Subgroups — C₃₁×#33253 #21490012

Order 7936.

922 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#499	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#495	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₁×#488	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2861	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1973	C₂	1	4	yes	maximal, complemented
6	C₃₁×#1972	C₂	1	2	yes	maximal
7	C₃₁×#1971	C₂	1	2	yes	maximal, complemented
8	C₄×C₃₁×#97	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
10	C₂²×SD₁₆×C₃₁	C₂²	1	2	yes	—
11	C₂²×D₈×C₃₁	C₂²	1	1	yes	characteristic
12	C₂×C₃₁×#125	C₂²	1	2	yes	—
13	C₂×C₃₁×#124	C₂²	1	2	yes	—
14	C₃₁×#499	—	2	4	no	—
15	C₃₁×#495	—	2	4	no	—
16	C₃₁×#488	—	2	8	no	—
17	C₂×C₃₁×#107	C₂²	1	2	yes	—
18	C₂×C₃₁×#106	C₂²	1	2	yes	—
19	C₂×C₃₁×#103	C₂²	1	4	yes	—
20	C₂×C₃₁×#102	C₂²	1	4	yes	—
21	C₂×C₃₁×#97	C₂²	1	2	yes	—
22	C₂×C₃₁×#97	C₂²	1	2	yes	—
23	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
25	C₃₁×#391	C₂²	1	4	yes	—
26	C₃₁×#390	C₂²	1	4	yes	—
27	C₃₁×#388	—	2	8	no	—
28	C₄³×C₃₁	C₂²	1	1	yes	characteristic
29	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
30	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
31	C₂×Q₁₆×C₃₁	—	2	2	no	—
32	C₂×Q₁₆×C₃₁	—	2	4	no	—
33	C₂×SD₁₆×C₃₁	—	2	4	no	—
34	C₂×SD₁₆×C₃₁	—	2	8	no	—
35	C₂×D₈×C₃₁	—	2	2	no	—
36	C₂×D₈×C₃₁	—	2	4	no	—
37	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
38	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
39	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
40	C₃₁×#125	—	2	4	no	—
41	C₃₁×#125	—	2	4	no	—
42	C₃₁×#124	—	2	4	no	—
43	C₃₁×#124	—	2	4	no	—
44	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
45	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
46	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
47	C₂×C₂²⋊C₄×C₃₁	—	2	8	no	—
48	C₂×C₄²×C₃₁	D₄	1	4	yes	—
49	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
50	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
51	C₃₁×#107	—	2	4	no	—
52	C₃₁×#106	—	2	4	no	—
53	C₃₁×#103	—	2	8	no	—
54	C₃₁×#102	—	2	8	no	—
55	C₃₁×#97	—	2	4	no	—
56	C₃₁×#97	—	2	4	no	—
57	C₄×C₈×C₃₁	D₄	1	4	yes	—
58	C₄×C₈×C₃₁	D₄	1	4	yes	—
59	C₃₁×#95	—	2	8	no	—
60	C₂⁴×C₃₁	—	2	2	no	—
61	C₂×Q₈×C₃₁	—	2	2	no	—
62	C₂×Q₈×C₃₁	—	2	2	no	—
63	C₂×Q₈×C₃₁	—	2	4	no	—
64	C₂×Q₈×C₃₁	—	2	4	no	—
65	C₂×D₄×C₃₁	—	2	2	no	—
66	C₂×D₄×C₃₁	—	2	2	no	—
67	C₂×D₄×C₃₁	—	2	4	no	—
68	C₂×D₄×C₃₁	—	2	4	no	—
69	C₂²×C₄×C₃₁	—	2	2	no	—
70	C₂²×C₄×C₃₁	—	2	4	no	—
71	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
72	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
73	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
74	Q₁₆×C₃₁	—	4	4	no	—
75	SD₁₆×C₃₁	—	4	8	no	—
76	D₈×C₃₁	—	4	4	no	—
77	C₂×C₈×C₃₁	—	2	2	no	—
78	C₂×C₈×C₃₁	—	2	2	no	—
79	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
80	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
81	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
82	C₂×C₈×C₃₁	#41	1	4	yes	—
83	C₂×C₈×C₃₁	#41	1	4	yes	—
84	C₄⋊C₄×C₃₁	—	2	4	no	—
85	C₄⋊C₄×C₃₁	—	2	4	no	—
86	C₄⋊C₄×C₃₁	—	4	8	no	—
87	C₂²⋊C₄×C₃₁	—	4	16	no	—
88	C₂²⋊C₄×C₃₁	—	4	8	no	—
89	C₄²×C₃₁	—	2	2	no	—
90	C₄²×C₃₁	—	2	2	no	—
91	C₄²×C₃₁	—	4	4	no	—
92	C₄²×C₃₁	C₂×D₄	1	2	yes	—
93	C₄²×C₃₁	C₂×D₄	1	2	yes	—
94	#33253	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
95	C₂³×C₃₁	—	4	2	no	—
96	C₂³×C₃₁	—	4	4	no	—
97	C₂³×C₃₁	—	4	8	no	—
98	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
99	Q₈×C₃₁	—	4	2	no	—
100	Q₈×C₃₁	—	4	2	no	—
101	Q₈×C₃₁	—	4	4	no	—
102	D₄×C₃₁	—	4	2	no	—
103	D₄×C₃₁	—	4	2	no	—
104	D₄×C₃₁	—	4	4	no	—
105	C₂×C₄×C₃₁	—	2	4	no	—
106	C₂×C₄×C₃₁	—	2	4	no	—
107	C₂×C₄×C₃₁	—	2	8	no	—
108	C₂×C₄×C₃₁	—	4	2	no	—
109	C₂×C₄×C₃₁	—	4	4	no	—
110	C₂×C₄×C₃₁	—	4	4	no	—
111	C₂×C₄×C₃₁	—	4	8	no	—
112	C₂×C₄×C₃₁	#127	1	4	yes	—
113	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
114	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
115	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
116	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
117	C₈×C₃₁	—	2	4	no	—
118	C₈×C₃₁	#124	1	4	yes	—
119	C₈×C₃₁	#124	1	4	yes	—
120	C₂×#499	C₂×C₃₁	1	1	yes	characteristic, complemented
121	C₂×#495	C₂×C₃₁	1	1	yes	characteristic
122	C₂×#488	C₂×C₃₁	1	2	yes	complemented
123	#2861	C₂×C₃₁	1	2	yes	complemented
124	#1973	C₂×C₃₁	1	4	yes	complemented
125	#1972	C₂×C₃₁	1	2	yes	—
126	#1971	C₂×C₃₁	1	2	yes	complemented
127	C₄×#97	C₂×C₃₁	1	1	yes	characteristic, complemented
128	C₂²×C₃₁	—	8	2	no	—
129	C₂²×C₃₁	—	8	4	no	—
130	C₂²×C₃₁	—	8	8	no	—
131	C₂²×C₃₁	C₂×#127	1	1	yes	characteristic
132	C₂²×C₃₁	#532	1	2	yes	—
133	C₂²×C₃₁	#575	1	4	yes	—
134	C₄×C₃₁	—	2	4	no	—
135	C₄×C₃₁	—	4	4	no	—
136	C₄×C₃₁	—	8	2	no	—
137	C₄×C₃₁	—	8	4	no	—
138	C₄×C₃₁	C₂×#124	1	1	yes	characteristic
139	C₄×C₃₁	C₂×#124	1	1	yes	characteristic
140	C₄×C₃₁	#532	1	2	yes	—
141	C₂²×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
142	C₂²×SD₁₆	C₂²×C₃₁	1	2	yes	—
143	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic
144	C₂×#125	C₂²×C₃₁	1	2	yes	—
145	C₂×#124	C₂²×C₃₁	1	2	yes	—
146	#499	—	2	4	no	—
147	#495	—	2	4	no	—
148	#488	—	2	8	no	—
149	C₂×#107	C₂²×C₃₁	1	2	yes	—
150	C₂×#106	C₂²×C₃₁	1	2	yes	—
151	C₂×#103	C₂²×C₃₁	1	4	yes	—
152	C₂×#102	C₂²×C₃₁	1	4	yes	—
153	C₂×#97	C₂²×C₃₁	1	2	yes	—
154	C₂×#97	C₂²×C₃₁	1	2	yes	—
155	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
156	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
157	#391	C₂²×C₃₁	1	4	yes	—
158	#390	C₂²×C₃₁	1	4	yes	—
159	#388	—	2	8	no	—
160	C₄³	C₂²×C₃₁	1	1	yes	characteristic
161	C₂×C₃₁	—	16	2	no	—
162	C₂×C₃₁	#2858	1	1	yes	characteristic
163	C₂×C₃₁	#3150	1	2	yes	—
164	C₂×C₃₁	#3183	1	4	yes	—
165	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
166	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
167	C₂×Q₁₆	—	2	2	no	—
168	C₂×Q₁₆	—	2	4	no	—
169	C₂×SD₁₆	—	2	4	no	—
170	C₂×SD₁₆	—	2	8	no	—
171	C₂×D₈	—	2	2	no	—
172	C₂×D₈	—	2	4	no	—
173	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
174	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
175	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
176	#125	—	2	4	no	—
177	#125	—	2	4	no	—
178	#124	—	2	4	no	—
179	#124	—	2	4	no	—
180	C₂×C₄⋊C₄	—	2	4	no	—
181	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
182	C₂×C₂²⋊C₄	—	2	4	no	—
183	C₂×C₂²⋊C₄	—	2	8	no	—
184	C₂×C₄²	D₄×C₃₁	1	4	yes	—
185	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
186	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
187	#107	—	2	4	no	—
188	#106	—	2	4	no	—
189	#103	—	2	8	no	—
190	#102	—	2	8	no	—
191	#97	—	2	4	no	—
192	#97	—	2	4	no	—
193	C₄×C₈	D₄×C₃₁	1	4	yes	—
194	C₄×C₈	D₄×C₃₁	1	4	yes	—
195	#95	—	2	8	no	—
196	C₃₁	#33253	1	1	yes	characteristic, direct product factor, Sylow
197	C₂⁴	—	2	2	no	—
198	C₂×Q₈	—	2	2	no	—
199	C₂×Q₈	—	2	2	no	—
200	C₂×Q₈	—	2	4	no	—
201	C₂×Q₈	—	2	4	no	—
202	C₂×D₄	—	2	2	no	—
203	C₂×D₄	—	2	2	no	—
204	C₂×D₄	—	2	4	no	—
205	C₂×D₄	—	2	4	no	—
206	C₂²×C₄	—	2	2	no	—
207	C₂²×C₄	—	2	4	no	—
208	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
209	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
210	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
211	Q₁₆	—	4	4	no	—
212	SD₁₆	—	4	8	no	—
213	D₈	—	4	4	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	2	2	no	—
216	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
217	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
218	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
219	C₂×C₈	#41×C₃₁	1	4	yes	—
220	C₂×C₈	#41×C₃₁	1	4	yes	—
221	C₄⋊C₄	—	2	4	no	—
222	C₄⋊C₄	—	2	4	no	—
223	C₄⋊C₄	—	4	8	no	—
224	C₂²⋊C₄	—	4	16	no	—
225	C₂²⋊C₄	—	4	8	no	—
226	C₄²	—	2	2	no	—
227	C₄²	—	2	2	no	—
228	C₄²	—	4	4	no	—
229	C₄²	C₂×D₄×C₃₁	1	2	yes	—
230	C₄²	C₂×D₄×C₃₁	1	2	yes	—
231	C₂³	—	4	2	no	—
232	C₂³	—	4	4	no	—
233	C₂³	—	4	8	no	—
234	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
235	Q₈	—	4	2	no	—
236	Q₈	—	4	2	no	—
237	Q₈	—	4	4	no	—
238	D₄	—	4	2	no	—
239	D₄	—	4	2	no	—
240	D₄	—	4	4	no	—
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	2	8	no	—
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	—	4	4	no	—
246	C₂×C₄	—	4	4	no	—
247	C₂×C₄	—	4	8	no	—
248	C₂×C₄	C₃₁×#127	1	4	yes	—
249	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
250	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
251	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
252	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
253	C₈	—	2	4	no	—
254	C₈	C₃₁×#124	1	4	yes	—
255	C₈	C₃₁×#124	1	4	yes	—
256	C₂²	—	8	2	no	—
257	C₂²	—	8	4	no	—
258	C₂²	—	8	8	no	—
259	C₂²	C₂×C₃₁×#127	1	1	yes	characteristic
260	C₂²	C₃₁×#532	1	2	yes	—
261	C₂²	C₃₁×#575	1	4	yes	—
262	C₄	—	2	4	no	—
263	C₄	—	4	4	no	—
264	C₄	—	8	2	no	—
265	C₄	—	8	4	no	—
266	C₄	C₂×C₃₁×#124	1	1	yes	characteristic
267	C₄	C₂×C₃₁×#124	1	1	yes	characteristic
268	C₄	C₃₁×#532	1	2	yes	—
269	C₂	—	16	2	no	—
270	C₂	C₃₁×#2858	1	1	yes	characteristic
271	C₂	C₃₁×#3150	1	2	yes	—
272	C₂	C₃₁×#3183	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂²×D₈×C₃₁	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₄³×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#499	C₂	semidirect
3. C₂×C₃₁×#488	C₂	semidirect
4. C₃₁×#2861	C₂	semidirect
5. C₃₁×#1973	C₂	semidirect
7. C₃₁×#1971	C₂	semidirect
8. C₄×C₃₁×#97	C₂	semidirect
94. #33253	C₃₁	direct (N × H)
120. C₂×#499	C₂×C₃₁	semidirect
122. C₂×#488	C₂×C₃₁	semidirect
123. #2861	C₂×C₃₁	semidirect
124. #1973	C₂×C₃₁	semidirect
126. #1971	C₂×C₃₁	semidirect
127. C₄×#97	C₂×C₃₁	semidirect
196. C₃₁	#33253	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#499	2
C₂×C₃₁×#495	2
C₂×C₃₁×#488	2
C₃₁×#2861	2
C₃₁×#1973	2
C₃₁×#1972	2
C₃₁×#1971	2
C₄×C₃₁×#97	2
#33253	31

Subgroups — C31×#33253 #21490012

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33253 #21490012