Subgroups — C₂×#20272855 #27304293

Order 13568.

306 conjugacy classes of subgroups.

The subgroup lattice has 200 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	2	yes	maximal, complemented
2	#20272855	C₂	1	4	yes	maximal, direct product factor
3	C₂²×M₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic
4	#16810457	—	2	2	no	—
5	#16810457	C₂²	1	4	yes	complemented
6	C₂×#13133123	—	2	2	no	—
7	C₂³×C₈×C₅₃	C₄	1	1	yes	characteristic
8	C₂×M₃₂×C₅₃	—	2	2	no	—
9	C₂×M₃₂×C₅₃	—	2	2	no	—
10	C₂×M₃₂×C₅₃	C₄	1	2	yes	—
11	C₂×M₃₂×C₅₃	C₂²	1	2	yes	—
12	C₂²×C₁₆×C₅₃	C₄	1	1	yes	characteristic
13	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
14	#13133123	—	2	4	no	—
15	C₂³×C₄×C₅₃	C₈	1	1	yes	characteristic
16	C₂²×C₈×C₅₃	—	2	1	no	—
17	C₂²×C₈×C₅₃	—	2	1	no	—
18	C₂²×C₈×C₅₃	—	2	1	no	—
19	C₂²×C₈×C₅₃	—	2	1	no	—
20	C₂²×C₈×C₅₃	C₈	1	2	yes	—
21	C₂²×C₈×C₅₃	C₈	1	1	yes	characteristic
22	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
23	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
24	M₃₂×C₅₃	—	2	4	no	—
25	M₃₂×C₅₃	—	2	4	no	—
26	C₂×C₁₆×C₅₃	—	2	1	no	—
27	C₂×C₁₆×C₅₃	—	2	1	no	—
28	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
29	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₁₆×C₅₃	D₄	1	2	yes	—
31	C₂×C₁₆×C₅₃	D₄	1	2	yes	—
32	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
33	C₂⁴×C₅₃	C₁₆	1	1	yes	characteristic
34	C₂²×C₄×C₅₃	—	2	1	no	—
35	C₂²×C₄×C₅₃	—	2	1	no	—
36	C₂²×C₄×C₅₃	—	2	1	no	—
37	C₂²×C₄×C₅₃	—	2	1	no	—
38	C₂²×C₄×C₅₃	C₁₆	1	1	yes	characteristic
39	C₂²×C₄×C₅₃	C₁₆	1	2	yes	—
40	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
41	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
42	C₂×C₈×C₅₃	—	2	1	no	—
43	C₂×C₈×C₅₃	—	2	1	no	—
44	C₂×C₈×C₅₃	—	2	2	no	—
45	C₂×C₈×C₅₃	—	2	2	no	—
46	C₂×C₈×C₅₃	—	2	2	no	—
47	C₂×C₈×C₅₃	—	2	2	no	—
48	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
51	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
52	C₂×C₈×C₅₃	M₁₆	1	1	yes	characteristic
53	C₂×C₈×C₅₃	M₁₆	1	1	yes	characteristic
54	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
55	C₁₆×C₅₃	—	2	2	no	—
56	C₁₆×C₅₃	C₂²⋊C₄	1	2	yes	—
57	C₁₆×C₅₃	C₂×D₄	1	2	yes	—
58	C₂³×C₅₃	—	2	1	no	—
59	C₂³×C₅₃	—	2	1	no	—
60	C₂³×C₅₃	—	4	2	no	—
61	C₂³×C₅₃	C₂×C₁₆	1	1	yes	characteristic
62	C₂³×C₅₃	C₂×C₁₆	1	2	yes	—
63	C₂×C₄×C₅₃	—	2	1	no	—
64	C₂×C₄×C₅₃	—	2	1	no	—
65	C₂×C₄×C₅₃	—	2	2	no	—
66	C₂×C₄×C₅₃	—	2	2	no	—
67	C₂×C₄×C₅₃	—	2	2	no	—
68	C₂×C₄×C₅₃	—	2	2	no	—
69	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
70	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
71	C₂×C₄×C₅₃	C₂×C₁₆	1	2	yes	—
72	C₂×C₄×C₅₃	C₂×C₁₆	1	1	yes	characteristic
73	C₂×C₄×C₅₃	M₃₂	1	1	yes	characteristic
74	C₂×C₄×C₅₃	M₃₂	1	1	yes	characteristic
75	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
76	C₈×C₅₃	—	2	2	no	—
77	C₈×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
78	C₈×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
79	C₈×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
80	C₈×C₅₃	C₂×M₁₆	1	1	yes	characteristic
81	C₂×#1400	—	53	1	no	maximal, Sylow
82	C₂²×C₅₃	—	2	2	no	—
83	C₂²×C₅₃	—	2	2	no	—
84	C₂²×C₅₃	—	4	2	no	—
85	C₂²×C₅₃	—	4	2	no	—
86	C₂²×C₅₃	—	4	2	no	—
87	C₂²×C₅₃	C₂²⋊C₁₆	1	1	yes	characteristic
88	C₂²×C₅₃	C₂²⋊C₁₆	1	1	yes	characteristic
89	C₂²×C₅₃	C₂²×C₁₆	1	1	yes	characteristic
90	C₄×C₅₃	—	2	2	no	—
91	C₄×C₅₃	C₂²⋊C₁₆	1	1	yes	characteristic
92	C₄×C₅₃	C₂²⋊C₁₆	1	1	yes	characteristic
93	C₄×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
94	C₄×C₅₃	C₂×M₃₂	1	1	yes	characteristic
95	C₂²×M₃₂	D₅₃	1	1	yes	characteristic
96	C₂×M₆₄	—	53	2	no	—
97	#1400	—	53	4	no	—
98	C₂×C₅₃	—	2	1	no	—
99	C₂×C₅₃	—	2	1	no	—
100	C₂×C₅₃	—	4	2	no	—
101	C₂×C₅₃	#1400	1	2	yes	complemented
102	C₂×C₅₃	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
103	C₂³×C₈	Dic₅₃	1	1	yes	characteristic
104	C₂×M₃₂	—	2	2	no	—
105	C₂×M₃₂	—	2	2	no	—
106	C₂×M₃₂	Dic₅₃	1	2	yes	—
107	C₂×M₃₂	C₂×D₅₃	1	2	yes	—
108	C₂²×C₁₆	Dic₅₃	1	1	yes	characteristic
109	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
110	M₆₄	—	53	4	no	—
111	M₆₄	—	106	2	no	—
112	C₂×C₃₂	—	106	2	no	—
113	C₅₃	C₂×#1400	1	1	yes	characteristic, complemented, Sylow
114	C₂³×C₄	#88322	1	1	yes	characteristic
115	C₂²×C₈	—	2	1	no	—
116	C₂²×C₈	—	2	1	no	—
117	C₂²×C₈	—	2	1	no	—
118	C₂²×C₈	—	2	1	no	—
119	C₂²×C₈	#88322	1	1	yes	characteristic
120	C₂²×C₈	#88322	1	2	yes	—
121	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
122	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
123	M₃₂	—	2	4	no	—
124	M₃₂	—	2	4	no	—
125	C₂×C₁₆	—	2	1	no	—
126	C₂×C₁₆	—	2	1	no	—
127	C₂×C₁₆	C₂×Dic₅₃	1	2	yes	—
128	C₂×C₁₆	C₂×Dic₅₃	1	1	yes	characteristic
129	C₂×C₁₆	C₂²⋊D₅₃	1	2	yes	—
130	C₂×C₁₆	C₂²⋊D₅₃	1	2	yes	—
131	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
132	C₃₂	—	106	4	no	—
133	C₂⁴	#11719908	1	1	yes	characteristic
134	C₂²×C₄	—	2	1	no	—
135	C₂²×C₄	—	2	1	no	—
136	C₂²×C₄	—	2	1	no	—
137	C₂²×C₄	—	2	1	no	—
138	C₂²×C₄	#11719908	1	1	yes	characteristic
139	C₂²×C₄	#11719908	1	2	yes	—
140	C₂²×C₄	C₂×#88322	1	2	yes	—
141	C₂²×C₄	C₂×#88322	1	1	yes	characteristic
142	C₂×C₈	—	2	1	no	—
143	C₂×C₈	—	2	1	no	—
144	C₂×C₈	—	2	2	no	—
145	C₂×C₈	—	2	2	no	—
146	C₂×C₈	—	2	2	no	—
147	C₂×C₈	—	2	2	no	—
148	C₂×C₈	C₂×#88322	1	2	yes	—
149	C₂×C₈	C₂×#88322	1	1	yes	characteristic
150	C₂×C₈	#11719917	1	1	yes	characteristic
151	C₂×C₈	#11719917	1	1	yes	characteristic
152	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
153	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
154	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
155	C₁₆	—	2	2	no	—
156	C₁₆	C₂²⋊Dic₅₃	1	2	yes	—
157	C₁₆	C₂×C₂²⋊D₅₃	1	2	yes	—
158	C₂³	—	2	1	no	—
159	C₂³	—	2	1	no	—
160	C₂³	—	4	2	no	—
161	C₂³	C₂×#11719908	1	1	yes	characteristic
162	C₂³	C₂×#11719908	1	2	yes	—
163	C₂×C₄	—	2	1	no	—
164	C₂×C₄	—	2	1	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	C₂²⋊#88322	1	1	yes	characteristic
170	C₂×C₄	C₂²⋊#88322	1	1	yes	characteristic
171	C₂×C₄	C₂×#11719908	1	2	yes	—
172	C₂×C₄	C₂×#11719908	1	1	yes	characteristic
173	C₂×C₄	#13133157	1	1	yes	characteristic
174	C₂×C₄	#13133157	1	1	yes	characteristic
175	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
176	C₈	—	2	2	no	—
177	C₈	C₂²⋊#88322	1	1	yes	characteristic
178	C₈	C₂²⋊#88322	1	1	yes	characteristic
179	C₈	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
180	C₈	C₂×#11719917	1	1	yes	characteristic
181	C₂²	—	2	2	no	—
182	C₂²	—	2	2	no	—
183	C₂²	—	4	2	no	—
184	C₂²	—	4	2	no	—
185	C₂²	—	4	2	no	—
186	C₂²	C₂²⋊#11719908	1	1	yes	characteristic
187	C₂²	C₂²⋊#11719908	1	1	yes	characteristic
188	C₂²	C₂²×#11719908	1	1	yes	characteristic
189	C₄	—	2	2	no	—
190	C₄	C₂²⋊#11719908	1	1	yes	characteristic
191	C₄	C₂²⋊#11719908	1	1	yes	characteristic
192	C₄	C₂×C₂²⋊#88322	1	1	yes	characteristic
193	C₄	C₂×#13133157	1	1	yes	characteristic
194	C₂	—	2	1	no	—
195	C₂	—	2	1	no	—
196	C₂	—	4	2	no	—
197	C₂	#20272855	1	2	yes	direct product factor
198	C₂	C₂×C₂²⋊#11719908	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂²⋊Dic₅₃
commutator	C₂²×C₅₃	C₂²×C₁₆
Frattini	C₂×C₁₆	C₂²×D₅₃
Fitting	C₂²×M₃₂×C₅₃	C₂
socle	C₂²×C₅₃	C₂²⋊C₁₆
prime radical	C₂⁴×C₅₃	C₁₆
cosocle	C₂×C₁₆×C₅₃	C₂³
hypercenter	C₂²×M₃₂	D₅₃
hypocenter	C₅₃	C₂×#1400
LCS	G ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₁₆ ◁ C₂²×M₃₂

Complements

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

N	H ≅ G/N	type
1. C₂×#16810457	C₂	semidirect
2. #20272855	C₂	direct (N × H)
5. #16810457	C₂²	semidirect
101. C₂×C₅₃	#1400	semidirect
113. C₅₃	C₂×#1400	semidirect
197. C₂	#20272855	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
#20272855	2
C₂²×M₃₂×C₅₃	2
C₂×#1400	53

Subgroups — C2×#20272855 #27304293

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20272855 #27304293