×Q₁₆×C₃₁	C₂²	1	2	yes	—
17	C₈×D₄×C₃₁	C₂²	1	1	yes	characteristic
18	C₂×C₃₁×#107	C₂²	1	4	yes	—
19	C₃₁×#366	C₂²	1	1	yes	characteristic
20	C₃₁×#366	C₂²	1	2	yes	—
21	C₃₁×#366	C₂²	1	4	yes	—
22	C₃₁×#366	C₂²	1	2	yes	—
23	C₃₁×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
24	C₃₁×#358	C₂²	1	2	yes	—
25	C₃₁×#358	C₂²	1	4	yes	—
26	C₃₁×C₂²⋊C₁₆	C₂²	1	2	yes	—
27	C₄×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
28	C₂×Q₁₆×C₃₁	C₂³	1	2	yes	—
29	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
30	C₃₁×C₄⋊Q₈	—	2	2	no	—
31	C₃₁×#125	—	4	2	no	—
32	C₃₁×C₂²⋊Q₈	—	2	4	no	—
33	C₃₁×C₂²⋊Q₈	—	4	4	no	—
34	C₄×Q₈×C₃₁	—	2	2	no	—
35	C₄×D₄×C₃₁	—	4	2	no	—
36	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
37	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
38	C₃₁×Q₃₂	—	2	2	no	—
39	C₂×C₁₆×C₃₁	—	2	4	no	—
40	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
41	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
42	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
43	C₃₁×#107	—	2	4	no	—
44	C₃₁×#107	—	2	2	no	—
45	C₃₁×#107	C₂³	1	2	yes	—
46	C₃₁×#107	C₂³	1	4	yes	—
47	C₃₁×C₄⋊C₈	D₄	1	1	yes	characteristic
48	C₃₁×#103	—	2	2	no	—
49	C₃₁×#103	—	2	4	no	—
50	C₃₁×C₂²⋊C₈	D₄	1	2	yes	—
51	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
52	C₂×Q₈×C₃₁	—	2	2	no	—
53	C₂×Q₈×C₃₁	—	4	2	no	—
54	C₂×D₄×C₃₁	D₈	1	1	yes	characteristic
55	C₂²×C₄×C₃₁	—	4	4	no	—
56	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
57	Q₁₆×C₃₁	—	2	2	no	—
58	Q₁₆×C₃₁	—	2	2	no	—
59	C₂×C₈×C₃₁	—	2	2	no	—
60	C₂×C₈×C₃₁	—	2	2	no	—
61	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
62	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
63	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
64	C₄⋊C₄×C₃₁	—	2	2	no	—
65	C₄⋊C₄×C₃₁	—	2	4	no	—
66	C₄⋊C₄×C₃₁	—	4	4	no	—
67	C₄⋊C₄×C₃₁	—	4	2	no	—
68	C₄⋊C₄×C₃₁	—	4	2	no	—
69	C₄⋊C₄×C₃₁	—	4	4	no	—
70	C₄⋊C₄×C₃₁	D₈	1	1	yes	characteristic
71	C₂²⋊C₄×C₃₁	—	4	4	no	—
72	C₂²⋊C₄×C₃₁	D₈	1	2	yes	—
73	C₄²×C₃₁	—	4	2	no	—
74	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
75	C₁₆×C₃₁	—	2	2	no	—
76	C₁₆×C₃₁	—	2	1	no	—
77	C₁₆×C₃₁	#41	1	2	yes	—
78	#33905	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
79	C₂³×C₃₁	C₂×D₈	1	2	yes	—
80	Q₈×C₃₁	—	4	2	no	—
81	Q₈×C₃₁	—	4	2	no	—
82	Q₈×C₃₁	—	16	2	no	—
83	D₄×C₃₁	Q₃₂	1	4	yes	complemented
84	C₂×C₄×C₃₁	—	2	2	no	—
85	C₂×C₄×C₃₁	—	2	2	no	—
86	C₂×C₄×C₃₁	—	4	2	no	—
87	C₂×C₄×C₃₁	—	4	2	no	—
88	C₂×C₄×C₃₁	—	4	4	no	—
89	C₂×C₄×C₃₁	—	16	4	no	—
90	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
91	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
92	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
93	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
94	C₈×C₃₁	—	2	2	no	—
95	C₈×C₃₁	—	2	1	no	—
96	C₈×C₃₁	C₂×#41	1	1	yes	characteristic
97	C₈×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
98	#3338	C₂×C₃₁	1	2	yes	—
99	#2252	C₂×C₃₁	1	1	yes	characteristic, complemented
100	#2236	C₂×C₃₁	1	4	yes	complemented
101	#2227	C₂×C₃₁	1	2	yes	complemented
102	#2216	C₂×C₃₁	1	2	yes	complemented
103	C₄×Q₃₂	C₂×C₃₁	1	1	yes	characteristic, complemented
104	D₄×C₁₆	C₂×C₃₁	1	1	yes	characteristic
105	C₂×#366	C₂×C₃₁	1	2	yes	complemented
106	C₂²×C₃₁	—	2	2	no	—
107	C₂²×C₃₁	—	2	2	no	—
108	C₂²×C₃₁	C₂×Q₃₂	1	4	yes	—
109	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
110	C₄×C₃₁	—	2	2	no	—
111	C₄×C₃₁	—	2	1	no	—
112	C₄×C₃₁	—	8	2	no	—
113	C₄×C₃₁	—	8	2	no	—
114	C₄×C₃₁	—	8	4	no	—
115	C₄×C₃₁	C₂×Q₃₂	1	2	yes	—
116	C₄×C₃₁	#547	1	1	yes	characteristic
117	C₄×C₃₁	#578	1	1	yes	characteristic
118	#554	—	2	2	no	—
119	C₂×Q₃₂	C₂²×C₃₁	1	1	yes	characteristic
120	C₂²×C₁₆	C₂²×C₃₁	1	2	yes	—
121	#500	C₂²×C₃₁	1	2	yes	—
122	#484	—	2	4	no	—
123	#477	—	2	2	no	—
124	#467	C₂²×C₃₁	1	4	yes	—
125	C₄×Q₁₆	C₂²×C₃₁	1	2	yes	—
126	C₈×D₄	C₂²×C₃₁	1	1	yes	characteristic
127	C₂×#107	C₂²×C₃₁	1	4	yes	—
128	#366	C₂²×C₃₁	1	1	yes	characteristic
129	#366	C₂²×C₃₁	1	2	yes	—
130	#366	C₂²×C₃₁	1	4	yes	—
131	#366	C₂²×C₃₁	1	2	yes	—
132	C₄⋊C₁₆	C₂²×C₃₁	1	1	yes	characteristic
133	#358	C₂²×C₃₁	1	2	yes	—
134	#358	C₂²×C₃₁	1	4	yes	—
135	C₂²⋊C₁₆	C₂²×C₃₁	1	2	yes	—
136	C₄×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
137	C₂×C₃₁	—	2	4	no	—
138	C₂×C₃₁	#3334	1	1	yes	characteristic
139	C₂×C₃₁	C₂²×Q₃₂	1	1	yes	characteristic
140	C₂×C₃₁	#3415	1	1	yes	characteristic
141	C₂×Q₁₆	C₂³×C₃₁	1	2	yes	—
142	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
143	C₄⋊Q₈	—	2	2	no	—
144	#125	—	4	2	no	—
145	C₂²⋊Q₈	—	2	4	no	—
146	C₂²⋊Q₈	—	4	4	no	—
147	C₄×Q₈	—	2	2	no	—
148	C₄×D₄	—	4	2	no	—
149	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
150	C₂×C₄⋊C₄	—	2	4	no	—
151	Q₃₂	—	2	2	no	—
152	C₂×C₁₆	—	2	4	no	—
153	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
154	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
155	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
156	#107	—	2	4	no	—
157	#107	—	2	2	no	—
158	#107	C₂³×C₃₁	1	2	yes	—
159	#107	C₂³×C₃₁	1	4	yes	—
160	C₄⋊C₈	D₄×C₃₁	1	1	yes	characteristic
161	#103	—	2	2	no	—
162	#103	—	2	4	no	—
163	C₂²⋊C₈	D₄×C₃₁	1	2	yes	—
164	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
165	C₃₁	#33905	1	1	yes	characteristic, direct product factor, Sylow
166	C₂×Q₈	—	2	2	no	—
167	C₂×Q₈	—	4	2	no	—
168	C₂×D₄	D₈×C₃₁	1	1	yes	characteristic
169	C₂²×C₄	—	4	4	no	—
170	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
171	Q₁₆	—	2	2	no	—
172	Q₁₆	—	2	2	no	—
173	C₂×C₈	—	2	2	no	—
174	C₂×C₈	—	2	2	no	—
175	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
176	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
177	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
178	C₄⋊C₄	—	2	2	no	—
179	C₄⋊C₄	—	2	4	no	—
180	C₄⋊C₄	—	4	4	no	—
181	C₄⋊C₄	—	4	2	no	—
182	C₄⋊C₄	—	4	2	no	—
183	C₄⋊C₄	—	4	4	no	—
184	C₄⋊C₄	D₈×C₃₁	1	1	yes	characteristic
185	C₂²⋊C₄	—	4	4	no	—
186	C₂²⋊C₄	D₈×C₃₁	1	2	yes	—
187	C₄²	—	4	2	no	—
188	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
189	C₁₆	—	2	2	no	—
190	C₁₆	—	2	1	no	—
191	C₁₆	#41×C₃₁	1	2	yes	—
192	C₂³	C₂×D₈×C₃₁	1	2	yes	—
193	Q₈	—	4	2	no	—
194	Q₈	—	4	2	no	—
195	Q₈	—	16	2	no	—
196	D₄	C₃₁×Q₃₂	1	4	yes	complemented
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	4	2	no	—
200	C₂×C₄	—	4	2	no	—
201	C₂×C₄	—	4	4	no	—
202	C₂×C₄	—	16	4	no	—
203	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
204	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
205	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
206	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
207	C₈	—	2	2	no	—
208	C₈	—	2	1	no	—
209	C₈	C₂×#41×C₃₁	1	1	yes	characteristic
210	C₈	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
211	C₂²	—	2	2	no	—
212	C₂²	—	2	2	no	—
213	C₂²	C₂×C₃₁×Q₃₂	1	4	yes	—
214	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
215	C₄	—	2	2	no	—
216	C₄	—	2	1	no	—
217	C₄	—	8	2	no	—
218	C₄	—	8	2	no	—
219	C₄	—	8	4	no	—
220	C₄	C₂×C₃₁×Q₃₂	1	2	yes	—
221	C₄	C₃₁×#547	1	1	yes	characteristic
222	C₄	C₃₁×#578	1	1	yes	characteristic
223	C₂	—	2	4	no	—
224	C₂	C₃₁×#3334	1	1	yes	characteristic
225	C₂	C₂²×C₃₁×Q₃₂	1	1	yes	characteristic
226	C₂	C₃₁×#3415	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
prime radical	C₂×D₄×C₃₁	D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₄×D₄×C₃₁ ◁ C₈×D₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2252	C₂	semidirect
3. C₃₁×#2236	C₂	semidirect
4. C₃₁×#2227	C₂	semidirect
5. C₃₁×#2216	C₂	semidirect
6. C₄×C₃₁×Q₃₂	C₂	semidirect
8. C₂×C₃₁×#366	C₂	semidirect
78. #33905	C₃₁	direct (N × H)
83. D₄×C₃₁	Q₃₂	semidirect
99. #2252	C₂×C₃₁	semidirect
100. #2236	C₂×C₃₁	semidirect
101. #2227	C₂×C₃₁	semidirect
102. #2216	C₂×C₃₁	semidirect
103. C₄×Q₃₂	C₂×C₃₁	semidirect
105. C₂×#366	C₂×C₃₁	semidirect
165. C₃₁	#33905	direct (N × H)
196. D₄	C₃₁×Q₃₂	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#3338	2
C₃₁×#2252	2
C₃₁×#2236	2
C₃₁×#2227	2
C₃₁×#2216	2
C₄×C₃₁×Q₃₂	2
D₄×C₁₆×C₃₁	2
C₂×C₃₁×#366	2
#33905	31

Subgroups — C31×#33905 #21490664

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33905 #21490664