Subgroups — C₃₁×#9769 #21466528

Order 7936.

450 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×#486	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1837	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#1549	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1548	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1545	C₂	1	1	yes	maximal, characteristic
6	C₃₁×#1545	C₂	1	1	yes	maximal, characteristic
7	C₃₁×#1450	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
10	C₃₁×#486	—	2	1	no	—
11	C₃₁×#486	—	2	1	no	—
12	C₃₁×#486	C₄	1	4	yes	—
13	C₃₁×#431	C₂²	1	1	yes	characteristic
14	C₃₁×#431	C₂²	1	1	yes	characteristic
15	C₃₁×#423	C₂²	1	1	yes	characteristic
16	C₃₁×#423	C₂²	1	1	yes	characteristic
17	C₂×C₃₁×#102	—	2	2	no	—
18	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
19	C₃₁×#341	—	2	2	no	—
20	C₃₁×#333	—	2	2	no	—
21	C₃₁×#332	—	2	2	no	—
22	C₃₁×#331	—	2	2	no	—
23	C₃₁×#331	—	2	2	no	—
24	C₃₁×#322	—	2	1	no	—
25	C₈²×C₃₁	—	2	1	no	—
26	C₂²×Q₈×C₃₁	—	2	1	no	—
27	C₂²×D₄×C₃₁	—	2	1	no	—
28	C₂²×D₄×C₃₁	—	2	1	no	—
29	C₂²×D₄×C₃₁	—	2	1	no	—
30	C₂×M₁₆×C₃₁	—	2	1	no	—
31	C₂×M₁₆×C₃₁	—	2	1	no	—
32	C₂²×C₈×C₃₁	D₄	1	2	yes	—
33	C₃₁×C₄⋊Q₈	—	2	1	no	—
34	C₃₁×C₄⋊Q₈	—	2	1	no	—
35	C₃₁×C₄⋊Q₈	—	2	1	no	—
36	C₃₁×C₄⋊Q₈	C₂×C₄	1	2	yes	—
37	C₃₁×#127	—	2	1	no	—
38	C₃₁×#127	—	2	1	no	—
39	C₃₁×#127	—	2	1	no	—
40	C₃₁×#127	C₂×C₄	1	2	yes	—
41	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
42	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
43	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
44	C₃₁×C₄⋊C₈	—	2	1	no	—
45	C₃₁×C₄⋊C₈	—	2	1	no	—
46	C₃₁×C₄⋊C₈	—	2	1	no	—
47	C₃₁×C₄⋊C₈	—	2	1	no	—
48	C₃₁×C₄⋊C₈	D₄	1	2	yes	—
49	C₃₁×C₄⋊C₈	D₄	1	2	yes	—
50	C₃₁×#102	—	2	4	no	—
51	C₃₁×#102	—	2	4	no	—
52	C₃₁×C₂²⋊C₈	—	2	1	no	—
53	C₃₁×C₂²⋊C₈	—	2	1	no	—
54	C₄×C₈×C₃₁	—	2	1	no	—
55	C₄×C₈×C₃₁	—	2	1	no	—
56	C₄×C₈×C₃₁	C₂×C₄	1	2	yes	—
57	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
58	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
59	C₂⁴×C₃₁	—	4	1	no	—
60	C₂×Q₈×C₃₁	—	2	2	no	—
61	C₂×Q₈×C₃₁	—	4	1	no	—
62	C₂×Q₈×C₃₁	—	8	2	no	—
63	C₂×D₄×C₃₁	—	2	2	no	—
64	C₂×D₄×C₃₁	—	2	2	no	—
65	C₂×D₄×C₃₁	—	2	2	no	—
66	C₂×D₄×C₃₁	—	4	1	no	—
67	C₂×D₄×C₃₁	—	4	1	no	—
68	C₂×D₄×C₃₁	—	4	1	no	—
69	C₂×D₄×C₃₁	—	4	2	no	—
70	C₂×D₄×C₃₁	—	4	2	no	—
71	C₂×D₄×C₃₁	—	8	2	no	—
72	C₂×D₄×C₃₁	—	8	2	no	—
73	C₂²×C₄×C₃₁	—	4	1	no	—
74	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
76	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
77	M₁₆×C₃₁	—	4	2	no	—
78	M₁₆×C₃₁	—	4	2	no	—
79	C₂×C₈×C₃₁	—	2	2	no	—
80	C₂×C₈×C₃₁	—	2	2	no	—
81	C₂×C₈×C₃₁	—	2	2	no	—
82	C₂×C₈×C₃₁	—	2	2	no	—
83	C₂×C₈×C₃₁	—	4	1	no	—
84	C₂×C₈×C₃₁	—	4	1	no	—
85	C₂×C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
86	C₂×C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
87	C₄⋊C₄×C₃₁	—	2	2	no	—
88	C₄⋊C₄×C₃₁	—	2	2	no	—
89	C₄⋊C₄×C₃₁	—	4	1	no	—
90	C₄⋊C₄×C₃₁	—	4	1	no	—
91	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
92	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
93	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
94	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
95	#9769	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
96	C₂³×C₃₁	—	4	2	no	—
97	C₂³×C₃₁	—	8	2	no	—
98	C₂³×C₃₁	—	8	2	no	—
99	C₂³×C₃₁	—	8	2	no	—
100	C₂³×C₃₁	#124	1	1	yes	characteristic
101	Q₈×C₃₁	—	8	2	no	—
102	Q₈×C₃₁	—	8	2	no	—
103	D₄×C₃₁	—	4	4	no	—
104	D₄×C₃₁	—	8	2	no	—
105	D₄×C₃₁	—	8	2	no	—
106	D₄×C₃₁	—	8	2	no	—
107	D₄×C₃₁	—	8	2	no	—
108	D₄×C₃₁	—	8	1	no	—
109	D₄×C₃₁	—	8	1	no	—
110	C₂×C₄×C₃₁	—	2	2	no	—
111	C₂×C₄×C₃₁	—	2	2	no	—
112	C₂×C₄×C₃₁	—	4	2	no	—
113	C₂×C₄×C₃₁	—	8	2	no	—
114	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
116	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
117	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
118	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
119	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
120	C₂×C₄×C₃₁	#135	1	2	yes	—
121	C₂×C₄×C₃₁	#135	1	2	yes	—
122	C₈×C₃₁	—	2	2	no	—
123	C₈×C₃₁	—	2	2	no	—
124	C₈×C₃₁	—	4	2	no	—
125	C₈×C₃₁	—	4	2	no	—
126	C₈×C₃₁	—	8	1	no	—
127	C₈×C₃₁	—	8	1	no	—
128	C₂×#486	C₂×C₃₁	1	1	yes	characteristic
129	#1837	C₂×C₃₁	1	1	yes	characteristic, complemented
130	#1549	C₂×C₃₁	1	1	yes	characteristic, complemented
131	#1548	C₂×C₃₁	1	1	yes	characteristic, complemented
132	#1545	C₂×C₃₁	1	1	yes	characteristic
133	#1545	C₂×C₃₁	1	1	yes	characteristic
134	#1450	C₂×C₃₁	1	1	yes	characteristic, complemented
135	C₂²×C₃₁	—	2	1	no	—
136	C₂²×C₃₁	—	2	1	no	—
137	C₂²×C₃₁	—	8	2	no	—
138	C₂²×C₃₁	—	8	2	no	—
139	C₂²×C₃₁	—	8	2	no	—
140	C₂²×C₃₁	—	16	4	no	—
141	C₂²×C₃₁	#390	1	1	yes	characteristic
142	C₂²×C₃₁	#488	1	2	yes	—
143	C₄×C₃₁	—	2	1	no	—
144	C₄×C₃₁	—	2	1	no	—
145	C₄×C₃₁	—	8	2	no	—
146	C₄×C₃₁	#416	1	2	yes	—
147	C₄×C₃₁	#471	1	2	yes	—
148	C₄×C₃₁	#472	1	2	yes	—
149	C₄×C₃₁	#495	1	1	yes	characteristic
150	C₄×C₃₁	#495	1	1	yes	characteristic
151	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
152	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
153	#486	—	2	1	no	—
154	#486	—	2	1	no	—
155	#486	C₄×C₃₁	1	4	yes	—
156	#431	C₂²×C₃₁	1	1	yes	characteristic
157	#431	C₂²×C₃₁	1	1	yes	characteristic
158	#423	C₂²×C₃₁	1	1	yes	characteristic
159	#423	C₂²×C₃₁	1	1	yes	characteristic
160	C₂×#102	—	2	2	no	—
161	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
162	#341	—	2	2	no	—
163	#333	—	2	2	no	—
164	#332	—	2	2	no	—
165	#331	—	2	2	no	—
166	#331	—	2	2	no	—
167	#322	—	2	1	no	—
168	C₈²	—	2	1	no	—
169	C₂×C₃₁	—	2	2	no	—
170	C₂×C₃₁	—	16	2	no	—
171	C₂×C₃₁	#1958	1	1	yes	characteristic
172	C₂×C₃₁	#1977	1	1	yes	characteristic
173	C₂×C₃₁	#1978	1	1	yes	characteristic
174	C₂²×Q₈	—	2	1	no	—
175	C₂²×D₄	—	2	1	no	—
176	C₂²×D₄	—	2	1	no	—
177	C₂²×D₄	—	2	1	no	—
178	C₂×M₁₆	—	2	1	no	—
179	C₂×M₁₆	—	2	1	no	—
180	C₂²×C₈	D₄×C₃₁	1	2	yes	—
181	C₄⋊Q₈	—	2	1	no	—
182	C₄⋊Q₈	—	2	1	no	—
183	C₄⋊Q₈	—	2	1	no	—
184	C₄⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
185	#127	—	2	1	no	—
186	#127	—	2	1	no	—
187	#127	—	2	1	no	—
188	#127	C₂×C₄×C₃₁	1	2	yes	—
189	C₂×C₄⋊C₄	—	2	1	no	—
190	C₂×C₄⋊C₄	—	2	1	no	—
191	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
192	C₄⋊C₈	—	2	1	no	—
193	C₄⋊C₈	—	2	1	no	—
194	C₄⋊C₈	—	2	1	no	—
195	C₄⋊C₈	—	2	1	no	—
196	C₄⋊C₈	D₄×C₃₁	1	2	yes	—
197	C₄⋊C₈	D₄×C₃₁	1	2	yes	—
198	#102	—	2	4	no	—
199	#102	—	2	4	no	—
200	C₂²⋊C₈	—	2	1	no	—
201	C₂²⋊C₈	—	2	1	no	—
202	C₄×C₈	—	2	1	no	—
203	C₄×C₈	—	2	1	no	—
204	C₄×C₈	C₂×C₄×C₃₁	1	2	yes	—
205	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
206	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
207	C₃₁	#9769	1	1	yes	characteristic, direct product factor, Sylow
208	C₂⁴	—	4	1	no	—
209	C₂×Q₈	—	2	2	no	—
210	C₂×Q₈	—	4	1	no	—
211	C₂×Q₈	—	8	2	no	—
212	C₂×D₄	—	2	2	no	—
213	C₂×D₄	—	2	2	no	—
214	C₂×D₄	—	2	2	no	—
215	C₂×D₄	—	4	1	no	—
216	C₂×D₄	—	4	1	no	—
217	C₂×D₄	—	4	1	no	—
218	C₂×D₄	—	4	2	no	—
219	C₂×D₄	—	4	2	no	—
220	C₂×D₄	—	8	2	no	—
221	C₂×D₄	—	8	2	no	—
222	C₂²×C₄	—	4	1	no	—
223	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
224	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
225	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
226	M₁₆	—	4	2	no	—
227	M₁₆	—	4	2	no	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	4	1	no	—
233	C₂×C₈	—	4	1	no	—
234	C₂×C₈	C₂²⋊C₄×C₃₁	1	2	yes	—
235	C₂×C₈	C₂²⋊C₄×C₃₁	1	2	yes	—
236	C₄⋊C₄	—	2	2	no	—
237	C₄⋊C₄	—	2	2	no	—
238	C₄⋊C₄	—	4	1	no	—
239	C₄⋊C₄	—	4	1	no	—
240	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
241	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
242	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
243	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
244	C₂³	—	4	2	no	—
245	C₂³	—	8	2	no	—
246	C₂³	—	8	2	no	—
247	C₂³	—	8	2	no	—
248	C₂³	C₃₁×#124	1	1	yes	characteristic
249	Q₈	—	8	2	no	—
250	Q₈	—	8	2	no	—
251	D₄	—	4	4	no	—
252	D₄	—	8	2	no	—
253	D₄	—	8	2	no	—
254	D₄	—	8	2	no	—
255	D₄	—	8	2	no	—
256	D₄	—	8	1	no	—
257	D₄	—	8	1	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	8	2	no	—
262	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
263	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
264	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
265	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
266	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
267	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
268	C₂×C₄	C₃₁×#135	1	2	yes	—
269	C₂×C₄	C₃₁×#135	1	2	yes	—
270	C₈	—	2	2	no	—
271	C₈	—	2	2	no	—
272	C₈	—	4	2	no	—
273	C₈	—	4	2	no	—
274	C₈	—	8	1	no	—
275	C₈	—	8	1	no	—
276	C₂²	—	2	1	no	—
277	C₂²	—	2	1	no	—
278	C₂²	—	8	2	no	—
279	C₂²	—	8	2	no	—
280	C₂²	—	8	2	no	—
281	C₂²	—	16	4	no	—
282	C₂²	C₃₁×#390	1	1	yes	characteristic
283	C₂²	C₃₁×#488	1	2	yes	—
284	C₄	—	2	1	no	—
285	C₄	—	2	1	no	—
286	C₄	—	8	2	no	—
287	C₄	C₃₁×#416	1	2	yes	—
288	C₄	C₃₁×#471	1	2	yes	—
289	C₄	C₃₁×#472	1	2	yes	—
290	C₄	C₃₁×#495	1	1	yes	characteristic
291	C₄	C₃₁×#495	1	1	yes	characteristic
292	C₂	—	2	2	no	—
293	C₂	—	16	2	no	—
294	C₂	C₃₁×#1958	1	1	yes	characteristic
295	C₂	C₃₁×#1977	1	1	yes	characteristic
296	C₂	C₃₁×#1978	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	#390
commutator	C₄²	C₂²×C₄×C₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
prime radical	C₂×C₃₁×#127	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ 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
2. C₃₁×#1837	C₂	semidirect
3. C₃₁×#1549	C₂	semidirect
4. C₃₁×#1548	C₂	semidirect
7. C₃₁×#1450	C₂	semidirect
95. #9769	C₃₁	direct (N × H)
129. #1837	C₂×C₃₁	semidirect
130. #1549	C₂×C₃₁	semidirect
131. #1548	C₂×C₃₁	semidirect
134. #1450	C₂×C₃₁	semidirect
207. C₃₁	#9769	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×#486	2
C₃₁×#1837	2
C₃₁×#1549	2
C₃₁×#1548	2
C₃₁×#1545	2
C₃₁×#1450	2
#9769	31

Subgroups — C31×#9769 #21466528

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#9769 #21466528