Subgroups — C₂²×#16810452 #27537143

Order 13568.

662 conjugacy classes of subgroups.

The subgroup lattice has 186 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#11719908	C₂	1	1	yes	maximal, characteristic
2	C₂²×#13133153	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810452	C₂	1	12	yes	maximal, direct product factor
4	C₂²×#108×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11719917	—	2	1	no	—
6	C₂²×#11719908	C₄	1	6	yes	—
7	C₂²×#11719908	C₄	1	2	yes	—
8	C₂²×#11719908	C₂²	1	6	yes	—
9	C₂×#13133153	—	2	3	no	—
10	C₂×#13133153	C₂²	1	6	yes	—
11	#16810452	C₂²	1	16	yes	direct product factor
12	C₂²×M₁₆×C₅₃	—	2	1	no	—
13	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×#108×C₅₃	—	2	3	no	—
15	C₂×#108×C₅₃	C₂²	1	6	yes	—
16	C₂×#11719917	—	2	6	no	—
17	C₂×#11719917	—	4	3	no	—
18	C₂²×#88322	—	4	1	no	—
19	C₂×#11719908	C₂×C₄	1	12	yes	—
20	C₂×#11719908	C₂×C₄	1	12	yes	—
21	C₂×#11719908	C₂³	1	4	yes	—
22	#13133153	—	2	6	no	—
23	#13133153	C₂³	1	4	yes	—
24	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
25	C₂×M₁₆×C₅₃	—	2	6	no	—
26	C₂×M₁₆×C₅₃	—	4	3	no	—
27	C₂²×C₈×C₅₃	—	4	1	no	—
28	C₂²×C₈×C₅₃	C₂×C₄	1	3	yes	—
29	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₅₃	D₄	1	3	yes	—
31	C₂²×C₈×C₅₃	Q₈	1	3	yes	—
32	C₂²×C₈×C₅₃	Q₈	1	1	yes	characteristic
33	C₂²×C₈×C₅₃	C₂³	1	3	yes	—
34	#108×C₅₃	—	2	6	no	—
35	#108×C₅₃	C₂³	1	4	yes	—
36	C₂⁴×C₅₃	Q₁₆	1	1	yes	characteristic
37	C₂²×C₄×C₅₃	C₄⋊C₄	1	3	yes	—
38	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
39	C₂²×C₄×C₅₃	D₈	1	3	yes	—
40	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
41	C₂²×C₄×C₅₃	Q₁₆	1	3	yes	—
42	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
43	M₁₆×C₅₃	—	2	4	no	—
44	M₁₆×C₅₃	—	4	6	no	—
45	C₂×C₈×C₅₃	—	4	6	no	—
46	C₂×C₈×C₅₃	C₄⋊C₄	1	6	yes	—
47	C₂×C₈×C₅₃	C₄⋊C₄	1	3	yes	—
48	C₂×C₈×C₅₃	C₄⋊C₄	1	3	yes	—
49	C₂×C₈×C₅₃	C₂²×C₄	1	3	yes	—
50	C₂×C₈×C₅₃	C₂²×C₄	1	3	yes	—
51	C₂×C₈×C₅₃	C₂×D₄	1	3	yes	—
52	C₂×C₈×C₅₃	C₂×Q₈	1	3	yes	—
53	C₂×C₈×C₅₃	C₂×Q₈	1	3	yes	—
54	C₂×C₈×C₅₃	C₂⁴	1	1	yes	characteristic
55	#11719917	—	2	4	no	—
56	#11719917	—	4	6	no	—
57	C₂×#88322	—	4	6	no	—
58	#11719908	C₂²×C₄	1	8	yes	—
59	C₂³×C₅₃	—	2	4	no	—
60	C₂³×C₅₃	#107	1	3	yes	—
61	C₂³×C₅₃	#107	1	1	yes	characteristic
62	C₂³×C₅₃	C₂×Q₁₆	1	3	yes	—
63	C₂×C₄×C₅₃	#107	1	6	yes	—
64	C₂×C₄×C₅₃	#107	1	3	yes	—
65	C₂×C₄×C₅₃	#107	1	3	yes	—
66	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	3	yes	—
67	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	3	yes	—
68	C₂×C₄×C₅₃	C₂×D₈	1	3	yes	—
69	C₂×C₄×C₅₃	C₂×D₈	1	3	yes	—
70	C₂×C₄×C₅₃	C₂×Q₁₆	1	3	yes	—
71	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
72	C₈×C₅₃	—	4	4	no	—
73	C₈×C₅₃	C₂×C₄⋊C₄	1	3	yes	—
74	C₈×C₅₃	C₂×C₄⋊C₄	1	3	yes	—
75	C₈×C₅₃	C₂³×C₄	1	1	yes	characteristic
76	C₈×C₅₃	C₂²×Q₈	1	1	yes	characteristic
77	#88322	—	4	4	no	—
78	C₂²×#368	—	53	1	no	maximal, Sylow
79	C₂²×C₅₃	—	2	6	no	—
80	C₂²×C₅₃	—	2	6	no	—
81	C₂²×C₅₃	#368	1	4	yes	complemented
82	C₂²×C₅₃	C₂×#107	1	3	yes	—
83	C₂²×C₅₃	C₂×#107	1	3	yes	—
84	C₂²×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
85	C₄×C₅₃	C₂×#107	1	3	yes	—
86	C₄×C₅₃	C₂×#107	1	3	yes	—
87	C₄×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
88	C₄×C₅₃	C₂²×D₈	1	1	yes	characteristic
89	C₂³×C₁₆	—	53	1	no	—
90	C₂²×#108	—	53	1	no	—
91	C₂²×#108	D₅₃	1	1	yes	characteristic
92	C₂×#368	—	53	12	no	—
93	C₂×C₅₃	—	2	1	no	—
94	C₂×C₅₃	—	2	3	no	—
95	C₂×C₅₃	C₂×#368	1	6	yes	complemented
96	C₂×C₅₃	C₂²×#107	1	1	yes	characteristic
97	C₂²×M₁₆	—	2	1	no	—
98	C₂²×M₁₆	—	106	1	no	—
99	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
100	C₂²×C₁₆	—	53	2	no	—
101	C₂²×C₁₆	—	53	6	no	—
102	C₂²×C₁₆	—	53	6	no	—
103	C₂×#108	—	2	3	no	—
104	C₂×#108	—	53	6	no	—
105	C₂×#108	—	106	3	no	—
106	C₂×#108	C₂×D₅₃	1	6	yes	—
107	#368	—	53	16	no	—
108	C₅₃	C₂²×#368	1	1	yes	characteristic, complemented, Sylow
109	C₂³×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
110	C₂×M₁₆	—	2	6	no	—
111	C₂×M₁₆	—	4	3	no	—
112	C₂×M₁₆	—	106	6	no	—
113	C₂×M₁₆	—	212	3	no	—
114	C₂²×C₈	—	4	1	no	—
115	C₂²×C₈	—	212	1	no	—
116	C₂²×C₈	Dic₁₀₆	1	1	yes	characteristic
117	C₂²×C₈	Dic₁₀₆	1	3	yes	—
118	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
119	C₂²×C₈	C₄×D₅₃	1	3	yes	—
120	C₂²×C₈	C₂²⋊D₅₃	1	3	yes	—
121	C₂²×C₈	C₂²×D₅₃	1	3	yes	—
122	C₂×C₁₆	—	53	12	no	—
123	C₂×C₁₆	—	53	12	no	—
124	C₂×C₁₆	—	53	4	no	—
125	#108	—	2	6	no	—
126	#108	—	53	4	no	—
127	#108	—	106	6	no	—
128	#108	C₂²×D₅₃	1	4	yes	—
129	C₂⁴	#11719925	1	1	yes	characteristic
130	C₂²×C₄	#11719919	1	1	yes	characteristic
131	C₂²×C₄	#11719919	1	3	yes	—
132	C₂²×C₄	#11719922	1	3	yes	—
133	C₂²×C₄	#11719922	1	1	yes	characteristic
134	C₂²×C₄	#11719925	1	3	yes	—
135	C₂²×C₄	C₂×C₂²⋊D₅₃	1	3	yes	—
136	M₁₆	—	2	4	no	—
137	M₁₆	—	4	6	no	—
138	M₁₆	—	106	4	no	—
139	M₁₆	—	212	6	no	—
140	C₂×C₈	—	4	6	no	—
141	C₂×C₈	—	212	6	no	—
142	C₂×C₈	#11719919	1	6	yes	—
143	C₂×C₈	#11719919	1	3	yes	—
144	C₂×C₈	#11719919	1	3	yes	—
145	C₂×C₈	C₂×Dic₁₀₆	1	3	yes	—
146	C₂×C₈	C₂×Dic₁₀₆	1	3	yes	—
147	C₂×C₈	C₂×C₄×D₅₃	1	3	yes	—
148	C₂×C₈	C₂×C₄×D₅₃	1	3	yes	—
149	C₂×C₈	C₂×C₂²⋊D₅₃	1	3	yes	—
150	C₂×C₈	C₂³×D₅₃	1	1	yes	characteristic
151	C₁₆	—	53	8	no	—
152	C₂³	—	2	4	no	—
153	C₂³	#13133150	1	3	yes	—
154	C₂³	#13133150	1	1	yes	characteristic
155	C₂³	C₂×#11719925	1	3	yes	—
156	C₂×C₄	#13133150	1	6	yes	—
157	C₂×C₄	#13133150	1	3	yes	—
158	C₂×C₄	#13133150	1	3	yes	—
159	C₂×C₄	C₂×#11719919	1	3	yes	—
160	C₂×C₄	C₂×#11719919	1	3	yes	—
161	C₂×C₄	C₂×#11719922	1	3	yes	—
162	C₂×C₄	C₂×#11719922	1	3	yes	—
163	C₂×C₄	C₂×#11719925	1	3	yes	—
164	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
165	C₈	—	4	4	no	—
166	C₈	—	212	4	no	—
167	C₈	C₂×#11719919	1	3	yes	—
168	C₈	C₂×#11719919	1	3	yes	—
169	C₈	C₂²×C₄×D₅₃	1	1	yes	characteristic
170	C₈	C₂²×Dic₁₀₆	1	1	yes	characteristic
171	C₂²	—	2	6	no	—
172	C₂²	—	2	6	no	—
173	C₂²	#16810452	1	4	yes	direct product factor
174	C₂²	C₂×#13133150	1	3	yes	—
175	C₂²	C₂×#13133150	1	3	yes	—
176	C₂²	C₂²×#11719925	1	1	yes	characteristic
177	C₄	C₂×#13133150	1	3	yes	—
178	C₄	C₂×#13133150	1	3	yes	—
179	C₄	C₂²×#11719919	1	1	yes	characteristic
180	C₄	C₂²×#11719922	1	1	yes	characteristic
181	C₂	—	2	1	no	—
182	C₂	—	2	3	no	—
183	C₂	C₂×#16810452	1	6	yes	direct product factor
184	C₂	C₂²×#13133150	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	#11719922
commutator	C₈×C₅₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₅₃
Fitting	C₂²×#108×C₅₃	C₂
socle	C₂³×C₅₃	#107
prime radical	C₂⁴×C₅₃	Q₁₆
cosocle	C₂×C₈×C₅₃	C₂⁴
hypercenter	C₂²×#108	D₅₃
hypocenter	C₅₃	C₂²×#368
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂²×#108

Complements

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

N	H ≅ G/N	type
3. C₂×#16810452	C₂	direct (N × H)
11. #16810452	C₂²	direct (N × H)
81. C₂²×C₅₃	#368	semidirect
95. C₂×C₅₃	C₂×#368	semidirect
108. C₅₃	C₂²×#368	semidirect
173. C₂²	#16810452	direct (N × H)
183. C₂	C₂×#16810452	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×#11719908	2
C₂²×#13133153	2
C₂×#16810452	2
C₂²×#108×C₅₃	2
C₂²×#368	53

Subgroups — C22×#16810452 #27537143

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16810452 #27537143