Subgroups — C₂×#13102757 #72253229

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×#11717227	C₂	1	2	yes	maximal, complemented
2	C₂×#11717224	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅²×#108	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#11716692	C₂	1	2	yes	maximal, complemented
5	C₂×#11716682	C₂	1	1	yes	maximal, characteristic, complemented
6	#13102757	C₂	1	8	yes	maximal, direct product factor
7	C₂×#87900	C₂²	1	2	yes	—
8	C₂×C₅²×M₁₆	C₂²	1	2	yes	—
9	C₂²×C₅²×C₈	C₂²	1	1	yes	characteristic
10	C₂×C₄×#1039	C₂²	1	1	yes	characteristic
11	C₂²×#6029	C₂²	1	1	yes	characteristic
12	#11717227	—	2	4	no	—
13	#11717227	C₂²	1	8	yes	complemented
14	C₂×#87802	C₄	1	1	yes	characteristic, complemented
15	#11717224	C₄	1	4	yes	complemented
16	#11717224	C₂²	1	4	yes	—
17	C₂×#87801	C₄	1	1	yes	characteristic, complemented
18	C₂×#87800	C₄	1	2	yes	complemented
19	C₂×#87799	—	2	2	no	—
20	C₂×C₈×C₅⋊D₅	—	2	2	no	—
21	C₅²×#108	C₂²	1	4	yes	complemented
22	#11716692	C₂²	1	8	yes	complemented
23	#11716682	C₂²	1	4	yes	complemented
24	C₂×#65183	—	5	6	no	maximal
25	C₂²×C₄×C₅²	C₂³	1	1	yes	characteristic
26	C₂×C₂²⋊(C₅⋊D₅)	—	2	2	no	—
27	C₂²×#1039	—	2	1	no	—
28	#87900	—	2	2	no	—
29	#87900	—	2	2	no	—
30	#87900	C₂×C₄	1	4	yes	—
31	#87900	C₂³	1	4	yes	—
32	C₂×C₄⋊(C₅⋊D₅)	C₂×C₄	1	2	yes	—
33	C₂×C₄×C₅⋊D₅	—	2	2	no	—
34	C₂×#6045	C₂×C₄	1	2	yes	—
35	C₅²×M₁₆	—	2	2	no	—
36	C₅²×M₁₆	C₂³	1	4	yes	—
37	C₂×C₅²×C₈	—	2	2	no	—
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	C₂×#6029	D₄	1	2	yes	—
45	C₂×#6029	D₄	1	2	yes	—
46	C₂×#6029	C₂³	1	2	yes	—
47	#87802	C₂×C₄	1	4	yes	complemented
48	#87801	C₂×C₄	1	4	yes	complemented
49	#87800	C₂×C₄	1	8	yes	complemented
50	#87799	—	2	8	no	—
51	C₈×C₅⋊D₅	—	2	8	no	—
52	C₂×#4053	—	5	12	no	—
53	C₂×#4050	—	5	6	no	—
54	C₂×C₅×#108	D₅	1	6	yes	complemented
55	C₂×#3957	—	5	12	no	—
56	C₂×#3947	—	5	6	no	—
57	#65183	—	5	48	no	—
58	C₂³×C₅²	#41	1	1	yes	characteristic
59	C₂²×C₅⋊D₅	—	2	2	no	—
60	C₂×C₄×C₅²	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₄×C₅²	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₄×C₅²	C₂×D₄	1	1	yes	characteristic
63	C₂×C₄×C₅²	C₂×D₄	1	1	yes	characteristic
64	C₂×C₄×C₅²	#41	1	1	yes	characteristic
65	C₂×C₄×C₅²	C₂⁴	1	1	yes	characteristic
66	C₂²⋊(C₅⋊D₅)	—	2	4	no	—
67	C₂²⋊(C₅⋊D₅)	—	2	4	no	—
68	C₂×#1039	—	2	2	no	—
69	C₂×#1039	—	2	2	no	—
70	C₂×#1039	—	4	2	no	—
71	C₄⋊(C₅⋊D₅)	—	2	2	no	—
72	C₄⋊(C₅⋊D₅)	C₂²×C₄	1	4	yes	—
73	C₄×C₅⋊D₅	—	2	4	no	—
74	C₄×C₅⋊D₅	—	2	4	no	—
75	#6045	—	2	2	no	—
76	#6045	C₂²×C₄	1	4	yes	—
77	C₅²×C₈	—	2	4	no	—
78	C₅²×C₈	C₂²×C₄	1	2	yes	—
79	C₅²×C₈	C₂²×C₄	1	2	yes	—
80	#6029	C₂×D₄	1	4	yes	—
81	C₂×#709	—	5	12	no	—
82	C₂×C₅×M₁₆	C₂×D₅	1	12	yes	—
83	C₂²×C₅×C₈	C₂×D₅	1	6	yes	—
84	C₂×C₄×Dic₅	—	5	6	no	—
85	C₂²×#168	—	5	6	no	—
86	#4053	—	5	48	no	—
87	#4053	—	10	24	no	—
88	C₂×Dic₂₀	—	5	6	no	—
89	#4050	—	5	24	no	—
90	#4050	—	5	24	no	—
91	C₂×D₄₀	—	5	6	no	—
92	C₂×#677	—	5	12	no	—
93	C₂×#676	—	10	12	no	—
94	C₂×C₈×D₅	—	10	12	no	—
95	C₅×#108	C₂×D₅	1	24	yes	complemented
96	#3957	—	5	48	no	—
97	#3947	—	5	24	no	—
98	C₂×#444	—	25	1	no	Sylow
99	C₂²×C₅²	—	2	2	no	—
100	C₂²×C₅²	C₄×D₄	1	1	yes	characteristic
101	C₂²×C₅²	C₄×D₄	1	1	yes	characteristic
102	C₂²×C₅²	C₂×#41	1	1	yes	characteristic
103	C₂×C₅⋊D₅	—	2	4	no	—
104	C₂×C₅⋊D₅	—	4	4	no	—
105	C₄×C₅²	C₄×D₄	1	1	yes	characteristic
106	C₄×C₅²	C₄×D₄	1	1	yes	characteristic
107	C₄×C₅²	C₂³×C₄	1	1	yes	characteristic
108	C₄×C₅²	C₂²×D₄	1	1	yes	characteristic
109	#1039	—	2	4	no	—
110	#1039	—	4	2	no	—
111	C₂²×C₄×C₅	C₂²×D₅	1	6	yes	—
112	C₂×C₂²⋊D₅	—	10	12	no	—
113	C₂²×Dic₅	—	10	6	no	—
114	#709	—	5	24	no	—
115	#709	—	5	24	no	—
116	#709	—	10	12	no	—
117	#709	—	10	12	no	—
118	C₂×D₂₀	—	5	12	no	—
119	C₂×C₄×D₅	—	10	12	no	—
120	C₂×Dic₁₀	—	5	12	no	—
121	C₅×M₁₆	—	2	12	no	—
122	C₅×M₁₆	C₂²×D₅	1	24	yes	—
123	C₂×C₅×C₈	—	2	12	no	—
124	C₂×C₅×C₈	C₄×D₅	1	6	yes	—
125	C₂×C₅×C₈	C₄×D₅	1	6	yes	—
126	C₂×C₅×C₈	C₄×D₅	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	C₂×#168	—	5	12	no	—
131	C₂×#168	—	5	12	no	—
132	C₂×#168	—	5	12	no	—
133	Dic₂₀	—	5	24	no	—
134	D₄₀	—	5	24	no	—
135	#677	—	5	48	no	—
136	#676	—	10	48	no	—
137	C₈×D₅	—	10	48	no	—
138	C₂×#135	—	25	1	no	—
139	C₂×#131	—	25	2	no	—
140	#444	—	25	8	no	—
141	C₂×#108	C₅⋊D₅	1	1	yes	characteristic, complemented
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	#444	1	2	yes	complemented
147	C₂×C₅²	C₂×C₄×D₄	1	1	yes	characteristic
148	C₅⋊D₅	—	4	4	no	—
149	C₂³×C₅	#713	1	6	yes	—
150	C₂²×D₅	—	10	12	no	—
151	C₂×C₄×C₅	C₂×C₄×D₅	1	6	yes	—
152	C₂×C₄×C₅	C₂×C₄×D₅	1	6	yes	—
153	C₂×C₄×C₅	D₄×D₅	1	6	yes	—
154	C₂×C₄×C₅	D₄×D₅	1	6	yes	—
155	C₂×C₄×C₅	#713	1	6	yes	—
156	C₂×C₄×C₅	C₂³×D₅	1	6	yes	—
157	C₂²⋊D₅	—	10	24	no	—
158	C₂²⋊D₅	—	10	24	no	—
159	C₂×Dic₅	—	10	12	no	—
160	C₂×Dic₅	—	10	12	no	—
161	C₂×Dic₅	—	20	12	no	—
162	D₂₀	—	5	24	no	—
163	D₂₀	—	10	12	no	—
164	C₄×D₅	—	10	24	no	—
165	C₄×D₅	—	10	24	no	—
166	Dic₁₀	—	5	24	no	—
167	Dic₁₀	—	10	12	no	—
168	C₅×C₈	—	2	24	no	—
169	C₅×C₈	C₂×C₄×D₅	1	12	yes	—
170	C₅×C₈	C₂×C₄×D₅	1	12	yes	—
171	#168	—	5	24	no	—
172	C₂×#41	—	25	2	no	—
173	#135	—	25	4	no	—
174	#135	—	25	4	no	—
175	C₂×Q₁₆	—	25	1	no	—
176	C₂×SD₁₆	—	25	2	no	—
177	C₂×D₈	—	25	1	no	—
178	#131	—	25	8	no	—
179	#131	—	50	4	no	—
180	C₂×M₁₆	—	50	2	no	—
181	C₂×M₁₆	C₂×C₅⋊D₅	1	2	yes	—
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	C₂×C₅⋊D₅	1	4	yes	complemented
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	#4039	1	6	yes	—
192	C₂²×C₅	#4039	1	6	yes	—
193	C₂²×C₅	C₂×#713	1	6	yes	—
194	C₂×D₅	—	10	24	no	—
195	C₂×D₅	—	20	24	no	—
196	C₄×C₅	#4039	1	6	yes	—
197	C₄×C₅	#4039	1	6	yes	—
198	C₄×C₅	C₂²×C₄×D₅	1	6	yes	—
199	C₄×C₅	C₂×D₄×D₅	1	6	yes	—
200	Dic₅	—	10	24	no	—
201	Dic₅	—	20	12	no	—
202	#41	—	25	4	no	—
203	#41	—	25	4	no	—
204	#41	—	50	2	no	—
205	#41	—	50	2	no	—
206	C₂×Q₈	—	25	2	no	—
207	C₂×D₄	—	25	2	no	—
208	C₂×D₄	—	50	2	no	—
209	C₂²×C₄	—	50	2	no	—
210	C₂²×C₄	—	50	1	no	—
211	C₂²×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
212	Q₁₆	—	25	4	no	—
213	SD₁₆	—	25	8	no	—
214	D₈	—	25	4	no	—
215	M₁₆	—	2	2	no	—
216	M₁₆	—	50	8	no	—
217	M₁₆	C₂²×C₅⋊D₅	1	4	yes	—
218	C₂×C₈	—	2	2	no	—
219	C₂×C₈	—	25	2	no	—
220	C₂×C₈	—	25	2	no	—
221	C₂×C₈	—	25	2	no	—
222	C₂×C₈	—	50	8	no	—
223	C₂×C₈	C₄×C₅⋊D₅	1	1	yes	characteristic
224	C₂×C₈	C₄×C₅⋊D₅	1	1	yes	characteristic
225	C₂×C₈	C₄×C₅⋊D₅	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	#65183	1	12	yes	complemented
232	C₂×C₅	C₂×#4039	1	6	yes	—
233	D₅	—	20	24	no	—
234	C₂³	—	50	2	no	—
235	C₂³	#87904	1	1	yes	characteristic
236	Q₈	—	25	4	no	—
237	Q₈	—	50	2	no	—
238	D₄	—	25	4	no	—
239	D₄	—	50	4	no	—
240	D₄	—	50	4	no	—
241	D₄	—	50	2	no	—
242	C₂×C₄	—	50	2	no	—
243	C₂×C₄	—	50	4	no	—
244	C₂×C₄	—	50	4	no	—
245	C₂×C₄	—	50	2	no	—
246	C₂×C₄	—	100	2	no	—
247	C₂×C₄	C₂×C₄×C₅⋊D₅	1	1	yes	characteristic
248	C₂×C₄	C₂×C₄×C₅⋊D₅	1	1	yes	characteristic
249	C₂×C₄	D₄×C₅⋊D₅	1	1	yes	characteristic
250	C₂×C₄	D₄×C₅⋊D₅	1	1	yes	characteristic
251	C₂×C₄	#87904	1	1	yes	characteristic
252	C₂×C₄	C₂³×C₅⋊D₅	1	1	yes	characteristic
253	C₈	—	2	4	no	—
254	C₈	—	25	4	no	—
255	C₈	C₂×C₄×C₅⋊D₅	1	2	yes	—
256	C₈	C₂×C₄×C₅⋊D₅	1	2	yes	—
257	C₅	C₂×#65183	1	6	yes	complemented
258	C₂²	—	2	2	no	—
259	C₂²	—	50	4	no	—
260	C₂²	—	100	4	no	—
261	C₂²	#11717213	1	1	yes	characteristic
262	C₂²	#11717213	1	1	yes	characteristic
263	C₂²	C₂×#87904	1	1	yes	characteristic
264	C₄	—	50	4	no	—
265	C₄	—	100	2	no	—
266	C₄	#11717213	1	1	yes	characteristic
267	C₄	#11717213	1	1	yes	characteristic
268	C₄	C₂²×C₄×C₅⋊D₅	1	1	yes	characteristic
269	C₄	C₂×D₄×C₅⋊D₅	1	1	yes	characteristic
270	C₂	—	2	1	no	—
271	C₂	—	2	1	no	—
272	C₂	—	100	4	no	—
273	C₂	#13102757	1	2	yes	direct product factor
274	C₂	C₂×#11717213	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	C₂×C₅²×#108	C₂
socle	C₂²×C₅²	C₄×D₄
prime radical	C₂×#11717224	C₂
cosocle	C₂×C₄×C₅²	C₂⁴
hypercenter	C₂×#108	C₅⋊D₅
hypocenter	C₅²	C₂×#444
LCS	G ▷ C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#108

Complements

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

N	H ≅ G/N	type
1. C₂×#11717227	C₂	semidirect
3. C₂×C₅²×#108	C₂	semidirect
4. C₂×#11716692	C₂	semidirect
5. C₂×#11716682	C₂	semidirect
6. #13102757	C₂	direct (N × H)
13. #11717227	C₂²	semidirect
14. C₂×#87802	C₄	semidirect
15. #11717224	C₄	semidirect
17. C₂×#87801	C₄	semidirect
18. C₂×#87800	C₄	semidirect
21. C₅²×#108	C₂²	semidirect
22. #11716692	C₂²	semidirect
23. #11716682	C₂²	semidirect
47. #87802	C₂×C₄	semidirect
48. #87801	C₂×C₄	semidirect
49. #87800	C₂×C₄	semidirect
54. C₂×C₅×#108	D₅	semidirect
95. C₅×#108	C₂×D₅	semidirect
141. C₂×#108	C₅⋊D₅	semidirect
146. C₂×C₅²	#444	semidirect
186. #108	C₂×C₅⋊D₅	semidirect
189. C₅²	C₂×#444	semidirect
231. C₂×C₅	#65183	semidirect
257. C₅	C₂×#65183	semidirect
273. C₂	#13102757	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×#11717227	2
C₂×#11717224	2
C₂×C₅²×#108	2
C₂×#11716692	2
C₂×#11716682	2
#13102757	2
C₂×#65183	5

Subgroups — C2×#13102757 #72253229

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13102757 #72253229