Subgroups — C₂×#20274468 #27381563

Order 13568.

1134 conjugacy classes of subgroups.

The subgroup lattice has 190 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×#88322	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810469	C₂	1	1	yes	maximal, characteristic
3	#20274468	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16810377	C₂	1	4	yes	maximal
5	C₂²×#97×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂³×#88322	C₂²	1	2	yes	—
7	C₂²×C₄×Dic₅₃	C₂²	1	2	yes	—
8	C₂×C₄×#88322	C₄	1	8	yes	complemented
9	C₂×C₄×#88322	C₂²	1	4	yes	—
10	#16810469	C₄	1	4	yes	—
11	#16810469	C₂²	1	4	yes	—
12	C₂×#13133124	C₄	1	4	yes	—
13	#16810377	C₂²	1	16	yes	—
14	C₂³×C₈×C₅₃	C₂²	1	2	yes	—
15	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×#97×C₅₃	C₄	1	8	yes	complemented
17	C₂×#97×C₅₃	C₂²	1	4	yes	—
18	C₂³×Dic₅₃	C₂×C₄	1	4	yes	—
19	C₂²×#88322	—	2	8	no	—
20	C₂²×#88322	C₂×C₄	1	8	yes	—
21	C₂²×#88322	C₂³	1	4	yes	—
22	C₂×C₄×Dic₅₃	—	2	8	no	—
23	C₂×C₄×Dic₅₃	C₂×C₄	1	4	yes	—
24	C₂×C₄×Dic₅₃	C₂³	1	4	yes	—
25	C₄×#88322	C₂×C₄	1	16	yes	complemented
26	#13133124	C₂×C₄	1	16	yes	—
27	C₂³×C₄×C₅₃	C₂×C₄	1	2	yes	—
28	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
29	C₂²×C₈×C₅₃	—	2	8	no	—
30	C₂²×C₈×C₅₃	C₂×C₄	1	8	yes	—
31	C₂²×C₈×C₅₃	C₂³	1	4	yes	—
32	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
33	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
34	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
35	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
36	#97×C₅₃	C₂×C₄	1	16	yes	complemented
37	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₅₃	—	2	8	no	—
39	C₂²×C₄×C₅₃	C₄²	1	8	yes	—
40	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
41	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
43	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
44	C₂²×C₄×C₅₃	#41	1	8	yes	—
45	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
46	C₂²×Dic₅₃	—	2	8	no	—
47	C₂²×Dic₅₃	—	2	16	no	—
48	C₂²×Dic₅₃	C₂²×C₄	1	8	yes	—
49	C₂×C₈×C₅₃	—	2	16	no	—
50	C₂×C₈×C₅₃	—	2	8	no	—
51	C₂×C₈×C₅₃	C₂²×C₄	1	8	yes	—
52	C₄²×C₅₃	C₄²	1	8	yes	—
53	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
54	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
55	C₄×Dic₅₃	—	2	8	no	—
56	C₄×Dic₅₃	—	2	8	no	—
57	C₂×#88322	—	2	16	no	—
58	C₂×#88322	—	2	8	no	—
59	C₂×#88322	C₂²×C₄	1	8	yes	—
60	C₂³×C₅₃	C₂×C₄²	1	2	yes	—
61	C₂³×C₅₃	#117	1	4	yes	—
62	C₂³×C₅₃	#131	1	8	yes	—
63	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
64	C₂×C₄×C₅₃	—	2	8	no	—
65	C₂×C₄×C₅₃	—	2	16	no	—
66	C₂×C₄×C₅₃	C₂×C₄²	1	8	yes	—
67	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
68	C₂×C₄×C₅₃	#117	1	8	yes	—
69	C₂×C₄×C₅₃	#117	1	4	yes	—
70	C₂×C₄×C₅₃	#117	1	8	yes	—
71	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
72	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
73	C₂×C₄×C₅₃	C₂×#41	1	4	yes	—
74	C₂×Dic₅₃	—	2	8	no	—
75	C₂×Dic₅₃	—	2	16	no	—
76	C₂×Dic₅₃	—	4	16	no	—
77	C₈×C₅₃	—	2	8	no	—
78	#88322	—	2	8	no	—
79	C₂×#1746	—	53	1	no	maximal, Sylow
80	C₂²×C₅₃	#376	1	4	yes	—
81	C₂²×C₅₃	#405	1	8	yes	—
82	C₂²×C₅₃	#433	1	16	yes	—
83	C₂²×C₅₃	C₂²×C₄²	1	1	yes	characteristic
84	C₂²×C₅₃	C₂×#117	1	2	yes	—
85	C₂²×C₅₃	C₂×#131	1	4	yes	—
86	C₄×C₅₃	—	2	8	no	—
87	C₄×C₅₃	#376	1	4	yes	—
88	C₄×C₅₃	C₂×#117	1	2	yes	—
89	C₄×C₅₃	C₂×#117	1	2	yes	—
90	Dic₅₃	—	4	8	no	—
91	C₂²×#97	D₅₃	1	1	yes	characteristic
92	C₂²×C₄×C₈	—	53	1	no	—
93	C₂×#376	—	53	1	no	—
94	#1746	—	53	8	no	—
95	C₂×#336	—	53	4	no	—
96	C₂×C₅₃	#1746	1	8	yes	complemented
97	C₂×C₅₃	C₂×#376	1	1	yes	characteristic
98	C₂×C₅₃	C₂×#405	1	2	yes	—
99	C₂×C₅₃	C₂×#433	1	4	yes	—
100	C₂³×C₈	—	53	2	no	—
101	C₂³×C₈	C₂×D₅₃	1	2	yes	—
102	C₂²×C₄²	—	53	2	no	—
103	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
104	C₂×#97	Dic₅₃	1	8	yes	complemented
105	C₂×#97	C₂×D₅₃	1	4	yes	—
106	C₂×C₄×C₈	—	53	8	no	—
107	C₂×C₄×C₈	—	53	4	no	—
108	#376	—	53	4	no	—
109	#376	—	53	4	no	—
110	C₂×#95	—	53	4	no	—
111	#336	—	53	16	no	—
112	C₅₃	C₂×#1746	1	1	yes	characteristic, complemented, Sylow
113	C₂³×C₄	—	53	4	no	—
114	C₂³×C₄	C₄×D₅₃	1	2	yes	—
115	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
116	C₂²×C₈	—	2	8	no	—
117	C₂²×C₈	—	53	8	no	—
118	C₂²×C₈	—	53	4	no	—
119	C₂²×C₈	—	106	8	no	—
120	C₂²×C₈	C₂×Dic₅₃	1	8	yes	—
121	C₂²×C₈	C₂²×D₅₃	1	4	yes	—
122	C₂×C₄²	—	53	4	no	—
123	C₂×C₄²	—	53	4	no	—
124	C₂×C₄²	—	106	8	no	—
125	C₂×C₄²	C₄×D₅₃	1	4	yes	—
126	C₂×C₄²	C₄×D₅₃	1	2	yes	—
127	C₂×C₄²	C₂×Dic₅₃	1	4	yes	—
128	C₂×C₄²	C₂²×D₅₃	1	2	yes	—
129	#97	C₂×Dic₅₃	1	16	yes	complemented
130	C₄×C₈	—	53	16	no	—
131	#95	—	53	16	no	—
132	C₂⁴	C₂×C₄×D₅₃	1	1	yes	characteristic
133	C₂²×C₄	—	2	8	no	—
134	C₂²×C₄	—	53	8	no	—
135	C₂²×C₄	—	106	8	no	—
136	C₂²×C₄	—	106	16	no	—
137	C₂²×C₄	C₄×Dic₅₃	1	8	yes	—
138	C₂²×C₄	C₂×C₄×D₅₃	1	4	yes	—
139	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
140	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
141	C₂²×C₄	#11719945	1	8	yes	—
142	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
143	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
144	C₂×C₈	—	2	16	no	—
145	C₂×C₈	—	2	8	no	—
146	C₂×C₈	—	53	8	no	—
147	C₂×C₈	—	106	16	no	—
148	C₂×C₈	—	106	8	no	—
149	C₂×C₈	C₂²×Dic₅₃	1	8	yes	—
150	C₄²	—	106	8	no	—
151	C₄²	—	106	8	no	—
152	C₄²	C₄×Dic₅₃	1	8	yes	—
153	C₄²	C₂×C₄×D₅₃	1	4	yes	—
154	C₄²	C₂²×Dic₅₃	1	4	yes	—
155	C₂³	C₂×C₄×Dic₅₃	1	2	yes	—
156	C₂³	#13133174	1	4	yes	—
157	C₂³	#13133225	1	8	yes	—
158	C₂³	C₂²×C₄×D₅₃	1	1	yes	characteristic
159	C₂×C₄	—	2	16	no	—
160	C₂×C₄	—	2	8	no	—
161	C₂×C₄	—	106	8	no	—
162	C₂×C₄	—	106	16	no	—
163	C₂×C₄	—	212	16	no	—
164	C₂×C₄	C₂×C₄×Dic₅₃	1	8	yes	—
165	C₂×C₄	C₂×C₄×Dic₅₃	1	2	yes	—
166	C₂×C₄	#13133174	1	8	yes	—
167	C₂×C₄	#13133174	1	4	yes	—
168	C₂×C₄	#13133176	1	8	yes	—
169	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
170	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
171	C₂×C₄	C₂×#11719945	1	4	yes	—
172	C₈	—	2	8	no	—
173	C₈	—	106	8	no	—
174	C₂²	#16810469	1	4	yes	—
175	C₂²	#16810581	1	8	yes	—
176	C₂²	#16810685	1	16	yes	—
177	C₂²	C₂²×C₄×Dic₅₃	1	1	yes	characteristic
178	C₂²	C₂×#13133174	1	2	yes	—
179	C₂²	C₂×#13133225	1	4	yes	—
180	C₄	—	2	8	no	—
181	C₄	—	212	8	no	—
182	C₄	#16810469	1	4	yes	—
183	C₄	C₂×#13133174	1	2	yes	—
184	C₄	C₂×#13133176	1	2	yes	—
185	C₂	#20274468	1	8	yes	direct product factor
186	C₂	C₂×#16810469	1	1	yes	characteristic
187	C₂	C₂×#16810581	1	2	yes	—
188	C₂	C₂×#16810685	1	4	yes	—

	subgroup	G/H
center	C₂³×C₄	C₂²×D₅₃
commutator	C₂²×C₅₃	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₂²×#97×C₅₃	C₂
socle, prime radical	C₂⁴×C₅₃	C₂²×C₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₂²×#97	D₅₃
hypocenter	C₅₃	C₂×#1746
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂²×#97

Complements

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

N	H ≅ G/N	type
3. #20274468	C₂	direct (N × H)
8. C₂×C₄×#88322	C₄	semidirect
16. C₂×#97×C₅₃	C₄	semidirect
25. C₄×#88322	C₂×C₄	semidirect
36. #97×C₅₃	C₂×C₄	semidirect
96. C₂×C₅₃	#1746	semidirect
104. C₂×#97	Dic₅₃	semidirect
112. C₅₃	C₂×#1746	semidirect
129. #97	C₂×Dic₅₃	semidirect
185. C₂	#20274468	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×C₄×#88322	2
C₂×#16810469	2
#20274468	2
C₂×#16810377	2
C₂²×#97×C₅₃	2
C₂×#1746	53

Subgroups — C2×#20274468 #27381563

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20274468 #27381563