Subgroups — C₃₁×#8625 #21465384

Order 7936.

342 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#446	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1568	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#1568	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1467	C₂	1	2	yes	maximal
5	C₃₁×#1290	C₂	1	2	yes	maximal
6	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
7	C₃₁×#446	—	2	1	no	—
8	C₃₁×#446	—	2	1	no	—
9	C₃₁×#446	—	2	1	no	—
10	C₃₁×#446	—	2	1	no	—
11	C₃₁×#432	C₂²	1	2	yes	—
12	C₃₁×#432	C₂²	1	2	yes	—
13	C₂×C₃₁×C₄⋊C₈	C₄	1	2	yes	—
14	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×#97	C₄	1	2	yes	—
16	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
17	C₃₁×#335	—	2	1	no	—
18	C₃₁×#335	—	2	1	no	—
19	C₃₁×#334	—	2	1	no	—
20	C₃₁×#334	—	2	1	no	—
21	C₃₁×#324	—	2	4	no	—
22	C₂²×Q₈×C₃₁	Q₈	1	1	yes	characteristic
23	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
25	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
26	C₄×Q₈×C₃₁	—	2	1	no	—
27	C₄×Q₈×C₃₁	—	2	1	no	—
28	C₄×Q₈×C₃₁	—	2	1	no	—
29	C₄×Q₈×C₃₁	—	2	1	no	—
30	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
31	C₂×C₄⋊C₄×C₃₁	Q₈	1	1	yes	characteristic
32	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
34	C₃₁×C₄⋊C₈	—	2	2	no	—
35	C₃₁×C₄⋊C₈	—	2	2	no	—
36	C₃₁×C₄⋊C₈	—	2	1	no	—
37	C₃₁×C₄⋊C₈	—	2	1	no	—
38	C₃₁×C₄⋊C₈	—	2	2	no	—
39	C₃₁×C₄⋊C₈	—	2	2	no	—
40	C₃₁×C₂²⋊C₈	—	2	2	no	—
41	C₃₁×C₂²⋊C₈	—	2	2	no	—
42	C₃₁×#97	—	2	2	no	—
43	C₃₁×#97	—	2	2	no	—
44	C₃₁×#97	—	2	2	no	—
45	C₃₁×#97	—	2	2	no	—
46	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
47	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
48	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
49	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
50	C₂×Q₈×C₃₁	—	2	1	no	—
51	C₂×Q₈×C₃₁	—	2	1	no	—
52	C₂×Q₈×C₃₁	—	4	2	no	—
53	C₂²×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
54	C₂²×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
55	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
58	C₂×C₈×C₃₁	—	2	2	no	—
59	C₂×C₈×C₃₁	—	2	2	no	—
60	C₂×C₈×C₃₁	—	4	1	no	—
61	C₂×C₈×C₃₁	—	4	2	no	—
62	C₂×C₈×C₃₁	—	4	2	no	—
63	C₂×C₈×C₃₁	—	4	2	no	—
64	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
68	C₄⋊C₄×C₃₁	—	2	2	no	—
69	C₄⋊C₄×C₃₁	—	2	2	no	—
70	C₄⋊C₄×C₃₁	—	2	1	no	—
71	C₄⋊C₄×C₃₁	—	2	1	no	—
72	C₄²×C₃₁	—	2	2	no	—
73	C₄²×C₃₁	—	2	2	no	—
74	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
75	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
76	C₄²×C₃₁	#41	1	1	yes	characteristic
77	C₄²×C₃₁	#41	1	1	yes	characteristic
78	#8625	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
79	C₂³×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
80	Q₈×C₃₁	—	4	2	no	—
81	Q₈×C₃₁	—	4	2	no	—
82	C₂×C₄×C₃₁	—	2	2	no	—
83	C₂×C₄×C₃₁	—	2	1	no	—
84	C₂×C₄×C₃₁	—	2	1	no	—
85	C₂×C₄×C₃₁	—	2	2	no	—
86	C₂×C₄×C₃₁	—	2	2	no	—
87	C₂×C₄×C₃₁	—	2	4	no	—
88	C₂×C₄×C₃₁	—	4	1	no	—
89	C₂×C₄×C₃₁	—	4	2	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	1	yes	characteristic
93	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
94	C₂×C₄×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
95	C₂×C₄×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
96	C₈×C₃₁	—	2	1	no	—
97	C₈×C₃₁	—	2	1	no	—
98	C₈×C₃₁	—	2	1	no	—
99	C₈×C₃₁	—	4	2	no	—
100	C₈×C₃₁	—	8	2	no	—
101	C₈×C₃₁	—	8	2	no	—
102	C₈×C₃₁	#101	1	2	yes	—
103	C₂×#446	C₂×C₃₁	1	1	yes	characteristic
104	#1568	C₂×C₃₁	1	1	yes	characteristic
105	#1568	C₂×C₃₁	1	1	yes	characteristic
106	#1467	C₂×C₃₁	1	2	yes	—
107	#1290	C₂×C₃₁	1	2	yes	—
108	C₂²×C₃₁	—	2	1	no	—
109	C₂²×C₃₁	—	2	1	no	—
110	C₂²×C₃₁	#380	1	1	yes	characteristic
111	C₂²×C₃₁	#430	1	2	yes	—
112	C₄×C₃₁	—	2	1	no	—
113	C₄×C₃₁	—	2	1	no	—
114	C₄×C₃₁	—	2	1	no	—
115	C₄×C₃₁	—	2	1	no	—
116	C₄×C₃₁	—	2	1	no	—
117	C₄×C₃₁	—	4	2	no	—
118	C₄×C₃₁	—	4	2	no	—
119	C₄×C₃₁	C₂×#101	1	1	yes	characteristic
120	C₄×C₃₁	#413	1	1	yes	characteristic
121	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
122	#446	—	2	1	no	—
123	#446	—	2	1	no	—
124	#446	—	2	1	no	—
125	#446	—	2	1	no	—
126	#432	C₂²×C₃₁	1	2	yes	—
127	#432	C₂²×C₃₁	1	2	yes	—
128	C₂×C₄⋊C₈	C₄×C₃₁	1	2	yes	—
129	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
130	C₂×#97	C₄×C₃₁	1	2	yes	—
131	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
132	#335	—	2	1	no	—
133	#335	—	2	1	no	—
134	#334	—	2	1	no	—
135	#334	—	2	1	no	—
136	#324	—	2	4	no	—
137	C₂×C₃₁	—	2	2	no	—
138	C₂×C₃₁	#1801	1	1	yes	characteristic
139	C₂×C₃₁	#1811	1	1	yes	characteristic
140	C₂×C₃₁	#1812	1	1	yes	characteristic
141	C₂²×Q₈	Q₈×C₃₁	1	1	yes	characteristic
142	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
143	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
144	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
145	C₄×Q₈	—	2	1	no	—
146	C₄×Q₈	—	2	1	no	—
147	C₄×Q₈	—	2	1	no	—
148	C₄×Q₈	—	2	1	no	—
149	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
150	C₂×C₄⋊C₄	Q₈×C₃₁	1	1	yes	characteristic
151	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
152	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
153	C₄⋊C₈	—	2	2	no	—
154	C₄⋊C₈	—	2	2	no	—
155	C₄⋊C₈	—	2	1	no	—
156	C₄⋊C₈	—	2	1	no	—
157	C₄⋊C₈	—	2	2	no	—
158	C₄⋊C₈	—	2	2	no	—
159	C₂²⋊C₈	—	2	2	no	—
160	C₂²⋊C₈	—	2	2	no	—
161	#97	—	2	2	no	—
162	#97	—	2	2	no	—
163	#97	—	2	2	no	—
164	#97	—	2	2	no	—
165	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
166	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
167	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
168	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
169	C₃₁	#8625	1	1	yes	characteristic, direct product factor, Sylow
170	C₂×Q₈	—	2	1	no	—
171	C₂×Q₈	—	2	1	no	—
172	C₂×Q₈	—	4	2	no	—
173	C₂²×C₄	C₄⋊C₄×C₃₁	1	2	yes	—
174	C₂²×C₄	C₄⋊C₄×C₃₁	1	2	yes	—
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	—	4	1	no	—
181	C₂×C₈	—	4	2	no	—
182	C₂×C₈	—	4	2	no	—
183	C₂×C₈	—	4	2	no	—
184	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
185	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
186	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
187	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
188	C₄⋊C₄	—	2	2	no	—
189	C₄⋊C₄	—	2	2	no	—
190	C₄⋊C₄	—	2	1	no	—
191	C₄⋊C₄	—	2	1	no	—
192	C₄²	—	2	2	no	—
193	C₄²	—	2	2	no	—
194	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
195	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
196	C₄²	#41×C₃₁	1	1	yes	characteristic
197	C₄²	#41×C₃₁	1	1	yes	characteristic
198	C₂³	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
199	Q₈	—	4	2	no	—
200	Q₈	—	4	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	2	4	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	4	1	no	—
208	C₂×C₄	—	4	2	no	—
209	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
210	C₂×C₄	C₃₁×#117	1	1	yes	characteristic
211	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
212	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
213	C₂×C₄	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
214	C₂×C₄	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
215	C₈	—	2	1	no	—
216	C₈	—	2	1	no	—
217	C₈	—	2	1	no	—
218	C₈	—	4	2	no	—
219	C₈	—	8	2	no	—
220	C₈	—	8	2	no	—
221	C₈	C₃₁×#101	1	2	yes	—
222	C₂²	—	2	1	no	—
223	C₂²	—	2	1	no	—
224	C₂²	C₃₁×#380	1	1	yes	characteristic
225	C₂²	C₃₁×#430	1	2	yes	—
226	C₄	—	2	1	no	—
227	C₄	—	2	1	no	—
228	C₄	—	2	1	no	—
229	C₄	—	2	1	no	—
230	C₄	—	2	1	no	—
231	C₄	—	4	2	no	—
232	C₄	—	4	2	no	—
233	C₄	C₂×C₃₁×#101	1	1	yes	characteristic
234	C₄	C₃₁×#413	1	1	yes	characteristic
235	C₂	—	2	2	no	—
236	C₂	C₃₁×#1801	1	1	yes	characteristic
237	C₂	C₃₁×#1811	1	1	yes	characteristic
238	C₂	C₃₁×#1812	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
78. #8625	C₃₁	direct (N × H)
169. C₃₁	#8625	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×#446	2
C₃₁×#1568	2
C₃₁×#1467	2
C₃₁×#1290	2
#8625	31

Subgroups — C31×#8625 #21465384

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#8625 #21465384