Subgroups — C₃₁×#12559 #21469318

Order 7936.

554 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃₁×M₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃₁×#108	C₂	1	2	yes	maximal
3	C₃₁×#1381	C₂	1	4	yes	maximal, complemented
4	C₂²×M₁₆×C₃₁	—	2	2	no	—
5	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
6	C₂×C₃₁×M₃₂	—	2	4	no	—
7	C₂×C₃₁×M₃₂	C₄	1	4	yes	—
8	C₂²×C₁₆×C₃₁	C₂²	1	2	yes	—
9	C₂×C₃₁×#108	—	2	2	no	—
10	C₂×C₃₁×#108	—	2	4	no	—
11	C₂×C₃₁×#108	C₄	1	8	yes	—
12	C₂×C₃₁×#108	C₂²	1	4	yes	—
13	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
14	C₂×M₁₆×C₃₁	—	2	8	no	—
15	C₂×M₁₆×C₃₁	—	2	4	no	—
16	C₂×M₁₆×C₃₁	—	4	2	no	—
17	C₂×M₁₆×C₃₁	—	4	4	no	—
18	C₂²×C₈×C₃₁	—	4	2	no	—
19	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
20	C₂²×C₈×C₃₁	C₂×C₄	1	4	yes	—
21	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
22	C₂²×C₈×C₃₁	D₄	1	4	yes	—
23	C₂²×C₈×C₃₁	Q₈	1	2	yes	—
24	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
25	C₃₁×M₃₂	—	2	8	no	—
26	C₂×C₁₆×C₃₁	—	2	4	no	—
27	C₂×C₁₆×C₃₁	C₂×C₄	1	4	yes	—
28	C₃₁×#108	—	2	8	no	—
29	C₃₁×#108	—	2	4	no	—
30	C₃₁×#108	C₂×C₄	1	8	yes	—
31	C₂⁴×C₃₁	Q₁₆	1	1	yes	characteristic
32	C₂²×C₄×C₃₁	C₂²⋊C₄	1	4	yes	—
33	C₂²×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
34	C₂²×C₄×C₃₁	D₈	1	2	yes	—
35	C₂²×C₄×C₃₁	SD₁₆	1	4	yes	—
36	C₂²×C₄×C₃₁	Q₁₆	1	1	yes	characteristic
37	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
38	M₁₆×C₃₁	—	2	8	no	—
39	M₁₆×C₃₁	—	4	8	no	—
40	M₁₆×C₃₁	—	4	4	no	—
41	C₂×C₈×C₃₁	—	4	8	no	—
42	C₂×C₈×C₃₁	—	4	4	no	—
43	C₂×C₈×C₃₁	C₄²	1	4	yes	—
44	C₂×C₈×C₃₁	C₂²⋊C₄	1	4	yes	—
45	C₂×C₈×C₃₁	C₂²⋊C₄	1	4	yes	—
46	C₂×C₈×C₃₁	C₄⋊C₄	1	2	yes	—
47	C₂×C₈×C₃₁	C₄⋊C₄	1	8	yes	—
48	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
50	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
51	C₂×C₈×C₃₁	C₂×Q₈	1	1	yes	characteristic
52	C₁₆×C₃₁	—	2	4	no	—
53	#12559	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
54	C₂³×C₃₁	—	4	2	no	—
55	C₂³×C₃₁	#103	1	4	yes	—
56	C₂³×C₃₁	#107	1	2	yes	—
57	C₂³×C₃₁	C₂×Q₁₆	1	1	yes	characteristic
58	C₂×C₄×C₃₁	#95	1	4	yes	—
59	C₂×C₄×C₃₁	#102	1	4	yes	—
60	C₂×C₄×C₃₁	#102	1	4	yes	—
61	C₂×C₄×C₃₁	#103	1	4	yes	—
62	C₂×C₄×C₃₁	#106	1	4	yes	—
63	C₂×C₄×C₃₁	#107	1	2	yes	—
64	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
65	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
67	C₂×C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
68	C₈×C₃₁	—	4	8	no	—
69	C₈×C₃₁	#95	1	4	yes	—
70	C₈×C₃₁	C₂×C₄²	1	1	yes	characteristic
71	C₈×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₈×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
73	C₂²×M₃₂	C₂×C₃₁	1	1	yes	characteristic
74	C₂²×#108	C₂×C₃₁	1	2	yes	—
75	#1381	C₂×C₃₁	1	4	yes	complemented
76	C₂²×C₃₁	—	2	4	no	—
77	C₂²×C₃₁	—	2	2	no	—
78	C₂²×C₃₁	—	4	4	no	—
79	C₂²×C₃₁	#340	1	4	yes	—
80	C₂²×C₃₁	C₂×#103	1	2	yes	—
81	C₂²×C₃₁	C₂×#107	1	1	yes	characteristic
82	C₄×C₃₁	#340	1	4	yes	—
83	C₄×C₃₁	C₂×#95	1	1	yes	characteristic
84	C₄×C₃₁	C₂×#102	1	2	yes	—
85	C₄×C₃₁	C₂×#106	1	1	yes	characteristic
86	C₂²×M₁₆	—	2	2	no	—
87	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
88	C₂×M₃₂	—	2	4	no	—
89	C₂×M₃₂	C₄×C₃₁	1	4	yes	—
90	C₂²×C₁₆	C₂²×C₃₁	1	2	yes	—
91	C₂×#108	—	2	2	no	—
92	C₂×#108	—	2	4	no	—
93	C₂×#108	C₄×C₃₁	1	8	yes	—
94	C₂×#108	C₂²×C₃₁	1	4	yes	—
95	C₂×C₃₁	—	2	4	no	—
96	C₂×C₃₁	—	2	2	no	—
97	C₂×C₃₁	C₂×#340	1	1	yes	characteristic
98	C₂×C₃₁	#2161	1	2	yes	—
99	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
100	C₂×M₁₆	—	2	8	no	—
101	C₂×M₁₆	—	2	4	no	—
102	C₂×M₁₆	—	4	2	no	—
103	C₂×M₁₆	—	4	4	no	—
104	C₂²×C₈	—	4	2	no	—
105	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
106	C₂²×C₈	C₂×C₄×C₃₁	1	4	yes	—
107	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
108	C₂²×C₈	D₄×C₃₁	1	4	yes	—
109	C₂²×C₈	Q₈×C₃₁	1	2	yes	—
110	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
111	M₃₂	—	2	8	no	—
112	C₂×C₁₆	—	2	4	no	—
113	C₂×C₁₆	C₂×C₄×C₃₁	1	4	yes	—
114	#108	—	2	8	no	—
115	#108	—	2	4	no	—
116	#108	C₂×C₄×C₃₁	1	8	yes	—
117	C₃₁	#12559	1	1	yes	characteristic, direct product factor, Sylow
118	C₂⁴	Q₁₆×C₃₁	1	1	yes	characteristic
119	C₂²×C₄	C₂²⋊C₄×C₃₁	1	4	yes	—
120	C₂²×C₄	C₄⋊C₄×C₃₁	1	2	yes	—
121	C₂²×C₄	D₈×C₃₁	1	2	yes	—
122	C₂²×C₄	SD₁₆×C₃₁	1	4	yes	—
123	C₂²×C₄	Q₁₆×C₃₁	1	1	yes	characteristic
124	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
125	M₁₆	—	2	8	no	—
126	M₁₆	—	4	8	no	—
127	M₁₆	—	4	4	no	—
128	C₂×C₈	—	4	8	no	—
129	C₂×C₈	—	4	4	no	—
130	C₂×C₈	C₄²×C₃₁	1	4	yes	—
131	C₂×C₈	C₂²⋊C₄×C₃₁	1	4	yes	—
132	C₂×C₈	C₂²⋊C₄×C₃₁	1	4	yes	—
133	C₂×C₈	C₄⋊C₄×C₃₁	1	2	yes	—
134	C₂×C₈	C₄⋊C₄×C₃₁	1	8	yes	—
135	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
136	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
137	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
138	C₂×C₈	C₂×Q₈×C₃₁	1	1	yes	characteristic
139	C₁₆	—	2	4	no	—
140	C₂³	—	4	2	no	—
141	C₂³	C₃₁×#103	1	4	yes	—
142	C₂³	C₃₁×#107	1	2	yes	—
143	C₂³	C₂×Q₁₆×C₃₁	1	1	yes	characteristic
144	C₂×C₄	C₃₁×#95	1	4	yes	—
145	C₂×C₄	C₃₁×#102	1	4	yes	—
146	C₂×C₄	C₃₁×#102	1	4	yes	—
147	C₂×C₄	C₃₁×#103	1	4	yes	—
148	C₂×C₄	C₃₁×#106	1	4	yes	—
149	C₂×C₄	C₃₁×#107	1	2	yes	—
150	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
151	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
152	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
153	C₂×C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
154	C₈	—	4	8	no	—
155	C₈	C₃₁×#95	1	4	yes	—
156	C₈	C₂×C₄²×C₃₁	1	1	yes	characteristic
157	C₈	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
158	C₈	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
159	C₂²	—	2	4	no	—
160	C₂²	—	2	2	no	—
161	C₂²	—	4	4	no	—
162	C₂²	C₃₁×#340	1	4	yes	—
163	C₂²	C₂×C₃₁×#103	1	2	yes	—
164	C₂²	C₂×C₃₁×#107	1	1	yes	characteristic
165	C₄	C₃₁×#340	1	4	yes	—
166	C₄	C₂×C₃₁×#95	1	1	yes	characteristic
167	C₄	C₂×C₃₁×#102	1	2	yes	—
168	C₄	C₂×C₃₁×#106	1	1	yes	characteristic
169	C₂	—	2	2	no	—
170	C₂	—	2	4	no	—
171	C₂	C₂×C₃₁×#340	1	1	yes	characteristic
172	C₂	C₃₁×#2161	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂×D₈
commutator	C₈	C₂×C₄²×C₃₁
Frattini, cosocle	C₂²×C₈	C₂³×C₃₁
socle	C₂²×C₃₁	C₂×#107
prime radical	C₂⁴×C₃₁	Q₁₆
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂³×C₄×C₃₁ ◁ C₂³×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃₁×#1381	C₂	semidirect
53. #12559	C₃₁	direct (N × H)
75. #1381	C₂×C₃₁	semidirect
117. C₃₁	#12559	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃₁×M₃₂	2
C₂²×C₃₁×#108	2
C₃₁×#1381	2
#12559	31

Subgroups — C31×#12559 #21469318

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#12559 #21469318