Subgroups — C₃₁×#20782 #21477541

Order 7936.

778 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₃₁×#117	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1840	C₂	1	2	yes	maximal, complemented
3	C₃₁×#1832	C₂	1	2	yes	maximal, complemented
4	C₃₁×#1818	C₂	1	2	yes	maximal
5	C₃₁×#1798	C₂	1	2	yes	maximal
6	C₄×C₃₁×C₄⋊C₈	C₂	1	2	yes	maximal, complemented
7	C₄×C₃₁×C₂²⋊C₈	C₂	1	2	yes	maximal
8	C₃₁×#1744	C₂	1	2	yes	maximal, complemented
9	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
10	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
11	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
12	C₂×C₃₁×C₄⋊C₈	C₂²	1	4	yes	—
13	C₂×C₃₁×C₂²⋊C₈	C₂²	1	4	yes	—
14	C₂×C₄×C₈×C₃₁	C₂²	1	4	yes	—
15	C₄×C₄⋊C₄×C₃₁	C₄	1	4	yes	—
16	C₄×C₂²⋊C₄×C₃₁	C₄	1	4	yes	—
17	C₃₁×#376	C₂²	1	1	yes	characteristic
18	C₃₁×#376	C₂²	1	1	yes	characteristic
19	C₄³×C₃₁	C₂²	1	1	yes	characteristic
20	C₄³×C₃₁	C₂²	1	1	yes	characteristic
21	C₃₁×#336	C₂²	1	4	yes	—
22	C₃₁×#336	C₂²	1	4	yes	—
23	C₃₁×#336	C₂²	1	4	yes	—
24	C₃₁×#336	C₂²	1	4	yes	—
25	C₂³×C₄×C₃₁	C₂×C₄	1	2	yes	—
26	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
27	C₂²×C₈×C₃₁	C₂³	1	8	yes	—
28	C₃₁×#117	—	2	4	no	—
29	C₃₁×#117	—	2	4	no	—
30	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
31	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
32	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
33	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
34	C₂×C₄²×C₃₁	—	2	4	no	—
35	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₃₁	C₂×C₄	1	4	yes	—
37	C₂×C₄²×C₃₁	C₂×C₄	1	4	yes	—
38	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
39	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
40	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
41	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
42	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
43	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
44	C₃₁×C₄⋊C₈	—	2	8	no	—
45	C₃₁×C₂²⋊C₈	—	2	8	no	—
46	C₄×C₈×C₃₁	—	2	8	no	—
47	C₃₁×#95	C₂×C₄	1	4	yes	—
48	C₃₁×#95	C₂×C₄	1	4	yes	—
49	C₂⁴×C₃₁	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₃₁	—	2	2	no	—
51	C₂²×C₄×C₃₁	—	2	2	no	—
52	C₂²×C₄×C₃₁	—	2	4	no	—
53	C₂²×C₄×C₃₁	—	2	8	no	—
54	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
55	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
56	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
57	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
58	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
59	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
61	C₂²×C₄×C₃₁	#41	1	4	yes	—
62	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
63	C₂×C₈×C₃₁	—	2	8	no	—
64	C₂×C₈×C₃₁	—	2	8	no	—
65	C₂×C₈×C₃₁	—	2	8	no	—
66	C₄⋊C₄×C₃₁	—	2	4	no	—
67	C₄⋊C₄×C₃₁	—	2	4	no	—
68	C₂²⋊C₄×C₃₁	—	2	4	no	—
69	C₂²⋊C₄×C₃₁	—	2	4	no	—
70	C₄²×C₃₁	—	2	4	no	—
71	C₄²×C₃₁	—	2	8	no	—
72	C₄²×C₃₁	—	2	8	no	—
73	C₄²×C₃₁	—	2	2	no	—
74	C₄²×C₃₁	—	2	2	no	—
75	C₄²×C₃₁	—	2	2	no	—
76	C₄²×C₃₁	—	2	2	no	—
77	C₄²×C₃₁	#41	1	4	yes	—
78	C₄²×C₃₁	#41	1	4	yes	—
79	#20782	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
80	C₂³×C₃₁	—	2	1	no	—
81	C₂³×C₃₁	—	2	1	no	—
82	C₂³×C₃₁	—	2	4	no	—
83	C₂³×C₃₁	#117	1	2	yes	—
84	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
85	C₂×C₄×C₃₁	—	2	1	no	—
86	C₂×C₄×C₃₁	—	2	1	no	—
87	C₂×C₄×C₃₁	—	2	4	no	—
88	C₂×C₄×C₃₁	—	2	8	no	—
89	C₂×C₄×C₃₁	—	2	2	no	—
90	C₂×C₄×C₃₁	—	2	2	no	—
91	C₂×C₄×C₃₁	—	2	2	no	—
92	C₂×C₄×C₃₁	—	2	2	no	—
93	C₂×C₄×C₃₁	—	2	4	no	—
94	C₂×C₄×C₃₁	—	2	4	no	—
95	C₂×C₄×C₃₁	—	2	4	no	—
96	C₂×C₄×C₃₁	—	2	4	no	—
97	C₂×C₄×C₃₁	—	4	8	no	—
98	C₂×C₄×C₃₁	#117	1	4	yes	—
99	C₂×C₄×C₃₁	#117	1	2	yes	—
100	C₂×C₄×C₃₁	#117	1	4	yes	—
101	C₂×C₄×C₃₁	#131	1	8	yes	—
102	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
103	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
104	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
105	C₈×C₃₁	—	4	8	no	—
106	C₄×#117	C₂×C₃₁	1	1	yes	characteristic
107	#1840	C₂×C₃₁	1	2	yes	complemented
108	#1832	C₂×C₃₁	1	2	yes	complemented
109	#1818	C₂×C₃₁	1	2	yes	—
110	#1798	C₂×C₃₁	1	2	yes	—
111	C₄×C₄⋊C₈	C₂×C₃₁	1	2	yes	complemented
112	C₄×C₂²⋊C₈	C₂×C₃₁	1	2	yes	—
113	#1744	C₂×C₃₁	1	2	yes	complemented
114	C₂²×C₃₁	—	2	4	no	—
115	C₂²×C₃₁	—	2	2	no	—
116	C₂²×C₃₁	—	4	2	no	—
117	C₂²×C₃₁	—	4	2	no	—
118	C₂²×C₃₁	C₂×#117	1	1	yes	characteristic
119	C₂²×C₃₁	C₂×#117	1	1	yes	characteristic
120	C₂²×C₃₁	C₂×#117	1	1	yes	characteristic
121	C₂²×C₃₁	C₂×#131	1	4	yes	—
122	C₄×C₃₁	—	2	4	no	—
123	C₄×C₃₁	—	2	2	no	—
124	C₄×C₃₁	—	4	2	no	—
125	C₄×C₃₁	—	4	2	no	—
126	C₄×C₃₁	—	4	2	no	—
127	C₄×C₃₁	—	4	2	no	—
128	C₄×C₃₁	#433	1	8	yes	—
129	C₄×C₃₁	#529	1	4	yes	—
130	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
131	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
132	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
133	C₂×C₄⋊C₈	C₂²×C₃₁	1	4	yes	—
134	C₂×C₂²⋊C₈	C₂²×C₃₁	1	4	yes	—
135	C₂×C₄×C₈	C₂²×C₃₁	1	4	yes	—
136	C₄×C₄⋊C₄	C₄×C₃₁	1	4	yes	—
137	C₄×C₂²⋊C₄	C₄×C₃₁	1	4	yes	—
138	#376	C₂²×C₃₁	1	1	yes	characteristic
139	#376	C₂²×C₃₁	1	1	yes	characteristic
140	C₄³	C₂²×C₃₁	1	1	yes	characteristic
141	C₄³	C₂²×C₃₁	1	1	yes	characteristic
142	#336	C₂²×C₃₁	1	4	yes	—
143	#336	C₂²×C₃₁	1	4	yes	—
144	#336	C₂²×C₃₁	1	4	yes	—
145	#336	C₂²×C₃₁	1	4	yes	—
146	C₂×C₃₁	—	4	2	no	—
147	C₂×C₃₁	#2283	1	1	yes	characteristic
148	C₂×C₃₁	C₂×#433	1	2	yes	—
149	C₂×C₃₁	#2922	1	2	yes	—
150	C₂×C₃₁	#2922	1	2	yes	—
151	C₂³×C₄	C₂×C₄×C₃₁	1	2	yes	—
152	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
153	C₂²×C₈	C₂³×C₃₁	1	8	yes	—
154	#117	—	2	4	no	—
155	#117	—	2	4	no	—
156	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
157	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
158	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
159	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
160	C₂×C₄²	—	2	4	no	—
161	C₂×C₄²	C₂×C₄×C₃₁	1	4	yes	—
162	C₂×C₄²	C₂×C₄×C₃₁	1	4	yes	—
163	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
164	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
165	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
166	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
167	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
168	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
169	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
170	C₄⋊C₈	—	2	8	no	—
171	C₂²⋊C₈	—	2	8	no	—
172	C₄×C₈	—	2	8	no	—
173	#95	C₂×C₄×C₃₁	1	4	yes	—
174	#95	C₂×C₄×C₃₁	1	4	yes	—
175	C₃₁	#20782	1	1	yes	characteristic, direct product factor, Sylow
176	C₂⁴	C₂²×C₄×C₃₁	1	1	yes	characteristic
177	C₂²×C₄	—	2	2	no	—
178	C₂²×C₄	—	2	2	no	—
179	C₂²×C₄	—	2	4	no	—
180	C₂²×C₄	—	2	8	no	—
181	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
182	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
183	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
184	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
185	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
186	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
187	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
188	C₂²×C₄	#41×C₃₁	1	4	yes	—
189	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
190	C₂×C₈	—	2	8	no	—
191	C₂×C₈	—	2	8	no	—
192	C₂×C₈	—	2	8	no	—
193	C₄⋊C₄	—	2	4	no	—
194	C₄⋊C₄	—	2	4	no	—
195	C₂²⋊C₄	—	2	4	no	—
196	C₂²⋊C₄	—	2	4	no	—
197	C₄²	—	2	4	no	—
198	C₄²	—	2	8	no	—
199	C₄²	—	2	8	no	—
200	C₄²	—	2	2	no	—
201	C₄²	—	2	2	no	—
202	C₄²	—	2	2	no	—
203	C₄²	—	2	2	no	—
204	C₄²	#41×C₃₁	1	4	yes	—
205	C₄²	#41×C₃₁	1	4	yes	—
206	C₂³	—	2	1	no	—
207	C₂³	—	2	1	no	—
208	C₂³	—	2	4	no	—
209	C₂³	C₃₁×#117	1	2	yes	—
210	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	4	no	—
214	C₂×C₄	—	2	8	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	4	no	—
220	C₂×C₄	—	2	4	no	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	2	4	no	—
223	C₂×C₄	—	4	8	no	—
224	C₂×C₄	C₃₁×#117	1	4	yes	—
225	C₂×C₄	C₃₁×#117	1	2	yes	—
226	C₂×C₄	C₃₁×#117	1	4	yes	—
227	C₂×C₄	C₃₁×#131	1	8	yes	—
228	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
229	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
230	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
231	C₈	—	4	8	no	—
232	C₂²	—	2	4	no	—
233	C₂²	—	2	2	no	—
234	C₂²	—	4	2	no	—
235	C₂²	—	4	2	no	—
236	C₂²	C₂×C₃₁×#117	1	1	yes	characteristic
237	C₂²	C₂×C₃₁×#117	1	1	yes	characteristic
238	C₂²	C₂×C₃₁×#117	1	1	yes	characteristic
239	C₂²	C₂×C₃₁×#131	1	4	yes	—
240	C₄	—	2	4	no	—
241	C₄	—	2	2	no	—
242	C₄	—	4	2	no	—
243	C₄	—	4	2	no	—
244	C₄	—	4	2	no	—
245	C₄	—	4	2	no	—
246	C₄	C₃₁×#433	1	8	yes	—
247	C₄	C₃₁×#529	1	4	yes	—
248	C₂	—	4	2	no	—
249	C₂	C₃₁×#2283	1	1	yes	characteristic
250	C₂	C₂×C₃₁×#433	1	2	yes	—
251	C₂	C₃₁×#2922	1	2	yes	—
252	C₂	C₃₁×#2922	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
2. C₃₁×#1840	C₂	semidirect
3. C₃₁×#1832	C₂	semidirect
6. C₄×C₃₁×C₄⋊C₈	C₂	semidirect
8. C₃₁×#1744	C₂	semidirect
79. #20782	C₃₁	direct (N × H)
107. #1840	C₂×C₃₁	semidirect
108. #1832	C₂×C₃₁	semidirect
111. C₄×C₄⋊C₈	C₂×C₃₁	semidirect
113. #1744	C₂×C₃₁	semidirect
175. C₃₁	#20782	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₃₁×#117	2
C₃₁×#1840	2
C₃₁×#1832	2
C₃₁×#1818	2
C₃₁×#1798	2
C₄×C₃₁×C₄⋊C₈	2
C₄×C₃₁×C₂²⋊C₈	2
C₃₁×#1744	2
#20782	31

Subgroups — C31×#20782 #21477541

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#20782 #21477541