Subgroups — C₂×C₃₁×#1381 #21469317

Order 7936.

574 conjugacy classes of subgroups.

The subgroup lattice has 212 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	C₂²×C₃₁×#108	C₂	1	2	yes	maximal
3	C₃₁×#1381	C₂	1	4	yes	maximal, direct product factor
4	C₂²×M₁₆×C₃₁	—	2	2	no	—
5	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
6	C₂²×C₁₆×C₃₁	—	2	4	no	—
7	C₂²×C₁₆×C₃₁	C₄	1	2	yes	—
8	C₂²×C₁₆×C₃₁	C₄	1	2	yes	—
9	C₂²×C₁₆×C₃₁	C₂²	1	2	yes	—
10	C₂×C₃₁×#108	—	2	2	no	—
11	C₂×C₃₁×#108	—	2	4	no	—
12	C₂×C₃₁×#108	C₄	1	8	yes	—
13	C₂×C₃₁×#108	C₂²	1	4	yes	—
14	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
15	C₂×M₁₆×C₃₁	—	2	8	no	—
16	C₂×M₁₆×C₃₁	—	2	4	no	—
17	C₂×M₁₆×C₃₁	—	4	2	no	—
18	C₂×M₁₆×C₃₁	—	4	4	no	—
19	C₂²×C₈×C₃₁	—	4	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	C₂×C₄	1	4	yes	—
23	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₃₁	D₄	1	4	yes	—
25	C₂²×C₈×C₃₁	Q₈	1	1	yes	characteristic
26	C₂²×C₈×C₃₁	Q₈	1	1	yes	characteristic
27	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
28	C₂×C₁₆×C₃₁	—	2	4	no	—
29	C₂×C₁₆×C₃₁	—	2	4	no	—
30	C₂×C₁₆×C₃₁	—	2	4	no	—
31	C₂×C₁₆×C₃₁	C₂×C₄	1	4	yes	—
32	C₃₁×#108	—	2	8	no	—
33	C₃₁×#108	—	2	4	no	—
34	C₃₁×#108	C₂×C₄	1	8	yes	—
35	C₂⁴×C₃₁	Q₁₆	1	1	yes	characteristic
36	C₂²×C₄×C₃₁	C₂²⋊C₄	1	4	yes	—
37	C₂²×C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
38	C₂²×C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
39	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
40	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
41	C₂²×C₄×C₃₁	SD₁₆	1	4	yes	—
42	C₂²×C₄×C₃₁	Q₁₆	1	1	yes	characteristic
43	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
44	M₁₆×C₃₁	—	2	8	no	—
45	M₁₆×C₃₁	—	4	8	no	—
46	M₁₆×C₃₁	—	4	4	no	—
47	C₂×C₈×C₃₁	—	4	8	no	—
48	C₂×C₈×C₃₁	—	4	4	no	—
49	C₂×C₈×C₃₁	C₄²	1	2	yes	—
50	C₂×C₈×C₃₁	C₄²	1	2	yes	—
51	C₂×C₈×C₃₁	C₂²⋊C₄	1	4	yes	—
52	C₂×C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
53	C₂×C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
54	C₂×C₈×C₃₁	C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₈×C₃₁	C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₈×C₃₁	C₄⋊C₄	1	4	yes	—
57	C₂×C₈×C₃₁	C₄⋊C₄	1	4	yes	—
58	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
60	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
61	C₂×C₈×C₃₁	C₂×Q₈	1	1	yes	characteristic
62	C₁₆×C₃₁	—	2	4	no	—
63	C₂×#1381	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
64	C₂³×C₃₁	—	2	4	no	—
65	C₂³×C₃₁	#103	1	4	yes	—
66	C₂³×C₃₁	#107	1	1	yes	characteristic
67	C₂³×C₃₁	#107	1	1	yes	characteristic
68	C₂³×C₃₁	C₂×Q₁₆	1	1	yes	characteristic
69	C₂×C₄×C₃₁	#95	1	2	yes	—
70	C₂×C₄×C₃₁	#95	1	2	yes	—
71	C₂×C₄×C₃₁	#102	1	4	yes	—
72	C₂×C₄×C₃₁	#102	1	4	yes	—
73	C₂×C₄×C₃₁	#103	1	4	yes	—
74	C₂×C₄×C₃₁	#106	1	2	yes	—
75	C₂×C₄×C₃₁	#106	1	2	yes	—
76	C₂×C₄×C₃₁	#107	1	1	yes	characteristic
77	C₂×C₄×C₃₁	#107	1	1	yes	characteristic
78	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
79	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
81	C₂×C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
82	C₈×C₃₁	—	4	8	no	—
83	C₈×C₃₁	#95	1	2	yes	—
84	C₈×C₃₁	#95	1	2	yes	—
85	C₈×C₃₁	C₂×C₄²	1	1	yes	characteristic
86	C₈×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
87	C₈×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
88	C₂³×C₁₆	C₂×C₃₁	1	1	yes	characteristic
89	C₂²×#108	C₂×C₃₁	1	2	yes	—
90	#1381	C₂×C₃₁	1	4	yes	direct product factor
91	C₂²×C₃₁	—	2	8	no	—
92	C₂²×C₃₁	—	2	2	no	—
93	C₂²×C₃₁	—	2	2	no	—
94	C₂²×C₃₁	#340	1	2	yes	—
95	C₂²×C₃₁	#340	1	2	yes	—
96	C₂²×C₃₁	#368	1	4	yes	—
97	C₂²×C₃₁	C₂×#103	1	2	yes	—
98	C₂²×C₃₁	C₂×#107	1	1	yes	characteristic
99	C₄×C₃₁	#340	1	2	yes	—
100	C₄×C₃₁	#340	1	2	yes	—
101	C₄×C₃₁	C₂×#95	1	1	yes	characteristic
102	C₄×C₃₁	C₂×#102	1	2	yes	—
103	C₄×C₃₁	C₂×#106	1	1	yes	characteristic
104	C₂²×M₁₆	—	2	2	no	—
105	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
106	C₂²×C₁₆	—	2	4	no	—
107	C₂²×C₁₆	C₄×C₃₁	1	2	yes	—
108	C₂²×C₁₆	C₄×C₃₁	1	2	yes	—
109	C₂²×C₁₆	C₂²×C₃₁	1	2	yes	—
110	C₂×#108	—	2	2	no	—
111	C₂×#108	—	2	4	no	—
112	C₂×#108	C₄×C₃₁	1	8	yes	—
113	C₂×#108	C₂²×C₃₁	1	4	yes	—
114	C₂×C₃₁	—	2	2	no	—
115	C₂×C₃₁	—	2	2	no	—
116	C₂×C₃₁	#1381	1	4	yes	direct product factor
117	C₂×C₃₁	C₂×#340	1	1	yes	characteristic
118	C₂×C₃₁	C₂×#368	1	2	yes	—
119	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
120	C₂×M₁₆	—	2	8	no	—
121	C₂×M₁₆	—	2	4	no	—
122	C₂×M₁₆	—	4	2	no	—
123	C₂×M₁₆	—	4	4	no	—
124	C₂²×C₈	—	4	2	no	—
125	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
126	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
127	C₂²×C₈	C₂×C₄×C₃₁	1	4	yes	—
128	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
129	C₂²×C₈	D₄×C₃₁	1	4	yes	—
130	C₂²×C₈	Q₈×C₃₁	1	1	yes	characteristic
131	C₂²×C₈	Q₈×C₃₁	1	1	yes	characteristic
132	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
133	C₂×C₁₆	—	2	4	no	—
134	C₂×C₁₆	—	2	4	no	—
135	C₂×C₁₆	—	2	4	no	—
136	C₂×C₁₆	C₂×C₄×C₃₁	1	4	yes	—
137	#108	—	2	8	no	—
138	#108	—	2	4	no	—
139	#108	C₂×C₄×C₃₁	1	8	yes	—
140	C₃₁	C₂×#1381	1	1	yes	characteristic, direct product factor, Sylow
141	C₂⁴	Q₁₆×C₃₁	1	1	yes	characteristic
142	C₂²×C₄	C₂²⋊C₄×C₃₁	1	4	yes	—
143	C₂²×C₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
144	C₂²×C₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
145	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
146	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
147	C₂²×C₄	SD₁₆×C₃₁	1	4	yes	—
148	C₂²×C₄	Q₁₆×C₃₁	1	1	yes	characteristic
149	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
150	M₁₆	—	2	8	no	—
151	M₁₆	—	4	8	no	—
152	M₁₆	—	4	4	no	—
153	C₂×C₈	—	4	8	no	—
154	C₂×C₈	—	4	4	no	—
155	C₂×C₈	C₄²×C₃₁	1	2	yes	—
156	C₂×C₈	C₄²×C₃₁	1	2	yes	—
157	C₂×C₈	C₂²⋊C₄×C₃₁	1	4	yes	—
158	C₂×C₈	C₂²⋊C₄×C₃₁	1	2	yes	—
159	C₂×C₈	C₂²⋊C₄×C₃₁	1	2	yes	—
160	C₂×C₈	C₄⋊C₄×C₃₁	1	1	yes	characteristic
161	C₂×C₈	C₄⋊C₄×C₃₁	1	1	yes	characteristic
162	C₂×C₈	C₄⋊C₄×C₃₁	1	4	yes	—
163	C₂×C₈	C₄⋊C₄×C₃₁	1	4	yes	—
164	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
165	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
166	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
167	C₂×C₈	C₂×Q₈×C₃₁	1	1	yes	characteristic
168	C₁₆	—	2	4	no	—
169	C₂³	—	2	4	no	—
170	C₂³	C₃₁×#103	1	4	yes	—
171	C₂³	C₃₁×#107	1	1	yes	characteristic
172	C₂³	C₃₁×#107	1	1	yes	characteristic
173	C₂³	C₂×Q₁₆×C₃₁	1	1	yes	characteristic
174	C₂×C₄	C₃₁×#95	1	2	yes	—
175	C₂×C₄	C₃₁×#95	1	2	yes	—
176	C₂×C₄	C₃₁×#102	1	4	yes	—
177	C₂×C₄	C₃₁×#102	1	4	yes	—
178	C₂×C₄	C₃₁×#103	1	4	yes	—
179	C₂×C₄	C₃₁×#106	1	2	yes	—
180	C₂×C₄	C₃₁×#106	1	2	yes	—
181	C₂×C₄	C₃₁×#107	1	1	yes	characteristic
182	C₂×C₄	C₃₁×#107	1	1	yes	characteristic
183	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
184	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
185	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
186	C₂×C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
187	C₈	—	4	8	no	—
188	C₈	C₃₁×#95	1	2	yes	—
189	C₈	C₃₁×#95	1	2	yes	—
190	C₈	C₂×C₄²×C₃₁	1	1	yes	characteristic
191	C₈	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
192	C₈	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
193	C₂²	—	2	8	no	—
194	C₂²	—	2	2	no	—
195	C₂²	—	2	2	no	—
196	C₂²	C₃₁×#340	1	2	yes	—
197	C₂²	C₃₁×#340	1	2	yes	—
198	C₂²	C₃₁×#368	1	4	yes	—
199	C₂²	C₂×C₃₁×#103	1	2	yes	—
200	C₂²	C₂×C₃₁×#107	1	1	yes	characteristic
201	C₄	C₃₁×#340	1	2	yes	—
202	C₄	C₃₁×#340	1	2	yes	—
203	C₄	C₂×C₃₁×#95	1	1	yes	characteristic
204	C₄	C₂×C₃₁×#102	1	2	yes	—
205	C₄	C₂×C₃₁×#106	1	1	yes	characteristic
206	C₂	—	2	2	no	—
207	C₂	—	2	2	no	—
208	C₂	C₃₁×#1381	1	4	yes	direct product factor
209	C₂	C₂×C₃₁×#340	1	1	yes	characteristic
210	C₂	C₂×C₃₁×#368	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₃₁	D₈
commutator	C₈	C₂×C₄²×C₃₁
Frattini, cosocle	C₂²×C₈	C₂³×C₃₁
socle	C₂³×C₃₁	#107
prime radical	C₂⁴×C₃₁	Q₁₆
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×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	direct (N × H)
63. C₂×#1381	C₃₁	direct (N × H)
90. #1381	C₂×C₃₁	direct (N × H)
116. C₂×C₃₁	#1381	direct (N × H)
140. C₃₁	C₂×#1381	direct (N × H)
208. C₂	C₃₁×#1381	direct (N × H)

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
C₂²×C₃₁×#108	2
C₃₁×#1381	2
C₂×#1381	31

Subgroups — C2×C31×#1381 #21469317

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#1381 #21469317