Subgroups — C₂×#13103057 #72253232

Order 3200.

1856 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅²×#135	C₂	1	1	yes	maximal, characteristic
2	C₂×#11717268	C₂	1	2	yes	maximal, complemented
3	C₂×#11716704	C₂	1	2	yes	maximal, complemented
4	#13103057	C₂	1	8	yes	maximal, direct product factor
5	C₂×#11716686	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11716682	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₅²×#41	C₂²	1	2	yes	—
8	C₅²×#135	C₄	1	4	yes	complemented
9	C₅²×#135	C₂²	1	4	yes	—
10	C₂×C₅²×Q₁₆	C₄	1	1	yes	characteristic, complemented
11	C₂×C₅²×SD₁₆	C₄	1	2	yes	complemented
12	C₂×C₅²×D₈	C₄	1	1	yes	characteristic, complemented
13	C₂²×C₅²×C₈	C₂²	1	1	yes	characteristic
14	#11717268	—	2	4	no	—
15	#11717268	C₂²	1	8	yes	complemented
16	C₂×C₄×#1039	C₂²	1	1	yes	characteristic
17	C₂×#87804	—	2	2	no	—
18	C₂×#87804	C₂²	1	2	yes	—
19	C₂²×#6029	—	2	2	no	—
20	C₂²×#6029	C₂²	1	1	yes	characteristic
21	#11716704	C₂²	1	8	yes	complemented
22	#11716686	C₂²	1	4	yes	complemented
23	#11716682	C₂²	1	4	yes	complemented
24	C₂×#65486	—	5	6	no	maximal
25	C₅²×#41	—	2	2	no	—
26	C₅²×#41	—	2	2	no	—
27	C₅²×#41	C₂×C₄	1	4	yes	—
28	C₅²×#41	C₂³	1	4	yes	—
29	C₂×C₅²×Q₈	C₂×C₄	1	2	yes	—
30	C₂×C₅²×D₄	—	2	2	no	—
31	C₂×C₅²×D₄	C₂×C₄	1	2	yes	—
32	C₂²×C₄×C₅²	—	2	2	no	—
33	C₂²×C₄×C₅²	C₂³	1	1	yes	characteristic
34	C₂²×#1039	—	2	1	no	—
35	C₅²×Q₁₆	C₂×C₄	1	4	yes	complemented
36	C₅²×SD₁₆	C₂×C₄	1	8	yes	complemented
37	C₅²×D₈	C₂×C₄	1	4	yes	complemented
38	C₂×C₅²×C₈	C₂×C₄	1	1	yes	characteristic
39	C₂×C₅²×C₈	C₂×C₄	1	1	yes	characteristic
40	C₂×C₅²×C₈	C₂×C₄	1	2	yes	—
41	C₂×C₅²×C₈	C₂³	1	2	yes	—
42	C₄×#1039	—	2	1	no	—
43	C₄×#1039	C₂³	1	2	yes	—
44	#87804	—	2	2	no	—
45	#87804	—	2	8	no	—
46	#87804	C₂³	1	4	yes	—
47	C₂×#6029	—	2	2	no	—
48	C₂×#6029	—	2	8	no	—
49	C₂×#6029	D₄	1	2	yes	—
50	C₂×#6029	D₄	1	2	yes	—
51	C₂×#6029	C₂³	1	2	yes	—
52	C₂×C₅×#135	D₅	1	6	yes	—
53	C₂×#4094	—	5	12	no	—
54	C₂×#3969	—	5	12	no	—
55	#65486	—	5	48	no	—
56	C₂×#3951	—	5	6	no	—
57	C₂×#3947	—	5	6	no	—
58	C₂³×C₅²	—	2	2	no	—
59	C₂³×C₅²	#41	1	1	yes	characteristic
60	C₅²×Q₈	—	2	2	no	—
61	C₅²×Q₈	C₂²×C₄	1	4	yes	—
62	C₅²×D₄	—	2	4	no	—
63	C₅²×D₄	—	2	4	no	—
64	C₅²×D₄	—	2	2	no	—
65	C₅²×D₄	C₂²×C₄	1	4	yes	—
66	C₂×C₄×C₅²	—	2	2	no	—
67	C₂×C₄×C₅²	—	2	4	no	—
68	C₂×C₄×C₅²	—	2	4	no	—
69	C₂×C₄×C₅²	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₄×C₅²	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₄×C₅²	C₂×D₄	1	1	yes	characteristic
72	C₂×C₄×C₅²	C₂×D₄	1	1	yes	characteristic
73	C₂×C₄×C₅²	#41	1	1	yes	characteristic
74	C₂×C₄×C₅²	C₂⁴	1	1	yes	characteristic
75	C₂×#1039	—	2	2	no	—
76	C₂×#1039	—	4	2	no	—
77	C₅²×C₈	C₂²×C₄	1	2	yes	—
78	C₅²×C₈	C₂²×C₄	1	2	yes	—
79	#6029	—	2	4	no	—
80	#6029	C₂×D₄	1	4	yes	—
81	C₂×C₅×#41	C₂×D₅	1	12	yes	—
82	C₅×#135	Dic₅	1	24	yes	complemented
83	C₅×#135	C₂×D₅	1	24	yes	—
84	C₂×C₅×Q₁₆	Dic₅	1	6	yes	complemented
85	C₂×C₅×SD₁₆	Dic₅	1	12	yes	complemented
86	C₂×C₅×D₈	Dic₅	1	6	yes	complemented
87	C₂²×C₅×C₈	C₂×D₅	1	6	yes	—
88	#4094	—	5	48	no	—
89	#4094	—	10	24	no	—
90	C₂×C₄×Dic₅	—	5	6	no	—
91	C₂×#681	—	5	12	no	—
92	C₂×#681	—	10	12	no	—
93	C₂²×#168	—	5	6	no	—
94	C₂²×#168	—	10	12	no	—
95	#3969	—	5	48	no	—
96	#3951	—	5	24	no	—
97	#3947	—	5	24	no	—
98	C₂×#444	—	25	1	no	Sylow
99	C₂²×C₅²	—	2	2	no	—
100	C₂²×C₅²	—	2	4	no	—
101	C₂²×C₅²	—	4	4	no	—
102	C₂²×C₅²	C₄×D₄	1	1	yes	characteristic
103	C₂²×C₅²	C₄×D₄	1	1	yes	characteristic
104	C₂²×C₅²	C₂×#41	1	1	yes	characteristic
105	C₄×C₅²	—	2	4	no	—
106	C₄×C₅²	C₄×D₄	1	1	yes	characteristic
107	C₄×C₅²	C₄×D₄	1	1	yes	characteristic
108	C₄×C₅²	C₂³×C₄	1	1	yes	characteristic
109	C₄×C₅²	C₂²×D₄	1	1	yes	characteristic
110	#1039	—	4	2	no	—
111	C₅×#41	—	2	12	no	—
112	C₅×#41	—	2	12	no	—
113	C₅×#41	C₂×Dic₅	1	24	yes	—
114	C₅×#41	C₂²×D₅	1	24	yes	—
115	C₂×C₅×Q₈	C₂×Dic₅	1	12	yes	—
116	C₂×C₅×D₄	—	2	12	no	—
117	C₂×C₅×D₄	C₂×Dic₅	1	12	yes	—
118	C₂²×C₄×C₅	—	2	12	no	—
119	C₂²×C₄×C₅	C₂²×D₅	1	6	yes	—
120	C₂²×Dic₅	—	10	6	no	—
121	C₅×Q₁₆	C₂×Dic₅	1	24	yes	complemented
122	C₅×SD₁₆	C₂×Dic₅	1	48	yes	complemented
123	C₅×D₈	C₂×Dic₅	1	24	yes	complemented
124	C₂×C₅×C₈	C₂×Dic₅	1	6	yes	—
125	C₂×C₅×C₈	C₂×Dic₅	1	6	yes	—
126	C₂×C₅×C₈	C₂×Dic₅	1	12	yes	—
127	C₂×C₅×C₈	C₂²×D₅	1	12	yes	—
128	C₄×Dic₅	—	5	12	no	—
129	C₄×Dic₅	—	10	6	no	—
130	#681	—	5	24	no	—
131	#681	—	10	48	no	—
132	#681	—	10	12	no	—
133	C₂×#168	—	5	12	no	—
134	C₂×#168	—	5	12	no	—
135	C₂×#168	—	5	12	no	—
136	C₂×#168	—	10	48	no	—
137	C₂×#168	—	10	12	no	—
138	C₂×#135	C₅⋊D₅	1	1	yes	characteristic
139	C₂×#131	—	25	2	no	—
140	#444	—	25	8	no	—
141	C₂×#108	—	25	1	no	—
142	C₂×#104	—	25	2	no	—
143	C₂×#97	—	25	1	no	—
144	C₂×C₅²	—	2	1	no	—
145	C₂×C₅²	—	2	1	no	—
146	C₂×C₅²	—	4	4	no	—
147	C₂×C₅²	#444	1	2	yes	complemented
148	C₂×C₅²	C₂×C₄×D₄	1	1	yes	characteristic
149	C₂³×C₅	—	2	12	no	—
150	C₂³×C₅	#711	1	6	yes	—
151	C₅×Q₈	—	2	12	no	—
152	C₅×Q₈	C₂²×Dic₅	1	24	yes	—
153	C₅×D₄	—	2	24	no	—
154	C₅×D₄	—	2	24	no	—
155	C₅×D₄	—	2	12	no	—
156	C₅×D₄	C₂²×Dic₅	1	24	yes	—
157	C₂×C₄×C₅	—	2	12	no	—
158	C₂×C₄×C₅	—	2	24	no	—
159	C₂×C₄×C₅	—	2	24	no	—
160	C₂×C₄×C₅	D₄×D₅	1	6	yes	—
161	C₂×C₄×C₅	D₄×D₅	1	6	yes	—
162	C₂×C₄×C₅	#711	1	6	yes	—
163	C₂×C₄×C₅	C₂²×Dic₅	1	6	yes	—
164	C₂×C₄×C₅	C₂²×Dic₅	1	6	yes	—
165	C₂×C₄×C₅	C₂³×D₅	1	6	yes	—
166	C₂×Dic₅	—	10	12	no	—
167	C₂×Dic₅	—	20	12	no	—
168	C₅×C₈	C₂²×Dic₅	1	12	yes	—
169	C₅×C₈	C₂²×Dic₅	1	12	yes	—
170	#168	—	5	24	no	—
171	#168	—	10	24	no	—
172	C₂×#41	C₂×C₅⋊D₅	1	2	yes	—
173	#135	#1039	1	4	yes	complemented
174	#135	C₂×C₅⋊D₅	1	4	yes	—
175	C₂×Q₁₆	#1039	1	1	yes	characteristic, complemented
176	C₂×SD₁₆	#1039	1	2	yes	complemented
177	C₂×D₈	#1039	1	1	yes	characteristic, complemented
178	#131	—	25	8	no	—
179	#131	—	50	4	no	—
180	C₂×M₁₆	—	25	2	no	—
181	C₂×M₁₆	—	50	2	no	—
182	C₂²×C₈	—	25	1	no	—
183	C₂²×C₈	—	50	2	no	—
184	C₂²×C₈	C₂×C₅⋊D₅	1	1	yes	characteristic
185	C₂×C₄²	—	25	1	no	—
186	#108	—	25	4	no	—
187	#104	—	25	8	no	—
188	#97	—	25	4	no	—
189	C₅²	C₂×#444	1	1	yes	characteristic, complemented, Sylow
190	C₂²×C₅	—	2	12	no	—
191	C₂²×C₅	—	2	24	no	—
192	C₂²×C₅	—	4	24	no	—
193	C₂²×C₅	D₄×Dic₅	1	6	yes	—
194	C₂²×C₅	D₄×Dic₅	1	6	yes	—
195	C₂²×C₅	C₂×#711	1	6	yes	—
196	C₄×C₅	—	2	24	no	—
197	C₄×C₅	D₄×Dic₅	1	6	yes	—
198	C₄×C₅	D₄×Dic₅	1	6	yes	—
199	C₄×C₅	C₂×D₄×D₅	1	6	yes	—
200	C₄×C₅	C₂³×Dic₅	1	6	yes	—
201	Dic₅	—	20	12	no	—
202	#41	—	2	2	no	—
203	#41	—	2	2	no	—
204	#41	C₂×#1039	1	4	yes	—
205	#41	C₂²×C₅⋊D₅	1	4	yes	—
206	C₂×Q₈	C₂×#1039	1	2	yes	—
207	C₂×D₄	—	2	2	no	—
208	C₂×D₄	C₂×#1039	1	2	yes	—
209	C₂²×C₄	—	2	2	no	—
210	C₂²×C₄	—	50	1	no	—
211	C₂²×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
212	Q₁₆	C₂×#1039	1	4	yes	complemented
213	SD₁₆	C₂×#1039	1	8	yes	complemented
214	D₈	C₂×#1039	1	4	yes	complemented
215	M₁₆	—	25	4	no	—
216	M₁₆	—	50	8	no	—
217	M₁₆	—	50	2	no	—
218	C₂×C₈	—	25	2	no	—
219	C₂×C₈	—	25	2	no	—
220	C₂×C₈	—	25	2	no	—
221	C₂×C₈	—	50	8	no	—
222	C₂×C₈	—	50	2	no	—
223	C₂×C₈	C₂×#1039	1	1	yes	characteristic
224	C₂×C₈	C₂×#1039	1	1	yes	characteristic
225	C₂×C₈	C₂×#1039	1	2	yes	—
226	C₂×C₈	C₂²×C₅⋊D₅	1	2	yes	—
227	C₄²	—	25	2	no	—
228	C₄²	—	50	1	no	—
229	C₂×C₅	—	2	6	no	—
230	C₂×C₅	—	2	6	no	—
231	C₂×C₅	—	4	24	no	—
232	C₂×C₅	#65486	1	12	yes	complemented
233	C₂×C₅	C₂×D₄×Dic₅	1	6	yes	—
234	C₂³	—	2	2	no	—
235	C₂³	#87902	1	1	yes	characteristic
236	Q₈	—	2	2	no	—
237	Q₈	C₂²×#1039	1	4	yes	—
238	D₄	—	2	4	no	—
239	D₄	—	2	4	no	—
240	D₄	—	2	2	no	—
241	D₄	C₂²×#1039	1	4	yes	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	4	no	—
244	C₂×C₄	—	2	4	no	—
245	C₂×C₄	—	50	2	no	—
246	C₂×C₄	—	100	2	no	—
247	C₂×C₄	D₄×C₅⋊D₅	1	1	yes	characteristic
248	C₂×C₄	D₄×C₅⋊D₅	1	1	yes	characteristic
249	C₂×C₄	#87902	1	1	yes	characteristic
250	C₂×C₄	C₂²×#1039	1	1	yes	characteristic
251	C₂×C₄	C₂²×#1039	1	1	yes	characteristic
252	C₂×C₄	C₂³×C₅⋊D₅	1	1	yes	characteristic
253	C₈	—	25	4	no	—
254	C₈	—	50	4	no	—
255	C₈	C₂²×#1039	1	2	yes	—
256	C₈	C₂²×#1039	1	2	yes	—
257	C₅	C₂×#65486	1	6	yes	complemented
258	C₂²	—	2	2	no	—
259	C₂²	—	2	4	no	—
260	C₂²	—	4	4	no	—
261	C₂²	D₄×#1039	1	1	yes	characteristic
262	C₂²	D₄×#1039	1	1	yes	characteristic
263	C₂²	C₂×#87902	1	1	yes	characteristic
264	C₄	—	2	4	no	—
265	C₄	—	100	2	no	—
266	C₄	D₄×#1039	1	1	yes	characteristic
267	C₄	D₄×#1039	1	1	yes	characteristic
268	C₄	C₂×D₄×C₅⋊D₅	1	1	yes	characteristic
269	C₄	C₂³×#1039	1	1	yes	characteristic
270	C₂	—	2	1	no	—
271	C₂	—	2	1	no	—
272	C₂	—	4	4	no	—
273	C₂	#13103057	1	2	yes	direct product factor
274	C₂	C₂×D₄×#1039	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×C₅⋊D₅
commutator	C₄×C₅²	C₂³×C₄
Frattini	C₂×C₄	C₂³×C₅⋊D₅
Fitting, prime radical	C₂×C₅²×#135	C₂
socle	C₂²×C₅²	C₄×D₄
cosocle	C₂×C₄×C₅²	C₂⁴
hypercenter	C₂×#135	C₅⋊D₅
hypocenter	C₅²	C₂×#444
LCS	G ▷ C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#135

Complements

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

N	H ≅ G/N	type
2. C₂×#11717268	C₂	semidirect
3. C₂×#11716704	C₂	semidirect
4. #13103057	C₂	direct (N × H)
5. C₂×#11716686	C₂	semidirect
6. C₂×#11716682	C₂	semidirect
8. C₅²×#135	C₄	semidirect
10. C₂×C₅²×Q₁₆	C₄	semidirect
11. C₂×C₅²×SD₁₆	C₄	semidirect
12. C₂×C₅²×D₈	C₄	semidirect
15. #11717268	C₂²	semidirect
21. #11716704	C₂²	semidirect
22. #11716686	C₂²	semidirect
23. #11716682	C₂²	semidirect
35. C₅²×Q₁₆	C₂×C₄	semidirect
36. C₅²×SD₁₆	C₂×C₄	semidirect
37. C₅²×D₈	C₂×C₄	semidirect
82. C₅×#135	Dic₅	semidirect
84. C₂×C₅×Q₁₆	Dic₅	semidirect
85. C₂×C₅×SD₁₆	Dic₅	semidirect
86. C₂×C₅×D₈	Dic₅	semidirect
121. C₅×Q₁₆	C₂×Dic₅	semidirect
122. C₅×SD₁₆	C₂×Dic₅	semidirect
123. C₅×D₈	C₂×Dic₅	semidirect
147. C₂×C₅²	#444	semidirect
173. #135	#1039	semidirect
175. C₂×Q₁₆	#1039	semidirect
176. C₂×SD₁₆	#1039	semidirect
177. C₂×D₈	#1039	semidirect
189. C₅²	C₂×#444	semidirect
212. Q₁₆	C₂×#1039	semidirect
213. SD₁₆	C₂×#1039	semidirect
214. D₈	C₂×#1039	semidirect
232. C₂×C₅	#65486	semidirect
257. C₅	C₂×#65486	semidirect
273. C₂	#13103057	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×C₅²×#135	2
C₂×#11717268	2
C₂×#11716704	2
#13103057	2
C₂×#11716686	2
C₂×#11716682	2
C₂×#65486	5

Subgroups — C2×#13103057 #72253232

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13103057 #72253232