Subgroups — C₃₁×#13190 #21469949

Order 7936.

522 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×D₄×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×C₂²⋊C₁₆	C₂	1	4	yes	maximal, complemented
4	C₂×C₃₁×#346	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂³×C₈×C₃₁	C₂²	1	2	yes	—
6	C₂×C₄×D₄×C₃₁	C₄	1	1	yes	characteristic
7	C₂²×C₁₆×C₃₁	C₂²	1	2	yes	—
8	C₂²×C₁₆×C₃₁	C₂²	1	2	yes	—
9	C₈×D₄×C₃₁	—	2	2	no	—
10	C₈×D₄×C₃₁	—	2	2	no	—
11	C₂×C₃₁×C₄⋊C₈	C₄	1	1	yes	characteristic
12	C₂×C₃₁×C₂²⋊C₈	C₄	1	2	yes	—
13	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
14	C₃₁×C₄⋊C₁₆	—	2	2	no	—
15	C₃₁×C₂²⋊C₁₆	—	2	8	no	—
16	C₃₁×#346	—	2	2	no	—
17	C₂²×D₄×C₃₁	C₈	1	1	yes	characteristic
18	C₂³×C₄×C₃₁	C₂×C₄	1	2	yes	—
19	C₂²×C₈×C₃₁	—	2	2	no	—
20	C₂²×C₈×C₃₁	—	2	2	no	—
21	C₂²×C₈×C₃₁	—	2	4	no	—
22	C₂²×C₈×C₃₁	—	2	4	no	—
23	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
24	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
26	C₄×D₄×C₃₁	—	2	2	no	—
27	C₄×D₄×C₃₁	—	2	2	no	—
28	C₂×C₄⋊C₄×C₃₁	C₈	1	1	yes	characteristic
29	C₂×C₂²⋊C₄×C₃₁	C₈	1	2	yes	—
30	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
31	C₂×C₁₆×C₃₁	—	2	2	no	—
32	C₂×C₁₆×C₃₁	—	2	2	no	—
33	C₂×C₁₆×C₃₁	—	2	4	no	—
34	C₂×C₁₆×C₃₁	—	2	2	no	—
35	C₂×C₁₆×C₃₁	D₄	1	4	yes	complemented
36	C₃₁×C₄⋊C₈	—	2	2	no	—
37	C₃₁×C₂²⋊C₈	—	2	4	no	—
38	C₄×C₈×C₃₁	D₄	1	2	yes	—
39	C₄×C₈×C₃₁	D₄	1	2	yes	—
40	C₂⁴×C₃₁	C₂×C₈	1	2	yes	—
41	C₂×D₄×C₃₁	—	2	4	no	—
42	C₂×D₄×C₃₁	—	2	2	no	—
43	C₂²×C₄×C₃₁	—	2	2	no	—
44	C₂²×C₄×C₃₁	—	2	4	no	—
45	C₂²×C₄×C₃₁	—	2	2	no	—
46	C₂²×C₄×C₃₁	—	2	4	no	—
47	C₂²×C₄×C₃₁	C₂×C₈	1	2	yes	—
48	C₂²×C₄×C₃₁	C₂×C₈	1	1	yes	characteristic
49	C₂²×C₄×C₃₁	C₂×C₈	1	1	yes	characteristic
50	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₈×C₃₁	—	2	2	no	—
52	C₂×C₈×C₃₁	—	2	2	no	—
53	C₂×C₈×C₃₁	—	2	4	no	—
54	C₂×C₈×C₃₁	—	4	4	no	—
55	C₂×C₈×C₃₁	—	4	2	no	—
56	C₂×C₈×C₃₁	—	4	2	no	—
57	C₂×C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
58	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
60	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
61	C₂×C₈×C₃₁	#41	1	2	yes	—
62	C₄⋊C₄×C₃₁	—	2	2	no	—
63	C₂²⋊C₄×C₃₁	—	2	4	no	—
64	C₄²×C₃₁	C₂²⋊C₄	1	2	yes	—
65	C₄²×C₃₁	M₁₆	1	2	yes	—
66	C₁₆×C₃₁	—	2	4	no	—
67	C₁₆×C₃₁	—	4	2	no	—
68	#13190	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
69	C₂³×C₃₁	—	2	2	no	—
70	C₂³×C₃₁	—	2	4	no	—
71	C₂³×C₃₁	—	2	2	no	—
72	C₂³×C₃₁	—	2	4	no	—
73	C₂³×C₃₁	M₃₂	1	4	yes	—
74	C₂³×C₃₁	C₂²×C₈	1	1	yes	characteristic
75	D₄×C₃₁	—	2	8	no	—
76	C₂×C₄×C₃₁	—	2	2	no	—
77	C₂×C₄×C₃₁	—	2	2	no	—
78	C₂×C₄×C₃₁	—	2	2	no	—
79	C₂×C₄×C₃₁	—	2	4	no	—
80	C₂×C₄×C₃₁	—	4	2	no	—
81	C₂×C₄×C₃₁	—	4	2	no	—
82	C₂×C₄×C₃₁	—	4	4	no	—
83	C₂×C₄×C₃₁	C₂²⋊C₈	1	2	yes	—
84	C₂×C₄×C₃₁	C₂²⋊C₈	1	2	yes	—
85	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
87	C₂×C₄×C₃₁	C₂×M₁₆	1	1	yes	characteristic
88	C₂×C₄×C₃₁	#131	1	2	yes	—
89	C₈×C₃₁	—	2	2	no	—
90	C₈×C₃₁	—	4	2	no	—
91	C₈×C₃₁	#121	1	2	yes	—
92	C₈×C₃₁	#124	1	1	yes	characteristic
93	C₈×C₃₁	#127	1	1	yes	characteristic
94	C₂×C₈×D₄	C₂×C₃₁	1	1	yes	characteristic
95	C₂×C₄⋊C₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
96	C₂×C₂²⋊C₁₆	C₂×C₃₁	1	4	yes	complemented
97	C₂×#346	C₂×C₃₁	1	1	yes	characteristic, complemented
98	C₂²×C₃₁	—	2	4	no	—
99	C₂²×C₃₁	—	2	8	no	—
100	C₂²×C₃₁	—	4	2	no	—
101	C₂²×C₃₁	—	4	4	no	—
102	C₂²×C₃₁	—	4	2	no	—
103	C₂²×C₃₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
104	C₂²×C₃₁	C₈×D₄	1	2	yes	—
105	C₂²×C₃₁	C₂×M₃₂	1	2	yes	—
106	C₂²×C₃₁	#504	1	2	yes	—
107	C₄×C₃₁	—	2	2	no	—
108	C₄×C₃₁	—	2	2	no	—
109	C₄×C₃₁	—	4	2	no	—
110	C₄×C₃₁	#390	1	1	yes	characteristic
111	C₄×C₃₁	#408	1	1	yes	characteristic
112	C₄×C₃₁	#436	1	2	yes	—
113	C₂³×C₈	C₂²×C₃₁	1	2	yes	—
114	C₂×C₄×D₄	C₄×C₃₁	1	1	yes	characteristic
115	C₂²×C₁₆	C₂²×C₃₁	1	2	yes	—
116	C₂²×C₁₆	C₂²×C₃₁	1	2	yes	—
117	C₈×D₄	—	2	2	no	—
118	C₈×D₄	—	2	2	no	—
119	C₂×C₄⋊C₈	C₄×C₃₁	1	1	yes	characteristic
120	C₂×C₂²⋊C₈	C₄×C₃₁	1	2	yes	—
121	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
122	C₄⋊C₁₆	—	2	2	no	—
123	C₂²⋊C₁₆	—	2	8	no	—
124	#346	—	2	2	no	—
125	C₂×C₃₁	—	4	4	no	—
126	C₂×C₃₁	#1954	1	1	yes	characteristic
127	C₂×C₃₁	#2112	1	1	yes	characteristic
128	C₂×C₃₁	#2113	1	1	yes	characteristic
129	C₂×C₃₁	#2168	1	4	yes	—
130	C₂²×D₄	C₈×C₃₁	1	1	yes	characteristic
131	C₂³×C₄	C₂×C₄×C₃₁	1	2	yes	—
132	C₂²×C₈	—	2	4	no	—
133	C₂²×C₈	—	2	2	no	—
134	C₂²×C₈	—	2	2	no	—
135	C₂²×C₈	—	2	4	no	—
136	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
137	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
138	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
139	C₄×D₄	—	2	2	no	—
140	C₄×D₄	—	2	2	no	—
141	C₂×C₄⋊C₄	C₈×C₃₁	1	1	yes	characteristic
142	C₂×C₂²⋊C₄	C₈×C₃₁	1	2	yes	—
143	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
144	C₂×C₁₆	—	2	2	no	—
145	C₂×C₁₆	—	2	2	no	—
146	C₂×C₁₆	—	2	4	no	—
147	C₂×C₁₆	—	2	2	no	—
148	C₂×C₁₆	D₄×C₃₁	1	4	yes	complemented
149	C₄⋊C₈	—	2	2	no	—
150	C₂²⋊C₈	—	2	4	no	—
151	C₄×C₈	D₄×C₃₁	1	2	yes	—
152	C₄×C₈	D₄×C₃₁	1	2	yes	—
153	C₃₁	#13190	1	1	yes	characteristic, direct product factor, Sylow
154	C₂⁴	C₂×C₈×C₃₁	1	2	yes	—
155	C₂×D₄	—	2	4	no	—
156	C₂×D₄	—	2	2	no	—
157	C₂²×C₄	—	2	2	no	—
158	C₂²×C₄	—	2	4	no	—
159	C₂²×C₄	—	2	2	no	—
160	C₂²×C₄	—	2	4	no	—
161	C₂²×C₄	C₂×C₈×C₃₁	1	2	yes	—
162	C₂²×C₄	C₂×C₈×C₃₁	1	1	yes	characteristic
163	C₂²×C₄	C₂×C₈×C₃₁	1	1	yes	characteristic
164	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	—	2	2	no	—
167	C₂×C₈	—	2	4	no	—
168	C₂×C₈	—	4	4	no	—
169	C₂×C₈	—	4	2	no	—
170	C₂×C₈	—	4	2	no	—
171	C₂×C₈	C₂²⋊C₄×C₃₁	1	2	yes	—
172	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
173	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
174	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
175	C₂×C₈	#41×C₃₁	1	2	yes	—
176	C₄⋊C₄	—	2	2	no	—
177	C₂²⋊C₄	—	2	4	no	—
178	C₄²	C₂²⋊C₄×C₃₁	1	2	yes	—
179	C₄²	M₁₆×C₃₁	1	2	yes	—
180	C₁₆	—	2	4	no	—
181	C₁₆	—	4	2	no	—
182	C₂³	—	2	2	no	—
183	C₂³	—	2	4	no	—
184	C₂³	—	2	2	no	—
185	C₂³	—	2	4	no	—
186	C₂³	C₃₁×M₃₂	1	4	yes	—
187	C₂³	C₂²×C₈×C₃₁	1	1	yes	characteristic
188	D₄	—	2	8	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	4	no	—
193	C₂×C₄	—	4	2	no	—
194	C₂×C₄	—	4	4	no	—
195	C₂×C₄	—	4	2	no	—
196	C₂×C₄	C₃₁×C₂²⋊C₈	1	2	yes	—
197	C₂×C₄	C₃₁×C₂²⋊C₈	1	2	yes	—
198	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
199	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
200	C₂×C₄	C₂×M₁₆×C₃₁	1	1	yes	characteristic
201	C₂×C₄	C₃₁×#131	1	2	yes	—
202	C₈	—	2	2	no	—
203	C₈	—	4	2	no	—
204	C₈	C₃₁×#121	1	2	yes	—
205	C₈	C₃₁×#124	1	1	yes	characteristic
206	C₈	C₃₁×#127	1	1	yes	characteristic
207	C₂²	—	2	4	no	—
208	C₂²	—	2	8	no	—
209	C₂²	—	4	2	no	—
210	C₂²	—	4	4	no	—
211	C₂²	—	4	2	no	—
212	C₂²	C₂×C₃₁×C₂²⋊C₈	1	1	yes	characteristic
213	C₂²	C₈×D₄×C₃₁	1	2	yes	—
214	C₂²	C₂×C₃₁×M₃₂	1	2	yes	—
215	C₂²	C₃₁×#504	1	2	yes	—
216	C₄	—	2	2	no	—
217	C₄	—	2	2	no	—
218	C₄	—	4	2	no	—
219	C₄	C₃₁×#390	1	1	yes	characteristic
220	C₄	C₃₁×#408	1	1	yes	characteristic
221	C₄	C₃₁×#436	1	2	yes	—
222	C₂	—	4	4	no	—
223	C₂	C₃₁×#1954	1	1	yes	characteristic
224	C₂	C₃₁×#2112	1	1	yes	characteristic
225	C₂	C₃₁×#2113	1	1	yes	characteristic
226	C₂	C₃₁×#2168	1	4	yes	—

	subgroup	G/H
center	C₂²×C₈×C₃₁	C₂³
commutator	C₂³	C₂²×C₈×C₃₁
Frattini, cosocle	C₂²×C₈	C₂³×C₃₁
socle	C₂³×C₃₁	C₂²×C₈
prime radical	C₂²×D₄×C₃₁	C₈
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃₁×C₄⋊C₁₆	C₂	semidirect
3. C₂×C₃₁×C₂²⋊C₁₆	C₂	semidirect
4. C₂×C₃₁×#346	C₂	semidirect
35. C₂×C₁₆×C₃₁	D₄	semidirect
68. #13190	C₃₁	direct (N × H)
95. C₂×C₄⋊C₁₆	C₂×C₃₁	semidirect
96. C₂×C₂²⋊C₁₆	C₂×C₃₁	semidirect
97. C₂×#346	C₂×C₃₁	semidirect
148. C₂×C₁₆	D₄×C₃₁	semidirect
153. C₃₁	#13190	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×C₈×D₄×C₃₁	2
C₂×C₃₁×C₄⋊C₁₆	2
C₂×C₃₁×C₂²⋊C₁₆	2
C₂×C₃₁×#346	2
#13190	31

Subgroups — C31×#13190 #21469949

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#13190 #21469949