Subgroups — C₂×#20273007 #27349695

Order 13568.

902 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810676	C₂	1	2	yes	maximal
2	#20273007	C₂	1	8	yes	maximal, direct product factor
3	C₂×#16810345	C₂	1	4	yes	maximal, complemented
4	C₂²×C₄×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
5	#16810676	C₄	1	8	yes	—
6	#16810676	C₂²	1	8	yes	—
7	C₂×C₄⋊#88322	C₄	1	4	yes	—
8	C₂×C₂²⋊#88322	C₄	1	4	yes	—
9	C₂×C₄×#88322	C₂²	1	4	yes	—
10	#16810345	C₂²	1	16	yes	complemented
11	C₂³×C₈×C₅₃	C₄	1	2	yes	—
12	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₅₃	C₄	1	4	yes	—
14	C₂×C₄×C₈×C₅₃	C₄	1	2	yes	—
15	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₅₃	C₂²	1	4	yes	—
17	C₂²×#88322	C₂×C₄	1	8	yes	—
18	C₄⋊#88322	C₂×C₄	1	16	yes	—
19	C₂²⋊#88322	C₂×C₄	1	16	yes	—
20	C₄×#88322	C₂×C₄	1	8	yes	—
21	C₄×#88322	C₂³	1	8	yes	—
22	C₂³×C₄×C₅₃	C₂×C₄	1	2	yes	—
23	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₅₃	—	2	8	no	—
25	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
26	C₂²×C₈×C₅₃	C₂×C₄	1	8	yes	—
27	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
29	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
32	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
33	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
34	C₄×C₈×C₅₃	C₂×C₄	1	8	yes	—
35	C₄×C₈×C₅₃	C₂³	1	4	yes	—
36	C₂⁴×C₅₃	C₄²	1	1	yes	characteristic
37	C₂²×C₄×C₅₃	—	2	8	no	—
38	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
39	C₂²×C₄×C₅₃	C₄²	1	4	yes	—
40	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
41	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
42	C₂²×C₄×C₅₃	M₁₆	1	8	yes	—
43	C₂²×C₄×C₅₃	C₂²×C₄	1	4	yes	—
44	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
45	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
47	C₂×C₈×C₅₃	—	2	16	no	—
48	C₂×C₈×C₅₃	—	2	8	no	—
49	C₂×C₈×C₅₃	C₂²×C₄	1	8	yes	—
50	C₄²×C₅₃	C₄²	1	2	yes	—
51	C₄²×C₅₃	C₄²	1	4	yes	—
52	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
53	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
54	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
55	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
56	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
57	C₂×#88322	—	2	16	no	—
58	C₂×#88322	C₂²×C₄	1	16	yes	—
59	C₂³×C₅₃	—	2	4	no	—
60	C₂³×C₅₃	#97	1	4	yes	—
61	C₂³×C₅₃	C₂×C₄²	1	2	yes	—
62	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
63	C₂×C₄×C₅₃	—	2	4	no	—
64	C₂×C₄×C₅₃	—	2	8	no	—
65	C₂×C₄×C₅₃	—	2	8	no	—
66	C₂×C₄×C₅₃	#97	1	8	yes	—
67	C₂×C₄×C₅₃	#97	1	4	yes	—
68	C₂×C₄×C₅₃	C₂×C₄²	1	4	yes	—
69	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
70	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
71	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
72	C₂×C₄×C₅₃	C₂×M₁₆	1	8	yes	—
73	C₂×C₄×C₅₃	C₂×M₁₆	1	4	yes	—
74	C₂×C₄×C₅₃	#131	1	8	yes	—
75	C₂×C₄×C₅₃	C₂³×C₄	1	2	yes	—
76	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
77	C₈×C₅₃	—	2	8	no	—
78	#88322	—	2	16	no	—
79	C₂×#1452	—	53	1	no	maximal, Sylow
80	C₂²×C₅₃	—	2	4	no	—
81	C₂²×C₅₃	—	2	8	no	—
82	C₂²×C₅₃	C₂×#97	1	4	yes	—
83	C₂²×C₅₃	C₂×#97	1	2	yes	—
84	C₂²×C₅₃	#405	1	4	yes	—
85	C₂²×C₅₃	C₂²×C₄²	1	1	yes	characteristic
86	C₄×C₅₃	—	2	4	no	—
87	C₄×C₅₃	C₂×#97	1	2	yes	—
88	C₄×C₅₃	C₂×#97	1	4	yes	—
89	C₄×C₅₃	#405	1	4	yes	—
90	C₄×C₅₃	C₂²×M₁₆	1	2	yes	—
91	C₄×C₅₃	C₂×#131	1	4	yes	—
92	C₂×#431	—	53	2	no	—
93	C₂²×C₄×C₈	D₅₃	1	1	yes	characteristic
94	#1452	—	53	8	no	—
95	C₂×#322	—	53	4	no	—
96	C₂×C₅₃	—	2	4	no	—
97	C₂×C₅₃	#1452	1	4	yes	complemented
98	C₂×C₅₃	C₂²×#97	1	1	yes	characteristic
99	C₂×C₅₃	C₂×#405	1	2	yes	—
100	C₂³×C₈	Dic₅₃	1	2	yes	—
101	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
102	#431	—	53	8	no	—
103	#431	—	53	8	no	—
104	C₂×C₄⋊C₈	—	53	4	no	—
105	C₂×C₂²⋊C₈	—	53	4	no	—
106	C₂×C₄×C₈	—	53	4	no	—
107	C₂×C₄×C₈	Dic₅₃	1	2	yes	—
108	C₂×C₄×C₈	Dic₅₃	1	4	yes	—
109	C₂×C₄×C₈	C₂×D₅₃	1	4	yes	—
110	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
111	#322	—	53	16	no	—
112	C₅₃	C₂×#1452	1	1	yes	characteristic, complemented, Sylow
113	C₂³×C₄	C₄×D₅₃	1	2	yes	—
114	C₂³×C₄	C₂×Dic₅₃	1	1	yes	characteristic
115	C₂²×C₈	—	2	8	no	—
116	C₂²×C₈	—	53	8	no	—
117	C₂²×C₈	C₂×Dic₅₃	1	8	yes	—
118	C₂²×C₈	C₂×Dic₅₃	1	4	yes	—
119	C₂×C₄²	C₄×D₅₃	1	2	yes	—
120	C₂×C₄²	C₄×D₅₃	1	4	yes	—
121	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
122	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
123	C₂×C₄²	C₂²×D₅₃	1	2	yes	—
124	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
125	C₄⋊C₈	—	53	16	no	—
126	C₂²⋊C₈	—	53	16	no	—
127	C₄×C₈	—	53	8	no	—
128	C₄×C₈	—	53	8	no	—
129	C₄×C₈	C₂×Dic₅₃	1	8	yes	—
130	C₄×C₈	C₂×Dic₅₃	1	4	yes	—
131	C₄×C₈	C₂²×D₅₃	1	4	yes	—
132	C₂⁴	C₄×Dic₅₃	1	1	yes	characteristic
133	C₂²×C₄	—	2	8	no	—
134	C₂²×C₄	#11719917	1	8	yes	—
135	C₂²×C₄	C₄×Dic₅₃	1	2	yes	—
136	C₂²×C₄	C₄×Dic₅₃	1	4	yes	—
137	C₂²×C₄	C₄×Dic₅₃	1	2	yes	—
138	C₂²×C₄	C₄×Dic₅₃	1	1	yes	characteristic
139	C₂²×C₄	C₂×C₄×D₅₃	1	4	yes	—
140	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
141	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
142	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
143	C₂×C₈	—	2	16	no	—
144	C₂×C₈	—	2	8	no	—
145	C₂×C₈	—	53	16	no	—
146	C₂×C₈	—	106	16	no	—
147	C₂×C₈	C₂²×Dic₅₃	1	8	yes	—
148	C₄²	C₄×Dic₅₃	1	2	yes	—
149	C₄²	C₄×Dic₅₃	1	4	yes	—
150	C₄²	C₂×C₄×D₅₃	1	2	yes	—
151	C₄²	C₂×C₄×D₅₃	1	4	yes	—
152	C₄²	C₂²×Dic₅₃	1	2	yes	—
153	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
154	C₄²	C₂³×D₅₃	1	1	yes	characteristic
155	C₂³	—	2	4	no	—
156	C₂³	#13133128	1	4	yes	—
157	C₂³	C₂×C₄×Dic₅₃	1	2	yes	—
158	C₂³	C₂×C₄×Dic₅₃	1	1	yes	characteristic
159	C₂×C₄	—	2	4	no	—
160	C₂×C₄	—	2	8	no	—
161	C₂×C₄	—	2	8	no	—
162	C₂×C₄	#13133128	1	8	yes	—
163	C₂×C₄	#13133128	1	4	yes	—
164	C₂×C₄	C₂×C₄×Dic₅₃	1	4	yes	—
165	C₂×C₄	C₂×C₄×Dic₅₃	1	1	yes	characteristic
166	C₂×C₄	C₂×C₄×Dic₅₃	1	2	yes	—
167	C₂×C₄	C₂×C₄×Dic₅₃	1	2	yes	—
168	C₂×C₄	C₂×#11719917	1	8	yes	—
169	C₂×C₄	C₂×#11719917	1	4	yes	—
170	C₂×C₄	#13133223	1	8	yes	—
171	C₂×C₄	C₂²×C₄×D₅₃	1	2	yes	—
172	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
173	C₈	—	2	8	no	—
174	C₈	—	106	16	no	—
175	C₂²	—	2	8	no	—
176	C₂²	—	2	4	no	—
177	C₂²	C₂×#13133128	1	4	yes	—
178	C₂²	C₂×#13133128	1	2	yes	—
179	C₂²	#16810578	1	4	yes	—
180	C₂²	C₂²×C₄×Dic₅₃	1	1	yes	characteristic
181	C₄	—	2	4	no	—
182	C₄	C₂×#13133128	1	2	yes	—
183	C₄	C₂×#13133128	1	4	yes	—
184	C₄	#16810578	1	4	yes	—
185	C₄	C₂²×#11719917	1	2	yes	—
186	C₄	C₂×#13133223	1	4	yes	—
187	C₂	—	2	4	no	—
188	C₂	#20273007	1	4	yes	direct product factor
189	C₂	C₂²×#13133128	1	1	yes	characteristic
190	C₂	C₂×#16810578	1	2	yes	—

	subgroup	G/H
center	C₂×C₄²	C₂²×D₅₃
commutator	C₂²×C₅₃	C₂²×C₄²
Frattini	C₄²	C₂³×D₅₃
Fitting	C₂²×C₄×C₈×C₅₃	C₂
socle	C₂³×C₅₃	C₂×C₄²
prime radical	C₂⁴×C₅₃	C₄²
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	C₂²×C₄×C₈	D₅₃
hypocenter	C₅₃	C₂×#1452
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #20273007	C₂	direct (N × H)
3. C₂×#16810345	C₂	semidirect
10. #16810345	C₂²	semidirect
97. C₂×C₅₃	#1452	semidirect
112. C₅₃	C₂×#1452	semidirect
188. C₂	#20273007	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×#16810676	2
#20273007	2
C₂×#16810345	2
C₂²×C₄×C₈×C₅₃	2
C₂×#1452	53

Subgroups — C2×#20273007 #27349695

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20273007 #27349695