Subgroups — C₂×#20277008 #27542200

Order 13568.

470 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811039	C₂	1	1	yes	maximal, characteristic
2	#20277008	C₂	1	8	yes	maximal, direct product factor
3	C₂×#16810457	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#16810456	C₂	1	2	yes	maximal, complemented
5	C₂×C₃₂×D₅₃	C₂	1	2	yes	maximal, complemented
6	C₂²×C₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#13133223	C₄	1	1	yes	characteristic
8	#16811039	C₄	1	4	yes	—
9	#16811039	C₂²	1	4	yes	—
10	C₂×#13133157	C₄	1	1	yes	characteristic
11	C₂×#13133156	C₄	1	2	yes	—
12	C₂×C₁₆×D₅₃	C₂²	1	2	yes	—
13	#16810457	C₂²	1	4	yes	complemented
14	#16810456	C₂²	1	8	yes	complemented
15	C₂×#13133123	C₂²	1	2	yes	—
16	C₃₂×D₅₃	C₂²	1	8	yes	complemented
17	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
18	C₂×C₃₂×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×C₃₂×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×C₃₂×C₅₃	C₂²	1	4	yes	complemented
21	C₂×#11719945	C₈	1	1	yes	characteristic
22	#13133223	C₈	1	4	yes	—
23	#13133223	C₂×C₄	1	4	yes	—
24	C₂×#11719917	C₈	1	1	yes	characteristic
25	C₂×#11719912	C₈	1	2	yes	—
26	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
27	#13133157	C₂×C₄	1	4	yes	—
28	#13133156	C₂×C₄	1	8	yes	—
29	C₂×#11719908	C₂×C₄	1	2	yes	—
30	C₁₆×D₅₃	C₂×C₄	1	4	yes	—
31	C₁₆×D₅₃	C₂³	1	4	yes	—
32	#13133123	C₂³	1	4	yes	—
33	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
36	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
37	C₂×C₁₆×C₅₃	C₂³	1	2	yes	—
38	C₃₂×C₅₃	C₂³	1	2	yes	—
39	C₃₂×C₅₃	C₂³	1	2	yes	—
40	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
41	C₂×C₂²⋊D₅₃	C₁₆	1	2	yes	—
42	#11719945	C₁₆	1	4	yes	—
43	#11719945	C₂×C₈	1	4	yes	—
44	C₂×D₂₁₂	C₁₆	1	1	yes	characteristic
45	C₂×C₄×D₅₃	C₂×C₈	1	2	yes	—
46	C₂×Dic₁₀₆	C₁₆	1	1	yes	characteristic
47	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
48	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
49	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
51	#11719917	C₂×C₈	1	4	yes	—
52	C₂×#88322	C₂×C₈	1	2	yes	—
53	#11719912	C₂×C₈	1	8	yes	—
54	C₈×D₅₃	C₂×C₈	1	4	yes	—
55	C₈×D₅₃	C₂²×C₄	1	4	yes	—
56	C₁₆×C₅₃	C₂²×C₄	1	1	yes	characteristic
57	C₁₆×C₅₃	C₂²×C₄	1	2	yes	—
58	C₁₆×C₅₃	C₂⁴	1	1	yes	characteristic
59	#11719908	C₂²×C₄	1	4	yes	—
60	C₂³×C₅₃	C₂×C₁₆	1	1	yes	characteristic
61	C₂²×D₅₃	C₂×C₁₆	1	2	yes	—
62	C₂×C₄×C₅₃	C₂×C₁₆	1	1	yes	characteristic
63	C₂×C₄×C₅₃	C₂×C₁₆	1	2	yes	—
64	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
65	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
66	C₂²⋊D₅₃	C₂×C₁₆	1	8	yes	—
67	C₂×Dic₅₃	C₂×C₁₆	1	2	yes	—
68	D₂₁₂	C₂×C₁₆	1	4	yes	—
69	C₄×D₅₃	C₂×C₁₆	1	4	yes	—
70	C₄×D₅₃	C₂²×C₈	1	4	yes	—
71	Dic₁₀₆	C₂×C₁₆	1	4	yes	—
72	C₈×C₅₃	C₂²×C₈	1	1	yes	characteristic
73	C₈×C₅₃	C₂²×C₈	1	2	yes	—
74	C₈×C₅₃	C₂³×C₄	1	1	yes	characteristic
75	#88322	C₂²×C₈	1	4	yes	—
76	C₂×#2258	—	53	1	no	maximal, Sylow
77	C₂²×C₅₃	—	2	2	no	—
78	C₂²×C₅₃	C₂²×C₁₆	1	2	yes	—
79	C₂²×C₅₃	C₂²×C₁₆	1	1	yes	characteristic
80	C₂×D₅₃	—	2	4	no	—
81	C₂×D₅₃	C₂²×C₁₆	1	4	yes	—
82	C₄×C₅₃	C₂²×C₁₆	1	2	yes	—
83	C₄×C₅₃	C₂²×C₁₆	1	1	yes	characteristic
84	C₄×C₅₃	C₂³×C₈	1	1	yes	characteristic
85	Dic₅₃	C₂²×C₁₆	1	4	yes	—
86	C₂×#504	—	53	1	no	—
87	#2258	—	53	8	no	—
88	C₂×M₆₄	—	53	1	no	—
89	C₂×M₆₄	—	53	2	no	—
90	C₂²×C₃₂	—	53	2	no	—
91	C₂²×C₃₂	D₅₃	1	1	yes	characteristic, complemented
92	C₂×C₅₃	—	2	2	no	—
93	C₂×C₅₃	#2258	1	2	yes	complemented
94	C₂×C₅₃	C₂³×C₁₆	1	1	yes	characteristic
95	D₅₃	—	2	4	no	—
96	C₂×#131	—	53	1	no	—
97	#504	—	53	4	no	—
98	#504	—	53	4	no	—
99	C₂×M₃₂	—	53	1	no	—
100	C₂×M₃₂	—	53	2	no	—
101	C₂²×C₁₆	—	53	2	no	—
102	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
103	M₆₄	—	53	4	no	—
104	M₆₄	—	53	8	no	—
105	C₂×C₃₂	—	53	2	no	—
106	C₂×C₃₂	—	53	8	no	—
107	C₂×C₃₂	C₂×D₅₃	1	1	yes	characteristic
108	C₂×C₃₂	C₂×D₅₃	1	1	yes	characteristic
109	C₂×C₃₂	C₂×D₅₃	1	4	yes	complemented
110	C₅₃	C₂×#2258	1	1	yes	characteristic, complemented, Sylow
111	C₂×#41	—	53	1	no	—
112	#131	—	53	4	no	—
113	#131	—	53	4	no	—
114	C₂×M₁₆	—	53	1	no	—
115	C₂×M₁₆	—	53	2	no	—
116	C₂²×C₈	—	53	2	no	—
117	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
118	M₃₂	—	53	4	no	—
119	M₃₂	—	53	8	no	—
120	C₂×C₁₆	—	53	2	no	—
121	C₂×C₁₆	—	53	4	no	—
122	C₂×C₁₆	—	53	4	no	—
123	C₂×C₁₆	C₄×D₅₃	1	1	yes	characteristic
124	C₂×C₁₆	C₄×D₅₃	1	2	yes	—
125	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
126	C₂×C₁₆	C₂²×D₅₃	1	2	yes	—
127	C₃₂	—	53	4	no	—
128	C₃₂	C₂²×D₅₃	1	2	yes	—
129	C₃₂	C₂²×D₅₃	1	2	yes	—
130	#41	—	53	4	no	—
131	#41	—	53	4	no	—
132	C₂×Q₈	—	53	1	no	—
133	C₂×D₄	—	53	2	no	—
134	C₂×D₄	—	53	1	no	—
135	C₂²×C₄	—	53	2	no	—
136	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
137	M₁₆	—	53	4	no	—
138	M₁₆	—	53	8	no	—
139	C₂×C₈	—	53	2	no	—
140	C₂×C₈	—	53	4	no	—
141	C₂×C₈	—	53	4	no	—
142	C₂×C₈	C₈×D₅₃	1	1	yes	characteristic
143	C₂×C₈	C₈×D₅₃	1	2	yes	—
144	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
145	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
146	C₁₆	—	53	4	no	—
147	C₁₆	C₂×C₄×D₅₃	1	1	yes	characteristic
148	C₁₆	C₂×C₄×D₅₃	1	2	yes	—
149	C₁₆	C₂³×D₅₃	1	1	yes	characteristic
150	C₂³	—	53	2	no	—
151	C₂³	C₁₆×D₅₃	1	1	yes	characteristic
152	Q₈	—	53	4	no	—
153	D₄	—	53	8	no	—
154	D₄	—	53	4	no	—
155	C₂×C₄	—	53	2	no	—
156	C₂×C₄	—	53	4	no	—
157	C₂×C₄	—	53	4	no	—
158	C₂×C₄	C₁₆×D₅₃	1	1	yes	characteristic
159	C₂×C₄	C₁₆×D₅₃	1	2	yes	—
160	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
161	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
162	C₈	—	53	4	no	—
163	C₈	C₂×C₈×D₅₃	1	1	yes	characteristic
164	C₈	C₂×C₈×D₅₃	1	2	yes	—
165	C₈	C₂²×C₄×D₅₃	1	1	yes	characteristic
166	C₂²	—	2	2	no	—
167	C₂²	—	53	4	no	—
168	C₂²	—	106	4	no	—
169	C₂²	C₂×C₁₆×D₅₃	1	2	yes	—
170	C₂²	C₂×C₁₆×D₅₃	1	1	yes	characteristic
171	C₄	—	53	4	no	—
172	C₄	C₂×C₁₆×D₅₃	1	1	yes	characteristic
173	C₄	C₂×C₁₆×D₅₃	1	2	yes	—
174	C₄	C₂²×C₈×D₅₃	1	1	yes	characteristic
175	C₂	—	2	2	no	—
176	C₂	—	106	4	no	—
177	C₂	#20277008	1	2	yes	direct product factor
178	C₂	C₂²×C₁₆×D₅₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃₂	C₂×D₅₃
commutator	C₂×C₅₃	C₂³×C₁₆
Frattini	C₁₆	C₂³×D₅₃
Fitting	C₂²×C₃₂×C₅₃	C₂
socle	C₂²×C₅₃	C₂²×C₁₆
prime radical	C₂×#11719945	C₈
cosocle	C₁₆×C₅₃	C₂⁴
hypercenter	C₂²×C₃₂	D₅₃
hypocenter	C₅₃	C₂×#2258
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₃₂ ◁ C₂²×C₃₂

Complements

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

N	H ≅ G/N	type
2. #20277008	C₂	direct (N × H)
3. C₂×#16810457	C₂	semidirect
4. C₂×#16810456	C₂	semidirect
5. C₂×C₃₂×D₅₃	C₂	semidirect
6. C₂²×C₃₂×C₅₃	C₂	semidirect
13. #16810457	C₂²	semidirect
14. #16810456	C₂²	semidirect
16. C₃₂×D₅₃	C₂²	semidirect
20. C₂×C₃₂×C₅₃	C₂²	semidirect
91. C₂²×C₃₂	D₅₃	semidirect
93. C₂×C₅₃	#2258	semidirect
109. C₂×C₃₂	C₂×D₅₃	semidirect
110. C₅₃	C₂×#2258	semidirect
177. C₂	#20277008	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×#16811039	2
#20277008	2
C₂×#16810457	2
C₂×#16810456	2
C₂×C₃₂×D₅₃	2
C₂²×C₃₂×C₅₃	2
C₂×#2258	53

Subgroups — C2×#20277008 #27542200

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20277008 #27542200