Subgroups — C₃₁×#12732 #21469491

Order 7936.

934 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×Q₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#358	C₂	1	4	yes	maximal, complemented
3	C₂×C₃₁×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1808	C₂	1	1	yes	maximal, characteristic
5	C₂³×Q₈×C₃₁	—	2	1	no	—
6	C₂²×Q₁₆×C₃₁	—	2	8	no	—
7	C₂²×Q₁₆×C₃₁	—	2	1	no	—
8	C₂²×Q₁₆×C₃₁	—	2	2	no	—
9	C₂²×Q₁₆×C₃₁	C₄	1	4	yes	complemented
10	C₂²×Q₁₆×C₃₁	C₂²	1	2	yes	—
11	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
12	C₂²×C₁₆×C₃₁	C₂²	1	2	yes	—
13	C₂×C₃₁×#107	C₂²	1	2	yes	—
14	C₃₁×#380	—	2	1	no	—
15	C₃₁×#358	—	2	8	no	—
16	C₃₁×C₂²⋊C₁₆	—	2	2	no	—
17	C₃₁×#340	—	2	1	no	—
18	C₂²×Q₈×C₃₁	—	2	8	no	—
19	C₂²×Q₈×C₃₁	—	2	4	no	—
20	C₂²×Q₈×C₃₁	—	2	2	no	—
21	C₂²×Q₈×C₃₁	—	4	4	no	—
22	C₂²×Q₈×C₃₁	—	4	1	no	—
23	C₂²×Q₈×C₃₁	—	4	2	no	—
24	C₂³×C₄×C₃₁	—	4	1	no	—
25	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
26	C₂×Q₁₆×C₃₁	—	2	8	no	—
27	C₂×Q₁₆×C₃₁	—	2	16	no	—
28	C₂×Q₁₆×C₃₁	—	2	16	no	—
29	C₂×Q₁₆×C₃₁	—	2	4	no	—
30	C₂×Q₁₆×C₃₁	—	2	4	no	—
31	C₂×Q₁₆×C₃₁	—	2	2	no	—
32	C₂×Q₁₆×C₃₁	C₂×C₄	1	4	yes	—
33	C₂×Q₁₆×C₃₁	D₄	1	8	yes	—
34	C₂²×C₈×C₃₁	—	2	4	no	—
35	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
36	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
37	C₂²×C₈×C₃₁	D₄	1	2	yes	—
38	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
39	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
40	C₂×C₂²⋊C₄×C₃₁	—	4	1	no	—
41	C₂×C₁₆×C₃₁	—	2	2	no	—
42	C₂×C₁₆×C₃₁	—	2	4	no	—
43	C₃₁×#107	—	2	4	no	—
44	C₃₁×#95	—	4	1	no	—
45	C₂⁴×C₃₁	D₈	1	1	yes	characteristic
46	C₂×Q₈×C₃₁	—	2	16	no	—
47	C₂×Q₈×C₃₁	—	2	4	no	—
48	C₂×Q₈×C₃₁	—	2	8	no	—
49	C₂×Q₈×C₃₁	—	4	8	no	—
50	C₂×Q₈×C₃₁	—	4	8	no	—
51	C₂×Q₈×C₃₁	—	4	16	no	—
52	C₂×Q₈×C₃₁	—	4	4	no	—
53	C₂×Q₈×C₃₁	—	4	2	no	—
54	C₂×Q₈×C₃₁	—	4	4	no	—
55	C₂²×C₄×C₃₁	—	2	2	no	—
56	C₂²×C₄×C₃₁	—	2	2	no	—
57	C₂²×C₄×C₃₁	—	4	8	no	—
58	C₂²×C₄×C₃₁	—	4	4	no	—
59	C₂²×C₄×C₃₁	—	4	2	no	—
60	C₂²×C₄×C₃₁	—	4	2	no	—
61	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
62	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
63	C₂²×C₄×C₃₁	SD₁₆	1	2	yes	—
64	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
65	Q₁₆×C₃₁	—	2	16	no	—
66	Q₁₆×C₃₁	—	2	4	no	—
67	Q₁₆×C₃₁	—	2	8	no	—
68	Q₁₆×C₃₁	C₂²⋊C₄	1	8	yes	complemented
69	C₂×C₈×C₃₁	—	2	2	no	—
70	C₂×C₈×C₃₁	—	2	4	no	—
71	C₂×C₈×C₃₁	—	2	4	no	—
72	C₂×C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
73	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
75	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
76	C₄⋊C₄×C₃₁	—	4	2	no	—
77	C₄⋊C₄×C₃₁	—	4	2	no	—
78	C₂²⋊C₄×C₃₁	—	8	2	no	—
79	C₁₆×C₃₁	—	4	2	no	—
80	#12732	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
81	C₂³×C₃₁	—	2	2	no	—
82	C₂³×C₃₁	—	2	2	no	—
83	C₂³×C₃₁	#102	1	2	yes	—
84	C₂³×C₃₁	SD₃₂	1	2	yes	—
85	C₂³×C₃₁	Q₃₂	1	2	yes	—
86	C₂³×C₃₁	C₂×D₈	1	1	yes	characteristic
87	Q₈×C₃₁	—	2	8	no	—
88	Q₈×C₃₁	—	4	16	no	—
89	Q₈×C₃₁	—	4	4	no	—
90	Q₈×C₃₁	—	4	8	no	—
91	C₂×C₄×C₃₁	—	2	2	no	—
92	C₂×C₄×C₃₁	—	2	2	no	—
93	C₂×C₄×C₃₁	—	2	2	no	—
94	C₂×C₄×C₃₁	—	2	4	no	—
95	C₂×C₄×C₃₁	—	4	16	no	—
96	C₂×C₄×C₃₁	—	4	4	no	—
97	C₂×C₄×C₃₁	—	4	8	no	—
98	C₂×C₄×C₃₁	—	8	2	no	—
99	C₂×C₄×C₃₁	—	8	2	no	—
100	C₂×C₄×C₃₁	—	8	2	no	—
101	C₂×C₄×C₃₁	#102	1	2	yes	—
102	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₃₁	#120	1	2	yes	—
104	C₂×C₄×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
105	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
106	C₈×C₃₁	—	2	2	no	—
107	C₈×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
108	C₈×C₃₁	#120	1	2	yes	—
109	C₂³×Q₁₆	C₂×C₃₁	1	1	yes	characteristic
110	C₂×#358	C₂×C₃₁	1	4	yes	complemented
111	C₂×C₂²⋊C₁₆	C₂×C₃₁	1	1	yes	characteristic
112	#1808	C₂×C₃₁	1	1	yes	characteristic
113	C₂²×C₃₁	—	2	2	no	—
114	C₂²×C₃₁	—	2	4	no	—
115	C₂²×C₃₁	—	2	4	no	—
116	C₂²×C₃₁	—	2	2	no	—
117	C₂²×C₃₁	#358	1	4	yes	complemented
118	C₂²×C₃₁	C₂×#102	1	1	yes	characteristic
119	C₂²×C₃₁	#447	1	2	yes	—
120	C₂²×C₃₁	C₂×SD₃₂	1	1	yes	characteristic
121	C₂²×C₃₁	C₂×Q₃₂	1	1	yes	characteristic
122	C₂²×C₃₁	#510	1	2	yes	—
123	C₄×C₃₁	—	2	2	no	—
124	C₄×C₃₁	—	4	8	no	—
125	C₄×C₃₁	—	16	2	no	—
126	C₄×C₃₁	#379	1	1	yes	characteristic
127	C₄×C₃₁	#418	1	1	yes	characteristic
128	C₄×C₃₁	#450	1	2	yes	—
129	C₂³×Q₈	—	2	1	no	—
130	C₂²×Q₁₆	—	2	8	no	—
131	C₂²×Q₁₆	—	2	1	no	—
132	C₂²×Q₁₆	—	2	2	no	—
133	C₂²×Q₁₆	C₄×C₃₁	1	4	yes	complemented
134	C₂²×Q₁₆	C₂²×C₃₁	1	2	yes	—
135	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
136	C₂²×C₁₆	C₂²×C₃₁	1	2	yes	—
137	C₂×#107	C₂²×C₃₁	1	2	yes	—
138	#380	—	2	1	no	—
139	#358	—	2	8	no	—
140	C₂²⋊C₁₆	—	2	2	no	—
141	#340	—	2	1	no	—
142	C₂×C₃₁	—	2	4	no	—
143	C₂×C₃₁	#1786	1	1	yes	characteristic
144	C₂×C₃₁	C₂×#358	1	1	yes	characteristic
145	C₂×C₃₁	#2141	1	1	yes	characteristic
146	C₂×C₃₁	#2185	1	2	yes	—
147	C₂×C₃₁	#2188	1	2	yes	—
148	C₂²×Q₈	—	2	8	no	—
149	C₂²×Q₈	—	2	4	no	—
150	C₂²×Q₈	—	2	2	no	—
151	C₂²×Q₈	—	4	4	no	—
152	C₂²×Q₈	—	4	1	no	—
153	C₂²×Q₈	—	4	2	no	—
154	C₂³×C₄	—	4	1	no	—
155	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
156	C₂×Q₁₆	—	2	8	no	—
157	C₂×Q₁₆	—	2	16	no	—
158	C₂×Q₁₆	—	2	16	no	—
159	C₂×Q₁₆	—	2	4	no	—
160	C₂×Q₁₆	—	2	4	no	—
161	C₂×Q₁₆	—	2	2	no	—
162	C₂×Q₁₆	C₂×C₄×C₃₁	1	4	yes	—
163	C₂×Q₁₆	D₄×C₃₁	1	8	yes	—
164	C₂²×C₈	—	2	4	no	—
165	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
166	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
167	C₂²×C₈	D₄×C₃₁	1	2	yes	—
168	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
169	C₂×C₄⋊C₄	—	2	2	no	—
170	C₂×C₂²⋊C₄	—	4	1	no	—
171	C₂×C₁₆	—	2	2	no	—
172	C₂×C₁₆	—	2	4	no	—
173	#107	—	2	4	no	—
174	#95	—	4	1	no	—
175	C₃₁	#12732	1	1	yes	characteristic, direct product factor, Sylow
176	C₂⁴	D₈×C₃₁	1	1	yes	characteristic
177	C₂×Q₈	—	2	16	no	—
178	C₂×Q₈	—	2	4	no	—
179	C₂×Q₈	—	2	8	no	—
180	C₂×Q₈	—	4	8	no	—
181	C₂×Q₈	—	4	8	no	—
182	C₂×Q₈	—	4	16	no	—
183	C₂×Q₈	—	4	4	no	—
184	C₂×Q₈	—	4	2	no	—
185	C₂×Q₈	—	4	4	no	—
186	C₂²×C₄	—	2	2	no	—
187	C₂²×C₄	—	2	2	no	—
188	C₂²×C₄	—	4	8	no	—
189	C₂²×C₄	—	4	4	no	—
190	C₂²×C₄	—	4	2	no	—
191	C₂²×C₄	—	4	2	no	—
192	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
193	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
194	C₂²×C₄	SD₁₆×C₃₁	1	2	yes	—
195	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
196	Q₁₆	—	2	16	no	—
197	Q₁₆	—	2	4	no	—
198	Q₁₆	—	2	8	no	—
199	Q₁₆	C₂²⋊C₄×C₃₁	1	8	yes	complemented
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	—	2	4	no	—
202	C₂×C₈	—	2	4	no	—
203	C₂×C₈	C₂²⋊C₄×C₃₁	1	2	yes	—
204	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
205	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
206	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
207	C₄⋊C₄	—	4	2	no	—
208	C₄⋊C₄	—	4	2	no	—
209	C₂²⋊C₄	—	8	2	no	—
210	C₁₆	—	4	2	no	—
211	C₂³	—	2	2	no	—
212	C₂³	—	2	2	no	—
213	C₂³	C₃₁×#102	1	2	yes	—
214	C₂³	C₃₁×SD₃₂	1	2	yes	—
215	C₂³	C₃₁×Q₃₂	1	2	yes	—
216	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
217	Q₈	—	2	8	no	—
218	Q₈	—	4	16	no	—
219	Q₈	—	4	4	no	—
220	Q₈	—	4	8	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	4	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	4	16	no	—
226	C₂×C₄	—	4	4	no	—
227	C₂×C₄	—	4	8	no	—
228	C₂×C₄	—	8	2	no	—
229	C₂×C₄	—	8	2	no	—
230	C₂×C₄	—	8	2	no	—
231	C₂×C₄	C₃₁×#102	1	2	yes	—
232	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
233	C₂×C₄	C₃₁×#120	1	2	yes	—
234	C₂×C₄	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
235	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
236	C₈	—	2	2	no	—
237	C₈	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
238	C₈	C₃₁×#120	1	2	yes	—
239	C₂²	—	2	2	no	—
240	C₂²	—	2	4	no	—
241	C₂²	—	2	4	no	—
242	C₂²	—	2	2	no	—
243	C₂²	C₃₁×#358	1	4	yes	complemented
244	C₂²	C₂×C₃₁×#102	1	1	yes	characteristic
245	C₂²	C₃₁×#447	1	2	yes	—
246	C₂²	C₂×C₃₁×SD₃₂	1	1	yes	characteristic
247	C₂²	C₂×C₃₁×Q₃₂	1	1	yes	characteristic
248	C₂²	C₃₁×#510	1	2	yes	—
249	C₄	—	2	2	no	—
250	C₄	—	4	8	no	—
251	C₄	—	16	2	no	—
252	C₄	C₃₁×#379	1	1	yes	characteristic
253	C₄	C₃₁×#418	1	1	yes	characteristic
254	C₄	C₃₁×#450	1	2	yes	—
255	C₂	—	2	4	no	—
256	C₂	C₃₁×#1786	1	1	yes	characteristic
257	C₂	C₂×C₃₁×#358	1	1	yes	characteristic
258	C₂	C₃₁×#2141	1	1	yes	characteristic
259	C₂	C₃₁×#2185	1	2	yes	—
260	C₂	C₃₁×#2188	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₃₁
Frattini, cosocle	C₂²×C₈	C₂³×C₃₁
prime radical	C₂⁴×C₃₁	D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂³×C₄×C₃₁ ◁ C₂³×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃₁×#358	C₂	semidirect
9. C₂²×Q₁₆×C₃₁	C₄	semidirect
68. Q₁₆×C₃₁	C₂²⋊C₄	semidirect
80. #12732	C₃₁	direct (N × H)
110. C₂×#358	C₂×C₃₁	semidirect
117. C₂²×C₃₁	#358	semidirect
133. C₂²×Q₁₆	C₄×C₃₁	semidirect
175. C₃₁	#12732	direct (N × H)
199. Q₁₆	C₂²⋊C₄×C₃₁	semidirect
243. C₂²	C₃₁×#358	semidirect

Maximal subgroups

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

subgroup	index
C₂³×Q₁₆×C₃₁	2
C₂×C₃₁×#358	2
C₂×C₃₁×C₂²⋊C₁₆	2
C₃₁×#1808	2
#12732	31

Subgroups — C31×#12732 #21469491

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#12732 #21469491