Subgroups — C₃₁×#12024 #21468783

Order 7936.

226 conjugacy classes of subgroups.

The subgroup lattice has 166 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1568	C₂	1	2	yes	maximal
3	C₃₁×#1372	C₂	1	2	yes	maximal, complemented
4	C₃₁×#1371	C₂	1	2	yes	maximal, complemented
5	C₂²×C₁₆×C₃₁	C₄	1	1	yes	characteristic
6	C₂²×C₁₆×C₃₁	C₄	1	1	yes	characteristic
7	C₃₁×#432	—	2	2	no	—
8	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
9	C₄×C₁₆×C₃₁	C₄	1	1	yes	characteristic
10	C₄×C₁₆×C₃₁	C₄	1	1	yes	characteristic
11	C₄×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
12	C₄×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
13	C₃₁×#335	C₂²	1	4	yes	—
14	C₃₁×#334	—	2	2	no	—
15	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈×C₃₁	Q₈	1	1	yes	characteristic
17	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
18	C₂×C₁₆×C₃₁	—	2	2	no	—
19	C₂×C₁₆×C₃₁	—	2	2	no	—
20	C₂×C₁₆×C₃₁	C₂×C₄	1	1	yes	characteristic
21	C₂×C₁₆×C₃₁	C₂×C₄	1	1	yes	characteristic
22	C₂×C₁₆×C₃₁	C₂×C₄	1	1	yes	characteristic
23	C₂×C₁₆×C₃₁	C₂×C₄	1	1	yes	characteristic
24	C₃₁×C₄⋊C₈	—	2	4	no	—
25	C₃₁×C₂²⋊C₈	—	4	2	no	—
26	C₃₁×#97	—	4	2	no	—
27	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
29	C₄×C₈×C₃₁	Q₈	1	1	yes	characteristic
30	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
31	C₂²×C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
32	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
33	C₂²×C₄×C₃₁	Q₁₆	1	1	yes	characteristic
34	C₂×C₈×C₃₁	—	2	2	no	—
35	C₂×C₈×C₃₁	—	2	1	no	—
36	C₂×C₈×C₃₁	—	2	1	no	—
37	C₂×C₈×C₃₁	—	4	4	no	—
38	C₂×C₈×C₃₁	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₈×C₃₁	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₈×C₃₁	C₂×Q₈	1	1	yes	characteristic
42	C₄²×C₃₁	C₄⋊C₄	1	1	yes	characteristic
43	C₄²×C₃₁	D₈	1	1	yes	characteristic
44	C₄²×C₃₁	Q₁₆	1	1	yes	characteristic
45	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
46	C₁₆×C₃₁	—	2	1	no	—
47	C₁₆×C₃₁	—	2	1	no	—
48	C₁₆×C₃₁	M₁₆	1	2	yes	—
49	C₁₆×C₃₁	M₁₆	1	2	yes	—
50	#12024	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
51	C₂³×C₃₁	#107	1	1	yes	characteristic
52	C₂×C₄×C₃₁	—	2	2	no	—
53	C₂×C₄×C₃₁	—	2	1	no	—
54	C₂×C₄×C₃₁	—	2	1	no	—
55	C₂×C₄×C₃₁	—	2	2	no	—
56	C₂×C₄×C₃₁	#107	1	1	yes	characteristic
57	C₂×C₄×C₃₁	#107	1	1	yes	characteristic
58	C₂×C₄×C₃₁	#107	1	1	yes	characteristic
59	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
61	C₂×C₄×C₃₁	C₂×Q₁₆	1	1	yes	characteristic
62	C₈×C₃₁	—	2	1	no	—
63	C₈×C₃₁	—	2	1	no	—
64	C₈×C₃₁	—	8	4	no	—
65	C₈×C₃₁	#108	1	2	yes	—
66	C₈×C₃₁	C₂×M₁₆	1	1	yes	characteristic
67	C₈×C₃₁	C₂×M₁₆	1	1	yes	characteristic
68	C₂×C₄×C₁₆	C₂×C₃₁	1	1	yes	characteristic
69	#1568	C₂×C₃₁	1	2	yes	—
70	#1372	C₂×C₃₁	1	2	yes	complemented
71	#1371	C₂×C₃₁	1	2	yes	complemented
72	C₂²×C₃₁	—	2	1	no	—
73	C₂²×C₃₁	—	2	1	no	—
74	C₂²×C₃₁	#368	1	2	yes	—
75	C₂²×C₃₁	C₂×#107	1	1	yes	characteristic
76	C₄×C₃₁	—	2	1	no	—
77	C₄×C₃₁	—	2	1	no	—
78	C₄×C₃₁	—	2	1	no	—
79	C₄×C₃₁	—	2	1	no	—
80	C₄×C₃₁	#423	1	1	yes	characteristic
81	C₄×C₃₁	C₂×#108	1	1	yes	characteristic
82	C₄×C₃₁	#508	1	2	yes	—
83	C₂²×C₁₆	C₄×C₃₁	1	1	yes	characteristic
84	C₂²×C₁₆	C₄×C₃₁	1	1	yes	characteristic
85	#432	—	2	2	no	—
86	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
87	C₄×C₁₆	C₄×C₃₁	1	1	yes	characteristic
88	C₄×C₁₆	C₄×C₃₁	1	1	yes	characteristic
89	C₄×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
90	C₄×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
91	#335	C₂²×C₃₁	1	4	yes	—
92	#334	—	2	2	no	—
93	C₂×C₃₁	—	2	2	no	—
94	C₂×C₃₁	#1567	1	1	yes	characteristic
95	C₂×C₃₁	#2158	1	1	yes	characteristic
96	C₂×C₃₁	C₂×#368	1	1	yes	characteristic
97	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
98	C₂²×C₈	Q₈×C₃₁	1	1	yes	characteristic
99	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
100	C₂×C₁₆	—	2	2	no	—
101	C₂×C₁₆	—	2	2	no	—
102	C₂×C₁₆	C₂×C₄×C₃₁	1	1	yes	characteristic
103	C₂×C₁₆	C₂×C₄×C₃₁	1	1	yes	characteristic
104	C₂×C₁₆	C₂×C₄×C₃₁	1	1	yes	characteristic
105	C₂×C₁₆	C₂×C₄×C₃₁	1	1	yes	characteristic
106	C₄⋊C₈	—	2	4	no	—
107	C₂²⋊C₈	—	4	2	no	—
108	#97	—	4	2	no	—
109	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
110	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
111	C₄×C₈	Q₈×C₃₁	1	1	yes	characteristic
112	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
113	C₃₁	#12024	1	1	yes	characteristic, direct product factor, Sylow
114	C₂²×C₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
115	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
116	C₂²×C₄	Q₁₆×C₃₁	1	1	yes	characteristic
117	C₂×C₈	—	2	2	no	—
118	C₂×C₈	—	2	1	no	—
119	C₂×C₈	—	2	1	no	—
120	C₂×C₈	—	4	4	no	—
121	C₂×C₈	C₄⋊C₄×C₃₁	1	1	yes	characteristic
122	C₂×C₈	C₄⋊C₄×C₃₁	1	1	yes	characteristic
123	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
124	C₂×C₈	C₂×Q₈×C₃₁	1	1	yes	characteristic
125	C₄²	C₄⋊C₄×C₃₁	1	1	yes	characteristic
126	C₄²	D₈×C₃₁	1	1	yes	characteristic
127	C₄²	Q₁₆×C₃₁	1	1	yes	characteristic
128	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
129	C₁₆	—	2	1	no	—
130	C₁₆	—	2	1	no	—
131	C₁₆	M₁₆×C₃₁	1	2	yes	—
132	C₁₆	M₁₆×C₃₁	1	2	yes	—
133	C₂³	C₃₁×#107	1	1	yes	characteristic
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	1	no	—
136	C₂×C₄	—	2	1	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	C₃₁×#107	1	1	yes	characteristic
139	C₂×C₄	C₃₁×#107	1	1	yes	characteristic
140	C₂×C₄	C₃₁×#107	1	1	yes	characteristic
141	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
142	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
143	C₂×C₄	C₂×Q₁₆×C₃₁	1	1	yes	characteristic
144	C₈	—	2	1	no	—
145	C₈	—	2	1	no	—
146	C₈	—	8	4	no	—
147	C₈	C₃₁×#108	1	2	yes	—
148	C₈	C₂×M₁₆×C₃₁	1	1	yes	characteristic
149	C₈	C₂×M₁₆×C₃₁	1	1	yes	characteristic
150	C₂²	—	2	1	no	—
151	C₂²	—	2	1	no	—
152	C₂²	C₃₁×#368	1	2	yes	—
153	C₂²	C₂×C₃₁×#107	1	1	yes	characteristic
154	C₄	—	2	1	no	—
155	C₄	—	2	1	no	—
156	C₄	—	2	1	no	—
157	C₄	—	2	1	no	—
158	C₄	C₃₁×#423	1	1	yes	characteristic
159	C₄	C₂×C₃₁×#108	1	1	yes	characteristic
160	C₄	C₃₁×#508	1	2	yes	—
161	C₂	—	2	2	no	—
162	C₂	C₃₁×#1567	1	1	yes	characteristic
163	C₂	C₃₁×#2158	1	1	yes	characteristic
164	C₂	C₂×C₃₁×#368	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₃₁
Frattini, cosocle	C₄×C₈	C₂³×C₃₁
socle	C₂²×C₃₁	C₂×#107
prime radical	C₂³×C₃₁	#107
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ C₂×C₄×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃₁×#1372	C₂	semidirect
4. C₃₁×#1371	C₂	semidirect
50. #12024	C₃₁	direct (N × H)
70. #1372	C₂×C₃₁	semidirect
71. #1371	C₂×C₃₁	semidirect
113. C₃₁	#12024	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄×C₁₆×C₃₁	2
C₃₁×#1568	2
C₃₁×#1372	2
C₃₁×#1371	2
#12024	31

Subgroups — C31×#12024 #21468783

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#12024 #21468783