Subgroups — C₃₁×#9624 #21466383

Order 7936.

298 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#1950	C₂	1	2	yes	maximal
2	C₃₁×#1567	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₁×#335	C₂	1	1	yes	maximal, characteristic
4	C₈×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#1301	C₂	1	2	yes	maximal
6	C₃₁×#431	C₂²	1	1	yes	characteristic
7	C₂×C₃₁×#108	—	2	2	no	—
8	C₃₁×#423	C₂²	1	2	yes	—
9	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
10	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
12	C₃₁×#341	—	2	2	no	—
13	C₃₁×#335	—	2	1	no	—
14	C₃₁×#335	—	2	1	no	—
15	C₃₁×#335	C₄	1	2	yes	—
16	C₃₁×#335	C₄	1	2	yes	—
17	C₃₁×#322	—	2	1	no	—
18	C₈²×C₃₁	—	2	1	no	—
19	C₂×M₁₆×C₃₁	—	2	2	no	—
20	C₂×M₁₆×C₃₁	D₄	1	1	yes	characteristic
21	C₂×M₁₆×C₃₁	D₄	1	1	yes	characteristic
22	C₂²×C₈×C₃₁	—	2	2	no	—
23	C₂²×C₈×C₃₁	Q₈	1	1	yes	characteristic
24	C₂²×C₈×C₃₁	Q₈	1	1	yes	characteristic
25	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
26	C₃₁×#108	—	4	4	no	—
27	C₃₁×C₄⋊C₈	—	2	2	no	—
28	C₃₁×C₄⋊C₈	—	2	2	no	—
29	C₃₁×C₄⋊C₈	—	2	1	no	—
30	C₃₁×C₄⋊C₈	—	2	2	no	—
31	C₃₁×C₄⋊C₈	C₂×C₄	1	4	yes	—
32	C₃₁×C₂²⋊C₈	Q₈	1	2	yes	—
33	C₃₁×#97	—	2	1	no	—
34	C₄×C₈×C₃₁	—	2	1	no	—
35	C₄×C₈×C₃₁	—	2	1	no	—
36	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
37	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
38	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
39	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
40	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
42	C₂²×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
43	M₁₆×C₃₁	—	2	1	no	—
44	M₁₆×C₃₁	—	2	1	no	—
45	M₁₆×C₃₁	—	4	2	no	—
46	M₁₆×C₃₁	—	4	2	no	—
47	M₁₆×C₃₁	D₈	1	2	yes	—
48	M₁₆×C₃₁	Q₁₆	1	2	yes	—
49	C₂×C₈×C₃₁	—	2	1	no	—
50	C₂×C₈×C₃₁	—	2	1	no	—
51	C₂×C₈×C₃₁	—	2	2	no	—
52	C₂×C₈×C₃₁	—	2	4	no	—
53	C₂×C₈×C₃₁	—	2	2	no	—
54	C₂×C₈×C₃₁	—	4	1	no	—
55	C₂×C₈×C₃₁	—	4	2	no	—
56	C₂×C₈×C₃₁	—	4	2	no	—
57	C₂×C₈×C₃₁	—	4	2	no	—
58	C₂×C₈×C₃₁	C₄⋊C₄	1	1	yes	characteristic
59	C₂×C₈×C₃₁	C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₈×C₃₁	C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₈×C₃₁	C₄⋊C₄	1	1	yes	characteristic
62	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
63	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
64	C₄²×C₃₁	#41	1	1	yes	characteristic
65	C₄²×C₃₁	#41	1	1	yes	characteristic
66	#9624	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
67	C₂³×C₃₁	C₄⋊Q₈	1	1	yes	characteristic
68	C₂×C₄×C₃₁	—	2	1	no	—
69	C₂×C₄×C₃₁	—	2	1	no	—
70	C₂×C₄×C₃₁	—	4	1	no	—
71	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
73	C₂×C₄×C₃₁	C₄×Q₈	1	1	yes	characteristic
74	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
75	C₂×C₄×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
76	C₂×C₄×C₃₁	#125	1	1	yes	characteristic
77	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
78	C₂×C₄×C₃₁	C₂×Q₁₆	1	1	yes	characteristic
79	C₂×C₄×C₃₁	#135	1	1	yes	characteristic
80	C₂×C₄×C₃₁	#135	1	1	yes	characteristic
81	C₈×C₃₁	—	2	1	no	—
82	C₈×C₃₁	—	2	1	no	—
83	C₈×C₃₁	—	2	1	no	—
84	C₈×C₃₁	—	2	2	no	—
85	C₈×C₃₁	—	2	2	no	—
86	C₈×C₃₁	—	4	2	no	—
87	C₈×C₃₁	—	4	4	no	—
88	C₈×C₃₁	—	8	2	no	—
89	C₈×C₃₁	#108	1	2	yes	complemented
90	#1950	C₂×C₃₁	1	2	yes	—
91	#1567	C₂×C₃₁	1	1	yes	characteristic
92	C₂×#335	C₂×C₃₁	1	1	yes	characteristic
93	C₈×M₁₆	C₂×C₃₁	1	1	yes	characteristic
94	#1301	C₂×C₃₁	1	2	yes	—
95	C₂²×C₃₁	—	2	1	no	—
96	C₂²×C₃₁	—	2	1	no	—
97	C₂²×C₃₁	#389	1	1	yes	characteristic
98	C₂²×C₃₁	#499	1	1	yes	characteristic
99	C₂²×C₃₁	#500	1	1	yes	characteristic
100	C₄×C₃₁	—	2	1	no	—
101	C₄×C₃₁	—	2	1	no	—
102	C₄×C₃₁	—	2	1	no	—
103	C₄×C₃₁	C₂×#108	1	1	yes	characteristic
104	C₄×C₃₁	#430	1	1	yes	characteristic
105	C₄×C₃₁	C₄×D₈	1	1	yes	characteristic
106	C₄×C₃₁	C₄×Q₁₆	1	1	yes	characteristic
107	C₄×C₃₁	#466	1	1	yes	characteristic
108	C₄×C₃₁	#467	1	1	yes	characteristic
109	#431	C₂²×C₃₁	1	1	yes	characteristic
110	C₂×#108	—	2	2	no	—
111	#423	C₂²×C₃₁	1	2	yes	—
112	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
113	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
114	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
115	#341	—	2	2	no	—
116	#335	—	2	1	no	—
117	#335	—	2	1	no	—
118	#335	C₄×C₃₁	1	2	yes	—
119	#335	C₄×C₃₁	1	2	yes	—
120	#322	—	2	1	no	—
121	C₈²	—	2	1	no	—
122	C₂×C₃₁	—	2	1	no	—
123	C₂×C₃₁	—	2	1	no	—
124	C₂×C₃₁	#1942	1	1	yes	characteristic
125	C₂×C₃₁	#1950	1	1	yes	characteristic
126	C₂×C₃₁	#1953	1	1	yes	characteristic
127	C₂×M₁₆	—	2	2	no	—
128	C₂×M₁₆	D₄×C₃₁	1	1	yes	characteristic
129	C₂×M₁₆	D₄×C₃₁	1	1	yes	characteristic
130	C₂²×C₈	—	2	2	no	—
131	C₂²×C₈	Q₈×C₃₁	1	1	yes	characteristic
132	C₂²×C₈	Q₈×C₃₁	1	1	yes	characteristic
133	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
134	#108	—	4	4	no	—
135	C₄⋊C₈	—	2	2	no	—
136	C₄⋊C₈	—	2	2	no	—
137	C₄⋊C₈	—	2	1	no	—
138	C₄⋊C₈	—	2	2	no	—
139	C₄⋊C₈	C₂×C₄×C₃₁	1	4	yes	—
140	C₂²⋊C₈	Q₈×C₃₁	1	2	yes	—
141	#97	—	2	1	no	—
142	C₄×C₈	—	2	1	no	—
143	C₄×C₈	—	2	1	no	—
144	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
145	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
146	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
147	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
148	C₃₁	#9624	1	1	yes	characteristic, direct product factor, Sylow
149	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
150	C₂²×C₄	C₂×Q₈×C₃₁	1	1	yes	characteristic
151	C₂²×C₄	C₂×Q₈×C₃₁	1	1	yes	characteristic
152	M₁₆	—	2	1	no	—
153	M₁₆	—	2	1	no	—
154	M₁₆	—	4	2	no	—
155	M₁₆	—	4	2	no	—
156	M₁₆	D₈×C₃₁	1	2	yes	—
157	M₁₆	Q₁₆×C₃₁	1	2	yes	—
158	C₂×C₈	—	2	1	no	—
159	C₂×C₈	—	2	1	no	—
160	C₂×C₈	—	2	2	no	—
161	C₂×C₈	—	2	4	no	—
162	C₂×C₈	—	2	2	no	—
163	C₂×C₈	—	4	1	no	—
164	C₂×C₈	—	4	2	no	—
165	C₂×C₈	—	4	2	no	—
166	C₂×C₈	—	4	2	no	—
167	C₂×C₈	C₄⋊C₄×C₃₁	1	1	yes	characteristic
168	C₂×C₈	C₄⋊C₄×C₃₁	1	1	yes	characteristic
169	C₂×C₈	C₄⋊C₄×C₃₁	1	1	yes	characteristic
170	C₂×C₈	C₄⋊C₄×C₃₁	1	1	yes	characteristic
171	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
172	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
173	C₄²	#41×C₃₁	1	1	yes	characteristic
174	C₄²	#41×C₃₁	1	1	yes	characteristic
175	C₂³	C₃₁×C₄⋊Q₈	1	1	yes	characteristic
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	2	1	no	—
178	C₂×C₄	—	4	1	no	—
179	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
180	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
181	C₂×C₄	C₄×Q₈×C₃₁	1	1	yes	characteristic
182	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
183	C₂×C₄	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
184	C₂×C₄	C₃₁×#125	1	1	yes	characteristic
185	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
186	C₂×C₄	C₂×Q₁₆×C₃₁	1	1	yes	characteristic
187	C₂×C₄	C₃₁×#135	1	1	yes	characteristic
188	C₂×C₄	C₃₁×#135	1	1	yes	characteristic
189	C₈	—	2	1	no	—
190	C₈	—	2	1	no	—
191	C₈	—	2	1	no	—
192	C₈	—	2	2	no	—
193	C₈	—	2	2	no	—
194	C₈	—	4	2	no	—
195	C₈	—	4	4	no	—
196	C₈	—	8	2	no	—
197	C₈	C₃₁×#108	1	2	yes	complemented
198	C₂²	—	2	1	no	—
199	C₂²	—	2	1	no	—
200	C₂²	C₃₁×#389	1	1	yes	characteristic
201	C₂²	C₃₁×#499	1	1	yes	characteristic
202	C₂²	C₃₁×#500	1	1	yes	characteristic
203	C₄	—	2	1	no	—
204	C₄	—	2	1	no	—
205	C₄	—	2	1	no	—
206	C₄	C₂×C₃₁×#108	1	1	yes	characteristic
207	C₄	C₃₁×#430	1	1	yes	characteristic
208	C₄	C₄×D₈×C₃₁	1	1	yes	characteristic
209	C₄	C₄×Q₁₆×C₃₁	1	1	yes	characteristic
210	C₄	C₃₁×#466	1	1	yes	characteristic
211	C₄	C₃₁×#467	1	1	yes	characteristic
212	C₂	—	2	1	no	—
213	C₂	—	2	1	no	—
214	C₂	C₃₁×#1942	1	1	yes	characteristic
215	C₂	C₃₁×#1950	1	1	yes	characteristic
216	C₂	C₃₁×#1953	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₁	#121
commutator	C₄²	C₂²×C₄×C₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
socle	C₂²×C₃₁	#389
prime radical	C₂³×C₃₁	C₄⋊Q₈
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
66. #9624	C₃₁	direct (N × H)
89. C₈×C₃₁	#108	semidirect
148. C₃₁	#9624	direct (N × H)
197. C₈	C₃₁×#108	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#1950	2
C₃₁×#1567	2
C₂×C₃₁×#335	2
C₈×M₁₆×C₃₁	2
C₃₁×#1301	2
#9624	31

Subgroups — C31×#9624 #21466383

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#9624 #21466383