Subgroups — #27304289

Order 13568.

298 conjugacy classes of subgroups.

The subgroup lattice has 168 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	C₂²×#13133123	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²⋊#13133123	C₂	1	4	yes	maximal, complemented
4	C₂×C₅₃×#504	C₂	1	1	yes	maximal, characteristic
5	#16810457	—	2	2	no	—
6	C₂×#13133123	—	2	2	no	—
7	C₂×#13133123	C₂²	1	2	yes	—
8	C₂×#13133123	C₂²	1	2	yes	—
9	C₂×#131×C₅₃	C₄	1	1	yes	characteristic
10	C₅₃×#504	—	2	2	no	—
11	C₅₃×#504	—	2	2	no	—
12	C₂×M₃₂×C₅₃	C₄	1	1	yes	characteristic
13	C₂×M₃₂×C₅₃	C₄	1	2	yes	—
14	C₂²×C₁₆×C₅₃	C₂²	1	2	yes	—
15	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
16	#13133123	—	2	2	no	—
17	#13133123	—	2	2	no	—
18	C₂×#41×C₅₃	C₈	1	1	yes	characteristic
19	#131×C₅₃	—	2	2	no	—
20	#131×C₅₃	—	2	2	no	—
21	C₂×M₁₆×C₅₃	C₈	1	2	yes	—
22	C₂×M₁₆×C₅₃	C₈	1	1	yes	characteristic
23	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
24	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
25	M₃₂×C₅₃	—	2	2	no	—
26	M₃₂×C₅₃	—	2	4	no	—
27	C₂×C₁₆×C₅₃	—	2	2	no	—
28	C₂×C₁₆×C₅₃	—	2	2	no	—
29	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
30	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₁₆×C₅₃	D₄	1	2	yes	—
32	C₂×C₁₆×C₅₃	D₄	1	2	yes	—
33	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
34	#41×C₅₃	—	2	2	no	—
35	#41×C₅₃	—	2	2	no	—
36	C₂×Q₈×C₅₃	C₁₆	1	1	yes	characteristic
37	C₂×D₄×C₅₃	C₁₆	1	2	yes	—
38	C₂×D₄×C₅₃	C₁₆	1	1	yes	characteristic
39	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
40	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
41	M₁₆×C₅₃	—	2	2	no	—
42	M₁₆×C₅₃	—	2	4	no	—
43	C₂×C₈×C₅₃	—	2	2	no	—
44	C₂×C₈×C₅₃	—	2	2	no	—
45	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
46	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
47	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
48	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
49	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
50	C₁₆×C₅₃	—	2	2	no	—
51	C₁₆×C₅₃	C₂²⋊C₄	1	2	yes	—
52	C₁₆×C₅₃	C₂×D₄	1	1	yes	characteristic
53	C₁₆×C₅₃	C₂×D₄	1	1	yes	characteristic
54	C₂³×C₅₃	C₂×C₁₆	1	2	yes	—
55	C₂³×C₅₃	C₂×C₁₆	1	1	yes	characteristic
56	Q₈×C₅₃	—	2	2	no	—
57	D₄×C₅₃	—	2	4	no	—
58	D₄×C₅₃	—	2	2	no	—
59	C₂×C₄×C₅₃	—	2	2	no	—
60	C₂×C₄×C₅₃	—	2	2	no	—
61	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
62	C₂×C₄×C₅₃	C₂×C₁₆	1	2	yes	—
63	C₂×C₄×C₅₃	C₂×C₁₆	1	1	yes	characteristic
64	C₂×C₄×C₅₃	M₃₂	1	2	yes	—
65	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
66	C₈×C₅₃	—	2	2	no	—
67	C₈×C₅₃	C₂²⋊C₈	1	2	yes	—
68	C₈×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
69	C₈×C₅₃	C₂×M₁₆	1	1	yes	characteristic
70	#13554	—	53	1	no	maximal, Sylow
71	C₂²×C₅₃	—	2	2	no	—
72	C₂²×C₅₃	—	2	2	no	—
73	C₂²×C₅₃	—	2	4	no	—
74	C₂²×C₅₃	C₂²⋊C₁₆	1	2	yes	—
75	C₂²×C₅₃	C₂²×C₁₆	1	1	yes	characteristic
76	C₄×C₅₃	—	2	2	no	—
77	C₄×C₅₃	C₂²⋊C₁₆	1	2	yes	—
78	C₄×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
79	C₄×C₅₃	C₂×M₃₂	1	1	yes	characteristic
80	C₂×#504	D₅₃	1	1	yes	characteristic
81	C₂×M₆₄	—	53	1	no	—
82	C₂²×C₃₂	—	53	1	no	—
83	C₂²⋊C₃₂	—	53	4	no	—
84	C₂×C₅₃	—	2	2	no	—
85	C₂×C₅₃	—	4	2	no	—
86	C₂×C₅₃	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
87	C₂×C₅₃	#2258	1	2	yes	—
88	C₂×#131	Dic₅₃	1	1	yes	characteristic
89	#504	—	2	2	no	—
90	#504	—	2	2	no	—
91	C₂×M₃₂	Dic₅₃	1	1	yes	characteristic
92	C₂×M₃₂	Dic₅₃	1	2	yes	—
93	C₂²×C₁₆	C₂×D₅₃	1	2	yes	—
94	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
95	M₆₄	—	106	2	no	—
96	C₂×C₃₂	—	53	2	no	—
97	C₂×C₃₂	—	53	2	no	—
98	C₂×C₃₂	—	106	2	no	—
99	C₅₃	#13554	1	1	yes	characteristic, complemented, Sylow
100	C₂×#41	#88322	1	1	yes	characteristic
101	#131	—	2	2	no	—
102	#131	—	2	2	no	—
103	C₂×M₁₆	#88322	1	1	yes	characteristic
104	C₂×M₁₆	#88322	1	2	yes	—
105	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
106	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
107	M₃₂	—	2	2	no	—
108	M₃₂	—	2	4	no	—
109	C₂×C₁₆	—	2	2	no	—
110	C₂×C₁₆	—	2	2	no	—
111	C₂×C₁₆	C₂×Dic₅₃	1	2	yes	—
112	C₂×C₁₆	C₂×Dic₅₃	1	1	yes	characteristic
113	C₂×C₁₆	C₂²⋊D₅₃	1	2	yes	—
114	C₂×C₁₆	C₂²⋊D₅₃	1	2	yes	—
115	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
116	C₃₂	—	106	2	no	—
117	C₃₂	—	106	2	no	—
118	#41	—	2	2	no	—
119	#41	—	2	2	no	—
120	C₂×Q₈	#11719908	1	1	yes	characteristic
121	C₂×D₄	#11719908	1	2	yes	—
122	C₂×D₄	#11719908	1	1	yes	characteristic
123	C₂²×C₄	C₂×#88322	1	2	yes	—
124	C₂²×C₄	C₂×#88322	1	1	yes	characteristic
125	M₁₆	—	2	2	no	—
126	M₁₆	—	2	4	no	—
127	C₂×C₈	—	2	2	no	—
128	C₂×C₈	—	2	2	no	—
129	C₂×C₈	C₂×#88322	1	2	yes	—
130	C₂×C₈	C₂×#88322	1	1	yes	characteristic
131	C₂×C₈	#11719917	1	2	yes	—
132	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
133	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
134	C₁₆	—	2	2	no	—
135	C₁₆	C₂²⋊Dic₅₃	1	2	yes	—
136	C₁₆	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
137	C₁₆	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
138	C₂³	C₂×#11719908	1	2	yes	—
139	C₂³	C₂×#11719908	1	1	yes	characteristic
140	Q₈	—	2	2	no	—
141	D₄	—	2	4	no	—
142	D₄	—	2	2	no	—
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	C₂²⋊#88322	1	2	yes	—
146	C₂×C₄	C₂×#11719908	1	2	yes	—
147	C₂×C₄	C₂×#11719908	1	1	yes	characteristic
148	C₂×C₄	#13133157	1	2	yes	—
149	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
150	C₈	—	2	2	no	—
151	C₈	C₂²⋊#88322	1	2	yes	—
152	C₈	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
153	C₈	C₂×#11719917	1	1	yes	characteristic
154	C₂²	—	2	2	no	—
155	C₂²	—	2	4	no	—
156	C₂²	—	2	2	no	—
157	C₂²	C₂²⋊#11719908	1	2	yes	—
158	C₂²	C₂²×#11719908	1	1	yes	characteristic
159	C₄	—	2	2	no	—
160	C₄	C₂²⋊#11719908	1	2	yes	—
161	C₄	C₂×C₂²⋊#88322	1	1	yes	characteristic
162	C₄	C₂×#13133157	1	1	yes	characteristic
163	C₂	—	2	2	no	—
164	C₂	—	4	2	no	—
165	C₂	C₂×C₂²⋊#11719908	1	1	yes	characteristic
166	C₂	#20277009	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₁₆	C₂²×D₅₃
commutator, socle	C₂²×C₅₃	C₂²×C₁₆
Fitting	C₂×C₅₃×#504	C₂
prime radical	C₂×#41×C₅₃	C₈
cosocle	C₂×C₁₆×C₅₃	C₂³
hypercenter	C₂×#504	D₅₃
hypocenter	C₅₃	#13554
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂×#504

Complements

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

N	H ≅ G/N	type
1. C₂×#16810457	C₂	semidirect
2. C₂²×#13133123	C₂	semidirect
3. C₂²⋊#13133123	C₂	semidirect
99. C₅₃	#13554	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#16810457	2
C₂²×#13133123	2
C₂²⋊#13133123	2
C₂×C₅₃×#504	2
#13554	53