Subgroups — C₂²×C₃₁×#480 #21517259

Order 7936.

3006 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	C₂²×C₃₁×#121	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₄⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#480	C₂	1	24	yes	maximal, direct product factor
4	C₂²×C₃₁×#107	C₂	1	2	yes	maximal, complemented
5	C₂²×C₃₁×#102	C₂	1	2	yes	maximal, complemented
6	C₂²×C₃₁×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×D₄×C₃₁	—	2	1	no	—
8	C₂³×D₄×C₃₁	C₂²	1	1	yes	characteristic
9	C₂⁴×C₄×C₃₁	—	2	1	no	—
10	C₂⁴×C₄×C₃₁	C₂²	1	1	yes	characteristic
11	C₂³×C₈×C₃₁	C₂²	1	2	yes	—
12	C₂×C₃₁×#121	—	2	6	no	—
13	C₂×C₃₁×#121	C₂²	1	12	yes	—
14	C₂²×C₄⋊C₄×C₃₁	—	2	6	no	—
15	C₂²×C₄⋊C₄×C₃₁	—	2	1	no	—
16	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
17	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
18	C₂²×C₄⋊C₄×C₃₁	C₂²	1	12	yes	complemented
19	C₂²×C₂²⋊C₄×C₃₁	—	2	1	no	—
20	C₃₁×#480	C₂²	1	64	yes	direct product factor
21	C₂×C₃₁×#107	C₂²	1	24	yes	complemented
22	C₂×C₃₁×#102	C₂²	1	24	yes	complemented
23	C₂×C₃₁×C₂²⋊C₈	C₂²	1	12	yes	complemented
24	C₂⁵×C₃₁	—	2	1	no	—
25	C₂⁵×C₃₁	D₄	1	1	yes	characteristic
26	C₂²×D₄×C₃₁	—	2	12	no	—
27	C₂²×D₄×C₃₁	—	2	3	no	—
28	C₂²×D₄×C₃₁	—	2	8	no	—
29	C₂²×D₄×C₃₁	—	4	8	no	—
30	C₂²×D₄×C₃₁	C₂³	1	6	yes	—
31	C₂³×C₄×C₃₁	—	2	8	no	—
32	C₂³×C₄×C₃₁	—	2	12	no	—
33	C₂³×C₄×C₃₁	—	2	1	no	—
34	C₂³×C₄×C₃₁	—	2	2	no	—
35	C₂³×C₄×C₃₁	—	4	8	no	—
36	C₂³×C₄×C₃₁	D₄	1	6	yes	—
37	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
38	C₂³×C₄×C₃₁	C₂³	1	6	yes	—
39	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
40	C₂²×C₈×C₃₁	—	2	8	no	—
41	C₂²×C₈×C₃₁	C₂³	1	12	yes	—
42	C₃₁×#121	—	2	24	no	—
43	C₃₁×#121	C₂³	1	16	yes	—
44	C₂×C₄⋊C₄×C₃₁	—	2	6	no	—
45	C₂×C₄⋊C₄×C₃₁	—	2	24	no	—
46	C₂×C₄⋊C₄×C₃₁	—	2	12	no	—
47	C₂×C₄⋊C₄×C₃₁	—	2	3	no	—
48	C₂×C₄⋊C₄×C₃₁	C₂³	1	12	yes	—
49	C₂×C₄⋊C₄×C₃₁	C₂³	1	6	yes	—
50	C₂×C₄⋊C₄×C₃₁	C₂³	1	16	yes	complemented
51	C₂×C₂²⋊C₄×C₃₁	—	2	12	no	—
52	C₃₁×#107	C₂³	1	32	yes	complemented
53	C₃₁×#102	C₂³	1	32	yes	complemented
54	C₃₁×C₂²⋊C₈	C₂³	1	16	yes	complemented
55	C₂⁴×C₃₁	—	2	4	no	—
56	C₂⁴×C₃₁	—	2	4	no	—
57	C₂⁴×C₃₁	—	2	6	no	—
58	C₂⁴×C₃₁	—	4	4	no	—
59	C₂⁴×C₃₁	—	4	4	no	—
60	C₂⁴×C₃₁	—	4	4	no	—
61	C₂⁴×C₃₁	D₈	1	8	yes	—
62	C₂⁴×C₃₁	C₂×D₄	1	6	yes	—
63	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
64	C₂×D₄×C₃₁	—	2	16	no	—
65	C₂×D₄×C₃₁	—	2	6	no	—
66	C₂×D₄×C₃₁	—	2	24	no	—
67	C₂×D₄×C₃₁	—	4	48	no	—
68	C₂×D₄×C₃₁	—	4	12	no	—
69	C₂×D₄×C₃₁	C₂⁴	1	4	yes	—
70	C₂²×C₄×C₃₁	—	2	4	no	—
71	C₂²×C₄×C₃₁	—	2	24	no	—
72	C₂²×C₄×C₃₁	—	2	24	no	—
73	C₂²×C₄×C₃₁	—	2	6	no	—
74	C₂²×C₄×C₃₁	—	2	12	no	—
75	C₂²×C₄×C₃₁	—	2	16	no	—
76	C₂²×C₄×C₃₁	—	4	48	no	—
77	C₂²×C₄×C₃₁	—	4	8	no	—
78	C₂²×C₄×C₃₁	—	4	4	no	—
79	C₂²×C₄×C₃₁	C₂×D₄	1	6	yes	—
80	C₂²×C₄×C₃₁	C₂×D₄	1	12	yes	—
81	C₂²×C₄×C₃₁	C₂×D₄	1	3	yes	—
82	C₂²×C₄×C₃₁	#41	1	8	yes	—
83	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
84	C₂²×C₄×C₃₁	C₂⁴	1	3	yes	—
85	C₂×C₈×C₃₁	—	2	24	no	—
86	C₂×C₈×C₃₁	C₂⁴	1	8	yes	—
87	C₄⋊C₄×C₃₁	—	2	12	no	—
88	C₄⋊C₄×C₃₁	—	2	16	no	—
89	C₄⋊C₄×C₃₁	—	2	6	no	—
90	C₄⋊C₄×C₃₁	C₂⁴	1	8	yes	—
91	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
92	C₂²⋊C₄×C₃₁	—	2	16	no	—
93	C₂²×#480	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
94	C₂³×C₃₁	—	2	12	no	—
95	C₂³×C₃₁	—	2	12	no	—
96	C₂³×C₃₁	—	2	32	no	—
97	C₂³×C₃₁	—	2	4	no	—
98	C₂³×C₃₁	—	4	12	no	—
99	C₂³×C₃₁	—	4	12	no	—
100	C₂³×C₃₁	—	4	12	no	—
101	C₂³×C₃₁	—	8	16	no	—
102	C₂³×C₃₁	#123	1	4	yes	—
103	C₂³×C₃₁	C₂×D₈	1	24	yes	—
104	C₂³×C₃₁	C₂×D₈	1	4	yes	—
105	C₂³×C₃₁	#137	1	4	yes	—
106	C₂³×C₃₁	C₂²×D₄	1	4	yes	—
107	C₂³×C₃₁	C₂²×D₄	1	3	yes	—
108	D₄×C₃₁	—	2	8	no	—
109	D₄×C₃₁	—	4	32	no	—
110	D₄×C₃₁	—	4	12	no	—
111	C₂×C₄×C₃₁	—	2	12	no	—
112	C₂×C₄×C₃₁	—	2	24	no	—
113	C₂×C₄×C₃₁	—	2	8	no	—
114	C₂×C₄×C₃₁	—	2	4	no	—
115	C₂×C₄×C₃₁	—	2	8	no	—
116	C₂×C₄×C₃₁	—	4	32	no	—
117	C₂×C₄×C₃₁	—	4	24	no	—
118	C₂×C₄×C₃₁	—	4	12	no	—
119	C₂×C₄×C₃₁	#123	1	12	yes	—
120	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
121	C₂×C₄×C₃₁	C₂²×D₄	1	3	yes	—
122	C₂×C₄×C₃₁	C₂×#41	1	12	yes	—
123	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
124	C₈×C₃₁	—	2	8	no	—
125	C₂²×#121	C₂×C₃₁	1	1	yes	characteristic
126	C₂³×C₄⋊C₄	C₂×C₃₁	1	1	yes	characteristic, complemented
127	C₂×#480	C₂×C₃₁	1	24	yes	direct product factor
128	C₂²×#107	C₂×C₃₁	1	2	yes	complemented
129	C₂²×#102	C₂×C₃₁	1	2	yes	complemented
130	C₂²×C₂²⋊C₈	C₂×C₃₁	1	1	yes	characteristic, complemented
131	C₂²×C₃₁	—	2	48	no	—
132	C₂²×C₃₁	—	2	4	no	—
133	C₂²×C₃₁	—	2	4	no	—
134	C₂²×C₃₁	—	4	4	no	—
135	C₂²×C₃₁	—	4	4	no	—
136	C₂²×C₃₁	—	4	4	no	—
137	C₂²×C₃₁	—	8	24	no	—
138	C₂²×C₃₁	#480	1	16	yes	direct product factor
139	C₂²×C₃₁	C₂×#123	1	6	yes	—
140	C₂²×C₃₁	C₂²×D₈	1	8	yes	—
141	C₂²×C₃₁	C₂²×D₈	1	6	yes	—
142	C₂²×C₃₁	C₂×#137	1	6	yes	—
143	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
144	C₄×C₃₁	—	2	4	no	—
145	C₄×C₃₁	—	4	8	no	—
146	C₄×C₃₁	—	4	4	no	—
147	C₄×C₃₁	C₂×#123	1	6	yes	—
148	C₄×C₃₁	C₂²×#41	1	2	yes	—
149	C₂³×D₄	—	2	1	no	—
150	C₂³×D₄	C₂²×C₃₁	1	1	yes	characteristic
151	C₂⁴×C₄	—	2	1	no	—
152	C₂⁴×C₄	C₂²×C₃₁	1	1	yes	characteristic
153	C₂³×C₈	C₂²×C₃₁	1	2	yes	—
154	C₂×#121	—	2	6	no	—
155	C₂×#121	C₂²×C₃₁	1	12	yes	—
156	C₂²×C₄⋊C₄	—	2	6	no	—
157	C₂²×C₄⋊C₄	—	2	1	no	—
158	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
159	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
160	C₂²×C₄⋊C₄	C₂²×C₃₁	1	12	yes	complemented
161	C₂²×C₂²⋊C₄	—	2	1	no	—
162	#480	C₂²×C₃₁	1	64	yes	direct product factor
163	C₂×#107	C₂²×C₃₁	1	24	yes	complemented
164	C₂×#102	C₂²×C₃₁	1	24	yes	complemented
165	C₂×C₂²⋊C₈	C₂²×C₃₁	1	12	yes	complemented
166	C₂×C₃₁	—	2	8	no	—
167	C₂×C₃₁	—	8	4	no	—
168	C₂×C₃₁	C₂×#480	1	12	yes	direct product factor
169	C₂×C₃₁	C₂²×#123	1	1	yes	characteristic
170	C₂×C₃₁	C₂³×D₈	1	1	yes	characteristic
171	C₂×C₃₁	C₂²×#137	1	1	yes	characteristic
172	C₂⁵	—	2	1	no	—
173	C₂⁵	D₄×C₃₁	1	1	yes	characteristic
174	C₂²×D₄	—	2	12	no	—
175	C₂²×D₄	—	2	3	no	—
176	C₂²×D₄	—	2	8	no	—
177	C₂²×D₄	—	4	8	no	—
178	C₂²×D₄	C₂³×C₃₁	1	6	yes	—
179	C₂³×C₄	—	2	8	no	—
180	C₂³×C₄	—	2	1	no	—
181	C₂³×C₄	—	2	2	no	—
182	C₂³×C₄	—	2	12	no	—
183	C₂³×C₄	—	4	8	no	—
184	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
185	C₂³×C₄	D₄×C₃₁	1	6	yes	—
186	C₂³×C₄	C₂³×C₃₁	1	6	yes	—
187	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
188	C₂²×C₈	—	2	8	no	—
189	C₂²×C₈	C₂³×C₃₁	1	12	yes	—
190	#121	—	2	24	no	—
191	#121	C₂³×C₃₁	1	16	yes	—
192	C₂×C₄⋊C₄	—	2	6	no	—
193	C₂×C₄⋊C₄	—	2	24	no	—
194	C₂×C₄⋊C₄	—	2	12	no	—
195	C₂×C₄⋊C₄	—	2	3	no	—
196	C₂×C₄⋊C₄	C₂³×C₃₁	1	12	yes	—
197	C₂×C₄⋊C₄	C₂³×C₃₁	1	6	yes	—
198	C₂×C₄⋊C₄	C₂³×C₃₁	1	16	yes	complemented
199	C₂×C₂²⋊C₄	—	2	12	no	—
200	#107	C₂³×C₃₁	1	32	yes	complemented
201	#102	C₂³×C₃₁	1	32	yes	complemented
202	C₂²⋊C₈	C₂³×C₃₁	1	16	yes	complemented
203	C₃₁	C₂²×#480	1	1	yes	characteristic, direct product factor, Sylow
204	C₂⁴	—	2	4	no	—
205	C₂⁴	—	2	4	no	—
206	C₂⁴	—	2	6	no	—
207	C₂⁴	—	4	4	no	—
208	C₂⁴	—	4	4	no	—
209	C₂⁴	—	4	4	no	—
210	C₂⁴	D₈×C₃₁	1	8	yes	—
211	C₂⁴	C₂×D₄×C₃₁	1	6	yes	—
212	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
213	C₂×D₄	—	2	16	no	—
214	C₂×D₄	—	2	6	no	—
215	C₂×D₄	—	2	24	no	—
216	C₂×D₄	—	4	48	no	—
217	C₂×D₄	—	4	12	no	—
218	C₂×D₄	C₂⁴×C₃₁	1	4	yes	—
219	C₂²×C₄	—	2	4	no	—
220	C₂²×C₄	—	2	24	no	—
221	C₂²×C₄	—	2	24	no	—
222	C₂²×C₄	—	2	6	no	—
223	C₂²×C₄	—	2	12	no	—
224	C₂²×C₄	—	2	16	no	—
225	C₂²×C₄	—	4	48	no	—
226	C₂²×C₄	—	4	8	no	—
227	C₂²×C₄	—	4	4	no	—
228	C₂²×C₄	C₂×D₄×C₃₁	1	6	yes	—
229	C₂²×C₄	C₂×D₄×C₃₁	1	12	yes	—
230	C₂²×C₄	C₂×D₄×C₃₁	1	3	yes	—
231	C₂²×C₄	#41×C₃₁	1	8	yes	—
232	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
233	C₂²×C₄	C₂⁴×C₃₁	1	3	yes	—
234	C₂×C₈	—	2	24	no	—
235	C₂×C₈	C₂⁴×C₃₁	1	8	yes	—
236	C₄⋊C₄	—	2	12	no	—
237	C₄⋊C₄	—	2	16	no	—
238	C₄⋊C₄	—	2	6	no	—
239	C₄⋊C₄	C₂⁴×C₃₁	1	8	yes	—
240	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
241	C₂²⋊C₄	—	2	16	no	—
242	C₂³	—	2	32	no	—
243	C₂³	—	2	12	no	—
244	C₂³	—	2	12	no	—
245	C₂³	—	2	4	no	—
246	C₂³	—	4	12	no	—
247	C₂³	—	4	12	no	—
248	C₂³	—	4	12	no	—
249	C₂³	—	8	16	no	—
250	C₂³	C₃₁×#123	1	4	yes	—
251	C₂³	C₂×D₈×C₃₁	1	24	yes	—
252	C₂³	C₂×D₈×C₃₁	1	4	yes	—
253	C₂³	C₃₁×#137	1	4	yes	—
254	C₂³	C₂²×D₄×C₃₁	1	4	yes	—
255	C₂³	C₂²×D₄×C₃₁	1	3	yes	—
256	D₄	—	2	8	no	—
257	D₄	—	4	32	no	—
258	D₄	—	4	12	no	—
259	C₂×C₄	—	2	12	no	—
260	C₂×C₄	—	2	24	no	—
261	C₂×C₄	—	2	8	no	—
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	2	8	no	—
264	C₂×C₄	—	4	32	no	—
265	C₂×C₄	—	4	24	no	—
266	C₂×C₄	—	4	12	no	—
267	C₂×C₄	C₃₁×#123	1	12	yes	—
268	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
269	C₂×C₄	C₂²×D₄×C₃₁	1	3	yes	—
270	C₂×C₄	C₂×#41×C₃₁	1	12	yes	—
271	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
272	C₈	—	2	8	no	—
273	C₂²	—	2	48	no	—
274	C₂²	—	2	4	no	—
275	C₂²	—	2	4	no	—
276	C₂²	—	4	4	no	—
277	C₂²	—	4	4	no	—
278	C₂²	—	4	4	no	—
279	C₂²	—	8	24	no	—
280	C₂²	C₃₁×#480	1	16	yes	direct product factor
281	C₂²	C₂×C₃₁×#123	1	6	yes	—
282	C₂²	C₂²×D₈×C₃₁	1	8	yes	—
283	C₂²	C₂²×D₈×C₃₁	1	6	yes	—
284	C₂²	C₂×C₃₁×#137	1	6	yes	—
285	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
286	C₄	—	2	4	no	—
287	C₄	—	4	8	no	—
288	C₄	—	4	4	no	—
289	C₄	C₂×C₃₁×#123	1	6	yes	—
290	C₄	C₂²×#41×C₃₁	1	2	yes	—
291	C₂	—	2	8	no	—
292	C₂	—	8	4	no	—
293	C₂	C₂×C₃₁×#480	1	12	yes	direct product factor
294	C₂	C₂²×C₃₁×#123	1	1	yes	characteristic
295	C₂	C₂³×D₈×C₃₁	1	1	yes	characteristic
296	C₂	C₂²×C₃₁×#137	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴×C₃₁	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
prime radical	C₂²×C₃₁×#121	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₃₁ ◁ C₂⁴×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂³×C₄⋊C₄×C₃₁	C₂	semidirect
3. C₂×C₃₁×#480	C₂	direct (N × H)
4. C₂²×C₃₁×#107	C₂	semidirect
5. C₂²×C₃₁×#102	C₂	semidirect
6. C₂²×C₃₁×C₂²⋊C₈	C₂	semidirect
18. C₂²×C₄⋊C₄×C₃₁	C₂²	semidirect
20. C₃₁×#480	C₂²	direct (N × H)
21. C₂×C₃₁×#107	C₂²	semidirect
22. C₂×C₃₁×#102	C₂²	semidirect
23. C₂×C₃₁×C₂²⋊C₈	C₂²	semidirect
50. C₂×C₄⋊C₄×C₃₁	C₂³	semidirect
52. C₃₁×#107	C₂³	semidirect
53. C₃₁×#102	C₂³	semidirect
54. C₃₁×C₂²⋊C₈	C₂³	semidirect
93. C₂²×#480	C₃₁	direct (N × H)
126. C₂³×C₄⋊C₄	C₂×C₃₁	semidirect
127. C₂×#480	C₂×C₃₁	direct (N × H)
128. C₂²×#107	C₂×C₃₁	semidirect
129. C₂²×#102	C₂×C₃₁	semidirect
130. C₂²×C₂²⋊C₈	C₂×C₃₁	semidirect
138. C₂²×C₃₁	#480	direct (N × H)
160. C₂²×C₄⋊C₄	C₂²×C₃₁	semidirect
162. #480	C₂²×C₃₁	direct (N × H)
163. C₂×#107	C₂²×C₃₁	semidirect
164. C₂×#102	C₂²×C₃₁	semidirect
165. C₂×C₂²⋊C₈	C₂²×C₃₁	semidirect
168. C₂×C₃₁	C₂×#480	direct (N × H)
198. C₂×C₄⋊C₄	C₂³×C₃₁	semidirect
200. #107	C₂³×C₃₁	semidirect
201. #102	C₂³×C₃₁	semidirect
202. C₂²⋊C₈	C₂³×C₃₁	semidirect
203. C₃₁	C₂²×#480	direct (N × H)
280. C₂²	C₃₁×#480	direct (N × H)
293. C₂	C₂×C₃₁×#480	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃₁×#121	2
C₂³×C₄⋊C₄×C₃₁	2
C₂×C₃₁×#480	2
C₂²×C₃₁×#107	2
C₂²×C₃₁×#102	2
C₂²×C₃₁×C₂²⋊C₈	2
C₂²×#480	31

Subgroups — C22×C31×#480 #21517259

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃₁×#480 #21517259