Subgroups — C₃₁×#20020 #21476779

Order 7936.

582 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3393	C₂	1	2	yes	maximal
2	C₃₁×#3342	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×#1670	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1650	C₂	1	4	yes	maximal, complemented
7	C₈×Q₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₈×D₈×C₃₁	C₂	1	1	yes	maximal, characteristic
9	C₃₁×#556	C₂²	1	4	yes	—
10	C₃₁×#500	C₂²	1	2	yes	—
11	C₃₁×#500	C₂²	1	2	yes	—
12	C₃₁×#495	—	2	2	no	—
13	C₃₁×#494	C₂²	1	2	yes	—
14	C₃₁×#485	—	2	4	no	—
15	C₃₁×#475	C₂²	1	4	yes	—
16	C₃₁×#467	—	2	2	no	—
17	C₃₁×#464	—	2	4	no	—
18	C₃₁×#461	C₂²	1	4	yes	—
19	C₈×Q₈×C₃₁	C₂²	1	2	yes	—
20	C₄×Q₁₆×C₃₁	—	2	2	no	—
21	C₄×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
22	C₄×D₈×C₃₁	C₂²	1	1	yes	characteristic
23	C₈×D₄×C₃₁	C₂²	1	2	yes	—
24	C₃₁×#427	—	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×Q₈⋊C₈	C₂²	1	2	yes	—
29	C₃₁×D₄⋊C₈	C₂²	1	2	yes	—
30	C₈²×C₃₁	C₂²	1	1	yes	characteristic
31	C₂×Q₁₆×C₃₁	—	2	2	no	—
32	C₂×Q₁₆×C₃₁	—	2	2	no	—
33	C₂×Q₁₆×C₃₁	D₄	1	1	yes	characteristic
34	C₂×SD₁₆×C₃₁	—	2	4	no	—
35	C₂×D₈×C₃₁	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₃₁	—	2	2	no	—
37	C₃₁×C₄⋊Q₈	C₂³	1	4	yes	—
38	C₃₁×#126	—	4	4	no	—
39	C₃₁×#124	—	2	4	no	—
40	C₃₁×C₂²⋊Q₈	—	2	4	no	—
41	C₄×Q₈×C₃₁	—	2	4	no	—
42	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
43	C₄×D₄×C₃₁	C₂³	1	2	yes	—
44	C₃₁×#117	—	2	4	no	—
45	C₃₁×#107	—	2	2	no	—
46	C₃₁×#107	—	2	2	no	—
47	C₃₁×#107	—	4	2	no	—
48	C₃₁×#107	D₄	1	1	yes	characteristic
49	C₃₁×#107	D₄	1	1	yes	characteristic
50	C₃₁×#106	—	2	4	no	—
51	C₃₁×C₄⋊C₈	—	2	2	no	—
52	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
53	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
54	C₃₁×#103	—	2	4	no	—
55	C₃₁×#103	—	2	4	no	—
56	C₃₁×#103	D₄	1	2	yes	—
57	C₃₁×#102	D₄	1	2	yes	—
58	C₃₁×C₂²⋊C₈	—	2	2	no	—
59	C₄×C₈×C₃₁	—	2	2	no	—
60	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
61	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
62	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
63	C₂×Q₈×C₃₁	—	2	4	no	—
64	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
65	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
66	C₂²×C₄×C₃₁	—	2	2	no	—
67	Q₁₆×C₃₁	—	2	2	no	—
68	Q₁₆×C₃₁	—	4	4	no	—
69	Q₁₆×C₃₁	—	8	2	no	—
70	SD₁₆×C₃₁	—	8	4	no	—
71	D₈×C₃₁	—	2	2	no	—
72	C₂×C₈×C₃₁	—	2	1	no	—
73	C₂×C₈×C₃₁	—	2	2	no	—
74	C₂×C₈×C₃₁	—	2	2	no	—
75	C₂×C₈×C₃₁	—	4	4	no	—
76	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
78	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
79	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
80	C₄⋊C₄×C₃₁	—	2	2	no	—
81	C₄⋊C₄×C₃₁	—	2	4	no	—
82	C₄⋊C₄×C₃₁	—	2	4	no	—
83	C₄⋊C₄×C₃₁	—	4	4	no	—
84	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
85	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
86	C₂²⋊C₄×C₃₁	—	2	2	no	—
87	C₂²⋊C₄×C₃₁	—	4	4	no	—
88	C₂²⋊C₄×C₃₁	—	4	4	no	—
89	C₄²×C₃₁	—	2	2	no	—
90	C₄²×C₃₁	—	4	4	no	—
91	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
92	#20020	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
93	C₂³×C₃₁	—	2	2	no	—
94	Q₈×C₃₁	—	2	2	no	—
95	Q₈×C₃₁	—	2	2	no	—
96	Q₈×C₃₁	—	4	4	no	—
97	Q₈×C₃₁	—	4	4	no	—
98	D₄×C₃₁	—	2	2	no	—
99	D₄×C₃₁	—	2	2	no	—
100	C₂×C₄×C₃₁	—	2	2	no	—
101	C₂×C₄×C₃₁	—	2	2	no	—
102	C₂×C₄×C₃₁	—	2	2	no	—
103	C₂×C₄×C₃₁	—	4	2	no	—
104	C₂×C₄×C₃₁	—	4	2	no	—
105	C₂×C₄×C₃₁	—	4	4	no	—
106	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
107	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
108	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
109	C₈×C₃₁	—	2	2	no	—
110	C₈×C₃₁	—	2	1	no	—
111	C₈×C₃₁	—	2	1	no	—
112	C₈×C₃₁	—	4	2	no	—
113	C₈×C₃₁	—	4	2	no	—
114	C₈×C₃₁	#135	1	2	yes	—
115	C₈×C₃₁	#135	1	2	yes	—
116	#3393	C₂×C₃₁	1	2	yes	—
117	#3342	C₂×C₃₁	1	2	yes	complemented
118	#1713	C₂×C₃₁	1	1	yes	characteristic, complemented
119	#1673	C₂×C₃₁	1	2	yes	complemented
120	#1670	C₂×C₃₁	1	2	yes	complemented
121	#1650	C₂×C₃₁	1	4	yes	complemented
122	C₈×Q₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
123	C₈×D₈	C₂×C₃₁	1	1	yes	characteristic
124	C₂²×C₃₁	—	4	2	no	—
125	C₂²×C₃₁	—	4	2	no	—
126	C₂²×C₃₁	—	4	2	no	—
127	C₂²×C₃₁	D₄²	1	1	yes	characteristic
128	C₄×C₃₁	—	2	1	no	—
129	C₄×C₃₁	—	4	2	no	—
130	C₄×C₃₁	—	4	2	no	—
131	C₄×C₃₁	—	4	2	no	—
132	C₄×C₃₁	—	8	4	no	—
133	C₄×C₃₁	C₂×#135	1	1	yes	characteristic
134	C₄×C₃₁	C₂×#135	1	1	yes	characteristic
135	C₄×C₃₁	#576	1	1	yes	characteristic
136	C₄×C₃₁	#578	1	1	yes	characteristic
137	#556	C₂²×C₃₁	1	4	yes	—
138	#500	C₂²×C₃₁	1	2	yes	—
139	#500	C₂²×C₃₁	1	2	yes	—
140	#495	—	2	2	no	—
141	#494	C₂²×C₃₁	1	2	yes	—
142	#485	—	2	4	no	—
143	#475	C₂²×C₃₁	1	4	yes	—
144	#467	—	2	2	no	—
145	#464	—	2	4	no	—
146	#461	C₂²×C₃₁	1	4	yes	—
147	C₈×Q₈	C₂²×C₃₁	1	2	yes	—
148	C₄×Q₁₆	—	2	2	no	—
149	C₄×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
150	C₄×D₈	C₂²×C₃₁	1	1	yes	characteristic
151	C₈×D₄	C₂²×C₃₁	1	2	yes	—
152	#427	—	2	2	no	—
153	#335	C₂²×C₃₁	1	1	yes	characteristic
154	#335	C₂²×C₃₁	1	1	yes	characteristic
155	#332	C₂²×C₃₁	1	4	yes	—
156	Q₈⋊C₈	C₂²×C₃₁	1	2	yes	—
157	D₄⋊C₈	C₂²×C₃₁	1	2	yes	—
158	C₈²	C₂²×C₃₁	1	1	yes	characteristic
159	C₂×C₃₁	—	8	2	no	—
160	C₂×C₃₁	#3283	1	1	yes	characteristic
161	C₂×C₃₁	#3287	1	1	yes	characteristic
162	C₂×C₃₁	#3293	1	1	yes	characteristic
163	C₂×Q₁₆	—	2	2	no	—
164	C₂×Q₁₆	—	2	2	no	—
165	C₂×Q₁₆	D₄×C₃₁	1	1	yes	characteristic
166	C₂×SD₁₆	—	2	4	no	—
167	C₂×D₈	D₄×C₃₁	1	1	yes	characteristic
168	C₂²×C₈	—	2	2	no	—
169	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	—
170	#126	—	4	4	no	—
171	#124	—	2	4	no	—
172	C₂²⋊Q₈	—	2	4	no	—
173	C₄×Q₈	—	2	4	no	—
174	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
175	C₄×D₄	C₂³×C₃₁	1	2	yes	—
176	#117	—	2	4	no	—
177	#107	—	2	2	no	—
178	#107	—	2	2	no	—
179	#107	—	4	2	no	—
180	#107	D₄×C₃₁	1	1	yes	characteristic
181	#107	D₄×C₃₁	1	1	yes	characteristic
182	#106	—	2	4	no	—
183	C₄⋊C₈	—	2	2	no	—
184	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
185	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
186	#103	—	2	4	no	—
187	#103	—	2	4	no	—
188	#103	D₄×C₃₁	1	2	yes	—
189	#102	D₄×C₃₁	1	2	yes	—
190	C₂²⋊C₈	—	2	2	no	—
191	C₄×C₈	—	2	2	no	—
192	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
193	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
194	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
195	C₃₁	#20020	1	1	yes	characteristic, direct product factor, Sylow
196	C₂×Q₈	—	2	4	no	—
197	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
198	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
199	C₂²×C₄	—	2	2	no	—
200	Q₁₆	—	2	2	no	—
201	Q₁₆	—	4	4	no	—
202	Q₁₆	—	8	2	no	—
203	SD₁₆	—	8	4	no	—
204	D₈	—	2	2	no	—
205	C₂×C₈	—	2	1	no	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	—	2	2	no	—
208	C₂×C₈	—	4	4	no	—
209	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
210	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
211	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
212	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
213	C₄⋊C₄	—	2	2	no	—
214	C₄⋊C₄	—	2	4	no	—
215	C₄⋊C₄	—	2	4	no	—
216	C₄⋊C₄	—	4	4	no	—
217	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
218	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
219	C₂²⋊C₄	—	2	2	no	—
220	C₂²⋊C₄	—	4	4	no	—
221	C₂²⋊C₄	—	4	4	no	—
222	C₄²	—	2	2	no	—
223	C₄²	—	4	4	no	—
224	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
225	C₂³	—	2	2	no	—
226	Q₈	—	2	2	no	—
227	Q₈	—	2	2	no	—
228	Q₈	—	4	4	no	—
229	Q₈	—	4	4	no	—
230	D₄	—	2	2	no	—
231	D₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	—	4	4	no	—
238	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
239	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
240	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
241	C₈	—	2	2	no	—
242	C₈	—	2	1	no	—
243	C₈	—	2	1	no	—
244	C₈	—	4	2	no	—
245	C₈	—	4	2	no	—
246	C₈	C₃₁×#135	1	2	yes	—
247	C₈	C₃₁×#135	1	2	yes	—
248	C₂²	—	4	2	no	—
249	C₂²	—	4	2	no	—
250	C₂²	—	4	2	no	—
251	C₂²	D₄²×C₃₁	1	1	yes	characteristic
252	C₄	—	2	1	no	—
253	C₄	—	4	2	no	—
254	C₄	—	4	2	no	—
255	C₄	—	4	2	no	—
256	C₄	—	8	4	no	—
257	C₄	C₂×C₃₁×#135	1	1	yes	characteristic
258	C₄	C₂×C₃₁×#135	1	1	yes	characteristic
259	C₄	C₃₁×#576	1	1	yes	characteristic
260	C₄	C₃₁×#578	1	1	yes	characteristic
261	C₂	—	8	2	no	—
262	C₂	C₃₁×#3283	1	1	yes	characteristic
263	C₂	C₃₁×#3287	1	1	yes	characteristic
264	C₂	C₃₁×#3293	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
prime radical	C₂×D₈×C₃₁	D₄
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
2. C₃₁×#3342	C₂	semidirect
3. C₃₁×#1713	C₂	semidirect
4. C₃₁×#1673	C₂	semidirect
5. C₃₁×#1670	C₂	semidirect
6. C₃₁×#1650	C₂	semidirect
7. C₈×Q₁₆×C₃₁	C₂	semidirect
92. #20020	C₃₁	direct (N × H)
117. #3342	C₂×C₃₁	semidirect
118. #1713	C₂×C₃₁	semidirect
119. #1673	C₂×C₃₁	semidirect
120. #1670	C₂×C₃₁	semidirect
121. #1650	C₂×C₃₁	semidirect
122. C₈×Q₁₆	C₂×C₃₁	semidirect
195. C₃₁	#20020	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×#3393	2
C₃₁×#3342	2
C₃₁×#1713	2
C₃₁×#1673	2
C₃₁×#1670	2
C₃₁×#1650	2
C₈×Q₁₆×C₃₁	2
C₈×D₈×C₃₁	2
#20020	31

Subgroups — C31×#20020 #21476779

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#20020 #21476779