Subgroups — C₃₁×#20006 #21476765

Order 7936.

586 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	C₃₁×#3380	C₂	1	2	yes	maximal
2	C₃₁×#3338	C₂	1	2	yes	maximal, complemented
3	C₃₁×#1713	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1676	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1675	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1632	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×#554	C₂²	1	4	yes	—
10	C₃₁×#500	C₂²	1	2	yes	—
11	C₃₁×#500	C₂²	1	2	yes	—
12	C₃₁×#499	—	2	2	no	—
13	C₃₁×#494	C₂²	1	2	yes	—
14	C₃₁×#484	—	2	4	no	—
15	C₃₁×#478	—	2	4	no	—
16	C₃₁×#477	C₂²	1	4	yes	—
17	C₃₁×#476	C₂²	1	4	yes	—
18	C₃₁×#467	—	2	2	no	—
19	C₈×Q₈×C₃₁	C₂²	1	2	yes	—
20	C₄×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
21	C₄×D₈×C₃₁	—	2	2	no	—
22	C₄×D₈×C₃₁	C₂²	1	1	yes	characteristic
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×#333	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	—	4	2	no	—
32	C₂×Q₁₆×C₃₁	D₄	1	1	yes	characteristic
33	C₂×D₈×C₃₁	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₃₁	—	2	2	no	—
35	C₃₁×C₄⋊Q₈	C₂³	1	4	yes	—
36	C₃₁×#125	—	2	4	no	—
37	C₃₁×C₂²⋊Q₈	—	2	4	no	—
38	C₃₁×C₂²⋊Q₈	—	4	4	no	—
39	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
40	C₄×D₄×C₃₁	—	2	4	no	—
41	C₄×D₄×C₃₁	C₂³	1	2	yes	—
42	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
43	C₃₁×#107	—	2	2	no	—
44	C₃₁×#107	—	2	2	no	—
45	C₃₁×#107	—	2	2	no	—
46	C₃₁×#107	—	2	2	no	—
47	C₃₁×#107	—	2	4	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	D₄	1	2	yes	—
56	C₃₁×#102	—	2	4	no	—
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	C₂²×C₄×C₃₁	—	4	4	no	—
68	Q₁₆×C₃₁	—	2	2	no	—
69	Q₁₆×C₃₁	—	8	4	no	—
70	D₈×C₃₁	Q₁₆	1	4	yes	complemented
71	C₂×C₈×C₃₁	—	2	1	no	—
72	C₂×C₈×C₃₁	—	2	2	no	—
73	C₂×C₈×C₃₁	—	2	2	no	—
74	C₂×C₈×C₃₁	—	4	4	no	—
75	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
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	C₄⋊C₄×C₃₁	—	2	2	no	—
80	C₄⋊C₄×C₃₁	—	2	4	no	—
81	C₄⋊C₄×C₃₁	—	2	4	no	—
82	C₄⋊C₄×C₃₁	—	4	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	C₄²×C₃₁	—	2	2	no	—
89	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
90	#20006	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
91	C₂³×C₃₁	—	2	2	no	—
92	Q₈×C₃₁	—	2	2	no	—
93	Q₈×C₃₁	—	2	2	no	—
94	Q₈×C₃₁	—	8	4	no	—
95	D₄×C₃₁	—	2	2	no	—
96	D₄×C₃₁	C₂×Q₁₆	1	4	yes	—
97	C₂×C₄×C₃₁	—	2	2	no	—
98	C₂×C₄×C₃₁	—	2	2	no	—
99	C₂×C₄×C₃₁	—	2	2	no	—
100	C₂×C₄×C₃₁	—	4	2	no	—
101	C₂×C₄×C₃₁	—	4	2	no	—
102	C₂×C₄×C₃₁	—	4	4	no	—
103	C₂×C₄×C₃₁	—	16	4	no	—
104	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
105	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
106	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
107	C₈×C₃₁	—	2	2	no	—
108	C₈×C₃₁	—	2	1	no	—
109	C₈×C₃₁	—	2	1	no	—
110	C₈×C₃₁	—	4	2	no	—
111	C₈×C₃₁	—	4	2	no	—
112	C₈×C₃₁	C₂×Q₁₆	1	2	yes	—
113	C₈×C₃₁	#135	1	2	yes	—
114	#3380	C₂×C₃₁	1	2	yes	—
115	#3338	C₂×C₃₁	1	2	yes	complemented
116	#1713	C₂×C₃₁	1	1	yes	characteristic, complemented
117	#1676	C₂×C₃₁	1	2	yes	complemented
118	#1675	C₂×C₃₁	1	2	yes	complemented
119	#1632	C₂×C₃₁	1	4	yes	complemented
120	C₈×Q₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
121	C₈×D₈	C₂×C₃₁	1	1	yes	characteristic
122	C₂²×C₃₁	—	2	4	no	—
123	C₂²×C₃₁	—	4	2	no	—
124	C₂²×C₃₁	—	4	2	no	—
125	C₂²×C₃₁	D₄²	1	1	yes	characteristic
126	C₄×C₃₁	—	2	1	no	—
127	C₄×C₃₁	—	4	2	no	—
128	C₄×C₃₁	—	4	2	no	—
129	C₄×C₃₁	—	4	2	no	—
130	C₄×C₃₁	—	8	4	no	—
131	C₄×C₃₁	C₂²×Q₁₆	1	1	yes	characteristic
132	C₄×C₃₁	C₂×#135	1	1	yes	characteristic
133	C₄×C₃₁	#578	1	1	yes	characteristic
134	C₄×C₃₁	#578	1	1	yes	characteristic
135	#554	C₂²×C₃₁	1	4	yes	—
136	#500	C₂²×C₃₁	1	2	yes	—
137	#500	C₂²×C₃₁	1	2	yes	—
138	#499	—	2	2	no	—
139	#494	C₂²×C₃₁	1	2	yes	—
140	#484	—	2	4	no	—
141	#478	—	2	4	no	—
142	#477	C₂²×C₃₁	1	4	yes	—
143	#476	C₂²×C₃₁	1	4	yes	—
144	#467	—	2	2	no	—
145	C₈×Q₈	C₂²×C₃₁	1	2	yes	—
146	C₄×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
147	C₄×D₈	—	2	2	no	—
148	C₄×D₈	C₂²×C₃₁	1	1	yes	characteristic
149	C₈×D₄	C₂²×C₃₁	1	2	yes	—
150	C₂×#107	—	2	2	no	—
151	#335	C₂²×C₃₁	1	1	yes	characteristic
152	#335	C₂²×C₃₁	1	1	yes	characteristic
153	#333	C₂²×C₃₁	1	4	yes	—
154	Q₈⋊C₈	C₂²×C₃₁	1	2	yes	—
155	D₄⋊C₈	C₂²×C₃₁	1	2	yes	—
156	C₈²	C₂²×C₃₁	1	1	yes	characteristic
157	C₂×C₃₁	—	4	4	no	—
158	C₂×C₃₁	#3283	1	1	yes	characteristic
159	C₂×C₃₁	D₄×Q₁₆	1	1	yes	characteristic
160	C₂×C₃₁	#3292	1	1	yes	characteristic
161	C₂×Q₁₆	—	4	2	no	—
162	C₂×Q₁₆	D₄×C₃₁	1	1	yes	characteristic
163	C₂×D₈	D₄×C₃₁	1	1	yes	characteristic
164	C₂²×C₈	—	2	2	no	—
165	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	—
166	#125	—	2	4	no	—
167	C₂²⋊Q₈	—	2	4	no	—
168	C₂²⋊Q₈	—	4	4	no	—
169	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
170	C₄×D₄	—	2	4	no	—
171	C₄×D₄	C₂³×C₃₁	1	2	yes	—
172	C₂×C₄⋊C₄	—	2	4	no	—
173	#107	—	2	2	no	—
174	#107	—	2	2	no	—
175	#107	—	2	2	no	—
176	#107	—	2	2	no	—
177	#107	—	2	4	no	—
178	#107	D₄×C₃₁	1	1	yes	characteristic
179	#107	D₄×C₃₁	1	1	yes	characteristic
180	#106	—	2	4	no	—
181	C₄⋊C₈	—	2	2	no	—
182	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
183	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
184	#103	—	2	4	no	—
185	#103	D₄×C₃₁	1	2	yes	—
186	#102	—	2	4	no	—
187	#102	D₄×C₃₁	1	2	yes	—
188	C₂²⋊C₈	—	2	2	no	—
189	C₄×C₈	—	2	2	no	—
190	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
191	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
192	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
193	C₃₁	#20006	1	1	yes	characteristic, direct product factor, Sylow
194	C₂×Q₈	—	2	4	no	—
195	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
196	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
197	C₂²×C₄	—	2	2	no	—
198	C₂²×C₄	—	4	4	no	—
199	Q₁₆	—	2	2	no	—
200	Q₁₆	—	8	4	no	—
201	D₈	Q₁₆×C₃₁	1	4	yes	complemented
202	C₂×C₈	—	2	1	no	—
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	—	4	4	no	—
206	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
207	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
208	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
209	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
210	C₄⋊C₄	—	2	2	no	—
211	C₄⋊C₄	—	2	4	no	—
212	C₄⋊C₄	—	2	4	no	—
213	C₄⋊C₄	—	4	4	no	—
214	C₄⋊C₄	—	4	4	no	—
215	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
216	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
217	C₂²⋊C₄	—	2	2	no	—
218	C₂²⋊C₄	—	4	4	no	—
219	C₄²	—	2	2	no	—
220	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
221	C₂³	—	2	2	no	—
222	Q₈	—	2	2	no	—
223	Q₈	—	2	2	no	—
224	Q₈	—	8	4	no	—
225	D₄	—	2	2	no	—
226	D₄	C₂×Q₁₆×C₃₁	1	4	yes	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	4	2	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	—	4	4	no	—
233	C₂×C₄	—	16	4	no	—
234	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
235	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
236	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
237	C₈	—	2	2	no	—
238	C₈	—	2	1	no	—
239	C₈	—	2	1	no	—
240	C₈	—	4	2	no	—
241	C₈	—	4	2	no	—
242	C₈	C₂×Q₁₆×C₃₁	1	2	yes	—
243	C₈	C₃₁×#135	1	2	yes	—
244	C₂²	—	2	4	no	—
245	C₂²	—	4	2	no	—
246	C₂²	—	4	2	no	—
247	C₂²	D₄²×C₃₁	1	1	yes	characteristic
248	C₄	—	2	1	no	—
249	C₄	—	4	2	no	—
250	C₄	—	4	2	no	—
251	C₄	—	4	2	no	—
252	C₄	—	8	4	no	—
253	C₄	C₂²×Q₁₆×C₃₁	1	1	yes	characteristic
254	C₄	C₂×C₃₁×#135	1	1	yes	characteristic
255	C₄	C₃₁×#578	1	1	yes	characteristic
256	C₄	C₃₁×#578	1	1	yes	characteristic
257	C₂	—	4	4	no	—
258	C₂	C₃₁×#3283	1	1	yes	characteristic
259	C₂	D₄×Q₁₆×C₃₁	1	1	yes	characteristic
260	C₂	C₃₁×#3292	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×#3338	C₂	semidirect
3. C₃₁×#1713	C₂	semidirect
4. C₃₁×#1676	C₂	semidirect
5. C₃₁×#1675	C₂	semidirect
6. C₃₁×#1632	C₂	semidirect
7. C₈×Q₁₆×C₃₁	C₂	semidirect
70. D₈×C₃₁	Q₁₆	semidirect
90. #20006	C₃₁	direct (N × H)
115. #3338	C₂×C₃₁	semidirect
116. #1713	C₂×C₃₁	semidirect
117. #1676	C₂×C₃₁	semidirect
118. #1675	C₂×C₃₁	semidirect
119. #1632	C₂×C₃₁	semidirect
120. C₈×Q₁₆	C₂×C₃₁	semidirect
193. C₃₁	#20006	direct (N × H)
201. D₈	Q₁₆×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#3380	2
C₃₁×#3338	2
C₃₁×#1713	2
C₃₁×#1676	2
C₃₁×#1675	2
C₃₁×#1632	2
C₈×Q₁₆×C₃₁	2
C₈×D₈×C₃₁	2
#20006	31

Subgroups — C31×#20006 #21476765

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#20006 #21476765