Subgroups — C₃₁×#8661 #21465420

Order 7936.

378 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×C₈×Q₈×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1837	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#1564	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1487	C₂	1	2	yes	maximal
5	C₃₁×#1301	C₂	1	2	yes	maximal
6	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
7	C₈×Q₈×C₃₁	—	2	1	no	—
8	C₈×Q₈×C₃₁	—	2	1	no	—
9	C₈×Q₈×C₃₁	C₄	1	2	yes	—
10	C₈×Q₈×C₃₁	C₄	1	2	yes	—
11	C₃₁×#431	C₂²	1	2	yes	—
12	C₃₁×#423	C₂²	1	2	yes	—
13	C₂×C₃₁×C₄⋊C₈	—	2	1	no	—
14	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₃₁	—	2	1	no	—
16	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
17	C₃₁×#341	—	2	1	no	—
18	C₃₁×#341	—	2	1	no	—
19	C₃₁×#335	—	2	1	no	—
20	C₃₁×#334	—	2	1	no	—
21	C₃₁×Q₈⋊C₈	—	2	4	no	—
22	C₂²×Q₈×C₃₁	Q₈	1	1	yes	characteristic
23	C₂×M₁₆×C₃₁	—	2	2	no	—
24	C₂²×C₈×C₃₁	—	2	1	no	—
25	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
27	C₄×Q₈×C₃₁	—	2	1	no	—
28	C₄×Q₈×C₃₁	—	2	1	no	—
29	C₄×Q₈×C₃₁	C₂×C₄	1	2	yes	—
30	C₄×Q₈×C₃₁	D₄	1	2	yes	—
31	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
32	C₂×C₄⋊C₄×C₃₁	Q₈	1	1	yes	characteristic
33	C₂×C₄²×C₃₁	—	2	1	no	—
34	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
35	C₃₁×C₄⋊C₈	—	2	2	no	—
36	C₃₁×C₄⋊C₈	—	2	2	no	—
37	C₃₁×C₄⋊C₈	—	2	1	no	—
38	C₃₁×C₄⋊C₈	—	2	2	no	—
39	C₃₁×C₄⋊C₈	—	2	2	no	—
40	C₃₁×C₄⋊C₈	—	2	2	no	—
41	C₃₁×C₄⋊C₈	C₂×C₄	1	2	yes	—
42	C₃₁×C₂²⋊C₈	—	2	2	no	—
43	C₄×C₈×C₃₁	—	2	2	no	—
44	C₄×C₈×C₃₁	—	2	2	no	—
45	C₄×C₈×C₃₁	—	2	2	no	—
46	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
47	C₄×C₈×C₃₁	C₂×C₄	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	4	no	—
51	C₂×Q₈×C₃₁	—	2	1	no	—
52	C₂×Q₈×C₃₁	C₄⋊C₄	1	2	yes	—
53	C₂²×C₄×C₃₁	—	2	1	no	—
54	C₂²×C₄×C₃₁	—	2	1	no	—
55	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
57	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
58	M₁₆×C₃₁	—	4	2	no	—
59	M₁₆×C₃₁	—	4	2	no	—
60	C₂×C₈×C₃₁	—	2	2	no	—
61	C₂×C₈×C₃₁	—	2	2	no	—
62	C₂×C₈×C₃₁	—	2	2	no	—
63	C₂×C₈×C₃₁	—	4	2	no	—
64	C₂×C₈×C₃₁	—	4	2	no	—
65	C₂×C₈×C₃₁	—	4	2	no	—
66	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
70	C₄⋊C₄×C₃₁	—	2	2	no	—
71	C₄⋊C₄×C₃₁	—	2	2	no	—
72	C₄⋊C₄×C₃₁	—	2	1	no	—
73	C₄⋊C₄×C₃₁	C₄⋊C₄	1	2	yes	—
74	C₄²×C₃₁	—	2	2	no	—
75	C₄²×C₃₁	—	2	2	no	—
76	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
77	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
78	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
79	C₄²×C₃₁	#41	1	1	yes	characteristic
80	#8661	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
81	C₂³×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
82	Q₈×C₃₁	—	2	4	no	—
83	Q₈×C₃₁	—	2	2	no	—
84	Q₈×C₃₁	#108	1	4	yes	—
85	C₂×C₄×C₃₁	—	2	4	no	—
86	C₂×C₄×C₃₁	—	2	2	no	—
87	C₂×C₄×C₃₁	—	2	2	no	—
88	C₂×C₄×C₃₁	—	2	2	no	—
89	C₂×C₄×C₃₁	—	4	1	no	—
90	C₂×C₄×C₃₁	—	4	2	no	—
91	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
93	C₂×C₄×C₃₁	#117	1	1	yes	characteristic
94	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
95	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
96	C₂×C₄×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
97	C₂×C₄×C₃₁	#135	1	2	yes	—
98	C₂×C₄×C₃₁	#137	1	2	yes	—
99	C₈×C₃₁	—	2	1	no	—
100	C₈×C₃₁	—	2	1	no	—
101	C₈×C₃₁	—	2	1	no	—
102	C₈×C₃₁	—	2	1	no	—
103	C₈×C₃₁	—	4	2	no	—
104	C₈×C₃₁	—	8	2	no	—
105	C₈×C₃₁	—	8	2	no	—
106	C₂×C₈×Q₈	C₂×C₃₁	1	1	yes	characteristic
107	#1837	C₂×C₃₁	1	1	yes	characteristic
108	#1564	C₂×C₃₁	1	1	yes	characteristic
109	#1487	C₂×C₃₁	1	2	yes	—
110	#1301	C₂×C₃₁	1	2	yes	—
111	C₂²×C₃₁	—	2	1	no	—
112	C₂²×C₃₁	—	2	1	no	—
113	C₂²×C₃₁	#380	1	1	yes	characteristic
114	C₂²×C₃₁	#479	1	2	yes	—
115	C₄×C₃₁	—	2	4	no	—
116	C₄×C₃₁	—	2	1	no	—
117	C₄×C₃₁	—	2	1	no	—
118	C₄×C₃₁	—	2	1	no	—
119	C₄×C₃₁	—	4	2	no	—
120	C₄×C₃₁	#416	1	1	yes	characteristic
121	C₄×C₃₁	#419	1	1	yes	characteristic
122	C₄×C₃₁	C₂×#108	1	2	yes	—
123	C₄×C₃₁	#464	1	2	yes	—
124	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
125	C₈×Q₈	—	2	1	no	—
126	C₈×Q₈	—	2	1	no	—
127	C₈×Q₈	C₄×C₃₁	1	2	yes	—
128	C₈×Q₈	C₄×C₃₁	1	2	yes	—
129	#431	C₂²×C₃₁	1	2	yes	—
130	#423	C₂²×C₃₁	1	2	yes	—
131	C₂×C₄⋊C₈	—	2	1	no	—
132	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
133	C₂×C₄×C₈	—	2	1	no	—
134	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
135	#341	—	2	1	no	—
136	#341	—	2	1	no	—
137	#335	—	2	1	no	—
138	#334	—	2	1	no	—
139	Q₈⋊C₈	—	2	4	no	—
140	C₂×C₃₁	—	2	2	no	—
141	C₂×C₃₁	#1805	1	1	yes	characteristic
142	C₂×C₃₁	#1810	1	1	yes	characteristic
143	C₂×C₃₁	#1814	1	1	yes	characteristic
144	C₂²×Q₈	Q₈×C₃₁	1	1	yes	characteristic
145	C₂×M₁₆	—	2	2	no	—
146	C₂²×C₈	—	2	1	no	—
147	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
148	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
149	C₄×Q₈	—	2	1	no	—
150	C₄×Q₈	—	2	1	no	—
151	C₄×Q₈	C₂×C₄×C₃₁	1	2	yes	—
152	C₄×Q₈	D₄×C₃₁	1	2	yes	—
153	C₂×C₄⋊C₄	—	2	1	no	—
154	C₂×C₄⋊C₄	Q₈×C₃₁	1	1	yes	characteristic
155	C₂×C₄²	—	2	1	no	—
156	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
157	C₄⋊C₈	—	2	2	no	—
158	C₄⋊C₈	—	2	2	no	—
159	C₄⋊C₈	—	2	1	no	—
160	C₄⋊C₈	—	2	2	no	—
161	C₄⋊C₈	—	2	2	no	—
162	C₄⋊C₈	—	2	2	no	—
163	C₄⋊C₈	C₂×C₄×C₃₁	1	2	yes	—
164	C₂²⋊C₈	—	2	2	no	—
165	C₄×C₈	—	2	2	no	—
166	C₄×C₈	—	2	2	no	—
167	C₄×C₈	—	2	2	no	—
168	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
169	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
170	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
171	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
172	C₃₁	#8661	1	1	yes	characteristic, direct product factor, Sylow
173	C₂×Q₈	—	2	4	no	—
174	C₂×Q₈	—	2	1	no	—
175	C₂×Q₈	C₄⋊C₄×C₃₁	1	2	yes	—
176	C₂²×C₄	—	2	1	no	—
177	C₂²×C₄	—	2	1	no	—
178	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
179	C₂²×C₄	C₂×Q₈×C₃₁	1	1	yes	characteristic
180	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
181	M₁₆	—	4	2	no	—
182	M₁₆	—	4	2	no	—
183	C₂×C₈	—	2	2	no	—
184	C₂×C₈	—	2	2	no	—
185	C₂×C₈	—	2	2	no	—
186	C₂×C₈	—	4	2	no	—
187	C₂×C₈	—	4	2	no	—
188	C₂×C₈	—	4	2	no	—
189	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
190	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
191	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
192	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
193	C₄⋊C₄	—	2	2	no	—
194	C₄⋊C₄	—	2	2	no	—
195	C₄⋊C₄	—	2	1	no	—
196	C₄⋊C₄	C₄⋊C₄×C₃₁	1	2	yes	—
197	C₄²	—	2	2	no	—
198	C₄²	—	2	2	no	—
199	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
200	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
201	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
202	C₄²	#41×C₃₁	1	1	yes	characteristic
203	C₂³	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
204	Q₈	—	2	4	no	—
205	Q₈	—	2	2	no	—
206	Q₈	C₃₁×#108	1	4	yes	—
207	C₂×C₄	—	2	4	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	4	1	no	—
212	C₂×C₄	—	4	2	no	—
213	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
214	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
215	C₂×C₄	C₃₁×#117	1	1	yes	characteristic
216	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
217	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
218	C₂×C₄	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
219	C₂×C₄	C₃₁×#135	1	2	yes	—
220	C₂×C₄	C₃₁×#137	1	2	yes	—
221	C₈	—	2	1	no	—
222	C₈	—	2	1	no	—
223	C₈	—	2	1	no	—
224	C₈	—	2	1	no	—
225	C₈	—	4	2	no	—
226	C₈	—	8	2	no	—
227	C₈	—	8	2	no	—
228	C₂²	—	2	1	no	—
229	C₂²	—	2	1	no	—
230	C₂²	C₃₁×#380	1	1	yes	characteristic
231	C₂²	C₃₁×#479	1	2	yes	—
232	C₄	—	2	4	no	—
233	C₄	—	2	1	no	—
234	C₄	—	2	1	no	—
235	C₄	—	2	1	no	—
236	C₄	—	4	2	no	—
237	C₄	C₃₁×#416	1	1	yes	characteristic
238	C₄	C₃₁×#419	1	1	yes	characteristic
239	C₄	C₂×C₃₁×#108	1	2	yes	—
240	C₄	C₃₁×#464	1	2	yes	—
241	C₂	—	2	2	no	—
242	C₂	C₃₁×#1805	1	1	yes	characteristic
243	C₂	C₃₁×#1810	1	1	yes	characteristic
244	C₂	C₃₁×#1814	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₁	#121
commutator	C₄²	C₂²×C₄×C₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
socle	C₂²×C₃₁	#380
prime radical	C₂³×C₃₁	C₂²⋊Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
80. #8661	C₃₁	direct (N × H)
172. C₃₁	#8661	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×C₈×Q₈×C₃₁	2
C₃₁×#1837	2
C₃₁×#1564	2
C₃₁×#1487	2
C₃₁×#1301	2
#8661	31

Subgroups — C31×#8661 #21465420

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#8661 #21465420