Subgroups — C₃₁×#9597 #21466356

Order 7936.

378 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#1939	C₂	1	2	yes	maximal
2	C₄×C₃₁×#106	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₁×#334	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃₁×#322	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#1279	C₂	1	2	yes	maximal
6	C₂×C₃₁×#106	C₄	1	2	yes	—
7	C₂×C₃₁×C₄⋊C₈	—	2	2	no	—
8	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
9	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
12	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
13	C₃₁×#340	C₄	1	1	yes	characteristic
14	C₃₁×#340	C₄	1	1	yes	characteristic
15	C₃₁×#336	—	2	2	no	—
16	C₃₁×#334	—	2	2	no	—
17	C₃₁×#322	—	2	2	no	—
18	C₂²×C₈×C₃₁	—	2	1	no	—
19	C₂²×C₈×C₃₁	—	2	2	no	—
20	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₃₁	D₄	1	2	yes	—
23	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
24	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
25	C₂×C₄²×C₃₁	—	2	2	no	—
26	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
27	C₃₁×#106	—	2	4	no	—
28	C₃₁×C₄⋊C₈	—	2	2	no	—
29	C₃₁×C₄⋊C₈	—	2	2	no	—
30	C₃₁×C₄⋊C₈	—	4	4	no	—
31	C₄×C₈×C₃₁	—	2	1	no	—
32	C₄×C₈×C₃₁	—	2	1	no	—
33	C₄×C₈×C₃₁	—	2	1	no	—
34	C₄×C₈×C₃₁	D₄	1	2	yes	—
35	C₄×C₈×C₃₁	Q₈	1	2	yes	—
36	C₄×C₈×C₃₁	Q₈	1	2	yes	—
37	C₃₁×#95	—	2	2	no	—
38	C₂²×C₄×C₃₁	—	2	2	no	—
39	C₂²×C₄×C₃₁	—	2	2	no	—
40	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
43	C₂×C₈×C₃₁	—	2	2	no	—
44	C₂×C₈×C₃₁	—	2	2	no	—
45	C₂×C₈×C₃₁	—	2	2	no	—
46	C₂×C₈×C₃₁	—	2	2	no	—
47	C₂×C₈×C₃₁	—	2	2	no	—
48	C₂×C₈×C₃₁	—	4	1	no	—
49	C₂×C₈×C₃₁	—	4	1	no	—
50	C₂×C₈×C₃₁	—	4	2	no	—
51	C₂×C₈×C₃₁	—	4	2	no	—
52	C₂×C₈×C₃₁	—	4	2	no	—
53	C₂×C₈×C₃₁	C₄⋊C₄	1	2	yes	—
54	C₂×C₈×C₃₁	C₄⋊C₄	1	2	yes	—
55	C₂×C₈×C₃₁	SD₁₆	1	4	yes	—
56	C₄⋊C₄×C₃₁	—	2	2	no	—
57	C₄⋊C₄×C₃₁	—	2	2	no	—
58	C₄⋊C₄×C₃₁	—	2	2	no	—
59	C₄⋊C₄×C₃₁	M₁₆	1	4	yes	—
60	C₄²×C₃₁	—	4	2	no	—
61	C₄²×C₃₁	—	4	2	no	—
62	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
63	C₄²×C₃₁	C₂×Q₈	1	1	yes	characteristic
64	C₄²×C₃₁	C₂×Q₈	1	1	yes	characteristic
65	C₄²×C₃₁	#41	1	1	yes	characteristic
66	#9597	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
67	C₂³×C₃₁	C₄×D₄	1	1	yes	characteristic
68	C₂×C₄×C₃₁	—	2	1	no	—
69	C₂×C₄×C₃₁	—	2	1	no	—
70	C₂×C₄×C₃₁	—	2	4	no	—
71	C₂×C₄×C₃₁	—	4	2	no	—
72	C₂×C₄×C₃₁	—	4	2	no	—
73	C₂×C₄×C₃₁	—	4	2	no	—
74	C₂×C₄×C₃₁	—	4	2	no	—
75	C₂×C₄×C₃₁	—	4	2	no	—
76	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₃₁	C₄×Q₈	1	1	yes	characteristic
78	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
79	C₂×C₄×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
80	C₂×C₄×C₃₁	#125	1	1	yes	characteristic
81	C₂×C₄×C₃₁	C₄⋊Q₈	1	1	yes	characteristic
82	C₂×C₄×C₃₁	C₂×M₁₆	1	2	yes	—
83	C₂×C₄×C₃₁	#131	1	2	yes	—
84	C₂×C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
85	C₂×C₄×C₃₁	#135	1	2	yes	—
86	C₈×C₃₁	—	2	2	no	—
87	C₈×C₃₁	—	2	2	no	—
88	C₈×C₃₁	—	2	4	no	—
89	C₈×C₃₁	—	4	2	no	—
90	C₈×C₃₁	—	8	2	no	—
91	#1939	C₂×C₃₁	1	2	yes	—
92	C₄×#106	C₂×C₃₁	1	1	yes	characteristic
93	C₂×#334	C₂×C₃₁	1	1	yes	characteristic
94	C₂×#322	C₂×C₃₁	1	1	yes	characteristic
95	#1279	C₂×C₃₁	1	2	yes	—
96	C₂²×C₃₁	#389	1	1	yes	characteristic
97	C₂²×C₃₁	#435	1	2	yes	—
98	C₂²×C₃₁	C₄×SD₁₆	1	2	yes	—
99	C₂²×C₃₁	#444	1	1	yes	characteristic
100	C₂²×C₃₁	#444	1	1	yes	characteristic
101	C₄×C₃₁	—	2	1	no	—
102	C₄×C₃₁	—	2	1	no	—
103	C₄×C₃₁	—	4	4	no	—
104	C₄×C₃₁	—	8	2	no	—
105	C₄×C₃₁	#423	1	1	yes	characteristic
106	C₄×C₃₁	#424	1	1	yes	characteristic
107	C₄×C₃₁	#446	1	2	yes	—
108	C₄×C₃₁	#465	1	2	yes	—
109	C₄×C₃₁	#498	1	1	yes	characteristic
110	C₄×C₃₁	#499	1	1	yes	characteristic
111	C₂×#106	C₄×C₃₁	1	2	yes	—
112	C₂×C₄⋊C₈	—	2	2	no	—
113	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
114	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
115	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
116	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
117	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
118	#340	C₄×C₃₁	1	1	yes	characteristic
119	#340	C₄×C₃₁	1	1	yes	characteristic
120	#336	—	2	2	no	—
121	#334	—	2	2	no	—
122	#322	—	2	2	no	—
123	C₂×C₃₁	#1582	1	2	yes	—
124	C₂×C₃₁	#1583	1	2	yes	—
125	C₂×C₃₁	#1940	1	1	yes	characteristic
126	C₂×C₃₁	#1941	1	1	yes	characteristic
127	C₂×C₃₁	#1946	1	1	yes	characteristic
128	C₂²×C₈	—	2	1	no	—
129	C₂²×C₈	—	2	2	no	—
130	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
131	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
132	C₂²×C₈	D₄×C₃₁	1	2	yes	—
133	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
134	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
135	C₂×C₄²	—	2	2	no	—
136	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
137	#106	—	2	4	no	—
138	C₄⋊C₈	—	2	2	no	—
139	C₄⋊C₈	—	2	2	no	—
140	C₄⋊C₈	—	4	4	no	—
141	C₄×C₈	—	2	1	no	—
142	C₄×C₈	—	2	1	no	—
143	C₄×C₈	—	2	1	no	—
144	C₄×C₈	D₄×C₃₁	1	2	yes	—
145	C₄×C₈	Q₈×C₃₁	1	2	yes	—
146	C₄×C₈	Q₈×C₃₁	1	2	yes	—
147	#95	—	2	2	no	—
148	C₃₁	#9597	1	1	yes	characteristic, direct product factor, Sylow
149	C₂²×C₄	—	2	2	no	—
150	C₂²×C₄	—	2	2	no	—
151	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
152	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
153	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
154	C₂×C₈	—	2	2	no	—
155	C₂×C₈	—	2	2	no	—
156	C₂×C₈	—	2	2	no	—
157	C₂×C₈	—	2	2	no	—
158	C₂×C₈	—	2	2	no	—
159	C₂×C₈	—	4	1	no	—
160	C₂×C₈	—	4	1	no	—
161	C₂×C₈	—	4	2	no	—
162	C₂×C₈	—	4	2	no	—
163	C₂×C₈	—	4	2	no	—
164	C₂×C₈	C₄⋊C₄×C₃₁	1	2	yes	—
165	C₂×C₈	C₄⋊C₄×C₃₁	1	2	yes	—
166	C₂×C₈	SD₁₆×C₃₁	1	4	yes	—
167	C₄⋊C₄	—	2	2	no	—
168	C₄⋊C₄	—	2	2	no	—
169	C₄⋊C₄	—	2	2	no	—
170	C₄⋊C₄	M₁₆×C₃₁	1	4	yes	—
171	C₄²	—	4	2	no	—
172	C₄²	—	4	2	no	—
173	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
174	C₄²	C₂×Q₈×C₃₁	1	1	yes	characteristic
175	C₄²	C₂×Q₈×C₃₁	1	1	yes	characteristic
176	C₄²	#41×C₃₁	1	1	yes	characteristic
177	C₂³	C₄×D₄×C₃₁	1	1	yes	characteristic
178	C₂×C₄	—	2	1	no	—
179	C₂×C₄	—	2	1	no	—
180	C₂×C₄	—	2	4	no	—
181	C₂×C₄	—	4	2	no	—
182	C₂×C₄	—	4	2	no	—
183	C₂×C₄	—	4	2	no	—
184	C₂×C₄	—	4	2	no	—
185	C₂×C₄	—	4	2	no	—
186	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
187	C₂×C₄	C₄×Q₈×C₃₁	1	1	yes	characteristic
188	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
189	C₂×C₄	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
190	C₂×C₄	C₃₁×#125	1	1	yes	characteristic
191	C₂×C₄	C₃₁×C₄⋊Q₈	1	1	yes	characteristic
192	C₂×C₄	C₂×M₁₆×C₃₁	1	2	yes	—
193	C₂×C₄	C₃₁×#131	1	2	yes	—
194	C₂×C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
195	C₂×C₄	C₃₁×#135	1	2	yes	—
196	C₈	—	2	2	no	—
197	C₈	—	2	2	no	—
198	C₈	—	2	4	no	—
199	C₈	—	4	2	no	—
200	C₈	—	8	2	no	—
201	C₂²	C₃₁×#389	1	1	yes	characteristic
202	C₂²	C₃₁×#435	1	2	yes	—
203	C₂²	C₄×SD₁₆×C₃₁	1	2	yes	—
204	C₂²	C₃₁×#444	1	1	yes	characteristic
205	C₂²	C₃₁×#444	1	1	yes	characteristic
206	C₄	—	2	1	no	—
207	C₄	—	2	1	no	—
208	C₄	—	4	4	no	—
209	C₄	—	8	2	no	—
210	C₄	C₃₁×#423	1	1	yes	characteristic
211	C₄	C₃₁×#424	1	1	yes	characteristic
212	C₄	C₃₁×#446	1	2	yes	—
213	C₄	C₃₁×#465	1	2	yes	—
214	C₄	C₃₁×#498	1	1	yes	characteristic
215	C₄	C₃₁×#499	1	1	yes	characteristic
216	C₂	C₃₁×#1582	1	2	yes	—
217	C₂	C₃₁×#1583	1	2	yes	—
218	C₂	C₃₁×#1940	1	1	yes	characteristic
219	C₂	C₃₁×#1941	1	1	yes	characteristic
220	C₂	C₃₁×#1946	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₃₁	C₂×D₄
commutator	C₂²×C₄	C₂²×C₄×C₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
socle, prime radical	C₂³×C₃₁	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₃₁ ◁ C₂×C₄×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
66. #9597	C₃₁	direct (N × H)
148. C₃₁	#9597	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×#1939	2
C₄×C₃₁×#106	2
C₂×C₃₁×#334	2
C₂×C₃₁×#322	2
C₃₁×#1279	2
#9597	31

Subgroups — C31×#9597 #21466356

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#9597 #21466356