Subgroups — C₂²×#16810453 #27537144

Order 13568.

662 conjugacy classes of subgroups.

The subgroup lattice has 160 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13133153	C₂	1	2	yes	maximal
2	C₂×#16810453	C₂	1	12	yes	maximal, direct product factor
3	C₂³×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11719917	—	2	2	no	—
5	C₂×#13133153	—	2	6	no	—
6	C₂×#13133153	C₂²	1	12	yes	—
7	#16810453	C₂²	1	16	yes	direct product factor
8	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
9	C₂²×C₁₆×C₅₃	C₄	1	6	yes	—
10	C₂²×C₁₆×C₅₃	C₄	1	2	yes	—
11	C₂²×C₁₆×C₅₃	C₂²	1	6	yes	—
12	C₂×#11719917	—	2	12	no	—
13	C₂×#11719917	—	4	6	no	—
14	C₂²×#88322	—	4	2	no	—
15	#13133153	—	2	12	no	—
16	#13133153	C₂³	1	8	yes	—
17	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
18	C₂²×C₈×C₅₃	C₂×C₄	1	3	yes	—
19	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₅₃	D₄	1	3	yes	—
21	C₂²×C₈×C₅₃	Q₈	1	3	yes	—
22	C₂²×C₈×C₅₃	Q₈	1	1	yes	characteristic
23	C₂²×C₈×C₅₃	C₂³	1	3	yes	—
24	C₂×C₁₆×C₅₃	C₂×C₄	1	12	yes	—
25	C₂×C₁₆×C₅₃	C₂×C₄	1	12	yes	—
26	C₂×C₁₆×C₅₃	C₂³	1	4	yes	—
27	C₂⁴×C₅₃	Q₁₆	1	1	yes	characteristic
28	C₂²×C₄×C₅₃	C₄⋊C₄	1	3	yes	—
29	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
30	C₂²×C₄×C₅₃	D₈	1	3	yes	—
31	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
32	C₂²×C₄×C₅₃	Q₁₆	1	3	yes	—
33	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
34	C₂×C₈×C₅₃	C₄⋊C₄	1	6	yes	—
35	C₂×C₈×C₅₃	C₄⋊C₄	1	3	yes	—
36	C₂×C₈×C₅₃	C₄⋊C₄	1	3	yes	—
37	C₂×C₈×C₅₃	C₂²×C₄	1	3	yes	—
38	C₂×C₈×C₅₃	C₂²×C₄	1	3	yes	—
39	C₂×C₈×C₅₃	C₂×D₄	1	3	yes	—
40	C₂×C₈×C₅₃	C₂×Q₈	1	3	yes	—
41	C₂×C₈×C₅₃	C₂×Q₈	1	3	yes	—
42	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
43	#11719917	—	2	8	no	—
44	#11719917	—	4	12	no	—
45	C₂×#88322	—	4	12	no	—
46	C₁₆×C₅₃	C₂²×C₄	1	8	yes	—
47	C₂³×C₅₃	—	2	4	no	—
48	C₂³×C₅₃	#107	1	3	yes	—
49	C₂³×C₅₃	#107	1	1	yes	characteristic
50	C₂³×C₅₃	C₂×Q₁₆	1	3	yes	—
51	C₂×C₄×C₅₃	#107	1	6	yes	—
52	C₂×C₄×C₅₃	#107	1	3	yes	—
53	C₂×C₄×C₅₃	#107	1	3	yes	—
54	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	3	yes	—
55	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	3	yes	—
56	C₂×C₄×C₅₃	C₂×D₈	1	3	yes	—
57	C₂×C₄×C₅₃	C₂×D₈	1	3	yes	—
58	C₂×C₄×C₅₃	C₂×Q₁₆	1	3	yes	—
59	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
60	C₈×C₅₃	C₂×C₄⋊C₄	1	3	yes	—
61	C₈×C₅₃	C₂×C₄⋊C₄	1	3	yes	—
62	C₈×C₅₃	C₂³×C₄	1	1	yes	characteristic
63	C₈×C₅₃	C₂²×Q₈	1	1	yes	characteristic
64	#88322	—	4	8	no	—
65	C₂²×#368	—	53	1	no	maximal, Sylow
66	C₂²×C₅₃	—	2	6	no	—
67	C₂²×C₅₃	—	2	6	no	—
68	C₂²×C₅₃	#368	1	4	yes	complemented
69	C₂²×C₅₃	C₂×#107	1	3	yes	—
70	C₂²×C₅₃	C₂×#107	1	3	yes	—
71	C₂²×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
72	C₄×C₅₃	C₂×#107	1	3	yes	—
73	C₄×C₅₃	C₂×#107	1	3	yes	—
74	C₄×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
75	C₄×C₅₃	C₂²×D₈	1	1	yes	characteristic
76	C₂³×C₁₆	D₅₃	1	1	yes	characteristic
77	C₂²×#108	—	53	2	no	—
78	C₂×#368	—	53	12	no	—
79	C₂×C₅₃	—	2	1	no	—
80	C₂×C₅₃	—	2	3	no	—
81	C₂×C₅₃	C₂×#368	1	6	yes	complemented
82	C₂×C₅₃	C₂²×#107	1	1	yes	characteristic
83	C₂²×M₁₆	—	106	2	no	—
84	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
85	C₂²×C₁₆	Dic₅₃	1	2	yes	—
86	C₂²×C₁₆	Dic₅₃	1	6	yes	—
87	C₂²×C₁₆	C₂×D₅₃	1	6	yes	—
88	C₂×#108	—	53	12	no	—
89	C₂×#108	—	106	6	no	—
90	#368	—	53	16	no	—
91	C₅₃	C₂²×#368	1	1	yes	characteristic, complemented, Sylow
92	C₂³×C₄	D₂₁₂	1	1	yes	characteristic
93	C₂×M₁₆	—	106	12	no	—
94	C₂×M₁₆	—	212	6	no	—
95	C₂²×C₈	—	212	2	no	—
96	C₂²×C₈	Dic₁₀₆	1	1	yes	characteristic
97	C₂²×C₈	Dic₁₀₆	1	3	yes	—
98	C₂²×C₈	D₂₁₂	1	3	yes	—
99	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
100	C₂²×C₈	C₂×Dic₅₃	1	3	yes	—
101	C₂²×C₈	C₂²×D₅₃	1	3	yes	—
102	C₂×C₁₆	C₂×Dic₅₃	1	12	yes	—
103	C₂×C₁₆	C₂×Dic₅₃	1	12	yes	—
104	C₂×C₁₆	C₂²×D₅₃	1	4	yes	—
105	#108	—	53	8	no	—
106	#108	—	106	12	no	—
107	C₂⁴	Dic₂₁₂	1	1	yes	characteristic
108	C₂²×C₄	D₄₂₄	1	3	yes	—
109	C₂²×C₄	D₄₂₄	1	1	yes	characteristic
110	C₂²×C₄	Dic₂₁₂	1	3	yes	—
111	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
112	C₂²×C₄	C₄⋊Dic₅₃	1	3	yes	—
113	C₂²×C₄	C₂×D₂₁₂	1	3	yes	—
114	M₁₆	—	106	8	no	—
115	M₁₆	—	212	12	no	—
116	C₂×C₈	—	212	12	no	—
117	C₂×C₈	C₄⋊Dic₅₃	1	6	yes	—
118	C₂×C₈	C₄⋊Dic₅₃	1	3	yes	—
119	C₂×C₈	C₄⋊Dic₅₃	1	3	yes	—
120	C₂×C₈	C₂×Dic₁₀₆	1	3	yes	—
121	C₂×C₈	C₂×Dic₁₀₆	1	3	yes	—
122	C₂×C₈	C₂×D₂₁₂	1	3	yes	—
123	C₂×C₈	C₂²×Dic₅₃	1	3	yes	—
124	C₂×C₈	C₂²×Dic₅₃	1	3	yes	—
125	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
126	C₁₆	C₂²×Dic₅₃	1	8	yes	—
127	C₂³	—	2	4	no	—
128	C₂³	#13133151	1	3	yes	—
129	C₂³	#13133151	1	1	yes	characteristic
130	C₂³	C₂×Dic₂₁₂	1	3	yes	—
131	C₂×C₄	#13133151	1	6	yes	—
132	C₂×C₄	#13133151	1	3	yes	—
133	C₂×C₄	#13133151	1	3	yes	—
134	C₂×C₄	C₂×C₄⋊Dic₅₃	1	3	yes	—
135	C₂×C₄	C₂×C₄⋊Dic₅₃	1	3	yes	—
136	C₂×C₄	C₂×D₄₂₄	1	3	yes	—
137	C₂×C₄	C₂×D₄₂₄	1	3	yes	—
138	C₂×C₄	C₂×Dic₂₁₂	1	3	yes	—
139	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
140	C₈	—	212	8	no	—
141	C₈	C₂×C₄⋊Dic₅₃	1	3	yes	—
142	C₈	C₂×C₄⋊Dic₅₃	1	3	yes	—
143	C₈	C₂³×Dic₅₃	1	1	yes	characteristic
144	C₈	C₂²×Dic₁₀₆	1	1	yes	characteristic
145	C₂²	—	2	6	no	—
146	C₂²	—	2	6	no	—
147	C₂²	#16810453	1	4	yes	direct product factor
148	C₂²	C₂×#13133151	1	3	yes	—
149	C₂²	C₂×#13133151	1	3	yes	—
150	C₂²	C₂²×Dic₂₁₂	1	1	yes	characteristic
151	C₄	C₂×#13133151	1	3	yes	—
152	C₄	C₂×#13133151	1	3	yes	—
153	C₄	C₂²×C₄⋊Dic₅₃	1	1	yes	characteristic
154	C₄	C₂²×D₄₂₄	1	1	yes	characteristic
155	C₂	—	2	1	no	—
156	C₂	—	2	3	no	—
157	C₂	C₂×#16810453	1	6	yes	direct product factor
158	C₂	C₂²×#13133151	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄₂₄
commutator	C₈×C₅₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂³×C₁₆×C₅₃	C₂
socle	C₂³×C₅₃	#107
prime radical	C₂⁴×C₅₃	Q₁₆
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₂³×C₁₆	D₅₃
hypocenter	C₅₃	C₂²×#368
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×C₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×#16810453	C₂	direct (N × H)
7. #16810453	C₂²	direct (N × H)
68. C₂²×C₅₃	#368	semidirect
81. C₂×C₅₃	C₂×#368	semidirect
91. C₅₃	C₂²×#368	semidirect
147. C₂²	#16810453	direct (N × H)
157. C₂	C₂×#16810453	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×#13133153	2
C₂×#16810453	2
C₂³×C₁₆×C₅₃	2
C₂²×#368	53

Subgroups — C22×#16810453 #27537144

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16810453 #27537144