Subgroups — C₄×C₃₁×#365 #21469428

Order 7936.

330 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#365	C₂	1	2	yes	maximal
2	C₄×C₃₁×M₃₂	C₂	1	1	yes	maximal, characteristic
3	C₄×C₃₁×#108	C₂	1	1	yes	maximal, characteristic
4	C₄×C₃₁×#106	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#1383	C₂	1	2	yes	maximal
6	C₂×C₃₁×M₃₂	C₂²	1	1	yes	characteristic
7	C₂×C₃₁×M₃₂	C₂²	1	1	yes	characteristic
8	C₂×C₃₁×#108	C₂²	1	2	yes	—
9	C₂×C₃₁×#106	C₂²	1	2	yes	—
10	C₄×M₁₆×C₃₁	—	2	1	no	—
11	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
12	C₄×C₄⋊C₄×C₃₁	—	2	1	no	—
13	C₃₁×#365	C₄	1	8	yes	direct product factor
14	C₃₁×#346	C₄	1	2	yes	complemented
15	C₄×C₁₆×C₃₁	C₄	1	2	yes	complemented
16	C₃₁×#341	—	2	1	no	—
17	C₃₁×#340	—	2	1	no	—
18	C₂×M₁₆×C₃₁	—	2	2	no	—
19	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
20	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
21	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
22	C₂×C₄²×C₃₁	—	4	1	no	—
23	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
24	C₃₁×M₃₂	—	2	2	no	—
25	C₃₁×M₃₂	C₂×C₄	1	4	yes	—
26	C₂×C₁₆×C₃₁	C₂×C₄	1	2	yes	—
27	C₂×C₁₆×C₃₁	C₂×C₄	1	2	yes	—
28	C₃₁×#108	—	2	2	no	—
29	C₃₁×#108	C₂×C₄	1	4	yes	—
30	C₃₁×#106	—	2	2	no	—
31	C₃₁×#106	C₂×C₄	1	4	yes	—
32	C₃₁×#97	—	4	1	no	—
33	C₄×C₈×C₃₁	—	4	1	no	—
34	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
35	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
36	C₄×C₈×C₃₁	Q₈	1	1	yes	characteristic
37	C₄×C₈×C₃₁	Q₈	1	1	yes	characteristic
38	C₃₁×#95	—	4	1	no	—
39	C₂²×C₄×C₃₁	—	4	2	no	—
40	C₂²×C₄×C₃₁	SD₁₆	1	1	yes	characteristic
41	C₂²×C₄×C₃₁	SD₁₆	1	1	yes	characteristic
42	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
43	M₁₆×C₃₁	—	2	4	no	—
44	M₁₆×C₃₁	—	4	2	no	—
45	C₂×C₈×C₃₁	—	2	2	no	—
46	C₂×C₈×C₃₁	—	4	2	no	—
47	C₂×C₈×C₃₁	C₄⋊C₄	1	1	yes	characteristic
48	C₂×C₈×C₃₁	C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
51	C₂×C₈×C₃₁	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₈×C₃₁	#41	1	2	yes	—
53	C₄⋊C₄×C₃₁	—	2	4	no	—
54	C₄⋊C₄×C₃₁	—	4	2	no	—
55	C₄²×C₃₁	—	8	2	no	—
56	C₄²×C₃₁	C₄⋊C₄	1	1	yes	characteristic
57	C₄²×C₃₁	C₄⋊C₄	1	1	yes	characteristic
58	C₄²×C₃₁	SD₁₆	1	1	yes	characteristic
59	C₄²×C₃₁	SD₁₆	1	1	yes	characteristic
60	C₁₆×C₃₁	—	2	2	no	—
61	C₁₆×C₃₁	C₄²	1	4	yes	complemented
62	C₄×#365	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
63	C₂³×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
64	C₂×C₄×C₃₁	—	2	2	no	—
65	C₂×C₄×C₃₁	—	4	1	no	—
66	C₂×C₄×C₃₁	—	4	4	no	—
67	C₂×C₄×C₃₁	—	8	2	no	—
68	C₂×C₄×C₃₁	—	8	2	no	—
69	C₂×C₄×C₃₁	#106	1	1	yes	characteristic
70	C₂×C₄×C₃₁	#106	1	1	yes	characteristic
71	C₂×C₄×C₃₁	#106	1	1	yes	characteristic
72	C₂×C₄×C₃₁	#106	1	1	yes	characteristic
73	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
75	C₂×C₄×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
76	C₂×C₄×C₃₁	#135	1	2	yes	—
77	C₈×C₃₁	—	2	1	no	—
78	C₈×C₃₁	—	2	1	no	—
79	C₈×C₃₁	—	4	4	no	—
80	C₈×C₃₁	C₂×C₄²	1	1	yes	characteristic
81	C₈×C₃₁	#117	1	1	yes	characteristic
82	C₈×C₃₁	C₄×Q₈	1	2	yes	—
83	C₂×#365	C₂×C₃₁	1	2	yes	—
84	C₄×M₃₂	C₂×C₃₁	1	1	yes	characteristic
85	C₄×#108	C₂×C₃₁	1	1	yes	characteristic
86	C₄×#106	C₂×C₃₁	1	1	yes	characteristic
87	#1383	C₂×C₃₁	1	2	yes	—
88	C₂²×C₃₁	—	2	1	no	—
89	C₂²×C₃₁	—	2	1	no	—
90	C₂²×C₃₁	C₂×#106	1	1	yes	characteristic
91	C₂²×C₃₁	C₄×SD₁₆	1	2	yes	—
92	C₄×C₃₁	—	2	1	no	—
93	C₄×C₃₁	—	2	1	no	—
94	C₄×C₃₁	—	2	1	no	—
95	C₄×C₃₁	—	8	4	no	—
96	C₄×C₃₁	#365	1	2	yes	direct product factor
97	C₄×C₃₁	C₄×C₄⋊C₄	1	1	yes	characteristic
98	C₄×C₃₁	#427	1	1	yes	characteristic
99	C₄×C₃₁	C₄×SD₁₆	1	2	yes	—
100	C₂×M₃₂	C₂²×C₃₁	1	1	yes	characteristic
101	C₂×M₃₂	C₂²×C₃₁	1	1	yes	characteristic
102	C₂×#108	C₂²×C₃₁	1	2	yes	—
103	C₂×#106	C₂²×C₃₁	1	2	yes	—
104	C₄×M₁₆	—	2	1	no	—
105	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
106	C₄×C₄⋊C₄	—	2	1	no	—
107	#365	C₄×C₃₁	1	8	yes	direct product factor
108	#346	C₄×C₃₁	1	2	yes	complemented
109	C₄×C₁₆	C₄×C₃₁	1	2	yes	complemented
110	#341	—	2	1	no	—
111	#340	—	2	1	no	—
112	C₂×C₃₁	—	2	2	no	—
113	C₂×C₃₁	C₄×#106	1	1	yes	characteristic
114	C₂×C₃₁	C₂×#365	1	1	yes	characteristic
115	C₂×C₃₁	#2155	1	1	yes	characteristic
116	C₂×M₁₆	—	2	2	no	—
117	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
118	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
119	C₂×C₄⋊C₄	—	2	2	no	—
120	C₂×C₄²	—	4	1	no	—
121	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
122	M₃₂	—	2	2	no	—
123	M₃₂	C₂×C₄×C₃₁	1	4	yes	—
124	C₂×C₁₆	C₂×C₄×C₃₁	1	2	yes	—
125	C₂×C₁₆	C₂×C₄×C₃₁	1	2	yes	—
126	#108	—	2	2	no	—
127	#108	C₂×C₄×C₃₁	1	4	yes	—
128	#106	—	2	2	no	—
129	#106	C₂×C₄×C₃₁	1	4	yes	—
130	#97	—	4	1	no	—
131	C₄×C₈	—	4	1	no	—
132	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
133	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
134	C₄×C₈	Q₈×C₃₁	1	1	yes	characteristic
135	C₄×C₈	Q₈×C₃₁	1	1	yes	characteristic
136	#95	—	4	1	no	—
137	C₃₁	C₄×#365	1	1	yes	characteristic, direct product factor, Sylow
138	C₂²×C₄	—	4	2	no	—
139	C₂²×C₄	SD₁₆×C₃₁	1	1	yes	characteristic
140	C₂²×C₄	SD₁₆×C₃₁	1	1	yes	characteristic
141	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
142	M₁₆	—	2	4	no	—
143	M₁₆	—	4	2	no	—
144	C₂×C₈	—	2	2	no	—
145	C₂×C₈	—	4	2	no	—
146	C₂×C₈	C₄⋊C₄×C₃₁	1	1	yes	characteristic
147	C₂×C₈	C₄⋊C₄×C₃₁	1	1	yes	characteristic
148	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
149	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
150	C₂×C₈	C₂×Q₈×C₃₁	1	1	yes	characteristic
151	C₂×C₈	#41×C₃₁	1	2	yes	—
152	C₄⋊C₄	—	2	4	no	—
153	C₄⋊C₄	—	4	2	no	—
154	C₄²	—	8	2	no	—
155	C₄²	C₄⋊C₄×C₃₁	1	1	yes	characteristic
156	C₄²	C₄⋊C₄×C₃₁	1	1	yes	characteristic
157	C₄²	SD₁₆×C₃₁	1	1	yes	characteristic
158	C₄²	SD₁₆×C₃₁	1	1	yes	characteristic
159	C₁₆	—	2	2	no	—
160	C₁₆	C₄²×C₃₁	1	4	yes	complemented
161	C₂³	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	4	1	no	—
164	C₂×C₄	—	4	4	no	—
165	C₂×C₄	—	8	2	no	—
166	C₂×C₄	—	8	2	no	—
167	C₂×C₄	C₃₁×#106	1	1	yes	characteristic
168	C₂×C₄	C₃₁×#106	1	1	yes	characteristic
169	C₂×C₄	C₃₁×#106	1	1	yes	characteristic
170	C₂×C₄	C₃₁×#106	1	1	yes	characteristic
171	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
172	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
173	C₂×C₄	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
174	C₂×C₄	C₃₁×#135	1	2	yes	—
175	C₈	—	2	1	no	—
176	C₈	—	2	1	no	—
177	C₈	—	4	4	no	—
178	C₈	C₂×C₄²×C₃₁	1	1	yes	characteristic
179	C₈	C₃₁×#117	1	1	yes	characteristic
180	C₈	C₄×Q₈×C₃₁	1	2	yes	—
181	C₂²	—	2	1	no	—
182	C₂²	—	2	1	no	—
183	C₂²	C₂×C₃₁×#106	1	1	yes	characteristic
184	C₂²	C₄×SD₁₆×C₃₁	1	2	yes	—
185	C₄	—	2	1	no	—
186	C₄	—	2	1	no	—
187	C₄	—	2	1	no	—
188	C₄	—	8	4	no	—
189	C₄	C₃₁×#365	1	2	yes	direct product factor
190	C₄	C₄×C₄⋊C₄×C₃₁	1	1	yes	characteristic
191	C₄	C₃₁×#427	1	1	yes	characteristic
192	C₄	C₄×SD₁₆×C₃₁	1	2	yes	—
193	C₂	—	2	2	no	—
194	C₂	C₄×C₃₁×#106	1	1	yes	characteristic
195	C₂	C₂×C₃₁×#365	1	1	yes	characteristic
196	C₂	C₃₁×#2155	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₁	#106
commutator	C₈	C₂×C₄²×C₃₁
Frattini, cosocle	C₂²×C₈	C₂³×C₃₁
socle	C₂²×C₃₁	C₂×#106
prime radical	C₂³×C₃₁	C₂×SD₁₆
LCS	G ▷ 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
13. C₃₁×#365	C₄	direct (N × H)
14. C₃₁×#346	C₄	semidirect
15. C₄×C₁₆×C₃₁	C₄	semidirect
61. C₁₆×C₃₁	C₄²	semidirect
62. C₄×#365	C₃₁	direct (N × H)
96. C₄×C₃₁	#365	direct (N × H)
107. #365	C₄×C₃₁	direct (N × H)
108. #346	C₄×C₃₁	semidirect
109. C₄×C₁₆	C₄×C₃₁	semidirect
137. C₃₁	C₄×#365	direct (N × H)
160. C₁₆	C₄²×C₃₁	semidirect
189. C₄	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
C₂×C₃₁×#365	2
C₄×C₃₁×M₃₂	2
C₄×C₃₁×#108	2
C₄×C₃₁×#106	2
C₃₁×#1383	2
C₄×#365	31

Subgroups — C4×C31×#365 #21469428

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₃₁×#365 #21469428