Subgroups — C₃₁×#22303 #21479062

Order 7936.

634 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₃₁×#125	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1940	C₂	1	4	yes	maximal
3	C₃₁×#1939	C₂	1	2	yes	maximal
4	C₃₁×#1917	C₂	1	4	yes	maximal
5	C₃₁×#1842	C₂	1	1	yes	maximal, characteristic
6	C₃₁×#1770	C₂	1	2	yes	maximal
7	C₃₁×#1744	C₂	1	1	yes	maximal, characteristic
8	C₂×C₃₁×#125	C₄	1	1	yes	characteristic
9	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
10	C₂×C₃₁×C₄⋊C₈	C₂²	1	4	yes	—
11	C₂×C₃₁×C₄⋊C₈	C₂²	1	4	yes	—
12	C₂×C₃₁×#97	C₂²	1	4	yes	—
13	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
15	C₃₁×#389	C₄	1	2	yes	—
16	C₃₁×#387	C₄	1	4	yes	—
17	C₃₁×#382	C₄	1	1	yes	characteristic
18	C₄×C₄⋊C₄×C₃₁	C₂²	1	4	yes	—
19	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
20	C₄³×C₃₁	C₂²	1	1	yes	characteristic
21	C₃₁×#336	C₂²	1	4	yes	—
22	C₃₁×#336	C₂²	1	2	yes	—
23	C₃₁×#336	C₂²	1	4	yes	—
24	C₃₁×#336	C₂²	1	2	yes	—
25	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
26	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
27	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
28	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
29	C₃₁×#125	—	2	4	no	—
30	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
31	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
32	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
33	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
34	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
36	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
38	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
39	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
40	C₃₁×C₄⋊C₈	—	2	4	no	—
41	C₃₁×C₄⋊C₈	—	4	4	no	—
42	C₃₁×C₄⋊C₈	—	4	4	no	—
43	C₃₁×#97	—	4	4	no	—
44	C₄×C₈×C₃₁	—	2	2	no	—
45	C₄×C₈×C₃₁	Q₈	1	4	yes	—
46	C₃₁×#95	C₂×C₄	1	4	yes	—
47	C₃₁×#95	C₂×C₄	1	4	yes	—
48	C₃₁×#95	C₂×C₄	1	4	yes	—
49	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
50	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
51	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
52	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
54	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
56	C₂×C₈×C₃₁	—	2	2	no	—
57	C₂×C₈×C₃₁	—	2	2	no	—
58	C₂×C₈×C₃₁	—	2	1	no	—
59	C₂×C₈×C₃₁	—	2	1	no	—
60	C₂×C₈×C₃₁	—	2	4	no	—
61	C₂×C₈×C₃₁	—	2	4	no	—
62	C₂×C₈×C₃₁	—	2	2	no	—
63	C₂×C₈×C₃₁	—	2	4	no	—
64	C₂×C₈×C₃₁	—	2	2	no	—
65	C₂×C₈×C₃₁	C₂×Q₈	1	4	yes	—
66	C₄⋊C₄×C₃₁	—	2	8	no	—
67	C₄⋊C₄×C₃₁	—	2	4	no	—
68	C₄⋊C₄×C₃₁	—	2	4	no	—
69	C₄⋊C₄×C₃₁	—	4	4	no	—
70	C₄²×C₃₁	—	2	2	no	—
71	C₄²×C₃₁	—	2	4	no	—
72	C₄²×C₃₁	—	2	2	no	—
73	C₄²×C₃₁	—	2	2	no	—
74	C₄²×C₃₁	—	2	4	no	—
75	C₄²×C₃₁	—	4	2	no	—
76	C₄²×C₃₁	M₁₆	1	4	yes	—
77	C₄²×C₃₁	C₂×Q₈	1	2	yes	—
78	C₄²×C₃₁	#41	1	2	yes	—
79	#22303	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
80	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
81	C₂×C₄×C₃₁	—	2	2	no	—
82	C₂×C₄×C₃₁	—	2	2	no	—
83	C₂×C₄×C₃₁	—	2	2	no	—
84	C₂×C₄×C₃₁	—	2	4	no	—
85	C₂×C₄×C₃₁	—	2	4	no	—
86	C₂×C₄×C₃₁	—	2	2	no	—
87	C₂×C₄×C₃₁	—	2	4	no	—
88	C₂×C₄×C₃₁	—	2	2	no	—
89	C₂×C₄×C₃₁	—	2	4	no	—
90	C₂×C₄×C₃₁	—	2	2	no	—
91	C₂×C₄×C₃₁	—	2	4	no	—
92	C₂×C₄×C₃₁	—	2	4	no	—
93	C₂×C₄×C₃₁	C₄×Q₈	1	2	yes	—
94	C₂×C₄×C₃₁	C₄×Q₈	1	2	yes	—
95	C₂×C₄×C₃₁	C₂×M₁₆	1	4	yes	—
96	C₂×C₄×C₃₁	C₂×M₁₆	1	2	yes	—
97	C₂×C₄×C₃₁	#131	1	2	yes	—
98	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
99	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
100	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
101	C₂×C₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
102	C₈×C₃₁	—	2	4	no	—
103	C₈×C₃₁	—	4	1	no	—
104	C₈×C₃₁	—	4	4	no	—
105	C₈×C₃₁	—	4	1	no	—
106	C₄×#125	C₂×C₃₁	1	1	yes	characteristic
107	#1940	C₂×C₃₁	1	4	yes	—
108	#1939	C₂×C₃₁	1	2	yes	—
109	#1917	C₂×C₃₁	1	4	yes	—
110	#1842	C₂×C₃₁	1	1	yes	characteristic
111	#1770	C₂×C₃₁	1	2	yes	—
112	#1744	C₂×C₃₁	1	1	yes	characteristic
113	C₂²×C₃₁	C₂×C₄×Q₈	1	1	yes	characteristic
114	C₂²×C₃₁	#520	1	2	yes	—
115	C₂²×C₃₁	C₂²×M₁₆	1	1	yes	characteristic
116	C₂²×C₃₁	C₂×#131	1	1	yes	characteristic
117	C₂²×C₃₁	#568	1	2	yes	—
118	C₄×C₃₁	—	2	2	no	—
119	C₄×C₃₁	—	2	4	no	—
120	C₄×C₃₁	—	4	4	no	—
121	C₄×C₃₁	—	4	2	no	—
122	C₄×C₃₁	—	4	4	no	—
123	C₄×C₃₁	#446	1	4	yes	—
124	C₄×C₃₁	#540	1	1	yes	characteristic
125	C₄×C₃₁	#541	1	1	yes	characteristic
126	C₄×C₃₁	#544	1	2	yes	—
127	C₂×#125	C₄×C₃₁	1	1	yes	characteristic
128	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
129	C₂×C₄⋊C₈	C₂²×C₃₁	1	4	yes	—
130	C₂×C₄⋊C₈	C₂²×C₃₁	1	4	yes	—
131	C₂×#97	C₂²×C₃₁	1	4	yes	—
132	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
133	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
134	#389	C₄×C₃₁	1	2	yes	—
135	#387	C₄×C₃₁	1	4	yes	—
136	#382	C₄×C₃₁	1	1	yes	characteristic
137	C₄×C₄⋊C₄	C₂²×C₃₁	1	4	yes	—
138	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
139	C₄³	C₂²×C₃₁	1	1	yes	characteristic
140	#336	C₂²×C₃₁	1	4	yes	—
141	#336	C₂²×C₃₁	1	2	yes	—
142	#336	C₂²×C₃₁	1	4	yes	—
143	#336	C₂²×C₃₁	1	2	yes	—
144	C₂×C₃₁	#2330	1	1	yes	characteristic
145	C₂×C₃₁	C₂×#446	1	1	yes	characteristic
146	C₂×C₃₁	#2961	1	1	yes	characteristic
147	C₂×C₃₁	#2989	1	2	yes	—
148	C₂×C₃₁	#2993	1	2	yes	—
149	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
150	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
151	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
152	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
153	#125	—	2	4	no	—
154	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	4	yes	—
155	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
156	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
157	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	4	yes	—
158	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
159	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
160	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
161	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
162	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
163	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
164	C₄⋊C₈	—	2	4	no	—
165	C₄⋊C₈	—	4	4	no	—
166	C₄⋊C₈	—	4	4	no	—
167	#97	—	4	4	no	—
168	C₄×C₈	—	2	2	no	—
169	C₄×C₈	Q₈×C₃₁	1	4	yes	—
170	#95	C₂×C₄×C₃₁	1	4	yes	—
171	#95	C₂×C₄×C₃₁	1	4	yes	—
172	#95	C₂×C₄×C₃₁	1	4	yes	—
173	C₃₁	#22303	1	1	yes	characteristic, direct product factor, Sylow
174	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
175	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
176	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
177	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
178	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
179	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
180	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
181	C₂×C₈	—	2	2	no	—
182	C₂×C₈	—	2	2	no	—
183	C₂×C₈	—	2	1	no	—
184	C₂×C₈	—	2	1	no	—
185	C₂×C₈	—	2	4	no	—
186	C₂×C₈	—	2	4	no	—
187	C₂×C₈	—	2	2	no	—
188	C₂×C₈	—	2	4	no	—
189	C₂×C₈	—	2	2	no	—
190	C₂×C₈	C₂×Q₈×C₃₁	1	4	yes	—
191	C₄⋊C₄	—	2	8	no	—
192	C₄⋊C₄	—	2	4	no	—
193	C₄⋊C₄	—	2	4	no	—
194	C₄⋊C₄	—	4	4	no	—
195	C₄²	—	2	2	no	—
196	C₄²	—	2	4	no	—
197	C₄²	—	2	2	no	—
198	C₄²	—	2	2	no	—
199	C₄²	—	2	4	no	—
200	C₄²	—	4	2	no	—
201	C₄²	M₁₆×C₃₁	1	4	yes	—
202	C₄²	C₂×Q₈×C₃₁	1	2	yes	—
203	C₄²	#41×C₃₁	1	2	yes	—
204	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	4	no	—
209	C₂×C₄	—	2	4	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	4	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	4	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	4	no	—
216	C₂×C₄	—	2	4	no	—
217	C₂×C₄	C₄×Q₈×C₃₁	1	2	yes	—
218	C₂×C₄	C₄×Q₈×C₃₁	1	2	yes	—
219	C₂×C₄	C₂×M₁₆×C₃₁	1	4	yes	—
220	C₂×C₄	C₂×M₁₆×C₃₁	1	2	yes	—
221	C₂×C₄	C₃₁×#131	1	2	yes	—
222	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
223	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
224	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
225	C₂×C₄	C₃₁×2₋¹⁺⁴	1	2	yes	—
226	C₈	—	2	4	no	—
227	C₈	—	4	1	no	—
228	C₈	—	4	4	no	—
229	C₈	—	4	1	no	—
230	C₂²	C₂×C₄×Q₈×C₃₁	1	1	yes	characteristic
231	C₂²	C₃₁×#520	1	2	yes	—
232	C₂²	C₂²×M₁₆×C₃₁	1	1	yes	characteristic
233	C₂²	C₂×C₃₁×#131	1	1	yes	characteristic
234	C₂²	C₃₁×#568	1	2	yes	—
235	C₄	—	2	2	no	—
236	C₄	—	2	4	no	—
237	C₄	—	4	4	no	—
238	C₄	—	4	2	no	—
239	C₄	—	4	4	no	—
240	C₄	C₃₁×#446	1	4	yes	—
241	C₄	C₃₁×#540	1	1	yes	characteristic
242	C₄	C₃₁×#541	1	1	yes	characteristic
243	C₄	C₃₁×#544	1	2	yes	—
244	C₂	C₃₁×#2330	1	1	yes	characteristic
245	C₂	C₂×C₃₁×#446	1	1	yes	characteristic
246	C₂	C₃₁×#2961	1	1	yes	characteristic
247	C₂	C₃₁×#2989	1	2	yes	—
248	C₂	C₃₁×#2993	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
79. #22303	C₃₁	direct (N × H)
173. C₃₁	#22303	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×#125	2
C₃₁×#1940	2
C₃₁×#1939	2
C₃₁×#1917	2
C₃₁×#1842	2
C₃₁×#1770	2
C₃₁×#1744	2
#22303	31

Subgroups — C31×#22303 #21479062

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22303 #21479062