Subgroups — C₃₁×#33191 #21489950

Order 7936.

978 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#494	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#473	C₂	1	4	yes	maximal
3	C₃₁×#3039	C₂	1	2	yes	maximal
4	C₄×C₃₁×#131	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#1979	C₂	1	4	yes	maximal, complemented
6	C₃₁×#1966	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1848	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₃₁×#137	C₂²	1	4	yes	—
9	C₂²×Q₁₆×C₃₁	C₂²	1	4	yes	—
10	C₂×C₃₁×#131	C₂²	1	2	yes	—
11	C₂×C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
12	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
13	C₃₁×#494	—	2	1	no	—
14	C₃₁×#494	—	2	1	no	—
15	C₃₁×#494	—	2	2	no	—
16	C₃₁×#473	—	2	16	no	—
17	C₃₁×#462	—	2	4	no	—
18	C₃₁×#461	—	2	4	no	—
19	C₂×C₃₁×#108	C₂²	1	4	yes	—
20	C₃₁×#423	C₂²	1	2	yes	—
21	C₃₁×#419	C₂²	1	4	yes	—
22	C₂×C₃₁×#101	C₂²	1	8	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	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
30	C₂²×Q₈×C₃₁	C₂³	1	4	yes	—
31	C₃₁×#137	—	2	16	no	—
32	C₂×Q₁₆×C₃₁	—	2	4	no	—
33	C₂×Q₁₆×C₃₁	—	2	8	no	—
34	C₂×Q₁₆×C₃₁	—	2	4	no	—
35	C₂×Q₁₆×C₃₁	—	2	4	no	—
36	C₂×Q₁₆×C₃₁	—	2	8	no	—
37	C₂×SD₁₆×C₃₁	—	2	4	no	—
38	C₃₁×#131	—	2	4	no	—
39	C₃₁×#131	D₄	1	8	yes	—
40	C₂×M₁₆×C₃₁	—	2	2	no	—
41	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
42	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
43	C₂²×C₈×C₃₁	—	2	2	no	—
44	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
45	C₃₁×C₄⋊Q₈	—	2	4	no	—
46	C₃₁×C₄⋊Q₈	—	2	2	no	—
47	C₃₁×C₄⋊Q₈	—	2	2	no	—
48	C₄×Q₈×C₃₁	D₄	1	2	yes	—
49	C₄×D₄×C₃₁	—	2	2	no	—
50	C₄×D₄×C₃₁	D₄	1	2	yes	—
51	C₃₁×#117	—	2	1	no	—
52	C₃₁×#117	C₂³	1	1	yes	characteristic
53	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
54	C₂×C₄²×C₃₁	—	2	1	no	—
55	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
56	C₃₁×#108	—	2	8	no	—
57	C₃₁×C₄⋊C₈	—	2	4	no	—
58	C₃₁×#103	—	2	8	no	—
59	C₃₁×#101	—	2	16	no	—
60	C₃₁×#97	—	2	2	no	—
61	C₃₁×#97	—	2	1	no	—
62	C₄×C₈×C₃₁	—	2	2	no	—
63	C₄×C₈×C₃₁	—	2	1	no	—
64	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
65	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
66	C₄×C₈×C₃₁	D₄	1	2	yes	—
67	#41×C₃₁	—	2	2	no	—
68	#41×C₃₁	C₂×D₄	1	4	yes	—
69	C₂×Q₈×C₃₁	—	2	8	no	—
70	C₂×Q₈×C₃₁	—	2	4	no	—
71	C₂×Q₈×C₃₁	—	2	4	no	—
72	C₂×Q₈×C₃₁	—	2	8	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	Q₁₆×C₃₁	—	4	8	no	—
82	Q₁₆×C₃₁	—	4	8	no	—
83	Q₁₆×C₃₁	—	4	8	no	—
84	SD₁₆×C₃₁	—	4	8	no	—
85	M₁₆×C₃₁	—	2	2	no	—
86	M₁₆×C₃₁	—	2	8	no	—
87	M₁₆×C₃₁	—	2	8	no	—
88	M₁₆×C₃₁	C₂×D₄	1	4	yes	—
89	C₂×C₈×C₃₁	—	2	2	no	—
90	C₂×C₈×C₃₁	—	2	8	no	—
91	C₂×C₈×C₃₁	—	2	4	no	—
92	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
93	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
94	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
95	C₂×C₈×C₃₁	#41	1	4	yes	—
96	C₄⋊C₄×C₃₁	—	2	2	no	—
97	C₄⋊C₄×C₃₁	—	4	4	no	—
98	C₄⋊C₄×C₃₁	—	4	4	no	—
99	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
100	C₂²⋊C₄×C₃₁	—	2	1	no	—
101	C₂²⋊C₄×C₃₁	—	2	2	no	—
102	C₄²×C₃₁	—	2	2	no	—
103	C₄²×C₃₁	—	2	1	no	—
104	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
105	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
106	C₄²×C₃₁	C₂×D₄	1	2	yes	—
107	#33191	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
108	C₂³×C₃₁	—	2	1	no	—
109	C₂³×C₃₁	C₂×#41	1	1	yes	characteristic
110	Q₈×C₃₁	—	2	1	no	—
111	Q₈×C₃₁	—	4	8	no	—
112	Q₈×C₃₁	—	4	4	no	—
113	Q₈×C₃₁	—	4	4	no	—
114	Q₈×C₃₁	#127	1	2	yes	—
115	D₄×C₃₁	—	2	4	no	—
116	D₄×C₃₁	—	2	1	no	—
117	D₄×C₃₁	#127	1	2	yes	—
118	C₂×C₄×C₃₁	—	2	1	no	—
119	C₂×C₄×C₃₁	—	2	4	no	—
120	C₂×C₄×C₃₁	—	2	4	no	—
121	C₂×C₄×C₃₁	—	2	2	no	—
122	C₂×C₄×C₃₁	—	4	4	no	—
123	C₂×C₄×C₃₁	—	4	4	no	—
124	C₂×C₄×C₃₁	—	4	8	no	—
125	C₂×C₄×C₃₁	#121	1	2	yes	—
126	C₂×C₄×C₃₁	#121	1	2	yes	—
127	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
128	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
129	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
130	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
131	C₈×C₃₁	—	2	4	no	—
132	C₈×C₃₁	—	4	4	no	—
133	C₈×C₃₁	#121	1	4	yes	—
134	C₈×C₃₁	#121	1	4	yes	—
135	C₂×#494	C₂×C₃₁	1	1	yes	characteristic, complemented
136	C₂×#473	C₂×C₃₁	1	4	yes	—
137	#3039	C₂×C₃₁	1	2	yes	—
138	C₄×#131	C₂×C₃₁	1	1	yes	characteristic
139	#1979	C₂×C₃₁	1	4	yes	complemented
140	#1966	C₂×C₃₁	1	2	yes	complemented
141	#1848	C₂×C₃₁	1	1	yes	characteristic, complemented
142	C₂²×C₃₁	—	2	2	no	—
143	C₂²×C₃₁	—	2	2	no	—
144	C₂²×C₃₁	—	4	2	no	—
145	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
146	C₂²×C₃₁	#532	1	2	yes	—
147	C₄×C₃₁	—	2	2	no	—
148	C₄×C₃₁	—	2	2	no	—
149	C₄×C₃₁	—	4	2	no	—
150	C₄×C₃₁	—	8	4	no	—
151	C₄×C₃₁	#473	1	4	yes	complemented
152	C₄×C₃₁	C₂×#121	1	2	yes	—
153	C₄×C₃₁	C₂×#127	1	1	yes	characteristic
154	C₄×C₃₁	#532	1	1	yes	characteristic
155	C₂×#137	C₂²×C₃₁	1	4	yes	—
156	C₂²×Q₁₆	C₂²×C₃₁	1	4	yes	—
157	C₂×#131	C₂²×C₃₁	1	2	yes	—
158	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
159	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
160	#494	—	2	1	no	—
161	#494	—	2	1	no	—
162	#494	—	2	2	no	—
163	#473	—	2	16	no	—
164	#462	—	2	4	no	—
165	#461	—	2	4	no	—
166	C₂×#108	C₂²×C₃₁	1	4	yes	—
167	#423	C₂²×C₃₁	1	2	yes	—
168	#419	C₂²×C₃₁	1	4	yes	—
169	C₂×#101	C₂²×C₃₁	1	8	yes	—
170	#405	—	2	2	no	—
171	#405	C₂²×C₃₁	1	2	yes	—
172	C₄×M₁₆	—	2	1	no	—
173	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
174	C₂×C₄×C₈	—	2	1	no	—
175	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
176	C₂×C₃₁	—	2	2	no	—
177	C₂×C₃₁	—	4	2	no	—
178	C₂×C₃₁	#2856	1	1	yes	characteristic
179	C₂×C₃₁	C₂×#473	1	2	yes	—
180	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
181	C₂²×Q₈	C₂³×C₃₁	1	4	yes	—
182	#137	—	2	16	no	—
183	C₂×Q₁₆	—	2	4	no	—
184	C₂×Q₁₆	—	2	8	no	—
185	C₂×Q₁₆	—	2	4	no	—
186	C₂×Q₁₆	—	2	4	no	—
187	C₂×Q₁₆	—	2	8	no	—
188	C₂×SD₁₆	—	2	4	no	—
189	#131	—	2	4	no	—
190	#131	D₄×C₃₁	1	8	yes	—
191	C₂×M₁₆	—	2	2	no	—
192	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
193	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
194	C₂²×C₈	—	2	2	no	—
195	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
196	C₄⋊Q₈	—	2	4	no	—
197	C₄⋊Q₈	—	2	2	no	—
198	C₄⋊Q₈	—	2	2	no	—
199	C₄×Q₈	D₄×C₃₁	1	2	yes	—
200	C₄×D₄	—	2	2	no	—
201	C₄×D₄	D₄×C₃₁	1	2	yes	—
202	#117	—	2	1	no	—
203	#117	C₂³×C₃₁	1	1	yes	characteristic
204	C₂×C₄⋊C₄	—	2	4	no	—
205	C₂×C₄²	—	2	1	no	—
206	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
207	#108	—	2	8	no	—
208	C₄⋊C₈	—	2	4	no	—
209	#103	—	2	8	no	—
210	#101	—	2	16	no	—
211	#97	—	2	2	no	—
212	#97	—	2	1	no	—
213	C₄×C₈	—	2	2	no	—
214	C₄×C₈	—	2	1	no	—
215	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
216	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
217	C₄×C₈	D₄×C₃₁	1	2	yes	—
218	C₃₁	#33191	1	1	yes	characteristic, direct product factor, Sylow
219	#41	—	2	2	no	—
220	#41	C₂×D₄×C₃₁	1	4	yes	—
221	C₂×Q₈	—	2	8	no	—
222	C₂×Q₈	—	2	4	no	—
223	C₂×Q₈	—	2	4	no	—
224	C₂×Q₈	—	2	8	no	—
225	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
226	C₂×D₄	—	2	1	no	—
227	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
228	C₂²×C₄	—	2	1	no	—
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	—	2	4	no	—
231	C₂²×C₄	#41×C₃₁	1	2	yes	—
232	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
233	Q₁₆	—	4	8	no	—
234	Q₁₆	—	4	8	no	—
235	Q₁₆	—	4	8	no	—
236	SD₁₆	—	4	8	no	—
237	M₁₆	—	2	2	no	—
238	M₁₆	—	2	8	no	—
239	M₁₆	—	2	8	no	—
240	M₁₆	C₂×D₄×C₃₁	1	4	yes	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	2	8	no	—
243	C₂×C₈	—	2	4	no	—
244	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
245	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
246	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
247	C₂×C₈	#41×C₃₁	1	4	yes	—
248	C₄⋊C₄	—	2	2	no	—
249	C₄⋊C₄	—	4	4	no	—
250	C₄⋊C₄	—	4	4	no	—
251	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
252	C₂²⋊C₄	—	2	1	no	—
253	C₂²⋊C₄	—	2	2	no	—
254	C₄²	—	2	2	no	—
255	C₄²	—	2	1	no	—
256	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
257	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
258	C₄²	C₂×D₄×C₃₁	1	2	yes	—
259	C₂³	—	2	1	no	—
260	C₂³	C₂×#41×C₃₁	1	1	yes	characteristic
261	Q₈	—	2	1	no	—
262	Q₈	—	4	8	no	—
263	Q₈	—	4	4	no	—
264	Q₈	—	4	4	no	—
265	Q₈	C₃₁×#127	1	2	yes	—
266	D₄	—	2	4	no	—
267	D₄	—	2	1	no	—
268	D₄	C₃₁×#127	1	2	yes	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	4	4	no	—
274	C₂×C₄	—	4	4	no	—
275	C₂×C₄	—	4	8	no	—
276	C₂×C₄	C₃₁×#121	1	2	yes	—
277	C₂×C₄	C₃₁×#121	1	2	yes	—
278	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
279	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
280	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
281	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
282	C₈	—	2	4	no	—
283	C₈	—	4	4	no	—
284	C₈	C₃₁×#121	1	4	yes	—
285	C₈	C₃₁×#121	1	4	yes	—
286	C₂²	—	2	2	no	—
287	C₂²	—	2	2	no	—
288	C₂²	—	4	2	no	—
289	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
290	C₂²	C₃₁×#532	1	2	yes	—
291	C₄	—	2	2	no	—
292	C₄	—	2	2	no	—
293	C₄	—	4	2	no	—
294	C₄	—	8	4	no	—
295	C₄	C₃₁×#473	1	4	yes	complemented
296	C₄	C₂×C₃₁×#121	1	2	yes	—
297	C₄	C₂×C₃₁×#127	1	1	yes	characteristic
298	C₄	C₃₁×#532	1	1	yes	characteristic
299	C₂	—	2	2	no	—
300	C₂	—	4	2	no	—
301	C₂	C₃₁×#2856	1	1	yes	characteristic
302	C₂	C₂×C₃₁×#473	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×#494	C₂	semidirect
5. C₃₁×#1979	C₂	semidirect
6. C₃₁×#1966	C₂	semidirect
7. C₃₁×#1848	C₂	semidirect
107. #33191	C₃₁	direct (N × H)
135. C₂×#494	C₂×C₃₁	semidirect
139. #1979	C₂×C₃₁	semidirect
140. #1966	C₂×C₃₁	semidirect
141. #1848	C₂×C₃₁	semidirect
151. C₄×C₃₁	#473	semidirect
218. C₃₁	#33191	direct (N × H)
295. C₄	C₃₁×#473	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#494	2
C₂×C₃₁×#473	2
C₃₁×#3039	2
C₄×C₃₁×#131	2
C₃₁×#1979	2
C₃₁×#1966	2
C₃₁×#1848	2
#33191	31

Subgroups — C31×#33191 #21489950

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33191 #21489950