Subgroups — C₃₁×#33230 #21489989

Order 7936.

1034 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#489	C₂	1	4	yes	maximal, complemented
2	C₃₁×#2868	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2858	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1959	C₂	1	4	yes	maximal, complemented
5	C₃₁×#1957	C₂	1	2	yes	maximal
6	C₃₁×#1956	C₂	1	2	yes	maximal, complemented
7	C₄×C₃₁×#97	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₃₁×C₄⋊Q₈	—	2	2	no	—
9	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
10	C₂×C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
11	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
12	C₂×C₃₁×#124	C₂²	1	2	yes	—
13	C₂×C₃₁×#124	C₂²	1	2	yes	—
14	C₃₁×#489	—	2	16	no	—
15	C₂×C₃₁×#103	C₂²	1	8	yes	—
16	C₂×C₃₁×#102	C₂²	1	8	yes	—
17	C₂×C₃₁×#97	C₂²	1	2	yes	—
18	C₂×C₃₁×#97	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
20	C₃₁×#390	—	2	2	no	—
21	C₃₁×#390	—	2	2	no	—
22	C₃₁×#384	—	2	2	no	—
23	C₃₁×#384	C₂²	1	2	yes	—
24	C₃₁×#384	C₂²	1	2	yes	—
25	C₄³×C₃₁	C₂²	1	1	yes	characteristic
26	C₂²×Q₈×C₃₁	—	2	2	no	—
27	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
28	C₂²×D₄×C₃₁	—	2	2	no	—
29	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
30	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
31	C₃₁×C₄⋊Q₈	—	2	8	no	—
32	C₃₁×C₄⋊Q₈	—	2	4	no	—
33	C₃₁×C₄⋊Q₈	—	4	8	no	—
34	C₃₁×#127	—	2	4	no	—
35	C₃₁×#124	—	2	8	no	—
36	C₃₁×#124	—	2	8	no	—
37	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
38	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
39	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
40	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
41	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
42	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
43	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
44	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
45	C₂×C₄²×C₃₁	D₄	1	2	yes	—
46	C₂×C₄²×C₃₁	D₄	1	2	yes	—
47	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
49	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
50	C₃₁×#103	—	2	16	no	—
51	C₃₁×#102	—	2	16	no	—
52	C₃₁×#97	—	2	4	no	—
53	C₃₁×#97	D₄	1	8	yes	complemented
54	C₄×C₈×C₃₁	—	2	4	no	—
55	C₂⁴×C₃₁	—	2	2	no	—
56	C₂×Q₈×C₃₁	—	2	4	no	—
57	C₂×Q₈×C₃₁	—	2	4	no	—
58	C₂×Q₈×C₃₁	—	2	4	no	—
59	C₂×Q₈×C₃₁	—	4	8	no	—
60	C₂×Q₈×C₃₁	—	4	4	no	—
61	C₂×D₄×C₃₁	—	2	4	no	—
62	C₂×D₄×C₃₁	—	2	4	no	—
63	C₂×D₄×C₃₁	—	2	4	no	—
64	C₂×D₄×C₃₁	—	4	4	no	—
65	C₂×D₄×C₃₁	—	4	4	no	—
66	C₂×D₄×C₃₁	—	4	4	no	—
67	C₂²×C₄×C₃₁	—	2	2	no	—
68	C₂²×C₄×C₃₁	—	2	4	no	—
69	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
70	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
73	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
74	C₂×C₈×C₃₁	—	2	4	no	—
75	C₂×C₈×C₃₁	—	2	4	no	—
76	C₂×C₈×C₃₁	C₂×D₄	1	8	yes	—
77	C₄⋊C₄×C₃₁	—	2	8	no	—
78	C₄⋊C₄×C₃₁	—	4	8	no	—
79	C₄⋊C₄×C₃₁	—	4	4	no	—
80	C₄⋊C₄×C₃₁	—	4	4	no	—
81	C₄⋊C₄×C₃₁	—	4	8	no	—
82	C₂²⋊C₄×C₃₁	—	4	8	no	—
83	C₂²⋊C₄×C₃₁	—	4	4	no	—
84	C₂²⋊C₄×C₃₁	—	4	4	no	—
85	C₄²×C₃₁	—	2	4	no	—
86	C₄²×C₃₁	—	2	4	no	—
87	C₄²×C₃₁	C₂×D₄	1	4	yes	—
88	C₄²×C₃₁	#41	1	4	yes	—
89	C₄²×C₃₁	#41	1	4	yes	—
90	#33230	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
91	C₂³×C₃₁	—	4	2	no	—
92	C₂³×C₃₁	—	4	2	no	—
93	C₂³×C₃₁	—	4	2	no	—
94	C₂³×C₃₁	—	4	4	no	—
95	C₂³×C₃₁	—	4	4	no	—
96	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
97	Q₈×C₃₁	—	4	8	no	—
98	Q₈×C₃₁	—	8	8	no	—
99	D₄×C₃₁	—	4	8	no	—
100	D₄×C₃₁	—	8	8	no	—
101	C₂×C₄×C₃₁	—	2	2	no	—
102	C₂×C₄×C₃₁	—	2	2	no	—
103	C₂×C₄×C₃₁	—	2	2	no	—
104	C₂×C₄×C₃₁	—	2	2	no	—
105	C₂×C₄×C₃₁	—	4	2	no	—
106	C₂×C₄×C₃₁	—	4	2	no	—
107	C₂×C₄×C₃₁	—	4	2	no	—
108	C₂×C₄×C₃₁	—	4	4	no	—
109	C₂×C₄×C₃₁	—	4	8	no	—
110	C₂×C₄×C₃₁	#124	1	4	yes	—
111	C₂×C₄×C₃₁	#124	1	4	yes	—
112	C₂×C₄×C₃₁	#127	1	2	yes	—
113	C₂×C₄×C₃₁	#127	1	2	yes	—
114	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	4	yes	—
115	C₂×C₄×C₃₁	#137	1	4	yes	—
116	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
117	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
118	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
119	C₈×C₃₁	—	2	8	no	—
120	C₂×#489	C₂×C₃₁	1	4	yes	complemented
121	#2868	C₂×C₃₁	1	1	yes	characteristic, complemented
122	#2858	C₂×C₃₁	1	1	yes	characteristic
123	#1959	C₂×C₃₁	1	4	yes	complemented
124	#1957	C₂×C₃₁	1	2	yes	—
125	#1956	C₂×C₃₁	1	2	yes	complemented
126	C₄×#97	C₂×C₃₁	1	1	yes	characteristic, complemented
127	C₂²×C₃₁	—	8	2	no	—
128	C₂²×C₃₁	—	8	4	no	—
129	C₂²×C₃₁	—	8	4	no	—
130	C₂²×C₃₁	—	8	2	no	—
131	C₂²×C₃₁	—	8	2	no	—
132	C₂²×C₃₁	C₂×#124	1	1	yes	characteristic
133	C₂²×C₃₁	C₂×#124	1	1	yes	characteristic
134	C₂²×C₃₁	C₂×#127	1	1	yes	characteristic
135	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
136	C₂²×C₃₁	C₂×#137	1	2	yes	—
137	C₄×C₃₁	—	2	4	no	—
138	C₄×C₃₁	—	2	2	no	—
139	C₄×C₃₁	—	2	2	no	—
140	C₄×C₃₁	—	8	2	no	—
141	C₄×C₃₁	—	8	4	no	—
142	C₄×C₃₁	#489	1	8	yes	complemented
143	C₄×C₃₁	#532	1	4	yes	—
144	C₂×C₄⋊Q₈	—	2	2	no	—
145	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
146	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
147	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
148	C₂×#124	C₂²×C₃₁	1	2	yes	—
149	C₂×#124	C₂²×C₃₁	1	2	yes	—
150	#489	—	2	16	no	—
151	C₂×#103	C₂²×C₃₁	1	8	yes	—
152	C₂×#102	C₂²×C₃₁	1	8	yes	—
153	C₂×#97	C₂²×C₃₁	1	2	yes	—
154	C₂×#97	C₂²×C₃₁	1	2	yes	—
155	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
156	#390	—	2	2	no	—
157	#390	—	2	2	no	—
158	#384	—	2	2	no	—
159	#384	C₂²×C₃₁	1	2	yes	—
160	#384	C₂²×C₃₁	1	2	yes	—
161	C₄³	C₂²×C₃₁	1	1	yes	characteristic
162	C₂×C₃₁	—	16	2	no	—
163	C₂×C₃₁	#2858	1	1	yes	characteristic
164	C₂×C₃₁	C₂×#489	1	2	yes	—
165	C₂×C₃₁	#3141	1	2	yes	—
166	C₂×C₃₁	#3151	1	1	yes	characteristic
167	C₂×C₃₁	#3152	1	1	yes	characteristic
168	C₂²×Q₈	—	2	2	no	—
169	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
170	C₂²×D₄	—	2	2	no	—
171	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
172	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
173	C₄⋊Q₈	—	2	8	no	—
174	C₄⋊Q₈	—	2	4	no	—
175	C₄⋊Q₈	—	4	8	no	—
176	#127	—	2	4	no	—
177	#124	—	2	8	no	—
178	#124	—	2	8	no	—
179	C₂×C₄⋊C₄	—	2	4	no	—
180	C₂×C₄⋊C₄	—	2	4	no	—
181	C₂×C₄⋊C₄	—	2	2	no	—
182	C₂×C₄⋊C₄	—	2	2	no	—
183	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
184	C₂×C₂²⋊C₄	—	2	4	no	—
185	C₂×C₂²⋊C₄	—	2	2	no	—
186	C₂×C₂²⋊C₄	—	2	2	no	—
187	C₂×C₄²	D₄×C₃₁	1	2	yes	—
188	C₂×C₄²	D₄×C₃₁	1	2	yes	—
189	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
190	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
191	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
192	#103	—	2	16	no	—
193	#102	—	2	16	no	—
194	#97	—	2	4	no	—
195	#97	D₄×C₃₁	1	8	yes	complemented
196	C₄×C₈	—	2	4	no	—
197	C₃₁	#33230	1	1	yes	characteristic, direct product factor, Sylow
198	C₂⁴	—	2	2	no	—
199	C₂×Q₈	—	2	4	no	—
200	C₂×Q₈	—	2	4	no	—
201	C₂×Q₈	—	2	4	no	—
202	C₂×Q₈	—	4	8	no	—
203	C₂×Q₈	—	4	4	no	—
204	C₂×D₄	—	2	4	no	—
205	C₂×D₄	—	2	4	no	—
206	C₂×D₄	—	2	4	no	—
207	C₂×D₄	—	4	4	no	—
208	C₂×D₄	—	4	4	no	—
209	C₂×D₄	—	4	4	no	—
210	C₂²×C₄	—	2	2	no	—
211	C₂²×C₄	—	2	4	no	—
212	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
213	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
214	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
215	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
216	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
217	C₂×C₈	—	2	4	no	—
218	C₂×C₈	—	2	4	no	—
219	C₂×C₈	C₂×D₄×C₃₁	1	8	yes	—
220	C₄⋊C₄	—	2	8	no	—
221	C₄⋊C₄	—	4	8	no	—
222	C₄⋊C₄	—	4	4	no	—
223	C₄⋊C₄	—	4	4	no	—
224	C₄⋊C₄	—	4	8	no	—
225	C₂²⋊C₄	—	4	8	no	—
226	C₂²⋊C₄	—	4	4	no	—
227	C₂²⋊C₄	—	4	4	no	—
228	C₄²	—	2	4	no	—
229	C₄²	—	2	4	no	—
230	C₄²	C₂×D₄×C₃₁	1	4	yes	—
231	C₄²	#41×C₃₁	1	4	yes	—
232	C₄²	#41×C₃₁	1	4	yes	—
233	C₂³	—	4	2	no	—
234	C₂³	—	4	2	no	—
235	C₂³	—	4	2	no	—
236	C₂³	—	4	4	no	—
237	C₂³	—	4	4	no	—
238	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
239	Q₈	—	4	8	no	—
240	Q₈	—	8	8	no	—
241	D₄	—	4	8	no	—
242	D₄	—	8	8	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	—	4	2	no	—
249	C₂×C₄	—	4	2	no	—
250	C₂×C₄	—	4	4	no	—
251	C₂×C₄	—	4	8	no	—
252	C₂×C₄	C₃₁×#124	1	4	yes	—
253	C₂×C₄	C₃₁×#124	1	4	yes	—
254	C₂×C₄	C₃₁×#127	1	2	yes	—
255	C₂×C₄	C₃₁×#127	1	2	yes	—
256	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	4	yes	—
257	C₂×C₄	C₃₁×#137	1	4	yes	—
258	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
259	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
260	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
261	C₈	—	2	8	no	—
262	C₂²	—	8	2	no	—
263	C₂²	—	8	4	no	—
264	C₂²	—	8	4	no	—
265	C₂²	—	8	2	no	—
266	C₂²	—	8	2	no	—
267	C₂²	C₂×C₃₁×#124	1	1	yes	characteristic
268	C₂²	C₂×C₃₁×#124	1	1	yes	characteristic
269	C₂²	C₂×C₃₁×#127	1	1	yes	characteristic
270	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	2	yes	—
271	C₂²	C₂×C₃₁×#137	1	2	yes	—
272	C₄	—	2	4	no	—
273	C₄	—	2	2	no	—
274	C₄	—	2	2	no	—
275	C₄	—	8	2	no	—
276	C₄	—	8	4	no	—
277	C₄	C₃₁×#489	1	8	yes	complemented
278	C₄	C₃₁×#532	1	4	yes	—
279	C₂	—	16	2	no	—
280	C₂	C₃₁×#2858	1	1	yes	characteristic
281	C₂	C₂×C₃₁×#489	1	2	yes	—
282	C₂	C₃₁×#3141	1	2	yes	—
283	C₂	C₃₁×#3151	1	1	yes	characteristic
284	C₂	C₃₁×#3152	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂×C₃₁×#127	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×#489	C₂	semidirect
2. C₃₁×#2868	C₂	semidirect
4. C₃₁×#1959	C₂	semidirect
6. C₃₁×#1956	C₂	semidirect
7. C₄×C₃₁×#97	C₂	semidirect
53. C₃₁×#97	D₄	semidirect
90. #33230	C₃₁	direct (N × H)
120. C₂×#489	C₂×C₃₁	semidirect
121. #2868	C₂×C₃₁	semidirect
123. #1959	C₂×C₃₁	semidirect
125. #1956	C₂×C₃₁	semidirect
126. C₄×#97	C₂×C₃₁	semidirect
142. C₄×C₃₁	#489	semidirect
195. #97	D₄×C₃₁	semidirect
197. C₃₁	#33230	direct (N × H)
277. C₄	C₃₁×#489	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#489	2
C₃₁×#2868	2
C₃₁×#2858	2
C₃₁×#1959	2
C₃₁×#1957	2
C₃₁×#1956	2
C₄×C₃₁×#97	2
#33230	31

Subgroups — C31×#33230 #21489989

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33230 #21489989