Subgroups — C₃₁×#9901 #21466660

Order 7936.

430 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2859	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1831	C₂	1	1	yes	maximal, characteristic
3	C₄×C₃₁×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1296	C₂	1	2	yes	maximal
5	C₃₁×#1296	C₂	1	2	yes	maximal
6	C₂×C₃₁×#125	C₄	1	2	yes	—
7	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
8	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
9	C₂×C₄×C₈×C₃₁	—	2	1	no	—
10	C₃₁×#389	—	2	1	no	—
11	C₃₁×#389	—	2	1	no	—
12	C₃₁×#389	—	2	1	no	—
13	C₃₁×#389	—	2	1	no	—
14	C₃₁×#384	C₂²	1	1	yes	characteristic
15	C₃₁×#384	C₂²	1	1	yes	characteristic
16	C₃₁×#382	C₄	1	2	yes	—
17	C₄³×C₃₁	C₂²	1	1	yes	characteristic
18	C₃₁×#336	—	2	1	no	—
19	C₃₁×#336	—	2	1	no	—
20	C₃₁×#336	—	2	1	no	—
21	C₂²×C₈×C₃₁	—	2	2	no	—
22	C₂²×C₈×C₃₁	—	2	2	no	—
23	C₃₁×#125	—	4	4	no	—
24	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
25	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
26	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
27	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
28	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
29	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
30	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
31	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
36	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
39	C₃₁×C₄⋊C₈	—	2	2	no	—
40	C₃₁×C₄⋊C₈	—	2	4	no	—
41	C₃₁×C₄⋊C₈	Q₈	1	4	yes	—
42	C₄×C₈×C₃₁	—	4	2	no	—
43	C₃₁×#95	—	2	2	no	—
44	C₃₁×#95	—	2	2	no	—
45	C₂²×C₄×C₃₁	—	2	2	no	—
46	C₂²×C₄×C₃₁	—	2	2	no	—
47	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
54	C₂×C₈×C₃₁	—	2	4	no	—
55	C₂×C₈×C₃₁	—	4	2	no	—
56	C₂×C₈×C₃₁	—	4	2	no	—
57	C₂×C₈×C₃₁	—	4	2	no	—
58	C₂×C₈×C₃₁	—	4	2	no	—
59	C₂×C₈×C₃₁	—	4	2	no	—
60	C₄⋊C₄×C₃₁	—	4	4	no	—
61	C₄⋊C₄×C₃₁	—	4	2	no	—
62	C₄⋊C₄×C₃₁	—	4	2	no	—
63	C₄⋊C₄×C₃₁	—	4	4	no	—
64	C₄⋊C₄×C₃₁	—	4	2	no	—
65	C₄⋊C₄×C₃₁	—	4	2	no	—
66	C₄⋊C₄×C₃₁	—	4	2	no	—
67	C₄⋊C₄×C₃₁	—	4	1	no	—
68	C₄⋊C₄×C₃₁	—	4	1	no	—
69	C₄²×C₃₁	—	2	2	no	—
70	C₄²×C₃₁	—	2	2	no	—
71	C₄²×C₃₁	—	2	2	no	—
72	C₄²×C₃₁	—	2	2	no	—
73	C₄²×C₃₁	—	2	2	no	—
74	C₄²×C₃₁	—	2	1	no	—
75	C₄²×C₃₁	—	2	1	no	—
76	C₄²×C₃₁	C₂×Q₈	1	2	yes	—
77	C₄²×C₃₁	#41	1	2	yes	—
78	#9901	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
79	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₃₁	—	2	1	no	—
81	C₂×C₄×C₃₁	—	2	1	no	—
82	C₂×C₄×C₃₁	—	2	1	no	—
83	C₂×C₄×C₃₁	—	2	1	no	—
84	C₂×C₄×C₃₁	—	2	1	no	—
85	C₂×C₄×C₃₁	—	2	1	no	—
86	C₂×C₄×C₃₁	—	2	1	no	—
87	C₂×C₄×C₃₁	—	2	1	no	—
88	C₂×C₄×C₃₁	—	2	1	no	—
89	C₂×C₄×C₃₁	—	2	1	no	—
90	C₂×C₄×C₃₁	—	2	1	no	—
91	C₂×C₄×C₃₁	—	2	1	no	—
92	C₂×C₄×C₃₁	—	4	2	no	—
93	C₂×C₄×C₃₁	—	4	2	no	—
94	C₂×C₄×C₃₁	—	4	2	no	—
95	C₂×C₄×C₃₁	—	4	2	no	—
96	C₂×C₄×C₃₁	—	4	2	no	—
97	C₂×C₄×C₃₁	—	4	2	no	—
98	C₂×C₄×C₃₁	#100	1	2	yes	—
99	C₂×C₄×C₃₁	#101	1	2	yes	—
100	C₂×C₄×C₃₁	C₄×Q₈	1	2	yes	—
101	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
102	C₂×C₄×C₃₁	#124	1	1	yes	characteristic
103	C₂×C₄×C₃₁	C₄⋊Q₈	1	1	yes	characteristic
104	C₂×C₄×C₃₁	#135	1	2	yes	—
105	C₂×C₄×C₃₁	#135	1	2	yes	—
106	C₂×C₄×C₃₁	#135	1	2	yes	—
107	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
108	C₂×C₄×C₃₁	#137	1	1	yes	characteristic
109	C₈×C₃₁	—	4	4	no	—
110	C₈×C₃₁	—	8	2	no	—
111	#2859	C₂×C₃₁	1	1	yes	characteristic
112	#1831	C₂×C₃₁	1	1	yes	characteristic
113	C₄×C₄⋊C₈	C₂×C₃₁	1	1	yes	characteristic
114	#1296	C₂×C₃₁	1	2	yes	—
115	#1296	C₂×C₃₁	1	2	yes	—
116	C₂²×C₃₁	#391	1	1	yes	characteristic
117	C₂²×C₃₁	C₂×#100	1	1	yes	characteristic
118	C₂²×C₃₁	C₂×#101	1	1	yes	characteristic
119	C₂²×C₃₁	#416	1	1	yes	characteristic
120	C₂²×C₃₁	#416	1	1	yes	characteristic
121	C₂²×C₃₁	#416	1	1	yes	characteristic
122	C₂²×C₃₁	#417	1	1	yes	characteristic
123	C₄×C₃₁	—	2	2	no	—
124	C₄×C₃₁	—	2	2	no	—
125	C₄×C₃₁	—	2	2	no	—
126	C₄×C₃₁	—	4	1	no	—
127	C₄×C₃₁	—	4	1	no	—
128	C₄×C₃₁	—	8	2	no	—
129	C₄×C₃₁	—	8	2	no	—
130	C₄×C₃₁	#478	1	2	yes	—
131	C₄×C₃₁	#479	1	2	yes	—
132	C₄×C₃₁	#488	1	2	yes	—
133	C₄×C₃₁	#499	1	2	yes	—
134	C₂×#125	C₄×C₃₁	1	2	yes	—
135	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
136	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
137	C₂×C₄×C₈	—	2	1	no	—
138	#389	—	2	1	no	—
139	#389	—	2	1	no	—
140	#389	—	2	1	no	—
141	#389	—	2	1	no	—
142	#384	C₂²×C₃₁	1	1	yes	characteristic
143	#384	C₂²×C₃₁	1	1	yes	characteristic
144	#382	C₄×C₃₁	1	2	yes	—
145	C₄³	C₂²×C₃₁	1	1	yes	characteristic
146	#336	—	2	1	no	—
147	#336	—	2	1	no	—
148	#336	—	2	1	no	—
149	C₂×C₃₁	#1543	1	1	yes	characteristic
150	C₂×C₃₁	#1544	1	1	yes	characteristic
151	C₂×C₃₁	#1551	1	1	yes	characteristic
152	C₂×C₃₁	#1552	1	1	yes	characteristic
153	C₂×C₃₁	#1989	1	1	yes	characteristic
154	C₂×C₃₁	#1991	1	1	yes	characteristic
155	C₂×C₃₁	#1997	1	1	yes	characteristic
156	C₂²×C₈	—	2	2	no	—
157	C₂²×C₈	—	2	2	no	—
158	#125	—	4	4	no	—
159	C₂×C₄⋊C₄	—	2	2	no	—
160	C₂×C₄⋊C₄	—	2	2	no	—
161	C₂×C₄⋊C₄	—	2	1	no	—
162	C₂×C₄⋊C₄	—	2	1	no	—
163	C₂×C₄⋊C₄	—	2	1	no	—
164	C₂×C₄⋊C₄	—	2	1	no	—
165	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
166	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
167	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
168	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
169	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
170	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
171	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
172	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
173	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
174	C₄⋊C₈	—	2	2	no	—
175	C₄⋊C₈	—	2	4	no	—
176	C₄⋊C₈	Q₈×C₃₁	1	4	yes	—
177	C₄×C₈	—	4	2	no	—
178	#95	—	2	2	no	—
179	#95	—	2	2	no	—
180	C₃₁	#9901	1	1	yes	characteristic, direct product factor, Sylow
181	C₂²×C₄	—	2	2	no	—
182	C₂²×C₄	—	2	2	no	—
183	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
184	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
185	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
186	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
187	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
188	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
189	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
190	C₂×C₈	—	2	4	no	—
191	C₂×C₈	—	4	2	no	—
192	C₂×C₈	—	4	2	no	—
193	C₂×C₈	—	4	2	no	—
194	C₂×C₈	—	4	2	no	—
195	C₂×C₈	—	4	2	no	—
196	C₄⋊C₄	—	4	2	no	—
197	C₄⋊C₄	—	4	2	no	—
198	C₄⋊C₄	—	4	4	no	—
199	C₄⋊C₄	—	4	4	no	—
200	C₄⋊C₄	—	4	2	no	—
201	C₄⋊C₄	—	4	2	no	—
202	C₄⋊C₄	—	4	2	no	—
203	C₄⋊C₄	—	4	1	no	—
204	C₄⋊C₄	—	4	1	no	—
205	C₄²	—	2	2	no	—
206	C₄²	—	2	2	no	—
207	C₄²	—	2	2	no	—
208	C₄²	—	2	2	no	—
209	C₄²	—	2	2	no	—
210	C₄²	—	2	1	no	—
211	C₄²	—	2	1	no	—
212	C₄²	C₂×Q₈×C₃₁	1	2	yes	—
213	C₄²	#41×C₃₁	1	2	yes	—
214	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	—	4	2	no	—
229	C₂×C₄	—	4	2	no	—
230	C₂×C₄	—	4	2	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	C₃₁×#100	1	2	yes	—
234	C₂×C₄	C₃₁×#101	1	2	yes	—
235	C₂×C₄	C₄×Q₈×C₃₁	1	2	yes	—
236	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
237	C₂×C₄	C₃₁×#124	1	1	yes	characteristic
238	C₂×C₄	C₃₁×C₄⋊Q₈	1	1	yes	characteristic
239	C₂×C₄	C₃₁×#135	1	2	yes	—
240	C₂×C₄	C₃₁×#135	1	2	yes	—
241	C₂×C₄	C₃₁×#135	1	2	yes	—
242	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
243	C₂×C₄	C₃₁×#137	1	1	yes	characteristic
244	C₈	—	4	4	no	—
245	C₈	—	8	2	no	—
246	C₂²	C₃₁×#391	1	1	yes	characteristic
247	C₂²	C₂×C₃₁×#100	1	1	yes	characteristic
248	C₂²	C₂×C₃₁×#101	1	1	yes	characteristic
249	C₂²	C₃₁×#416	1	1	yes	characteristic
250	C₂²	C₃₁×#416	1	1	yes	characteristic
251	C₂²	C₃₁×#416	1	1	yes	characteristic
252	C₂²	C₃₁×#417	1	1	yes	characteristic
253	C₄	—	2	2	no	—
254	C₄	—	2	2	no	—
255	C₄	—	2	2	no	—
256	C₄	—	4	1	no	—
257	C₄	—	4	1	no	—
258	C₄	—	8	2	no	—
259	C₄	—	8	2	no	—
260	C₄	C₃₁×#478	1	2	yes	—
261	C₄	C₃₁×#479	1	2	yes	—
262	C₄	C₃₁×#488	1	2	yes	—
263	C₄	C₃₁×#499	1	2	yes	—
264	C₂	C₃₁×#1543	1	1	yes	characteristic
265	C₂	C₃₁×#1544	1	1	yes	characteristic
266	C₂	C₃₁×#1551	1	1	yes	characteristic
267	C₂	C₃₁×#1552	1	1	yes	characteristic
268	C₂	C₃₁×#1989	1	1	yes	characteristic
269	C₂	C₃₁×#1991	1	1	yes	characteristic
270	C₂	C₃₁×#1997	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₃₁	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₄×C₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₄³×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
78. #9901	C₃₁	direct (N × H)
180. C₃₁	#9901	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×#2859	2
C₃₁×#1831	2
C₄×C₃₁×C₄⋊C₈	2
C₃₁×#1296	2
#9901	31

Subgroups — C31×#9901 #21466660

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#9901 #21466660