Subgroups — C₂×#20276508 #27537146

Order 13568.

642 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13133156	C₂	1	1	yes	maximal, characteristic
2	C₂×#108×D₅₃	C₂	1	2	yes	maximal
3	#20276508	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16810453	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#16810452	C₂	1	2	yes	maximal, complemented
6	C₂×C₅₃×#368	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×M₁₆×D₅₃	—	2	2	no	—
8	C₂²×C₈×D₅₃	C₂²	1	1	yes	characteristic
9	C₂×#13133156	C₄	1	4	yes	—
10	C₂×#13133156	C₄	1	4	yes	—
11	C₂×#13133156	C₂²	1	4	yes	—
12	C₂²×#11719908	C₂²	1	1	yes	characteristic
13	C₂×#13133153	C₂²	1	2	yes	—
14	C₂×#13133152	—	2	2	no	—
15	#108×D₅₃	—	2	4	no	—
16	#108×D₅₃	C₂²	1	8	yes	—
17	#16810453	C₂²	1	4	yes	complemented
18	#16810452	C₂²	1	8	yes	complemented
19	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×#108×C₅₃	C₂²	1	2	yes	—
21	C₅₃×#368	C₂²	1	4	yes	complemented
22	C₂²×C₄×D₅₃	D₄	1	1	yes	characteristic
23	C₂×#11719917	—	2	2	no	—
24	C₂×#11719912	—	4	2	no	—
25	M₁₆×D₅₃	—	2	8	no	—
26	M₁₆×D₅₃	—	4	4	no	—
27	C₂²×#88322	D₄	1	1	yes	characteristic
28	C₂×C₈×D₅₃	—	4	2	no	—
29	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
30	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
31	C₂×C₈×D₅₃	D₄	1	2	yes	—
32	C₂×C₈×D₅₃	Q₈	1	2	yes	—
33	C₂×C₈×D₅₃	Q₈	1	2	yes	—
34	C₂×C₈×D₅₃	C₂³	1	2	yes	—
35	#13133156	C₂×C₄	1	16	yes	—
36	C₂×#11719908	C₂×C₄	1	2	yes	—
37	C₂×#11719908	C₂×C₄	1	2	yes	—
38	C₂×#11719908	C₂³	1	2	yes	—
39	#13133153	—	2	2	no	—
40	#13133153	C₂³	1	4	yes	—
41	#13133152	—	2	8	no	—
42	C₂×M₁₆×C₅₃	—	2	2	no	—
43	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
44	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
45	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
46	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
47	#108×C₅₃	—	2	2	no	—
48	#108×C₅₃	C₂³	1	4	yes	—
49	C₂³×D₅₃	Q₁₆	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
51	C₂²×Dic₅₃	Q₁₆	1	1	yes	characteristic
52	C₂×C₄×D₅₃	C₄⋊C₄	1	2	yes	—
53	C₂×C₄×D₅₃	C₄⋊C₄	1	2	yes	—
54	C₂×C₄×D₅₃	D₈	1	2	yes	—
55	C₂×C₄×D₅₃	D₈	1	2	yes	—
56	C₂×C₄×D₅₃	Q₁₆	1	2	yes	—
57	C₂×C₄×D₅₃	C₂×D₄	1	2	yes	—
58	M₁₆×C₅₃	—	2	4	no	—
59	M₁₆×C₅₃	—	4	2	no	—
60	C₂×C₈×C₅₃	—	4	2	no	—
61	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
64	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
65	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
66	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
67	#11719917	—	2	4	no	—
68	#11719917	—	4	2	no	—
69	C₂×#88322	—	4	2	no	—
70	C₂×#88322	C₄⋊C₄	1	2	yes	—
71	C₂×#88322	C₄⋊C₄	1	2	yes	—
72	C₂×#88322	C₂×D₄	1	2	yes	—
73	#11719912	—	4	8	no	—
74	C₈×D₅₃	—	4	8	no	—
75	C₈×D₅₃	C₄⋊C₄	1	4	yes	—
76	C₈×D₅₃	C₄⋊C₄	1	4	yes	—
77	C₈×D₅₃	C₂²×C₄	1	4	yes	—
78	C₈×D₅₃	C₂×Q₈	1	4	yes	—
79	C₁₆×C₅₃	C₂²×C₄	1	4	yes	—
80	#11719908	C₂²×C₄	1	4	yes	—
81	C₂³×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
82	C₂²×D₅₃	—	4	2	no	—
83	C₂²×D₅₃	#107	1	2	yes	—
84	C₂²×D₅₃	#107	1	2	yes	—
85	C₂²×D₅₃	C₂×Q₁₆	1	2	yes	—
86	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
87	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
88	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
89	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
90	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
91	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
92	C₂×Dic₅₃	#107	1	2	yes	—
93	C₂×Dic₅₃	#107	1	2	yes	—
94	C₂×Dic₅₃	C₂×Q₁₆	1	2	yes	—
95	C₄×D₅₃	#107	1	4	yes	—
96	C₄×D₅₃	#107	1	4	yes	—
97	C₄×D₅₃	C₂×C₄⋊C₄	1	4	yes	—
98	C₄×D₅₃	C₂×D₈	1	4	yes	—
99	C₈×C₅₃	—	4	4	no	—
100	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
102	C₈×C₅₃	C₂³×C₄	1	1	yes	characteristic
103	C₈×C₅₃	C₂²×Q₈	1	1	yes	characteristic
104	#88322	—	4	4	no	—
105	#88322	C₂×C₄⋊C₄	1	4	yes	—
106	C₂×#2161	—	53	1	no	maximal, Sylow
107	C₂²×C₅₃	—	2	2	no	—
108	C₂²×C₅₃	C₂×#107	1	1	yes	characteristic
109	C₂²×C₅₃	C₂×#107	1	1	yes	characteristic
110	C₂²×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
111	C₂×D₅₃	—	2	4	no	—
112	C₂×D₅₃	—	4	2	no	—
113	C₂×D₅₃	—	4	2	no	—
114	C₂×D₅₃	C₂×#107	1	4	yes	—
115	C₄×C₅₃	C₂×#107	1	1	yes	characteristic
116	C₄×C₅₃	C₂×#107	1	1	yes	characteristic
117	C₄×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
118	C₄×C₅₃	C₂²×D₈	1	1	yes	characteristic
119	Dic₅₃	C₂×#107	1	4	yes	—
120	C₂²×M₃₂	—	53	1	no	—
121	C₂²×#108	—	53	2	no	—
122	#2161	—	53	8	no	—
123	C₂×#368	—	53	1	no	—
124	C₂×#368	—	53	2	no	—
125	C₂×#368	D₅₃	1	1	yes	characteristic, complemented
126	C₂×C₅₃	—	2	1	no	—
127	C₂×C₅₃	—	2	1	no	—
128	C₂×C₅₃	#2161	1	2	yes	complemented
129	C₂×C₅₃	C₂²×#107	1	1	yes	characteristic
130	D₅₃	—	2	4	no	—
131	C₂²×M₁₆	—	106	2	no	—
132	C₂³×C₈	—	53	1	no	—
133	C₂×M₃₂	—	53	4	no	—
134	C₂×M₃₂	—	53	4	no	—
135	C₂×M₃₂	—	53	4	no	—
136	C₂²×C₁₆	—	53	1	no	—
137	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
138	C₂×#108	—	53	2	no	—
139	C₂×#108	—	53	8	no	—
140	C₂×#108	—	106	4	no	—
141	C₂×#108	—	106	2	no	—
142	C₂×#108	C₂×D₅₃	1	2	yes	—
143	#368	—	53	4	no	—
144	#368	—	53	8	no	—
145	#368	C₂×D₅₃	1	4	yes	complemented
146	C₅₃	C₂×#2161	1	1	yes	characteristic, complemented, Sylow
147	C₂³×C₄	—	53	1	no	—
148	C₂×M₁₆	—	2	2	no	—
149	C₂×M₁₆	—	106	8	no	—
150	C₂×M₁₆	—	106	2	no	—
151	C₂×M₁₆	—	212	2	no	—
152	C₂×M₁₆	—	212	4	no	—
153	C₂²×C₈	—	53	1	no	—
154	C₂²×C₈	—	53	2	no	—
155	C₂²×C₈	—	53	2	no	—
156	C₂²×C₈	—	53	2	no	—
157	C₂²×C₈	—	53	2	no	—
158	C₂²×C₈	—	53	2	no	—
159	C₂²×C₈	—	53	2	no	—
160	C₂²×C₈	—	212	2	no	—
161	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
162	M₃₂	—	53	16	no	—
163	C₂×C₁₆	—	53	2	no	—
164	C₂×C₁₆	—	53	2	no	—
165	C₂×C₁₆	—	53	2	no	—
166	C₂×C₁₆	C₄×D₅₃	1	2	yes	—
167	C₂×C₁₆	C₄×D₅₃	1	2	yes	—
168	C₂×C₁₆	C₂²×D₅₃	1	2	yes	—
169	#108	—	2	2	no	—
170	#108	—	53	4	no	—
171	#108	—	106	8	no	—
172	#108	—	106	2	no	—
173	#108	C₂²×D₅₃	1	4	yes	—
174	C₂⁴	—	53	1	no	—
175	C₂²×C₄	—	53	1	no	—
176	C₂²×C₄	—	53	2	no	—
177	C₂²×C₄	—	53	2	no	—
178	C₂²×C₄	—	53	2	no	—
179	C₂²×C₄	—	53	2	no	—
180	C₂²×C₄	—	53	2	no	—
181	C₂²×C₄	—	53	2	no	—
182	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
183	M₁₆	—	2	4	no	—
184	M₁₆	—	4	2	no	—
185	M₁₆	—	106	4	no	—
186	M₁₆	—	212	8	no	—
187	M₁₆	—	212	2	no	—
188	C₂×C₈	—	4	2	no	—
189	C₂×C₈	—	53	2	no	—
190	C₂×C₈	—	53	2	no	—
191	C₂×C₈	—	53	2	no	—
192	C₂×C₈	—	53	4	no	—
193	C₂×C₈	—	53	4	no	—
194	C₂×C₈	—	53	4	no	—
195	C₂×C₈	—	53	4	no	—
196	C₂×C₈	—	212	8	no	—
197	C₂×C₈	—	212	2	no	—
198	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
199	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
200	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
201	C₂×C₈	Q₈×D₅₃	1	1	yes	characteristic
202	C₂×C₈	Q₈×D₅₃	1	1	yes	characteristic
203	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
204	C₁₆	—	53	4	no	—
205	C₁₆	C₂×C₄×D₅₃	1	4	yes	—
206	C₂³	—	53	2	no	—
207	C₂³	—	53	2	no	—
208	C₂³	—	53	2	no	—
209	C₂³	—	212	2	no	—
210	C₂³	Q₁₆×D₅₃	1	1	yes	characteristic
211	C₂×C₄	—	53	2	no	—
212	C₂×C₄	—	53	2	no	—
213	C₂×C₄	—	53	2	no	—
214	C₂×C₄	—	53	4	no	—
215	C₂×C₄	—	53	4	no	—
216	C₂×C₄	—	53	4	no	—
217	C₂×C₄	—	53	4	no	—
218	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
219	C₂×C₄	C₄⋊C₄×D₅₃	1	1	yes	characteristic
220	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
221	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
222	C₂×C₄	Q₁₆×D₅₃	1	1	yes	characteristic
223	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
224	C₈	—	4	4	no	—
225	C₈	—	53	4	no	—
226	C₈	—	212	4	no	—
227	C₈	C₄⋊C₄×D₅₃	1	1	yes	characteristic
228	C₈	C₄⋊C₄×D₅₃	1	1	yes	characteristic
229	C₈	C₂²×C₄×D₅₃	1	1	yes	characteristic
230	C₈	C₂×Q₈×D₅₃	1	1	yes	characteristic
231	C₂²	—	2	2	no	—
232	C₂²	—	53	4	no	—
233	C₂²	—	106	4	no	—
234	C₂²	—	212	2	no	—
235	C₂²	—	212	2	no	—
236	C₂²	#107×D₅₃	1	1	yes	characteristic
237	C₂²	#107×D₅₃	1	1	yes	characteristic
238	C₂²	C₂×Q₁₆×D₅₃	1	1	yes	characteristic
239	C₄	—	53	4	no	—
240	C₄	#107×D₅₃	1	1	yes	characteristic
241	C₄	#107×D₅₃	1	1	yes	characteristic
242	C₄	C₂×C₄⋊C₄×D₅₃	1	1	yes	characteristic
243	C₄	C₂×D₈×D₅₃	1	1	yes	characteristic
244	C₂	—	2	1	no	—
245	C₂	—	2	1	no	—
246	C₂	—	106	4	no	—
247	C₂	#20276508	1	2	yes	direct product factor
248	C₂	C₂×#107×D₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₈×D₅₃
commutator	C₈×C₅₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂×C₅₃×#368	C₂
socle	C₂²×C₅₃	C₂×#107
prime radical	C₂³×D₅₃	Q₁₆
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₂×#368	D₅₃
hypocenter	C₅₃	C₂×#2161
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#368

Complements

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

N	H ≅ G/N	type
3. #20276508	C₂	direct (N × H)
4. C₂×#16810453	C₂	semidirect
5. C₂×#16810452	C₂	semidirect
6. C₂×C₅₃×#368	C₂	semidirect
17. #16810453	C₂²	semidirect
18. #16810452	C₂²	semidirect
21. C₅₃×#368	C₂²	semidirect
125. C₂×#368	D₅₃	semidirect
128. C₂×C₅₃	#2161	semidirect
145. #368	C₂×D₅₃	semidirect
146. C₅₃	C₂×#2161	semidirect
247. C₂	#20276508	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×#13133156	2
C₂×#108×D₅₃	2
#20276508	2
C₂×#16810453	2
C₂×#16810452	2
C₂×C₅₃×#368	2
C₂×#2161	53

Subgroups — C2×#20276508 #27537146

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20276508 #27537146