Subgroups — C₂×#20272902 #27304551

Order 13568.

302 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810457	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16810451	C₂	1	1	yes	maximal, characteristic, complemented
3	#20272902	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₅₃×#508	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133151	—	2	1	no	—
6	#16810457	—	2	1	no	—
7	#16810457	C₂²	1	2	yes	complemented
8	C₂×#13133123	—	2	1	no	—
9	#16810451	—	2	1	no	—
10	#16810451	C₂²	1	2	yes	complemented
11	C₂×#135×C₅₃	—	2	1	no	—
12	C₅₃×#508	—	2	1	no	—
13	C₅₃×#508	—	2	1	no	—
14	C₅₃×#508	C₄	1	2	yes	complemented
15	C₅₃×#508	C₂²	1	2	yes	—
16	C₂×Q₃₂×C₅₃	C₄	1	1	yes	characteristic, complemented
17	C₂×SD₃₂×C₅₃	—	2	1	no	—
18	C₂×D₁₆×C₅₃	C₄	1	1	yes	characteristic, complemented
19	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊Dic₅₃	—	4	1	no	—
21	#13133151	—	2	2	no	—
22	#13133151	—	4	1	no	—
23	#13133123	—	2	2	no	—
24	C₂×#41×C₅₃	—	4	1	no	—
25	#135×C₅₃	—	2	2	no	—
26	#135×C₅₃	—	2	2	no	—
27	#135×C₅₃	—	4	1	no	—
28	#135×C₅₃	—	4	1	no	—
29	C₂×Q₁₆×C₅₃	—	2	1	no	—
30	C₂×SD₁₆×C₅₃	—	4	1	no	—
31	C₂×D₈×C₅₃	—	2	1	no	—
32	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
33	Q₃₂×C₅₃	—	2	1	no	—
34	Q₃₂×C₅₃	C₂×C₄	1	2	yes	complemented
35	SD₃₂×C₅₃	—	2	2	no	—
36	SD₃₂×C₅₃	—	2	2	no	—
37	D₁₆×C₅₃	—	2	1	no	—
38	D₁₆×C₅₃	C₂×C₄	1	2	yes	complemented
39	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
40	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₁₆×C₅₃	D₄	1	1	yes	characteristic
42	C₂×C₁₆×C₅₃	D₄	1	1	yes	characteristic
43	C₂×C₁₆×C₅₃	D₄	1	1	yes	characteristic
44	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
45	#41×C₅₃	—	4	2	no	—
46	#41×C₅₃	—	4	2	no	—
47	#41×C₅₃	—	8	1	no	—
48	#41×C₅₃	—	8	1	no	—
49	C₂×Q₈×C₅₃	—	4	1	no	—
50	C₂×D₄×C₅₃	—	4	1	no	—
51	C₂×D₄×C₅₃	—	8	1	no	—
52	C₂²×C₄×C₅₃	—	8	1	no	—
53	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
54	C₂²×Dic₅₃	—	8	1	no	—
55	Q₁₆×C₅₃	—	2	2	no	—
56	Q₁₆×C₅₃	—	4	1	no	—
57	SD₁₆×C₅₃	—	4	2	no	—
58	SD₁₆×C₅₃	—	4	2	no	—
59	D₈×C₅₃	—	2	2	no	—
60	D₈×C₅₃	—	4	1	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	D₈	1	1	yes	characteristic
64	C₂×C₈×C₅₃	SD₁₆	1	1	yes	characteristic
65	C₂×C₈×C₅₃	SD₁₆	1	1	yes	characteristic
66	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
67	C₄⋊Dic₅₃	—	4	2	no	—
68	C₄⋊Dic₅₃	—	8	1	no	—
69	C₁₆×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
70	C₁₆×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
71	C₁₆×C₅₃	C₂²×C₄	1	1	yes	characteristic
72	C₁₆×C₅₃	C₂×D₄	1	1	yes	characteristic
73	C₂³×C₅₃	—	8	1	no	—
74	C₂³×C₅₃	D₁₆	1	1	yes	characteristic
75	Q₈×C₅₃	—	4	2	no	—
76	Q₈×C₅₃	—	8	1	no	—
77	D₄×C₅₃	—	4	2	no	—
78	D₄×C₅₃	—	8	2	no	—
79	D₄×C₅₃	—	8	2	no	—
80	D₄×C₅₃	—	8	1	no	—
81	C₂×C₄×C₅₃	—	8	1	no	—
82	C₂×C₄×C₅₃	—	8	2	no	—
83	C₂×C₄×C₅₃	—	8	2	no	—
84	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
85	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
86	C₂×C₄×C₅₃	D₁₆	1	1	yes	characteristic
87	C₂×C₄×C₅₃	SD₃₂	1	1	yes	characteristic
88	C₂×C₄×C₅₃	SD₃₂	1	1	yes	characteristic
89	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
90	C₂×Dic₅₃	—	8	2	no	—
91	C₂×Dic₅₃	—	16	2	no	—
92	C₈×C₅₃	#102	1	1	yes	characteristic
93	C₈×C₅₃	#102	1	1	yes	characteristic
94	C₈×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₈×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
96	C₂×#1418	—	53	1	no	maximal, Sylow
97	C₂²×C₅₃	—	2	2	no	—
98	C₂²×C₅₃	—	8	2	no	—
99	C₂²×C₅₃	—	16	2	no	—
100	C₂²×C₅₃	#357	1	1	yes	characteristic
101	C₂²×C₅₃	#357	1	1	yes	characteristic
102	C₂²×C₅₃	C₂×D₁₆	1	1	yes	characteristic
103	C₄×C₅₃	—	8	2	no	—
104	C₄×C₅₃	#357	1	1	yes	characteristic
105	C₄×C₅₃	#357	1	1	yes	characteristic
106	C₄×C₅₃	C₂×#102	1	1	yes	characteristic
107	C₄×C₅₃	C₂×SD₃₂	1	1	yes	characteristic
108	Dic₅₃	—	16	2	no	—
109	C₂×#508	D₅₃	1	1	yes	characteristic
110	C₂×M₆₄	—	53	1	no	—
111	C₂×#367	—	53	1	no	—
112	#1418	—	53	4	no	—
113	C₂×C₅₃	—	2	1	no	—
114	C₂×C₅₃	—	2	1	no	—
115	C₂×C₅₃	—	16	2	no	—
116	C₂×C₅₃	#1418	1	2	yes	complemented
117	C₂×C₅₃	C₂×#357	1	1	yes	characteristic
118	C₂×#135	—	2	1	no	—
119	#508	—	2	1	no	—
120	#508	—	2	1	no	—
121	#508	Dic₅₃	1	2	yes	complemented
122	#508	C₂×D₅₃	1	2	yes	—
123	C₂×Q₃₂	Dic₅₃	1	1	yes	characteristic, complemented
124	C₂×SD₃₂	—	2	1	no	—
125	C₂×D₁₆	Dic₅₃	1	1	yes	characteristic, complemented
126	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
127	C₂×#107	—	106	1	no	—
128	M₆₄	—	53	2	no	—
129	M₆₄	—	106	1	no	—
130	C₂×C₃₂	—	106	1	no	—
131	#367	—	53	2	no	—
132	#367	—	106	1	no	—
133	C₅₃	C₂×#1418	1	1	yes	characteristic, complemented, Sylow
134	C₂×#41	—	4	1	no	—
135	#135	—	2	2	no	—
136	#135	—	2	2	no	—
137	#135	—	4	1	no	—
138	#135	—	4	1	no	—
139	C₂×Q₁₆	—	2	1	no	—
140	C₂×SD₁₆	—	4	1	no	—
141	C₂×D₈	—	2	1	no	—
142	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
143	C₂×C₄⋊C₄	—	212	1	no	—
144	Q₃₂	—	2	1	no	—
145	Q₃₂	C₂×Dic₅₃	1	2	yes	complemented
146	SD₃₂	—	2	2	no	—
147	SD₃₂	—	2	2	no	—
148	D₁₆	—	2	1	no	—
149	D₁₆	C₂×Dic₅₃	1	2	yes	complemented
150	C₂×C₁₆	C₂×Dic₅₃	1	1	yes	characteristic
151	C₂×C₁₆	C₂×Dic₅₃	1	1	yes	characteristic
152	C₂×C₁₆	C₂²⋊D₅₃	1	1	yes	characteristic
153	C₂×C₁₆	C₂²⋊D₅₃	1	1	yes	characteristic
154	C₂×C₁₆	C₂²⋊D₅₃	1	1	yes	characteristic
155	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
156	#107	—	106	2	no	—
157	#107	—	212	1	no	—
158	C₃₂	—	106	2	no	—
159	#41	—	4	2	no	—
160	#41	—	4	2	no	—
161	#41	—	8	1	no	—
162	#41	—	8	1	no	—
163	C₂×Q₈	—	4	1	no	—
164	C₂×D₄	—	4	1	no	—
165	C₂×D₄	—	8	1	no	—
166	C₂²×C₄	—	8	1	no	—
167	C₂²×C₄	—	424	1	no	—
168	C₂²×C₄	#11719922	1	1	yes	characteristic
169	Q₁₆	—	2	2	no	—
170	Q₁₆	—	4	1	no	—
171	SD₁₆	—	4	2	no	—
172	SD₁₆	—	4	2	no	—
173	D₈	—	2	2	no	—
174	D₈	—	4	1	no	—
175	C₂×C₈	#11719922	1	1	yes	characteristic
176	C₂×C₈	#11719923	1	1	yes	characteristic
177	C₂×C₈	#11719923	1	1	yes	characteristic
178	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
179	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
180	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
181	C₄⋊C₄	—	424	1	no	—
182	C₄⋊C₄	—	212	2	no	—
183	C₁₆	C₂²⋊Dic₅₃	1	1	yes	characteristic
184	C₁₆	C₂²⋊Dic₅₃	1	1	yes	characteristic
185	C₁₆	C₂²×Dic₅₃	1	1	yes	characteristic
186	C₁₆	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
187	C₂³	—	8	1	no	—
188	C₂³	#13133158	1	1	yes	characteristic
189	Q₈	—	4	2	no	—
190	Q₈	—	8	1	no	—
191	D₄	—	4	2	no	—
192	D₄	—	8	2	no	—
193	D₄	—	8	2	no	—
194	D₄	—	8	1	no	—
195	C₂×C₄	—	8	1	no	—
196	C₂×C₄	—	8	2	no	—
197	C₂×C₄	—	8	2	no	—
198	C₂×C₄	—	848	2	no	—
199	C₂×C₄	—	424	2	no	—
200	C₂×C₄	#13133138	1	1	yes	characteristic
201	C₂×C₄	#13133138	1	1	yes	characteristic
202	C₂×C₄	#13133158	1	1	yes	characteristic
203	C₂×C₄	#13133161	1	1	yes	characteristic
204	C₂×C₄	#13133161	1	1	yes	characteristic
205	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
206	C₈	#13133138	1	1	yes	characteristic
207	C₈	#13133138	1	1	yes	characteristic
208	C₈	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
209	C₈	C₂×#11719923	1	1	yes	characteristic
210	C₂²	—	2	2	no	—
211	C₂²	—	8	2	no	—
212	C₂²	—	16	2	no	—
213	C₂²	#16810424	1	1	yes	characteristic
214	C₂²	#16810424	1	1	yes	characteristic
215	C₂²	C₂×#13133158	1	1	yes	characteristic
216	C₄	—	8	2	no	—
217	C₄	—	848	2	no	—
218	C₄	#16810424	1	1	yes	characteristic
219	C₄	#16810424	1	1	yes	characteristic
220	C₄	C₂×#13133138	1	1	yes	characteristic
221	C₄	C₂×#13133161	1	1	yes	characteristic
222	C₂	—	2	1	no	—
223	C₂	—	2	1	no	—
224	C₂	—	16	2	no	—
225	C₂	#20272902	1	2	yes	direct product factor
226	C₂	C₂×#16810424	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16810424
commutator	C₁₆×C₅₃	C₂²×C₄
Frattini	C₂×C₁₆	C₂²×D₅₃
Fitting, prime radical	C₂×C₅₃×#508	C₂
socle	C₂²×C₅₃	#357
cosocle	C₂×C₁₆×C₅₃	C₂³
hypercenter	C₂×#508	D₅₃
hypocenter	C₅₃	C₂×#1418
LCS	G ▷ C₁₆×C₅₃ ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆ ◁ C₂×#508

Complements

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

N	H ≅ G/N	type
1. C₂×#16810457	C₂	semidirect
2. C₂×#16810451	C₂	semidirect
3. #20272902	C₂	direct (N × H)
7. #16810457	C₂²	semidirect
10. #16810451	C₂²	semidirect
14. C₅₃×#508	C₄	semidirect
16. C₂×Q₃₂×C₅₃	C₄	semidirect
18. C₂×D₁₆×C₅₃	C₄	semidirect
34. Q₃₂×C₅₃	C₂×C₄	semidirect
38. D₁₆×C₅₃	C₂×C₄	semidirect
116. C₂×C₅₃	#1418	semidirect
121. #508	Dic₅₃	semidirect
123. C₂×Q₃₂	Dic₅₃	semidirect
125. C₂×D₁₆	Dic₅₃	semidirect
133. C₅₃	C₂×#1418	semidirect
145. Q₃₂	C₂×Dic₅₃	semidirect
149. D₁₆	C₂×Dic₅₃	semidirect
225. C₂	#20272902	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×#16810457	2
C₂×#16810451	2
#20272902	2
C₂×C₅₃×#508	2
C₂×#1418	53

Subgroups — C2×#20272902 #27304551

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20272902 #27304551