Subgroups — C₃₁×#8687 #21465446

Order 7936.

346 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2960	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1837	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₁×#334	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1550	C₂	1	2	yes	maximal
5	C₃₁×#1292	C₂	1	2	yes	maximal
6	C₃₁×#533	C₂²	1	1	yes	characteristic
7	C₃₁×#446	—	2	1	no	—
8	C₃₁×#446	C₄	1	2	yes	—
9	C₈×Q₈×C₃₁	—	2	1	no	—
10	C₈×Q₈×C₃₁	C₄	1	2	yes	—
11	C₃₁×#424	—	2	1	no	—
12	C₃₁×#423	C₂²	1	2	yes	—
13	C₃₁×#423	C₂²	1	1	yes	characteristic
14	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
15	C₃₁×#405	—	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×#334	—	2	1	no	—
20	C₃₁×#334	—	2	1	no	—
21	C₃₁×#333	—	2	2	no	—
22	C₃₁×#332	—	2	2	no	—
23	C₂×M₁₆×C₃₁	—	2	1	no	—
24	C₂×M₁₆×C₃₁	—	2	2	no	—
25	C₂²×C₈×C₃₁	—	2	2	no	—
26	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
28	C₃₁×C₄⋊Q₈	D₄	1	1	yes	characteristic
29	C₃₁×C₄⋊Q₈	D₄	1	1	yes	characteristic
30	C₃₁×#125	—	2	1	no	—
31	C₃₁×C₂²⋊Q₈	—	2	1	no	—
32	C₃₁×C₂²⋊Q₈	Q₈	1	2	yes	—
33	C₄×Q₈×C₃₁	—	2	1	no	—
34	C₄×Q₈×C₃₁	C₂×C₄	1	2	yes	—
35	C₃₁×#117	—	2	1	no	—
36	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
37	C₃₁×C₄⋊C₈	—	2	2	no	—
38	C₃₁×C₄⋊C₈	—	2	2	no	—
39	C₃₁×C₄⋊C₈	—	2	1	no	—
40	C₃₁×C₄⋊C₈	—	2	2	no	—
41	C₃₁×C₄⋊C₈	—	2	2	no	—
42	C₃₁×C₄⋊C₈	—	2	2	no	—
43	C₃₁×C₄⋊C₈	—	2	2	no	—
44	C₃₁×C₄⋊C₈	C₂×C₄	1	2	yes	—
45	C₃₁×#97	—	2	2	no	—
46	C₄×C₈×C₃₁	—	2	2	no	—
47	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
48	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
49	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
50	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
51	C₂×Q₈×C₃₁	—	2	1	no	—
52	C₂×Q₈×C₃₁	C₄⋊C₄	1	2	yes	—
53	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
55	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
56	M₁₆×C₃₁	—	4	2	no	—
57	M₁₆×C₃₁	—	4	2	no	—
58	M₁₆×C₃₁	—	4	2	no	—
59	C₂×C₈×C₃₁	—	2	2	no	—
60	C₂×C₈×C₃₁	—	2	2	no	—
61	C₂×C₈×C₃₁	—	2	2	no	—
62	C₂×C₈×C₃₁	—	4	2	no	—
63	C₂×C₈×C₃₁	—	4	2	no	—
64	C₂×C₈×C₃₁	—	8	2	no	—
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	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
69	C₄⋊C₄×C₃₁	—	2	2	no	—
70	C₄⋊C₄×C₃₁	—	2	1	no	—
71	C₄⋊C₄×C₃₁	—	2	1	no	—
72	C₄⋊C₄×C₃₁	—	2	1	no	—
73	C₄⋊C₄×C₃₁	—	2	2	no	—
74	C₄⋊C₄×C₃₁	C₄⋊C₄	1	2	yes	—
75	C₂²⋊C₄×C₃₁	—	2	2	no	—
76	C₄²×C₃₁	—	2	2	no	—
77	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
78	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
79	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
80	C₄²×C₃₁	#41	1	1	yes	characteristic
81	#8687	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
82	C₂³×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
83	Q₈×C₃₁	—	4	2	no	—
84	Q₈×C₃₁	—	4	2	no	—
85	C₂×C₄×C₃₁	—	2	2	no	—
86	C₂×C₄×C₃₁	—	2	2	no	—
87	C₂×C₄×C₃₁	—	2	2	no	—
88	C₂×C₄×C₃₁	—	4	1	no	—
89	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₃₁	C₂×C₄⋊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	#135	1	2	yes	—
96	C₂×C₄×C₃₁	#137	1	2	yes	—
97	C₈×C₃₁	—	2	1	no	—
98	C₈×C₃₁	—	2	1	no	—
99	C₈×C₃₁	—	2	1	no	—
100	C₈×C₃₁	—	2	1	no	—
101	C₈×C₃₁	—	4	2	no	—
102	C₈×C₃₁	—	8	2	no	—
103	C₈×C₃₁	—	8	2	no	—
104	#2960	C₂×C₃₁	1	1	yes	characteristic
105	#1837	C₂×C₃₁	1	1	yes	characteristic
106	C₂×#334	C₂×C₃₁	1	1	yes	characteristic
107	#1550	C₂×C₃₁	1	2	yes	—
108	#1292	C₂×C₃₁	1	2	yes	—
109	C₂²×C₃₁	—	2	1	no	—
110	C₂²×C₃₁	—	2	1	no	—
111	C₂²×C₃₁	#380	1	1	yes	characteristic
112	C₂²×C₃₁	#479	1	2	yes	—
113	C₄×C₃₁	—	2	1	no	—
114	C₄×C₃₁	—	2	1	no	—
115	C₄×C₃₁	—	2	1	no	—
116	C₄×C₃₁	—	4	2	no	—
117	C₄×C₃₁	—	4	2	no	—
118	C₄×C₃₁	#416	1	1	yes	characteristic
119	C₄×C₃₁	#419	1	1	yes	characteristic
120	C₄×C₃₁	#464	1	2	yes	—
121	C₄×C₃₁	#471	1	2	yes	—
122	#533	C₂²×C₃₁	1	1	yes	characteristic
123	#446	—	2	1	no	—
124	#446	C₄×C₃₁	1	2	yes	—
125	C₈×Q₈	—	2	1	no	—
126	C₈×Q₈	C₄×C₃₁	1	2	yes	—
127	#424	—	2	1	no	—
128	#423	C₂²×C₃₁	1	2	yes	—
129	#423	C₂²×C₃₁	1	1	yes	characteristic
130	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
131	#405	—	2	1	no	—
132	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
133	#341	—	2	1	no	—
134	#341	—	2	1	no	—
135	#334	—	2	1	no	—
136	#334	—	2	1	no	—
137	#333	—	2	2	no	—
138	#332	—	2	2	no	—
139	C₂×C₃₁	—	2	2	no	—
140	C₂×C₃₁	#1805	1	1	yes	characteristic
141	C₂×C₃₁	#1813	1	1	yes	characteristic
142	C₂×C₃₁	#1815	1	1	yes	characteristic
143	C₂×M₁₆	—	2	1	no	—
144	C₂×M₁₆	—	2	2	no	—
145	C₂²×C₈	—	2	2	no	—
146	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
147	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
148	C₄⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
149	C₄⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
150	#125	—	2	1	no	—
151	C₂²⋊Q₈	—	2	1	no	—
152	C₂²⋊Q₈	Q₈×C₃₁	1	2	yes	—
153	C₄×Q₈	—	2	1	no	—
154	C₄×Q₈	C₂×C₄×C₃₁	1	2	yes	—
155	#117	—	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	—	2	2	no	—
164	C₄⋊C₈	C₂×C₄×C₃₁	1	2	yes	—
165	#97	—	2	2	no	—
166	C₄×C₈	—	2	2	no	—
167	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
168	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
169	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
170	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
171	C₃₁	#8687	1	1	yes	characteristic, direct product factor, Sylow
172	C₂×Q₈	—	2	1	no	—
173	C₂×Q₈	C₄⋊C₄×C₃₁	1	2	yes	—
174	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
175	C₂²×C₄	C₂×Q₈×C₃₁	1	1	yes	characteristic
176	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
177	M₁₆	—	4	2	no	—
178	M₁₆	—	4	2	no	—
179	M₁₆	—	4	2	no	—
180	C₂×C₈	—	2	2	no	—
181	C₂×C₈	—	2	2	no	—
182	C₂×C₈	—	2	2	no	—
183	C₂×C₈	—	4	2	no	—
184	C₂×C₈	—	4	2	no	—
185	C₂×C₈	—	8	2	no	—
186	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
187	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
188	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
189	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
190	C₄⋊C₄	—	2	2	no	—
191	C₄⋊C₄	—	2	1	no	—
192	C₄⋊C₄	—	2	1	no	—
193	C₄⋊C₄	—	2	1	no	—
194	C₄⋊C₄	—	2	2	no	—
195	C₄⋊C₄	C₄⋊C₄×C₃₁	1	2	yes	—
196	C₂²⋊C₄	—	2	2	no	—
197	C₄²	—	2	2	no	—
198	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
199	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
200	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
201	C₄²	#41×C₃₁	1	1	yes	characteristic
202	C₂³	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
203	Q₈	—	4	2	no	—
204	Q₈	—	4	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	4	1	no	—
209	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
210	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
211	C₂×C₄	C₃₁×#117	1	1	yes	characteristic
212	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
213	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
214	C₂×C₄	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
215	C₂×C₄	C₃₁×#135	1	2	yes	—
216	C₂×C₄	C₃₁×#137	1	2	yes	—
217	C₈	—	2	1	no	—
218	C₈	—	2	1	no	—
219	C₈	—	2	1	no	—
220	C₈	—	2	1	no	—
221	C₈	—	4	2	no	—
222	C₈	—	8	2	no	—
223	C₈	—	8	2	no	—
224	C₂²	—	2	1	no	—
225	C₂²	—	2	1	no	—
226	C₂²	C₃₁×#380	1	1	yes	characteristic
227	C₂²	C₃₁×#479	1	2	yes	—
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	C₃₁×#416	1	1	yes	characteristic
234	C₄	C₃₁×#419	1	1	yes	characteristic
235	C₄	C₃₁×#464	1	2	yes	—
236	C₄	C₃₁×#471	1	2	yes	—
237	C₂	—	2	2	no	—
238	C₂	C₃₁×#1805	1	1	yes	characteristic
239	C₂	C₃₁×#1813	1	1	yes	characteristic
240	C₂	C₃₁×#1815	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	#380
commutator	C₄²	C₂²×C₄×C₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
prime radical	C₂³×C₃₁	C₂²⋊Q₈
LCS	G ▷ C₄² ▷ 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
81. #8687	C₃₁	direct (N × H)
171. C₃₁	#8687	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃₁×#2960	2
C₃₁×#1837	2
C₂×C₃₁×#334	2
C₃₁×#1550	2
C₃₁×#1292	2
#8687	31

Subgroups — C31×#8687 #21465446

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#8687 #21465446