Subgroups — C₃₁×#8603 #21465362

Order 7936.

368 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#433	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1564	C₂	1	2	yes	maximal, complemented
3	C₃₁×#1458	C₂	1	2	yes	maximal
4	C₃₁×#1275	C₂	1	2	yes	maximal, complemented
5	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
6	C₃₁×#433	—	2	2	no	—
7	C₃₁×#433	—	2	2	no	—
8	C₃₁×#431	C₂²	1	4	yes	—
9	C₂×C₃₁×C₄⋊C₈	C₄	1	1	yes	characteristic
10	C₂×C₃₁×C₄⋊C₈	C₄	1	1	yes	characteristic
11	C₂×C₃₁×C₂²⋊C₈	C₄	1	1	yes	characteristic
12	C₂×C₃₁×C₂²⋊C₈	C₄	1	1	yes	characteristic
13	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
15	C₃₁×#335	—	2	2	no	—
16	C₃₁×#334	—	2	2	no	—
17	C₃₁×#324	—	2	2	no	—
18	C₃₁×#323	—	2	2	no	—
19	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
24	C₃₁×#117	—	2	2	no	—
25	C₃₁×#117	—	2	2	no	—
26	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
27	C₂×C₄⋊C₄×C₃₁	Q₈	1	1	yes	characteristic
28	C₂×C₂²⋊C₄×C₃₁	D₄	1	1	yes	characteristic
29	C₂×C₂²⋊C₄×C₃₁	Q₈	1	1	yes	characteristic
30	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
32	C₃₁×C₄⋊C₈	—	2	1	no	—
33	C₃₁×C₄⋊C₈	—	2	2	no	—
34	C₃₁×C₄⋊C₈	—	2	1	no	—
35	C₃₁×C₄⋊C₈	—	2	4	no	—
36	C₃₁×C₂²⋊C₈	—	2	1	no	—
37	C₃₁×C₂²⋊C₈	—	2	2	no	—
38	C₃₁×C₂²⋊C₈	—	2	1	no	—
39	C₃₁×C₂²⋊C₈	—	2	4	no	—
40	C₃₁×#97	—	2	2	no	—
41	C₄×C₈×C₃₁	—	2	4	no	—
42	C₄×C₈×C₃₁	D₄	1	2	yes	—
43	C₄×C₈×C₃₁	D₄	1	2	yes	—
44	C₂⁴×C₃₁	C₄⋊C₄	1	1	yes	characteristic
45	C₂²×C₄×C₃₁	—	2	1	no	—
46	C₂²×C₄×C₃₁	—	2	1	no	—
47	C₂²×C₄×C₃₁	—	2	2	no	—
48	C₂²×C₄×C₃₁	—	2	2	no	—
49	C₂²×C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
55	C₂×C₈×C₃₁	—	2	2	no	—
56	C₂×C₈×C₃₁	—	2	2	no	—
57	C₂×C₈×C₃₁	—	2	1	no	—
58	C₂×C₈×C₃₁	—	2	1	no	—
59	C₂×C₈×C₃₁	—	2	2	no	—
60	C₂×C₈×C₃₁	—	2	2	no	—
61	C₂×C₈×C₃₁	—	4	4	no	—
62	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
64	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
65	C₄⋊C₄×C₃₁	—	2	2	no	—
66	C₄⋊C₄×C₃₁	—	2	2	no	—
67	C₂²⋊C₄×C₃₁	—	2	2	no	—
68	C₂²⋊C₄×C₃₁	—	2	2	no	—
69	C₄²×C₃₁	—	2	1	no	—
70	C₄²×C₃₁	—	2	1	no	—
71	C₄²×C₃₁	C₂×D₄	1	2	yes	—
72	C₄²×C₃₁	#41	1	2	yes	—
73	#8603	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
74	C₂³×C₃₁	—	2	1	no	—
75	C₂³×C₃₁	—	2	2	no	—
76	C₂³×C₃₁	—	4	1	no	—
77	C₂³×C₃₁	—	4	1	no	—
78	C₂³×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₃₁	—	2	1	no	—
80	C₂×C₄×C₃₁	—	2	1	no	—
81	C₂×C₄×C₃₁	—	2	1	no	—
82	C₂×C₄×C₃₁	—	2	2	no	—
83	C₂×C₄×C₃₁	—	4	1	no	—
84	C₂×C₄×C₃₁	—	4	1	no	—
85	C₂×C₄×C₃₁	—	4	1	no	—
86	C₂×C₄×C₃₁	—	4	1	no	—
87	C₂×C₄×C₃₁	—	4	2	no	—
88	C₂×C₄×C₃₁	—	4	1	no	—
89	C₂×C₄×C₃₁	—	4	1	no	—
90	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₃₁	#117	1	1	yes	characteristic
92	C₂×C₄×C₃₁	#121	1	2	yes	—
93	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
94	C₂×C₄×C₃₁	#131	1	2	yes	—
95	C₂×C₄×C₃₁	#131	1	2	yes	—
96	C₈×C₃₁	—	2	1	no	—
97	C₈×C₃₁	—	2	2	no	—
98	C₈×C₃₁	—	2	1	no	—
99	C₈×C₃₁	—	4	1	no	—
100	C₈×C₃₁	—	4	1	no	—
101	C₈×C₃₁	—	8	4	no	—
102	C₂×#433	C₂×C₃₁	1	1	yes	characteristic
103	#1564	C₂×C₃₁	1	2	yes	complemented
104	#1458	C₂×C₃₁	1	2	yes	—
105	#1275	C₂×C₃₁	1	2	yes	complemented
106	C₂²×C₃₁	—	2	1	no	—
107	C₂²×C₃₁	—	2	1	no	—
108	C₂²×C₃₁	—	4	1	no	—
109	C₂²×C₃₁	—	4	1	no	—
110	C₂²×C₃₁	—	4	1	no	—
111	C₂²×C₃₁	—	8	2	no	—
112	C₂²×C₃₁	#380	1	1	yes	characteristic
113	C₂²×C₃₁	#424	1	2	yes	—
114	C₄×C₃₁	—	2	2	no	—
115	C₄×C₃₁	—	2	2	no	—
116	C₄×C₃₁	—	4	1	no	—
117	C₄×C₃₁	—	4	1	no	—
118	C₄×C₃₁	—	4	1	no	—
119	C₄×C₃₁	#408	1	1	yes	characteristic
120	C₄×C₃₁	#408	1	1	yes	characteristic
121	C₄×C₃₁	#433	1	2	yes	—
122	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
123	#433	—	2	2	no	—
124	#433	—	2	2	no	—
125	#431	C₂²×C₃₁	1	4	yes	—
126	C₂×C₄⋊C₈	C₄×C₃₁	1	1	yes	characteristic
127	C₂×C₄⋊C₈	C₄×C₃₁	1	1	yes	characteristic
128	C₂×C₂²⋊C₈	C₄×C₃₁	1	1	yes	characteristic
129	C₂×C₂²⋊C₈	C₄×C₃₁	1	1	yes	characteristic
130	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
131	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
132	#335	—	2	2	no	—
133	#334	—	2	2	no	—
134	#324	—	2	2	no	—
135	#323	—	2	2	no	—
136	C₂×C₃₁	—	2	2	no	—
137	C₂×C₃₁	—	8	1	no	—
138	C₂×C₃₁	#1798	1	1	yes	characteristic
139	C₂×C₃₁	#1812	1	1	yes	characteristic
140	C₂×C₃₁	#1813	1	1	yes	characteristic
141	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
142	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
143	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
144	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
145	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
146	#117	—	2	2	no	—
147	#117	—	2	2	no	—
148	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
149	C₂×C₄⋊C₄	Q₈×C₃₁	1	1	yes	characteristic
150	C₂×C₂²⋊C₄	D₄×C₃₁	1	1	yes	characteristic
151	C₂×C₂²⋊C₄	Q₈×C₃₁	1	1	yes	characteristic
152	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
153	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
154	C₄⋊C₈	—	2	1	no	—
155	C₄⋊C₈	—	2	2	no	—
156	C₄⋊C₈	—	2	1	no	—
157	C₄⋊C₈	—	2	4	no	—
158	C₂²⋊C₈	—	2	1	no	—
159	C₂²⋊C₈	—	2	2	no	—
160	C₂²⋊C₈	—	2	1	no	—
161	C₂²⋊C₈	—	2	4	no	—
162	#97	—	2	2	no	—
163	C₄×C₈	—	2	4	no	—
164	C₄×C₈	D₄×C₃₁	1	2	yes	—
165	C₄×C₈	D₄×C₃₁	1	2	yes	—
166	C₃₁	#8603	1	1	yes	characteristic, direct product factor, Sylow
167	C₂⁴	C₄⋊C₄×C₃₁	1	1	yes	characteristic
168	C₂²×C₄	—	2	1	no	—
169	C₂²×C₄	—	2	1	no	—
170	C₂²×C₄	—	2	2	no	—
171	C₂²×C₄	—	2	2	no	—
172	C₂²×C₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
173	C₂²×C₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
174	C₂²×C₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
175	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
176	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
177	C₂²×C₄	C₂×Q₈×C₃₁	1	1	yes	characteristic
178	C₂×C₈	—	2	2	no	—
179	C₂×C₈	—	2	2	no	—
180	C₂×C₈	—	2	1	no	—
181	C₂×C₈	—	2	1	no	—
182	C₂×C₈	—	2	2	no	—
183	C₂×C₈	—	2	2	no	—
184	C₂×C₈	—	4	4	no	—
185	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
186	C₂×C₈	C₂²⋊C₄×C₃₁	1	2	yes	—
187	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
188	C₄⋊C₄	—	2	2	no	—
189	C₄⋊C₄	—	2	2	no	—
190	C₂²⋊C₄	—	2	2	no	—
191	C₂²⋊C₄	—	2	2	no	—
192	C₄²	—	2	1	no	—
193	C₄²	—	2	1	no	—
194	C₄²	C₂×D₄×C₃₁	1	2	yes	—
195	C₄²	#41×C₃₁	1	2	yes	—
196	C₂³	—	2	1	no	—
197	C₂³	—	2	2	no	—
198	C₂³	—	4	1	no	—
199	C₂³	—	4	1	no	—
200	C₂³	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	4	1	no	—
206	C₂×C₄	—	4	1	no	—
207	C₂×C₄	—	4	1	no	—
208	C₂×C₄	—	4	1	no	—
209	C₂×C₄	—	4	2	no	—
210	C₂×C₄	—	4	1	no	—
211	C₂×C₄	—	4	1	no	—
212	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
213	C₂×C₄	C₃₁×#117	1	1	yes	characteristic
214	C₂×C₄	C₃₁×#121	1	2	yes	—
215	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
216	C₂×C₄	C₃₁×#131	1	2	yes	—
217	C₂×C₄	C₃₁×#131	1	2	yes	—
218	C₈	—	2	1	no	—
219	C₈	—	2	2	no	—
220	C₈	—	2	1	no	—
221	C₈	—	4	1	no	—
222	C₈	—	4	1	no	—
223	C₈	—	8	4	no	—
224	C₂²	—	2	1	no	—
225	C₂²	—	2	1	no	—
226	C₂²	—	4	1	no	—
227	C₂²	—	4	1	no	—
228	C₂²	—	4	1	no	—
229	C₂²	—	8	2	no	—
230	C₂²	C₃₁×#380	1	1	yes	characteristic
231	C₂²	C₃₁×#424	1	2	yes	—
232	C₄	—	2	2	no	—
233	C₄	—	2	2	no	—
234	C₄	—	4	1	no	—
235	C₄	—	4	1	no	—
236	C₄	—	4	1	no	—
237	C₄	C₃₁×#408	1	1	yes	characteristic
238	C₄	C₃₁×#408	1	1	yes	characteristic
239	C₄	C₃₁×#433	1	2	yes	—
240	C₂	—	2	2	no	—
241	C₂	—	8	1	no	—
242	C₂	C₃₁×#1798	1	1	yes	characteristic
243	C₂	C₃₁×#1812	1	1	yes	characteristic
244	C₂	C₃₁×#1813	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₁	#117
commutator	C₂²×C₄	C₂²×C₄×C₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
socle	C₂²×C₃₁	#380
prime radical	C₂⁴×C₃₁	C₄⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂×C₃₁×#97 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#1564	C₂	semidirect
4. C₃₁×#1275	C₂	semidirect
73. #8603	C₃₁	direct (N × H)
103. #1564	C₂×C₃₁	semidirect
105. #1275	C₂×C₃₁	semidirect
166. C₃₁	#8603	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#433	2
C₃₁×#1564	2
C₃₁×#1458	2
C₃₁×#1275	2
#8603	31

Subgroups — C31×#8603 #21465362

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#8603 #21465362