Subgroups — C₃₁×#9841 #21466600

Order 7936.

510 conjugacy classes of subgroups.

The subgroup lattice has 206 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#498	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1990	C₂	1	2	yes	maximal
3	C₃₁×#1956	C₂	1	2	yes	maximal
4	C₃₁×#1845	C₂	1	1	yes	maximal, characteristic
5	C₂×C₃₁×#322	C₂	1	1	yes	maximal, characteristic
6	C₂×C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
7	C₃₁×#498	—	2	2	no	—
8	C₃₁×#498	C₄	1	4	yes	complemented
9	C₂×C₃₁×#107	—	2	2	no	—
10	C₂×C₃₁×#106	—	2	2	no	—
11	C₂×C₃₁×#103	—	2	4	no	—
12	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
14	C₃₁×#384	C₂²	1	1	yes	characteristic
15	C₃₁×#384	C₂²	1	1	yes	characteristic
16	C₃₁×#340	—	2	4	no	—
17	C₃₁×#322	—	2	2	no	—
18	C₂²×Q₈×C₃₁	—	2	2	no	—
19	C₂²×C₈×C₃₁	D₄	1	4	yes	—
20	C₂²×C₈×C₃₁	D₄	1	2	yes	—
21	C₃₁×C₄⋊Q₈	—	2	2	no	—
22	C₃₁×C₄⋊Q₈	—	2	2	no	—
23	C₃₁×C₄⋊Q₈	—	2	2	no	—
24	C₃₁×C₄⋊Q₈	C₂×C₄	1	4	yes	—
25	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
26	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
27	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
28	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
29	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
30	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
31	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
32	C₃₁×#107	—	4	4	no	—
33	C₃₁×#106	—	2	8	no	—
34	C₃₁×#103	—	4	8	no	—
35	C₄×C₈×C₃₁	—	2	2	no	—
36	C₄×C₈×C₃₁	—	2	2	no	—
37	C₄×C₈×C₃₁	C₂×C₄	1	2	yes	—
38	C₄×C₈×C₃₁	D₄	1	2	yes	—
39	C₂×Q₈×C₃₁	—	2	4	no	—
40	C₂×Q₈×C₃₁	—	4	2	no	—
41	C₂×Q₈×C₃₁	—	4	4	no	—
42	C₂×Q₈×C₃₁	—	4	4	no	—
43	C₂²×C₄×C₃₁	—	4	2	no	—
44	C₂²×C₄×C₃₁	—	4	1	no	—
45	C₂²×C₄×C₃₁	—	4	1	no	—
46	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
48	C₂×C₈×C₃₁	—	2	4	no	—
49	C₂×C₈×C₃₁	—	2	4	no	—
50	C₂×C₈×C₃₁	—	2	4	no	—
51	C₂×C₈×C₃₁	C₂²⋊C₄	1	4	yes	—
52	C₂×C₈×C₃₁	SD₁₆	1	4	yes	—
53	C₂×C₈×C₃₁	Q₁₆	1	4	yes	—
54	C₄⋊C₄×C₃₁	—	2	4	no	—
55	C₄⋊C₄×C₃₁	—	2	4	no	—
56	C₄⋊C₄×C₃₁	—	4	2	no	—
57	C₄⋊C₄×C₃₁	—	4	2	no	—
58	C₄⋊C₄×C₃₁	—	4	2	no	—
59	C₄⋊C₄×C₃₁	—	4	2	no	—
60	C₄⋊C₄×C₃₁	—	4	4	no	—
61	C₄⋊C₄×C₃₁	—	4	4	no	—
62	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
63	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
64	C₄²×C₃₁	#41	1	2	yes	—
65	#9841	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
66	C₂³×C₃₁	#127	1	1	yes	characteristic
67	Q₈×C₃₁	—	4	8	no	—
68	Q₈×C₃₁	—	8	4	no	—
69	C₂×C₄×C₃₁	—	4	4	no	—
70	C₂×C₄×C₃₁	—	8	2	no	—
71	C₂×C₄×C₃₁	—	8	2	no	—
72	C₂×C₄×C₃₁	—	8	1	no	—
73	C₂×C₄×C₃₁	—	8	1	no	—
74	C₂×C₄×C₃₁	—	8	2	no	—
75	C₂×C₄×C₃₁	—	8	2	no	—
76	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
78	C₂×C₄×C₃₁	#121	1	2	yes	—
79	C₂×C₄×C₃₁	#124	1	1	yes	characteristic
80	C₂×C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
81	C₂×C₄×C₃₁	C₂×Q₁₆	1	2	yes	—
82	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	4	yes	—
83	C₂×C₄×C₃₁	#137	1	4	yes	—
84	C₈×C₃₁	—	4	4	no	—
85	C₈×C₃₁	#103	1	8	yes	complemented
86	C₂×#498	C₂×C₃₁	1	1	yes	characteristic
87	#1990	C₂×C₃₁	1	2	yes	—
88	#1956	C₂×C₃₁	1	2	yes	—
89	#1845	C₂×C₃₁	1	1	yes	characteristic
90	C₂×#322	C₂×C₃₁	1	1	yes	characteristic
91	C₂²×C₃₁	#390	1	1	yes	characteristic
92	C₂²×C₃₁	#492	1	1	yes	characteristic
93	C₂²×C₃₁	#494	1	1	yes	characteristic
94	C₂²×C₃₁	#496	1	2	yes	—
95	C₂²×C₃₁	#497	1	2	yes	—
96	C₄×C₃₁	—	8	4	no	—
97	C₄×C₃₁	—	16	1	no	—
98	C₄×C₃₁	—	16	1	no	—
99	C₄×C₃₁	C₂×#103	1	2	yes	—
100	C₄×C₃₁	#440	1	4	yes	—
101	C₄×C₃₁	#468	1	4	yes	—
102	C₄×C₃₁	#487	1	2	yes	—
103	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
104	#498	—	2	2	no	—
105	#498	C₄×C₃₁	1	4	yes	complemented
106	C₂×#107	—	2	2	no	—
107	C₂×#106	—	2	2	no	—
108	C₂×#103	—	2	4	no	—
109	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
110	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
111	#384	C₂²×C₃₁	1	1	yes	characteristic
112	#384	C₂²×C₃₁	1	1	yes	characteristic
113	#340	—	2	4	no	—
114	#322	—	2	2	no	—
115	C₂×C₃₁	#1715	1	1	yes	characteristic
116	C₂×C₃₁	#1716	1	1	yes	characteristic
117	C₂×C₃₁	#1723	1	2	yes	—
118	C₂×C₃₁	#1956	1	1	yes	characteristic
119	C₂×C₃₁	#1973	1	2	yes	—
120	C₂²×Q₈	—	2	2	no	—
121	C₂²×C₈	D₄×C₃₁	1	4	yes	—
122	C₂²×C₈	D₄×C₃₁	1	2	yes	—
123	C₄⋊Q₈	—	2	2	no	—
124	C₄⋊Q₈	—	2	2	no	—
125	C₄⋊Q₈	—	2	2	no	—
126	C₄⋊Q₈	C₂×C₄×C₃₁	1	4	yes	—
127	C₂×C₄⋊C₄	—	2	2	no	—
128	C₂×C₄⋊C₄	—	2	2	no	—
129	C₂×C₄⋊C₄	—	2	2	no	—
130	C₂×C₄⋊C₄	—	2	2	no	—
131	C₂×C₄⋊C₄	—	2	1	no	—
132	C₂×C₄⋊C₄	—	2	1	no	—
133	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
134	#107	—	4	4	no	—
135	#106	—	2	8	no	—
136	#103	—	4	8	no	—
137	C₄×C₈	—	2	2	no	—
138	C₄×C₈	—	2	2	no	—
139	C₄×C₈	C₂×C₄×C₃₁	1	2	yes	—
140	C₄×C₈	D₄×C₃₁	1	2	yes	—
141	C₃₁	#9841	1	1	yes	characteristic, direct product factor, Sylow
142	C₂×Q₈	—	2	4	no	—
143	C₂×Q₈	—	4	2	no	—
144	C₂×Q₈	—	4	4	no	—
145	C₂×Q₈	—	4	4	no	—
146	C₂²×C₄	—	4	2	no	—
147	C₂²×C₄	—	4	1	no	—
148	C₂²×C₄	—	4	1	no	—
149	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
150	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
151	C₂×C₈	—	2	4	no	—
152	C₂×C₈	—	2	4	no	—
153	C₂×C₈	—	2	4	no	—
154	C₂×C₈	C₂²⋊C₄×C₃₁	1	4	yes	—
155	C₂×C₈	SD₁₆×C₃₁	1	4	yes	—
156	C₂×C₈	Q₁₆×C₃₁	1	4	yes	—
157	C₄⋊C₄	—	2	4	no	—
158	C₄⋊C₄	—	2	4	no	—
159	C₄⋊C₄	—	4	2	no	—
160	C₄⋊C₄	—	4	2	no	—
161	C₄⋊C₄	—	4	2	no	—
162	C₄⋊C₄	—	4	2	no	—
163	C₄⋊C₄	—	4	4	no	—
164	C₄⋊C₄	—	4	4	no	—
165	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
166	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
167	C₄²	#41×C₃₁	1	2	yes	—
168	C₂³	C₃₁×#127	1	1	yes	characteristic
169	Q₈	—	4	8	no	—
170	Q₈	—	8	4	no	—
171	C₂×C₄	—	4	4	no	—
172	C₂×C₄	—	8	2	no	—
173	C₂×C₄	—	8	2	no	—
174	C₂×C₄	—	8	1	no	—
175	C₂×C₄	—	8	1	no	—
176	C₂×C₄	—	8	2	no	—
177	C₂×C₄	—	8	2	no	—
178	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
179	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
180	C₂×C₄	C₃₁×#121	1	2	yes	—
181	C₂×C₄	C₃₁×#124	1	1	yes	characteristic
182	C₂×C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
183	C₂×C₄	C₂×Q₁₆×C₃₁	1	2	yes	—
184	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	4	yes	—
185	C₂×C₄	C₃₁×#137	1	4	yes	—
186	C₈	—	4	4	no	—
187	C₈	C₃₁×#103	1	8	yes	complemented
188	C₂²	C₃₁×#390	1	1	yes	characteristic
189	C₂²	C₃₁×#492	1	1	yes	characteristic
190	C₂²	C₃₁×#494	1	1	yes	characteristic
191	C₂²	C₃₁×#496	1	2	yes	—
192	C₂²	C₃₁×#497	1	2	yes	—
193	C₄	—	8	4	no	—
194	C₄	—	16	1	no	—
195	C₄	—	16	1	no	—
196	C₄	C₂×C₃₁×#103	1	2	yes	—
197	C₄	C₃₁×#440	1	4	yes	—
198	C₄	C₃₁×#468	1	4	yes	—
199	C₄	C₃₁×#487	1	2	yes	—
200	C₂	C₃₁×#1715	1	1	yes	characteristic
201	C₂	C₃₁×#1716	1	1	yes	characteristic
202	C₂	C₃₁×#1723	1	2	yes	—
203	C₂	C₃₁×#1956	1	1	yes	characteristic
204	C₂	C₃₁×#1973	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
8. C₃₁×#498	C₄	semidirect
65. #9841	C₃₁	direct (N × H)
85. C₈×C₃₁	#103	semidirect
105. #498	C₄×C₃₁	semidirect
141. C₃₁	#9841	direct (N × H)
187. C₈	C₃₁×#103	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#498	2
C₃₁×#1990	2
C₃₁×#1956	2
C₃₁×#1845	2
C₂×C₃₁×#322	2
#9841	31

Subgroups — C31×#9841 #21466600

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#9841 #21466600