Subgroups — C₂²×C₃₁×#365 #21490161

Order 7936.

702 conjugacy classes of subgroups.

The subgroup lattice has 192 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	1	yes	maximal, characteristic
3	C₂²×C₃₁×#106	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃₁×#365	C₂	1	12	yes	maximal, direct product factor
5	C₂²×M₁₆×C₃₁	—	2	1	no	—
6	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
7	C₂²×C₄⋊C₄×C₃₁	—	2	1	no	—
8	C₂×C₃₁×M₃₂	C₄	1	6	yes	complemented
9	C₂×C₃₁×M₃₂	C₂²	1	6	yes	—
10	C₂²×C₁₆×C₃₁	C₄	1	2	yes	complemented
11	C₂×C₃₁×#108	—	2	3	no	—
12	C₂×C₃₁×#108	C₂²	1	6	yes	—
13	C₂×C₃₁×#106	—	2	3	no	—
14	C₂×C₃₁×#106	C₂²	1	6	yes	—
15	C₃₁×#365	C₂²	1	16	yes	direct product factor
16	C₂³×C₄×C₃₁	—	4	1	no	—
17	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
18	C₂×M₁₆×C₃₁	—	2	6	no	—
19	C₂×M₁₆×C₃₁	—	4	3	no	—
20	C₂²×C₈×C₃₁	—	4	1	no	—
21	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₃₁	C₂×C₄	1	3	yes	—
23	C₂²×C₈×C₃₁	D₄	1	3	yes	—
24	C₂²×C₈×C₃₁	Q₈	1	3	yes	—
25	C₂²×C₈×C₃₁	Q₈	1	1	yes	characteristic
26	C₂²×C₈×C₃₁	C₂³	1	3	yes	—
27	C₂×C₄⋊C₄×C₃₁	—	2	6	no	—
28	C₂×C₄⋊C₄×C₃₁	—	4	3	no	—
29	C₃₁×M₃₂	C₂×C₄	1	12	yes	complemented
30	C₃₁×M₃₂	C₂³	1	4	yes	—
31	C₂×C₁₆×C₃₁	C₂×C₄	1	12	yes	complemented
32	C₃₁×#108	—	2	6	no	—
33	C₃₁×#108	C₂³	1	4	yes	—
34	C₃₁×#106	—	2	6	no	—
35	C₃₁×#106	C₂³	1	4	yes	—
36	C₂⁴×C₃₁	SD₁₆	1	1	yes	characteristic
37	C₂²×C₄×C₃₁	—	4	6	no	—
38	C₂²×C₄×C₃₁	—	8	4	no	—
39	C₂²×C₄×C₃₁	C₄⋊C₄	1	3	yes	—
40	C₂²×C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
41	C₂²×C₄×C₃₁	SD₁₆	1	3	yes	—
42	C₂²×C₄×C₃₁	SD₁₆	1	3	yes	—
43	C₂²×C₄×C₃₁	SD₁₆	1	1	yes	characteristic
44	C₂²×C₄×C₃₁	C₂×D₄	1	3	yes	—
45	M₁₆×C₃₁	—	2	4	no	—
46	M₁₆×C₃₁	—	4	6	no	—
47	C₂×C₈×C₃₁	—	4	6	no	—
48	C₂×C₈×C₃₁	C₄⋊C₄	1	3	yes	—
49	C₂×C₈×C₃₁	C₄⋊C₄	1	6	yes	—
50	C₂×C₈×C₃₁	C₄⋊C₄	1	3	yes	—
51	C₂×C₈×C₃₁	C₂²×C₄	1	3	yes	—
52	C₂×C₈×C₃₁	C₂²×C₄	1	3	yes	—
53	C₂×C₈×C₃₁	C₂×D₄	1	3	yes	—
54	C₂×C₈×C₃₁	C₂×Q₈	1	3	yes	—
55	C₂×C₈×C₃₁	C₂×Q₈	1	3	yes	—
56	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
57	C₄⋊C₄×C₃₁	—	2	4	no	—
58	C₄⋊C₄×C₃₁	—	4	6	no	—
59	C₁₆×C₃₁	C₂²×C₄	1	8	yes	complemented
60	C₂²×#365	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
61	C₂³×C₃₁	—	2	4	no	—
62	C₂³×C₃₁	#106	1	3	yes	—
63	C₂³×C₃₁	#106	1	1	yes	characteristic
64	C₂³×C₃₁	C₂×SD₁₆	1	3	yes	—
65	C₂×C₄×C₃₁	—	4	4	no	—
66	C₂×C₄×C₃₁	—	8	12	no	—
67	C₂×C₄×C₃₁	#106	1	6	yes	—
68	C₂×C₄×C₃₁	#106	1	3	yes	—
69	C₂×C₄×C₃₁	#106	1	3	yes	—
70	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	3	yes	—
71	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	3	yes	—
72	C₂×C₄×C₃₁	C₂×SD₁₆	1	3	yes	—
73	C₂×C₄×C₃₁	C₂×SD₁₆	1	3	yes	—
74	C₂×C₄×C₃₁	C₂×SD₁₆	1	3	yes	—
75	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
76	C₈×C₃₁	—	4	4	no	—
77	C₈×C₃₁	C₂×C₄⋊C₄	1	3	yes	—
78	C₈×C₃₁	C₂×C₄⋊C₄	1	3	yes	—
79	C₈×C₃₁	C₂³×C₄	1	1	yes	characteristic
80	C₈×C₃₁	C₂²×Q₈	1	1	yes	characteristic
81	C₂²×M₃₂	C₂×C₃₁	1	1	yes	characteristic
82	C₂²×#108	C₂×C₃₁	1	1	yes	characteristic
83	C₂²×#106	C₂×C₃₁	1	1	yes	characteristic
84	C₂×#365	C₂×C₃₁	1	12	yes	direct product factor
85	C₂²×C₃₁	—	2	6	no	—
86	C₂²×C₃₁	—	2	6	no	—
87	C₂²×C₃₁	#365	1	4	yes	direct product factor
88	C₂²×C₃₁	C₂×#106	1	3	yes	—
89	C₂²×C₃₁	C₂×#106	1	3	yes	—
90	C₂²×C₃₁	C₂²×SD₁₆	1	1	yes	characteristic
91	C₄×C₃₁	—	8	4	no	—
92	C₄×C₃₁	C₂×#106	1	3	yes	—
93	C₄×C₃₁	C₂×#106	1	3	yes	—
94	C₄×C₃₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
95	C₄×C₃₁	C₂²×SD₁₆	1	1	yes	characteristic
96	C₂²×M₁₆	—	2	1	no	—
97	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
98	C₂²×C₄⋊C₄	—	2	1	no	—
99	C₂×M₃₂	C₄×C₃₁	1	6	yes	complemented
100	C₂×M₃₂	C₂²×C₃₁	1	6	yes	—
101	C₂²×C₁₆	C₄×C₃₁	1	2	yes	complemented
102	C₂×#108	—	2	3	no	—
103	C₂×#108	C₂²×C₃₁	1	6	yes	—
104	C₂×#106	—	2	3	no	—
105	C₂×#106	C₂²×C₃₁	1	6	yes	—
106	#365	C₂²×C₃₁	1	16	yes	direct product factor
107	C₂×C₃₁	—	2	1	no	—
108	C₂×C₃₁	—	2	3	no	—
109	C₂×C₃₁	C₂×#365	1	6	yes	direct product factor
110	C₂×C₃₁	C₂²×#106	1	1	yes	characteristic
111	C₂³×C₄	—	4	1	no	—
112	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
113	C₂×M₁₆	—	2	6	no	—
114	C₂×M₁₆	—	4	3	no	—
115	C₂²×C₈	—	4	1	no	—
116	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
117	C₂²×C₈	C₂×C₄×C₃₁	1	3	yes	—
118	C₂²×C₈	D₄×C₃₁	1	3	yes	—
119	C₂²×C₈	Q₈×C₃₁	1	3	yes	—
120	C₂²×C₈	Q₈×C₃₁	1	1	yes	characteristic
121	C₂²×C₈	C₂³×C₃₁	1	3	yes	—
122	C₂×C₄⋊C₄	—	2	6	no	—
123	C₂×C₄⋊C₄	—	4	3	no	—
124	M₃₂	C₂×C₄×C₃₁	1	12	yes	complemented
125	M₃₂	C₂³×C₃₁	1	4	yes	—
126	C₂×C₁₆	C₂×C₄×C₃₁	1	12	yes	complemented
127	#108	—	2	6	no	—
128	#108	C₂³×C₃₁	1	4	yes	—
129	#106	—	2	6	no	—
130	#106	C₂³×C₃₁	1	4	yes	—
131	C₃₁	C₂²×#365	1	1	yes	characteristic, direct product factor, Sylow
132	C₂⁴	SD₁₆×C₃₁	1	1	yes	characteristic
133	C₂²×C₄	—	4	6	no	—
134	C₂²×C₄	—	8	4	no	—
135	C₂²×C₄	C₄⋊C₄×C₃₁	1	3	yes	—
136	C₂²×C₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
137	C₂²×C₄	SD₁₆×C₃₁	1	3	yes	—
138	C₂²×C₄	SD₁₆×C₃₁	1	1	yes	characteristic
139	C₂²×C₄	SD₁₆×C₃₁	1	3	yes	—
140	C₂²×C₄	C₂×D₄×C₃₁	1	3	yes	—
141	M₁₆	—	2	4	no	—
142	M₁₆	—	4	6	no	—
143	C₂×C₈	—	4	6	no	—
144	C₂×C₈	C₄⋊C₄×C₃₁	1	3	yes	—
145	C₂×C₈	C₄⋊C₄×C₃₁	1	6	yes	—
146	C₂×C₈	C₄⋊C₄×C₃₁	1	3	yes	—
147	C₂×C₈	C₂²×C₄×C₃₁	1	3	yes	—
148	C₂×C₈	C₂²×C₄×C₃₁	1	3	yes	—
149	C₂×C₈	C₂×D₄×C₃₁	1	3	yes	—
150	C₂×C₈	C₂×Q₈×C₃₁	1	3	yes	—
151	C₂×C₈	C₂×Q₈×C₃₁	1	3	yes	—
152	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
153	C₄⋊C₄	—	2	4	no	—
154	C₄⋊C₄	—	4	6	no	—
155	C₁₆	C₂²×C₄×C₃₁	1	8	yes	complemented
156	C₂³	—	2	4	no	—
157	C₂³	C₃₁×#106	1	3	yes	—
158	C₂³	C₃₁×#106	1	1	yes	characteristic
159	C₂³	C₂×SD₁₆×C₃₁	1	3	yes	—
160	C₂×C₄	—	4	4	no	—
161	C₂×C₄	—	8	12	no	—
162	C₂×C₄	C₃₁×#106	1	6	yes	—
163	C₂×C₄	C₃₁×#106	1	3	yes	—
164	C₂×C₄	C₃₁×#106	1	3	yes	—
165	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	3	yes	—
166	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	3	yes	—
167	C₂×C₄	C₂×SD₁₆×C₃₁	1	3	yes	—
168	C₂×C₄	C₂×SD₁₆×C₃₁	1	3	yes	—
169	C₂×C₄	C₂×SD₁₆×C₃₁	1	3	yes	—
170	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
171	C₈	—	4	4	no	—
172	C₈	C₂×C₄⋊C₄×C₃₁	1	3	yes	—
173	C₈	C₂×C₄⋊C₄×C₃₁	1	3	yes	—
174	C₈	C₂³×C₄×C₃₁	1	1	yes	characteristic
175	C₈	C₂²×Q₈×C₃₁	1	1	yes	characteristic
176	C₂²	—	2	6	no	—
177	C₂²	—	2	6	no	—
178	C₂²	C₃₁×#365	1	4	yes	direct product factor
179	C₂²	C₂×C₃₁×#106	1	3	yes	—
180	C₂²	C₂×C₃₁×#106	1	3	yes	—
181	C₂²	C₂²×SD₁₆×C₃₁	1	1	yes	characteristic
182	C₄	—	8	4	no	—
183	C₄	C₂×C₃₁×#106	1	3	yes	—
184	C₄	C₂×C₃₁×#106	1	3	yes	—
185	C₄	C₂²×C₄⋊C₄×C₃₁	1	1	yes	characteristic
186	C₄	C₂²×SD₁₆×C₃₁	1	1	yes	characteristic
187	C₂	—	2	1	no	—
188	C₂	—	2	3	no	—
189	C₂	C₂×C₃₁×#365	1	6	yes	direct product factor
190	C₂	C₂²×C₃₁×#106	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	#106
commutator	C₈	C₂³×C₄×C₃₁
Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
prime radical	C₂⁴×C₃₁	SD₁₆
LCS	G ▷ 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
4. C₂×C₃₁×#365	C₂	direct (N × H)
8. C₂×C₃₁×M₃₂	C₄	semidirect
10. C₂²×C₁₆×C₃₁	C₄	semidirect
15. C₃₁×#365	C₂²	direct (N × H)
29. C₃₁×M₃₂	C₂×C₄	semidirect
31. C₂×C₁₆×C₃₁	C₂×C₄	semidirect
59. C₁₆×C₃₁	C₂²×C₄	semidirect
60. C₂²×#365	C₃₁	direct (N × H)
84. C₂×#365	C₂×C₃₁	direct (N × H)
87. C₂²×C₃₁	#365	direct (N × H)
99. C₂×M₃₂	C₄×C₃₁	semidirect
101. C₂²×C₁₆	C₄×C₃₁	semidirect
106. #365	C₂²×C₃₁	direct (N × H)
109. C₂×C₃₁	C₂×#365	direct (N × H)
124. M₃₂	C₂×C₄×C₃₁	semidirect
126. C₂×C₁₆	C₂×C₄×C₃₁	semidirect
131. C₃₁	C₂²×#365	direct (N × H)
155. C₁₆	C₂²×C₄×C₃₁	semidirect
178. C₂²	C₃₁×#365	direct (N × H)
189. C₂	C₂×C₃₁×#365	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
C₂²×C₃₁×#106	2
C₂×C₃₁×#365	2
C₂²×#365	31

Subgroups — C22×C31×#365 #21490161

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃₁×#365 #21490161