Subgroups — C₂×C₃₁×#1365 #21468697

Order 7936.

350 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#500	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₁×#335	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1365	C₂	1	4	yes	maximal, direct product factor
5	C₂×C₃₁×C₄⋊Q₈	—	2	1	no	—
6	C₂²×C₁₆×C₃₁	—	2	1	no	—
7	C₃₁×#500	—	2	1	no	—
8	C₃₁×#500	—	2	1	no	—
9	C₃₁×#500	C₄	1	2	yes	—
10	C₃₁×#500	C₂²	1	2	yes	—
11	C₂×C₃₁×#107	—	2	1	no	—
12	C₂×C₃₁×#107	C₄	1	2	yes	—
13	C₂×C₃₁×C₄⋊C₈	—	2	1	no	—
14	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
15	C₃₁×C₄⋊C₁₆	—	2	1	no	—
16	C₃₁×C₄⋊C₁₆	C₂²	1	2	yes	—
17	C₃₁×#335	—	2	1	no	—
18	C₃₁×#335	C₂²	1	2	yes	—
19	C₂²×Q₈×C₃₁	—	4	1	no	—
20	C₂²×C₈×C₃₁	—	4	1	no	—
21	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
23	C₃₁×C₄⋊Q₈	—	2	2	no	—
24	C₃₁×C₄⋊Q₈	—	2	2	no	—
25	C₃₁×C₄⋊Q₈	—	4	1	no	—
26	C₃₁×C₄⋊Q₈	—	4	1	no	—
27	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
28	C₂×C₄⋊C₄×C₃₁	—	4	1	no	—
29	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
30	C₂×C₁₆×C₃₁	—	2	2	no	—
31	C₂×C₁₆×C₃₁	—	4	2	no	—
32	C₃₁×#107	—	2	2	no	—
33	C₃₁×#107	—	2	2	no	—
34	C₃₁×#107	—	2	2	no	—
35	C₃₁×#107	C₂×C₄	1	4	yes	—
36	C₃₁×C₄⋊C₈	—	2	2	no	—
37	C₃₁×C₄⋊C₈	—	4	1	no	—
38	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
39	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
40	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
41	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
42	C₂×Q₈×C₃₁	—	4	2	no	—
43	C₂×Q₈×C₃₁	—	4	2	no	—
44	C₂×Q₈×C₃₁	—	8	4	no	—
45	C₂²×C₄×C₃₁	—	4	2	no	—
46	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
47	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
48	C₂²×C₄×C₃₁	SD₁₆	1	1	yes	characteristic
49	C₂×C₈×C₃₁	—	2	2	no	—
50	C₂×C₈×C₃₁	—	4	2	no	—
51	C₂×C₈×C₃₁	—	8	2	no	—
52	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×C₈×C₃₁	SD₁₆	1	2	yes	—
55	C₂×C₈×C₃₁	Q₁₆	1	2	yes	—
56	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
58	C₄⋊C₄×C₃₁	—	2	4	no	—
59	C₄⋊C₄×C₃₁	—	4	2	no	—
60	C₄⋊C₄×C₃₁	—	4	2	no	—
61	C₄⋊C₄×C₃₁	—	4	2	no	—
62	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
63	C₄²×C₃₁	D₈	1	1	yes	characteristic
64	C₄²×C₃₁	SD₁₆	1	1	yes	characteristic
65	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
66	C₁₆×C₃₁	—	4	2	no	—
67	C₂×#1365	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
68	C₂³×C₃₁	#102	1	1	yes	characteristic
69	Q₈×C₃₁	—	8	4	no	—
70	Q₈×C₃₁	—	8	4	no	—
71	C₂×C₄×C₃₁	—	2	2	no	—
72	C₂×C₄×C₃₁	—	4	4	no	—
73	C₂×C₄×C₃₁	—	8	4	no	—
74	C₂×C₄×C₃₁	#101	1	2	yes	—
75	C₂×C₄×C₃₁	#102	1	1	yes	characteristic
76	C₂×C₄×C₃₁	#102	1	1	yes	characteristic
77	C₂×C₄×C₃₁	#102	1	1	yes	characteristic
78	C₂×C₄×C₃₁	#103	1	2	yes	—
79	C₂×C₄×C₃₁	SD₃₂	1	2	yes	—
80	C₂×C₄×C₃₁	Q₃₂	1	2	yes	—
81	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
83	C₂×C₄×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
84	C₈×C₃₁	—	2	1	no	—
85	C₈×C₃₁	—	2	1	no	—
86	C₈×C₃₁	—	8	2	no	—
87	C₈×C₃₁	#103	1	2	yes	—
88	C₈×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
89	C₈×C₃₁	C₂×Q₁₆	1	1	yes	characteristic
90	C₂×#500	C₂×C₃₁	1	1	yes	characteristic
91	C₂×C₄⋊C₁₆	C₂×C₃₁	1	1	yes	characteristic
92	C₂×#335	C₂×C₃₁	1	1	yes	characteristic
93	#1365	C₂×C₃₁	1	4	yes	direct product factor
94	C₂²×C₃₁	#332	1	2	yes	—
95	C₂²×C₃₁	#358	1	2	yes	—
96	C₂²×C₃₁	#361	1	2	yes	—
97	C₂²×C₃₁	C₂×#102	1	1	yes	characteristic
98	C₄×C₃₁	—	2	1	no	—
99	C₄×C₃₁	—	2	1	no	—
100	C₄×C₃₁	—	8	4	no	—
101	C₄×C₃₁	#332	1	2	yes	—
102	C₄×C₃₁	#358	1	2	yes	—
103	C₄×C₃₁	C₂×#101	1	1	yes	characteristic
104	C₄×C₃₁	C₂×#103	1	1	yes	characteristic
105	C₄×C₃₁	C₂×SD₃₂	1	1	yes	characteristic
106	C₄×C₃₁	C₂×Q₃₂	1	1	yes	characteristic
107	C₂×C₄⋊Q₈	—	2	1	no	—
108	C₂²×C₁₆	—	2	1	no	—
109	#500	—	2	1	no	—
110	#500	—	2	1	no	—
111	#500	C₄×C₃₁	1	2	yes	—
112	#500	C₂²×C₃₁	1	2	yes	—
113	C₂×#107	—	2	1	no	—
114	C₂×#107	C₄×C₃₁	1	2	yes	—
115	C₂×C₄⋊C₈	—	2	1	no	—
116	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
117	C₄⋊C₁₆	—	2	1	no	—
118	C₄⋊C₁₆	C₂²×C₃₁	1	2	yes	—
119	#335	—	2	1	no	—
120	#335	C₂²×C₃₁	1	2	yes	—
121	C₂×C₃₁	#1365	1	4	yes	direct product factor
122	C₂×C₃₁	C₂×#332	1	1	yes	characteristic
123	C₂×C₃₁	C₂×#358	1	1	yes	characteristic
124	C₂×C₃₁	C₂×#361	1	1	yes	characteristic
125	C₂²×Q₈	—	4	1	no	—
126	C₂²×C₈	—	4	1	no	—
127	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
128	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
129	C₄⋊Q₈	—	2	2	no	—
130	C₄⋊Q₈	—	2	2	no	—
131	C₄⋊Q₈	—	4	1	no	—
132	C₄⋊Q₈	—	4	1	no	—
133	C₂×C₄⋊C₄	—	2	2	no	—
134	C₂×C₄⋊C₄	—	4	1	no	—
135	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
136	C₂×C₁₆	—	2	2	no	—
137	C₂×C₁₆	—	4	2	no	—
138	#107	—	2	2	no	—
139	#107	—	2	2	no	—
140	#107	—	2	2	no	—
141	#107	C₂×C₄×C₃₁	1	4	yes	—
142	C₄⋊C₈	—	2	2	no	—
143	C₄⋊C₈	—	4	1	no	—
144	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
145	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
146	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
147	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
148	C₃₁	C₂×#1365	1	1	yes	characteristic, direct product factor, Sylow
149	C₂×Q₈	—	4	2	no	—
150	C₂×Q₈	—	4	2	no	—
151	C₂×Q₈	—	8	4	no	—
152	C₂²×C₄	—	4	2	no	—
153	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
154	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
155	C₂²×C₄	SD₁₆×C₃₁	1	1	yes	characteristic
156	C₂×C₈	—	2	2	no	—
157	C₂×C₈	—	4	2	no	—
158	C₂×C₈	—	8	2	no	—
159	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
160	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
161	C₂×C₈	SD₁₆×C₃₁	1	2	yes	—
162	C₂×C₈	Q₁₆×C₃₁	1	2	yes	—
163	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
164	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
165	C₄⋊C₄	—	2	4	no	—
166	C₄⋊C₄	—	4	2	no	—
167	C₄⋊C₄	—	4	2	no	—
168	C₄⋊C₄	—	4	2	no	—
169	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
170	C₄²	D₈×C₃₁	1	1	yes	characteristic
171	C₄²	SD₁₆×C₃₁	1	1	yes	characteristic
172	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
173	C₁₆	—	4	2	no	—
174	C₂³	C₃₁×#102	1	1	yes	characteristic
175	Q₈	—	8	4	no	—
176	Q₈	—	8	4	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	4	4	no	—
179	C₂×C₄	—	8	4	no	—
180	C₂×C₄	C₃₁×#101	1	2	yes	—
181	C₂×C₄	C₃₁×#102	1	1	yes	characteristic
182	C₂×C₄	C₃₁×#102	1	1	yes	characteristic
183	C₂×C₄	C₃₁×#102	1	1	yes	characteristic
184	C₂×C₄	C₃₁×#103	1	2	yes	—
185	C₂×C₄	C₃₁×SD₃₂	1	2	yes	—
186	C₂×C₄	C₃₁×Q₃₂	1	2	yes	—
187	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
188	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
189	C₂×C₄	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
190	C₈	—	2	1	no	—
191	C₈	—	2	1	no	—
192	C₈	—	8	2	no	—
193	C₈	C₃₁×#103	1	2	yes	—
194	C₈	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
195	C₈	C₂×Q₁₆×C₃₁	1	1	yes	characteristic
196	C₂²	C₃₁×#332	1	2	yes	—
197	C₂²	C₃₁×#358	1	2	yes	—
198	C₂²	C₃₁×#361	1	2	yes	—
199	C₂²	C₂×C₃₁×#102	1	1	yes	characteristic
200	C₄	—	2	1	no	—
201	C₄	—	2	1	no	—
202	C₄	—	8	4	no	—
203	C₄	C₃₁×#332	1	2	yes	—
204	C₄	C₃₁×#358	1	2	yes	—
205	C₄	C₂×C₃₁×#101	1	1	yes	characteristic
206	C₄	C₂×C₃₁×#103	1	1	yes	characteristic
207	C₄	C₂×C₃₁×SD₃₂	1	1	yes	characteristic
208	C₄	C₂×C₃₁×Q₃₂	1	1	yes	characteristic
209	C₂	C₃₁×#1365	1	4	yes	direct product factor
210	C₂	C₂×C₃₁×#332	1	1	yes	characteristic
211	C₂	C₂×C₃₁×#358	1	1	yes	characteristic
212	C₂	C₂×C₃₁×#361	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₃₁	#102
commutator	C₂×C₈	C₂²×C₄×C₃₁
Frattini, cosocle	C₄×C₈	C₂³×C₃₁
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×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
4. C₃₁×#1365	C₂	direct (N × H)
67. C₂×#1365	C₃₁	direct (N × H)
93. #1365	C₂×C₃₁	direct (N × H)
121. C₂×C₃₁	#1365	direct (N × H)
148. C₃₁	C₂×#1365	direct (N × H)
209. C₂	C₃₁×#1365	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×#500	2
C₂×C₃₁×C₄⋊C₁₆	2
C₂×C₃₁×#335	2
C₃₁×#1365	2
C₂×#1365	31

Subgroups — C2×C31×#1365 #21468697

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#1365 #21468697