Subgroups — C₃₁×#33220 #21489979

Order 7936.

738 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#498	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3079	C₂	1	2	yes	maximal
3	C₄×C₃₁×#131	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1997	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1951	C₂	1	4	yes	maximal, complemented
6	C₃₁×#1848	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1815	C₂	1	4	yes	maximal
8	C₂×C₃₁×#131	C₂²	1	2	yes	—
9	C₂×C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
10	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
11	C₃₁×#498	—	2	1	no	—
12	C₃₁×#498	—	2	1	no	—
13	C₃₁×#498	—	2	2	no	—
14	C₃₁×#477	—	2	2	no	—
15	C₃₁×#476	—	2	2	no	—
16	C₃₁×#475	—	2	2	no	—
17	C₃₁×#474	—	2	2	no	—
18	C₂×C₃₁×#108	C₂²	1	4	yes	—
19	C₃₁×#428	C₂²	1	4	yes	—
20	C₂×C₃₁×#106	C₂²	1	4	yes	—
21	C₃₁×#423	C₂²	1	2	yes	—
22	C₃₁×#417	C₂²	1	4	yes	—
23	C₃₁×#405	—	2	2	no	—
24	C₃₁×#405	C₂²	1	2	yes	—
25	C₄×M₁₆×C₃₁	—	2	1	no	—
26	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₈×C₃₁	—	2	1	no	—
28	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
29	C₃₁×#343	C₂²	1	8	yes	—
30	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
31	C₂²×Q₈×C₃₁	—	2	2	no	—
32	C₃₁×#131	—	2	4	no	—
33	C₃₁×#131	Q₈	1	8	yes	complemented
34	C₂×M₁₆×C₃₁	—	2	2	no	—
35	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
36	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
37	C₂²×C₈×C₃₁	—	2	2	no	—
38	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
39	C₃₁×C₄⋊Q₈	—	2	2	no	—
40	C₃₁×C₄⋊Q₈	—	2	2	no	—
41	C₃₁×C₄⋊Q₈	—	2	2	no	—
42	C₃₁×C₄⋊Q₈	—	2	2	no	—
43	C₄×Q₈×C₃₁	D₄	1	2	yes	—
44	C₄×D₄×C₃₁	—	2	2	no	—
45	C₄×D₄×C₃₁	D₄	1	2	yes	—
46	C₃₁×#117	—	2	1	no	—
47	C₃₁×#117	C₂³	1	1	yes	characteristic
48	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
49	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
50	C₂×C₄²×C₃₁	—	2	1	no	—
51	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
52	C₃₁×#108	—	2	8	no	—
53	C₃₁×#107	—	2	4	no	—
54	C₃₁×#106	—	2	4	no	—
55	C₃₁×#106	—	2	4	no	—
56	C₃₁×#106	—	2	4	no	—
57	C₃₁×C₄⋊C₈	—	2	2	no	—
58	C₃₁×C₄⋊C₈	—	2	2	no	—
59	C₃₁×#103	—	2	4	no	—
60	C₃₁×#102	—	2	4	no	—
61	C₃₁×#97	—	2	2	no	—
62	C₃₁×#97	—	2	1	no	—
63	C₄×C₈×C₃₁	—	2	2	no	—
64	C₄×C₈×C₃₁	—	2	1	no	—
65	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
66	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
67	C₄×C₈×C₃₁	D₄	1	2	yes	—
68	#41×C₃₁	—	2	2	no	—
69	#41×C₃₁	C₂×Q₈	1	4	yes	—
70	C₂×Q₈×C₃₁	—	4	2	no	—
71	C₂×Q₈×C₃₁	—	4	2	no	—
72	C₂×Q₈×C₃₁	—	8	4	no	—
73	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
74	C₂×D₄×C₃₁	—	2	1	no	—
75	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₃₁	—	2	1	no	—
77	C₂²×C₄×C₃₁	—	2	2	no	—
78	C₂²×C₄×C₃₁	—	2	4	no	—
79	C₂²×C₄×C₃₁	#41	1	2	yes	—
80	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
81	M₁₆×C₃₁	—	2	2	no	—
82	M₁₆×C₃₁	—	2	8	no	—
83	M₁₆×C₃₁	—	2	8	no	—
84	M₁₆×C₃₁	C₂×Q₈	1	4	yes	—
85	C₂×C₈×C₃₁	—	2	2	no	—
86	C₂×C₈×C₃₁	—	2	8	no	—
87	C₂×C₈×C₃₁	—	2	4	no	—
88	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
89	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
90	C₂×C₈×C₃₁	C₂×Q₈	1	4	yes	—
91	C₂×C₈×C₃₁	#41	1	4	yes	—
92	C₄⋊C₄×C₃₁	—	2	2	no	—
93	C₄⋊C₄×C₃₁	—	2	4	no	—
94	C₄⋊C₄×C₃₁	—	2	4	no	—
95	C₄⋊C₄×C₃₁	—	4	2	no	—
96	C₄⋊C₄×C₃₁	—	4	2	no	—
97	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
98	C₂²⋊C₄×C₃₁	—	2	1	no	—
99	C₂²⋊C₄×C₃₁	—	2	2	no	—
100	C₄²×C₃₁	—	2	2	no	—
101	C₄²×C₃₁	—	2	1	no	—
102	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
103	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
104	C₄²×C₃₁	C₂×D₄	1	2	yes	—
105	#33220	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
106	C₂³×C₃₁	—	2	1	no	—
107	C₂³×C₃₁	C₂×#41	1	1	yes	characteristic
108	Q₈×C₃₁	—	2	1	no	—
109	Q₈×C₃₁	—	8	4	no	—
110	Q₈×C₃₁	—	8	4	no	—
111	Q₈×C₃₁	C₄⋊Q₈	1	2	yes	—
112	D₄×C₃₁	—	2	4	no	—
113	D₄×C₃₁	—	2	1	no	—
114	D₄×C₃₁	C₄⋊Q₈	1	2	yes	—
115	C₂×C₄×C₃₁	—	2	1	no	—
116	C₂×C₄×C₃₁	—	2	4	no	—
117	C₂×C₄×C₃₁	—	2	4	no	—
118	C₂×C₄×C₃₁	—	2	2	no	—
119	C₂×C₄×C₃₁	—	4	4	no	—
120	C₂×C₄×C₃₁	—	4	4	no	—
121	C₂×C₄×C₃₁	—	8	4	no	—
122	C₂×C₄×C₃₁	#121	1	2	yes	—
123	C₂×C₄×C₃₁	#121	1	2	yes	—
124	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
125	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
126	C₂×C₄×C₃₁	C₂²×Q₈	1	2	yes	—
127	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
128	C₈×C₃₁	—	2	4	no	—
129	C₈×C₃₁	—	4	4	no	—
130	C₈×C₃₁	C₂²⋊Q₈	1	4	yes	—
131	C₈×C₃₁	C₂²⋊Q₈	1	4	yes	—
132	C₂×#498	C₂×C₃₁	1	1	yes	characteristic, complemented
133	#3079	C₂×C₃₁	1	2	yes	—
134	C₄×#131	C₂×C₃₁	1	1	yes	characteristic
135	#1997	C₂×C₃₁	1	2	yes	complemented
136	#1951	C₂×C₃₁	1	4	yes	complemented
137	#1848	C₂×C₃₁	1	1	yes	characteristic, complemented
138	#1815	C₂×C₃₁	1	4	yes	—
139	C₂²×C₃₁	—	2	2	no	—
140	C₂²×C₃₁	—	2	2	no	—
141	C₂²×C₃₁	—	4	2	no	—
142	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
143	C₂²×C₃₁	#533	1	2	yes	—
144	C₄×C₃₁	—	2	2	no	—
145	C₄×C₃₁	—	2	2	no	—
146	C₄×C₃₁	—	4	2	no	—
147	C₄×C₃₁	—	8	4	no	—
148	C₄×C₃₁	#471	1	4	yes	—
149	C₄×C₃₁	C₂×C₂²⋊Q₈	1	2	yes	—
150	C₄×C₃₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
151	C₄×C₃₁	#532	1	1	yes	characteristic
152	C₂×#131	C₂²×C₃₁	1	2	yes	—
153	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
154	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
155	#498	—	2	1	no	—
156	#498	—	2	1	no	—
157	#498	—	2	2	no	—
158	#477	—	2	2	no	—
159	#476	—	2	2	no	—
160	#475	—	2	2	no	—
161	#474	—	2	2	no	—
162	C₂×#108	C₂²×C₃₁	1	4	yes	—
163	#428	C₂²×C₃₁	1	4	yes	—
164	C₂×#106	C₂²×C₃₁	1	4	yes	—
165	#423	C₂²×C₃₁	1	2	yes	—
166	#417	C₂²×C₃₁	1	4	yes	—
167	#405	—	2	2	no	—
168	#405	C₂²×C₃₁	1	2	yes	—
169	C₄×M₁₆	—	2	1	no	—
170	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
171	C₂×C₄×C₈	—	2	1	no	—
172	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
173	#343	C₂²×C₃₁	1	8	yes	—
174	C₂×C₃₁	—	2	2	no	—
175	C₂×C₃₁	—	4	2	no	—
176	C₂×C₃₁	#2857	1	1	yes	characteristic
177	C₂×C₃₁	C₂×#471	1	2	yes	—
178	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
179	C₂²×Q₈	—	2	2	no	—
180	#131	—	2	4	no	—
181	#131	Q₈×C₃₁	1	8	yes	complemented
182	C₂×M₁₆	—	2	2	no	—
183	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
184	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
185	C₂²×C₈	—	2	2	no	—
186	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
187	C₄⋊Q₈	—	2	2	no	—
188	C₄⋊Q₈	—	2	2	no	—
189	C₄⋊Q₈	—	2	2	no	—
190	C₄⋊Q₈	—	2	2	no	—
191	C₄×Q₈	D₄×C₃₁	1	2	yes	—
192	C₄×D₄	—	2	2	no	—
193	C₄×D₄	D₄×C₃₁	1	2	yes	—
194	#117	—	2	1	no	—
195	#117	C₂³×C₃₁	1	1	yes	characteristic
196	C₂×C₄⋊C₄	—	2	2	no	—
197	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
198	C₂×C₄²	—	2	1	no	—
199	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
200	#108	—	2	8	no	—
201	#107	—	2	4	no	—
202	#106	—	2	4	no	—
203	#106	—	2	4	no	—
204	#106	—	2	4	no	—
205	C₄⋊C₈	—	2	2	no	—
206	C₄⋊C₈	—	2	2	no	—
207	#103	—	2	4	no	—
208	#102	—	2	4	no	—
209	#97	—	2	2	no	—
210	#97	—	2	1	no	—
211	C₄×C₈	—	2	2	no	—
212	C₄×C₈	—	2	1	no	—
213	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
214	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
215	C₄×C₈	D₄×C₃₁	1	2	yes	—
216	C₃₁	#33220	1	1	yes	characteristic, direct product factor, Sylow
217	#41	—	2	2	no	—
218	#41	C₂×Q₈×C₃₁	1	4	yes	—
219	C₂×Q₈	—	4	2	no	—
220	C₂×Q₈	—	4	2	no	—
221	C₂×Q₈	—	8	4	no	—
222	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
223	C₂×D₄	—	2	1	no	—
224	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
225	C₂²×C₄	—	2	1	no	—
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	—	2	4	no	—
228	C₂²×C₄	#41×C₃₁	1	2	yes	—
229	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
230	M₁₆	—	2	2	no	—
231	M₁₆	—	2	8	no	—
232	M₁₆	—	2	8	no	—
233	M₁₆	C₂×Q₈×C₃₁	1	4	yes	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	2	8	no	—
236	C₂×C₈	—	2	4	no	—
237	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
238	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
239	C₂×C₈	C₂×Q₈×C₃₁	1	4	yes	—
240	C₂×C₈	#41×C₃₁	1	4	yes	—
241	C₄⋊C₄	—	2	2	no	—
242	C₄⋊C₄	—	2	4	no	—
243	C₄⋊C₄	—	2	4	no	—
244	C₄⋊C₄	—	4	2	no	—
245	C₄⋊C₄	—	4	2	no	—
246	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
247	C₂²⋊C₄	—	2	1	no	—
248	C₂²⋊C₄	—	2	2	no	—
249	C₄²	—	2	2	no	—
250	C₄²	—	2	1	no	—
251	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
252	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
253	C₄²	C₂×D₄×C₃₁	1	2	yes	—
254	C₂³	—	2	1	no	—
255	C₂³	C₂×#41×C₃₁	1	1	yes	characteristic
256	Q₈	—	2	1	no	—
257	Q₈	—	8	4	no	—
258	Q₈	—	8	4	no	—
259	Q₈	C₃₁×C₄⋊Q₈	1	2	yes	—
260	D₄	—	2	4	no	—
261	D₄	—	2	1	no	—
262	D₄	C₃₁×C₄⋊Q₈	1	2	yes	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	4	4	no	—
268	C₂×C₄	—	4	4	no	—
269	C₂×C₄	—	8	4	no	—
270	C₂×C₄	C₃₁×#121	1	2	yes	—
271	C₂×C₄	C₃₁×#121	1	2	yes	—
272	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
273	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
274	C₂×C₄	C₂²×Q₈×C₃₁	1	2	yes	—
275	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
276	C₈	—	2	4	no	—
277	C₈	—	4	4	no	—
278	C₈	C₃₁×C₂²⋊Q₈	1	4	yes	—
279	C₈	C₃₁×C₂²⋊Q₈	1	4	yes	—
280	C₂²	—	2	2	no	—
281	C₂²	—	2	2	no	—
282	C₂²	—	4	2	no	—
283	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
284	C₂²	C₃₁×#533	1	2	yes	—
285	C₄	—	2	2	no	—
286	C₄	—	2	2	no	—
287	C₄	—	4	2	no	—
288	C₄	—	8	4	no	—
289	C₄	C₃₁×#471	1	4	yes	—
290	C₄	C₂×C₃₁×C₂²⋊Q₈	1	2	yes	—
291	C₄	C₂×C₃₁×C₄⋊Q₈	1	1	yes	characteristic
292	C₄	C₃₁×#532	1	1	yes	characteristic
293	C₂	—	2	2	no	—
294	C₂	—	4	2	no	—
295	C₂	C₃₁×#2857	1	1	yes	characteristic
296	C₂	C₂×C₃₁×#471	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂×#41×C₃₁	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#498	C₂	semidirect
4. C₃₁×#1997	C₂	semidirect
5. C₃₁×#1951	C₂	semidirect
6. C₃₁×#1848	C₂	semidirect
33. C₃₁×#131	Q₈	semidirect
105. #33220	C₃₁	direct (N × H)
132. C₂×#498	C₂×C₃₁	semidirect
135. #1997	C₂×C₃₁	semidirect
136. #1951	C₂×C₃₁	semidirect
137. #1848	C₂×C₃₁	semidirect
181. #131	Q₈×C₃₁	semidirect
216. C₃₁	#33220	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×#498	2
C₃₁×#3079	2
C₄×C₃₁×#131	2
C₃₁×#1997	2
C₃₁×#1951	2
C₃₁×#1848	2
C₃₁×#1815	2
#33220	31

Subgroups — C31×#33220 #21489979

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33220 #21489979