Subgroups — C₂×#10601498 #12102713

Order 1280.

934 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×#108	C₂	1	1	yes	maximal, characteristic
2	C₂×#65408	C₂	1	2	yes	maximal
3	C₂×#65395	C₂	1	1	yes	maximal, characteristic
4	C₂²×#3947	C₂	1	1	yes	maximal, characteristic
5	C₂×#64775	C₂	1	2	yes	maximal
6	#10601498	C₂	1	8	yes	maximal, direct product factor
7	C₂²×C₅×M₁₆	C₂²	1	2	yes	—
8	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
9	C₂×#4072	C₂²	1	2	yes	—
10	C₂²×C₄×Dic₅	C₂²	1	1	yes	characteristic
11	C₂³×#168	C₂²	1	1	yes	characteristic
12	C₂×C₅×#108	—	2	4	no	—
13	C₂×C₅×#108	C₂²	1	4	yes	—
14	#65408	—	2	4	no	—
15	#65408	C₂²	1	8	yes	—
16	#65395	C₄	1	4	yes	complemented
17	#65395	C₂²	1	4	yes	—
18	C₂×#3950	C₄	1	2	yes	complemented
19	C₂×#3949	C₄	1	2	yes	complemented
20	C₂×#3947	—	2	4	no	—
21	C₂×#3947	C₂²	1	4	yes	—
22	C₂×#3946	—	2	4	no	—
23	#64775	C₂²	1	8	yes	—
24	C₂×#1946	—	5	1	no	maximal, Sylow
25	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
26	C₂³×Dic₅	—	2	1	no	—
27	C₂×C₅×M₁₆	—	2	2	no	—
28	C₂×C₅×M₁₆	—	2	4	no	—
29	C₂×C₅×M₁₆	—	2	4	no	—
30	C₂×C₅×M₁₆	C₂³	1	4	yes	—
31	C₂²×C₅×C₈	—	2	2	no	—
32	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
35	C₂²×C₅×C₈	D₄	1	4	yes	—
36	C₂²×C₅×C₈	Q₈	1	4	yes	—
37	C₂²×C₅×C₈	C₂³	1	2	yes	—
38	C₂×C₂²⋊Dic₅	—	2	2	no	—
39	#4072	—	2	2	no	—
40	#4072	—	2	2	no	—
41	#4072	C₂×C₄	1	4	yes	—
42	#4072	C₂³	1	4	yes	—
43	C₂×C₄⋊Dic₅	C₂×C₄	1	4	yes	—
44	C₂×C₄×Dic₅	—	2	2	no	—
45	C₂×C₄×Dic₅	—	2	2	no	—
46	C₂×C₄×Dic₅	—	2	1	no	—
47	C₂×C₄×Dic₅	Q₈	1	4	yes	—
48	C₂×C₄×Dic₅	C₂³	1	2	yes	—
49	C₂²×#168	—	2	2	no	—
50	C₂²×#168	—	2	2	no	—
51	C₂²×#168	D₄	1	2	yes	—
52	C₂²×#168	D₄	1	2	yes	—
53	C₂²×#168	C₂³	1	2	yes	—
54	C₅×#108	—	2	8	no	—
55	#3950	C₂×C₄	1	8	yes	complemented
56	#3949	C₂×C₄	1	8	yes	complemented
57	#3947	—	2	8	no	—
58	#3946	—	2	16	no	—
59	C₂²×#108	D₅	1	1	yes	characteristic
60	C₂×#427	—	5	1	no	—
61	C₂×#424	—	5	2	no	—
62	C₂²×#97	—	5	1	no	—
63	#1946	—	5	8	no	—
64	C₂×#339	—	5	2	no	—
65	C₂⁴×C₅	#41	1	1	yes	characteristic
66	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
71	C₂²×C₄×C₅	C₂×Q₈	1	2	yes	—
72	C₂²×C₄×C₅	C₂×Q₈	1	2	yes	—
73	C₂²×C₄×C₅	#41	1	2	yes	—
74	C₂²×C₄×C₅	#41	1	1	yes	characteristic
75	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
76	C₂²×Dic₅	—	2	4	no	—
77	C₂²×Dic₅	—	2	4	no	—
78	C₂²×Dic₅	—	2	2	no	—
79	C₂²×Dic₅	—	4	4	no	—
80	C₅×M₁₆	—	2	4	no	—
81	C₅×M₁₆	—	2	4	no	—
82	C₅×M₁₆	—	4	4	no	—
83	C₂×C₅×C₈	—	2	4	no	—
84	C₂×C₅×C₈	—	4	4	no	—
85	C₂×C₅×C₈	C₄⋊C₄	1	4	yes	—
86	C₂×C₅×C₈	C₄⋊C₄	1	4	yes	—
87	C₂×C₅×C₈	C₄⋊C₄	1	8	yes	—
88	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
89	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
90	C₂×C₅×C₈	C₂×D₄	1	4	yes	—
91	C₂×C₅×C₈	C₂×Q₈	1	4	yes	—
92	C₂²⋊Dic₅	—	2	4	no	—
93	C₂²⋊Dic₅	—	2	4	no	—
94	C₄⋊Dic₅	—	2	4	no	—
95	C₄⋊Dic₅	C₂²×C₄	1	8	yes	—
96	C₄×Dic₅	—	2	4	no	—
97	C₄×Dic₅	—	2	4	no	—
98	C₄×Dic₅	—	2	4	no	—
99	C₄×Dic₅	—	2	2	no	—
100	C₄×Dic₅	C₂×Q₈	1	4	yes	—
101	C₂×#168	—	2	4	no	—
102	C₂×#168	—	2	4	no	—
103	C₂×#168	—	2	2	no	—
104	C₂×#168	—	2	2	no	—
105	C₂×#168	C₂×D₄	1	4	yes	—
106	C₂²×M₁₆	C₂×D₅	1	2	yes	—
107	C₂³×C₈	—	5	1	no	—
108	C₂³×C₈	C₂×D₅	1	1	yes	characteristic
109	C₂×#117	—	5	2	no	—
110	C₂²×C₄²	—	5	1	no	—
111	C₂×#108	—	2	4	no	—
112	C₂×#108	C₂×D₅	1	4	yes	—
113	#427	—	5	4	no	—
114	#427	—	5	4	no	—
115	C₂×#107	—	5	2	no	—
116	C₂×#106	—	5	2	no	—
117	#424	—	5	8	no	—
118	#424	—	10	4	no	—
119	C₂×C₄⋊C₈	—	10	4	no	—
120	C₂×#97	—	5	4	no	—
121	C₂×#97	—	10	4	no	—
122	#339	—	5	8	no	—
123	C₂³×C₅	—	2	4	no	—
124	C₂³×C₅	C₄×D₄	1	1	yes	characteristic
125	C₂³×C₅	C₄×D₄	1	1	yes	characteristic
126	C₂³×C₅	C₂²⋊Q₈	1	2	yes	—
127	C₂³×C₅	#125	1	2	yes	—
128	C₂³×C₅	C₂×#41	1	1	yes	characteristic
129	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
130	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
131	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
132	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
133	C₂×C₄×C₅	C₄×Q₈	1	2	yes	—
134	C₂×C₄×C₅	C₄×Q₈	1	2	yes	—
135	C₂×C₄×C₅	#121	1	2	yes	—
136	C₂×C₄×C₅	#121	1	2	yes	—
137	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
138	C₂×C₄×C₅	#125	1	2	yes	—
139	C₂×C₄×C₅	C₄⋊Q₈	1	2	yes	—
140	C₂×C₄×C₅	C₄⋊Q₈	1	2	yes	—
141	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
142	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
143	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
144	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
145	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
146	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
147	C₂×Dic₅	—	2	8	no	—
148	C₂×Dic₅	—	2	4	no	—
149	C₂×Dic₅	—	4	4	no	—
150	C₂×Dic₅	—	4	8	no	—
151	C₂×Dic₅	—	4	8	no	—
152	C₅×C₈	—	4	4	no	—
153	C₅×C₈	C₂×C₄⋊C₄	1	4	yes	—
154	C₅×C₈	C₂×C₄⋊C₄	1	4	yes	—
155	#168	—	2	4	no	—
156	C₂³×C₄	—	10	1	no	—
157	C₂³×C₄	C₂²×D₅	1	1	yes	characteristic
158	C₂×M₁₆	—	2	2	no	—
159	C₂×M₁₆	—	2	4	no	—
160	C₂×M₁₆	—	2	4	no	—
161	C₂×M₁₆	C₂²×D₅	1	4	yes	—
162	C₂²×C₈	—	2	2	no	—
163	C₂²×C₈	—	5	2	no	—
164	C₂²×C₈	—	5	2	no	—
165	C₂²×C₈	—	5	2	no	—
166	C₂²×C₈	—	10	2	no	—
167	C₂²×C₈	—	10	2	no	—
168	C₂²×C₈	Dic₁₀	1	4	yes	—
169	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
170	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
171	C₂²×C₈	C₄×D₅	1	2	yes	—
172	C₂²×C₈	C₂²⋊D₅	1	4	yes	—
173	C₂²×C₈	C₂²×D₅	1	2	yes	—
174	#117	—	5	4	no	—
175	#117	—	5	4	no	—
176	#117	—	10	2	no	—
177	#117	—	10	2	no	—
178	C₂×C₄⋊C₄	—	5	4	no	—
179	C₂×C₂²⋊C₄	—	10	2	no	—
180	C₂×C₄²	—	5	4	no	—
181	C₂×C₄²	—	5	2	no	—
182	C₂×C₄²	—	10	2	no	—
183	C₂×C₄²	—	10	1	no	—
184	C₂×C₄²	—	10	2	no	—
185	#108	—	2	8	no	—
186	#107	—	5	8	no	—
187	#106	—	5	8	no	—
188	C₄⋊C₈	—	10	16	no	—
189	#97	—	10	8	no	—
190	C₂²×C₅	—	2	2	no	—
191	C₂²×C₅	—	2	2	no	—
192	C₂²×C₅	—	2	4	no	—
193	C₂²×C₅	—	2	4	no	—
194	C₂²×C₅	#389	1	2	yes	—
195	C₂²×C₅	#389	1	2	yes	—
196	C₂²×C₅	#444	1	4	yes	—
197	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
198	C₂²×C₅	C₂×C₂²⋊Q₈	1	1	yes	characteristic
199	C₂²×C₅	C₂×#125	1	1	yes	characteristic
200	C₄×C₅	#389	1	2	yes	—
201	C₄×C₅	#389	1	2	yes	—
202	C₄×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
203	C₄×C₅	C₂×C₄×Q₈	1	1	yes	characteristic
204	C₄×C₅	C₂×#121	1	1	yes	characteristic
205	C₄×C₅	C₂×C₄⋊Q₈	1	1	yes	characteristic
206	Dic₅	—	4	4	no	—
207	Dic₅	—	4	8	no	—
208	C₂⁴	#713	1	1	yes	characteristic
209	C₂²×C₄	—	10	2	no	—
210	C₂²×C₄	—	10	4	no	—
211	C₂²×C₄	—	10	4	no	—
212	C₂²×C₄	—	20	4	no	—
213	C₂²×C₄	C₂×Dic₁₀	1	2	yes	—
214	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
215	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
216	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
217	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
218	C₂²×C₄	#711	1	2	yes	—
219	C₂²×C₄	Q₈×D₅	1	2	yes	—
220	C₂²×C₄	#713	1	1	yes	characteristic
221	C₂²×C₄	C₂×C₂²⋊D₅	1	2	yes	—
222	C₂²×C₄	C₂³×D₅	1	1	yes	characteristic
223	M₁₆	—	2	4	no	—
224	M₁₆	—	2	4	no	—
225	M₁₆	—	4	4	no	—
226	C₂×C₈	—	2	4	no	—
227	C₂×C₈	—	4	4	no	—
228	C₂×C₈	—	5	4	no	—
229	C₂×C₈	—	10	2	no	—
230	C₂×C₈	—	10	2	no	—
231	C₂×C₈	—	10	4	no	—
232	C₂×C₈	—	10	4	no	—
233	C₂×C₈	#683	1	4	yes	—
234	C₂×C₈	#683	1	4	yes	—
235	C₂×C₈	#683	1	8	yes	—
236	C₂×C₈	C₂×Dic₁₀	1	4	yes	—
237	C₂×C₈	C₂×C₄×D₅	1	2	yes	—
238	C₂×C₈	C₂×C₄×D₅	1	2	yes	—
239	C₂×C₈	C₂×C₂²⋊D₅	1	4	yes	—
240	C₄⋊C₄	—	5	8	no	—
241	C₄⋊C₄	—	10	4	no	—
242	C₂²⋊C₄	—	10	4	no	—
243	C₂²⋊C₄	—	10	4	no	—
244	C₄²	—	5	4	no	—
245	C₄²	—	10	4	no	—
246	C₄²	—	10	2	no	—
247	C₄²	—	10	4	no	—
248	C₄²	—	10	4	no	—
249	C₂×C₅	—	2	1	no	—
250	C₂×C₅	—	2	1	no	—
251	C₂×C₅	—	2	2	no	—
252	C₂×C₅	#1946	1	4	yes	complemented
253	C₂×C₅	C₂×#389	1	1	yes	characteristic
254	C₂×C₅	C₂×#444	1	2	yes	—
255	C₂³	—	2	4	no	—
256	C₂³	#4036	1	2	yes	—
257	C₂³	#4039	1	1	yes	characteristic
258	C₂³	#4039	1	1	yes	characteristic
259	C₂³	#4092	1	2	yes	—
260	C₂³	C₂×#713	1	1	yes	characteristic
261	C₂×C₄	—	10	8	no	—
262	C₂×C₄	—	10	4	no	—
263	C₂×C₄	—	20	8	no	—
264	C₂×C₄	—	20	4	no	—
265	C₂×C₄	—	20	8	no	—
266	C₂×C₄	#4033	1	2	yes	—
267	C₂×C₄	#4033	1	2	yes	—
268	C₂×C₄	#4034	1	2	yes	—
269	C₂×C₄	#4034	1	2	yes	—
270	C₂×C₄	#4036	1	2	yes	—
271	C₂×C₄	#4039	1	1	yes	characteristic
272	C₂×C₄	#4039	1	1	yes	characteristic
273	C₂×C₄	C₂×#683	1	2	yes	—
274	C₂×C₄	C₂×#683	1	2	yes	—
275	C₂×C₄	#4084	1	2	yes	—
276	C₂×C₄	#4084	1	2	yes	—
277	C₂×C₄	#4092	1	2	yes	—
278	C₂×C₄	C₂²×Dic₁₀	1	1	yes	characteristic
279	C₂×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
280	C₂×C₄	C₂×D₄×D₅	1	1	yes	characteristic
281	C₂×C₄	C₂×#711	1	1	yes	characteristic
282	C₂×C₄	C₂×Q₈×D₅	1	1	yes	characteristic
283	C₂×C₄	C₂²×C₂²⋊D₅	1	1	yes	characteristic
284	C₈	—	4	4	no	—
285	C₈	—	10	4	no	—
286	C₈	C₂×#683	1	4	yes	—
287	C₈	C₂×#683	1	4	yes	—
288	C₅	C₂×#1946	1	1	yes	characteristic, complemented, Sylow
289	C₂²	—	2	4	no	—
290	C₂²	—	2	4	no	—
291	C₂²	—	2	2	no	—
292	C₂²	—	2	2	no	—
293	C₂²	#65183	1	4	yes	—
294	C₂²	#65262	1	2	yes	—
295	C₂²	#65262	1	2	yes	—
296	C₂²	C₂×#4036	1	1	yes	characteristic
297	C₂²	C₂×#4039	1	1	yes	characteristic
298	C₂²	C₂×#4092	1	1	yes	characteristic
299	C₄	—	20	4	no	—
300	C₄	—	20	8	no	—
301	C₄	#65262	1	2	yes	—
302	C₄	#65262	1	2	yes	—
303	C₄	C₂×#4033	1	1	yes	characteristic
304	C₄	C₂×#4034	1	1	yes	characteristic
305	C₄	C₂²×#683	1	1	yes	characteristic
306	C₄	C₂×#4084	1	1	yes	characteristic
307	C₂	—	2	1	no	—
308	C₂	—	2	1	no	—
309	C₂	—	2	2	no	—
310	C₂	#10601498	1	4	yes	direct product factor
311	C₂	C₂×#65183	1	2	yes	—
312	C₂	C₂×#65262	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₅
commutator	C₂×C₄×C₅	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅
Fitting	C₂²×C₅×#108	C₂
socle	C₂³×C₅	C₄×D₄
prime radical	C₂⁴×C₅	#41
cosocle	C₂²×C₄×C₅	C₂⁴
hypercenter	C₂²×#108	D₅
hypocenter	C₅	C₂×#1946
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂²×#108

Complements

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

N	H ≅ G/N	type
6. #10601498	C₂	direct (N × H)
16. #65395	C₄	semidirect
18. C₂×#3950	C₄	semidirect
19. C₂×#3949	C₄	semidirect
55. #3950	C₂×C₄	semidirect
56. #3949	C₂×C₄	semidirect
252. C₂×C₅	#1946	semidirect
288. C₅	C₂×#1946	semidirect
310. C₂	#10601498	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×#108	2
C₂×#65408	2
C₂×#65395	2
C₂²×#3947	2
C₂×#64775	2
#10601498	2
C₂×#1946	5

Subgroups — C2×#10601498 #12102713

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10601498 #12102713