Subgroups — C₃₁×#19971 #21476730

Order 7936.

706 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3334	C₂	1	2	yes	maximal, complemented
2	C₃₁×#3283	C₂	1	2	yes	maximal, complemented
3	C₃₁×#1713	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1673	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1669	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1647	C₂	1	4	yes	maximal, complemented
7	C₈×D₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₈×D₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₃₁×#135	—	2	2	no	—
10	C₃₁×#547	C₂²	1	4	yes	—
11	C₃₁×#500	C₂²	1	2	yes	—
12	C₃₁×#500	C₂²	1	2	yes	—
13	C₃₁×#494	C₂²	1	2	yes	—
14	C₃₁×#480	—	2	4	no	—
15	C₃₁×#474	C₂²	1	4	yes	—
16	C₃₁×#467	—	2	2	no	—
17	C₃₁×#466	—	2	2	no	—
18	C₃₁×#461	C₂²	1	4	yes	—
19	C₃₁×#452	—	2	4	no	—
20	C₄×D₈×C₃₁	C₂²	1	1	yes	characteristic
21	C₄×D₈×C₃₁	C₂²	1	1	yes	characteristic
22	C₈×D₄×C₃₁	C₂²	1	2	yes	—
23	C₈×D₄×C₃₁	C₂²	1	2	yes	—
24	C₂×C₃₁×#107	—	2	2	no	—
25	C₃₁×#335	C₂²	1	1	yes	characteristic
26	C₃₁×#335	C₂²	1	1	yes	characteristic
27	C₃₁×#332	C₂²	1	4	yes	—
28	C₃₁×D₄⋊C₈	C₂²	1	2	yes	—
29	C₃₁×D₄⋊C₈	C₂²	1	2	yes	—
30	C₈²×C₃₁	C₂²	1	1	yes	characteristic
31	C₂×#41×C₃₁	—	2	4	no	—
32	C₃₁×#135	—	4	8	no	—
33	C₂×Q₁₆×C₃₁	—	2	2	no	—
34	C₂×Q₁₆×C₃₁	—	2	2	no	—
35	C₂×SD₁₆×C₃₁	—	2	4	no	—
36	C₂×D₈×C₃₁	D₄	1	1	yes	characteristic
37	C₂×D₈×C₃₁	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₃₁	—	2	2	no	—
39	C₂²×C₈×C₃₁	—	2	2	no	—
40	C₃₁×C₄⋊Q₈	C₂³	1	4	yes	—
41	C₃₁×#123	—	4	4	no	—
42	C₃₁×C₂²⋊Q₈	—	2	4	no	—
43	C₃₁×#121	—	2	4	no	—
44	C₄×D₄×C₃₁	C₂³	1	2	yes	—
45	C₄×D₄×C₃₁	C₂³	1	2	yes	—
46	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
47	C₃₁×#107	—	2	2	no	—
48	C₃₁×#107	—	2	2	no	—
49	C₃₁×#107	—	2	4	no	—
50	C₃₁×#107	—	4	2	no	—
51	C₃₁×#107	D₄	1	1	yes	characteristic
52	C₃₁×#107	D₄	1	1	yes	characteristic
53	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
54	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
55	C₃₁×#103	—	2	4	no	—
56	C₃₁×#102	—	2	4	no	—
57	C₃₁×#102	D₄	1	2	yes	—
58	C₃₁×#102	D₄	1	2	yes	—
59	C₃₁×C₂²⋊C₈	—	2	2	no	—
60	C₃₁×C₂²⋊C₈	—	2	2	no	—
61	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
62	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
63	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
64	#41×C₃₁	—	4	8	no	—
65	#41×C₃₁	—	8	4	no	—
66	C₂×Q₈×C₃₁	—	2	4	no	—
67	C₂×D₄×C₃₁	—	4	4	no	—
68	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
69	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
70	C₂²×C₄×C₃₁	—	2	2	no	—
71	C₂²×C₄×C₃₁	—	2	2	no	—
72	C₂²×C₄×C₃₁	—	4	4	no	—
73	Q₁₆×C₃₁	—	4	4	no	—
74	Q₁₆×C₃₁	—	8	2	no	—
75	SD₁₆×C₃₁	—	4	8	no	—
76	D₈×C₃₁	—	2	2	no	—
77	D₈×C₃₁	D₈	1	4	yes	complemented
78	C₂×C₈×C₃₁	—	2	1	no	—
79	C₂×C₈×C₃₁	—	2	2	no	—
80	C₂×C₈×C₃₁	—	2	2	no	—
81	C₂×C₈×C₃₁	—	4	4	no	—
82	C₂×C₈×C₃₁	—	4	4	no	—
83	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
84	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
85	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
87	C₄⋊C₄×C₃₁	—	2	4	no	—
88	C₄⋊C₄×C₃₁	—	2	4	no	—
89	C₄⋊C₄×C₃₁	—	4	4	no	—
90	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
91	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
92	C₂²⋊C₄×C₃₁	—	2	2	no	—
93	C₂²⋊C₄×C₃₁	—	2	2	no	—
94	C₂²⋊C₄×C₃₁	—	4	4	no	—
95	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
96	#19971	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
97	C₂³×C₃₁	—	2	2	no	—
98	C₂³×C₃₁	—	2	2	no	—
99	Q₈×C₃₁	—	4	4	no	—
100	Q₈×C₃₁	—	4	4	no	—
101	D₄×C₃₁	—	2	2	no	—
102	D₄×C₃₁	—	2	2	no	—
103	D₄×C₃₁	—	2	2	no	—
104	D₄×C₃₁	—	8	8	no	—
105	D₄×C₃₁	C₂×D₈	1	4	yes	—
106	C₂×C₄×C₃₁	—	2	2	no	—
107	C₂×C₄×C₃₁	—	2	2	no	—
108	C₂×C₄×C₃₁	—	4	2	no	—
109	C₂×C₄×C₃₁	—	4	2	no	—
110	C₂×C₄×C₃₁	—	4	2	no	—
111	C₂×C₄×C₃₁	—	4	2	no	—
112	C₂×C₄×C₃₁	—	4	4	no	—
113	C₂×C₄×C₃₁	—	8	8	no	—
114	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
115	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
116	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
117	C₈×C₃₁	—	2	2	no	—
118	C₈×C₃₁	—	2	1	no	—
119	C₈×C₃₁	—	2	1	no	—
120	C₈×C₃₁	—	4	2	no	—
121	C₈×C₃₁	—	4	2	no	—
122	C₈×C₃₁	C₂×D₈	1	2	yes	—
123	C₈×C₃₁	#135	1	2	yes	complemented
124	#3334	C₂×C₃₁	1	2	yes	complemented
125	#3283	C₂×C₃₁	1	2	yes	complemented
126	#1713	C₂×C₃₁	1	1	yes	characteristic, complemented
127	#1673	C₂×C₃₁	1	2	yes	complemented
128	#1669	C₂×C₃₁	1	2	yes	complemented
129	#1647	C₂×C₃₁	1	4	yes	complemented
130	C₈×D₈	C₂×C₃₁	1	1	yes	characteristic, complemented
131	C₈×D₈	C₂×C₃₁	1	1	yes	characteristic, complemented
132	C₂²×C₃₁	—	2	4	no	—
133	C₂²×C₃₁	—	4	2	no	—
134	C₂²×C₃₁	—	4	2	no	—
135	C₂²×C₃₁	—	4	2	no	—
136	C₂²×C₃₁	—	4	2	no	—
137	C₂²×C₃₁	—	4	2	no	—
138	C₂²×C₃₁	D₄²	1	1	yes	characteristic
139	C₄×C₃₁	—	2	1	no	—
140	C₄×C₃₁	—	4	2	no	—
141	C₄×C₃₁	—	4	2	no	—
142	C₄×C₃₁	—	8	4	no	—
143	C₄×C₃₁	C₂²×D₈	1	1	yes	characteristic
144	C₄×C₃₁	C₂×#135	1	1	yes	characteristic
145	C₄×C₃₁	#576	1	1	yes	characteristic
146	C₄×C₃₁	#578	1	1	yes	characteristic
147	C₂×#135	—	2	2	no	—
148	#547	C₂²×C₃₁	1	4	yes	—
149	#500	C₂²×C₃₁	1	2	yes	—
150	#500	C₂²×C₃₁	1	2	yes	—
151	#494	C₂²×C₃₁	1	2	yes	—
152	#480	—	2	4	no	—
153	#474	C₂²×C₃₁	1	4	yes	—
154	#467	—	2	2	no	—
155	#466	—	2	2	no	—
156	#461	C₂²×C₃₁	1	4	yes	—
157	#452	—	2	4	no	—
158	C₄×D₈	C₂²×C₃₁	1	1	yes	characteristic
159	C₄×D₈	C₂²×C₃₁	1	1	yes	characteristic
160	C₈×D₄	C₂²×C₃₁	1	2	yes	—
161	C₈×D₄	C₂²×C₃₁	1	2	yes	—
162	C₂×#107	—	2	2	no	—
163	#335	C₂²×C₃₁	1	1	yes	characteristic
164	#335	C₂²×C₃₁	1	1	yes	characteristic
165	#332	C₂²×C₃₁	1	4	yes	—
166	D₄⋊C₈	C₂²×C₃₁	1	2	yes	—
167	D₄⋊C₈	C₂²×C₃₁	1	2	yes	—
168	C₈²	C₂²×C₃₁	1	1	yes	characteristic
169	C₂×C₃₁	—	4	4	no	—
170	C₂×C₃₁	—	8	2	no	—
171	C₂×C₃₁	D₄×D₈	1	1	yes	characteristic
172	C₂×C₃₁	#3283	1	1	yes	characteristic
173	C₂×C₃₁	#3293	1	1	yes	characteristic
174	C₂×#41	—	2	4	no	—
175	#135	—	4	8	no	—
176	C₂×Q₁₆	—	2	2	no	—
177	C₂×Q₁₆	—	2	2	no	—
178	C₂×SD₁₆	—	2	4	no	—
179	C₂×D₈	D₄×C₃₁	1	1	yes	characteristic
180	C₂×D₈	D₄×C₃₁	1	1	yes	characteristic
181	C₂²×C₈	—	2	2	no	—
182	C₂²×C₈	—	2	2	no	—
183	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	—
184	#123	—	4	4	no	—
185	C₂²⋊Q₈	—	2	4	no	—
186	#121	—	2	4	no	—
187	C₄×D₄	C₂³×C₃₁	1	2	yes	—
188	C₄×D₄	C₂³×C₃₁	1	2	yes	—
189	C₂×C₄⋊C₄	—	2	4	no	—
190	#107	—	2	2	no	—
191	#107	—	2	2	no	—
192	#107	—	2	4	no	—
193	#107	—	4	2	no	—
194	#107	D₄×C₃₁	1	1	yes	characteristic
195	#107	D₄×C₃₁	1	1	yes	characteristic
196	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
197	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
198	#103	—	2	4	no	—
199	#102	—	2	4	no	—
200	#102	D₄×C₃₁	1	2	yes	—
201	#102	D₄×C₃₁	1	2	yes	—
202	C₂²⋊C₈	—	2	2	no	—
203	C₂²⋊C₈	—	2	2	no	—
204	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
205	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
206	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
207	C₃₁	#19971	1	1	yes	characteristic, direct product factor, Sylow
208	#41	—	4	8	no	—
209	#41	—	8	4	no	—
210	C₂×Q₈	—	2	4	no	—
211	C₂×D₄	—	4	4	no	—
212	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
213	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
214	C₂²×C₄	—	2	2	no	—
215	C₂²×C₄	—	2	2	no	—
216	C₂²×C₄	—	4	4	no	—
217	Q₁₆	—	4	4	no	—
218	Q₁₆	—	8	2	no	—
219	SD₁₆	—	4	8	no	—
220	D₈	—	2	2	no	—
221	D₈	D₈×C₃₁	1	4	yes	complemented
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	4	4	no	—
226	C₂×C₈	—	4	4	no	—
227	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
228	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
229	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
230	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
231	C₄⋊C₄	—	2	4	no	—
232	C₄⋊C₄	—	2	4	no	—
233	C₄⋊C₄	—	4	4	no	—
234	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
235	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
236	C₂²⋊C₄	—	2	2	no	—
237	C₂²⋊C₄	—	2	2	no	—
238	C₂²⋊C₄	—	4	4	no	—
239	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
240	C₂³	—	2	2	no	—
241	C₂³	—	2	2	no	—
242	Q₈	—	4	4	no	—
243	Q₈	—	4	4	no	—
244	D₄	—	2	2	no	—
245	D₄	—	2	2	no	—
246	D₄	—	2	2	no	—
247	D₄	—	8	8	no	—
248	D₄	C₂×D₈×C₃₁	1	4	yes	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	4	2	no	—
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	—	4	2	no	—
255	C₂×C₄	—	4	4	no	—
256	C₂×C₄	—	8	8	no	—
257	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
258	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
259	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
260	C₈	—	2	2	no	—
261	C₈	—	2	1	no	—
262	C₈	—	2	1	no	—
263	C₈	—	4	2	no	—
264	C₈	—	4	2	no	—
265	C₈	C₂×D₈×C₃₁	1	2	yes	—
266	C₈	C₃₁×#135	1	2	yes	complemented
267	C₂²	—	2	4	no	—
268	C₂²	—	4	2	no	—
269	C₂²	—	4	2	no	—
270	C₂²	—	4	2	no	—
271	C₂²	—	4	2	no	—
272	C₂²	—	4	2	no	—
273	C₂²	D₄²×C₃₁	1	1	yes	characteristic
274	C₄	—	2	1	no	—
275	C₄	—	4	2	no	—
276	C₄	—	4	2	no	—
277	C₄	—	8	4	no	—
278	C₄	C₂²×D₈×C₃₁	1	1	yes	characteristic
279	C₄	C₂×C₃₁×#135	1	1	yes	characteristic
280	C₄	C₃₁×#576	1	1	yes	characteristic
281	C₄	C₃₁×#578	1	1	yes	characteristic
282	C₂	—	4	4	no	—
283	C₂	—	8	2	no	—
284	C₂	D₄×D₈×C₃₁	1	1	yes	characteristic
285	C₂	C₃₁×#3283	1	1	yes	characteristic
286	C₂	C₃₁×#3293	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3334	C₂	semidirect
2. C₃₁×#3283	C₂	semidirect
3. C₃₁×#1713	C₂	semidirect
4. C₃₁×#1673	C₂	semidirect
5. C₃₁×#1669	C₂	semidirect
6. C₃₁×#1647	C₂	semidirect
7. C₈×D₈×C₃₁	C₂	semidirect
8. C₈×D₈×C₃₁	C₂	semidirect
77. D₈×C₃₁	D₈	semidirect
96. #19971	C₃₁	direct (N × H)
123. C₈×C₃₁	#135	semidirect
124. #3334	C₂×C₃₁	semidirect
125. #3283	C₂×C₃₁	semidirect
126. #1713	C₂×C₃₁	semidirect
127. #1673	C₂×C₃₁	semidirect
128. #1669	C₂×C₃₁	semidirect
129. #1647	C₂×C₃₁	semidirect
130. C₈×D₈	C₂×C₃₁	semidirect
131. C₈×D₈	C₂×C₃₁	semidirect
207. C₃₁	#19971	direct (N × H)
221. D₈	D₈×C₃₁	semidirect
266. C₈	C₃₁×#135	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#3334	2
C₃₁×#3283	2
C₃₁×#1713	2
C₃₁×#1673	2
C₃₁×#1669	2
C₃₁×#1647	2
C₈×D₈×C₃₁	2
#19971	31

Subgroups — C31×#19971 #21476730

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19971 #21476730