Subgroups — C₃₁×#10814 #21467573

Order 7936.

658 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#405	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1740	C₂	1	2	yes	maximal
3	C₃₁×#1735	C₂	1	2	yes	maximal
4	C₂×C₃₁×H(3,ℤ/4)	C₂	1	2	yes	maximal
5	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
6	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
7	C₂×C₃₁×#117	C₂²	1	4	yes	—
8	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
9	C₃₁×#405	—	2	4	no	—
10	C₂×C₃₁×#97	C₄	1	2	yes	complemented
11	C₂×C₄×C₈×C₃₁	C₄	1	2	yes	complemented
12	C₃₁×#342	—	2	4	no	—
13	C₃₁×#340	—	2	4	no	—
14	C₃₁×H(3,ℤ/4)	C₄	1	8	yes	complemented
15	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
16	C₂×M₁₆×C₃₁	—	2	2	no	—
17	C₂×M₁₆×C₃₁	—	2	2	no	—
18	C₂×M₁₆×C₃₁	C₂×C₄	1	4	yes	—
19	C₂²×C₈×C₃₁	—	2	2	no	—
20	C₂²×C₈×C₃₁	—	2	2	no	—
21	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
22	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₃₁	D₄	1	4	yes	—
25	C₃₁×#117	—	2	2	no	—
26	C₃₁×#117	—	2	2	no	—
27	C₃₁×#117	—	2	4	no	—
28	C₃₁×#117	—	2	4	no	—
29	C₃₁×#117	—	2	4	no	—
30	C₃₁×#117	C₂×C₄	1	8	yes	—
31	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
32	C₂×C₄⋊C₄×C₃₁	Q₈	1	2	yes	—
33	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
34	C₂×C₂²⋊C₄×C₃₁	D₄	1	2	yes	—
35	C₂×C₄²×C₃₁	—	2	4	no	—
36	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
37	C₃₁×#97	—	2	4	no	—
38	C₄×C₈×C₃₁	—	2	4	no	—
39	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄×C₃₁	—	4	4	no	—
41	C₂²×C₄×C₃₁	C₄⋊C₄	1	4	yes	—
42	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
44	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
45	C₂²×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
46	C₂²×C₄×C₃₁	#41	1	2	yes	—
47	C₂²×C₄×C₃₁	#41	1	2	yes	—
48	C₂²×C₄×C₃₁	#41	1	2	yes	—
49	C₂²×C₄×C₃₁	#41	1	2	yes	—
50	M₁₆×C₃₁	—	2	8	no	—
51	C₂×C₈×C₃₁	—	2	2	no	—
52	C₂×C₈×C₃₁	—	2	1	no	—
53	C₂×C₈×C₃₁	—	2	1	no	—
54	C₂×C₈×C₃₁	—	2	1	no	—
55	C₂×C₈×C₃₁	—	2	1	no	—
56	C₂×C₈×C₃₁	—	2	2	no	—
57	C₂×C₈×C₃₁	—	2	8	no	—
58	C₂×C₈×C₃₁	C₄²	1	4	yes	—
59	C₂×C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
60	C₂×C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
61	C₄⋊C₄×C₃₁	—	2	4	no	—
62	C₄⋊C₄×C₃₁	—	2	8	no	—
63	C₄⋊C₄×C₃₁	—	4	4	no	—
64	C₂²⋊C₄×C₃₁	—	2	4	no	—
65	C₂²⋊C₄×C₃₁	—	2	8	no	—
66	C₂²⋊C₄×C₃₁	—	4	4	no	—
67	C₄²×C₃₁	—	2	2	no	—
68	C₄²×C₃₁	—	2	2	no	—
69	C₄²×C₃₁	—	2	8	no	—
70	C₄²×C₃₁	—	4	4	no	—
71	#10814	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
72	C₂³×C₃₁	—	4	2	no	—
73	C₂³×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₂³×C₃₁	C₄×D₄	1	2	yes	—
75	C₂³×C₃₁	#121	1	2	yes	—
76	C₂³×C₃₁	#124	1	2	yes	—
77	C₂×C₄×C₃₁	—	2	4	no	—
78	C₂×C₄×C₃₁	—	4	4	no	—
79	C₂×C₄×C₃₁	—	4	8	no	—
80	C₂×C₄×C₃₁	—	8	8	no	—
81	C₂×C₄×C₃₁	C₂×C₄²	1	1	yes	characteristic
82	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₃₁	#117	1	1	yes	characteristic
84	C₂×C₄×C₃₁	#117	1	4	yes	—
85	C₂×C₄×C₃₁	#117	1	1	yes	characteristic
86	C₂×C₄×C₃₁	#117	1	1	yes	characteristic
87	C₂×C₄×C₃₁	#117	1	1	yes	characteristic
88	C₂×C₄×C₃₁	C₄×D₄	1	4	yes	—
89	C₂×C₄×C₃₁	C₄×Q₈	1	2	yes	—
90	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
91	C₂×C₄×C₃₁	#123	1	4	yes	—
92	C₂×C₄×C₃₁	#125	1	2	yes	—
93	C₂×C₄×C₃₁	#126	1	4	yes	—
94	C₈×C₃₁	—	2	4	no	—
95	C₈×C₃₁	—	2	2	no	—
96	C₈×C₃₁	—	2	2	no	—
97	C₂×#405	C₂×C₃₁	1	1	yes	characteristic
98	#1740	C₂×C₃₁	1	2	yes	—
99	#1735	C₂×C₃₁	1	2	yes	—
100	C₂×H(3,ℤ/4)	C₂×C₃₁	1	2	yes	—
101	C₂²×C₃₁	—	2	2	no	—
102	C₂²×C₃₁	—	2	4	no	—
103	C₂²×C₃₁	—	4	2	no	—
104	C₂²×C₃₁	—	4	2	no	—
105	C₂²×C₃₁	C₄×C₄⋊C₄	1	1	yes	characteristic
106	C₂²×C₃₁	#380	1	1	yes	characteristic
107	C₂²×C₃₁	#382	1	1	yes	characteristic
108	C₂²×C₃₁	#388	1	4	yes	—
109	C₄×C₃₁	—	4	4	no	—
110	C₄×C₃₁	—	8	8	no	—
111	C₄×C₃₁	#376	1	1	yes	characteristic
112	C₄×C₃₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
113	C₄×C₃₁	#381	1	1	yes	characteristic
114	C₄×C₃₁	#383	1	1	yes	characteristic
115	C₄×C₃₁	#387	1	4	yes	—
116	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
117	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
118	C₂×#117	C₂²×C₃₁	1	4	yes	—
119	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
120	#405	—	2	4	no	—
121	C₂×#97	C₄×C₃₁	1	2	yes	complemented
122	C₂×C₄×C₈	C₄×C₃₁	1	2	yes	complemented
123	#342	—	2	4	no	—
124	#340	—	2	4	no	—
125	H(3,ℤ/4)	C₄×C₃₁	1	8	yes	complemented
126	C₂×C₃₁	—	2	4	no	—
127	C₂×C₃₁	—	2	1	no	—
128	C₂×C₃₁	—	2	1	no	—
129	C₂×C₃₁	#1434	1	1	yes	characteristic
130	C₂×C₃₁	#1812	1	2	yes	—
131	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
132	C₂×M₁₆	—	2	2	no	—
133	C₂×M₁₆	—	2	2	no	—
134	C₂×M₁₆	C₂×C₄×C₃₁	1	4	yes	—
135	C₂²×C₈	—	2	2	no	—
136	C₂²×C₈	—	2	2	no	—
137	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
138	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
139	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
140	C₂²×C₈	D₄×C₃₁	1	4	yes	—
141	#117	—	2	2	no	—
142	#117	—	2	2	no	—
143	#117	—	2	4	no	—
144	#117	—	2	4	no	—
145	#117	—	2	4	no	—
146	#117	C₂×C₄×C₃₁	1	8	yes	—
147	C₂×C₄⋊C₄	—	2	4	no	—
148	C₂×C₄⋊C₄	Q₈×C₃₁	1	2	yes	—
149	C₂×C₂²⋊C₄	—	2	4	no	—
150	C₂×C₂²⋊C₄	D₄×C₃₁	1	2	yes	—
151	C₂×C₄²	—	2	4	no	—
152	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
153	#97	—	2	4	no	—
154	C₄×C₈	—	2	4	no	—
155	C₃₁	#10814	1	1	yes	characteristic, direct product factor, Sylow
156	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
157	C₂²×C₄	—	4	4	no	—
158	C₂²×C₄	C₄⋊C₄×C₃₁	1	4	yes	—
159	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
160	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
161	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
162	C₂²×C₄	C₂×Q₈×C₃₁	1	1	yes	characteristic
163	C₂²×C₄	#41×C₃₁	1	2	yes	—
164	C₂²×C₄	#41×C₃₁	1	2	yes	—
165	C₂²×C₄	#41×C₃₁	1	2	yes	—
166	C₂²×C₄	#41×C₃₁	1	2	yes	—
167	M₁₆	—	2	8	no	—
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	—	2	1	no	—
170	C₂×C₈	—	2	1	no	—
171	C₂×C₈	—	2	1	no	—
172	C₂×C₈	—	2	1	no	—
173	C₂×C₈	—	2	2	no	—
174	C₂×C₈	—	2	8	no	—
175	C₂×C₈	C₄²×C₃₁	1	4	yes	—
176	C₂×C₈	C₂²⋊C₄×C₃₁	1	2	yes	—
177	C₂×C₈	C₂²⋊C₄×C₃₁	1	2	yes	—
178	C₄⋊C₄	—	2	4	no	—
179	C₄⋊C₄	—	2	8	no	—
180	C₄⋊C₄	—	4	4	no	—
181	C₂²⋊C₄	—	2	4	no	—
182	C₂²⋊C₄	—	2	8	no	—
183	C₂²⋊C₄	—	4	4	no	—
184	C₄²	—	2	2	no	—
185	C₄²	—	2	2	no	—
186	C₄²	—	2	8	no	—
187	C₄²	—	4	4	no	—
188	C₂³	—	4	2	no	—
189	C₂³	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
190	C₂³	C₄×D₄×C₃₁	1	2	yes	—
191	C₂³	C₃₁×#121	1	2	yes	—
192	C₂³	C₃₁×#124	1	2	yes	—
193	C₂×C₄	—	2	4	no	—
194	C₂×C₄	—	4	4	no	—
195	C₂×C₄	—	4	8	no	—
196	C₂×C₄	—	8	8	no	—
197	C₂×C₄	C₂×C₄²×C₃₁	1	1	yes	characteristic
198	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
199	C₂×C₄	C₃₁×#117	1	1	yes	characteristic
200	C₂×C₄	C₃₁×#117	1	1	yes	characteristic
201	C₂×C₄	C₃₁×#117	1	4	yes	—
202	C₂×C₄	C₃₁×#117	1	1	yes	characteristic
203	C₂×C₄	C₃₁×#117	1	1	yes	characteristic
204	C₂×C₄	C₄×D₄×C₃₁	1	4	yes	—
205	C₂×C₄	C₄×Q₈×C₃₁	1	2	yes	—
206	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
207	C₂×C₄	C₃₁×#123	1	4	yes	—
208	C₂×C₄	C₃₁×#125	1	2	yes	—
209	C₂×C₄	C₃₁×#126	1	4	yes	—
210	C₈	—	2	4	no	—
211	C₈	—	2	2	no	—
212	C₈	—	2	2	no	—
213	C₂²	—	2	2	no	—
214	C₂²	—	2	4	no	—
215	C₂²	—	4	2	no	—
216	C₂²	—	4	2	no	—
217	C₂²	C₄×C₄⋊C₄×C₃₁	1	1	yes	characteristic
218	C₂²	C₃₁×#380	1	1	yes	characteristic
219	C₂²	C₃₁×#382	1	1	yes	characteristic
220	C₂²	C₃₁×#388	1	4	yes	—
221	C₄	—	4	4	no	—
222	C₄	—	8	8	no	—
223	C₄	C₃₁×#376	1	1	yes	characteristic
224	C₄	C₄×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
225	C₄	C₃₁×#381	1	1	yes	characteristic
226	C₄	C₃₁×#383	1	1	yes	characteristic
227	C₄	C₃₁×#387	1	4	yes	—
228	C₂	—	2	4	no	—
229	C₂	—	2	1	no	—
230	C₂	—	2	1	no	—
231	C₂	C₃₁×#1434	1	1	yes	characteristic
232	C₂	C₃₁×#1812	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₃₁	#117
commutator	C₂×C₄	C₂×C₄²×C₃₁
Frattini, cosocle	C₂³×C₄	C₂³×C₃₁
socle	C₂²×C₃₁	#380
prime radical	C₂⁴×C₃₁	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂²×M₁₆×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
10. C₂×C₃₁×#97	C₄	semidirect
11. C₂×C₄×C₈×C₃₁	C₄	semidirect
14. C₃₁×H(3,ℤ/4)	C₄	semidirect
71. #10814	C₃₁	direct (N × H)
121. C₂×#97	C₄×C₃₁	semidirect
122. C₂×C₄×C₈	C₄×C₃₁	semidirect
125. H(3,ℤ/4)	C₄×C₃₁	semidirect
155. C₃₁	#10814	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×#405	2
C₃₁×#1740	2
C₃₁×#1735	2
C₂×C₃₁×H(3,ℤ/4)	2
#10814	31

Subgroups — C31×#10814 #21467573

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#10814 #21467573