Subgroups — #13043654

Order 1280.

558 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	#41×#168	C₂	1	1	yes	maximal, characteristic
2	#10604448	C₂	1	2	yes	maximal, complemented
3	#10604414	C₂	1	2	yes	maximal, complemented
4	#10604413	C₂	1	4	yes	maximal, complemented
5	#10604151	C₂	1	2	yes	maximal, complemented
6	#10604151	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×#3998	C₂	1	2	yes	maximal, complemented
8	C₂×C₄×#674	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×C₅×#41	C₄	1	1	yes	characteristic
10	C₂×#3998	C₂²	1	4	yes	—
11	C₂²×#674	C₂²	1	2	yes	—
12	Q₈×#168	C₄	1	1	yes	characteristic
13	Q₈×#168	C₂²	1	1	yes	characteristic
14	D₄×#168	C₄	1	2	yes	—
15	D₄×#168	C₄	1	1	yes	characteristic
16	D₄×#168	C₂²	1	2	yes	—
17	D₄×#168	C₂²	1	1	yes	characteristic
18	#65214	C₄	1	2	yes	—
19	#65214	C₄	1	1	yes	characteristic
20	C₂×C₄×#168	C₂²	1	1	yes	characteristic
21	C₂×C₄×#168	C₂²	1	2	yes	—
22	C₂²⋊#674	C₂²	1	2	yes	—
23	C₂²⋊#674	C₂²	1	4	yes	—
24	#64859	C₂²	1	4	yes	—
25	C₄⋊#674	C₂²	1	4	yes	—
26	C₄⋊#674	C₂²	1	2	yes	—
27	C₄×#674	C₂²	1	1	yes	characteristic
28	C₄×#674	C₂²	1	1	yes	characteristic
29	C₄×#674	C₂²	1	2	yes	—
30	C₄×#674	C₂²	1	2	yes	—
31	#33479	—	5	1	no	maximal, Sylow
32	C₂×C₅×#41	C₈	1	1	yes	characteristic
33	C₄×C₅×Q₈	C₈	1	1	yes	characteristic
34	C₄×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
35	C₄×C₅×D₄	C₈	1	2	yes	—
36	C₄×C₅×D₄	C₈	1	1	yes	characteristic
37	C₄×C₅×D₄	C₂×C₄	1	2	yes	—
38	C₄×C₅×D₄	C₂×C₄	1	1	yes	characteristic
39	C₅×#117	C₈	1	2	yes	—
40	C₅×#117	C₈	1	1	yes	characteristic
41	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
43	C₂²×#168	C₂×C₄	1	1	yes	characteristic
44	C₂²×#168	C₂×C₄	1	2	yes	—
45	C₂²×#168	C₂³	1	1	yes	characteristic
46	C₂²×#168	C₂³	1	2	yes	—
47	#3998	—	2	8	no	—
48	C₂×#674	—	2	4	no	—
49	C₂×#674	C₂³	1	2	yes	—
50	C₂×#674	C₂³	1	4	yes	—
51	C₂×#674	C₂³	1	2	yes	—
52	C₂²⋊#168	C₂×C₄	1	2	yes	—
53	C₂²⋊#168	C₂×C₄	1	2	yes	—
54	C₂²⋊#168	C₂×C₄	1	2	yes	—
55	C₄⋊#168	C₂×C₄	1	2	yes	—
56	C₄⋊#168	C₂×C₄	1	1	yes	characteristic
57	C₄⋊#168	C₂×C₄	1	2	yes	—
58	C₄⋊#168	C₂×C₄	1	1	yes	characteristic
59	C₄×#168	C₂×C₄	1	1	yes	characteristic
60	C₄×#168	C₂×C₄	1	2	yes	—
61	C₄×#168	C₂×C₄	1	1	yes	characteristic
62	C₄×#168	C₂×C₄	1	2	yes	—
63	C₄×#168	C₂³	1	1	yes	characteristic
64	C₄×#168	C₂³	1	2	yes	—
65	C₄×#168	C₂³	1	1	yes	characteristic
66	C₈×#41	—	5	1	no	—
67	#2183	—	5	2	no	—
68	#2169	—	5	2	no	—
69	#2168	—	5	4	no	—
70	#2162	—	5	2	no	—
71	#2162	—	5	1	no	—
72	C₄×M₃₂	—	5	2	no	—
73	C₂×C₄×C₁₆	—	5	1	no	—
74	C₅×#41	—	2	4	no	—
75	C₂×C₅×Q₈	C₂×C₈	1	1	yes	characteristic
76	C₂×C₅×D₄	C₂×C₈	1	2	yes	—
77	C₂×C₅×D₄	C₂×C₈	1	1	yes	characteristic
78	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
79	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
80	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
81	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
82	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
83	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
84	C₅×C₄⋊C₄	C₂×C₈	1	2	yes	—
85	C₅×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
86	C₅×C₄⋊C₄	C₂×C₈	1	2	yes	—
87	C₅×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
88	C₅×C₂²⋊C₄	C₂×C₈	1	2	yes	—
89	C₅×C₂²⋊C₄	C₂×C₈	1	2	yes	—
90	C₅×C₂²⋊C₄	C₂×C₈	1	2	yes	—
91	C₄²×C₅	C₂×C₈	1	2	yes	—
92	C₄²×C₅	C₂×C₈	1	1	yes	characteristic
93	C₄²×C₅	C₂×C₈	1	2	yes	—
94	C₄²×C₅	C₂×C₈	1	1	yes	characteristic
95	C₄²×C₅	C₂²×C₄	1	2	yes	—
96	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
97	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
98	C₂×#168	—	2	2	no	—
99	C₂×#168	—	2	4	no	—
100	C₂×#168	—	2	2	no	—
101	C₂×#168	—	2	4	no	—
102	C₂×#168	C₂²×C₄	1	1	yes	characteristic
103	C₂×#168	C₂²×C₄	1	2	yes	—
104	C₂×#168	C₂²×C₄	1	1	yes	characteristic
105	C₂×#168	C₂²×C₄	1	2	yes	—
106	C₂×#168	C₂²×C₄	1	1	yes	characteristic
107	C₂×#168	C₂⁴	1	1	yes	characteristic
108	#674	—	2	2	no	—
109	#674	—	2	4	no	—
110	#674	#41	1	4	yes	complemented
111	C₄×#41	AGL(1,𝔽₅)	1	1	yes	characteristic
112	C₂×M₃₂	—	5	4	no	—
113	C₂²×C₁₆	—	5	2	no	—
114	C₈×Q₈	—	5	1	no	—
115	C₈×Q₈	—	5	1	no	—
116	C₈×D₄	—	5	1	no	—
117	C₈×D₄	—	5	2	no	—
118	C₈×D₄	—	5	2	no	—
119	C₈×D₄	—	5	1	no	—
120	#431	—	5	2	no	—
121	#431	—	5	1	no	—
122	C₂×C₄×C₈	—	5	1	no	—
123	C₂×C₄×C₈	—	5	2	no	—
124	C₄⋊C₁₆	—	5	4	no	—
125	C₄⋊C₁₆	—	5	2	no	—
126	C₂²⋊C₁₆	—	5	2	no	—
127	C₂²⋊C₁₆	—	5	4	no	—
128	#346	—	5	4	no	—
129	C₄×C₁₆	—	5	1	no	—
130	C₄×C₁₆	—	5	1	no	—
131	C₄×C₁₆	—	5	2	no	—
132	C₄×C₁₆	—	5	2	no	—
133	C₂³×C₅	C₂²×C₈	1	2	yes	—
134	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
135	C₅×Q₈	—	2	2	no	—
136	C₅×D₄	—	2	4	no	—
137	C₅×D₄	—	2	2	no	—
138	C₂×C₄×C₅	—	2	1	no	—
139	C₂×C₄×C₅	—	2	2	no	—
140	C₂×C₄×C₅	—	2	1	no	—
141	C₂×C₄×C₅	—	2	2	no	—
142	C₂×C₄×C₅	—	2	4	no	—
143	C₂×C₄×C₅	—	2	2	no	—
144	C₂×C₄×C₅	—	2	4	no	—
145	C₂×C₄×C₅	M₃₂	1	4	yes	complemented
146	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
147	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
148	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
149	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
150	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
151	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
152	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
153	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
154	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
155	#168	—	2	1	no	—
156	#168	—	2	2	no	—
157	#168	—	2	1	no	—
158	#168	—	2	2	no	—
159	#168	#131	1	2	yes	—
160	#168	C₂×#41	1	2	yes	—
161	C₂×#41	#170	1	1	yes	characteristic
162	C₂²×C₈	—	5	1	no	—
163	C₂²×C₈	—	5	2	no	—
164	C₂²×C₈	—	5	1	no	—
165	C₂²×C₈	—	5	2	no	—
166	C₄×Q₈	#170	1	1	yes	characteristic
167	C₄×Q₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
168	C₄×D₄	#170	1	2	yes	—
169	C₄×D₄	#170	1	1	yes	characteristic
170	C₄×D₄	C₂×AGL(1,𝔽₅)	1	2	yes	—
171	C₄×D₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
172	#117	#170	1	2	yes	—
173	#117	#170	1	1	yes	characteristic
174	C₂×C₄²	C₂×AGL(1,𝔽₅)	1	2	yes	—
175	C₂×C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
176	M₃₂	—	10	8	no	—
177	C₂×C₁₆	—	5	2	no	—
178	C₂×C₁₆	—	5	2	no	—
179	C₂×C₁₆	—	5	4	no	—
180	C₂×C₁₆	—	10	4	no	—
181	C₄⋊C₈	—	5	2	no	—
182	C₄⋊C₈	—	5	1	no	—
183	C₄⋊C₈	—	5	2	no	—
184	C₄⋊C₈	—	5	1	no	—
185	C₂²⋊C₈	—	5	2	no	—
186	C₂²⋊C₈	—	5	2	no	—
187	C₂²⋊C₈	—	5	2	no	—
188	C₄×C₈	—	5	1	no	—
189	C₄×C₈	—	5	2	no	—
190	C₄×C₈	—	5	1	no	—
191	C₄×C₈	—	5	2	no	—
192	C₄×C₈	—	5	1	no	—
193	C₄×C₈	—	5	1	no	—
194	C₄×C₈	—	5	2	no	—
195	C₂²×C₅	—	2	1	no	—
196	C₂²×C₅	—	2	1	no	—
197	C₂²×C₅	—	2	2	no	—
198	C₂²×C₅	—	2	2	no	—
199	C₂²×C₅	—	2	2	no	—
200	C₂²×C₅	C₂×M₃₂	1	2	yes	—
201	C₂²×C₅	C₂³×C₈	1	1	yes	characteristic
202	C₄×C₅	—	2	2	no	—
203	C₄×C₅	—	2	1	no	—
204	C₄×C₅	—	2	2	no	—
205	C₄×C₅	—	2	1	no	—
206	C₄×C₅	—	2	2	no	—
207	C₄×C₅	C₂×M₃₂	1	2	yes	—
208	C₄×C₅	C₂×M₃₂	1	2	yes	—
209	C₄×C₅	#504	1	2	yes	—
210	C₄×C₅	C₄×#41	1	1	yes	characteristic
211	C₄×C₅	C₂×#131	1	1	yes	characteristic
212	#41	—	2	4	no	—
213	C₂×Q₈	C₂×#170	1	1	yes	characteristic
214	C₂×D₄	C₂×#170	1	2	yes	—
215	C₂×D₄	C₂×#170	1	1	yes	characteristic
216	C₂²×C₄	C₂×#170	1	2	yes	—
217	C₂²×C₄	C₂×#170	1	2	yes	—
218	C₂²×C₄	C₂×#170	1	1	yes	characteristic
219	C₂²×C₄	C₂×#170	1	1	yes	characteristic
220	C₂²×C₄	C₂²×AGL(1,𝔽₅)	1	2	yes	—
221	C₂²×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
222	C₂×C₈	—	5	1	no	—
223	C₂×C₈	—	5	1	no	—
224	C₂×C₈	—	5	1	no	—
225	C₂×C₈	—	5	2	no	—
226	C₂×C₈	—	5	1	no	—
227	C₂×C₈	—	5	2	no	—
228	C₂×C₈	—	10	2	no	—
229	C₂×C₈	—	10	4	no	—
230	C₂×C₈	—	10	2	no	—
231	C₂×C₈	—	10	4	no	—
232	C₄⋊C₄	C₂×#170	1	2	yes	—
233	C₄⋊C₄	C₂×#170	1	1	yes	characteristic
234	C₄⋊C₄	C₂×#170	1	2	yes	—
235	C₄⋊C₄	C₂×#170	1	1	yes	characteristic
236	C₂²⋊C₄	C₂×#170	1	2	yes	—
237	C₂²⋊C₄	C₂×#170	1	2	yes	—
238	C₂²⋊C₄	C₂×#170	1	2	yes	—
239	C₄²	C₂×#170	1	2	yes	—
240	C₄²	C₂×#170	1	1	yes	characteristic
241	C₄²	C₂×#170	1	2	yes	—
242	C₄²	C₂×#170	1	1	yes	characteristic
243	C₄²	C₂²×AGL(1,𝔽₅)	1	2	yes	—
244	C₄²	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
245	C₄²	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
246	C₁₆	—	5	4	no	—
247	C₁₆	—	10	2	no	—
248	C₁₆	—	10	4	no	—
249	C₂×C₅	—	2	2	no	—
250	C₂×C₅	—	4	2	no	—
251	C₂×C₅	C₈×#41	1	1	yes	characteristic
252	C₂×C₅	C₂²×M₃₂	1	1	yes	characteristic
253	C₂×C₅	C₂×#504	1	1	yes	characteristic
254	C₂³	C₂²×#170	1	2	yes	—
255	C₂³	C₂²×#170	1	1	yes	characteristic
256	Q₈	—	2	2	no	—
257	D₄	—	2	4	no	—
258	D₄	—	2	2	no	—
259	C₂×C₄	—	2	4	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	#3998	1	4	yes	complemented
267	C₂×C₄	C₂²×#170	1	2	yes	—
268	C₂×C₄	C₂²×#170	1	1	yes	characteristic
269	C₂×C₄	C₂²×#170	1	2	yes	—
270	C₂×C₄	C₂²×#170	1	1	yes	characteristic
271	C₂×C₄	C₂²×#170	1	2	yes	—
272	C₂×C₄	C₂²×#170	1	1	yes	characteristic
273	C₂×C₄	C₂²×#170	1	1	yes	characteristic
274	C₂×C₄	C₂²×#170	1	1	yes	characteristic
275	C₂×C₄	C₂³×AGL(1,𝔽₅)	1	1	yes	characteristic
276	C₈	—	5	2	no	—
277	C₈	—	5	2	no	—
278	C₈	—	10	1	no	—
279	C₈	—	10	2	no	—
280	C₈	—	10	1	no	—
281	C₈	—	10	2	no	—
282	C₅	#33479	1	1	yes	characteristic, complemented, Sylow
283	C₂²	—	2	1	no	—
284	C₂²	—	2	1	no	—
285	C₂²	—	2	2	no	—
286	C₂²	—	2	2	no	—
287	C₂²	—	2	2	no	—
288	C₂²	C₂×#3998	1	2	yes	—
289	C₂²	C₂³×#170	1	1	yes	characteristic
290	C₄	—	2	2	no	—
291	C₄	—	2	1	no	—
292	C₄	—	2	2	no	—
293	C₄	—	2	1	no	—
294	C₄	—	2	2	no	—
295	C₄	C₂×#3998	1	2	yes	—
296	C₄	C₂×#3998	1	2	yes	—
297	C₄	#65791	1	2	yes	—
298	C₄	D₅⋊#131	1	1	yes	characteristic
299	C₄	#41×AGL(1,𝔽₅)	1	1	yes	characteristic
300	C₂	—	2	2	no	—
301	C₂	—	4	2	no	—
302	C₂	C₂²×#3998	1	1	yes	characteristic
303	C₂	C₂×#65791	1	1	yes	characteristic
304	C₂	#41×#170	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	C₂²×AGL(1,𝔽₅)
commutator, socle	C₂²×C₅	C₂³×C₈
Frattini	C₂×C₄	C₂³×AGL(1,𝔽₅)
Fitting	C₄×C₅×#41	C₄
prime radical	C₂×C₅×#41	C₈
cosocle	C₂×#168	C₂⁴
hypercenter	C₄×#41	AGL(1,𝔽₅)
hypocenter	C₅	#33479
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₄² ◁ C₄×#41

Complements

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

N	H ≅ G/N	type
2. #10604448	C₂	semidirect
3. #10604414	C₂	semidirect
4. #10604413	C₂	semidirect
5. #10604151	C₂	semidirect
6. #10604151	C₂	semidirect
7. C₄×#3998	C₂	semidirect
8. C₂×C₄×#674	C₂	semidirect
110. #674	#41	semidirect
145. C₂×C₄×C₅	M₃₂	semidirect
266. C₂×C₄	#3998	semidirect
282. C₅	#33479	semidirect

Maximal subgroups

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

subgroup	index
#41×#168	2
#10604448	2
#10604414	2
#10604413	2
#10604151	2
C₄×#3998	2
C₂×C₄×#674	2
#33479	5