Subgroups — C₃₁×#9852 #21466611

Order 7936.

498 conjugacy classes of subgroups.

The subgroup lattice has 248 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	1	yes	maximal, characteristic
2	C₃₁×#1962	C₂	1	2	yes	maximal, complemented
3	C₃₁×#1744	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1566	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1492	C₂	1	2	yes	maximal, complemented
6	C₃₁×#532	C₂²	1	2	yes	—
7	C₃₁×#500	C₄	1	1	yes	characteristic, complemented
8	C₃₁×#493	C₄	1	1	yes	characteristic, complemented
9	C₃₁×#486	C₄	1	2	yes	complemented
10	C₃₁×#466	—	2	2	no	—
11	C₄×D₈×C₃₁	—	2	2	no	—
12	C₃₁×#423	C₂²	1	2	yes	—
13	C₂×C₃₁×#104	—	2	4	no	—
14	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
16	C₄³×C₃₁	C₂²	1	1	yes	characteristic
17	C₃₁×#336	—	2	2	no	—
18	C₃₁×#334	—	2	2	no	—
19	C₃₁×D₄⋊C₈	—	2	4	no	—
20	C₂×#41×C₃₁	—	2	2	no	—
21	C₂×D₈×C₃₁	—	2	2	no	—
22	C₂×M₁₆×C₃₁	—	2	2	no	—
23	C₂²×C₈×C₃₁	—	2	1	no	—
24	C₂²×C₈×C₃₁	D₄	1	2	yes	—
25	C₃₁×C₄⋊Q₈	C₂×C₄	1	2	yes	—
26	C₃₁×#127	C₂×C₄	1	2	yes	—
27	C₃₁×#124	—	2	2	no	—
28	C₃₁×#121	—	2	2	no	—
29	C₃₁×#121	—	2	2	no	—
30	C₄×D₄×C₃₁	—	2	2	no	—
31	C₄×D₄×C₃₁	—	2	2	no	—
32	C₂×C₄²×C₃₁	—	2	1	no	—
33	C₂×C₄²×C₃₁	—	2	1	no	—
34	C₂×C₄²×C₃₁	D₄	1	2	yes	—
35	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
36	C₃₁×#107	—	2	2	no	—
37	C₃₁×C₄⋊C₈	—	2	2	no	—
38	C₃₁×C₄⋊C₈	—	2	2	no	—
39	C₃₁×#104	—	4	8	no	—
40	C₃₁×#102	—	2	4	no	—
41	C₄×C₈×C₃₁	—	2	1	no	—
42	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
44	C₄×C₈×C₃₁	D₄	1	2	yes	—
45	C₄×C₈×C₃₁	D₄	1	2	yes	—
46	#41×C₃₁	—	4	4	no	—
47	#41×C₃₁	—	4	4	no	—
48	C₂×Q₈×C₃₁	—	2	2	no	—
49	C₂×D₄×C₃₁	—	2	2	no	—
50	C₂×D₄×C₃₁	—	2	2	no	—
51	C₂×D₄×C₃₁	—	2	2	no	—
52	C₂×D₄×C₃₁	—	4	2	no	—
53	C₂²×C₄×C₃₁	—	2	2	no	—
54	C₂²×C₄×C₃₁	—	4	2	no	—
55	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
58	D₈×C₃₁	—	4	4	no	—
59	M₁₆×C₃₁	—	4	4	no	—
60	C₂×C₈×C₃₁	—	2	2	no	—
61	C₂×C₈×C₃₁	—	4	2	no	—
62	C₂×C₈×C₃₁	—	4	2	no	—
63	C₂×C₈×C₃₁	—	4	2	no	—
64	C₂×C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
65	C₂×C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
66	C₄⋊C₄×C₃₁	—	2	2	no	—
67	C₄⋊C₄×C₃₁	—	2	2	no	—
68	C₂²⋊C₄×C₃₁	—	4	2	no	—
69	C₂²⋊C₄×C₃₁	—	4	2	no	—
70	C₄²×C₃₁	—	2	4	no	—
71	C₄²×C₃₁	—	2	4	no	—
72	C₄²×C₃₁	—	4	2	no	—
73	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
74	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
75	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
76	C₄²×C₃₁	#41	1	1	yes	characteristic
77	#9852	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
78	C₂³×C₃₁	—	4	2	no	—
79	C₂³×C₃₁	#121	1	1	yes	characteristic
80	Q₈×C₃₁	—	4	4	no	—
81	D₄×C₃₁	—	4	4	no	—
82	D₄×C₃₁	—	4	4	no	—
83	D₄×C₃₁	—	4	4	no	—
84	D₄×C₃₁	—	8	4	no	—
85	C₂×C₄×C₃₁	—	2	2	no	—
86	C₂×C₄×C₃₁	—	2	2	no	—
87	C₂×C₄×C₃₁	—	2	4	no	—
88	C₂×C₄×C₃₁	—	4	2	no	—
89	C₂×C₄×C₃₁	—	4	2	no	—
90	C₂×C₄×C₃₁	—	4	2	no	—
91	C₂×C₄×C₃₁	—	8	2	no	—
92	C₂×C₄×C₃₁	—	8	2	no	—
93	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
95	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
96	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
97	C₂×C₄×C₃₁	#124	1	1	yes	characteristic
98	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
99	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	4	yes	—
100	C₈×C₃₁	—	2	2	no	—
101	C₈×C₃₁	—	2	2	no	—
102	C₈×C₃₁	—	4	2	no	—
103	C₈×C₃₁	—	8	2	no	—
104	#3169	C₂×C₃₁	1	1	yes	characteristic
105	#1962	C₂×C₃₁	1	2	yes	complemented
106	#1744	C₂×C₃₁	1	1	yes	characteristic, complemented
107	#1566	C₂×C₃₁	1	1	yes	characteristic, complemented
108	#1492	C₂×C₃₁	1	2	yes	complemented
109	C₂²×C₃₁	—	2	1	no	—
110	C₂²×C₃₁	—	2	1	no	—
111	C₂²×C₃₁	—	8	2	no	—
112	C₂²×C₃₁	—	8	2	no	—
113	C₂²×C₃₁	—	8	2	no	—
114	C₂²×C₃₁	#390	1	1	yes	characteristic
115	C₂²×C₃₁	#469	1	2	yes	—
116	C₄×C₃₁	—	2	1	no	—
117	C₄×C₃₁	—	2	1	no	—
118	C₄×C₃₁	—	4	4	no	—
119	C₄×C₃₁	—	8	2	no	—
120	C₄×C₃₁	#421	1	2	yes	—
121	C₄×C₃₁	#442	1	2	yes	—
122	C₄×C₃₁	#443	1	2	yes	—
123	C₄×C₃₁	#496	1	2	yes	—
124	#532	C₂²×C₃₁	1	2	yes	—
125	#500	C₄×C₃₁	1	1	yes	characteristic, complemented
126	#493	C₄×C₃₁	1	1	yes	characteristic, complemented
127	#486	C₄×C₃₁	1	2	yes	complemented
128	#466	—	2	2	no	—
129	C₄×D₈	—	2	2	no	—
130	#423	C₂²×C₃₁	1	2	yes	—
131	C₂×#104	—	2	4	no	—
132	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
133	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
134	C₄³	C₂²×C₃₁	1	1	yes	characteristic
135	#336	—	2	2	no	—
136	#334	—	2	2	no	—
137	D₄⋊C₈	—	2	4	no	—
138	C₂×C₃₁	—	2	2	no	—
139	C₂×C₃₁	—	16	2	no	—
140	C₂×C₃₁	#1963	1	1	yes	characteristic
141	C₂×C₃₁	#1972	1	1	yes	characteristic
142	C₂×C₃₁	#1974	1	1	yes	characteristic
143	C₂×#41	—	2	2	no	—
144	C₂×D₈	—	2	2	no	—
145	C₂×M₁₆	—	2	2	no	—
146	C₂²×C₈	—	2	1	no	—
147	C₂²×C₈	D₄×C₃₁	1	2	yes	—
148	C₄⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
149	#127	C₂×C₄×C₃₁	1	2	yes	—
150	#124	—	2	2	no	—
151	#121	—	2	2	no	—
152	#121	—	2	2	no	—
153	C₄×D₄	—	2	2	no	—
154	C₄×D₄	—	2	2	no	—
155	C₂×C₄²	—	2	1	no	—
156	C₂×C₄²	—	2	1	no	—
157	C₂×C₄²	D₄×C₃₁	1	2	yes	—
158	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
159	#107	—	2	2	no	—
160	C₄⋊C₈	—	2	2	no	—
161	C₄⋊C₈	—	2	2	no	—
162	#104	—	4	8	no	—
163	#102	—	2	4	no	—
164	C₄×C₈	—	2	1	no	—
165	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
166	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
167	C₄×C₈	D₄×C₃₁	1	2	yes	—
168	C₄×C₈	D₄×C₃₁	1	2	yes	—
169	C₃₁	#9852	1	1	yes	characteristic, direct product factor, Sylow
170	#41	—	4	4	no	—
171	#41	—	4	4	no	—
172	C₂×Q₈	—	2	2	no	—
173	C₂×D₄	—	2	2	no	—
174	C₂×D₄	—	2	2	no	—
175	C₂×D₄	—	2	2	no	—
176	C₂×D₄	—	4	2	no	—
177	C₂²×C₄	—	2	2	no	—
178	C₂²×C₄	—	4	2	no	—
179	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
180	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
181	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
182	D₈	—	4	4	no	—
183	M₁₆	—	4	4	no	—
184	C₂×C₈	—	2	2	no	—
185	C₂×C₈	—	4	2	no	—
186	C₂×C₈	—	4	2	no	—
187	C₂×C₈	—	4	2	no	—
188	C₂×C₈	C₂²⋊C₄×C₃₁	1	2	yes	—
189	C₂×C₈	C₂²⋊C₄×C₃₁	1	2	yes	—
190	C₄⋊C₄	—	2	2	no	—
191	C₄⋊C₄	—	2	2	no	—
192	C₂²⋊C₄	—	4	2	no	—
193	C₂²⋊C₄	—	4	2	no	—
194	C₄²	—	2	4	no	—
195	C₄²	—	2	4	no	—
196	C₄²	—	4	2	no	—
197	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
198	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
199	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
200	C₄²	#41×C₃₁	1	1	yes	characteristic
201	C₂³	—	4	2	no	—
202	C₂³	C₃₁×#121	1	1	yes	characteristic
203	Q₈	—	4	4	no	—
204	D₄	—	4	4	no	—
205	D₄	—	4	4	no	—
206	D₄	—	4	4	no	—
207	D₄	—	8	4	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	4	no	—
211	C₂×C₄	—	4	2	no	—
212	C₂×C₄	—	4	2	no	—
213	C₂×C₄	—	4	2	no	—
214	C₂×C₄	—	8	2	no	—
215	C₂×C₄	—	8	2	no	—
216	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
217	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
218	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
219	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
220	C₂×C₄	C₃₁×#124	1	1	yes	characteristic
221	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
222	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	4	yes	—
223	C₈	—	2	2	no	—
224	C₈	—	2	2	no	—
225	C₈	—	4	2	no	—
226	C₈	—	8	2	no	—
227	C₂²	—	2	1	no	—
228	C₂²	—	2	1	no	—
229	C₂²	—	8	2	no	—
230	C₂²	—	8	2	no	—
231	C₂²	—	8	2	no	—
232	C₂²	C₃₁×#390	1	1	yes	characteristic
233	C₂²	C₃₁×#469	1	2	yes	—
234	C₄	—	2	1	no	—
235	C₄	—	2	1	no	—
236	C₄	—	4	4	no	—
237	C₄	—	8	2	no	—
238	C₄	C₃₁×#421	1	2	yes	—
239	C₄	C₃₁×#442	1	2	yes	—
240	C₄	C₃₁×#443	1	2	yes	—
241	C₄	C₃₁×#496	1	2	yes	—
242	C₂	—	2	2	no	—
243	C₂	—	16	2	no	—
244	C₂	C₃₁×#1963	1	1	yes	characteristic
245	C₂	C₃₁×#1972	1	1	yes	characteristic
246	C₂	C₃₁×#1974	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₁	#127
commutator	C₄²	C₂²×C₄×C₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
socle	C₂²×C₃₁	#390
prime radical	C₃₁×#3169	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#1962	C₂	semidirect
3. C₃₁×#1744	C₂	semidirect
4. C₃₁×#1566	C₂	semidirect
5. C₃₁×#1492	C₂	semidirect
7. C₃₁×#500	C₄	semidirect
8. C₃₁×#493	C₄	semidirect
9. C₃₁×#486	C₄	semidirect
77. #9852	C₃₁	direct (N × H)
105. #1962	C₂×C₃₁	semidirect
106. #1744	C₂×C₃₁	semidirect
107. #1566	C₂×C₃₁	semidirect
108. #1492	C₂×C₃₁	semidirect
125. #500	C₄×C₃₁	semidirect
126. #493	C₄×C₃₁	semidirect
127. #486	C₄×C₃₁	semidirect
169. C₃₁	#9852	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×#3169	2
C₃₁×#1962	2
C₃₁×#1744	2
C₃₁×#1566	2
C₃₁×#1492	2
#9852	31

Subgroups — C31×#9852 #21466611

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#9852 #21466611