Subgroups — C₃₁×#33741 #21490500

Order 7936.

484 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3177	C₂	1	2	yes	maximal
2	C₃₁×#2253	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2247	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2245	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2219	C₂	1	2	yes	maximal
6	C₃₁×#2216	C₂	1	2	yes	maximal
7	C₃₁×#2176	C₂	1	2	yes	maximal, complemented
8	C₄×C₃₁×Q₃₂	C₂	1	2	yes	maximal, complemented
9	C₃₁×#2108	C₂	1	1	yes	maximal, characteristic
10	C₃₁×#541	—	2	2	no	—
11	C₂×C₃₁×Q₃₂	C₂²	1	2	yes	—
12	C₂×C₃₁×Q₃₂	C₂²	1	2	yes	—
13	C₂×C₃₁×M₃₂	C₂²	1	1	yes	characteristic
14	C₂²×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
15	C₃₁×#499	C₂²	1	2	yes	—
16	C₃₁×#494	C₂²	1	2	yes	—
17	C₃₁×#491	—	2	2	no	—
18	C₃₁×#470	—	2	2	no	—
19	C₃₁×#467	C₂²	1	4	yes	—
20	C₃₁×#441	—	2	2	no	—
21	C₄×Q₁₆×C₃₁	C₂²	1	4	yes	—
22	C₃₁×#405	C₂²	1	1	yes	characteristic
23	C₃₁×#367	C₂²	1	2	yes	—
24	C₃₁×#366	C₂²	1	2	yes	—
25	C₃₁×#358	C₂²	1	4	yes	—
26	C₃₁×#358	C₂²	1	4	yes	—
27	C₃₁×#346	C₂²	1	1	yes	characteristic
28	C₃₁×#346	C₂²	1	1	yes	characteristic
29	C₄×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
30	C₄×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
31	C₂×Q₁₆×C₃₁	—	2	2	no	—
32	C₂×Q₁₆×C₃₁	C₂³	1	4	yes	—
33	C₂×M₁₆×C₃₁	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
35	C₃₁×C₄⋊Q₈	—	2	2	no	—
36	C₃₁×#126	—	4	4	no	—
37	C₃₁×#125	—	2	2	no	—
38	C₃₁×#125	—	4	4	no	—
39	C₃₁×C₂²⋊Q₈	—	2	4	no	—
40	C₄×Q₈×C₃₁	—	2	4	no	—
41	C₃₁×#117	D₄	1	1	yes	characteristic
42	C₃₁×Q₃₂	—	2	4	no	—
43	C₃₁×Q₃₂	—	2	4	no	—
44	C₃₁×M₃₂	—	2	2	no	—
45	C₂×C₁₆×C₃₁	—	2	2	no	—
46	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
47	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
48	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
49	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
50	C₃₁×#107	C₂³	1	4	yes	—
51	C₃₁×#106	—	2	2	no	—
52	C₃₁×#103	—	2	4	no	—
53	C₃₁×#103	—	2	4	no	—
54	C₃₁×#97	D₄	1	1	yes	characteristic
55	C₃₁×#97	D₄	1	1	yes	characteristic
56	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
57	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
58	C₂×Q₈×C₃₁	—	2	4	no	—
59	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
60	Q₁₆×C₃₁	—	2	4	no	—
61	Q₁₆×C₃₁	—	2	4	no	—
62	Q₁₆×C₃₁	—	4	4	no	—
63	M₁₆×C₃₁	—	2	2	no	—
64	C₂×C₈×C₃₁	—	2	1	no	—
65	C₂×C₈×C₃₁	—	2	1	no	—
66	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
67	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
68	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
69	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
70	C₄⋊C₄×C₃₁	—	2	4	no	—
71	C₄⋊C₄×C₃₁	—	4	4	no	—
72	C₄⋊C₄×C₃₁	—	4	4	no	—
73	C₄⋊C₄×C₃₁	—	4	4	no	—
74	C₄⋊C₄×C₃₁	—	4	4	no	—
75	C₄⋊C₄×C₃₁	D₈	1	2	yes	—
76	C₂²⋊C₄×C₃₁	—	4	4	no	—
77	C₂²⋊C₄×C₃₁	D₈	1	2	yes	—
78	C₄²×C₃₁	—	4	4	no	—
79	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
80	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
81	C₁₆×C₃₁	—	2	1	no	—
82	C₁₆×C₃₁	—	2	1	no	—
83	C₁₆×C₃₁	#41	1	2	yes	—
84	C₁₆×C₃₁	#41	1	2	yes	—
85	#33741	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
86	C₂³×C₃₁	C₂×D₈	1	1	yes	characteristic
87	Q₈×C₃₁	—	4	4	no	—
88	Q₈×C₃₁	—	4	4	no	—
89	C₂×C₄×C₃₁	—	2	1	no	—
90	C₂×C₄×C₃₁	—	2	1	no	—
91	C₂×C₄×C₃₁	—	4	4	no	—
92	C₂×C₄×C₃₁	—	4	4	no	—
93	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
94	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
95	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
96	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
97	C₈×C₃₁	—	2	1	no	—
98	C₈×C₃₁	—	2	1	no	—
99	C₈×C₃₁	#135	1	2	yes	—
100	C₈×C₃₁	C₂×#41	1	1	yes	characteristic
101	C₈×C₃₁	C₂×#41	1	1	yes	characteristic
102	#3177	C₂×C₃₁	1	2	yes	—
103	#2253	C₂×C₃₁	1	1	yes	characteristic, complemented
104	#2247	C₂×C₃₁	1	1	yes	characteristic, complemented
105	#2245	C₂×C₃₁	1	2	yes	complemented
106	#2219	C₂×C₃₁	1	2	yes	—
107	#2216	C₂×C₃₁	1	2	yes	—
108	#2176	C₂×C₃₁	1	2	yes	complemented
109	C₄×Q₃₂	C₂×C₃₁	1	2	yes	complemented
110	#2108	C₂×C₃₁	1	1	yes	characteristic
111	C₂²×C₃₁	—	2	2	no	—
112	C₂²×C₃₁	—	2	1	no	—
113	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
114	C₄×C₃₁	—	2	2	no	—
115	C₄×C₃₁	—	2	2	no	—
116	C₄×C₃₁	—	2	1	no	—
117	C₄×C₃₁	—	8	4	no	—
118	C₄×C₃₁	—	8	4	no	—
119	C₄×C₃₁	#532	1	1	yes	characteristic
120	C₄×C₃₁	C₂×#135	1	1	yes	characteristic
121	#541	—	2	2	no	—
122	C₂×Q₃₂	C₂²×C₃₁	1	2	yes	—
123	C₂×Q₃₂	C₂²×C₃₁	1	2	yes	—
124	C₂×M₃₂	C₂²×C₃₁	1	1	yes	characteristic
125	C₂²×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
126	#499	C₂²×C₃₁	1	2	yes	—
127	#494	C₂²×C₃₁	1	2	yes	—
128	#491	—	2	2	no	—
129	#470	—	2	2	no	—
130	#467	C₂²×C₃₁	1	4	yes	—
131	#441	—	2	2	no	—
132	C₄×Q₁₆	C₂²×C₃₁	1	4	yes	—
133	#405	C₂²×C₃₁	1	1	yes	characteristic
134	#367	C₂²×C₃₁	1	2	yes	—
135	#366	C₂²×C₃₁	1	2	yes	—
136	#358	C₂²×C₃₁	1	4	yes	—
137	#358	C₂²×C₃₁	1	4	yes	—
138	#346	C₂²×C₃₁	1	1	yes	characteristic
139	#346	C₂²×C₃₁	1	1	yes	characteristic
140	C₄×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
141	C₄×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
142	C₂×C₃₁	—	4	1	no	—
143	C₂×C₃₁	#3169	1	1	yes	characteristic
144	C₂×C₃₁	#3417	1	2	yes	—
145	C₂×Q₁₆	—	2	2	no	—
146	C₂×Q₁₆	C₂³×C₃₁	1	4	yes	—
147	C₂×M₁₆	D₄×C₃₁	1	1	yes	characteristic
148	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
149	C₄⋊Q₈	—	2	2	no	—
150	#126	—	4	4	no	—
151	#125	—	2	2	no	—
152	#125	—	4	4	no	—
153	C₂²⋊Q₈	—	2	4	no	—
154	C₄×Q₈	—	2	4	no	—
155	#117	D₄×C₃₁	1	1	yes	characteristic
156	Q₃₂	—	2	4	no	—
157	Q₃₂	—	2	4	no	—
158	M₃₂	—	2	2	no	—
159	C₂×C₁₆	—	2	2	no	—
160	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
161	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
162	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
163	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
164	#107	C₂³×C₃₁	1	4	yes	—
165	#106	—	2	2	no	—
166	#103	—	2	4	no	—
167	#103	—	2	4	no	—
168	#97	D₄×C₃₁	1	1	yes	characteristic
169	#97	D₄×C₃₁	1	1	yes	characteristic
170	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
171	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
172	C₃₁	#33741	1	1	yes	characteristic, direct product factor, Sylow
173	C₂×Q₈	—	2	4	no	—
174	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
175	Q₁₆	—	2	4	no	—
176	Q₁₆	—	2	4	no	—
177	Q₁₆	—	4	4	no	—
178	M₁₆	—	2	2	no	—
179	C₂×C₈	—	2	1	no	—
180	C₂×C₈	—	2	1	no	—
181	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
182	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
183	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
184	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
185	C₄⋊C₄	—	2	4	no	—
186	C₄⋊C₄	—	4	4	no	—
187	C₄⋊C₄	—	4	4	no	—
188	C₄⋊C₄	—	4	4	no	—
189	C₄⋊C₄	—	4	4	no	—
190	C₄⋊C₄	D₈×C₃₁	1	2	yes	—
191	C₂²⋊C₄	—	4	4	no	—
192	C₂²⋊C₄	D₈×C₃₁	1	2	yes	—
193	C₄²	—	4	4	no	—
194	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
195	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
196	C₁₆	—	2	1	no	—
197	C₁₆	—	2	1	no	—
198	C₁₆	#41×C₃₁	1	2	yes	—
199	C₁₆	#41×C₃₁	1	2	yes	—
200	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
201	Q₈	—	4	4	no	—
202	Q₈	—	4	4	no	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	4	4	no	—
206	C₂×C₄	—	4	4	no	—
207	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
208	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
209	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
210	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
211	C₈	—	2	1	no	—
212	C₈	—	2	1	no	—
213	C₈	C₃₁×#135	1	2	yes	—
214	C₈	C₂×#41×C₃₁	1	1	yes	characteristic
215	C₈	C₂×#41×C₃₁	1	1	yes	characteristic
216	C₂²	—	2	2	no	—
217	C₂²	—	2	1	no	—
218	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
219	C₄	—	2	2	no	—
220	C₄	—	2	2	no	—
221	C₄	—	2	1	no	—
222	C₄	—	8	4	no	—
223	C₄	—	8	4	no	—
224	C₄	C₃₁×#532	1	1	yes	characteristic
225	C₄	C₂×C₃₁×#135	1	1	yes	characteristic
226	C₂	—	4	1	no	—
227	C₂	C₃₁×#3169	1	1	yes	characteristic
228	C₂	C₃₁×#3417	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
prime radical	C₂³×C₃₁	C₂×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₃₁×#117 ◁ C₃₁×#405 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2253	C₂	semidirect
3. C₃₁×#2247	C₂	semidirect
4. C₃₁×#2245	C₂	semidirect
7. C₃₁×#2176	C₂	semidirect
8. C₄×C₃₁×Q₃₂	C₂	semidirect
85. #33741	C₃₁	direct (N × H)
103. #2253	C₂×C₃₁	semidirect
104. #2247	C₂×C₃₁	semidirect
105. #2245	C₂×C₃₁	semidirect
108. #2176	C₂×C₃₁	semidirect
109. C₄×Q₃₂	C₂×C₃₁	semidirect
172. C₃₁	#33741	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×#3177	2
C₃₁×#2253	2
C₃₁×#2247	2
C₃₁×#2245	2
C₃₁×#2219	2
C₃₁×#2216	2
C₃₁×#2176	2
C₄×C₃₁×Q₃₂	2
C₃₁×#2108	2
#33741	31

Subgroups — C31×#33741 #21490500

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33741 #21490500