Subgroups — C₃₁×#3228 #16843161

Order 3968.

370 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#533	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#532	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#500	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#498	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#486	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#486	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#482	C₂	1	4	yes	maximal, complemented
8	C₃₁×#478	C₂	1	4	yes	maximal, complemented
9	C₃₁×#431	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#41×C₃₁	—	2	1	no	—
11	C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
12	C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
13	C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
14	C₃₁×#127	C₂²	1	1	yes	characteristic
15	C₃₁×#125	C₂²	1	2	yes	—
16	C₃₁×#124	—	2	1	no	—
17	C₃₁×C₂²⋊Q₈	—	2	1	no	—
18	C₃₁×C₂²⋊Q₈	—	2	1	no	—
19	C₃₁×#121	—	2	1	no	—
20	C₃₁×#121	C₂²	1	2	yes	—
21	C₄×Q₈×C₃₁	—	2	1	no	—
22	C₄×Q₈×C₃₁	—	2	1	no	—
23	C₄×D₄×C₃₁	—	2	1	no	—
24	C₄×D₄×C₃₁	C₂²	1	2	yes	—
25	C₃₁×#117	C₂²	1	2	yes	—
26	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
27	C₃₁×#107	C₂²	1	4	yes	—
28	C₃₁×#106	C₂²	1	4	yes	—
29	C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
30	C₃₁×#102	C₂²	1	4	yes	—
31	C₃₁×#102	C₂²	1	4	yes	—
32	C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
33	C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
34	C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
35	#41×C₃₁	—	4	4	no	—
36	C₂×Q₈×C₃₁	—	2	1	no	—
37	C₂×Q₈×C₃₁	—	2	1	no	—
38	C₂×Q₈×C₃₁	—	2	1	no	—
39	C₂×D₄×C₃₁	—	2	2	no	—
40	C₂×D₄×C₃₁	—	2	1	no	—
41	C₂×D₄×C₃₁	—	2	1	no	—
42	C₂×D₄×C₃₁	C₂³	1	2	yes	—
43	C₂²×C₄×C₃₁	—	2	2	no	—
44	C₂²×C₄×C₃₁	D₄	1	1	yes	characteristic
45	C₂²×C₄×C₃₁	D₄	1	1	yes	characteristic
46	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
47	C₂×C₈×C₃₁	C₂³	1	2	yes	—
48	C₂×C₈×C₃₁	C₂³	1	2	yes	—
49	C₄⋊C₄×C₃₁	—	2	1	no	—
50	C₄⋊C₄×C₃₁	—	2	2	no	—
51	C₄⋊C₄×C₃₁	—	2	2	no	—
52	C₄⋊C₄×C₃₁	—	2	1	no	—
53	C₄⋊C₄×C₃₁	—	2	1	no	—
54	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
55	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
56	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
57	C₂²⋊C₄×C₃₁	—	2	2	no	—
58	C₂²⋊C₄×C₃₁	—	2	2	no	—
59	C₂²⋊C₄×C₃₁	—	2	2	no	—
60	C₄²×C₃₁	—	2	2	no	—
61	C₄²×C₃₁	D₄	1	1	yes	characteristic
62	C₄²×C₃₁	D₄	1	1	yes	characteristic
63	C₄²×C₃₁	C₂³	1	1	yes	characteristic
64	C₄²×C₃₁	C₂³	1	1	yes	characteristic
65	C₂³×C₃₁	—	2	2	no	—
66	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
67	Q₈×C₃₁	—	4	2	no	—
68	Q₈×C₃₁	—	4	2	no	—
69	Q₈×C₃₁	—	4	2	no	—
70	D₄×C₃₁	—	2	4	no	—
71	D₄×C₃₁	—	4	4	no	—
72	D₄×C₃₁	—	4	2	no	—
73	D₄×C₃₁	—	4	2	no	—
74	C₂×C₄×C₃₁	—	2	2	no	—
75	C₂×C₄×C₃₁	—	2	2	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	2	no	—
80	C₂×C₄×C₃₁	—	4	4	no	—
81	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
82	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
83	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
84	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
85	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
86	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
87	C₈×C₃₁	—	2	2	no	—
88	C₈×C₃₁	—	2	2	no	—
89	#3228	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
90	C₂²×C₃₁	—	2	1	no	—
91	C₂²×C₃₁	—	2	1	no	—
92	C₂²×C₃₁	—	2	1	no	—
93	C₂²×C₃₁	—	4	2	no	—
94	C₂²×C₃₁	—	4	2	no	—
95	C₂²×C₃₁	—	4	2	no	—
96	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
97	C₄×C₃₁	—	2	1	no	—
98	C₄×C₃₁	—	2	1	no	—
99	C₄×C₃₁	—	2	1	no	—
100	C₄×C₃₁	—	4	2	no	—
101	C₄×C₃₁	—	4	2	no	—
102	C₄×C₃₁	—	4	2	no	—
103	C₄×C₃₁	#135	1	2	yes	—
104	C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
105	C₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
106	#533	C₂×C₃₁	1	1	yes	characteristic, complemented
107	#532	C₂×C₃₁	1	1	yes	characteristic
108	#500	C₂×C₃₁	1	1	yes	characteristic, complemented
109	#498	C₂×C₃₁	1	1	yes	characteristic, complemented
110	#486	C₂×C₃₁	1	1	yes	characteristic, complemented
111	#486	C₂×C₃₁	1	1	yes	characteristic, complemented
112	#482	C₂×C₃₁	1	4	yes	complemented
113	#478	C₂×C₃₁	1	4	yes	complemented
114	#431	C₂×C₃₁	1	1	yes	characteristic, complemented
115	C₂×C₃₁	—	4	1	no	—
116	C₂×C₃₁	—	8	2	no	—
117	C₂×C₃₁	#536	1	1	yes	characteristic
118	C₂×C₃₁	C₂×#135	1	1	yes	characteristic
119	C₂×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
120	C₂×#41	—	2	1	no	—
121	C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
122	C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
123	C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
124	#127	C₂²×C₃₁	1	1	yes	characteristic
125	#125	C₂²×C₃₁	1	2	yes	—
126	#124	—	2	1	no	—
127	C₂²⋊Q₈	—	2	1	no	—
128	C₂²⋊Q₈	—	2	1	no	—
129	#121	—	2	1	no	—
130	#121	C₂²×C₃₁	1	2	yes	—
131	C₄×Q₈	—	2	1	no	—
132	C₄×Q₈	—	2	1	no	—
133	C₄×D₄	—	2	1	no	—
134	C₄×D₄	C₂²×C₃₁	1	2	yes	—
135	#117	C₂²×C₃₁	1	2	yes	—
136	C₂×C₄²	C₂²×C₃₁	1	1	yes	characteristic
137	#107	C₂²×C₃₁	1	4	yes	—
138	#106	C₂²×C₃₁	1	4	yes	—
139	C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
140	#102	C₂²×C₃₁	1	4	yes	—
141	#102	C₂²×C₃₁	1	4	yes	—
142	C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
143	C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
144	C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
145	C₃₁	#3228	1	1	yes	characteristic, direct product factor, Sylow
146	#41	—	4	4	no	—
147	C₂×Q₈	—	2	1	no	—
148	C₂×Q₈	—	2	1	no	—
149	C₂×Q₈	—	2	1	no	—
150	C₂×D₄	—	2	2	no	—
151	C₂×D₄	—	2	1	no	—
152	C₂×D₄	—	2	1	no	—
153	C₂×D₄	C₂³×C₃₁	1	2	yes	—
154	C₂²×C₄	—	2	2	no	—
155	C₂²×C₄	D₄×C₃₁	1	1	yes	characteristic
156	C₂²×C₄	D₄×C₃₁	1	1	yes	characteristic
157	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
158	C₂×C₈	C₂³×C₃₁	1	2	yes	—
159	C₂×C₈	C₂³×C₃₁	1	2	yes	—
160	C₄⋊C₄	—	2	1	no	—
161	C₄⋊C₄	—	2	2	no	—
162	C₄⋊C₄	—	2	2	no	—
163	C₄⋊C₄	—	2	1	no	—
164	C₄⋊C₄	—	2	1	no	—
165	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
166	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
167	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
168	C₂²⋊C₄	—	2	2	no	—
169	C₂²⋊C₄	—	2	2	no	—
170	C₂²⋊C₄	—	2	2	no	—
171	C₄²	—	2	2	no	—
172	C₄²	D₄×C₃₁	1	1	yes	characteristic
173	C₄²	D₄×C₃₁	1	1	yes	characteristic
174	C₄²	C₂³×C₃₁	1	1	yes	characteristic
175	C₄²	C₂³×C₃₁	1	1	yes	characteristic
176	C₂³	—	2	2	no	—
177	C₂³	C₂×D₄×C₃₁	1	1	yes	characteristic
178	Q₈	—	4	2	no	—
179	Q₈	—	4	2	no	—
180	Q₈	—	4	2	no	—
181	D₄	—	2	4	no	—
182	D₄	—	4	4	no	—
183	D₄	—	4	2	no	—
184	D₄	—	4	2	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	4	4	no	—
192	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
193	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
194	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
195	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
196	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
197	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
198	C₈	—	2	2	no	—
199	C₈	—	2	2	no	—
200	C₂²	—	2	1	no	—
201	C₂²	—	2	1	no	—
202	C₂²	—	2	1	no	—
203	C₂²	—	4	2	no	—
204	C₂²	—	4	2	no	—
205	C₂²	—	4	2	no	—
206	C₂²	C₂²×D₄×C₃₁	1	1	yes	characteristic
207	C₄	—	2	1	no	—
208	C₄	—	2	1	no	—
209	C₄	—	2	1	no	—
210	C₄	—	4	2	no	—
211	C₄	—	4	2	no	—
212	C₄	—	4	2	no	—
213	C₄	C₃₁×#135	1	2	yes	—
214	C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
215	C₄	C₃₁×2₋¹⁺⁴	1	2	yes	—
216	C₂	—	4	1	no	—
217	C₂	—	8	2	no	—
218	C₂	C₃₁×#536	1	1	yes	characteristic
219	C₂	C₂×C₃₁×#135	1	1	yes	characteristic
220	C₂	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃₁
prime radical	C₃₁×#532	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#533	C₂	semidirect
3. C₃₁×#500	C₂	semidirect
4. C₃₁×#498	C₂	semidirect
5. C₃₁×#486	C₂	semidirect
6. C₃₁×#486	C₂	semidirect
7. C₃₁×#482	C₂	semidirect
8. C₃₁×#478	C₂	semidirect
9. C₃₁×#431	C₂	semidirect
89. #3228	C₃₁	direct (N × H)
106. #533	C₂×C₃₁	semidirect
108. #500	C₂×C₃₁	semidirect
109. #498	C₂×C₃₁	semidirect
110. #486	C₂×C₃₁	semidirect
111. #486	C₂×C₃₁	semidirect
112. #482	C₂×C₃₁	semidirect
113. #478	C₂×C₃₁	semidirect
114. #431	C₂×C₃₁	semidirect
145. C₃₁	#3228	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₃₁×#533	2
C₃₁×#532	2
C₃₁×#500	2
C₃₁×#498	2
C₃₁×#486	2
C₃₁×#482	2
C₃₁×#478	2
C₃₁×#431	2
#3228	31

Subgroups — C31×#3228 #16843161

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#3228 #16843161