Subgroups — C₂×#20276938 #27539375

Order 13568.

726 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811028	C₂	1	1	yes	maximal, characteristic, complemented
2	#20276938	C₂	1	8	yes	maximal, direct product factor
3	C₂×#16810424	C₂	1	4	yes	maximal, complemented
4	C₂×C₄×#11719908	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₅₃×#493	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133217	—	2	1	no	—
7	C₂²×#11719908	C₂²	1	2	yes	—
8	#16811028	—	2	2	no	—
9	#16811028	C₂²	1	4	yes	complemented
10	C₂×#13133151	—	2	1	no	—
11	C₂×#13133151	C₂²	1	2	yes	—
12	#16810424	C₂²	1	16	yes	complemented
13	C₄×#11719908	C₂²	1	4	yes	complemented
14	C₂²×D₈×C₅₃	—	2	1	no	—
15	C₂²×D₈×C₅₃	C₂²	1	2	yes	—
16	C₂×#127×C₅₃	—	2	1	no	—
17	C₅₃×#493	—	2	2	no	—
18	C₅₃×#493	C₂²	1	4	yes	—
19	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₂²×Dic₁₀₆	—	4	1	no	—
21	#13133217	—	2	4	no	—
22	#13133217	—	4	2	no	—
23	C₂×C₄⋊Dic₅₃	—	2	2	no	—
24	C₂×C₄⋊Dic₅₃	—	4	1	no	—
25	C₂×#11719908	—	2	4	no	—
26	C₂×#11719908	C₂³	1	4	yes	—
27	#13133151	—	2	2	no	—
28	#13133151	—	2	4	no	—
29	#13133151	C₂³	1	4	yes	—
30	C₂²×D₄×C₅₃	—	2	2	no	—
31	C₂²×D₄×C₅₃	—	4	1	no	—
32	C₂²×D₄×C₅₃	—	4	1	no	—
33	C₂×D₈×C₅₃	—	2	2	no	—
34	C₂×D₈×C₅₃	—	2	4	no	—
35	C₂×D₈×C₅₃	—	2	8	no	—
36	C₂×D₈×C₅₃	—	4	4	no	—
37	C₂×D₈×C₅₃	C₂³	1	4	yes	—
38	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
40	#127×C₅₃	—	2	4	no	—
41	#127×C₅₃	—	4	2	no	—
42	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
43	C₄×C₈×C₅₃	D₄	1	2	yes	—
44	C₄×C₈×C₅₃	C₂³	1	2	yes	—
45	C₂⁴×C₅₃	—	4	2	no	—
46	C₂×D₄×C₅₃	—	2	4	no	—
47	C₂×D₄×C₅₃	—	4	4	no	—
48	C₂×D₄×C₅₃	—	4	4	no	—
49	C₂×D₄×C₅₃	—	4	4	no	—
50	C₂×D₄×C₅₃	—	4	2	no	—
51	C₂×D₄×C₅₃	—	4	4	no	—
52	C₂×D₄×C₅₃	—	8	4	no	—
53	C₂×D₄×C₅₃	—	8	4	no	—
54	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
56	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
57	C₂²×Dic₅₃	—	4	2	no	—
58	C₂×Dic₁₀₆	—	4	4	no	—
59	C₂×Dic₁₀₆	—	8	4	no	—
60	D₈×C₅₃	—	2	8	no	—
61	D₈×C₅₃	—	4	4	no	—
62	D₈×C₅₃	—	4	8	no	—
63	C₂×C₈×C₅₃	SD₁₆	1	4	yes	—
64	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
65	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
66	C₂×C₈×C₅₃	#41	1	4	yes	—
67	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
68	C₄²×C₅₃	D₈	1	2	yes	—
69	C₄²×C₅₃	C₂×D₄	1	2	yes	—
70	C₄⋊Dic₅₃	—	2	4	no	—
71	C₄⋊Dic₅₃	—	4	2	no	—
72	C₄⋊Dic₅₃	—	4	4	no	—
73	#11719908	—	2	4	no	—
74	C₂³×C₅₃	—	4	4	no	—
75	C₂³×C₅₃	—	8	4	no	—
76	C₂³×C₅₃	—	8	4	no	—
77	C₂³×C₅₃	—	8	4	no	—
78	C₂³×C₅₃	C₂×D₈	1	1	yes	characteristic
79	D₄×C₅₃	—	4	4	no	—
80	D₄×C₅₃	—	4	4	no	—
81	D₄×C₅₃	—	8	4	no	—
82	D₄×C₅₃	—	8	8	no	—
83	D₄×C₅₃	—	8	8	no	—
84	C₂×C₄×C₅₃	D₁₆	1	4	yes	—
85	C₂×C₄×C₅₃	SD₃₂	1	4	yes	—
86	C₂×C₄×C₅₃	#124	1	2	yes	—
87	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
88	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
89	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
90	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
91	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
92	C₂×Dic₅₃	—	4	4	no	—
93	C₂×Dic₅₃	—	8	4	no	—
94	Dic₁₀₆	—	8	8	no	—
95	C₈×C₅₃	#124	1	2	yes	—
96	C₈×C₅₃	C₂×SD₁₆	1	4	yes	—
97	C₈×C₅₃	C₂×#41	1	2	yes	—
98	C₂×#2240	—	53	1	no	maximal, Sylow
99	C₂²×C₅₃	—	8	4	no	—
100	C₂²×C₅₃	—	8	4	no	—
101	C₂²×C₅₃	—	8	4	no	—
102	C₂²×C₅₃	—	16	8	no	—
103	C₂²×C₅₃	#486	1	2	yes	—
104	C₂²×C₅₃	C₂×D₁₆	1	2	yes	—
105	C₂²×C₅₃	C₂×SD₃₂	1	2	yes	—
106	C₂²×C₅₃	C₂²×D₈	1	1	yes	characteristic
107	C₄×C₅₃	#486	1	2	yes	—
108	C₄×C₅₃	C₂×D₁₆	1	4	yes	—
109	C₄×C₅₃	C₂×SD₃₂	1	4	yes	—
110	C₄×C₅₃	C₂×#124	1	1	yes	characteristic
111	C₄×C₅₃	C₂²×SD₁₆	1	1	yes	characteristic
112	Dic₅₃	—	8	4	no	—
113	C₂×#500	—	53	1	no	—
114	C₂×#493	D₅₃	1	1	yes	characteristic
115	#2240	—	53	8	no	—
116	C₂×#357	—	53	4	no	—
117	C₂×C₄×C₁₆	—	53	1	no	—
118	C₂×C₅₃	—	16	4	no	—
119	C₂×C₅₃	#2240	1	4	yes	complemented
120	C₂×C₅₃	C₂×#486	1	1	yes	characteristic
121	C₂×C₅₃	C₂²×D₁₆	1	1	yes	characteristic
122	C₂×C₅₃	C₂²×SD₃₂	1	1	yes	characteristic
123	C₂²×D₈	—	2	1	no	—
124	C₂²×D₈	C₂×D₅₃	1	2	yes	—
125	C₂×C₄⋊Q₈	—	106	1	no	—
126	C₂×#127	—	2	1	no	—
127	C₂²×C₁₆	—	53	2	no	—
128	#500	—	53	4	no	—
129	#500	—	106	2	no	—
130	#493	—	2	2	no	—
131	#493	C₂×D₅₃	1	4	yes	—
132	C₂×#107	—	53	2	no	—
133	C₂×#107	—	106	1	no	—
134	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
135	#357	—	53	16	no	—
136	C₄×C₁₆	—	53	4	no	—
137	C₅₃	C₂×#2240	1	1	yes	characteristic, complemented, Sylow
138	C₂²×Q₈	—	212	1	no	—
139	C₂²×D₄	—	2	2	no	—
140	C₂²×D₄	—	4	1	no	—
141	C₂²×D₄	—	4	1	no	—
142	C₂×D₈	—	2	8	no	—
143	C₂×D₈	—	2	4	no	—
144	C₂×D₈	—	2	2	no	—
145	C₂×D₈	—	4	4	no	—
146	C₂×D₈	C₂²×D₅₃	1	4	yes	—
147	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
148	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
149	C₄⋊Q₈	—	106	4	no	—
150	C₄⋊Q₈	—	212	2	no	—
151	#127	—	2	4	no	—
152	#127	—	4	2	no	—
153	C₂×C₄⋊C₄	—	106	2	no	—
154	C₂×C₄⋊C₄	—	212	1	no	—
155	C₂×C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
156	C₂×C₁₆	—	53	4	no	—
157	C₂×C₁₆	—	106	4	no	—
158	#107	—	53	4	no	—
159	#107	—	106	2	no	—
160	#107	—	106	4	no	—
161	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
162	C₄×C₈	C₂²×D₅₃	1	2	yes	—
163	C₂⁴	—	4	2	no	—
164	C₂×Q₈	—	424	4	no	—
165	C₂×Q₈	—	212	4	no	—
166	C₂×D₄	—	2	4	no	—
167	C₂×D₄	—	4	4	no	—
168	C₂×D₄	—	4	4	no	—
169	C₂×D₄	—	4	4	no	—
170	C₂×D₄	—	4	2	no	—
171	C₂×D₄	—	4	4	no	—
172	C₂×D₄	—	8	4	no	—
173	C₂×D₄	—	8	4	no	—
174	C₂²×C₄	—	212	2	no	—
175	C₂²×C₄	#11719922	1	1	yes	characteristic
176	C₂²×C₄	#11719922	1	1	yes	characteristic
177	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
178	D₈	—	2	8	no	—
179	D₈	—	4	4	no	—
180	D₈	—	4	8	no	—
181	C₂×C₈	#11719923	1	4	yes	—
182	C₂×C₈	#11719947	1	4	yes	—
183	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
184	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
185	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
186	C₄⋊C₄	—	106	4	no	—
187	C₄⋊C₄	—	212	2	no	—
188	C₄⋊C₄	—	212	4	no	—
189	C₄²	#11719922	1	2	yes	—
190	C₄²	C₂×C₂²⋊D₅₃	1	2	yes	—
191	C₁₆	—	106	4	no	—
192	C₂³	—	4	4	no	—
193	C₂³	—	8	4	no	—
194	C₂³	—	8	4	no	—
195	C₂³	—	8	4	no	—
196	C₂³	C₂×#11719922	1	1	yes	characteristic
197	Q₈	—	424	8	no	—
198	D₄	—	4	4	no	—
199	D₄	—	4	4	no	—
200	D₄	—	8	8	no	—
201	D₄	—	8	8	no	—
202	D₄	—	8	4	no	—
203	C₂×C₄	—	424	4	no	—
204	C₂×C₄	—	212	4	no	—
205	C₂×C₄	#13133158	1	4	yes	—
206	C₂×C₄	#13133161	1	4	yes	—
207	C₂×C₄	#13133204	1	2	yes	—
208	C₂×C₄	C₂×#11719922	1	2	yes	—
209	C₂×C₄	C₂×#11719922	1	2	yes	—
210	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
211	C₂×C₄	C₂×#11719923	1	2	yes	—
212	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
213	C₈	#13133204	1	2	yes	—
214	C₈	C₂×#11719923	1	4	yes	—
215	C₈	C₂×#11719947	1	2	yes	—
216	C₂²	—	8	4	no	—
217	C₂²	—	8	4	no	—
218	C₂²	—	8	4	no	—
219	C₂²	—	16	8	no	—
220	C₂²	#16810969	1	2	yes	—
221	C₂²	C₂×#13133158	1	2	yes	—
222	C₂²	C₂×#13133161	1	2	yes	—
223	C₂²	C₂²×#11719922	1	1	yes	characteristic
224	C₄	—	424	4	no	—
225	C₄	#16810969	1	2	yes	—
226	C₄	C₂×#13133158	1	4	yes	—
227	C₄	C₂×#13133161	1	4	yes	—
228	C₄	C₂×#13133204	1	1	yes	characteristic
229	C₄	C₂²×#11719923	1	1	yes	characteristic
230	C₂	—	16	4	no	—
231	C₂	#20276938	1	4	yes	direct product factor
232	C₂	C₂×#16810969	1	1	yes	characteristic
233	C₂	C₂²×#13133158	1	1	yes	characteristic
234	C₂	C₂²×#13133161	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#11719922
commutator, cosocle	C₂×C₈×C₅₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₃
Fitting, prime radical	C₂×C₅₃×#493	C₂
socle	C₂³×C₅₃	C₂×D₈
hypercenter	C₂×#493	D₅₃
hypocenter	C₅₃	C₂×#2240
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#493

Complements

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

N	H ≅ G/N	type
1. C₂×#16811028	C₂	semidirect
2. #20276938	C₂	direct (N × H)
3. C₂×#16810424	C₂	semidirect
4. C₂×C₄×#11719908	C₂	semidirect
9. #16811028	C₂²	semidirect
12. #16810424	C₂²	semidirect
13. C₄×#11719908	C₂²	semidirect
119. C₂×C₅₃	#2240	semidirect
137. C₅₃	C₂×#2240	semidirect
231. C₂	#20276938	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×#16811028	2
#20276938	2
C₂×#16810424	2
C₂×C₄×#11719908	2
C₂×C₅₃×#493	2
C₂×#2240	53

Subgroups — C2×#20276938 #27539375

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20276938 #27539375