Subgroups — C₃₁×#33901 #21490660

Order 7936.

548 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3334	C₂	1	2	yes	maximal, complemented
2	C₃₁×#2252	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2232	C₂	1	4	yes	maximal, complemented
4	C₃₁×#2224	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2215	C₂	1	2	yes	maximal, complemented
6	C₄×C₃₁×D₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	D₄×C₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₃₁×#366	C₂	1	2	yes	maximal, complemented
9	C₃₁×#547	—	2	2	no	—
10	C₂×C₃₁×D₁₆	C₂²	1	1	yes	characteristic
11	C₂²×C₁₆×C₃₁	C₂²	1	2	yes	—
12	C₃₁×#500	C₂²	1	2	yes	—
13	C₃₁×#480	—	2	4	no	—
14	C₃₁×#474	—	2	2	no	—
15	C₃₁×#466	C₂²	1	4	yes	—
16	C₄×D₈×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×#357	C₂²	1	2	yes	—
25	C₃₁×#357	C₂²	1	4	yes	—
26	C₃₁×C₂²⋊C₁₆	C₂²	1	2	yes	—
27	C₄×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
28	C₂×#41×C₃₁	—	4	2	no	—
29	C₂×D₈×C₃₁	C₂³	1	2	yes	—
30	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
31	C₃₁×C₄⋊Q₈	—	2	2	no	—
32	C₃₁×#123	—	4	4	no	—
33	C₃₁×C₂²⋊Q₈	—	4	2	no	—
34	C₃₁×#121	—	2	4	no	—
35	C₄×D₄×C₃₁	—	2	2	no	—
36	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
37	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
38	C₃₁×D₁₆	—	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×#102	—	2	2	no	—
49	C₃₁×#102	—	2	4	no	—
50	C₃₁×C₂²⋊C₈	D₄	1	2	yes	—
51	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
52	#41×C₃₁	—	8	8	no	—
53	C₂×Q₈×C₃₁	—	4	2	no	—
54	C₂×D₄×C₃₁	—	2	2	no	—
55	C₂×D₄×C₃₁	—	4	4	no	—
56	C₂×D₄×C₃₁	D₈	1	1	yes	characteristic
57	C₂²×C₄×C₃₁	—	4	4	no	—
58	C₂²×C₄×C₃₁	—	4	2	no	—
59	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
60	D₈×C₃₁	—	2	2	no	—
61	D₈×C₃₁	—	2	2	no	—
62	C₂×C₈×C₃₁	—	2	2	no	—
63	C₂×C₈×C₃₁	—	2	2	no	—
64	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
65	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
66	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
67	C₄⋊C₄×C₃₁	—	2	2	no	—
68	C₄⋊C₄×C₃₁	—	2	4	no	—
69	C₄⋊C₄×C₃₁	—	4	4	no	—
70	C₄⋊C₄×C₃₁	—	4	2	no	—
71	C₄⋊C₄×C₃₁	D₈	1	1	yes	characteristic
72	C₂²⋊C₄×C₃₁	—	4	2	no	—
73	C₂²⋊C₄×C₃₁	—	4	4	no	—
74	C₂²⋊C₄×C₃₁	D₈	1	2	yes	—
75	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
76	C₁₆×C₃₁	—	2	2	no	—
77	C₁₆×C₃₁	—	2	1	no	—
78	C₁₆×C₃₁	#41	1	2	yes	complemented
79	#33901	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
80	C₂³×C₃₁	—	4	2	no	—
81	C₂³×C₃₁	C₂×D₈	1	2	yes	—
82	Q₈×C₃₁	—	8	4	no	—
83	D₄×C₃₁	—	4	2	no	—
84	D₄×C₃₁	—	4	2	no	—
85	D₄×C₃₁	—	8	8	no	—
86	D₄×C₃₁	D₁₆	1	4	yes	complemented
87	C₂×C₄×C₃₁	—	2	2	no	—
88	C₂×C₄×C₃₁	—	2	2	no	—
89	C₂×C₄×C₃₁	—	4	2	no	—
90	C₂×C₄×C₃₁	—	4	4	no	—
91	C₂×C₄×C₃₁	—	8	8	no	—
92	C₂×C₄×C₃₁	—	8	4	no	—
93	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
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	2	no	—
98	C₈×C₃₁	—	2	1	no	—
99	C₈×C₃₁	C₂×#41	1	1	yes	characteristic
100	C₈×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
101	#3334	C₂×C₃₁	1	2	yes	complemented
102	#2252	C₂×C₃₁	1	1	yes	characteristic, complemented
103	#2232	C₂×C₃₁	1	4	yes	complemented
104	#2224	C₂×C₃₁	1	2	yes	complemented
105	#2215	C₂×C₃₁	1	2	yes	complemented
106	C₄×D₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
107	D₄×C₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
108	C₂×#366	C₂×C₃₁	1	2	yes	complemented
109	C₂²×C₃₁	—	2	2	no	—
110	C₂²×C₃₁	—	2	2	no	—
111	C₂²×C₃₁	—	8	2	no	—
112	C₂²×C₃₁	—	8	2	no	—
113	C₂²×C₃₁	—	8	2	no	—
114	C₂²×C₃₁	C₂×D₁₆	1	4	yes	—
115	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
116	C₄×C₃₁	—	2	2	no	—
117	C₄×C₃₁	—	2	1	no	—
118	C₄×C₃₁	—	8	2	no	—
119	C₄×C₃₁	—	8	4	no	—
120	C₄×C₃₁	C₂×D₁₆	1	2	yes	—
121	C₄×C₃₁	#547	1	1	yes	characteristic
122	C₄×C₃₁	#578	1	1	yes	characteristic
123	#547	—	2	2	no	—
124	C₂×D₁₆	C₂²×C₃₁	1	1	yes	characteristic
125	C₂²×C₁₆	C₂²×C₃₁	1	2	yes	—
126	#500	C₂²×C₃₁	1	2	yes	—
127	#480	—	2	4	no	—
128	#474	—	2	2	no	—
129	#466	C₂²×C₃₁	1	4	yes	—
130	C₄×D₈	C₂²×C₃₁	1	2	yes	—
131	C₈×D₄	C₂²×C₃₁	1	1	yes	characteristic
132	C₂×#107	C₂²×C₃₁	1	4	yes	—
133	#366	C₂²×C₃₁	1	1	yes	characteristic
134	#366	C₂²×C₃₁	1	2	yes	—
135	#366	C₂²×C₃₁	1	4	yes	—
136	#366	C₂²×C₃₁	1	2	yes	—
137	C₄⋊C₁₆	C₂²×C₃₁	1	1	yes	characteristic
138	#357	C₂²×C₃₁	1	2	yes	—
139	#357	C₂²×C₃₁	1	4	yes	—
140	C₂²⋊C₁₆	C₂²×C₃₁	1	2	yes	—
141	C₄×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
142	C₂×C₃₁	—	2	4	no	—
143	C₂×C₃₁	—	16	2	no	—
144	C₂×C₃₁	#3334	1	1	yes	characteristic
145	C₂×C₃₁	C₂²×D₁₆	1	1	yes	characteristic
146	C₂×C₃₁	#3417	1	1	yes	characteristic
147	C₂×#41	—	4	2	no	—
148	C₂×D₈	C₂³×C₃₁	1	2	yes	—
149	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
150	C₄⋊Q₈	—	2	2	no	—
151	#123	—	4	4	no	—
152	C₂²⋊Q₈	—	4	2	no	—
153	#121	—	2	4	no	—
154	C₄×D₄	—	2	2	no	—
155	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
156	C₂×C₄⋊C₄	—	2	4	no	—
157	D₁₆	—	2	2	no	—
158	C₂×C₁₆	—	2	4	no	—
159	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
160	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
161	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
162	#107	—	2	4	no	—
163	#107	—	2	2	no	—
164	#107	C₂³×C₃₁	1	2	yes	—
165	#107	C₂³×C₃₁	1	4	yes	—
166	C₄⋊C₈	D₄×C₃₁	1	1	yes	characteristic
167	#102	—	2	2	no	—
168	#102	—	2	4	no	—
169	C₂²⋊C₈	D₄×C₃₁	1	2	yes	—
170	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
171	C₃₁	#33901	1	1	yes	characteristic, direct product factor, Sylow
172	#41	—	8	8	no	—
173	C₂×Q₈	—	4	2	no	—
174	C₂×D₄	—	2	2	no	—
175	C₂×D₄	—	4	4	no	—
176	C₂×D₄	D₈×C₃₁	1	1	yes	characteristic
177	C₂²×C₄	—	4	4	no	—
178	C₂²×C₄	—	4	2	no	—
179	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
180	D₈	—	2	2	no	—
181	D₈	—	2	2	no	—
182	C₂×C₈	—	2	2	no	—
183	C₂×C₈	—	2	2	no	—
184	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
185	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
186	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
187	C₄⋊C₄	—	2	2	no	—
188	C₄⋊C₄	—	2	4	no	—
189	C₄⋊C₄	—	4	4	no	—
190	C₄⋊C₄	—	4	2	no	—
191	C₄⋊C₄	D₈×C₃₁	1	1	yes	characteristic
192	C₂²⋊C₄	—	4	2	no	—
193	C₂²⋊C₄	—	4	4	no	—
194	C₂²⋊C₄	D₈×C₃₁	1	2	yes	—
195	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
196	C₁₆	—	2	2	no	—
197	C₁₆	—	2	1	no	—
198	C₁₆	#41×C₃₁	1	2	yes	complemented
199	C₂³	—	4	2	no	—
200	C₂³	C₂×D₈×C₃₁	1	2	yes	—
201	Q₈	—	8	4	no	—
202	D₄	—	4	2	no	—
203	D₄	—	4	2	no	—
204	D₄	—	8	8	no	—
205	D₄	C₃₁×D₁₆	1	4	yes	complemented
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	4	2	no	—
209	C₂×C₄	—	4	4	no	—
210	C₂×C₄	—	8	8	no	—
211	C₂×C₄	—	8	4	no	—
212	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
213	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
214	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
215	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
216	C₈	—	2	2	no	—
217	C₈	—	2	1	no	—
218	C₈	C₂×#41×C₃₁	1	1	yes	characteristic
219	C₈	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
220	C₂²	—	2	2	no	—
221	C₂²	—	2	2	no	—
222	C₂²	—	8	2	no	—
223	C₂²	—	8	2	no	—
224	C₂²	—	8	2	no	—
225	C₂²	C₂×C₃₁×D₁₆	1	4	yes	—
226	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
227	C₄	—	2	2	no	—
228	C₄	—	2	1	no	—
229	C₄	—	8	2	no	—
230	C₄	—	8	4	no	—
231	C₄	C₂×C₃₁×D₁₆	1	2	yes	—
232	C₄	C₃₁×#547	1	1	yes	characteristic
233	C₄	C₃₁×#578	1	1	yes	characteristic
234	C₂	—	2	4	no	—
235	C₂	—	16	2	no	—
236	C₂	C₃₁×#3334	1	1	yes	characteristic
237	C₂	C₂²×C₃₁×D₁₆	1	1	yes	characteristic
238	C₂	C₃₁×#3417	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
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
1. C₃₁×#3334	C₂	semidirect
2. C₃₁×#2252	C₂	semidirect
3. C₃₁×#2232	C₂	semidirect
4. C₃₁×#2224	C₂	semidirect
5. C₃₁×#2215	C₂	semidirect
6. C₄×C₃₁×D₁₆	C₂	semidirect
7. D₄×C₁₆×C₃₁	C₂	semidirect
8. C₂×C₃₁×#366	C₂	semidirect
78. C₁₆×C₃₁	#41	semidirect
79. #33901	C₃₁	direct (N × H)
86. D₄×C₃₁	D₁₆	semidirect
101. #3334	C₂×C₃₁	semidirect
102. #2252	C₂×C₃₁	semidirect
103. #2232	C₂×C₃₁	semidirect
104. #2224	C₂×C₃₁	semidirect
105. #2215	C₂×C₃₁	semidirect
106. C₄×D₁₆	C₂×C₃₁	semidirect
107. D₄×C₁₆	C₂×C₃₁	semidirect
108. C₂×#366	C₂×C₃₁	semidirect
171. C₃₁	#33901	direct (N × H)
198. C₁₆	#41×C₃₁	semidirect
205. D₄	C₃₁×D₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#3334	2
C₃₁×#2252	2
C₃₁×#2232	2
C₃₁×#2224	2
C₃₁×#2215	2
C₄×C₃₁×D₁₆	2
D₄×C₁₆×C₃₁	2
C₂×C₃₁×#366	2
#33901	31

Subgroups — C31×#33901 #21490660

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33901 #21490660