Subgroups — C₃₁×#22269 #21479028

Order 7936.

1222 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#493	C₂	1	1	yes	maximal, characteristic
2	C₄×C₃₁×#127	C₂	1	2	yes	maximal, complemented
3	C₃₁×#1957	C₂	1	2	yes	maximal, complemented
4	C₃₁×#1879	C₂	1	8	yes	maximal, complemented
5	C₃₁×#1834	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1744	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₈×C₃₁	C₂²	1	4	yes	—
8	C₂×C₃₁×#127	C₂²	1	2	yes	—
9	C₂×C₄×D₄×C₃₁	—	2	2	no	—
10	C₂×C₄×D₄×C₃₁	—	2	2	no	—
11	C₂×C₄×D₄×C₃₁	C₂²	1	4	yes	—
12	C₃₁×#493	C₄	1	8	yes	complemented
13	C₂×C₃₁×#102	C₂²	1	8	yes	—
14	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
16	C₃₁×#390	—	2	2	no	—
17	C₃₁×#390	—	2	2	no	—
18	C₃₁×#390	C₂²	1	4	yes	—
19	C₃₁×#384	C₂²	1	2	yes	—
20	C₄³×C₃₁	C₂²	1	1	yes	characteristic
21	C₃₁×#340	C₂²	1	4	yes	—
22	C₃₁×#336	C₂²	1	4	yes	—
23	C₂²×D₄×C₃₁	—	2	2	no	—
24	C₂²×D₄×C₃₁	—	2	2	no	—
25	C₂²×D₄×C₃₁	C₂³	1	4	yes	—
26	C₂³×C₄×C₃₁	—	2	4	no	—
27	C₂×D₈×C₃₁	—	4	8	no	—
28	C₂×D₈×C₃₁	C₂×C₄	1	16	yes	complemented
29	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
30	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
31	C₃₁×#127	—	2	4	no	—
32	C₃₁×#127	C₂×C₄	1	8	yes	complemented
33	C₄×D₄×C₃₁	—	4	8	no	—
34	C₄×D₄×C₃₁	—	4	8	no	—
35	C₄×D₄×C₃₁	—	4	8	no	—
36	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
37	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
38	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
39	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
40	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
41	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
42	C₂×C₄²×C₃₁	D₄	1	2	yes	—
43	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
46	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
47	C₃₁×#102	—	4	8	no	—
48	C₄×C₈×C₃₁	—	2	2	no	—
49	C₄×C₈×C₃₁	C₂×C₄	1	4	yes	complemented
50	C₂⁴×C₃₁	—	2	4	no	—
51	C₂×D₄×C₃₁	—	2	8	no	—
52	C₂×D₄×C₃₁	—	2	8	no	—
53	C₂×D₄×C₃₁	—	2	4	no	—
54	C₂×D₄×C₃₁	—	2	8	no	—
55	C₂×D₄×C₃₁	—	2	8	no	—
56	C₂×D₄×C₃₁	—	4	4	no	—
57	C₂×D₄×C₃₁	—	4	4	no	—
58	C₂×D₄×C₃₁	—	4	4	no	—
59	C₂×D₄×C₃₁	—	4	4	no	—
60	C₂×D₄×C₃₁	C₂²×C₄	1	8	yes	—
61	C₂²×C₄×C₃₁	—	2	4	no	—
62	C₂²×C₄×C₃₁	—	4	8	no	—
63	C₂²×C₄×C₃₁	—	4	8	no	—
64	C₂²×C₄×C₃₁	—	4	8	no	—
65	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
66	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
67	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
70	D₈×C₃₁	—	4	16	no	—
71	C₂×C₈×C₃₁	—	2	4	no	—
72	C₂×C₈×C₃₁	—	2	2	no	—
73	C₂×C₈×C₃₁	—	2	4	no	—
74	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
75	C₄⋊C₄×C₃₁	—	2	8	no	—
76	C₄⋊C₄×C₃₁	—	4	4	no	—
77	C₄⋊C₄×C₃₁	—	4	4	no	—
78	C₂²⋊C₄×C₃₁	—	4	8	no	—
79	C₂²⋊C₄×C₃₁	—	4	8	no	—
80	C₂²⋊C₄×C₃₁	—	4	8	no	—
81	C₄²×C₃₁	—	2	2	no	—
82	C₄²×C₃₁	—	2	4	no	—
83	C₄²×C₃₁	—	2	4	no	—
84	C₄²×C₃₁	D₈	1	4	yes	complemented
85	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
86	C₄²×C₃₁	#41	1	2	yes	—
87	#22269	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
88	C₂³×C₃₁	—	2	8	no	—
89	C₂³×C₃₁	—	4	4	no	—
90	C₂³×C₃₁	—	4	4	no	—
91	C₂³×C₃₁	—	4	4	no	—
92	C₂³×C₃₁	—	4	4	no	—
93	C₂³×C₃₁	—	4	4	no	—
94	C₂³×C₃₁	—	4	4	no	—
95	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
96	D₄×C₃₁	—	2	16	no	—
97	D₄×C₃₁	—	4	8	no	—
98	D₄×C₃₁	—	4	16	no	—
99	D₄×C₃₁	—	4	16	no	—
100	C₂×C₄×C₃₁	—	2	2	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	4	no	—
106	C₂×C₄×C₃₁	—	4	4	no	—
107	C₂×C₄×C₃₁	—	4	4	no	—
108	C₂×C₄×C₃₁	—	8	16	no	—
109	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
110	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
111	C₂×C₄×C₃₁	C₂×D₈	1	4	yes	—
112	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
113	C₂×C₄×C₃₁	#135	1	2	yes	—
114	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	4	yes	—
115	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
116	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
117	C₂×C₄×C₃₁	H(5,𝔽₂)	1	4	yes	—
118	C₈×C₃₁	—	2	4	no	—
119	C₈×C₃₁	—	4	2	no	—
120	C₂×#493	C₂×C₃₁	1	1	yes	characteristic
121	C₄×#127	C₂×C₃₁	1	2	yes	complemented
122	#1957	C₂×C₃₁	1	2	yes	complemented
123	#1879	C₂×C₃₁	1	8	yes	complemented
124	#1834	C₂×C₃₁	1	1	yes	characteristic, complemented
125	#1744	C₂×C₃₁	1	1	yes	characteristic, complemented
126	C₂²×C₃₁	—	4	8	no	—
127	C₂²×C₃₁	—	4	8	no	—
128	C₂²×C₃₁	—	4	8	no	—
129	C₂²×C₃₁	—	8	4	no	—
130	C₂²×C₃₁	—	8	4	no	—
131	C₂²×C₃₁	—	8	4	no	—
132	C₂²×C₃₁	—	8	4	no	—
133	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
134	C₂²×C₃₁	#535	1	2	yes	—
135	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
136	C₂²×C₃₁	C₂×#135	1	1	yes	characteristic
137	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
138	C₄×C₃₁	—	2	4	no	—
139	C₄×C₃₁	—	4	2	no	—
140	C₄×C₃₁	—	8	4	no	—
141	C₄×C₃₁	C₄×D₈	1	4	yes	—
142	C₄×C₃₁	#442	1	4	yes	—
143	C₄×C₃₁	#518	1	2	yes	—
144	C₄×C₃₁	#540	1	2	yes	—
145	C₂²×D₈	C₂²×C₃₁	1	4	yes	—
146	C₂×#127	C₂²×C₃₁	1	2	yes	—
147	C₂×C₄×D₄	—	2	2	no	—
148	C₂×C₄×D₄	—	2	2	no	—
149	C₂×C₄×D₄	C₂²×C₃₁	1	4	yes	—
150	#493	C₄×C₃₁	1	8	yes	complemented
151	C₂×#102	C₂²×C₃₁	1	8	yes	—
152	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
153	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
154	#390	—	2	2	no	—
155	#390	—	2	2	no	—
156	#390	C₂²×C₃₁	1	4	yes	—
157	#384	C₂²×C₃₁	1	2	yes	—
158	C₄³	C₂²×C₃₁	1	1	yes	characteristic
159	#340	C₂²×C₃₁	1	4	yes	—
160	#336	C₂²×C₃₁	1	4	yes	—
161	C₂×C₃₁	—	8	8	no	—
162	C₂×C₃₁	#2328	1	1	yes	characteristic
163	C₂×C₃₁	C₂×C₄×D₈	1	1	yes	characteristic
164	C₂×C₃₁	C₂×#442	1	1	yes	characteristic
165	C₂×C₃₁	#3205	1	2	yes	—
166	C₂×C₃₁	#3207	1	2	yes	—
167	C₂²×D₄	—	2	2	no	—
168	C₂²×D₄	—	2	2	no	—
169	C₂²×D₄	C₂³×C₃₁	1	4	yes	—
170	C₂³×C₄	—	2	4	no	—
171	C₂×D₈	—	4	8	no	—
172	C₂×D₈	C₂×C₄×C₃₁	1	16	yes	complemented
173	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
174	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
175	#127	—	2	4	no	—
176	#127	C₂×C₄×C₃₁	1	8	yes	complemented
177	C₄×D₄	—	4	8	no	—
178	C₄×D₄	—	4	8	no	—
179	C₄×D₄	—	4	8	no	—
180	C₂×C₄⋊C₄	—	2	2	no	—
181	C₂×C₄⋊C₄	—	2	2	no	—
182	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
183	C₂×C₂²⋊C₄	—	2	4	no	—
184	C₂×C₂²⋊C₄	—	2	4	no	—
185	C₂×C₂²⋊C₄	—	2	4	no	—
186	C₂×C₄²	D₄×C₃₁	1	2	yes	—
187	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
188	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
189	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
190	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
191	#102	—	4	8	no	—
192	C₄×C₈	—	2	2	no	—
193	C₄×C₈	C₂×C₄×C₃₁	1	4	yes	complemented
194	C₃₁	#22269	1	1	yes	characteristic, direct product factor, Sylow
195	C₂⁴	—	2	4	no	—
196	C₂×D₄	—	2	8	no	—
197	C₂×D₄	—	2	8	no	—
198	C₂×D₄	—	2	4	no	—
199	C₂×D₄	—	2	8	no	—
200	C₂×D₄	—	2	8	no	—
201	C₂×D₄	—	4	4	no	—
202	C₂×D₄	—	4	4	no	—
203	C₂×D₄	—	4	4	no	—
204	C₂×D₄	—	4	4	no	—
205	C₂×D₄	C₂²×C₄×C₃₁	1	8	yes	—
206	C₂²×C₄	—	2	4	no	—
207	C₂²×C₄	—	4	8	no	—
208	C₂²×C₄	—	4	8	no	—
209	C₂²×C₄	—	4	8	no	—
210	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
211	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
212	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
213	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
214	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
215	D₈	—	4	16	no	—
216	C₂×C₈	—	2	4	no	—
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	—	2	4	no	—
219	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
220	C₄⋊C₄	—	2	8	no	—
221	C₄⋊C₄	—	4	4	no	—
222	C₄⋊C₄	—	4	4	no	—
223	C₂²⋊C₄	—	4	8	no	—
224	C₂²⋊C₄	—	4	8	no	—
225	C₂²⋊C₄	—	4	8	no	—
226	C₄²	—	2	2	no	—
227	C₄²	—	2	4	no	—
228	C₄²	—	2	4	no	—
229	C₄²	D₈×C₃₁	1	4	yes	complemented
230	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
231	C₄²	#41×C₃₁	1	2	yes	—
232	C₂³	—	2	8	no	—
233	C₂³	—	4	4	no	—
234	C₂³	—	4	4	no	—
235	C₂³	—	4	4	no	—
236	C₂³	—	4	4	no	—
237	C₂³	—	4	4	no	—
238	C₂³	—	4	4	no	—
239	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
240	D₄	—	2	16	no	—
241	D₄	—	4	16	no	—
242	D₄	—	4	16	no	—
243	D₄	—	4	8	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	4	4	no	—
250	C₂×C₄	—	4	4	no	—
251	C₂×C₄	—	4	4	no	—
252	C₂×C₄	—	8	16	no	—
253	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
254	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
255	C₂×C₄	C₂×D₈×C₃₁	1	4	yes	—
256	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
257	C₂×C₄	C₃₁×#135	1	2	yes	—
258	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	4	yes	—
259	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
260	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
261	C₂×C₄	C₃₁×H(5,𝔽₂)	1	4	yes	—
262	C₈	—	2	4	no	—
263	C₈	—	4	2	no	—
264	C₂²	—	4	8	no	—
265	C₂²	—	4	8	no	—
266	C₂²	—	4	8	no	—
267	C₂²	—	8	4	no	—
268	C₂²	—	8	4	no	—
269	C₂²	—	8	4	no	—
270	C₂²	—	8	4	no	—
271	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
272	C₂²	C₃₁×#535	1	2	yes	—
273	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
274	C₂²	C₂×C₃₁×#135	1	1	yes	characteristic
275	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	2	yes	—
276	C₄	—	2	4	no	—
277	C₄	—	4	2	no	—
278	C₄	—	8	4	no	—
279	C₄	C₄×D₈×C₃₁	1	4	yes	—
280	C₄	C₃₁×#442	1	4	yes	—
281	C₄	C₃₁×#518	1	2	yes	—
282	C₄	C₃₁×#540	1	2	yes	—
283	C₂	—	8	8	no	—
284	C₂	C₃₁×#2328	1	1	yes	characteristic
285	C₂	C₂×C₄×D₈×C₃₁	1	1	yes	characteristic
286	C₂	C₂×C₃₁×#442	1	1	yes	characteristic
287	C₂	C₃₁×#3205	1	2	yes	—
288	C₂	C₃₁×#3207	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂×C₃₁×#493	C₂
LCS	G ▷ C₂×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
2. C₄×C₃₁×#127	C₂	semidirect
3. C₃₁×#1957	C₂	semidirect
4. C₃₁×#1879	C₂	semidirect
5. C₃₁×#1834	C₂	semidirect
6. C₃₁×#1744	C₂	semidirect
12. C₃₁×#493	C₄	semidirect
28. C₂×D₈×C₃₁	C₂×C₄	semidirect
32. C₃₁×#127	C₂×C₄	semidirect
49. C₄×C₈×C₃₁	C₂×C₄	semidirect
84. C₄²×C₃₁	D₈	semidirect
87. #22269	C₃₁	direct (N × H)
121. C₄×#127	C₂×C₃₁	semidirect
122. #1957	C₂×C₃₁	semidirect
123. #1879	C₂×C₃₁	semidirect
124. #1834	C₂×C₃₁	semidirect
125. #1744	C₂×C₃₁	semidirect
150. #493	C₄×C₃₁	semidirect
172. C₂×D₈	C₂×C₄×C₃₁	semidirect
176. #127	C₂×C₄×C₃₁	semidirect
193. C₄×C₈	C₂×C₄×C₃₁	semidirect
194. C₃₁	#22269	direct (N × H)
229. C₄²	D₈×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×#493	2
C₄×C₃₁×#127	2
C₃₁×#1957	2
C₃₁×#1879	2
C₃₁×#1834	2
C₃₁×#1744	2
#22269	31

Subgroups — C31×#22269 #21479028

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22269 #21479028