Subgroups — D₄×#713 #10618643

Order 640.

1648 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#713	C₂	1	2	yes	maximal, complemented
2	C₅×D₄×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#66151	C₂	1	2	yes	maximal, complemented
4	#66060	C₂	1	3	yes	maximal, complemented
5	#66059	C₂	1	3	yes	maximal, complemented
6	#65964	C₂	1	6	yes	maximal, complemented
7	#65961	C₂	1	6	yes	maximal, complemented
8	#65911	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#713	C₂	1	1	yes	maximal, characteristic, complemented
10	D₄×D₂₀	C₂	1	3	yes	maximal, complemented
11	C₄×D₄×D₅	C₂	1	3	yes	maximal, complemented
12	C₂²×C₅×Q₈	C₂²	1	2	yes	complemented
13	C₂×#713	—	2	16	no	—
14	C₂×#713	C₂²	1	1	yes	characteristic
15	C₂×#713	C₂²	1	6	yes	complemented
16	C₂×#713	C₂²	1	6	yes	complemented
17	C₂×#713	C₂²	1	2	yes	complemented
18	C₂×D₄×D₅	C₂²	1	3	yes	—
19	C₂²×D₂₀	C₂²	1	6	yes	complemented
20	C₂²×C₄×D₅	C₂²	1	6	yes	complemented
21	C₅×C₄⋊Q₈	C₂²	1	3	yes	complemented
22	C₅×C₂²⋊Q₈	C₂²	1	6	yes	complemented
23	C₄×C₅×Q₈	C₂²	1	1	yes	characteristic, complemented
24	C₄×C₅×D₄	C₂²	1	3	yes	—
25	#4093	C₂²	1	6	yes	complemented
26	#4092	C₂²	1	6	yes	complemented
27	Q₈×Dic₅	C₂²	1	1	yes	characteristic, complemented
28	D₄×Dic₅	C₂²	1	1	yes	characteristic
29	#4076	C₂²	1	6	yes	complemented
30	C₄×C₂²⋊D₅	C₂²	1	6	yes	complemented
31	#4041	C₂²	1	6	yes	complemented
32	#4041	C₂²	1	6	yes	complemented
33	#4041	C₂²	1	6	yes	complemented
34	#4040	C₂²	1	12	yes	complemented
35	#4039	C₂²	1	3	yes	complemented
36	#4039	C₂²	1	6	yes	complemented
37	#4039	C₂²	1	3	yes	complemented
38	#4038	C₂²	1	3	yes	complemented
39	D₅×C₄⋊C₄	C₂²	1	3	yes	complemented
40	#4028	C₂²	1	12	yes	complemented
41	#4027	C₂²	1	6	yes	complemented
42	D₅×C₂²⋊C₄	C₂²	1	6	yes	complemented
43	#4020	C₂²	1	3	yes	complemented
44	C₄×D₂₀	C₂²	1	3	yes	complemented
45	C₄×D₂₀	C₂²	1	3	yes	complemented
46	C₄²×D₅	C₂²	1	3	yes	complemented
47	D₄×#41	—	5	1	no	maximal, Sylow
48	C₂³×D₅	C₂³	1	6	yes	—
49	C₂×C₅×Q₈	—	2	8	no	—
50	C₂×C₅×Q₈	C₂³	1	6	yes	—
51	C₂×C₅×Q₈	C₂³	1	1	yes	characteristic
52	C₂×C₅×D₄	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₅	C₂³	1	6	yes	—
54	C₂×C₂²⋊D₅	C₂³	1	6	yes	—
55	C₂²×Dic₅	C₂³	1	2	yes	—
56	#713	—	2	8	no	—
57	#713	—	2	24	no	—
58	#713	—	2	24	no	—
59	#713	D₄	1	8	yes	direct product factor
60	D₄×D₅	—	2	12	no	—
61	C₂×D₂₀	—	2	24	no	—
62	C₂×D₂₀	C₂³	1	6	yes	—
63	C₂×D₂₀	C₂³	1	12	yes	—
64	C₂×D₂₀	C₂³	1	3	yes	—
65	C₂×D₂₀	C₂³	1	6	yes	—
66	C₂×D₂₀	C₂³	1	3	yes	—
67	C₂×C₄×D₅	—	2	24	no	—
68	C₂×C₄×D₅	C₂³	1	6	yes	—
69	C₂×C₄×D₅	C₂³	1	6	yes	—
70	C₂×C₄×D₅	C₂³	1	6	yes	—
71	C₂×C₄×D₅	C₂³	1	3	yes	—
72	C₂×C₄×D₅	C₂³	1	3	yes	—
73	C₂×C₄×D₅	C₂³	1	3	yes	—
74	C₅×C₄⋊C₄	C₂³	1	3	yes	—
75	C₅×C₄⋊C₄	C₂³	1	3	yes	—
76	C₅×C₄⋊C₄	C₂³	1	6	yes	—
77	C₅×C₂²⋊C₄	C₂³	1	6	yes	—
78	C₄²×C₅	C₂³	1	3	yes	—
79	C₂²⋊Dic₅	C₂³	1	2	yes	—
80	#685	C₂³	1	6	yes	—
81	#685	C₂³	1	6	yes	—
82	#685	C₂³	1	6	yes	—
83	#685	C₂³	1	12	yes	—
84	C₄⋊Dic₅	C₂³	1	1	yes	characteristic
85	C₄⋊Dic₅	C₂³	1	3	yes	—
86	#683	C₂³	1	6	yes	—
87	C₄×Dic₅	C₂³	1	1	yes	characteristic
88	C₄×Dic₅	C₂³	1	3	yes	—
89	C₄×Dic₅	C₂³	1	3	yes	—
90	C₂²×#41	—	5	2	no	—
91	#550	—	5	1	no	—
92	D₄×Q₈	D₅	1	1	yes	characteristic, complemented
93	#548	—	5	2	no	—
94	#547	—	5	3	no	—
95	#546	—	5	6	no	—
96	D₄²	—	5	3	no	—
97	#532	—	5	3	no	—
98	#525	—	5	6	no	—
99	C₄×#41	—	5	1	no	—
100	C₂×C₄×D₄	—	5	3	no	—
101	C₂³×C₅	C₂⁴	1	2	yes	—
102	C₂²×D₅	—	2	12	no	—
103	C₂²×D₅	—	2	12	no	—
104	C₂²×D₅	—	4	6	no	—
105	C₂²×D₅	C₂⁴	1	6	yes	—
106	C₂²×D₅	C₂⁴	1	3	yes	—
107	C₅×Q₈	—	2	12	no	—
108	C₅×Q₈	C₂×D₄	1	4	yes	complemented
109	C₅×D₄	#41	1	4	yes	complemented
110	C₂×C₄×C₅	—	2	12	no	—
111	C₂×C₄×C₅	C₂⁴	1	3	yes	—
112	C₂×C₄×C₅	C₂⁴	1	6	yes	—
113	C₂×C₄×C₅	C₂⁴	1	3	yes	—
114	C₂×C₄×C₅	C₂⁴	1	1	yes	characteristic
115	C₂²⋊D₅	—	2	12	no	—
116	C₂×Dic₅	—	2	4	no	—
117	C₂×Dic₅	C₂⁴	1	1	yes	characteristic
118	C₂×Dic₅	C₂⁴	1	3	yes	—
119	C₂×Dic₅	C₂⁴	1	2	yes	—
120	C₂×Dic₅	C₂⁴	1	1	yes	characteristic
121	D₂₀	—	2	24	no	—
122	D₂₀	—	2	12	no	—
123	D₂₀	—	2	6	no	—
124	D₂₀	—	4	6	no	—
125	D₂₀	C₂×D₄	1	12	yes	complemented
126	C₄×D₅	—	2	12	no	—
127	C₄×D₅	—	2	12	no	—
128	C₄×D₅	—	2	6	no	—
129	C₄×D₅	—	2	12	no	—
130	C₄×D₅	—	4	3	no	—
131	C₄×D₅	C₂×D₄	1	12	yes	complemented
132	C₂×#41	—	5	1	no	—
133	C₂×#41	—	5	6	no	—
134	C₂×#41	—	5	6	no	—
135	C₂×#41	—	5	2	no	—
136	C₂×#41	—	10	16	no	—
137	C₂²×Q₈	C₂×D₅	1	2	yes	complemented
138	C₂²×D₄	—	5	3	no	—
139	C₂²×D₄	—	5	6	no	—
140	C₂³×C₄	—	5	6	no	—
141	C₄⋊Q₈	C₂×D₅	1	3	yes	complemented
142	#127	—	5	3	no	—
143	#124	—	5	6	no	—
144	#123	—	5	12	no	—
145	C₂²⋊Q₈	—	5	6	no	—
146	C₂²⋊Q₈	C₂×D₅	1	6	yes	complemented
147	#121	—	5	6	no	—
148	#121	—	5	6	no	—
149	#121	—	5	6	no	—
150	#121	—	5	6	no	—
151	#120	—	5	12	no	—
152	C₄×Q₈	—	5	1	no	—
153	C₄×Q₈	C₂×D₅	1	1	yes	characteristic, complemented
154	C₄×D₄	—	5	1	no	—
155	C₄×D₄	—	5	6	no	—
156	C₄×D₄	—	5	3	no	—
157	C₄×D₄	—	5	3	no	—
158	C₄×D₄	—	5	6	no	—
159	C₄×D₄	—	5	6	no	—
160	C₄×D₄	—	5	3	no	—
161	C₄×D₄	—	5	3	no	—
162	C₄×D₄	C₂×D₅	1	3	yes	—
163	#117	—	5	3	no	—
164	C₂×C₄⋊C₄	—	5	3	no	—
165	C₂×C₂²⋊C₄	—	5	6	no	—
166	C₂×C₄²	—	5	3	no	—
167	C₂²×C₅	—	2	2	no	—
168	C₂²×C₅	—	2	2	no	—
169	C₂²×C₅	C₂×#41	1	4	yes	complemented
170	C₂²×C₅	C₂⁵	1	1	yes	characteristic
171	C₂×D₅	—	2	3	no	—
172	C₂×D₅	—	2	3	no	—
173	C₂×D₅	—	2	6	no	—
174	C₂×D₅	—	2	6	no	—
175	C₂×D₅	—	2	6	no	—
176	C₂×D₅	—	4	24	no	—
177	C₂×D₅	C₂²×D₄	1	6	yes	—
178	C₄×C₅	—	2	6	no	—
179	C₄×C₅	—	2	3	no	—
180	C₄×C₅	C₂²×D₄	1	6	yes	—
181	C₄×C₅	C₂×#41	1	2	yes	complemented
182	Dic₅	—	2	2	no	—
183	Dic₅	—	2	3	no	—
184	Dic₅	—	2	1	no	—
185	Dic₅	C₂²×D₄	1	2	yes	—
186	C₂⁴	—	5	6	no	—
187	#41	—	5	8	no	—
188	#41	—	10	24	no	—
189	#41	—	10	24	no	—
190	#41	—	10	8	no	—
191	C₂×Q₈	—	2	8	no	—
192	C₂×Q₈	C₂²×D₅	1	6	yes	—
193	C₂×Q₈	C₂²×D₅	1	1	yes	characteristic
194	C₂×D₄	—	5	6	no	—
195	C₂×D₄	—	5	3	no	—
196	C₂×D₄	—	5	3	no	—
197	C₂×D₄	—	5	12	no	—
198	C₂×D₄	—	5	6	no	—
199	C₂×D₄	—	5	6	no	—
200	C₂×D₄	—	10	24	no	—
201	C₂×D₄	—	10	12	no	—
202	C₂×D₄	C₂²×D₅	1	1	yes	characteristic
203	C₂²×C₄	—	5	3	no	—
204	C₂²×C₄	—	5	2	no	—
205	C₂²×C₄	—	5	3	no	—
206	C₂²×C₄	—	5	6	no	—
207	C₂²×C₄	—	5	6	no	—
208	C₂²×C₄	—	5	6	no	—
209	C₂²×C₄	—	5	3	no	—
210	C₂²×C₄	—	10	24	no	—
211	C₂²×C₄	C₂²×D₅	1	6	yes	—
212	C₄⋊C₄	—	5	1	no	—
213	C₄⋊C₄	—	5	6	no	—
214	C₄⋊C₄	—	5	3	no	—
215	C₄⋊C₄	C₂²×D₅	1	3	yes	—
216	C₄⋊C₄	C₂²×D₅	1	3	yes	—
217	C₄⋊C₄	C₂²×D₅	1	6	yes	—
218	C₂²⋊C₄	—	5	2	no	—
219	C₂²⋊C₄	—	5	6	no	—
220	C₂²⋊C₄	—	5	6	no	—
221	C₂²⋊C₄	—	5	6	no	—
222	C₂²⋊C₄	—	5	12	no	—
223	C₂²⋊C₄	C₂²×D₅	1	6	yes	—
224	C₄²	—	5	1	no	—
225	C₄²	—	5	3	no	—
226	C₄²	—	5	3	no	—
227	C₄²	C₂²×D₅	1	3	yes	—
228	C₂×C₅	—	2	4	no	—
229	C₂×C₅	C₂³×D₄	1	1	yes	characteristic
230	C₂×C₅	C₂²×#41	1	1	yes	characteristic
231	C₂×C₅	#585	1	1	yes	characteristic
232	D₅	—	2	6	no	—
233	D₅	—	4	6	no	—
234	C₂³	—	5	3	no	—
235	C₂³	—	5	6	no	—
236	C₂³	—	10	12	no	—
237	C₂³	—	10	12	no	—
238	C₂³	—	20	6	no	—
239	C₂³	C₂³×D₅	1	2	yes	—
240	Q₈	—	2	12	no	—
241	Q₈	D₄×D₅	1	4	yes	complemented
242	D₄	—	5	12	no	—
243	D₄	—	10	24	no	—
244	D₄	—	10	12	no	—
245	D₄	—	10	12	no	—
246	D₄	—	10	6	no	—
247	D₄	—	20	6	no	—
248	D₄	#713	1	4	yes	direct product factor
249	C₂×C₄	—	2	12	no	—
250	C₂×C₄	—	5	1	no	—
251	C₂×C₄	—	5	12	no	—
252	C₂×C₄	—	5	1	no	—
253	C₂×C₄	—	5	3	no	—
254	C₂×C₄	—	5	2	no	—
255	C₂×C₄	—	10	12	no	—
256	C₂×C₄	—	10	12	no	—
257	C₂×C₄	—	10	4	no	—
258	C₂×C₄	—	10	12	no	—
259	C₂×C₄	—	10	6	no	—
260	C₂×C₄	—	20	3	no	—
261	C₂×C₄	C₂³×D₅	1	3	yes	—
262	C₂×C₄	C₂³×D₅	1	6	yes	—
263	C₂×C₄	C₂³×D₅	1	3	yes	—
264	C₂×C₄	C₂³×D₅	1	1	yes	characteristic
265	C₅	D₄×#41	1	1	yes	characteristic, complemented, Sylow
266	C₂²	—	2	2	no	—
267	C₂²	—	2	2	no	—
268	C₂²	—	5	6	no	—
269	C₂²	—	10	3	no	—
270	C₂²	—	10	3	no	—
271	C₂²	—	10	6	no	—
272	C₂²	—	10	6	no	—
273	C₂²	—	10	6	no	—
274	C₂²	—	20	24	no	—
275	C₂²	C₂×#713	1	4	yes	complemented
276	C₂²	C₂⁴×D₅	1	1	yes	characteristic
277	C₄	—	2	6	no	—
278	C₄	—	2	3	no	—
279	C₄	—	5	2	no	—
280	C₄	—	10	2	no	—
281	C₄	—	10	3	no	—
282	C₄	—	10	1	no	—
283	C₄	C₂×D₄×D₅	1	6	yes	—
284	C₄	C₂×#713	1	2	yes	complemented
285	C₂	—	2	4	no	—
286	C₂	—	10	6	no	—
287	C₂	—	20	6	no	—
288	C₂	C₂²×D₄×D₅	1	1	yes	characteristic
289	C₂	C₂²×#713	1	1	yes	characteristic
290	C₂	#66287	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₅
commutator, socle, cosocle	C₂²×C₅	C₂⁵
Fitting	C₅×D₄×Q₈	C₂
hypercenter	D₄×Q₈	D₅
hypocenter	C₅	D₄×#41
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂² ◁ D₄×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂²×#713	C₂	semidirect
2. C₅×D₄×Q₈	C₂	semidirect
3. #66151	C₂	semidirect
4. #66060	C₂	semidirect
5. #66059	C₂	semidirect
6. #65964	C₂	semidirect
7. #65961	C₂	semidirect
8. #65911	C₂	semidirect
9. C₄×#713	C₂	semidirect
10. D₄×D₂₀	C₂	semidirect
11. C₄×D₄×D₅	C₂	semidirect
12. C₂²×C₅×Q₈	C₂²	semidirect
15. C₂×#713	C₂²	semidirect
16. C₂×#713	C₂²	semidirect
17. C₂×#713	C₂²	semidirect
19. C₂²×D₂₀	C₂²	semidirect
20. C₂²×C₄×D₅	C₂²	semidirect
21. C₅×C₄⋊Q₈	C₂²	semidirect
22. C₅×C₂²⋊Q₈	C₂²	semidirect
23. C₄×C₅×Q₈	C₂²	semidirect
25. #4093	C₂²	semidirect
26. #4092	C₂²	semidirect
27. Q₈×Dic₅	C₂²	semidirect
29. #4076	C₂²	semidirect
30. C₄×C₂²⋊D₅	C₂²	semidirect
31. #4041	C₂²	semidirect
32. #4041	C₂²	semidirect
33. #4041	C₂²	semidirect
34. #4040	C₂²	semidirect
35. #4039	C₂²	semidirect
36. #4039	C₂²	semidirect
37. #4039	C₂²	semidirect
38. #4038	C₂²	semidirect
39. D₅×C₄⋊C₄	C₂²	semidirect
40. #4028	C₂²	semidirect
41. #4027	C₂²	semidirect
42. D₅×C₂²⋊C₄	C₂²	semidirect
43. #4020	C₂²	semidirect
44. C₄×D₂₀	C₂²	semidirect
45. C₄×D₂₀	C₂²	semidirect
46. C₄²×D₅	C₂²	semidirect
59. #713	D₄	direct (N × H)
92. D₄×Q₈	D₅	semidirect
108. C₅×Q₈	C₂×D₄	semidirect
109. C₅×D₄	#41	semidirect
125. D₂₀	C₂×D₄	semidirect
131. C₄×D₅	C₂×D₄	semidirect
137. C₂²×Q₈	C₂×D₅	semidirect
141. C₄⋊Q₈	C₂×D₅	semidirect
146. C₂²⋊Q₈	C₂×D₅	semidirect
153. C₄×Q₈	C₂×D₅	semidirect
169. C₂²×C₅	C₂×#41	semidirect
181. C₄×C₅	C₂×#41	semidirect
241. Q₈	D₄×D₅	semidirect
248. D₄	#713	direct (N × H)
265. C₅	D₄×#41	semidirect
275. C₂²	C₂×#713	semidirect
284. C₄	C₂×#713	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#713	2
C₅×D₄×Q₈	2
#66151	2
#66060	2
#66059	2
#65964	2
#65961	2
#65911	2
C₄×#713	2
D₄×D₂₀	2
C₄×D₄×D₅	2
D₄×#41	5

Subgroups — D4×#713 #10618643

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄×#713 #10618643