Subgroups — C₂²×#16810375 #27353060

Order 13568.

942 conjugacy classes of subgroups.

The subgroup lattice has 206 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊#88322	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×#88322	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810375	C₂	1	12	yes	maximal, direct product factor
4	C₂²×C₄⋊C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂³×#88322	—	2	1	no	—
6	C₂³×#88322	C₄	1	1	yes	characteristic
7	C₂³×#88322	C₄	1	1	yes	characteristic
8	C₂×C₄⋊#88322	—	2	3	no	—
9	C₂×C₄⋊#88322	C₂²	1	6	yes	—
10	C₂×C₄×#88322	C₄	1	6	yes	—
11	C₂×C₄×#88322	C₂²	1	6	yes	—
12	#16810375	C₂²	1	16	yes	direct product factor
13	C₂³×C₈×C₅₃	—	2	1	no	—
14	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₈×C₅₃	—	2	3	no	—
16	C₂×C₄⋊C₈×C₅₃	C₂²	1	6	yes	—
17	C₂²×#88322	—	2	4	no	—
18	C₂²×#88322	—	2	6	no	—
19	C₂²×#88322	—	4	4	no	—
20	C₂²×#88322	C₈	1	8	yes	complemented
21	C₂²×#88322	C₂×C₄	1	6	yes	—
22	C₂²×#88322	C₂×C₄	1	6	yes	—
23	C₄⋊#88322	—	2	6	no	—
24	C₄⋊#88322	C₂³	1	4	yes	—
25	C₄×#88322	C₂×C₄	1	12	yes	—
26	C₄×#88322	C₂³	1	4	yes	—
27	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
29	C₂³×C₄×C₅₃	Q₈	1	1	yes	characteristic
30	C₂²×C₈×C₅₃	—	2	6	no	—
31	C₂²×C₈×C₅₃	—	4	4	no	—
32	C₂×C₄²×C₅₃	C₂×C₄	1	3	yes	—
33	C₂×C₄²×C₅₃	D₄	1	3	yes	—
34	C₂×C₄²×C₅₃	Q₈	1	3	yes	—
35	C₂×C₄²×C₅₃	C₂³	1	3	yes	—
36	C₄⋊C₈×C₅₃	—	2	6	no	—
37	C₄⋊C₈×C₅₃	C₂³	1	4	yes	—
38	C₂⁴×C₅₃	C₄⋊C₄	1	1	yes	characteristic
39	C₂²×C₄×C₅₃	—	2	4	no	—
40	C₂²×C₄×C₅₃	C₄⋊C₄	1	3	yes	—
41	C₂²×C₄×C₅₃	C₄⋊C₄	1	3	yes	—
42	C₂²×C₄×C₅₃	C₄⋊C₄	1	3	yes	—
43	C₂²×C₄×C₅₃	C₂×C₈	1	4	yes	—
44	C₂²×C₄×C₅₃	M₁₆	1	4	yes	—
45	C₂²×C₄×C₅₃	D₈	1	4	yes	—
46	C₂²×C₄×C₅₃	Q₁₆	1	4	yes	—
47	C₂²×C₄×C₅₃	C₂²×C₄	1	3	yes	—
48	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
49	C₂²×C₄×C₅₃	C₂×Q₈	1	3	yes	—
50	C₂×C₈×C₅₃	—	2	4	no	—
51	C₂×C₈×C₅₃	—	4	12	no	—
52	C₄²×C₅₃	C₄⋊C₄	1	6	yes	—
53	C₄²×C₅₃	C₂²×C₄	1	3	yes	—
54	C₄²×C₅₃	C₂×D₄	1	3	yes	—
55	C₄²×C₅₃	C₂×Q₈	1	3	yes	—
56	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
57	C₂×#88322	—	2	12	no	—
58	C₂×#88322	—	2	4	no	—
59	C₂×#88322	—	4	12	no	—
60	C₂×#88322	C₂×C₈	1	24	yes	complemented
61	C₂×#88322	C₂²×C₄	1	4	yes	—
62	C₂×#88322	C₂²×C₄	1	4	yes	—
63	C₂³×C₅₃	C₄⋊C₈	1	4	yes	—
64	C₂³×C₅₃	#107	1	4	yes	—
65	C₂³×C₅₃	#108	1	4	yes	—
66	C₂³×C₅₃	C₂×C₄⋊C₄	1	3	yes	—
67	C₂×C₄×C₅₃	—	2	6	no	—
68	C₂×C₄×C₅₃	—	2	6	no	—
69	C₂×C₄×C₅₃	C₄⋊C₈	1	12	yes	—
70	C₂×C₄×C₅₃	#107	1	12	yes	—
71	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	3	yes	—
72	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	3	yes	—
73	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	3	yes	—
74	C₂×C₄×C₅₃	C₂²×C₈	1	6	yes	—
75	C₂×C₄×C₅₃	C₂×M₁₆	1	6	yes	—
76	C₂×C₄×C₅₃	C₂×D₈	1	6	yes	—
77	C₂×C₄×C₅₃	C₂×Q₁₆	1	6	yes	—
78	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
79	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
80	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
81	C₈×C₅₃	—	4	4	no	—
82	#88322	—	2	4	no	—
83	#88322	—	4	4	no	—
84	#88322	C₂²×C₈	1	8	yes	complemented
85	C₂²×#335	—	53	1	no	maximal, Sylow
86	C₂²×C₅₃	#335	1	16	yes	complemented
87	C₂²×C₅₃	C₂×C₄⋊C₈	1	6	yes	—
88	C₂²×C₅₃	C₂×#107	1	6	yes	—
89	C₂²×C₅₃	C₂×#108	1	6	yes	—
90	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
91	C₄×C₅₃	—	2	3	no	—
92	C₄×C₅₃	—	2	1	no	—
93	C₄×C₅₃	C₂×C₄⋊C₈	1	6	yes	—
94	C₄×C₅₃	C₂×#107	1	6	yes	—
95	C₄×C₅₃	C₂³×C₈	1	1	yes	characteristic
96	C₄×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
97	C₄×C₅₃	C₂²×D₈	1	1	yes	characteristic
98	C₄×C₅₃	C₂²×Q₁₆	1	1	yes	characteristic
99	C₂²×C₄⋊C₈	—	53	1	no	—
100	C₂²×C₄⋊C₈	D₅₃	1	1	yes	characteristic
101	C₂²×C₄×C₈	—	53	1	no	—
102	C₂×#335	—	53	12	no	—
103	C₂×C₅₃	C₂×#335	1	12	yes	complemented
104	C₂×C₅₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
105	C₂×C₅₃	C₂²×#107	1	1	yes	characteristic
106	C₂×C₅₃	C₂²×#108	1	1	yes	characteristic
107	C₂³×C₈	—	2	1	no	—
108	C₂³×C₈	—	53	1	no	—
109	C₂³×C₈	—	53	1	no	—
110	C₂³×C₈	—	106	1	no	—
111	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
112	C₂×C₄⋊C₈	—	2	3	no	—
113	C₂×C₄⋊C₈	—	53	6	no	—
114	C₂×C₄⋊C₈	—	106	3	no	—
115	C₂×C₄⋊C₈	C₂×D₅₃	1	6	yes	—
116	C₂×C₄×C₈	—	53	6	no	—
117	C₂×C₄×C₈	—	53	6	no	—
118	#335	—	53	16	no	—
119	C₅₃	C₂²×#335	1	1	yes	characteristic, complemented, Sylow
120	C₂³×C₄	Dic₁₀₆	1	1	yes	characteristic
121	C₂³×C₄	C₄×D₅₃	1	1	yes	characteristic
122	C₂³×C₄	C₂²⋊D₅₃	1	1	yes	characteristic
123	C₂²×C₈	—	2	6	no	—
124	C₂²×C₈	—	4	4	no	—
125	C₂²×C₈	—	53	6	no	—
126	C₂²×C₈	—	53	8	no	—
127	C₂²×C₈	—	53	6	no	—
128	C₂²×C₈	—	106	4	no	—
129	C₂²×C₈	—	106	6	no	—
130	C₂²×C₈	—	212	4	no	—
131	C₂×C₄²	Dic₁₀₆	1	3	yes	—
132	C₂×C₄²	C₄×D₅₃	1	3	yes	—
133	C₂×C₄²	C₂²⋊D₅₃	1	3	yes	—
134	C₂×C₄²	C₂²×D₅₃	1	3	yes	—
135	C₄⋊C₈	—	2	6	no	—
136	C₄⋊C₈	—	53	4	no	—
137	C₄⋊C₈	—	106	6	no	—
138	C₄⋊C₈	C₂²×D₅₃	1	4	yes	—
139	C₄×C₈	—	53	12	no	—
140	C₄×C₈	—	53	4	no	—
141	C₂⁴	#11719919	1	1	yes	characteristic
142	C₂²×C₄	—	2	4	no	—
143	C₂²×C₄	C₈×D₅₃	1	4	yes	—
144	C₂²×C₄	#11719912	1	4	yes	—
145	C₂²×C₄	#11719919	1	3	yes	—
146	C₂²×C₄	#11719919	1	3	yes	—
147	C₂²×C₄	#11719919	1	3	yes	—
148	C₂²×C₄	#11719922	1	4	yes	—
149	C₂²×C₄	#11719925	1	4	yes	—
150	C₂²×C₄	C₂×Dic₁₀₆	1	3	yes	—
151	C₂²×C₄	C₂×C₄×D₅₃	1	3	yes	—
152	C₂²×C₄	C₂×C₂²⋊D₅₃	1	3	yes	—
153	C₂×C₈	—	2	4	no	—
154	C₂×C₈	—	4	12	no	—
155	C₂×C₈	—	53	24	no	—
156	C₂×C₈	—	53	4	no	—
157	C₂×C₈	—	53	4	no	—
158	C₂×C₈	—	106	12	no	—
159	C₂×C₈	—	106	4	no	—
160	C₂×C₈	—	212	12	no	—
161	C₄²	#11719919	1	6	yes	—
162	C₄²	C₂×Dic₁₀₆	1	3	yes	—
163	C₄²	C₂×C₄×D₅₃	1	3	yes	—
164	C₄²	C₂×C₂²⋊D₅₃	1	3	yes	—
165	C₄²	C₂³×D₅₃	1	1	yes	characteristic
166	C₂³	#13133146	1	4	yes	—
167	C₂³	#13133150	1	4	yes	—
168	C₂³	#13133152	1	4	yes	—
169	C₂³	C₂×#11719919	1	3	yes	—
170	C₂×C₄	—	2	6	no	—
171	C₂×C₄	—	2	6	no	—
172	C₂×C₄	#13133146	1	12	yes	—
173	C₂×C₄	#13133150	1	12	yes	—
174	C₂×C₄	C₂×#11719919	1	3	yes	—
175	C₂×C₄	C₂×#11719919	1	3	yes	—
176	C₂×C₄	C₂×#11719919	1	3	yes	—
177	C₂×C₄	C₂×C₈×D₅₃	1	6	yes	—
178	C₂×C₄	C₂×#11719912	1	6	yes	—
179	C₂×C₄	C₂×#11719922	1	6	yes	—
180	C₂×C₄	C₂×#11719925	1	6	yes	—
181	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
182	C₂×C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
183	C₂×C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
184	C₈	—	4	4	no	—
185	C₈	—	53	8	no	—
186	C₈	—	106	4	no	—
187	C₈	—	212	4	no	—
188	C₂²	#16810375	1	16	yes	direct product factor
189	C₂²	C₂×#13133146	1	6	yes	—
190	C₂²	C₂×#13133150	1	6	yes	—
191	C₂²	C₂×#13133152	1	6	yes	—
192	C₂²	C₂²×#11719919	1	1	yes	characteristic
193	C₄	—	2	3	no	—
194	C₄	—	2	1	no	—
195	C₄	C₂×#13133146	1	6	yes	—
196	C₄	C₂×#13133150	1	6	yes	—
197	C₄	C₂²×C₈×D₅₃	1	1	yes	characteristic
198	C₄	C₂²×#11719912	1	1	yes	characteristic
199	C₄	C₂²×#11719922	1	1	yes	characteristic
200	C₄	C₂²×#11719925	1	1	yes	characteristic
201	C₂	C₂×#16810375	1	12	yes	direct product factor
202	C₂	C₂²×#13133146	1	1	yes	characteristic
203	C₂	C₂²×#13133150	1	1	yes	characteristic
204	C₂	C₂²×#13133152	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄	C₂²⋊D₅₃
commutator	C₄×C₅₃	C₂³×C₈
Frattini	C₄²	C₂³×D₅₃
Fitting	C₂²×C₄⋊C₈×C₅₃	C₂
socle, prime radical	C₂⁴×C₅₃	C₄⋊C₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	C₂²×C₄⋊C₈	D₅₃
hypocenter	C₅₃	C₂²×#335
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
3. C₂×#16810375	C₂	direct (N × H)
12. #16810375	C₂²	direct (N × H)
20. C₂²×#88322	C₈	semidirect
60. C₂×#88322	C₂×C₈	semidirect
84. #88322	C₂²×C₈	semidirect
86. C₂²×C₅₃	#335	semidirect
103. C₂×C₅₃	C₂×#335	semidirect
119. C₅₃	C₂²×#335	semidirect
188. C₂²	#16810375	direct (N × H)
201. C₂	C₂×#16810375	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×C₄⋊#88322	2
C₂²×C₄×#88322	2
C₂×#16810375	2
C₂²×C₄⋊C₈×C₅₃	2
C₂²×#335	53

Subgroups — C22×#16810375 #27353060

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16810375 #27353060