Subgroups — C₂²×#66017 #13006172

Order 1280.

10246 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	C₂²×C₅×#124	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#4093	C₂	1	2	yes	maximal, complemented
3	C₂²×#4083	C₂	1	2	yes	maximal, complemented
4	C₂×#66017	C₂	1	48	yes	maximal, direct product factor
5	C₂²×#4028	C₂	1	4	yes	maximal, complemented
6	C₂²×#4031	C₂	1	4	yes	maximal, complemented
7	C₂²×#4021	C₂	1	2	yes	maximal, complemented
8	C₂⁵×D₅	C₂²	1	2	yes	complemented
9	C₂³×C₅×Q₈	C₂²	1	1	yes	characteristic, complemented
10	C₂³×C₅×D₄	C₂²	1	1	yes	characteristic
11	C₂³×C₂²⋊D₅	C₂²	1	4	yes	—
12	C₂³×D₂₀	C₂²	1	2	yes	—
13	C₂³×D₂₀	C₂²	1	2	yes	—
14	C₂³×Dic₁₀	C₂²	1	2	yes	complemented
15	C₂×C₅×#124	C₂²	1	24	yes	complemented
16	C₂²×C₅×C₂²⋊C₄	C₂²	1	4	yes	complemented
17	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic, complemented
18	C₂²×C₂²⋊Dic₅	C₂²	1	2	yes	complemented
19	C₂×#4093	C₂²	1	48	yes	complemented
20	C₂×#4083	C₂²	1	48	yes	complemented
21	C₂²×#685	C₂²	1	4	yes	complemented
22	C₂²×#685	C₂²	1	4	yes	complemented
23	C₂²×#685	C₂²	1	4	yes	complemented
24	C₂²×C₄×Dic₅	C₂²	1	2	yes	complemented
25	#66017	C₂²	1	256	yes	direct product factor
26	C₂×#4031	C₂²	1	96	yes	complemented
27	C₂×#4028	C₂²	1	96	yes	complemented
28	C₂×#4021	C₂²	1	48	yes	complemented
29	C₂²×H(3,𝔽₄)	—	5	1	no	maximal, Sylow
30	C₂⁵×C₅	C₂³	1	2	yes	—
31	C₂⁴×D₅	—	4	8	no	—
32	C₂⁴×D₅	—	4	16	no	—
33	C₂⁴×D₅	C₂³	1	4	yes	—
34	C₂⁴×D₅	C₂³	1	24	yes	complemented
35	C₂²×C₅×Q₈	—	4	4	no	—
36	C₂²×C₅×Q₈	C₂³	1	12	yes	complemented
37	C₂²×C₅×D₄	—	4	4	no	—
38	C₂²×C₅×D₄	C₂³	1	12	yes	—
39	C₂³×C₄×C₅	C₂³	1	2	yes	—
40	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
41	C₂³×C₄×C₅	C₂³	1	2	yes	—
42	C₂²×C₂²⋊D₅	—	4	16	no	—
43	C₂²×C₂²⋊D₅	C₂³	1	48	yes	—
44	C₂³×Dic₅	C₂³	1	4	yes	—
45	C₂²×D₂₀	—	4	8	no	—
46	C₂²×D₂₀	—	4	8	no	—
47	C₂²×D₂₀	C₂³	1	24	yes	—
48	C₂²×D₂₀	C₂³	1	24	yes	—
49	C₂²×Dic₁₀	—	4	8	no	—
50	C₂²×Dic₁₀	C₂³	1	24	yes	complemented
51	C₅×#124	C₂³	1	64	yes	complemented
52	C₂×C₅×C₂²⋊C₄	C₂³	1	48	yes	complemented
53	C₂×C₄²×C₅	C₂³	1	12	yes	complemented
54	C₂×C₂²⋊Dic₅	C₂³	1	24	yes	complemented
55	#4093	C₂³	1	128	yes	complemented
56	#4083	C₂³	1	128	yes	complemented
57	C₂×#685	C₂³	1	48	yes	complemented
58	C₂×#685	C₂³	1	48	yes	complemented
59	C₂×#685	C₂³	1	48	yes	complemented
60	C₂×C₄×Dic₅	C₂³	1	24	yes	complemented
61	#4031	C₂³	1	256	yes	complemented
62	#4028	C₂³	1	256	yes	complemented
63	#4021	C₂³	1	128	yes	complemented
64	C₂×H(3,𝔽₄)	—	5	48	no	—
65	C₂²×#124	—	5	2	no	—
66	C₂²×#124	—	5	2	no	—
67	C₂²×#124	—	5	4	no	—
68	C₂²×#124	D₅	1	1	yes	characteristic, complemented
69	C₂²×#120	—	5	2	no	—
70	C₂²×#120	—	5	4	no	—
71	C₂⁴×C₅	—	2	16	no	—
72	C₂⁴×C₅	—	2	8	no	—
73	C₂⁴×C₅	C₂⁴	1	12	yes	—
74	C₂⁴×C₅	C₂⁴	1	1	yes	characteristic
75	C₂³×D₅	—	2	16	no	—
76	C₂³×D₅	—	2	16	no	—
77	C₂³×D₅	—	2	16	no	—
78	C₂³×D₅	—	4	48	no	—
79	C₂³×D₅	—	4	96	no	—
80	C₂³×D₅	—	4	128	no	—
81	C₂³×D₅	C₂⁴	1	24	yes	—
82	C₂³×D₅	C₂⁴	1	32	yes	complemented
83	C₂×C₅×Q₈	—	4	24	no	—
84	C₂×C₅×Q₈	C₂⁴	1	16	yes	complemented
85	C₂×C₅×D₄	—	4	24	no	—
86	C₂×C₅×D₄	C₂⁴	1	16	yes	—
87	C₂²×C₄×C₅	—	2	8	no	—
88	C₂²×C₄×C₅	—	2	8	no	—
89	C₂²×C₄×C₅	—	2	4	no	—
90	C₂²×C₄×C₅	C₂⁴	1	12	yes	—
91	C₂²×C₄×C₅	C₂⁴	1	6	yes	—
92	C₂²×C₄×C₅	C₂⁴	1	12	yes	—
93	C₂×C₂²⋊D₅	—	4	96	no	—
94	C₂×C₂²⋊D₅	C₂⁴	1	64	yes	—
95	C₂²×Dic₅	—	2	16	no	—
96	C₂²×Dic₅	C₂⁴	1	24	yes	—
97	C₂×D₂₀	—	4	48	no	—
98	C₂×D₂₀	—	4	48	no	—
99	C₂×D₂₀	C₂⁴	1	32	yes	—
100	C₂×D₂₀	C₂⁴	1	32	yes	—
101	C₂×Dic₁₀	—	4	48	no	—
102	C₂×Dic₁₀	C₂⁴	1	32	yes	complemented
103	C₅×C₂²⋊C₄	C₂⁴	1	64	yes	complemented
104	C₄²×C₅	C₂⁴	1	16	yes	complemented
105	C₂²⋊Dic₅	C₂⁴	1	32	yes	complemented
106	#685	C₂⁴	1	64	yes	complemented
107	#685	C₂⁴	1	64	yes	complemented
108	#685	C₂⁴	1	64	yes	complemented
109	C₄×Dic₅	C₂⁴	1	32	yes	complemented
110	C₂⁶	—	5	2	no	—
111	C₂³×Q₈	—	5	2	no	—
112	C₂³×Q₈	C₂×D₅	1	1	yes	characteristic, complemented
113	C₂³×D₄	—	5	4	no	—
114	C₂³×D₄	—	5	2	no	—
115	C₂³×D₄	—	5	2	no	—
116	C₂³×D₄	C₂×D₅	1	1	yes	characteristic
117	H(3,𝔽₄)	—	5	256	no	—
118	C₂×#124	—	5	48	no	—
119	C₂×#124	—	5	96	no	—
120	C₂×#124	—	5	48	no	—
121	C₂×#124	C₂×D₅	1	24	yes	complemented
122	C₂×#120	—	5	48	no	—
123	C₂×#120	—	5	96	no	—
124	C₂²×C₂²⋊C₄	—	5	2	no	—
125	C₂²×C₂²⋊C₄	—	5	4	no	—
126	C₂²×C₂²⋊C₄	—	5	4	no	—
127	C₂²×C₂²⋊C₄	—	5	4	no	—
128	C₂²×C₂²⋊C₄	C₂×D₅	1	4	yes	complemented
129	C₂²×C₄²	—	5	2	no	—
130	C₂²×C₄²	C₂×D₅	1	1	yes	characteristic, complemented
131	C₂³×C₅	—	2	48	no	—
132	C₂³×C₅	—	2	24	no	—
133	C₂³×C₅	—	4	32	no	—
134	C₂³×C₅	H(5,𝔽₂)	1	4	yes	—
135	C₂³×C₅	H(5,𝔽₂)	1	8	yes	—
136	C₂³×C₅	C₂⁵	1	8	yes	—
137	C₂³×C₅	C₂⁵	1	3	yes	—
138	C₂²×D₅	—	2	48	no	—
139	C₂²×D₅	—	2	48	no	—
140	C₂²×D₅	—	2	48	no	—
141	C₂²×D₅	—	4	32	no	—
142	C₂²×D₅	—	4	64	no	—
143	C₂²×D₅	—	4	384	no	—
144	C₂²×D₅	—	4	64	no	—
145	C₂²×D₅	C₂⁵	1	16	yes	—
146	C₅×Q₈	—	4	16	no	—
147	C₅×D₄	—	4	16	no	—
148	C₂×C₄×C₅	—	2	24	no	—
149	C₂×C₄×C₅	—	2	24	no	—
150	C₂×C₄×C₅	—	2	12	no	—
151	C₂×C₄×C₅	C₂⁵	1	8	yes	—
152	C₂×C₄×C₅	C₂⁵	1	4	yes	—
153	C₂×C₄×C₅	C₂⁵	1	8	yes	—
154	C₂²⋊D₅	—	4	64	no	—
155	C₂×Dic₅	—	2	48	no	—
156	C₂×Dic₅	C₂⁵	1	16	yes	—
157	D₂₀	—	4	32	no	—
158	D₂₀	—	4	32	no	—
159	Dic₁₀	—	4	32	no	—
160	C₂⁵	—	5	4	no	—
161	C₂⁵	—	5	24	no	—
162	C₂⁵	—	20	8	no	—
163	C₂⁵	—	20	16	no	—
164	C₂⁵	C₂²×D₅	1	2	yes	—
165	C₂²×Q₈	—	4	4	no	—
166	C₂²×Q₈	—	5	24	no	—
167	C₂²×Q₈	—	20	8	no	—
168	C₂²×Q₈	C₂²×D₅	1	12	yes	complemented
169	C₂²×D₄	—	4	4	no	—
170	C₂²×D₄	—	5	48	no	—
171	C₂²×D₄	—	5	24	no	—
172	C₂²×D₄	—	5	24	no	—
173	C₂²×D₄	—	20	8	no	—
174	C₂²×D₄	—	20	8	no	—
175	C₂²×D₄	—	20	16	no	—
176	C₂²×D₄	C₂²×D₅	1	12	yes	—
177	C₂³×C₄	—	5	4	no	—
178	C₂³×C₄	C₂²×D₅	1	1	yes	characteristic
179	C₂³×C₄	C₂²×D₅	1	2	yes	—
180	C₂³×C₄	C₂²×D₅	1	2	yes	—
181	#124	—	5	128	no	—
182	#124	—	5	256	no	—
183	#124	—	5	128	no	—
184	#124	C₂²×D₅	1	64	yes	complemented
185	#120	—	5	128	no	—
186	#120	—	5	256	no	—
187	C₂×C₂²⋊C₄	—	5	24	no	—
188	C₂×C₂²⋊C₄	—	5	48	no	—
189	C₂×C₂²⋊C₄	—	5	48	no	—
190	C₂×C₂²⋊C₄	—	5	48	no	—
191	C₂×C₂²⋊C₄	C₂²×D₅	1	48	yes	complemented
192	C₂×C₄²	—	5	24	no	—
193	C₂×C₄²	C₂²×D₅	1	12	yes	complemented
194	C₂²×C₅	—	2	16	no	—
195	C₂²×C₅	—	2	8	no	—
196	C₂²×C₅	—	4	48	no	—
197	C₂²×C₅	H(3,𝔽₄)	1	16	yes	complemented
198	C₂²×C₅	C₂×H(5,𝔽₂)	1	6	yes	—
199	C₂²×C₅	C₂×H(5,𝔽₂)	1	12	yes	—
200	C₂²×C₅	C₂⁶	1	1	yes	characteristic
201	C₂×D₅	—	2	16	no	—
202	C₂×D₅	—	2	16	no	—
203	C₂×D₅	—	2	16	no	—
204	C₂×D₅	—	4	128	no	—
205	C₂×D₅	—	4	96	no	—
206	C₄×C₅	—	2	8	no	—
207	C₄×C₅	—	2	8	no	—
208	C₄×C₅	—	2	4	no	—
209	Dic₅	—	2	16	no	—
210	C₂⁴	—	2	8	no	—
211	C₂⁴	—	2	16	no	—
212	C₂⁴	—	5	24	no	—
213	C₂⁴	—	5	32	no	—
214	C₂⁴	—	10	16	no	—
215	C₂⁴	—	10	16	no	—
216	C₂⁴	—	10	16	no	—
217	C₂⁴	—	20	48	no	—
218	C₂⁴	—	20	96	no	—
219	C₂⁴	—	20	128	no	—
220	C₂⁴	C₂³×D₅	1	12	yes	—
221	C₂⁴	C₂³×D₅	1	1	yes	characteristic
222	C₂×Q₈	—	4	24	no	—
223	C₂×Q₈	—	5	32	no	—
224	C₂×Q₈	—	20	48	no	—
225	C₂×Q₈	C₂³×D₅	1	16	yes	complemented
226	C₂×D₄	—	4	24	no	—
227	C₂×D₄	—	5	64	no	—
228	C₂×D₄	—	5	32	no	—
229	C₂×D₄	—	5	32	no	—
230	C₂×D₄	—	20	48	no	—
231	C₂×D₄	—	20	48	no	—
232	C₂×D₄	—	20	96	no	—
233	C₂×D₄	C₂³×D₅	1	16	yes	—
234	C₂²×C₄	—	2	4	no	—
235	C₂²×C₄	—	2	8	no	—
236	C₂²×C₄	—	2	8	no	—
237	C₂²×C₄	—	5	24	no	—
238	C₂²×C₄	—	10	16	no	—
239	C₂²×C₄	C₂³×D₅	1	12	yes	—
240	C₂²×C₄	C₂³×D₅	1	6	yes	—
241	C₂²×C₄	C₂³×D₅	1	12	yes	—
242	C₂²⋊C₄	—	5	32	no	—
243	C₂²⋊C₄	—	5	64	no	—
244	C₂²⋊C₄	—	5	64	no	—
245	C₂²⋊C₄	—	5	64	no	—
246	C₂²⋊C₄	C₂³×D₅	1	64	yes	complemented
247	C₄²	—	5	32	no	—
248	C₄²	C₂³×D₅	1	16	yes	complemented
249	C₂×C₅	—	4	8	no	—
250	C₂×C₅	C₂×H(3,𝔽₄)	1	12	yes	complemented
251	C₂×C₅	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
252	C₂×C₅	C₂²×H(5,𝔽₂)	1	2	yes	—
253	D₅	—	4	16	no	—
254	C₂³	—	2	24	no	—
255	C₂³	—	2	48	no	—
256	C₂³	—	4	32	no	—
257	C₂³	—	5	16	no	—
258	C₂³	—	10	48	no	—
259	C₂³	—	10	48	no	—
260	C₂³	—	10	48	no	—
261	C₂³	—	20	32	no	—
262	C₂³	—	20	64	no	—
263	C₂³	—	20	384	no	—
264	C₂³	—	20	64	no	—
265	C₂³	#4144	1	4	yes	—
266	C₂³	#4149	1	8	yes	—
267	C₂³	C₂⁴×D₅	1	8	yes	—
268	C₂³	C₂⁴×D₅	1	3	yes	—
269	Q₈	—	4	16	no	—
270	Q₈	—	20	32	no	—
271	D₄	—	4	16	no	—
272	D₄	—	20	32	no	—
273	D₄	—	20	32	no	—
274	D₄	—	20	64	no	—
275	C₂×C₄	—	2	12	no	—
276	C₂×C₄	—	2	24	no	—
277	C₂×C₄	—	2	24	no	—
278	C₂×C₄	—	5	16	no	—
279	C₂×C₄	—	10	48	no	—
280	C₂×C₄	C₂⁴×D₅	1	8	yes	—
281	C₂×C₄	C₂⁴×D₅	1	4	yes	—
282	C₂×C₄	C₂⁴×D₅	1	8	yes	—
283	C₅	C₂²×H(3,𝔽₄)	1	1	yes	characteristic, complemented, Sylow
284	C₂²	—	2	8	no	—
285	C₂²	—	2	16	no	—
286	C₂²	—	4	48	no	—
287	C₂²	—	10	16	no	—
288	C₂²	—	10	16	no	—
289	C₂²	—	10	16	no	—
290	C₂²	—	20	128	no	—
291	C₂²	—	20	96	no	—
292	C₂²	#66017	1	16	yes	direct product factor
293	C₂²	C₂×#4144	1	6	yes	—
294	C₂²	C₂×#4149	1	12	yes	—
295	C₂²	C₂⁵×D₅	1	1	yes	characteristic
296	C₄	—	2	4	no	—
297	C₄	—	2	8	no	—
298	C₄	—	2	8	no	—
299	C₄	—	10	16	no	—
300	C₂	—	4	8	no	—
301	C₂	—	20	16	no	—
302	C₂	C₂×#66017	1	12	yes	direct product factor
303	C₂	C₂²×#4144	1	1	yes	characteristic
304	C₂	C₂²×#4149	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂³×D₅
commutator, cosocle	C₂²×C₅	C₂⁶
Frattini	C₂²	C₂⁵×D₅
Fitting	C₂²×C₅×#124	C₂
socle	C₂⁴×C₅	C₂⁴
hypercenter	C₂²×#124	D₅
hypocenter	C₅	C₂²×H(3,𝔽₄)
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂⁴ ◁ C₂²×#124

Complements

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

N	H ≅ G/N	type
1. C₂²×C₅×#124	C₂	semidirect
2. C₂²×#4093	C₂	semidirect
3. C₂²×#4083	C₂	semidirect
4. C₂×#66017	C₂	direct (N × H)
5. C₂²×#4028	C₂	semidirect
6. C₂²×#4031	C₂	semidirect
7. C₂²×#4021	C₂	semidirect
8. C₂⁵×D₅	C₂²	semidirect
9. C₂³×C₅×Q₈	C₂²	semidirect
14. C₂³×Dic₁₀	C₂²	semidirect
15. C₂×C₅×#124	C₂²	semidirect
16. C₂²×C₅×C₂²⋊C₄	C₂²	semidirect
17. C₂²×C₄²×C₅	C₂²	semidirect
18. C₂²×C₂²⋊Dic₅	C₂²	semidirect
19. C₂×#4093	C₂²	semidirect
20. C₂×#4083	C₂²	semidirect
21. C₂²×#685	C₂²	semidirect
22. C₂²×#685	C₂²	semidirect
23. C₂²×#685	C₂²	semidirect
24. C₂²×C₄×Dic₅	C₂²	semidirect
25. #66017	C₂²	direct (N × H)
26. C₂×#4031	C₂²	semidirect
27. C₂×#4028	C₂²	semidirect
28. C₂×#4021	C₂²	semidirect
34. C₂⁴×D₅	C₂³	semidirect
36. C₂²×C₅×Q₈	C₂³	semidirect
50. C₂²×Dic₁₀	C₂³	semidirect
51. C₅×#124	C₂³	semidirect
52. C₂×C₅×C₂²⋊C₄	C₂³	semidirect
53. C₂×C₄²×C₅	C₂³	semidirect
54. C₂×C₂²⋊Dic₅	C₂³	semidirect
55. #4093	C₂³	semidirect
56. #4083	C₂³	semidirect
57. C₂×#685	C₂³	semidirect
58. C₂×#685	C₂³	semidirect
59. C₂×#685	C₂³	semidirect
60. C₂×C₄×Dic₅	C₂³	semidirect
61. #4031	C₂³	semidirect
62. #4028	C₂³	semidirect
63. #4021	C₂³	semidirect
68. C₂²×#124	D₅	semidirect
82. C₂³×D₅	C₂⁴	semidirect
84. C₂×C₅×Q₈	C₂⁴	semidirect
102. C₂×Dic₁₀	C₂⁴	semidirect
103. C₅×C₂²⋊C₄	C₂⁴	semidirect
104. C₄²×C₅	C₂⁴	semidirect
105. C₂²⋊Dic₅	C₂⁴	semidirect
106. #685	C₂⁴	semidirect
107. #685	C₂⁴	semidirect
108. #685	C₂⁴	semidirect
109. C₄×Dic₅	C₂⁴	semidirect
112. C₂³×Q₈	C₂×D₅	semidirect
121. C₂×#124	C₂×D₅	semidirect
128. C₂²×C₂²⋊C₄	C₂×D₅	semidirect
130. C₂²×C₄²	C₂×D₅	semidirect
168. C₂²×Q₈	C₂²×D₅	semidirect
184. #124	C₂²×D₅	semidirect
191. C₂×C₂²⋊C₄	C₂²×D₅	semidirect
193. C₂×C₄²	C₂²×D₅	semidirect
197. C₂²×C₅	H(3,𝔽₄)	semidirect
225. C₂×Q₈	C₂³×D₅	semidirect
246. C₂²⋊C₄	C₂³×D₅	semidirect
248. C₄²	C₂³×D₅	semidirect
250. C₂×C₅	C₂×H(3,𝔽₄)	semidirect
283. C₅	C₂²×H(3,𝔽₄)	semidirect
292. C₂²	#66017	direct (N × H)
302. C₂	C₂×#66017	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×C₅×#124	2
C₂²×#4093	2
C₂²×#4083	2
C₂×#66017	2
C₂²×#4028	2
C₂²×#4031	2
C₂²×#4021	2
C₂²×H(3,𝔽₄)	5

Subgroups — C22×#66017 #13006172

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#66017 #13006172