Subgroups — C₃₁×#12834 #21469593

Order 7936.

438 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#366	C₂	1	2	yes	maximal, complemented
3	C₃₁×#1808	C₂	1	2	yes	maximal
4	C₃₁×#1380	C₂	1	2	yes	maximal, complemented
5	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
6	C₂²×C₁₆×C₃₁	—	2	4	no	—
7	C₂²×C₁₆×C₃₁	C₄	1	4	yes	complemented
8	C₂²×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
9	C₂²×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
10	C₂×C₃₁×#107	C₂²	1	4	yes	—
11	C₃₁×#380	—	2	2	no	—
12	C₃₁×#366	—	2	4	no	—
13	C₃₁×#340	—	2	2	no	—
14	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
15	C₂²×C₈×C₃₁	—	2	2	no	—
16	C₂²×C₈×C₃₁	—	2	2	no	—
17	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
18	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
19	C₂²×C₈×C₃₁	Q₈	1	2	yes	—
20	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
21	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
22	C₂×C₂²⋊C₄×C₃₁	—	4	2	no	—
23	C₂×C₁₆×C₃₁	—	2	2	no	—
24	C₂×C₁₆×C₃₁	—	2	8	no	—
25	C₂×C₁₆×C₃₁	C₂×C₄	1	2	yes	—
26	C₂×C₁₆×C₃₁	C₂×C₄	1	2	yes	—
27	C₂×C₁₆×C₃₁	D₄	1	4	yes	—
28	C₃₁×#107	—	2	4	no	—
29	C₃₁×#107	—	2	4	no	—
30	C₃₁×#95	—	4	2	no	—
31	C₂⁴×C₃₁	D₈	1	1	yes	characteristic
32	C₂²×C₄×C₃₁	—	2	2	no	—
33	C₂²×C₄×C₃₁	—	2	2	no	—
34	C₂²×C₄×C₃₁	—	4	4	no	—
35	C₂²×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
36	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
37	C₂²×C₄×C₃₁	Q₁₆	1	2	yes	—
38	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
39	C₂×C₈×C₃₁	—	2	2	no	—
40	C₂×C₈×C₃₁	—	2	2	no	—
41	C₂×C₈×C₃₁	—	2	2	no	—
42	C₂×C₈×C₃₁	—	2	4	no	—
43	C₂×C₈×C₃₁	C₄⋊C₄	1	2	yes	—
44	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
46	C₂×C₈×C₃₁	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₈×C₃₁	#41	1	2	yes	—
48	C₄⋊C₄×C₃₁	—	4	4	no	—
49	C₄⋊C₄×C₃₁	—	4	4	no	—
50	C₂²⋊C₄×C₃₁	—	8	4	no	—
51	C₁₆×C₃₁	—	2	2	no	—
52	C₁₆×C₃₁	C₂²⋊C₄	1	4	yes	complemented
53	#12834	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
54	C₂³×C₃₁	—	2	2	no	—
55	C₂³×C₃₁	—	2	2	no	—
56	C₂³×C₃₁	#107	1	2	yes	—
57	C₂³×C₃₁	D₁₆	1	2	yes	—
58	C₂³×C₃₁	Q₃₂	1	2	yes	—
59	C₂³×C₃₁	C₂×D₈	1	1	yes	characteristic
60	C₂×C₄×C₃₁	—	2	2	no	—
61	C₂×C₄×C₃₁	—	2	2	no	—
62	C₂×C₄×C₃₁	—	2	2	no	—
63	C₂×C₄×C₃₁	—	2	4	no	—
64	C₂×C₄×C₃₁	—	8	4	no	—
65	C₂×C₄×C₃₁	—	8	4	no	—
66	C₂×C₄×C₃₁	—	8	4	no	—
67	C₂×C₄×C₃₁	#107	1	2	yes	—
68	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
69	C₂×C₄×C₃₁	#121	1	2	yes	—
70	C₂×C₄×C₃₁	C₂×Q₁₆	1	1	yes	characteristic
71	C₂×C₄×C₃₁	#135	1	2	yes	—
72	C₈×C₃₁	—	2	2	no	—
73	C₈×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
74	C₈×C₃₁	#117	1	1	yes	characteristic
75	C₈×C₃₁	C₂²⋊Q₈	1	2	yes	—
76	C₂³×C₁₆	C₂×C₃₁	1	1	yes	characteristic
77	C₂×#366	C₂×C₃₁	1	2	yes	complemented
78	#1808	C₂×C₃₁	1	2	yes	—
79	#1380	C₂×C₃₁	1	2	yes	complemented
80	C₂²×C₃₁	—	2	2	no	—
81	C₂²×C₃₁	—	2	2	no	—
82	C₂²×C₃₁	—	2	4	no	—
83	C₂²×C₃₁	—	2	4	no	—
84	C₂²×C₃₁	#366	1	4	yes	complemented
85	C₂²×C₃₁	C₂×#107	1	1	yes	characteristic
86	C₂²×C₃₁	#466	1	2	yes	—
87	C₂²×C₃₁	C₂×D₁₆	1	1	yes	characteristic
88	C₂²×C₃₁	C₂×Q₃₂	1	1	yes	characteristic
89	C₂²×C₃₁	#508	1	2	yes	—
90	C₄×C₃₁	—	2	2	no	—
91	C₄×C₃₁	—	16	4	no	—
92	C₄×C₃₁	#380	1	1	yes	characteristic
93	C₄×C₃₁	#427	1	1	yes	characteristic
94	C₄×C₃₁	#467	1	2	yes	—
95	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
96	C₂²×C₁₆	—	2	4	no	—
97	C₂²×C₁₆	C₄×C₃₁	1	4	yes	complemented
98	C₂²×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
99	C₂²×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
100	C₂×#107	C₂²×C₃₁	1	4	yes	—
101	#380	—	2	2	no	—
102	#366	—	2	4	no	—
103	#340	—	2	2	no	—
104	C₂×C₃₁	—	2	4	no	—
105	C₂×C₃₁	#1808	1	1	yes	characteristic
106	C₂×C₃₁	C₂×#366	1	1	yes	characteristic
107	C₂×C₃₁	#2158	1	1	yes	characteristic
108	C₂×C₃₁	#2215	1	2	yes	—
109	C₂×C₃₁	#2216	1	2	yes	—
110	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
111	C₂²×C₈	—	2	2	no	—
112	C₂²×C₈	—	2	2	no	—
113	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
114	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
115	C₂²×C₈	Q₈×C₃₁	1	2	yes	—
116	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
117	C₂×C₄⋊C₄	—	2	4	no	—
118	C₂×C₂²⋊C₄	—	4	2	no	—
119	C₂×C₁₆	—	2	2	no	—
120	C₂×C₁₆	—	2	8	no	—
121	C₂×C₁₆	C₂×C₄×C₃₁	1	2	yes	—
122	C₂×C₁₆	C₂×C₄×C₃₁	1	2	yes	—
123	C₂×C₁₆	D₄×C₃₁	1	4	yes	—
124	#107	—	2	4	no	—
125	#107	—	2	4	no	—
126	#95	—	4	2	no	—
127	C₃₁	#12834	1	1	yes	characteristic, direct product factor, Sylow
128	C₂⁴	D₈×C₃₁	1	1	yes	characteristic
129	C₂²×C₄	—	2	2	no	—
130	C₂²×C₄	—	2	2	no	—
131	C₂²×C₄	—	4	4	no	—
132	C₂²×C₄	C₄⋊C₄×C₃₁	1	2	yes	—
133	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
134	C₂²×C₄	Q₁₆×C₃₁	1	2	yes	—
135	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
136	C₂×C₈	—	2	2	no	—
137	C₂×C₈	—	2	2	no	—
138	C₂×C₈	—	2	2	no	—
139	C₂×C₈	—	2	4	no	—
140	C₂×C₈	C₄⋊C₄×C₃₁	1	2	yes	—
141	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
142	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
143	C₂×C₈	C₂×Q₈×C₃₁	1	1	yes	characteristic
144	C₂×C₈	#41×C₃₁	1	2	yes	—
145	C₄⋊C₄	—	4	4	no	—
146	C₄⋊C₄	—	4	4	no	—
147	C₂²⋊C₄	—	8	4	no	—
148	C₁₆	—	2	2	no	—
149	C₁₆	C₂²⋊C₄×C₃₁	1	4	yes	complemented
150	C₂³	—	2	2	no	—
151	C₂³	—	2	2	no	—
152	C₂³	C₃₁×#107	1	2	yes	—
153	C₂³	C₃₁×D₁₆	1	2	yes	—
154	C₂³	C₃₁×Q₃₂	1	2	yes	—
155	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	4	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	8	4	no	—
161	C₂×C₄	—	8	4	no	—
162	C₂×C₄	—	8	4	no	—
163	C₂×C₄	C₃₁×#107	1	2	yes	—
164	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
165	C₂×C₄	C₃₁×#121	1	2	yes	—
166	C₂×C₄	C₂×Q₁₆×C₃₁	1	1	yes	characteristic
167	C₂×C₄	C₃₁×#135	1	2	yes	—
168	C₈	—	2	2	no	—
169	C₈	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
170	C₈	C₃₁×#117	1	1	yes	characteristic
171	C₈	C₃₁×C₂²⋊Q₈	1	2	yes	—
172	C₂²	—	2	2	no	—
173	C₂²	—	2	2	no	—
174	C₂²	—	2	4	no	—
175	C₂²	—	2	4	no	—
176	C₂²	C₃₁×#366	1	4	yes	complemented
177	C₂²	C₂×C₃₁×#107	1	1	yes	characteristic
178	C₂²	C₃₁×#466	1	2	yes	—
179	C₂²	C₂×C₃₁×D₁₆	1	1	yes	characteristic
180	C₂²	C₂×C₃₁×Q₃₂	1	1	yes	characteristic
181	C₂²	C₃₁×#508	1	2	yes	—
182	C₄	—	2	2	no	—
183	C₄	—	16	4	no	—
184	C₄	C₃₁×#380	1	1	yes	characteristic
185	C₄	C₃₁×#427	1	1	yes	characteristic
186	C₄	C₃₁×#467	1	2	yes	—
187	C₂	—	2	4	no	—
188	C₂	C₃₁×#1808	1	1	yes	characteristic
189	C₂	C₂×C₃₁×#366	1	1	yes	characteristic
190	C₂	C₃₁×#2158	1	1	yes	characteristic
191	C₂	C₃₁×#2215	1	2	yes	—
192	C₂	C₃₁×#2216	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₃₁
Frattini, cosocle	C₂²×C₈	C₂³×C₃₁
prime radical	C₂⁴×C₃₁	D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂³×C₄×C₃₁ ◁ C₂³×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃₁×#366	C₂	semidirect
4. C₃₁×#1380	C₂	semidirect
7. C₂²×C₁₆×C₃₁	C₄	semidirect
52. C₁₆×C₃₁	C₂²⋊C₄	semidirect
53. #12834	C₃₁	direct (N × H)
77. C₂×#366	C₂×C₃₁	semidirect
79. #1380	C₂×C₃₁	semidirect
84. C₂²×C₃₁	#366	semidirect
97. C₂²×C₁₆	C₄×C₃₁	semidirect
127. C₃₁	#12834	direct (N × H)
149. C₁₆	C₂²⋊C₄×C₃₁	semidirect
176. C₂²	C₃₁×#366	semidirect

Maximal subgroups

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

subgroup	index
C₂³×C₁₆×C₃₁	2
C₂×C₃₁×#366	2
C₃₁×#1808	2
C₃₁×#1380	2
#12834	31

Subgroups — C31×#12834 #21469593

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#12834 #21469593