Subgroups — C₃₁×#11966 #21468725

Order 7936.

400 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3147	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2163	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#1567	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1362	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1361	C₂	1	2	yes	maximal, complemented
6	C₂×C₃₁×#135	—	2	1	no	—
7	C₂×C₃₁×#135	C₄	1	2	yes	—
8	C₃₁×#532	—	2	1	no	—
9	C₃₁×#495	—	2	1	no	—
10	C₃₁×#495	C₄	1	2	yes	—
11	C₃₁×#494	C₂²	1	1	yes	characteristic
12	C₃₁×#493	C₂²	1	1	yes	characteristic
13	C₃₁×#492	—	2	1	no	—
14	C₃₁×#423	—	2	1	no	—
15	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
16	C₃₁×C₄⋊C₁₆	C₂²	1	2	yes	—
17	C₃₁×C₂²⋊C₁₆	—	2	1	no	—
18	C₃₁×#346	—	2	1	no	—
19	C₃₁×#335	—	2	1	no	—
20	C₃₁×#335	C₂²	1	2	yes	—
21	C₂×#41×C₃₁	—	2	2	no	—
22	C₃₁×#135	—	4	4	no	—
23	C₃₁×#135	—	4	4	no	—
24	C₂×Q₁₆×C₃₁	—	2	1	no	—
25	C₂×Q₁₆×C₃₁	C₂×C₄	1	2	yes	—
26	C₂×SD₁₆×C₃₁	—	2	2	no	—
27	C₂×SD₁₆×C₃₁	—	2	2	no	—
28	C₂×D₈×C₃₁	—	2	1	no	—
29	C₂×D₈×C₃₁	C₂×C₄	1	2	yes	—
30	C₂×M₁₆×C₃₁	—	4	1	no	—
31	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
33	C₃₁×C₄⋊Q₈	—	2	1	no	—
34	C₃₁×#127	—	2	1	no	—
35	C₃₁×#124	—	2	2	no	—
36	C₃₁×#121	—	4	1	no	—
37	C₃₁×#121	—	4	1	no	—
38	C₄×D₄×C₃₁	—	4	1	no	—
39	C₄×D₄×C₃₁	—	4	1	no	—
40	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
41	C₂×C₁₆×C₃₁	—	2	2	no	—
42	C₃₁×C₄⋊C₈	—	2	2	no	—
43	C₃₁×C₄⋊C₈	—	4	1	no	—
44	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
45	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
46	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
47	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
48	#41×C₃₁	—	4	4	no	—
49	#41×C₃₁	—	4	4	no	—
50	C₂×Q₈×C₃₁	—	2	2	no	—
51	C₂×D₄×C₃₁	—	2	2	no	—
52	C₂×D₄×C₃₁	—	4	2	no	—
53	C₂×D₄×C₃₁	—	4	1	no	—
54	C₂×D₄×C₃₁	—	4	1	no	—
55	C₂²×C₄×C₃₁	—	4	2	no	—
56	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
58	C₂²×C₄×C₃₁	SD₁₆	1	1	yes	characteristic
59	Q₁₆×C₃₁	—	4	2	no	—
60	Q₁₆×C₃₁	—	4	2	no	—
61	SD₁₆×C₃₁	—	4	4	no	—
62	SD₁₆×C₃₁	—	4	4	no	—
63	D₈×C₃₁	—	4	2	no	—
64	D₈×C₃₁	—	4	2	no	—
65	M₁₆×C₃₁	—	8	2	no	—
66	C₂×C₈×C₃₁	—	2	2	no	—
67	C₂×C₈×C₃₁	—	2	2	no	—
68	C₂×C₈×C₃₁	—	4	2	no	—
69	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
73	C₄⋊C₄×C₃₁	—	4	1	no	—
74	C₄⋊C₄×C₃₁	—	4	1	no	—
75	C₂²⋊C₄×C₃₁	—	4	2	no	—
76	C₂²⋊C₄×C₃₁	—	4	2	no	—
77	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
78	C₄²×C₃₁	D₈	1	1	yes	characteristic
79	C₄²×C₃₁	SD₁₆	1	1	yes	characteristic
80	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
81	C₁₆×C₃₁	—	4	2	no	—
82	#11966	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
83	C₂³×C₃₁	—	4	2	no	—
84	C₂³×C₃₁	#102	1	1	yes	characteristic
85	Q₈×C₃₁	—	4	2	no	—
86	Q₈×C₃₁	—	4	2	no	—
87	D₄×C₃₁	—	4	2	no	—
88	D₄×C₃₁	—	4	2	no	—
89	D₄×C₃₁	—	8	2	no	—
90	D₄×C₃₁	—	8	4	no	—
91	D₄×C₃₁	—	8	1	no	—
92	D₄×C₃₁	—	8	1	no	—
93	C₂×C₄×C₃₁	—	2	1	no	—
94	C₂×C₄×C₃₁	—	2	1	no	—
95	C₂×C₄×C₃₁	—	2	1	no	—
96	C₂×C₄×C₃₁	—	2	1	no	—
97	C₂×C₄×C₃₁	—	4	1	no	—
98	C₂×C₄×C₃₁	—	4	2	no	—
99	C₂×C₄×C₃₁	—	8	4	no	—
100	C₂×C₄×C₃₁	#102	1	1	yes	characteristic
101	C₂×C₄×C₃₁	#102	1	1	yes	characteristic
102	C₂×C₄×C₃₁	#102	1	1	yes	characteristic
103	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
105	C₂×C₄×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
106	C₈×C₃₁	—	2	1	no	—
107	C₈×C₃₁	—	2	1	no	—
108	C₈×C₃₁	—	8	2	no	—
109	C₈×C₃₁	#100	1	2	yes	—
110	C₈×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
111	#3147	C₂×C₃₁	1	1	yes	characteristic
112	#2163	C₂×C₃₁	1	1	yes	characteristic, complemented
113	#1567	C₂×C₃₁	1	1	yes	characteristic, complemented
114	#1362	C₂×C₃₁	1	2	yes	complemented
115	#1361	C₂×C₃₁	1	2	yes	complemented
116	C₂²×C₃₁	—	2	1	no	—
117	C₂²×C₃₁	—	2	1	no	—
118	C₂²×C₃₁	—	2	1	no	—
119	C₂²×C₃₁	—	8	2	no	—
120	C₂²×C₃₁	—	8	2	no	—
121	C₂²×C₃₁	—	8	2	no	—
122	C₂²×C₃₁	C₂×#102	1	1	yes	characteristic
123	C₄×C₃₁	—	2	1	no	—
124	C₄×C₃₁	—	2	1	no	—
125	C₄×C₃₁	—	2	1	no	—
126	C₄×C₃₁	—	2	1	no	—
127	C₄×C₃₁	—	8	2	no	—
128	C₄×C₃₁	C₂×#100	1	1	yes	characteristic
129	C₄×C₃₁	#418	1	1	yes	characteristic
130	C₄×C₃₁	#509	1	1	yes	characteristic
131	C₄×C₃₁	#510	1	1	yes	characteristic
132	C₂×#135	—	2	1	no	—
133	C₂×#135	C₄×C₃₁	1	2	yes	—
134	#532	—	2	1	no	—
135	#495	—	2	1	no	—
136	#495	C₄×C₃₁	1	2	yes	—
137	#494	C₂²×C₃₁	1	1	yes	characteristic
138	#493	C₂²×C₃₁	1	1	yes	characteristic
139	#492	—	2	1	no	—
140	#423	—	2	1	no	—
141	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
142	C₄⋊C₁₆	C₂²×C₃₁	1	2	yes	—
143	C₂²⋊C₁₆	—	2	1	no	—
144	#346	—	2	1	no	—
145	#335	—	2	1	no	—
146	#335	C₂²×C₃₁	1	2	yes	—
147	C₂×C₃₁	—	4	1	no	—
148	C₂×C₃₁	—	16	2	no	—
149	C₂×C₃₁	#1545	1	1	yes	characteristic
150	C₂×C₃₁	#2139	1	1	yes	characteristic
151	C₂×C₃₁	#2148	1	1	yes	characteristic
152	C₂×#41	—	2	2	no	—
153	#135	—	4	4	no	—
154	#135	—	4	4	no	—
155	C₂×Q₁₆	—	2	1	no	—
156	C₂×Q₁₆	C₂×C₄×C₃₁	1	2	yes	—
157	C₂×SD₁₆	—	2	2	no	—
158	C₂×SD₁₆	—	2	2	no	—
159	C₂×D₈	—	2	1	no	—
160	C₂×D₈	C₂×C₄×C₃₁	1	2	yes	—
161	C₂×M₁₆	—	4	1	no	—
162	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
163	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
164	C₄⋊Q₈	—	2	1	no	—
165	#127	—	2	1	no	—
166	#124	—	2	2	no	—
167	#121	—	4	1	no	—
168	#121	—	4	1	no	—
169	C₄×D₄	—	4	1	no	—
170	C₄×D₄	—	4	1	no	—
171	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
172	C₂×C₁₆	—	2	2	no	—
173	C₄⋊C₈	—	2	2	no	—
174	C₄⋊C₈	—	4	1	no	—
175	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
176	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
177	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
178	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
179	C₃₁	#11966	1	1	yes	characteristic, direct product factor, Sylow
180	#41	—	4	4	no	—
181	#41	—	4	4	no	—
182	C₂×Q₈	—	2	2	no	—
183	C₂×D₄	—	2	2	no	—
184	C₂×D₄	—	4	2	no	—
185	C₂×D₄	—	4	1	no	—
186	C₂×D₄	—	4	1	no	—
187	C₂²×C₄	—	4	2	no	—
188	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
189	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
190	C₂²×C₄	SD₁₆×C₃₁	1	1	yes	characteristic
191	Q₁₆	—	4	2	no	—
192	Q₁₆	—	4	2	no	—
193	SD₁₆	—	4	4	no	—
194	SD₁₆	—	4	4	no	—
195	D₈	—	4	2	no	—
196	D₈	—	4	2	no	—
197	M₁₆	—	8	2	no	—
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	4	2	no	—
201	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
202	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
203	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
204	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
205	C₄⋊C₄	—	4	1	no	—
206	C₄⋊C₄	—	4	1	no	—
207	C₂²⋊C₄	—	4	2	no	—
208	C₂²⋊C₄	—	4	2	no	—
209	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
210	C₄²	D₈×C₃₁	1	1	yes	characteristic
211	C₄²	SD₁₆×C₃₁	1	1	yes	characteristic
212	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
213	C₁₆	—	4	2	no	—
214	C₂³	—	4	2	no	—
215	C₂³	C₃₁×#102	1	1	yes	characteristic
216	Q₈	—	4	2	no	—
217	Q₈	—	4	2	no	—
218	D₄	—	4	2	no	—
219	D₄	—	4	2	no	—
220	D₄	—	8	2	no	—
221	D₄	—	8	4	no	—
222	D₄	—	8	1	no	—
223	D₄	—	8	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	4	1	no	—
229	C₂×C₄	—	4	2	no	—
230	C₂×C₄	—	8	4	no	—
231	C₂×C₄	C₃₁×#102	1	1	yes	characteristic
232	C₂×C₄	C₃₁×#102	1	1	yes	characteristic
233	C₂×C₄	C₃₁×#102	1	1	yes	characteristic
234	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
235	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
236	C₂×C₄	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
237	C₈	—	2	1	no	—
238	C₈	—	2	1	no	—
239	C₈	—	8	2	no	—
240	C₈	C₃₁×#100	1	2	yes	—
241	C₈	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
242	C₂²	—	2	1	no	—
243	C₂²	—	2	1	no	—
244	C₂²	—	2	1	no	—
245	C₂²	—	8	2	no	—
246	C₂²	—	8	2	no	—
247	C₂²	—	8	2	no	—
248	C₂²	C₂×C₃₁×#102	1	1	yes	characteristic
249	C₄	—	2	1	no	—
250	C₄	—	2	1	no	—
251	C₄	—	2	1	no	—
252	C₄	—	2	1	no	—
253	C₄	—	8	2	no	—
254	C₄	C₂×C₃₁×#100	1	1	yes	characteristic
255	C₄	C₃₁×#418	1	1	yes	characteristic
256	C₄	C₃₁×#509	1	1	yes	characteristic
257	C₄	C₃₁×#510	1	1	yes	characteristic
258	C₂	—	4	1	no	—
259	C₂	—	16	2	no	—
260	C₂	C₃₁×#1545	1	1	yes	characteristic
261	C₂	C₃₁×#2139	1	1	yes	characteristic
262	C₂	C₃₁×#2148	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₃₁
Frattini, cosocle	C₄×C₈	C₂³×C₃₁
prime radical	C₃₁×#3147	C₂
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ C₂×C₄×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2163	C₂	semidirect
3. C₃₁×#1567	C₂	semidirect
4. C₃₁×#1362	C₂	semidirect
5. C₃₁×#1361	C₂	semidirect
82. #11966	C₃₁	direct (N × H)
112. #2163	C₂×C₃₁	semidirect
113. #1567	C₂×C₃₁	semidirect
114. #1362	C₂×C₃₁	semidirect
115. #1361	C₂×C₃₁	semidirect
179. C₃₁	#11966	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×#3147	2
C₃₁×#2163	2
C₃₁×#1567	2
C₃₁×#1362	2
C₃₁×#1361	2
#11966	31

Subgroups — C31×#11966 #21468725

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#11966 #21468725