Subgroups — C₃₁×#18090 #21474849

Order 7936.

564 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3170	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2986	C₂	1	2	yes	maximal
3	C₃₁×#1593	C₂	1	2	yes	maximal, complemented
4	C₃₁×#1584	C₂	1	4	yes	maximal, complemented
5	C₃₁×#1579	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1564	C₂	1	1	yes	maximal, characteristic
7	C₃₁×#1485	C₂	1	2	yes	maximal
8	C₃₁×#1451	C₂	1	1	yes	maximal, characteristic
9	C₃₁×#533	C₂²	1	2	yes	—
10	C₃₁×#500	C₄	1	1	yes	characteristic
11	C₃₁×#494	C₄	1	1	yes	characteristic
12	C₃₁×#487	C₄	1	2	yes	—
13	C₃₁×#467	C₄	1	2	yes	—
14	C₃₁×#467	C₄	1	2	yes	—
15	C₃₁×#446	—	2	2	no	—
16	C₃₁×#446	C₂²	1	4	yes	—
17	C₈×Q₈×C₃₁	—	2	2	no	—
18	C₈×Q₈×C₃₁	C₂²	1	4	yes	—
19	C₄×Q₁₆×C₃₁	C₂²	1	2	yes	—
20	C₄×Q₁₆×C₃₁	C₂²	1	2	yes	—
21	C₃₁×#433	—	2	4	no	—
22	C₃₁×#431	C₂²	1	2	yes	—
23	C₃₁×#431	C₂²	1	1	yes	characteristic
24	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
26	C₃₁×#335	C₂²	1	2	yes	—
27	C₃₁×#334	C₂²	1	1	yes	characteristic
28	C₃₁×#334	C₂²	1	1	yes	characteristic
29	C₃₁×Q₈⋊C₈	C₂²	1	4	yes	—
30	C₃₁×Q₈⋊C₈	C₂²	1	4	yes	—
31	C₃₁×#322	C₂²	1	2	yes	—
32	C₃₁×#322	C₂²	1	2	yes	—
33	C₂×Q₁₆×C₃₁	C₂×C₄	1	2	yes	—
34	C₂×Q₁₆×C₃₁	C₂×C₄	1	2	yes	—
35	C₂×M₁₆×C₃₁	—	2	1	no	—
36	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
38	C₃₁×C₄⋊Q₈	C₂×C₄	1	2	yes	—
39	C₃₁×C₄⋊Q₈	C₂×C₄	1	2	yes	—
40	C₃₁×#125	—	2	2	no	—
41	C₃₁×C₂²⋊Q₈	—	2	2	no	—
42	C₃₁×C₂²⋊Q₈	C₂×C₄	1	4	yes	—
43	C₄×Q₈×C₃₁	—	2	2	no	—
44	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
45	C₃₁×#117	—	2	2	no	—
46	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
47	C₃₁×#107	C₂×C₄	1	2	yes	—
48	C₃₁×#107	C₂×C₄	1	2	yes	—
49	C₃₁×C₄⋊C₈	—	2	4	no	—
50	C₃₁×C₄⋊C₈	—	2	4	no	—
51	C₃₁×C₄⋊C₈	D₄	1	2	yes	—
52	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
53	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
54	C₃₁×#103	C₂×C₄	1	4	yes	—
55	C₃₁×#103	C₂×C₄	1	4	yes	—
56	C₃₁×C₂²⋊C₈	—	2	2	no	—
57	C₃₁×C₂²⋊C₈	D₄	1	2	yes	—
58	C₃₁×#97	—	2	4	no	—
59	C₃₁×#97	—	2	1	no	—
60	C₄×C₈×C₃₁	—	2	4	no	—
61	C₄×C₈×C₃₁	—	2	2	no	—
62	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
63	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
64	C₄×C₈×C₃₁	C₂³	1	2	yes	—
65	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
66	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
67	C₄×C₈×C₃₁	C₂³	1	2	yes	—
68	C₂×Q₈×C₃₁	—	2	2	no	—
69	C₂×Q₈×C₃₁	C₂²×C₄	1	4	yes	—
70	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
73	Q₁₆×C₃₁	—	2	4	no	—
74	Q₁₆×C₃₁	—	4	2	no	—
75	M₁₆×C₃₁	—	4	2	no	—
76	C₂×C₈×C₃₁	—	2	2	no	—
77	C₂×C₈×C₃₁	—	2	2	no	—
78	C₂×C₈×C₃₁	—	2	2	no	—
79	C₂×C₈×C₃₁	—	2	2	no	—
80	C₂×C₈×C₃₁	—	4	1	no	—
81	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
86	C₄⋊C₄×C₃₁	—	2	2	no	—
87	C₄⋊C₄×C₃₁	—	2	2	no	—
88	C₄⋊C₄×C₃₁	—	2	2	no	—
89	C₄⋊C₄×C₃₁	—	2	4	no	—
90	C₄⋊C₄×C₃₁	—	2	4	no	—
91	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
92	C₂²⋊C₄×C₃₁	—	2	4	no	—
93	C₄²×C₃₁	—	2	4	no	—
94	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
95	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
96	C₄²×C₃₁	#41	1	1	yes	characteristic
97	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
98	#18090	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
99	C₂³×C₃₁	C₄×D₄	1	1	yes	characteristic
100	Q₈×C₃₁	—	2	4	no	—
101	Q₈×C₃₁	—	2	4	no	—
102	Q₈×C₃₁	—	4	4	no	—
103	C₂×C₄×C₃₁	—	2	1	no	—
104	C₂×C₄×C₃₁	—	2	1	no	—
105	C₂×C₄×C₃₁	—	2	2	no	—
106	C₂×C₄×C₃₁	—	2	2	no	—
107	C₂×C₄×C₃₁	—	2	4	no	—
108	C₂×C₄×C₃₁	—	2	4	no	—
109	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
110	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
111	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
112	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
113	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
114	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
115	C₈×C₃₁	—	2	1	no	—
116	C₈×C₃₁	—	2	1	no	—
117	C₈×C₃₁	—	2	1	no	—
118	C₈×C₃₁	—	2	4	no	—
119	C₈×C₃₁	—	4	2	no	—
120	C₈×C₃₁	—	4	2	no	—
121	C₈×C₃₁	—	4	2	no	—
122	C₈×C₃₁	#131	1	2	yes	—
123	#3170	C₂×C₃₁	1	1	yes	characteristic
124	#2986	C₂×C₃₁	1	2	yes	—
125	#1593	C₂×C₃₁	1	2	yes	complemented
126	#1584	C₂×C₃₁	1	4	yes	complemented
127	#1579	C₂×C₃₁	1	2	yes	complemented
128	#1564	C₂×C₃₁	1	1	yes	characteristic
129	#1485	C₂×C₃₁	1	2	yes	—
130	#1451	C₂×C₃₁	1	1	yes	characteristic
131	C₂²×C₃₁	—	2	1	no	—
132	C₂²×C₃₁	—	2	1	no	—
133	C₂²×C₃₁	—	2	1	no	—
134	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
135	C₄×C₃₁	—	2	1	no	—
136	C₄×C₃₁	—	2	1	no	—
137	C₄×C₃₁	—	2	1	no	—
138	C₄×C₃₁	—	4	4	no	—
139	C₄×C₃₁	—	4	4	no	—
140	C₄×C₃₁	#444	1	2	yes	—
141	C₄×C₃₁	C₂×#131	1	1	yes	characteristic
142	C₄×C₃₁	#568	1	1	yes	characteristic
143	C₄×C₃₁	#578	1	2	yes	—
144	#533	C₂²×C₃₁	1	2	yes	—
145	#500	C₄×C₃₁	1	1	yes	characteristic
146	#494	C₄×C₃₁	1	1	yes	characteristic
147	#487	C₄×C₃₁	1	2	yes	—
148	#467	C₄×C₃₁	1	2	yes	—
149	#467	C₄×C₃₁	1	2	yes	—
150	#446	—	2	2	no	—
151	#446	C₂²×C₃₁	1	4	yes	—
152	C₈×Q₈	—	2	2	no	—
153	C₈×Q₈	C₂²×C₃₁	1	4	yes	—
154	C₄×Q₁₆	C₂²×C₃₁	1	2	yes	—
155	C₄×Q₁₆	C₂²×C₃₁	1	2	yes	—
156	#433	—	2	4	no	—
157	#431	C₂²×C₃₁	1	2	yes	—
158	#431	C₂²×C₃₁	1	1	yes	characteristic
159	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
160	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
161	#335	C₂²×C₃₁	1	2	yes	—
162	#334	C₂²×C₃₁	1	1	yes	characteristic
163	#334	C₂²×C₃₁	1	1	yes	characteristic
164	Q₈⋊C₈	C₂²×C₃₁	1	4	yes	—
165	Q₈⋊C₈	C₂²×C₃₁	1	4	yes	—
166	#322	C₂²×C₃₁	1	2	yes	—
167	#322	C₂²×C₃₁	1	2	yes	—
168	C₂×C₃₁	—	4	1	no	—
169	C₂×C₃₁	#2935	1	1	yes	characteristic
170	C₂×C₃₁	#2950	1	1	yes	characteristic
171	C₂×C₃₁	C₂×#444	1	1	yes	characteristic
172	C₂×Q₁₆	C₂×C₄×C₃₁	1	2	yes	—
173	C₂×Q₁₆	C₂×C₄×C₃₁	1	2	yes	—
174	C₂×M₁₆	—	2	1	no	—
175	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
176	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
177	C₄⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
178	C₄⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
179	#125	—	2	2	no	—
180	C₂²⋊Q₈	—	2	2	no	—
181	C₂²⋊Q₈	C₂×C₄×C₃₁	1	4	yes	—
182	C₄×Q₈	—	2	2	no	—
183	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
184	#117	—	2	2	no	—
185	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
186	#107	C₂×C₄×C₃₁	1	2	yes	—
187	#107	C₂×C₄×C₃₁	1	2	yes	—
188	C₄⋊C₈	—	2	4	no	—
189	C₄⋊C₈	—	2	4	no	—
190	C₄⋊C₈	D₄×C₃₁	1	2	yes	—
191	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
192	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
193	#103	C₂×C₄×C₃₁	1	4	yes	—
194	#103	C₂×C₄×C₃₁	1	4	yes	—
195	C₂²⋊C₈	—	2	2	no	—
196	C₂²⋊C₈	D₄×C₃₁	1	2	yes	—
197	#97	—	2	4	no	—
198	#97	—	2	1	no	—
199	C₄×C₈	—	2	4	no	—
200	C₄×C₈	—	2	2	no	—
201	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
202	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
203	C₄×C₈	C₂³×C₃₁	1	2	yes	—
204	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
205	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
206	C₄×C₈	C₂³×C₃₁	1	2	yes	—
207	C₃₁	#18090	1	1	yes	characteristic, direct product factor, Sylow
208	C₂×Q₈	—	2	2	no	—
209	C₂×Q₈	C₂²×C₄×C₃₁	1	4	yes	—
210	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
211	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
212	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
213	Q₁₆	—	2	4	no	—
214	Q₁₆	—	4	2	no	—
215	M₁₆	—	4	2	no	—
216	C₂×C₈	—	2	2	no	—
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	—	2	2	no	—
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	—	4	1	no	—
221	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
222	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
223	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
224	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
225	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
226	C₄⋊C₄	—	2	2	no	—
227	C₄⋊C₄	—	2	2	no	—
228	C₄⋊C₄	—	2	2	no	—
229	C₄⋊C₄	—	2	4	no	—
230	C₄⋊C₄	—	2	4	no	—
231	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
232	C₂²⋊C₄	—	2	4	no	—
233	C₄²	—	2	4	no	—
234	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
235	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
236	C₄²	#41×C₃₁	1	1	yes	characteristic
237	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
238	C₂³	C₄×D₄×C₃₁	1	1	yes	characteristic
239	Q₈	—	2	4	no	—
240	Q₈	—	2	4	no	—
241	Q₈	—	4	4	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	4	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
249	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
250	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
251	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
252	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
253	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
254	C₈	—	2	1	no	—
255	C₈	—	2	1	no	—
256	C₈	—	2	1	no	—
257	C₈	—	2	4	no	—
258	C₈	—	4	2	no	—
259	C₈	—	4	2	no	—
260	C₈	—	4	2	no	—
261	C₈	C₃₁×#131	1	2	yes	—
262	C₂²	—	2	1	no	—
263	C₂²	—	2	1	no	—
264	C₂²	—	2	1	no	—
265	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
266	C₄	—	2	1	no	—
267	C₄	—	2	1	no	—
268	C₄	—	2	1	no	—
269	C₄	—	4	4	no	—
270	C₄	—	4	4	no	—
271	C₄	C₃₁×#444	1	2	yes	—
272	C₄	C₂×C₃₁×#131	1	1	yes	characteristic
273	C₄	C₃₁×#568	1	1	yes	characteristic
274	C₄	C₃₁×#578	1	2	yes	—
275	C₂	—	4	1	no	—
276	C₂	C₃₁×#2935	1	1	yes	characteristic
277	C₂	C₃₁×#2950	1	1	yes	characteristic
278	C₂	C₂×C₃₁×#444	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₄²	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₂×C₄×D₄
prime radical	C₂³×C₃₁	C₄×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₃₁×#431 ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃₁×#1593	C₂	semidirect
4. C₃₁×#1584	C₂	semidirect
5. C₃₁×#1579	C₂	semidirect
98. #18090	C₃₁	direct (N × H)
125. #1593	C₂×C₃₁	semidirect
126. #1584	C₂×C₃₁	semidirect
127. #1579	C₂×C₃₁	semidirect
207. C₃₁	#18090	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×#3170	2
C₃₁×#2986	2
C₃₁×#1593	2
C₃₁×#1584	2
C₃₁×#1579	2
C₃₁×#1564	2
C₃₁×#1485	2
C₃₁×#1451	2
#18090	31

Subgroups — C31×#18090 #21474849

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#18090 #21474849