Subgroups — C₃₁×#33733 #21490492

Order 7936.

602 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3169	C₂	1	2	yes	maximal, complemented
2	C₃₁×#2252	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2246	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2240	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2215	C₂	1	4	yes	maximal, complemented
6	C₄×C₃₁×D₁₆	C₂	1	4	yes	maximal, complemented
7	C₂×C₄×C₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#532	—	2	2	no	—
9	C₂×C₃₁×D₁₆	C₂²	1	4	yes	—
10	C₂²×C₁₆×C₃₁	C₂²	1	2	yes	—
11	C₃₁×#500	C₂²	1	2	yes	—
12	C₃₁×#493	C₂²	1	2	yes	—
13	C₃₁×#486	—	2	2	no	—
14	C₃₁×#466	—	2	2	no	—
15	C₃₁×#466	C₂²	1	4	yes	—
16	C₄×D₈×C₃₁	—	2	2	no	—
17	C₄×D₈×C₃₁	C₂²	1	4	yes	—
18	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
19	C₃₁×#366	C₂²	1	4	yes	—
20	C₃₁×#357	C₂²	1	8	yes	—
21	C₄×C₁₆×C₃₁	C₂²	1	2	yes	—
22	C₄×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
23	C₄×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
24	C₂×#41×C₃₁	—	4	2	no	—
25	C₂×D₈×C₃₁	—	2	2	no	—
26	C₂×D₈×C₃₁	C₂³	1	4	yes	—
27	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
29	C₃₁×C₄⋊Q₈	—	2	2	no	—
30	C₃₁×#127	—	2	2	no	—
31	C₃₁×#124	—	4	2	no	—
32	C₃₁×#121	—	2	4	no	—
33	C₃₁×#121	—	4	2	no	—
34	C₄×D₄×C₃₁	—	2	4	no	—
35	C₄×D₄×C₃₁	—	4	2	no	—
36	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
37	C₃₁×D₁₆	—	2	8	no	—
38	C₂×C₁₆×C₃₁	—	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×#102	—	2	4	no	—
44	C₃₁×#102	—	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	#41×C₃₁	—	8	8	no	—
50	C₂×Q₈×C₃₁	—	4	2	no	—
51	C₂×D₄×C₃₁	—	2	4	no	—
52	C₂×D₄×C₃₁	—	4	4	no	—
53	C₂×D₄×C₃₁	—	4	2	no	—
54	C₂×D₄×C₃₁	—	4	2	no	—
55	C₂²×C₄×C₃₁	—	4	4	no	—
56	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
57	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
58	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
59	D₈×C₃₁	—	2	8	no	—
60	D₈×C₃₁	—	4	4	no	—
61	C₂×C₈×C₃₁	—	2	2	no	—
62	C₂×C₈×C₃₁	—	2	2	no	—
63	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
64	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
65	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
66	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
67	C₄⋊C₄×C₃₁	—	2	4	no	—
68	C₄⋊C₄×C₃₁	—	4	2	no	—
69	C₂²⋊C₄×C₃₁	—	4	4	no	—
70	C₂²⋊C₄×C₃₁	—	4	4	no	—
71	C₄²×C₃₁	D₈	1	1	yes	characteristic
72	C₄²×C₃₁	D₈	1	1	yes	characteristic
73	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
74	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
75	C₁₆×C₃₁	—	2	2	no	—
76	C₁₆×C₃₁	#41	1	4	yes	complemented
77	#33733	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
78	C₂³×C₃₁	—	4	4	no	—
79	C₂³×C₃₁	C₂×D₈	1	1	yes	characteristic
80	Q₈×C₃₁	—	8	4	no	—
81	D₄×C₃₁	—	4	4	no	—
82	D₄×C₃₁	—	4	4	no	—
83	D₄×C₃₁	—	8	8	no	—
84	D₄×C₃₁	—	8	4	no	—
85	D₄×C₃₁	—	8	4	no	—
86	C₂×C₄×C₃₁	—	2	1	no	—
87	C₂×C₄×C₃₁	—	2	1	no	—
88	C₂×C₄×C₃₁	—	2	2	no	—
89	C₂×C₄×C₃₁	—	4	4	no	—
90	C₂×C₄×C₃₁	—	8	8	no	—
91	C₂×C₄×C₃₁	D₁₆	1	4	yes	complemented
92	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
93	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
94	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
95	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
96	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
97	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
98	C₈×C₃₁	—	2	1	no	—
99	C₈×C₃₁	—	2	1	no	—
100	C₈×C₃₁	#135	1	2	yes	—
101	C₈×C₃₁	C₂×#41	1	2	yes	—
102	#3169	C₂×C₃₁	1	2	yes	complemented
103	#2252	C₂×C₃₁	1	1	yes	characteristic, complemented
104	#2246	C₂×C₃₁	1	1	yes	characteristic, complemented
105	#2240	C₂×C₃₁	1	2	yes	complemented
106	#2215	C₂×C₃₁	1	4	yes	complemented
107	C₄×D₁₆	C₂×C₃₁	1	4	yes	complemented
108	C₂×C₄×C₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
109	C₂²×C₃₁	—	2	1	no	—
110	C₂²×C₃₁	—	2	1	no	—
111	C₂²×C₃₁	—	8	4	no	—
112	C₂²×C₃₁	—	8	4	no	—
113	C₂²×C₃₁	—	8	4	no	—
114	C₂²×C₃₁	C₂×D₁₆	1	2	yes	—
115	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
116	C₄×C₃₁	—	2	1	no	—
117	C₄×C₃₁	—	2	1	no	—
118	C₄×C₃₁	—	8	4	no	—
119	C₄×C₃₁	C₂×D₁₆	1	2	yes	—
120	C₄×C₃₁	C₂×D₁₆	1	2	yes	—
121	C₄×C₃₁	#508	1	2	yes	—
122	C₄×C₃₁	#532	1	1	yes	characteristic
123	C₄×C₃₁	C₂×#135	1	1	yes	characteristic
124	#532	—	2	2	no	—
125	C₂×D₁₆	C₂²×C₃₁	1	4	yes	—
126	C₂²×C₁₆	C₂²×C₃₁	1	2	yes	—
127	#500	C₂²×C₃₁	1	2	yes	—
128	#493	C₂²×C₃₁	1	2	yes	—
129	#486	—	2	2	no	—
130	#466	—	2	2	no	—
131	#466	C₂²×C₃₁	1	4	yes	—
132	C₄×D₈	—	2	2	no	—
133	C₄×D₈	C₂²×C₃₁	1	4	yes	—
134	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
135	#366	C₂²×C₃₁	1	4	yes	—
136	#357	C₂²×C₃₁	1	8	yes	—
137	C₄×C₁₆	C₂²×C₃₁	1	2	yes	—
138	C₄×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
139	C₄×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
140	C₂×C₃₁	—	2	2	no	—
141	C₂×C₃₁	—	16	4	no	—
142	C₂×C₃₁	#3169	1	1	yes	characteristic
143	C₂×C₃₁	C₂²×D₁₆	1	1	yes	characteristic
144	C₂×C₃₁	C₂×#508	1	1	yes	characteristic
145	C₂×#41	—	4	2	no	—
146	C₂×D₈	—	2	2	no	—
147	C₂×D₈	C₂³×C₃₁	1	4	yes	—
148	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
149	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
150	C₄⋊Q₈	—	2	2	no	—
151	#127	—	2	2	no	—
152	#124	—	4	2	no	—
153	#121	—	2	4	no	—
154	#121	—	4	2	no	—
155	C₄×D₄	—	2	4	no	—
156	C₄×D₄	—	4	2	no	—
157	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
158	D₁₆	—	2	8	no	—
159	C₂×C₁₆	—	2	4	no	—
160	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
161	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
162	#107	—	2	2	no	—
163	#107	C₂³×C₃₁	1	4	yes	—
164	#102	—	2	4	no	—
165	#102	—	2	4	no	—
166	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
167	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
168	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
169	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
170	C₃₁	#33733	1	1	yes	characteristic, direct product factor, Sylow
171	#41	—	8	8	no	—
172	C₂×Q₈	—	4	2	no	—
173	C₂×D₄	—	2	4	no	—
174	C₂×D₄	—	4	4	no	—
175	C₂×D₄	—	4	2	no	—
176	C₂×D₄	—	4	2	no	—
177	C₂²×C₄	—	4	4	no	—
178	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
179	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
180	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
181	D₈	—	2	8	no	—
182	D₈	—	4	4	no	—
183	C₂×C₈	—	2	2	no	—
184	C₂×C₈	—	2	2	no	—
185	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
186	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
187	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
188	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
189	C₄⋊C₄	—	2	4	no	—
190	C₄⋊C₄	—	4	2	no	—
191	C₂²⋊C₄	—	4	4	no	—
192	C₂²⋊C₄	—	4	4	no	—
193	C₄²	D₈×C₃₁	1	1	yes	characteristic
194	C₄²	D₈×C₃₁	1	1	yes	characteristic
195	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
196	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
197	C₁₆	—	2	2	no	—
198	C₁₆	#41×C₃₁	1	4	yes	complemented
199	C₂³	—	4	4	no	—
200	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
201	Q₈	—	8	4	no	—
202	D₄	—	4	4	no	—
203	D₄	—	4	4	no	—
204	D₄	—	8	8	no	—
205	D₄	—	8	4	no	—
206	D₄	—	8	4	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	4	4	no	—
211	C₂×C₄	—	8	8	no	—
212	C₂×C₄	C₃₁×D₁₆	1	4	yes	complemented
213	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
214	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
215	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
216	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
217	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
218	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
219	C₈	—	2	1	no	—
220	C₈	—	2	1	no	—
221	C₈	C₃₁×#135	1	2	yes	—
222	C₈	C₂×#41×C₃₁	1	2	yes	—
223	C₂²	—	2	1	no	—
224	C₂²	—	2	1	no	—
225	C₂²	—	8	4	no	—
226	C₂²	—	8	4	no	—
227	C₂²	—	8	4	no	—
228	C₂²	C₂×C₃₁×D₁₆	1	2	yes	—
229	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
230	C₄	—	2	1	no	—
231	C₄	—	2	1	no	—
232	C₄	—	8	4	no	—
233	C₄	C₂×C₃₁×D₁₆	1	2	yes	—
234	C₄	C₂×C₃₁×D₁₆	1	2	yes	—
235	C₄	C₃₁×#508	1	2	yes	—
236	C₄	C₃₁×#532	1	1	yes	characteristic
237	C₄	C₂×C₃₁×#135	1	1	yes	characteristic
238	C₂	—	2	2	no	—
239	C₂	—	16	4	no	—
240	C₂	C₃₁×#3169	1	1	yes	characteristic
241	C₂	C₂²×C₃₁×D₁₆	1	1	yes	characteristic
242	C₂	C₂×C₃₁×#508	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₂²×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×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
1. C₃₁×#3169	C₂	semidirect
2. C₃₁×#2252	C₂	semidirect
3. C₃₁×#2246	C₂	semidirect
4. C₃₁×#2240	C₂	semidirect
5. C₃₁×#2215	C₂	semidirect
6. C₄×C₃₁×D₁₆	C₂	semidirect
7. C₂×C₄×C₁₆×C₃₁	C₂	semidirect
76. C₁₆×C₃₁	#41	semidirect
77. #33733	C₃₁	direct (N × H)
91. C₂×C₄×C₃₁	D₁₆	semidirect
102. #3169	C₂×C₃₁	semidirect
103. #2252	C₂×C₃₁	semidirect
104. #2246	C₂×C₃₁	semidirect
105. #2240	C₂×C₃₁	semidirect
106. #2215	C₂×C₃₁	semidirect
107. C₄×D₁₆	C₂×C₃₁	semidirect
108. C₂×C₄×C₁₆	C₂×C₃₁	semidirect
170. C₃₁	#33733	direct (N × H)
198. C₁₆	#41×C₃₁	semidirect
212. C₂×C₄	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×#3169	2
C₃₁×#2252	2
C₃₁×#2246	2
C₃₁×#2240	2
C₃₁×#2215	2
C₄×C₃₁×D₁₆	2
C₂×C₄×C₁₆×C₃₁	2
#33733	31

Subgroups — C31×#33733 #21490492

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33733 #21490492