Subgroups — C₃₁×#9064 #21465823

Order 7936.

738 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#492	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2366	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#1987	C₂	1	2	yes	maximal, complemented
4	C₂×C₃₁×#331	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#1279	C₂	1	2	yes	maximal, complemented
6	C₂²×SD₁₆×C₃₁	—	2	2	no	—
7	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
8	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
9	C₂×C₄×D₄×C₃₁	—	2	2	no	—
10	C₃₁×#492	—	2	2	no	—
11	C₃₁×#492	C₄	1	4	yes	complemented
12	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
14	C₃₁×#390	—	2	2	no	—
15	C₃₁×#386	—	2	2	no	—
16	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
17	C₃₁×#340	—	2	2	no	—
18	C₃₁×#340	—	2	2	no	—
19	C₃₁×#331	—	2	2	no	—
20	C₂²×Q₈×C₃₁	—	2	1	no	—
21	C₂²×D₄×C₃₁	—	2	2	no	—
22	C₂²×D₄×C₃₁	D₄	1	2	yes	—
23	C₂³×C₄×C₃₁	—	2	2	no	—
24	C₂×SD₁₆×C₃₁	—	2	8	no	—
25	C₂×SD₁₆×C₃₁	—	4	4	no	—
26	C₂×SD₁₆×C₃₁	—	4	4	no	—
27	C₂²×C₈×C₃₁	—	2	1	no	—
28	C₂²×C₈×C₃₁	—	2	2	no	—
29	C₃₁×C₄⋊Q₈	—	2	1	no	—
30	C₃₁×C₄⋊Q₈	—	2	2	no	—
31	C₃₁×C₄⋊Q₈	C₂×C₄	1	2	yes	complemented
32	C₃₁×#127	—	2	2	no	—
33	C₃₁×#127	C₂×C₄	1	2	yes	—
34	C₃₁×#127	D₄	1	2	yes	—
35	C₄×D₄×C₃₁	—	4	4	no	—
36	C₄×D₄×C₃₁	—	4	4	no	—
37	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
38	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
39	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
40	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
41	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
42	C₂×C₄²×C₃₁	—	2	2	no	—
43	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
44	C₃₁×C₄⋊C₈	—	2	2	no	—
45	C₃₁×C₄⋊C₈	—	2	2	no	—
46	C₄×C₈×C₃₁	—	2	1	no	—
47	C₄×C₈×C₃₁	C₂×C₄	1	2	yes	complemented
48	C₃₁×#95	—	2	2	no	—
49	C₂⁴×C₃₁	—	2	2	no	—
50	C₂×Q₈×C₃₁	—	2	2	no	—
51	C₂×Q₈×C₃₁	—	4	2	no	—
52	C₂×Q₈×C₃₁	—	4	2	no	—
53	C₂×Q₈×C₃₁	—	4	1	no	—
54	C₂×D₄×C₃₁	—	2	4	no	—
55	C₂×D₄×C₃₁	—	2	4	no	—
56	C₂×D₄×C₃₁	—	2	2	no	—
57	C₂×D₄×C₃₁	—	4	2	no	—
58	C₂×D₄×C₃₁	—	4	2	no	—
59	C₂×D₄×C₃₁	—	4	2	no	—
60	C₂×D₄×C₃₁	—	4	2	no	—
61	C₂×D₄×C₃₁	—	4	2	no	—
62	C₂×D₄×C₃₁	—	4	2	no	—
63	C₂×D₄×C₃₁	C₂²⋊C₄	1	4	yes	—
64	C₂²×C₄×C₃₁	—	2	2	no	—
65	C₂²×C₄×C₃₁	—	2	2	no	—
66	C₂²×C₄×C₃₁	—	4	8	no	—
67	C₂²×C₄×C₃₁	—	4	1	no	—
68	C₂²×C₄×C₃₁	—	8	2	no	—
69	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
71	SD₁₆×C₃₁	—	4	16	no	—
72	C₂×C₈×C₃₁	—	2	2	no	—
73	C₂×C₈×C₃₁	—	2	2	no	—
74	C₂×C₈×C₃₁	—	2	2	no	—
75	C₂×C₈×C₃₁	—	4	2	no	—
76	C₂×C₈×C₃₁	—	4	2	no	—
77	C₂×C₈×C₃₁	—	4	2	no	—
78	C₄⋊C₄×C₃₁	—	2	2	no	—
79	C₄⋊C₄×C₃₁	—	2	4	no	—
80	C₄⋊C₄×C₃₁	—	2	2	no	—
81	C₄⋊C₄×C₃₁	—	2	2	no	—
82	C₄⋊C₄×C₃₁	—	4	2	no	—
83	C₄⋊C₄×C₃₁	SD₁₆	1	4	yes	complemented
84	C₂²⋊C₄×C₃₁	—	4	8	no	—
85	C₂²⋊C₄×C₃₁	—	8	2	no	—
86	C₄²×C₃₁	—	4	2	no	—
87	C₄²×C₃₁	—	4	2	no	—
88	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
89	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
90	C₄²×C₃₁	#41	1	2	yes	—
91	#9064	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
92	C₂³×C₃₁	—	2	4	no	—
93	C₂³×C₃₁	—	4	4	no	—
94	C₂³×C₃₁	—	4	4	no	—
95	C₂³×C₃₁	—	4	2	no	—
96	C₂³×C₃₁	—	4	2	no	—
97	C₂³×C₃₁	#120	1	1	yes	characteristic
98	Q₈×C₃₁	—	4	4	no	—
99	Q₈×C₃₁	—	8	2	no	—
100	D₄×C₃₁	—	4	4	no	—
101	D₄×C₃₁	—	4	4	no	—
102	D₄×C₃₁	—	4	4	no	—
103	D₄×C₃₁	—	4	4	no	—
104	D₄×C₃₁	—	4	4	no	—
105	D₄×C₃₁	—	8	2	no	—
106	C₂×C₄×C₃₁	—	2	1	no	—
107	C₂×C₄×C₃₁	—	2	1	no	—
108	C₂×C₄×C₃₁	—	2	4	no	—
109	C₂×C₄×C₃₁	—	4	2	no	—
110	C₂×C₄×C₃₁	—	4	2	no	—
111	C₂×C₄×C₃₁	—	4	2	no	—
112	C₂×C₄×C₃₁	—	4	2	no	—
113	C₂×C₄×C₃₁	—	4	2	no	—
114	C₂×C₄×C₃₁	—	4	2	no	—
115	C₂×C₄×C₃₁	—	8	8	no	—
116	C₂×C₄×C₃₁	—	8	1	no	—
117	C₂×C₄×C₃₁	—	8	1	no	—
118	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
120	C₂×C₄×C₃₁	#121	1	2	yes	—
121	C₂×C₄×C₃₁	#123	1	1	yes	characteristic
122	C₂×C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
123	C₂×C₄×C₃₁	#135	1	2	yes	—
124	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
125	C₂×C₄×C₃₁	#137	1	2	yes	—
126	C₈×C₃₁	—	2	4	no	—
127	C₈×C₃₁	—	8	2	no	—
128	C₂×#492	C₂×C₃₁	1	1	yes	characteristic
129	#2366	C₂×C₃₁	1	1	yes	characteristic
130	#1987	C₂×C₃₁	1	2	yes	complemented
131	C₂×#331	C₂×C₃₁	1	1	yes	characteristic
132	#1279	C₂×C₃₁	1	2	yes	complemented
133	C₂²×C₃₁	—	4	8	no	—
134	C₂²×C₃₁	—	4	4	no	—
135	C₂²×C₃₁	—	8	2	no	—
136	C₂²×C₃₁	—	8	2	no	—
137	C₂²×C₃₁	—	8	4	no	—
138	C₂²×C₃₁	#386	1	1	yes	characteristic
139	C₂²×C₃₁	#450	1	2	yes	—
140	C₂²×C₃₁	#452	1	2	yes	—
141	C₂²×C₃₁	#453	1	1	yes	characteristic
142	C₂²×C₃₁	#453	1	1	yes	characteristic
143	C₄×C₃₁	—	2	1	no	—
144	C₄×C₃₁	—	2	1	no	—
145	C₄×C₃₁	—	4	4	no	—
146	C₄×C₃₁	—	8	2	no	—
147	C₄×C₃₁	—	8	2	no	—
148	C₄×C₃₁	C₄×SD₁₆	1	2	yes	—
149	C₄×C₃₁	#440	1	2	yes	—
150	C₄×C₃₁	#461	1	2	yes	—
151	C₄×C₃₁	#463	1	2	yes	—
152	C₂²×SD₁₆	—	2	2	no	—
153	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
154	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
155	C₂×C₄×D₄	—	2	2	no	—
156	#492	—	2	2	no	—
157	#492	C₄×C₃₁	1	4	yes	complemented
158	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
159	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
160	#390	—	2	2	no	—
161	#386	—	2	2	no	—
162	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
163	#340	—	2	2	no	—
164	#340	—	2	2	no	—
165	#331	—	2	2	no	—
166	C₂×C₃₁	—	8	4	no	—
167	C₂×C₃₁	#1629	1	1	yes	characteristic
168	C₂×C₃₁	#1630	1	1	yes	characteristic
169	C₂×C₃₁	#1631	1	2	yes	—
170	C₂×C₃₁	#1877	1	2	yes	—
171	C₂×C₃₁	#1887	1	1	yes	characteristic
172	C₂²×Q₈	—	2	1	no	—
173	C₂²×D₄	—	2	2	no	—
174	C₂²×D₄	D₄×C₃₁	1	2	yes	—
175	C₂³×C₄	—	2	2	no	—
176	C₂×SD₁₆	—	2	8	no	—
177	C₂×SD₁₆	—	4	4	no	—
178	C₂×SD₁₆	—	4	4	no	—
179	C₂²×C₈	—	2	1	no	—
180	C₂²×C₈	—	2	2	no	—
181	C₄⋊Q₈	—	2	1	no	—
182	C₄⋊Q₈	—	2	2	no	—
183	C₄⋊Q₈	C₂×C₄×C₃₁	1	2	yes	complemented
184	#127	—	2	2	no	—
185	#127	C₂×C₄×C₃₁	1	2	yes	—
186	#127	D₄×C₃₁	1	2	yes	—
187	C₄×D₄	—	4	4	no	—
188	C₄×D₄	—	4	4	no	—
189	C₂×C₄⋊C₄	—	2	2	no	—
190	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
191	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
192	C₂×C₂²⋊C₄	—	2	4	no	—
193	C₂×C₂²⋊C₄	—	2	2	no	—
194	C₂×C₄²	—	2	2	no	—
195	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
196	C₄⋊C₈	—	2	2	no	—
197	C₄⋊C₈	—	2	2	no	—
198	C₄×C₈	—	2	1	no	—
199	C₄×C₈	C₂×C₄×C₃₁	1	2	yes	complemented
200	#95	—	2	2	no	—
201	C₃₁	#9064	1	1	yes	characteristic, direct product factor, Sylow
202	C₂⁴	—	2	2	no	—
203	C₂×Q₈	—	2	2	no	—
204	C₂×Q₈	—	4	2	no	—
205	C₂×Q₈	—	4	2	no	—
206	C₂×Q₈	—	4	1	no	—
207	C₂×D₄	—	2	4	no	—
208	C₂×D₄	—	2	4	no	—
209	C₂×D₄	—	2	2	no	—
210	C₂×D₄	—	4	2	no	—
211	C₂×D₄	—	4	2	no	—
212	C₂×D₄	—	4	2	no	—
213	C₂×D₄	—	4	2	no	—
214	C₂×D₄	—	4	2	no	—
215	C₂×D₄	—	4	2	no	—
216	C₂×D₄	C₂²⋊C₄×C₃₁	1	4	yes	—
217	C₂²×C₄	—	2	2	no	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	4	8	no	—
220	C₂²×C₄	—	4	1	no	—
221	C₂²×C₄	—	8	2	no	—
222	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
223	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
224	SD₁₆	—	4	16	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	4	2	no	—
229	C₂×C₈	—	4	2	no	—
230	C₂×C₈	—	4	2	no	—
231	C₄⋊C₄	—	2	2	no	—
232	C₄⋊C₄	—	2	4	no	—
233	C₄⋊C₄	—	2	2	no	—
234	C₄⋊C₄	—	2	2	no	—
235	C₄⋊C₄	—	4	2	no	—
236	C₄⋊C₄	SD₁₆×C₃₁	1	4	yes	complemented
237	C₂²⋊C₄	—	4	8	no	—
238	C₂²⋊C₄	—	8	2	no	—
239	C₄²	—	4	2	no	—
240	C₄²	—	4	2	no	—
241	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
242	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
243	C₄²	#41×C₃₁	1	2	yes	—
244	C₂³	—	2	4	no	—
245	C₂³	—	4	4	no	—
246	C₂³	—	4	4	no	—
247	C₂³	—	4	2	no	—
248	C₂³	—	4	2	no	—
249	C₂³	C₃₁×#120	1	1	yes	characteristic
250	Q₈	—	4	4	no	—
251	Q₈	—	8	2	no	—
252	D₄	—	4	4	no	—
253	D₄	—	4	4	no	—
254	D₄	—	4	4	no	—
255	D₄	—	4	4	no	—
256	D₄	—	4	4	no	—
257	D₄	—	8	2	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	4	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	4	2	no	—
264	C₂×C₄	—	4	2	no	—
265	C₂×C₄	—	4	2	no	—
266	C₂×C₄	—	4	2	no	—
267	C₂×C₄	—	8	8	no	—
268	C₂×C₄	—	8	1	no	—
269	C₂×C₄	—	8	1	no	—
270	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
271	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
272	C₂×C₄	C₃₁×#121	1	2	yes	—
273	C₂×C₄	C₃₁×#123	1	1	yes	characteristic
274	C₂×C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
275	C₂×C₄	C₃₁×#135	1	2	yes	—
276	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
277	C₂×C₄	C₃₁×#137	1	2	yes	—
278	C₈	—	2	4	no	—
279	C₈	—	8	2	no	—
280	C₂²	—	4	8	no	—
281	C₂²	—	4	4	no	—
282	C₂²	—	8	2	no	—
283	C₂²	—	8	2	no	—
284	C₂²	—	8	4	no	—
285	C₂²	C₃₁×#386	1	1	yes	characteristic
286	C₂²	C₃₁×#450	1	2	yes	—
287	C₂²	C₃₁×#452	1	2	yes	—
288	C₂²	C₃₁×#453	1	1	yes	characteristic
289	C₂²	C₃₁×#453	1	1	yes	characteristic
290	C₄	—	2	1	no	—
291	C₄	—	2	1	no	—
292	C₄	—	4	4	no	—
293	C₄	—	8	2	no	—
294	C₄	—	8	2	no	—
295	C₄	C₄×SD₁₆×C₃₁	1	2	yes	—
296	C₄	C₃₁×#440	1	2	yes	—
297	C₄	C₃₁×#461	1	2	yes	—
298	C₄	C₃₁×#463	1	2	yes	—
299	C₂	—	8	4	no	—
300	C₂	C₃₁×#1629	1	1	yes	characteristic
301	C₂	C₃₁×#1630	1	1	yes	characteristic
302	C₂	C₃₁×#1631	1	2	yes	—
303	C₂	C₃₁×#1877	1	2	yes	—
304	C₂	C₃₁×#1887	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	#120
commutator	C₄²	C₂²×C₄×C₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
prime radical	C₂×C₃₁×#127	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃₁×#1987	C₂	semidirect
5. C₃₁×#1279	C₂	semidirect
11. C₃₁×#492	C₄	semidirect
31. C₃₁×C₄⋊Q₈	C₂×C₄	semidirect
47. C₄×C₈×C₃₁	C₂×C₄	semidirect
83. C₄⋊C₄×C₃₁	SD₁₆	semidirect
91. #9064	C₃₁	direct (N × H)
130. #1987	C₂×C₃₁	semidirect
132. #1279	C₂×C₃₁	semidirect
157. #492	C₄×C₃₁	semidirect
183. C₄⋊Q₈	C₂×C₄×C₃₁	semidirect
199. C₄×C₈	C₂×C₄×C₃₁	semidirect
201. C₃₁	#9064	direct (N × H)
236. C₄⋊C₄	SD₁₆×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#492	2
C₃₁×#2366	2
C₃₁×#1987	2
C₂×C₃₁×#331	2
C₃₁×#1279	2
#9064	31

Subgroups — C31×#9064 #21465823

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#9064 #21465823