Subgroups — C₃₁×#11958 #21468717

Order 7936.

404 conjugacy classes of subgroups.

The subgroup lattice has 256 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×#2162	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#1566	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1360	C₂	1	4	yes	maximal, complemented
5	C₂×C₃₁×#135	—	2	1	no	—
6	C₂×C₃₁×#135	C₄	1	2	yes	—
7	C₃₁×#532	—	2	1	no	—
8	C₃₁×#495	—	2	1	no	—
9	C₃₁×#495	C₂²	1	2	yes	—
10	C₃₁×#494	C₄	1	1	yes	characteristic
11	C₃₁×#493	C₄	1	1	yes	characteristic
12	C₃₁×#492	—	2	1	no	—
13	C₃₁×#423	—	2	1	no	—
14	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
15	C₃₁×C₄⋊C₁₆	C₂²	1	2	yes	—
16	C₃₁×C₂²⋊C₁₆	—	2	1	no	—
17	C₄×C₁₆×C₃₁	—	2	1	no	—
18	C₃₁×#334	—	2	1	no	—
19	C₃₁×#334	C₂²	1	2	yes	—
20	C₂×#41×C₃₁	—	2	2	no	—
21	C₃₁×#135	—	4	4	no	—
22	C₃₁×#135	—	4	4	no	—
23	C₂×Q₁₆×C₃₁	—	2	1	no	—
24	C₂×Q₁₆×C₃₁	C₂×C₄	1	2	yes	—
25	C₂×SD₁₆×C₃₁	—	2	2	no	—
26	C₂×SD₁₆×C₃₁	—	2	2	no	—
27	C₂×D₈×C₃₁	—	2	1	no	—
28	C₂×D₈×C₃₁	C₂×C₄	1	2	yes	—
29	C₂×M₁₆×C₃₁	—	4	1	no	—
30	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
32	C₃₁×C₄⋊Q₈	—	2	1	no	—
33	C₃₁×#127	—	2	1	no	—
34	C₃₁×#124	—	2	2	no	—
35	C₃₁×#121	—	4	1	no	—
36	C₃₁×#121	—	4	1	no	—
37	C₄×D₄×C₃₁	—	4	1	no	—
38	C₄×D₄×C₃₁	—	4	1	no	—
39	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
40	C₂×C₁₆×C₃₁	—	2	2	no	—
41	C₃₁×C₄⋊C₈	—	2	2	no	—
42	C₃₁×C₄⋊C₈	—	4	1	no	—
43	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
44	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
45	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
46	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
47	#41×C₃₁	—	4	4	no	—
48	#41×C₃₁	—	4	4	no	—
49	C₂×Q₈×C₃₁	—	2	2	no	—
50	C₂×D₄×C₃₁	—	2	2	no	—
51	C₂×D₄×C₃₁	—	4	2	no	—
52	C₂×D₄×C₃₁	—	4	1	no	—
53	C₂×D₄×C₃₁	—	4	1	no	—
54	C₂²×C₄×C₃₁	—	4	2	no	—
55	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
57	C₂²×C₄×C₃₁	SD₁₆	1	1	yes	characteristic
58	Q₁₆×C₃₁	—	4	2	no	—
59	Q₁₆×C₃₁	—	4	2	no	—
60	SD₁₆×C₃₁	—	4	4	no	—
61	SD₁₆×C₃₁	—	4	4	no	—
62	D₈×C₃₁	—	4	2	no	—
63	D₈×C₃₁	—	4	2	no	—
64	M₁₆×C₃₁	—	8	2	no	—
65	C₂×C₈×C₃₁	—	2	2	no	—
66	C₂×C₈×C₃₁	—	2	2	no	—
67	C₂×C₈×C₃₁	—	4	2	no	—
68	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
72	C₄⋊C₄×C₃₁	—	4	1	no	—
73	C₄⋊C₄×C₃₁	—	4	1	no	—
74	C₂²⋊C₄×C₃₁	—	4	2	no	—
75	C₂²⋊C₄×C₃₁	—	4	2	no	—
76	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
77	C₄²×C₃₁	D₈	1	1	yes	characteristic
78	C₄²×C₃₁	SD₁₆	1	1	yes	characteristic
79	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
80	C₁₆×C₃₁	—	4	2	no	—
81	#11958	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
82	C₂³×C₃₁	—	4	2	no	—
83	C₂³×C₃₁	#102	1	1	yes	characteristic
84	Q₈×C₃₁	—	4	2	no	—
85	Q₈×C₃₁	—	4	2	no	—
86	D₄×C₃₁	—	4	2	no	—
87	D₄×C₃₁	—	4	2	no	—
88	D₄×C₃₁	—	8	4	no	—
89	D₄×C₃₁	—	8	2	no	—
90	D₄×C₃₁	—	8	2	no	—
91	C₂×C₄×C₃₁	—	2	2	no	—
92	C₂×C₄×C₃₁	—	2	2	no	—
93	C₂×C₄×C₃₁	—	2	1	no	—
94	C₂×C₄×C₃₁	—	2	1	no	—
95	C₂×C₄×C₃₁	—	4	2	no	—
96	C₂×C₄×C₃₁	—	8	4	no	—
97	C₂×C₄×C₃₁	#102	1	1	yes	characteristic
98	C₂×C₄×C₃₁	#102	1	1	yes	characteristic
99	C₂×C₄×C₃₁	#102	1	1	yes	characteristic
100	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
102	C₂×C₄×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
103	C₈×C₃₁	—	2	1	no	—
104	C₈×C₃₁	—	2	1	no	—
105	C₈×C₃₁	—	8	2	no	—
106	C₈×C₃₁	#100	1	2	yes	—
107	C₈×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
108	#3147	C₂×C₃₁	1	1	yes	characteristic
109	#2162	C₂×C₃₁	1	1	yes	characteristic, complemented
110	#1566	C₂×C₃₁	1	1	yes	characteristic, complemented
111	#1360	C₂×C₃₁	1	4	yes	complemented
112	C₂²×C₃₁	—	2	1	no	—
113	C₂²×C₃₁	—	2	1	no	—
114	C₂²×C₃₁	—	2	1	no	—
115	C₂²×C₃₁	—	8	2	no	—
116	C₂²×C₃₁	—	8	2	no	—
117	C₂²×C₃₁	—	8	2	no	—
118	C₂²×C₃₁	C₂×#102	1	1	yes	characteristic
119	C₄×C₃₁	—	2	1	no	—
120	C₄×C₃₁	—	2	1	no	—
121	C₄×C₃₁	—	2	1	no	—
122	C₄×C₃₁	—	8	2	no	—
123	C₄×C₃₁	#359	1	2	yes	—
124	C₄×C₃₁	#360	1	2	yes	—
125	C₄×C₃₁	C₂×#100	1	1	yes	characteristic
126	C₄×C₃₁	#418	1	1	yes	characteristic
127	C₂×#135	—	2	1	no	—
128	C₂×#135	C₄×C₃₁	1	2	yes	—
129	#532	—	2	1	no	—
130	#495	—	2	1	no	—
131	#495	C₂²×C₃₁	1	2	yes	—
132	#494	C₄×C₃₁	1	1	yes	characteristic
133	#493	C₄×C₃₁	1	1	yes	characteristic
134	#492	—	2	1	no	—
135	#423	—	2	1	no	—
136	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
137	C₄⋊C₁₆	C₂²×C₃₁	1	2	yes	—
138	C₂²⋊C₁₆	—	2	1	no	—
139	C₄×C₁₆	—	2	1	no	—
140	#334	—	2	1	no	—
141	#334	C₂²×C₃₁	1	2	yes	—
142	C₂×C₃₁	—	4	1	no	—
143	C₂×C₃₁	—	16	2	no	—
144	C₂×C₃₁	#1545	1	1	yes	characteristic
145	C₂×C₃₁	C₂×#359	1	1	yes	characteristic
146	C₂×C₃₁	C₂×#360	1	1	yes	characteristic
147	C₂×#41	—	2	2	no	—
148	#135	—	4	4	no	—
149	#135	—	4	4	no	—
150	C₂×Q₁₆	—	2	1	no	—
151	C₂×Q₁₆	C₂×C₄×C₃₁	1	2	yes	—
152	C₂×SD₁₆	—	2	2	no	—
153	C₂×SD₁₆	—	2	2	no	—
154	C₂×D₈	—	2	1	no	—
155	C₂×D₈	C₂×C₄×C₃₁	1	2	yes	—
156	C₂×M₁₆	—	4	1	no	—
157	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
158	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
159	C₄⋊Q₈	—	2	1	no	—
160	#127	—	2	1	no	—
161	#124	—	2	2	no	—
162	#121	—	4	1	no	—
163	#121	—	4	1	no	—
164	C₄×D₄	—	4	1	no	—
165	C₄×D₄	—	4	1	no	—
166	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
167	C₂×C₁₆	—	2	2	no	—
168	C₄⋊C₈	—	2	2	no	—
169	C₄⋊C₈	—	4	1	no	—
170	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
171	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
172	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
173	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
174	C₃₁	#11958	1	1	yes	characteristic, direct product factor, Sylow
175	#41	—	4	4	no	—
176	#41	—	4	4	no	—
177	C₂×Q₈	—	2	2	no	—
178	C₂×D₄	—	2	2	no	—
179	C₂×D₄	—	4	2	no	—
180	C₂×D₄	—	4	1	no	—
181	C₂×D₄	—	4	1	no	—
182	C₂²×C₄	—	4	2	no	—
183	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
184	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
185	C₂²×C₄	SD₁₆×C₃₁	1	1	yes	characteristic
186	Q₁₆	—	4	2	no	—
187	Q₁₆	—	4	2	no	—
188	SD₁₆	—	4	4	no	—
189	SD₁₆	—	4	4	no	—
190	D₈	—	4	2	no	—
191	D₈	—	4	2	no	—
192	M₁₆	—	8	2	no	—
193	C₂×C₈	—	2	2	no	—
194	C₂×C₈	—	2	2	no	—
195	C₂×C₈	—	4	2	no	—
196	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
197	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
198	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
199	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
200	C₄⋊C₄	—	4	1	no	—
201	C₄⋊C₄	—	4	1	no	—
202	C₂²⋊C₄	—	4	2	no	—
203	C₂²⋊C₄	—	4	2	no	—
204	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
205	C₄²	D₈×C₃₁	1	1	yes	characteristic
206	C₄²	SD₁₆×C₃₁	1	1	yes	characteristic
207	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
208	C₁₆	—	4	2	no	—
209	C₂³	—	4	2	no	—
210	C₂³	C₃₁×#102	1	1	yes	characteristic
211	Q₈	—	4	2	no	—
212	Q₈	—	4	2	no	—
213	D₄	—	4	2	no	—
214	D₄	—	4	2	no	—
215	D₄	—	8	4	no	—
216	D₄	—	8	2	no	—
217	D₄	—	8	2	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	4	2	no	—
223	C₂×C₄	—	8	4	no	—
224	C₂×C₄	C₃₁×#102	1	1	yes	characteristic
225	C₂×C₄	C₃₁×#102	1	1	yes	characteristic
226	C₂×C₄	C₃₁×#102	1	1	yes	characteristic
227	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
228	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
229	C₂×C₄	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
230	C₈	—	2	1	no	—
231	C₈	—	2	1	no	—
232	C₈	—	8	2	no	—
233	C₈	C₃₁×#100	1	2	yes	—
234	C₈	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
235	C₂²	—	2	1	no	—
236	C₂²	—	2	1	no	—
237	C₂²	—	2	1	no	—
238	C₂²	—	8	2	no	—
239	C₂²	—	8	2	no	—
240	C₂²	—	8	2	no	—
241	C₂²	C₂×C₃₁×#102	1	1	yes	characteristic
242	C₄	—	2	1	no	—
243	C₄	—	2	1	no	—
244	C₄	—	2	1	no	—
245	C₄	—	8	2	no	—
246	C₄	C₃₁×#359	1	2	yes	—
247	C₄	C₃₁×#360	1	2	yes	—
248	C₄	C₂×C₃₁×#100	1	1	yes	characteristic
249	C₄	C₃₁×#418	1	1	yes	characteristic
250	C₂	—	4	1	no	—
251	C₂	—	16	2	no	—
252	C₂	C₃₁×#1545	1	1	yes	characteristic
253	C₂	C₂×C₃₁×#359	1	1	yes	characteristic
254	C₂	C₂×C₃₁×#360	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×#2162	C₂	semidirect
3. C₃₁×#1566	C₂	semidirect
4. C₃₁×#1360	C₂	semidirect
81. #11958	C₃₁	direct (N × H)
109. #2162	C₂×C₃₁	semidirect
110. #1566	C₂×C₃₁	semidirect
111. #1360	C₂×C₃₁	semidirect
174. C₃₁	#11958	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×#2162	2
C₃₁×#1566	2
C₃₁×#1360	2
#11958	31

Subgroups — C31×#11958 #21468717

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#11958 #21468717