Subgroups — C₃₁×#33657 #21490416

Order 7936.

566 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3058	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2916	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2238	C₂	1	8	yes	maximal, complemented
4	C₃₁×#2155	C₂	1	2	yes	maximal, complemented
5	C₂×C₃₁×#361	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2115	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₈×C₃₁	C₂²	1	1	yes	characteristic
8	C₃₁×#568	—	2	1	no	—
9	C₂×C₃₁×#131	C₂²	1	1	yes	characteristic
10	C₃₁×#535	—	2	1	no	—
11	C₂×C₃₁×M₃₂	C₂²	1	2	yes	—
12	C₃₁×#469	C₂²	1	4	yes	—
13	C₃₁×#466	—	2	2	no	—
14	C₃₁×#430	—	2	1	no	—
15	C₃₁×#430	C₂²	1	4	yes	complemented
16	C₂×C₃₁×#108	—	2	2	no	—
17	C₂×C₃₁×#108	C₂²	1	2	yes	—
18	C₃₁×#427	C₂²	1	1	yes	characteristic
19	C₃₁×#418	—	2	1	no	—
20	C₃₁×#365	C₂²	1	8	yes	complemented
21	C₃₁×#361	C₂²	1	8	yes	—
22	C₃₁×#349	C₂²	1	4	yes	complemented
23	C₂×#41×C₃₁	D₄	1	1	yes	characteristic
24	C₂²×D₄×C₃₁	—	2	1	no	—
25	C₂×D₈×C₃₁	—	2	1	no	—
26	C₂×D₈×C₃₁	—	2	2	no	—
27	C₂×D₈×C₃₁	—	2	2	no	—
28	C₂×D₈×C₃₁	C₂³	1	2	yes	—
29	C₃₁×#131	—	2	2	no	—
30	C₃₁×#131	—	2	2	no	—
31	C₃₁×#131	—	4	4	no	—
32	C₂×M₁₆×C₃₁	—	2	4	no	—
33	C₂×M₁₆×C₃₁	—	2	2	no	—
34	C₂×M₁₆×C₃₁	D₄	1	1	yes	characteristic
35	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
36	C₂²×C₈×C₃₁	D₄	1	2	yes	—
37	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
38	C₃₁×#127	—	4	1	no	—
39	C₃₁×#124	—	4	1	no	—
40	C₃₁×#121	—	2	4	no	—
41	C₃₁×#120	—	4	2	no	—
42	C₃₁×#117	—	2	1	no	—
43	C₃₁×M₃₂	—	2	2	no	—
44	C₃₁×M₃₂	C₂³	1	4	yes	—
45	C₂×C₁₆×C₃₁	—	2	2	no	—
46	C₃₁×#108	—	2	2	no	—
47	C₃₁×#108	—	2	4	no	—
48	C₃₁×#108	C₂³	1	4	yes	—
49	C₃₁×#107	—	2	1	no	—
50	C₃₁×#106	C₂³	1	2	yes	—
51	C₃₁×#102	—	2	4	no	—
52	C₂⁴×C₃₁	—	4	1	no	—
53	#41×C₃₁	—	2	2	no	—
54	#41×C₃₁	—	2	2	no	—
55	C₂×Q₈×C₃₁	SD₁₆	1	1	yes	characteristic
56	C₂×D₄×C₃₁	—	2	2	no	—
57	C₂×D₄×C₃₁	—	4	4	no	—
58	C₂×D₄×C₃₁	—	4	2	no	—
59	C₂×D₄×C₃₁	—	4	1	no	—
60	C₂×D₄×C₃₁	—	4	2	no	—
61	C₂×D₄×C₃₁	SD₁₆	1	2	yes	—
62	C₂×D₄×C₃₁	SD₁₆	1	1	yes	characteristic
63	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
64	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
65	D₈×C₃₁	—	2	4	no	—
66	D₈×C₃₁	—	4	1	no	—
67	D₈×C₃₁	—	4	1	no	—
68	M₁₆×C₃₁	—	2	2	no	—
69	M₁₆×C₃₁	—	2	4	no	—
70	M₁₆×C₃₁	—	2	4	no	—
71	M₁₆×C₃₁	—	4	4	no	—
72	M₁₆×C₃₁	—	4	2	no	—
73	C₂×C₈×C₃₁	—	2	2	no	—
74	C₂×C₈×C₃₁	—	2	2	no	—
75	C₂×C₈×C₃₁	—	4	4	no	—
76	C₂×C₈×C₃₁	—	4	2	no	—
77	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
78	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
79	C₂×C₈×C₃₁	#41	1	2	yes	—
80	C₂×C₈×C₃₁	#41	1	2	yes	—
81	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
82	C₄⋊C₄×C₃₁	—	2	2	no	—
83	C₂²⋊C₄×C₃₁	—	4	1	no	—
84	C₂²⋊C₄×C₃₁	—	4	4	no	—
85	C₄²×C₃₁	—	4	1	no	—
86	C₁₆×C₃₁	—	2	4	no	—
87	#33657	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
88	C₂³×C₃₁	—	4	2	no	—
89	C₂³×C₃₁	—	8	2	no	—
90	C₂³×C₃₁	—	16	2	no	—
91	C₂³×C₃₁	C₂×SD₁₆	1	2	yes	—
92	C₂³×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
93	Q₈×C₃₁	—	2	2	no	—
94	D₄×C₃₁	—	2	4	no	—
95	D₄×C₃₁	—	2	2	no	—
96	D₄×C₃₁	—	4	2	no	—
97	D₄×C₃₁	—	4	2	no	—
98	D₄×C₃₁	—	8	2	no	—
99	D₄×C₃₁	—	16	4	no	—
100	C₂×C₄×C₃₁	—	2	2	no	—
101	C₂×C₄×C₃₁	—	2	2	no	—
102	C₂×C₄×C₃₁	—	4	2	no	—
103	C₂×C₄×C₃₁	#123	1	2	yes	—
104	C₂×C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
105	C₂×C₄×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
106	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
107	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
108	C₈×C₃₁	—	2	2	no	—
109	C₈×C₃₁	—	4	4	no	—
110	C₈×C₃₁	#123	1	1	yes	characteristic
111	C₈×C₃₁	#123	1	1	yes	characteristic
112	C₈×C₃₁	C₂×#41	1	2	yes	—
113	#3058	C₂×C₃₁	1	1	yes	characteristic
114	#2916	C₂×C₃₁	1	1	yes	characteristic, complemented
115	#2238	C₂×C₃₁	1	8	yes	complemented
116	#2155	C₂×C₃₁	1	2	yes	complemented
117	C₂×#361	C₂×C₃₁	1	2	yes	complemented
118	#2115	C₂×C₃₁	1	1	yes	characteristic, complemented
119	C₂²×C₃₁	—	2	2	no	—
120	C₂²×C₃₁	—	4	1	no	—
121	C₂²×C₃₁	—	4	2	no	—
122	C₂²×C₃₁	—	8	4	no	—
123	C₂²×C₃₁	—	8	2	no	—
124	C₂²×C₃₁	—	16	4	no	—
125	C₂²×C₃₁	#481	1	2	yes	—
126	C₂²×C₃₁	C₂²×SD₁₆	1	1	yes	characteristic
127	C₄×C₃₁	—	2	2	no	—
128	C₄×C₃₁	—	8	2	no	—
129	C₄×C₃₁	#481	1	2	yes	—
130	C₄×C₃₁	C₂×#123	1	1	yes	characteristic
131	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
132	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic
133	#568	—	2	1	no	—
134	C₂×#131	C₂²×C₃₁	1	1	yes	characteristic
135	#535	—	2	1	no	—
136	C₂×M₃₂	C₂²×C₃₁	1	2	yes	—
137	#469	C₂²×C₃₁	1	4	yes	—
138	#466	—	2	2	no	—
139	#430	—	2	1	no	—
140	#430	C₂²×C₃₁	1	4	yes	complemented
141	C₂×#108	—	2	2	no	—
142	C₂×#108	C₂²×C₃₁	1	2	yes	—
143	#427	C₂²×C₃₁	1	1	yes	characteristic
144	#418	—	2	1	no	—
145	#365	C₂²×C₃₁	1	8	yes	complemented
146	#361	C₂²×C₃₁	1	8	yes	—
147	#349	C₂²×C₃₁	1	4	yes	complemented
148	C₂×C₃₁	—	2	1	no	—
149	C₂×C₃₁	—	2	2	no	—
150	C₂×C₃₁	—	4	2	no	—
151	C₂×C₃₁	—	16	2	no	—
152	C₂×C₃₁	C₂×#481	1	1	yes	characteristic
153	C₂×#41	D₄×C₃₁	1	1	yes	characteristic
154	C₂²×D₄	—	2	1	no	—
155	C₂×D₈	—	2	1	no	—
156	C₂×D₈	—	2	2	no	—
157	C₂×D₈	—	2	2	no	—
158	C₂×D₈	C₂³×C₃₁	1	2	yes	—
159	#131	—	2	2	no	—
160	#131	—	2	2	no	—
161	#131	—	4	4	no	—
162	C₂×M₁₆	—	2	4	no	—
163	C₂×M₁₆	—	2	2	no	—
164	C₂×M₁₆	D₄×C₃₁	1	1	yes	characteristic
165	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
166	C₂²×C₈	D₄×C₃₁	1	2	yes	—
167	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
168	#127	—	4	1	no	—
169	#124	—	4	1	no	—
170	#121	—	2	4	no	—
171	#120	—	4	2	no	—
172	#117	—	2	1	no	—
173	M₃₂	—	2	2	no	—
174	M₃₂	C₂³×C₃₁	1	4	yes	—
175	C₂×C₁₆	—	2	2	no	—
176	#108	—	2	2	no	—
177	#108	—	2	4	no	—
178	#108	C₂³×C₃₁	1	4	yes	—
179	#107	—	2	1	no	—
180	#106	C₂³×C₃₁	1	2	yes	—
181	#102	—	2	4	no	—
182	C₃₁	#33657	1	1	yes	characteristic, direct product factor, Sylow
183	C₂⁴	—	4	1	no	—
184	#41	—	2	2	no	—
185	#41	—	2	2	no	—
186	C₂×Q₈	SD₁₆×C₃₁	1	1	yes	characteristic
187	C₂×D₄	—	2	2	no	—
188	C₂×D₄	—	4	4	no	—
189	C₂×D₄	—	4	2	no	—
190	C₂×D₄	—	4	1	no	—
191	C₂×D₄	—	4	2	no	—
192	C₂×D₄	SD₁₆×C₃₁	1	2	yes	—
193	C₂×D₄	SD₁₆×C₃₁	1	1	yes	characteristic
194	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
195	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
196	D₈	—	2	4	no	—
197	D₈	—	4	1	no	—
198	D₈	—	4	1	no	—
199	M₁₆	—	2	2	no	—
200	M₁₆	—	2	4	no	—
201	M₁₆	—	2	4	no	—
202	M₁₆	—	4	4	no	—
203	M₁₆	—	4	2	no	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	—	4	4	no	—
207	C₂×C₈	—	4	2	no	—
208	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
209	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
210	C₂×C₈	#41×C₃₁	1	2	yes	—
211	C₂×C₈	#41×C₃₁	1	2	yes	—
212	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
213	C₄⋊C₄	—	2	2	no	—
214	C₂²⋊C₄	—	4	1	no	—
215	C₂²⋊C₄	—	4	4	no	—
216	C₄²	—	4	1	no	—
217	C₁₆	—	2	4	no	—
218	C₂³	—	4	2	no	—
219	C₂³	—	8	2	no	—
220	C₂³	—	16	2	no	—
221	C₂³	C₂×SD₁₆×C₃₁	1	2	yes	—
222	C₂³	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
223	Q₈	—	2	2	no	—
224	D₄	—	2	4	no	—
225	D₄	—	2	2	no	—
226	D₄	—	4	2	no	—
227	D₄	—	4	2	no	—
228	D₄	—	8	2	no	—
229	D₄	—	16	4	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	C₃₁×#123	1	2	yes	—
234	C₂×C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
235	C₂×C₄	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
236	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
237	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
238	C₈	—	2	2	no	—
239	C₈	—	4	4	no	—
240	C₈	C₃₁×#123	1	1	yes	characteristic
241	C₈	C₃₁×#123	1	1	yes	characteristic
242	C₈	C₂×#41×C₃₁	1	2	yes	—
243	C₂²	—	2	2	no	—
244	C₂²	—	4	1	no	—
245	C₂²	—	4	2	no	—
246	C₂²	—	8	4	no	—
247	C₂²	—	8	2	no	—
248	C₂²	—	16	4	no	—
249	C₂²	C₃₁×#481	1	2	yes	—
250	C₂²	C₂²×SD₁₆×C₃₁	1	1	yes	characteristic
251	C₄	—	2	2	no	—
252	C₄	—	8	2	no	—
253	C₄	C₃₁×#481	1	2	yes	—
254	C₄	C₂×C₃₁×#123	1	1	yes	characteristic
255	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
256	C₂	—	2	1	no	—
257	C₂	—	2	2	no	—
258	C₂	—	4	2	no	—
259	C₂	—	16	2	no	—
260	C₂	C₂×C₃₁×#481	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃₁	C₂×#481
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
prime radical	C₃₁×#3058	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃₁ ◁ C₂²×C₄×C₃₁ ◁ C₂×C₃₁×#131 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2916	C₂	semidirect
3. C₃₁×#2238	C₂	semidirect
4. C₃₁×#2155	C₂	semidirect
5. C₂×C₃₁×#361	C₂	semidirect
6. C₃₁×#2115	C₂	semidirect
15. C₃₁×#430	C₂²	semidirect
20. C₃₁×#365	C₂²	semidirect
22. C₃₁×#349	C₂²	semidirect
87. #33657	C₃₁	direct (N × H)
114. #2916	C₂×C₃₁	semidirect
115. #2238	C₂×C₃₁	semidirect
116. #2155	C₂×C₃₁	semidirect
117. C₂×#361	C₂×C₃₁	semidirect
118. #2115	C₂×C₃₁	semidirect
140. #430	C₂²×C₃₁	semidirect
145. #365	C₂²×C₃₁	semidirect
147. #349	C₂²×C₃₁	semidirect
182. C₃₁	#33657	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×#3058	2
C₃₁×#2916	2
C₃₁×#2238	2
C₃₁×#2155	2
C₂×C₃₁×#361	2
C₃₁×#2115	2
#33657	31

Subgroups — C31×#33657 #21490416

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33657 #21490416