Subgroups — C₃₁×#33749 #21490508

Order 7936.

504 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3170	C₂	1	2	yes	maximal
2	C₃₁×#2254	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2247	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2241	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2219	C₂	1	4	yes	maximal
6	C₃₁×#2176	C₂	1	4	yes	maximal, complemented
7	C₄×C₃₁×M₃₂	C₂	1	1	yes	maximal, characteristic
8	C₃₁×#533	—	2	2	no	—
9	C₂×C₃₁×Q₃₂	C₂²	1	4	yes	—
10	C₂×C₃₁×M₃₂	C₂²	1	2	yes	—
11	C₃₁×#500	C₂²	1	2	yes	—
12	C₃₁×#494	C₂²	1	2	yes	—
13	C₃₁×#487	—	2	2	no	—
14	C₃₁×#467	—	2	2	no	—
15	C₃₁×#467	C₂²	1	4	yes	—
16	C₄×Q₁₆×C₃₁	—	2	2	no	—
17	C₄×Q₁₆×C₃₁	C₂²	1	4	yes	—
18	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
19	C₃₁×#367	C₂²	1	4	yes	—
20	C₃₁×#358	C₂²	1	8	yes	—
21	C₃₁×#346	C₂²	1	2	yes	—
22	C₄×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
23	C₄×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
24	C₂×Q₁₆×C₃₁	—	2	2	no	—
25	C₂×Q₁₆×C₃₁	C₂³	1	4	yes	—
26	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
28	C₃₁×C₄⋊Q₈	—	2	2	no	—
29	C₃₁×C₄⋊Q₈	—	2	2	no	—
30	C₃₁×#125	—	4	2	no	—
31	C₃₁×C₂²⋊Q₈	—	2	4	no	—
32	C₃₁×C₂²⋊Q₈	—	4	2	no	—
33	C₄×Q₈×C₃₁	—	2	4	no	—
34	C₄×Q₈×C₃₁	—	4	2	no	—
35	C₃₁×#117	—	4	2	no	—
36	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
37	C₃₁×Q₃₂	—	2	8	no	—
38	C₃₁×M₃₂	—	2	4	no	—
39	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
40	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
41	C₃₁×#107	—	2	2	no	—
42	C₃₁×#107	C₂³	1	4	yes	—
43	C₃₁×#103	—	2	4	no	—
44	C₃₁×#103	—	2	4	no	—
45	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
46	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
47	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
48	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
49	C₂×Q₈×C₃₁	—	2	4	no	—
50	C₂×Q₈×C₃₁	—	4	2	no	—
51	C₂²×C₄×C₃₁	D₈	1	2	yes	—
52	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
53	Q₁₆×C₃₁	—	2	8	no	—
54	Q₁₆×C₃₁	—	4	4	no	—
55	C₂×C₈×C₃₁	—	2	2	no	—
56	C₂×C₈×C₃₁	—	2	2	no	—
57	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
58	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
61	C₄⋊C₄×C₃₁	—	2	4	no	—
62	C₄⋊C₄×C₃₁	—	4	2	no	—
63	C₄⋊C₄×C₃₁	—	4	2	no	—
64	C₄⋊C₄×C₃₁	—	4	4	no	—
65	C₄⋊C₄×C₃₁	—	4	4	no	—
66	C₄⋊C₄×C₃₁	—	4	2	no	—
67	C₂²⋊C₄×C₃₁	—	4	4	no	—
68	C₄²×C₃₁	—	4	4	no	—
69	C₄²×C₃₁	D₈	1	2	yes	—
70	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
71	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
72	C₁₆×C₃₁	—	2	2	no	—
73	C₁₆×C₃₁	#41	1	4	yes	—
74	#33749	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
75	C₂³×C₃₁	C₂×D₈	1	1	yes	characteristic
76	Q₈×C₃₁	—	4	4	no	—
77	Q₈×C₃₁	—	4	4	no	—
78	Q₈×C₃₁	—	8	4	no	—
79	C₂×C₄×C₃₁	—	2	1	no	—
80	C₂×C₄×C₃₁	—	2	1	no	—
81	C₂×C₄×C₃₁	—	2	4	no	—
82	C₂×C₄×C₃₁	—	4	4	no	—
83	C₂×C₄×C₃₁	—	4	4	no	—
84	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
85	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
86	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
87	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
88	C₈×C₃₁	—	2	1	no	—
89	C₈×C₃₁	—	2	1	no	—
90	C₈×C₃₁	#135	1	2	yes	—
91	C₈×C₃₁	C₂×#41	1	2	yes	—
92	#3170	C₂×C₃₁	1	2	yes	—
93	#2254	C₂×C₃₁	1	1	yes	characteristic, complemented
94	#2247	C₂×C₃₁	1	1	yes	characteristic, complemented
95	#2241	C₂×C₃₁	1	2	yes	complemented
96	#2219	C₂×C₃₁	1	4	yes	—
97	#2176	C₂×C₃₁	1	4	yes	complemented
98	C₄×M₃₂	C₂×C₃₁	1	1	yes	characteristic
99	C₂²×C₃₁	—	2	2	no	—
100	C₂²×C₃₁	—	2	1	no	—
101	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
102	C₄×C₃₁	—	2	2	no	—
103	C₄×C₃₁	—	2	1	no	—
104	C₄×C₃₁	—	8	4	no	—
105	C₄×C₃₁	—	8	4	no	—
106	C₄×C₃₁	#510	1	4	yes	—
107	C₄×C₃₁	#532	1	1	yes	characteristic
108	C₄×C₃₁	C₂×#135	1	1	yes	characteristic
109	#533	—	2	2	no	—
110	C₂×Q₃₂	C₂²×C₃₁	1	4	yes	—
111	C₂×M₃₂	C₂²×C₃₁	1	2	yes	—
112	#500	C₂²×C₃₁	1	2	yes	—
113	#494	C₂²×C₃₁	1	2	yes	—
114	#487	—	2	2	no	—
115	#467	—	2	2	no	—
116	#467	C₂²×C₃₁	1	4	yes	—
117	C₄×Q₁₆	—	2	2	no	—
118	C₄×Q₁₆	C₂²×C₃₁	1	4	yes	—
119	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
120	#367	C₂²×C₃₁	1	4	yes	—
121	#358	C₂²×C₃₁	1	8	yes	—
122	#346	C₂²×C₃₁	1	2	yes	—
123	C₄×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
124	C₄×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
125	C₂×C₃₁	—	4	1	no	—
126	C₂×C₃₁	#3169	1	1	yes	characteristic
127	C₂×C₃₁	C₂×#510	1	2	yes	—
128	C₂×Q₁₆	—	2	2	no	—
129	C₂×Q₁₆	C₂³×C₃₁	1	4	yes	—
130	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
131	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
132	C₄⋊Q₈	—	2	2	no	—
133	C₄⋊Q₈	—	2	2	no	—
134	#125	—	4	2	no	—
135	C₂²⋊Q₈	—	2	4	no	—
136	C₂²⋊Q₈	—	4	2	no	—
137	C₄×Q₈	—	2	4	no	—
138	C₄×Q₈	—	4	2	no	—
139	#117	—	4	2	no	—
140	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
141	Q₃₂	—	2	8	no	—
142	M₃₂	—	2	4	no	—
143	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
144	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
145	#107	—	2	2	no	—
146	#107	C₂³×C₃₁	1	4	yes	—
147	#103	—	2	4	no	—
148	#103	—	2	4	no	—
149	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
150	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
151	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
152	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
153	C₃₁	#33749	1	1	yes	characteristic, direct product factor, Sylow
154	C₂×Q₈	—	2	4	no	—
155	C₂×Q₈	—	4	2	no	—
156	C₂²×C₄	D₈×C₃₁	1	2	yes	—
157	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
158	Q₁₆	—	2	8	no	—
159	Q₁₆	—	4	4	no	—
160	C₂×C₈	—	2	2	no	—
161	C₂×C₈	—	2	2	no	—
162	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
163	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
164	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
165	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
166	C₄⋊C₄	—	2	4	no	—
167	C₄⋊C₄	—	4	2	no	—
168	C₄⋊C₄	—	4	2	no	—
169	C₄⋊C₄	—	4	4	no	—
170	C₄⋊C₄	—	4	4	no	—
171	C₄⋊C₄	—	4	2	no	—
172	C₂²⋊C₄	—	4	4	no	—
173	C₄²	—	4	4	no	—
174	C₄²	D₈×C₃₁	1	2	yes	—
175	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
176	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
177	C₁₆	—	2	2	no	—
178	C₁₆	#41×C₃₁	1	4	yes	—
179	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
180	Q₈	—	4	4	no	—
181	Q₈	—	4	4	no	—
182	Q₈	—	8	4	no	—
183	C₂×C₄	—	2	4	no	—
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	2	1	no	—
186	C₂×C₄	—	4	4	no	—
187	C₂×C₄	—	4	4	no	—
188	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
189	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
190	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
191	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
192	C₈	—	2	1	no	—
193	C₈	—	2	1	no	—
194	C₈	C₃₁×#135	1	2	yes	—
195	C₈	C₂×#41×C₃₁	1	2	yes	—
196	C₂²	—	2	2	no	—
197	C₂²	—	2	1	no	—
198	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
199	C₄	—	2	2	no	—
200	C₄	—	2	1	no	—
201	C₄	—	8	4	no	—
202	C₄	—	8	4	no	—
203	C₄	C₃₁×#510	1	4	yes	—
204	C₄	C₃₁×#532	1	1	yes	characteristic
205	C₄	C₂×C₃₁×#135	1	1	yes	characteristic
206	C₂	—	4	1	no	—
207	C₂	C₃₁×#3169	1	1	yes	characteristic
208	C₂	C₂×C₃₁×#510	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 ◁ C₂×C₄²×C₃₁ ◁ C₂×C₄×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2254	C₂	semidirect
3. C₃₁×#2247	C₂	semidirect
4. C₃₁×#2241	C₂	semidirect
6. C₃₁×#2176	C₂	semidirect
74. #33749	C₃₁	direct (N × H)
93. #2254	C₂×C₃₁	semidirect
94. #2247	C₂×C₃₁	semidirect
95. #2241	C₂×C₃₁	semidirect
97. #2176	C₂×C₃₁	semidirect
153. C₃₁	#33749	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×#3170	2
C₃₁×#2254	2
C₃₁×#2247	2
C₃₁×#2241	2
C₃₁×#2219	2
C₃₁×#2176	2
C₄×C₃₁×M₃₂	2
#33749	31

Subgroups — C31×#33749 #21490508

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33749 #21490508