Subgroups — C₂×#65978 #10617878

Order 640.

1336 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#4093	C₂	1	2	yes	maximal, complemented
3	C₂×#4076	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×#685	C₂	1	1	yes	maximal, characteristic, complemented
5	#65978	C₂	1	16	yes	maximal, direct product factor
6	C₂×#4044	C₂	1	2	yes	maximal, complemented
7	C₂×#4041	C₂	1	2	yes	maximal, complemented
8	C₂×#4040	C₂	1	2	yes	maximal, complemented
9	C₂×#4028	C₂	1	2	yes	maximal, complemented
10	C₂×#4027	C₂	1	2	yes	maximal, complemented
11	C₂⁴×D₅	C₂²	1	1	yes	characteristic
12	C₂²×C₅×Q₈	C₂²	1	1	yes	characteristic, complemented
13	C₂³×C₄×C₅	C₂²	1	1	yes	characteristic
14	C₂²×C₂²⋊D₅	C₂²	1	2	yes	—
15	C₂³×Dic₅	C₂²	1	1	yes	characteristic
16	C₂²×D₂₀	C₂²	1	2	yes	—
17	C₂²×D₂₀	C₂²	1	2	yes	—
18	C₂²×D₂₀	C₂²	1	1	yes	characteristic, complemented
19	C₂²×C₄×D₅	C₂²	1	2	yes	complemented
20	C₅×C₂²⋊Q₈	C₂²	1	8	yes	complemented
21	C₂×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic, complemented
22	C₂×C₅×C₄⋊C₄	C₂²	1	2	yes	complemented
23	C₂×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
24	#4093	C₂²	1	16	yes	complemented
25	#4076	C₂²	1	8	yes	complemented
26	C₂×#685	C₂²	1	2	yes	—
27	C₂×#685	C₂²	1	2	yes	—
28	C₂×#685	C₂²	1	2	yes	—
29	C₂×#685	C₂²	1	2	yes	—
30	C₂×#685	C₂²	1	8	yes	complemented
31	C₂×#685	C₂²	1	2	yes	complemented
32	C₂×#685	C₂²	1	2	yes	complemented
33	C₂×C₄⋊Dic₅	C₂²	1	1	yes	characteristic, complemented
34	C₂×#683	C₂²	1	2	yes	complemented
35	C₂×C₄×Dic₅	C₂²	1	2	yes	complemented
36	#4044	C₂²	1	16	yes	complemented
37	#4041	C₂²	1	16	yes	complemented
38	#4040	C₂²	1	16	yes	complemented
39	#4028	C₂²	1	16	yes	complemented
40	#4027	C₂²	1	16	yes	complemented
41	C₂×#538	—	5	1	no	maximal, Sylow
42	C₂⁴×C₅	C₂³	1	1	yes	characteristic
43	C₂³×D₅	—	2	4	no	—
44	C₂³×D₅	—	4	2	no	—
45	C₂³×D₅	—	4	2	no	—
46	C₂³×D₅	C₂³	1	2	yes	—
47	C₂³×D₅	C₂³	1	2	yes	—
48	C₂³×D₅	C₂³	1	4	yes	—
49	C₂×C₅×Q₈	—	4	2	no	—
50	C₂×C₅×Q₈	C₂³	1	4	yes	complemented
51	C₂²×C₄×C₅	—	4	2	no	—
52	C₂²×C₄×C₅	C₂³	1	2	yes	—
53	C₂²×C₄×C₅	C₂³	1	2	yes	—
54	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
55	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
56	C₂²×C₄×C₅	C₂³	1	4	yes	—
57	C₂×C₂²⋊D₅	—	2	8	no	—
58	C₂×C₂²⋊D₅	C₂³	1	8	yes	—
59	C₂²×Dic₅	—	4	2	no	—
60	C₂²×Dic₅	C₂³	1	2	yes	—
61	C₂²×Dic₅	C₂³	1	2	yes	—
62	C₂²×Dic₅	C₂³	1	4	yes	—
63	C₂×D₂₀	—	4	4	no	—
64	C₂×D₂₀	—	4	4	no	—
65	C₂×D₂₀	—	4	2	no	—
66	C₂×D₂₀	C₂³	1	8	yes	—
67	C₂×D₂₀	C₂³	1	8	yes	—
68	C₂×D₂₀	C₂³	1	4	yes	complemented
69	C₂×C₄×D₅	—	4	4	no	—
70	C₂×C₄×D₅	C₂³	1	8	yes	complemented
71	C₅×C₄⋊C₄	C₂³	1	4	yes	complemented
72	C₅×C₄⋊C₄	C₂³	1	8	yes	complemented
73	C₅×C₂²⋊C₄	C₂³	1	8	yes	—
74	#685	C₂³	1	8	yes	—
75	#685	C₂³	1	8	yes	—
76	#685	C₂³	1	8	yes	—
77	#685	C₂³	1	8	yes	—
78	#685	C₂³	1	8	yes	complemented
79	#685	C₂³	1	8	yes	complemented
80	C₄⋊Dic₅	C₂³	1	4	yes	complemented
81	#683	C₂³	1	8	yes	complemented
82	C₄×Dic₅	C₂³	1	8	yes	complemented
83	#538	—	5	16	no	—
84	C₂×#126	—	5	2	no	—
85	C₂×#124	—	5	2	no	—
86	C₂×#123	—	5	2	no	—
87	C₂×C₂²⋊Q₈	D₅	1	1	yes	characteristic, complemented
88	C₂×#121	—	5	1	no	—
89	C₂×#121	—	5	2	no	—
90	C₂×#120	—	5	2	no	—
91	C₂×C₄×D₄	—	5	2	no	—
92	C₂²×C₂²⋊C₄	—	5	1	no	—
93	C₂³×C₅	—	2	2	no	—
94	C₂³×C₅	—	2	2	no	—
95	C₂³×C₅	#41	1	4	yes	—
96	C₂³×C₅	C₂⁴	1	2	yes	—
97	C₂³×C₅	C₂⁴	1	1	yes	characteristic
98	C₂²×D₅	—	2	4	no	—
99	C₂²×D₅	—	2	8	no	—
100	C₂²×D₅	—	2	4	no	—
101	C₂²×D₅	—	2	4	no	—
102	C₂²×D₅	—	2	4	no	—
103	C₂²×D₅	—	2	4	no	—
104	C₂²×D₅	—	2	4	no	—
105	C₂²×D₅	—	4	4	no	—
106	C₂²×D₅	—	4	4	no	—
107	C₂²×D₅	—	4	16	no	—
108	C₂²×D₅	C₂⁴	1	4	yes	—
109	C₂²×D₅	C₂⁴	1	4	yes	—
110	C₅×Q₈	—	4	4	no	—
111	C₂×C₄×C₅	—	2	4	no	—
112	C₂×C₄×C₅	—	2	2	no	—
113	C₂×C₄×C₅	—	2	4	no	—
114	C₂×C₄×C₅	—	2	2	no	—
115	C₂×C₄×C₅	—	4	4	no	—
116	C₂×C₄×C₅	C₂⁴	1	4	yes	—
117	C₂×C₄×C₅	C₂⁴	1	4	yes	—
118	C₂×C₄×C₅	C₂⁴	1	2	yes	—
119	C₂×C₄×C₅	C₂⁴	1	2	yes	—
120	C₂²⋊D₅	—	2	16	no	—
121	C₂×Dic₅	—	2	4	no	—
122	C₂×Dic₅	—	2	4	no	—
123	C₂×Dic₅	—	4	4	no	—
124	C₂×Dic₅	C₂⁴	1	4	yes	—
125	C₂×Dic₅	C₂⁴	1	4	yes	—
126	D₂₀	—	4	8	no	—
127	D₂₀	—	4	8	no	—
128	D₂₀	—	4	4	no	—
129	C₄×D₅	—	4	8	no	—
130	C₂⁵	—	5	1	no	—
131	C₂²×Q₈	C₂×D₅	1	1	yes	characteristic, complemented
132	C₂²×D₄	—	5	2	no	—
133	C₂²×D₄	—	5	2	no	—
134	C₂²×D₄	—	5	2	no	—
135	C₂²×D₄	—	5	1	no	—
136	C₂³×C₄	—	5	1	no	—
137	C₂³×C₄	—	5	2	no	—
138	C₂³×C₄	C₂×D₅	1	1	yes	characteristic
139	#126	—	5	16	no	—
140	#124	—	5	16	no	—
141	#123	—	5	16	no	—
142	C₂²⋊Q₈	C₂×D₅	1	8	yes	complemented
143	#121	—	5	8	no	—
144	#121	—	5	16	no	—
145	#120	—	5	16	no	—
146	C₄×D₄	—	5	16	no	—
147	C₂×C₄⋊C₄	—	5	2	no	—
148	C₂×C₄⋊C₄	—	5	1	no	—
149	C₂×C₄⋊C₄	C₂×D₅	1	2	yes	complemented
150	C₂×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic, complemented
151	C₂×C₂²⋊C₄	—	5	2	no	—
152	C₂×C₂²⋊C₄	—	5	2	no	—
153	C₂×C₂²⋊C₄	—	5	8	no	—
154	C₂×C₂²⋊C₄	—	5	2	no	—
155	C₂×C₂²⋊C₄	—	5	2	no	—
156	C₂×C₂²⋊C₄	—	5	2	no	—
157	C₂×C₂²⋊C₄	—	5	2	no	—
158	C₂×C₂²⋊C₄	C₂×D₅	1	2	yes	—
159	C₂×C₄²	—	5	2	no	—
160	C₂²×C₅	—	2	8	no	—
161	C₂²×C₅	—	2	2	no	—
162	C₂²×C₅	—	2	2	no	—
163	C₂²×C₅	C₂×#41	1	4	yes	—
164	C₂²×C₅	C₂×#41	1	2	yes	—
165	C₂²×C₅	H(5,𝔽₂)	1	2	yes	—
166	C₂²×C₅	H(5,𝔽₂)	1	2	yes	—
167	C₂²×C₅	C₂⁵	1	1	yes	characteristic
168	C₂×D₅	—	2	4	no	—
169	C₂×D₅	—	2	4	no	—
170	C₂×D₅	—	2	4	no	—
171	C₂×D₅	—	2	4	no	—
172	C₂×D₅	—	2	4	no	—
173	C₂×D₅	—	2	4	no	—
174	C₂×D₅	—	4	16	no	—
175	C₂×D₅	—	4	8	no	—
176	C₂×D₅	—	4	8	no	—
177	C₄×C₅	—	2	4	no	—
178	C₄×C₅	—	2	2	no	—
179	C₄×C₅	—	2	4	no	—
180	C₄×C₅	—	2	2	no	—
181	Dic₅	—	2	4	no	—
182	Dic₅	—	2	4	no	—
183	C₂⁴	—	5	2	no	—
184	C₂⁴	—	5	2	no	—
185	C₂⁴	—	5	4	no	—
186	C₂⁴	—	10	4	no	—
187	C₂⁴	—	20	2	no	—
188	C₂⁴	—	20	2	no	—
189	C₂⁴	C₂²×D₅	1	1	yes	characteristic
190	C₂×Q₈	—	4	2	no	—
191	C₂×Q₈	C₂²×D₅	1	4	yes	complemented
192	C₂×D₄	—	5	8	no	—
193	C₂×D₄	—	5	8	no	—
194	C₂×D₄	—	5	8	no	—
195	C₂×D₄	—	5	4	no	—
196	C₂×D₄	—	10	8	no	—
197	C₂×D₄	—	20	2	no	—
198	C₂×D₄	—	20	4	no	—
199	C₂×D₄	—	20	4	no	—
200	C₂²×C₄	—	4	2	no	—
201	C₂²×C₄	—	5	2	no	—
202	C₂²×C₄	—	5	2	no	—
203	C₂²×C₄	—	5	4	no	—
204	C₂²×C₄	—	5	8	no	—
205	C₂²×C₄	—	20	4	no	—
206	C₂²×C₄	—	20	2	no	—
207	C₂²×C₄	C₂²×D₅	1	2	yes	—
208	C₂²×C₄	C₂²×D₅	1	1	yes	characteristic
209	C₂²×C₄	C₂²×D₅	1	1	yes	characteristic
210	C₂²×C₄	C₂²×D₅	1	2	yes	—
211	C₂²×C₄	C₂²×D₅	1	4	yes	—
212	C₄⋊C₄	—	5	8	no	—
213	C₄⋊C₄	—	5	4	no	—
214	C₄⋊C₄	C₂²×D₅	1	8	yes	complemented
215	C₄⋊C₄	C₂²×D₅	1	4	yes	complemented
216	C₂²⋊C₄	—	5	8	no	—
217	C₂²⋊C₄	—	5	8	no	—
218	C₂²⋊C₄	—	5	8	no	—
219	C₂²⋊C₄	—	5	8	no	—
220	C₂²⋊C₄	—	5	8	no	—
221	C₂²⋊C₄	—	5	8	no	—
222	C₂²⋊C₄	C₂²×D₅	1	8	yes	—
223	C₄²	—	5	8	no	—
224	C₂×C₅	—	2	4	no	—
225	C₂×C₅	#538	1	4	yes	complemented
226	C₂×C₅	C₂²×#41	1	1	yes	characteristic
227	C₂×C₅	C₂×H(5,𝔽₂)	1	1	yes	characteristic
228	C₂×C₅	C₂×H(5,𝔽₂)	1	1	yes	characteristic
229	D₅	—	4	4	no	—
230	D₅	—	4	4	no	—
231	C₂³	—	2	2	no	—
232	C₂³	—	2	2	no	—
233	C₂³	—	5	4	no	—
234	C₂³	—	5	4	no	—
235	C₂³	—	10	4	no	—
236	C₂³	—	10	4	no	—
237	C₂³	—	10	4	no	—
238	C₂³	—	10	8	no	—
239	C₂³	—	10	4	no	—
240	C₂³	—	10	4	no	—
241	C₂³	—	10	4	no	—
242	C₂³	—	20	4	no	—
243	C₂³	—	20	4	no	—
244	C₂³	—	20	16	no	—
245	C₂³	#713	1	4	yes	—
246	C₂³	C₂³×D₅	1	2	yes	—
247	C₂³	C₂³×D₅	1	1	yes	characteristic
248	Q₈	—	4	4	no	—
249	D₄	—	10	16	no	—
250	D₄	—	20	4	no	—
251	D₄	—	20	8	no	—
252	D₄	—	20	8	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	4	no	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	4	4	no	—
258	C₂×C₄	—	5	4	no	—
259	C₂×C₄	—	5	4	no	—
260	C₂×C₄	—	10	4	no	—
261	C₂×C₄	—	10	4	no	—
262	C₂×C₄	—	20	8	no	—
263	C₂×C₄	—	20	4	no	—
264	C₂×C₄	C₂³×D₅	1	4	yes	—
265	C₂×C₄	C₂³×D₅	1	2	yes	—
266	C₂×C₄	C₂³×D₅	1	2	yes	—
267	C₂×C₄	C₂³×D₅	1	4	yes	—
268	C₅	C₂×#538	1	1	yes	characteristic, complemented, Sylow
269	C₂²	—	2	8	no	—
270	C₂²	—	2	2	no	—
271	C₂²	—	2	2	no	—
272	C₂²	—	10	4	no	—
273	C₂²	—	10	4	no	—
274	C₂²	—	10	4	no	—
275	C₂²	—	10	4	no	—
276	C₂²	—	10	4	no	—
277	C₂²	—	10	4	no	—
278	C₂²	—	20	16	no	—
279	C₂²	—	20	8	no	—
280	C₂²	—	20	8	no	—
281	C₂²	#4144	1	2	yes	—
282	C₂²	C₂×#713	1	4	yes	—
283	C₂²	C₂×#713	1	2	yes	—
284	C₂²	#4149	1	2	yes	—
285	C₂²	C₂⁴×D₅	1	1	yes	characteristic
286	C₄	—	2	2	no	—
287	C₄	—	2	2	no	—
288	C₄	—	2	4	no	—
289	C₄	—	2	4	no	—
290	C₄	—	10	4	no	—
291	C₄	—	10	4	no	—
292	C₂	—	2	4	no	—
293	C₂	—	20	4	no	—
294	C₂	—	20	4	no	—
295	C₂	#65978	1	4	yes	direct product factor
296	C₂	C₂×#4144	1	1	yes	characteristic
297	C₂	C₂²×#713	1	1	yes	characteristic
298	C₂	C₂×#4149	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₅
commutator, cosocle	C₂²×C₅	C₂⁵
Frattini	C₂²	C₂⁴×D₅
Fitting	C₂×C₅×C₂²⋊Q₈	C₂
socle	C₂³×C₅	C₂⁴
hypercenter	C₂×C₂²⋊Q₈	D₅
hypocenter	C₅	C₂×#538
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂×C₂²⋊Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×C₅×C₂²⋊Q₈	C₂	semidirect
2. C₂×#4093	C₂	semidirect
3. C₂×#4076	C₂	semidirect
4. C₂²×#685	C₂	semidirect
5. #65978	C₂	direct (N × H)
6. C₂×#4044	C₂	semidirect
7. C₂×#4041	C₂	semidirect
8. C₂×#4040	C₂	semidirect
9. C₂×#4028	C₂	semidirect
10. C₂×#4027	C₂	semidirect
12. C₂²×C₅×Q₈	C₂²	semidirect
18. C₂²×D₂₀	C₂²	semidirect
19. C₂²×C₄×D₅	C₂²	semidirect
20. C₅×C₂²⋊Q₈	C₂²	semidirect
21. C₂×C₅×C₄⋊C₄	C₂²	semidirect
22. C₂×C₅×C₄⋊C₄	C₂²	semidirect
24. #4093	C₂²	semidirect
25. #4076	C₂²	semidirect
30. C₂×#685	C₂²	semidirect
31. C₂×#685	C₂²	semidirect
32. C₂×#685	C₂²	semidirect
33. C₂×C₄⋊Dic₅	C₂²	semidirect
34. C₂×#683	C₂²	semidirect
35. C₂×C₄×Dic₅	C₂²	semidirect
36. #4044	C₂²	semidirect
37. #4041	C₂²	semidirect
38. #4040	C₂²	semidirect
39. #4028	C₂²	semidirect
40. #4027	C₂²	semidirect
50. C₂×C₅×Q₈	C₂³	semidirect
68. C₂×D₂₀	C₂³	semidirect
70. C₂×C₄×D₅	C₂³	semidirect
71. C₅×C₄⋊C₄	C₂³	semidirect
72. C₅×C₄⋊C₄	C₂³	semidirect
78. #685	C₂³	semidirect
79. #685	C₂³	semidirect
80. C₄⋊Dic₅	C₂³	semidirect
81. #683	C₂³	semidirect
82. C₄×Dic₅	C₂³	semidirect
87. C₂×C₂²⋊Q₈	D₅	semidirect
131. C₂²×Q₈	C₂×D₅	semidirect
142. C₂²⋊Q₈	C₂×D₅	semidirect
149. C₂×C₄⋊C₄	C₂×D₅	semidirect
150. C₂×C₄⋊C₄	C₂×D₅	semidirect
191. C₂×Q₈	C₂²×D₅	semidirect
214. C₄⋊C₄	C₂²×D₅	semidirect
215. C₄⋊C₄	C₂²×D₅	semidirect
225. C₂×C₅	#538	semidirect
268. C₅	C₂×#538	semidirect
295. C₂	#65978	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅×C₂²⋊Q₈	2
C₂×#4093	2
C₂×#4076	2
C₂²×#685	2
#65978	2
C₂×#4044	2
C₂×#4041	2
C₂×#4040	2
C₂×#4028	2
C₂×#4027	2
C₂×#538	5

Subgroups — C2×#65978 #10617878

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#65978 #10617878