Subgroups — C₃₁×#9848 #21466607

Order 7936.

422 conjugacy classes of subgroups.

The subgroup lattice has 232 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	1	yes	maximal, characteristic
2	C₃₁×#1994	C₂	1	2	yes	maximal
3	C₃₁×#1744	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1567	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#1493	C₂	1	2	yes	maximal
6	C₃₁×#533	C₂²	1	2	yes	—
7	C₃₁×#500	C₄	1	1	yes	characteristic, complemented
8	C₃₁×#494	C₄	1	1	yes	characteristic, complemented
9	C₃₁×#487	C₄	1	2	yes	complemented
10	C₃₁×#467	—	2	2	no	—
11	C₄×Q₁₆×C₃₁	—	2	2	no	—
12	C₃₁×#423	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
15	C₄³×C₃₁	C₂²	1	1	yes	characteristic
16	C₃₁×#339	—	2	4	no	—
17	C₃₁×#336	—	2	2	no	—
18	C₃₁×#335	—	2	2	no	—
19	C₃₁×Q₈⋊C₈	—	2	4	no	—
20	C₂×Q₁₆×C₃₁	—	2	2	no	—
21	C₂×M₁₆×C₃₁	—	2	2	no	—
22	C₂²×C₈×C₃₁	—	2	1	no	—
23	C₂²×C₈×C₃₁	D₄	1	2	yes	—
24	C₃₁×C₄⋊Q₈	C₂×C₄	1	2	yes	—
25	C₃₁×C₄⋊Q₈	C₂×C₄	1	2	yes	—
26	C₃₁×#125	—	2	2	no	—
27	C₃₁×C₂²⋊Q₈	—	2	2	no	—
28	C₃₁×C₂²⋊Q₈	—	2	2	no	—
29	C₄×Q₈×C₃₁	—	2	2	no	—
30	C₄×Q₈×C₃₁	—	2	2	no	—
31	C₃₁×#117	—	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×#103	—	2	4	no	—
40	C₄×C₈×C₃₁	—	2	1	no	—
41	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
42	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈×C₃₁	D₄	1	2	yes	—
44	C₄×C₈×C₃₁	D₄	1	2	yes	—
45	C₂×Q₈×C₃₁	—	2	2	no	—
46	C₂×Q₈×C₃₁	—	2	2	no	—
47	C₂²×C₄×C₃₁	—	2	2	no	—
48	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
51	Q₁₆×C₃₁	—	4	4	no	—
52	M₁₆×C₃₁	—	4	4	no	—
53	C₂×C₈×C₃₁	—	2	2	no	—
54	C₂×C₈×C₃₁	—	4	2	no	—
55	C₂×C₈×C₃₁	—	4	2	no	—
56	C₂×C₈×C₃₁	—	4	2	no	—
57	C₂×C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
58	C₂×C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
59	C₄⋊C₄×C₃₁	—	2	2	no	—
60	C₄⋊C₄×C₃₁	—	2	2	no	—
61	C₄⋊C₄×C₃₁	—	2	2	no	—
62	C₄⋊C₄×C₃₁	—	2	2	no	—
63	C₄⋊C₄×C₃₁	—	4	2	no	—
64	C₄⋊C₄×C₃₁	—	4	2	no	—
65	C₂²⋊C₄×C₃₁	—	4	2	no	—
66	C₄²×C₃₁	—	2	2	no	—
67	C₄²×C₃₁	—	4	2	no	—
68	C₄²×C₃₁	—	4	2	no	—
69	C₄²×C₃₁	—	4	2	no	—
70	C₄²×C₃₁	Q₁₆	1	4	yes	complemented
71	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
72	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
73	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
74	C₄²×C₃₁	#41	1	1	yes	characteristic
75	#9848	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
76	C₂³×C₃₁	#121	1	1	yes	characteristic
77	Q₈×C₃₁	—	4	4	no	—
78	Q₈×C₃₁	—	4	4	no	—
79	C₂×C₄×C₃₁	—	2	2	no	—
80	C₂×C₄×C₃₁	—	2	2	no	—
81	C₂×C₄×C₃₁	—	2	4	no	—
82	C₂×C₄×C₃₁	—	4	2	no	—
83	C₂×C₄×C₃₁	—	4	2	no	—
84	C₂×C₄×C₃₁	—	4	2	no	—
85	C₂×C₄×C₃₁	—	4	2	no	—
86	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
88	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
89	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
90	C₂×C₄×C₃₁	#124	1	1	yes	characteristic
91	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
92	C₂×C₄×C₃₁	C₂×Q₁₆	1	2	yes	—
93	C₂×C₄×C₃₁	#135	1	2	yes	—
94	C₈×C₃₁	—	2	2	no	—
95	C₈×C₃₁	—	2	2	no	—
96	C₈×C₃₁	—	4	2	no	—
97	C₈×C₃₁	—	8	2	no	—
98	#3170	C₂×C₃₁	1	1	yes	characteristic
99	#1994	C₂×C₃₁	1	2	yes	—
100	#1744	C₂×C₃₁	1	1	yes	characteristic
101	#1567	C₂×C₃₁	1	1	yes	characteristic
102	#1493	C₂×C₃₁	1	2	yes	—
103	C₂²×C₃₁	—	2	1	no	—
104	C₂²×C₃₁	—	2	1	no	—
105	C₂²×C₃₁	#390	1	1	yes	characteristic
106	C₂²×C₃₁	#467	1	2	yes	—
107	C₄×C₃₁	—	2	1	no	—
108	C₄×C₃₁	—	2	1	no	—
109	C₄×C₃₁	—	4	4	no	—
110	C₄×C₃₁	—	8	2	no	—
111	C₄×C₃₁	—	8	2	no	—
112	C₄×C₃₁	#421	1	2	yes	—
113	C₄×C₃₁	C₄×Q₁₆	1	2	yes	—
114	C₄×C₃₁	#444	1	2	yes	—
115	C₄×C₃₁	#494	1	1	yes	characteristic
116	C₄×C₃₁	#495	1	1	yes	characteristic
117	#533	C₂²×C₃₁	1	2	yes	—
118	#500	C₄×C₃₁	1	1	yes	characteristic, complemented
119	#494	C₄×C₃₁	1	1	yes	characteristic, complemented
120	#487	C₄×C₃₁	1	2	yes	complemented
121	#467	—	2	2	no	—
122	C₄×Q₁₆	—	2	2	no	—
123	#423	C₂²×C₃₁	1	2	yes	—
124	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
125	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
126	C₄³	C₂²×C₃₁	1	1	yes	characteristic
127	#339	—	2	4	no	—
128	#336	—	2	2	no	—
129	#335	—	2	2	no	—
130	Q₈⋊C₈	—	2	4	no	—
131	C₂×C₃₁	—	2	2	no	—
132	C₂×C₃₁	#1963	1	1	yes	characteristic
133	C₂×C₃₁	#1969	1	1	yes	characteristic
134	C₂×C₃₁	#1975	1	1	yes	characteristic
135	C₂×Q₁₆	—	2	2	no	—
136	C₂×M₁₆	—	2	2	no	—
137	C₂²×C₈	—	2	1	no	—
138	C₂²×C₈	D₄×C₃₁	1	2	yes	—
139	C₄⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
140	C₄⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
141	#125	—	2	2	no	—
142	C₂²⋊Q₈	—	2	2	no	—
143	C₂²⋊Q₈	—	2	2	no	—
144	C₄×Q₈	—	2	2	no	—
145	C₄×Q₈	—	2	2	no	—
146	#117	—	2	2	no	—
147	C₂×C₄²	—	2	1	no	—
148	C₂×C₄²	—	2	1	no	—
149	C₂×C₄²	D₄×C₃₁	1	2	yes	—
150	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
151	#107	—	2	2	no	—
152	C₄⋊C₈	—	2	2	no	—
153	C₄⋊C₈	—	2	2	no	—
154	#103	—	2	4	no	—
155	C₄×C₈	—	2	1	no	—
156	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
157	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
158	C₄×C₈	D₄×C₃₁	1	2	yes	—
159	C₄×C₈	D₄×C₃₁	1	2	yes	—
160	C₃₁	#9848	1	1	yes	characteristic, direct product factor, Sylow
161	C₂×Q₈	—	2	2	no	—
162	C₂×Q₈	—	2	2	no	—
163	C₂²×C₄	—	2	2	no	—
164	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
165	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
166	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
167	Q₁₆	—	4	4	no	—
168	M₁₆	—	4	4	no	—
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	—	4	2	no	—
171	C₂×C₈	—	4	2	no	—
172	C₂×C₈	—	4	2	no	—
173	C₂×C₈	C₂²⋊C₄×C₃₁	1	2	yes	—
174	C₂×C₈	C₂²⋊C₄×C₃₁	1	2	yes	—
175	C₄⋊C₄	—	2	2	no	—
176	C₄⋊C₄	—	2	2	no	—
177	C₄⋊C₄	—	2	2	no	—
178	C₄⋊C₄	—	2	2	no	—
179	C₄⋊C₄	—	4	2	no	—
180	C₄⋊C₄	—	4	2	no	—
181	C₂²⋊C₄	—	4	2	no	—
182	C₄²	—	2	2	no	—
183	C₄²	—	4	2	no	—
184	C₄²	—	4	2	no	—
185	C₄²	—	4	2	no	—
186	C₄²	Q₁₆×C₃₁	1	4	yes	complemented
187	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
188	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
189	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
190	C₄²	#41×C₃₁	1	1	yes	characteristic
191	C₂³	C₃₁×#121	1	1	yes	characteristic
192	Q₈	—	4	4	no	—
193	Q₈	—	4	4	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	4	no	—
197	C₂×C₄	—	4	2	no	—
198	C₂×C₄	—	4	2	no	—
199	C₂×C₄	—	4	2	no	—
200	C₂×C₄	—	4	2	no	—
201	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
202	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
203	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
204	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
205	C₂×C₄	C₃₁×#124	1	1	yes	characteristic
206	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
207	C₂×C₄	C₂×Q₁₆×C₃₁	1	2	yes	—
208	C₂×C₄	C₃₁×#135	1	2	yes	—
209	C₈	—	2	2	no	—
210	C₈	—	2	2	no	—
211	C₈	—	4	2	no	—
212	C₈	—	8	2	no	—
213	C₂²	—	2	1	no	—
214	C₂²	—	2	1	no	—
215	C₂²	C₃₁×#390	1	1	yes	characteristic
216	C₂²	C₃₁×#467	1	2	yes	—
217	C₄	—	2	1	no	—
218	C₄	—	2	1	no	—
219	C₄	—	4	4	no	—
220	C₄	—	8	2	no	—
221	C₄	—	8	2	no	—
222	C₄	C₃₁×#421	1	2	yes	—
223	C₄	C₄×Q₁₆×C₃₁	1	2	yes	—
224	C₄	C₃₁×#444	1	2	yes	—
225	C₄	C₃₁×#494	1	1	yes	characteristic
226	C₄	C₃₁×#495	1	1	yes	characteristic
227	C₂	—	2	2	no	—
228	C₂	C₃₁×#1963	1	1	yes	characteristic
229	C₂	C₃₁×#1969	1	1	yes	characteristic
230	C₂	C₃₁×#1975	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	C₂³×C₃₁	#121
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
7. C₃₁×#500	C₄	semidirect
8. C₃₁×#494	C₄	semidirect
9. C₃₁×#487	C₄	semidirect
70. C₄²×C₃₁	Q₁₆	semidirect
75. #9848	C₃₁	direct (N × H)
118. #500	C₄×C₃₁	semidirect
119. #494	C₄×C₃₁	semidirect
120. #487	C₄×C₃₁	semidirect
160. C₃₁	#9848	direct (N × H)
186. C₄²	Q₁₆×C₃₁	semidirect

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×#1994	2
C₃₁×#1744	2
C₃₁×#1567	2
C₃₁×#1493	2
#9848	31

Subgroups — C31×#9848 #21466607

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#9848 #21466607