Subgroups — C₂²×#16811023 #28253684

Order 13568.

1966 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13133208	C₂	1	2	yes	maximal
2	C₂×#16811023	C₂	1	24	yes	maximal, direct product factor
3	C₂²×#13133150	C₂	1	2	yes	maximal
4	C₂²×#13133149	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×#106×C₅₃	C₂	1	1	yes	maximal, characteristic
7	C₂³×#88322	C₂²	1	1	yes	characteristic
8	C₂×#13133208	—	2	12	no	—
9	C₂×#13133208	C₂²	1	24	yes	—
10	C₂²×C₄⋊Dic₅₃	—	2	2	no	—
11	C₂²×C₄⋊Dic₅₃	C₂²	1	2	yes	—
12	C₂²×#11719919	—	2	2	no	—
13	C₂²×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
14	#16811023	C₂²	1	64	yes	direct product factor
15	C₂×#13133150	C₂²	1	24	yes	—
16	C₂×#13133149	C₂²	1	12	yes	—
17	C₂×C₈×Dic₅₃	C₂²	1	12	yes	—
18	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₂²×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
20	C₂×#106×C₅₃	C₂²	1	12	yes	—
21	C₂³×Dic₅₃	—	2	2	no	—
22	C₂³×Dic₅₃	D₄	1	2	yes	—
23	C₂²×#88322	Q₈	1	8	yes	—
24	C₂²×#88322	C₂³	1	6	yes	—
25	#13133208	—	2	48	no	—
26	#13133208	C₂³	1	32	yes	—
27	C₂×C₄⋊Dic₅₃	—	2	6	no	—
28	C₂×C₄⋊Dic₅₃	—	2	24	no	—
29	C₂×C₄⋊Dic₅₃	C₂³	1	12	yes	—
30	C₂×#11719919	—	2	24	no	—
31	C₂×C₄×Dic₅₃	D₄	1	6	yes	—
32	C₂×C₄×Dic₅₃	C₂³	1	6	yes	—
33	#13133150	C₂³	1	32	yes	—
34	#13133149	C₂³	1	16	yes	—
35	C₈×Dic₅₃	C₂³	1	16	yes	—
36	C₂³×C₄×C₅₃	—	2	2	no	—
37	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
38	C₂²×C₈×C₅₃	Q₈	1	8	yes	—
39	C₂²×C₈×C₅₃	C₂³	1	6	yes	—
40	C₂×C₄⋊C₄×C₅₃	—	2	6	no	—
41	C₂×C₄⋊C₄×C₅₃	C₂³	1	12	yes	—
42	#106×C₅₃	C₂³	1	16	yes	—
43	C₂⁴×C₅₃	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₄×C₅₃	—	2	12	no	—
45	C₂²×C₄×C₅₃	—	4	8	no	—
46	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
47	C₂²×C₄×C₅₃	C₂×Q₈	1	4	yes	—
48	C₂²×C₄×C₅₃	C₂×Q₈	1	4	yes	—
49	C₂²×C₄×C₅₃	C₂⁴	1	3	yes	—
50	C₂²×Dic₅₃	—	2	8	no	—
51	C₂²×Dic₅₃	—	2	12	no	—
52	C₂²×Dic₅₃	—	4	8	no	—
53	C₂²×Dic₅₃	C₂×D₄	1	12	yes	—
54	C₂×C₈×C₅₃	C₂×Q₈	1	24	yes	—
55	C₂×C₈×C₅₃	C₂⁴	1	4	yes	—
56	C₄⋊C₄×C₅₃	—	2	12	no	—
57	C₄⋊C₄×C₅₃	C₂⁴	1	8	yes	—
58	C₄⋊Dic₅₃	—	2	12	no	—
59	C₄⋊Dic₅₃	—	2	32	no	—
60	C₄⋊Dic₅₃	C₂⁴	1	8	yes	—
61	#11719919	—	2	32	no	—
62	C₄×Dic₅₃	C₂×D₄	1	12	yes	—
63	C₄×Dic₅₃	C₂⁴	1	4	yes	—
64	C₂×#88322	C₂×Q₈	1	24	yes	—
65	C₂×#88322	C₂⁴	1	4	yes	—
66	C₂³×C₅₃	C₄⋊Q₈	1	4	yes	—
67	C₂³×C₅₃	#135	1	8	yes	—
68	C₂³×C₅₃	C₂²×D₄	1	3	yes	—
69	C₂×C₄×C₅₃	—	2	8	no	—
70	C₂×C₄×C₅₃	—	4	24	no	—
71	C₂×C₄×C₅₃	C₄⋊Q₈	1	12	yes	—
72	C₂×C₄×C₅₃	C₂²×D₄	1	3	yes	—
73	C₂×C₄×C₅₃	C₂²×Q₈	1	6	yes	—
74	C₂×C₄×C₅₃	C₂²×Q₈	1	6	yes	—
75	C₂×C₄×C₅₃	C₂⁵	1	1	yes	characteristic
76	C₂×Dic₅₃	—	2	24	no	—
77	C₂×Dic₅₃	—	2	8	no	—
78	C₂×Dic₅₃	—	4	24	no	—
79	C₂×Dic₅₃	C₂²×D₄	1	8	yes	—
80	C₈×C₅₃	C₂²×Q₈	1	8	yes	—
81	#88322	C₂²×Q₈	1	8	yes	—
82	C₂²×#499	—	53	1	no	maximal, Sylow
83	C₂²×C₅₃	#499	1	16	yes	complemented
84	C₂²×C₅₃	C₂×C₄⋊Q₈	1	6	yes	—
85	C₂²×C₅₃	C₂×#135	1	12	yes	—
86	C₂²×C₅₃	C₂³×D₄	1	1	yes	characteristic
87	C₄×C₅₃	—	4	8	no	—
88	C₄×C₅₃	C₂×C₄⋊Q₈	1	6	yes	—
89	C₄×C₅₃	C₂³×Q₈	1	1	yes	characteristic
90	C₄×C₅₃	C₂³×Q₈	1	1	yes	characteristic
91	Dic₅₃	—	2	8	no	—
92	Dic₅₃	—	4	8	no	—
93	C₂²×#125	—	53	2	no	—
94	C₂×#499	—	53	24	no	—
95	C₂²×#107	—	53	2	no	—
96	C₂²×#106	—	53	1	no	—
97	C₂²×#106	D₅₃	1	1	yes	characteristic
98	C₂²×C₄×C₈	—	53	1	no	—
99	C₂×C₅₃	C₂×#499	1	12	yes	complemented
100	C₂×C₅₃	C₂²×C₄⋊Q₈	1	1	yes	characteristic
101	C₂×C₅₃	C₂²×#135	1	2	yes	—
102	C₂³×C₈	—	53	1	no	—
103	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
104	C₂×#125	—	53	24	no	—
105	C₂×#125	—	106	12	no	—
106	C₂²×C₄⋊C₄	—	53	2	no	—
107	C₂²×C₄⋊C₄	—	106	2	no	—
108	C₂²×C₄⋊C₄	—	106	2	no	—
109	C₂²×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
110	C₂²×C₄²	—	53	1	no	—
111	#499	—	53	64	no	—
112	C₂×#107	—	53	24	no	—
113	C₂×#106	—	53	12	no	—
114	C₂×#106	C₂×D₅₃	1	12	yes	—
115	C₂×C₄×C₈	—	53	12	no	—
116	C₅₃	C₂²×#499	1	1	yes	characteristic, complemented, Sylow
117	C₂³×C₄	—	2	2	no	—
118	C₂³×C₄	—	53	2	no	—
119	C₂³×C₄	—	106	2	no	—
120	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
121	C₂²×C₈	—	53	8	no	—
122	C₂²×C₈	—	53	6	no	—
123	C₂²×C₈	Dic₁₀₆	1	8	yes	—
124	C₂²×C₈	C₂²×D₅₃	1	6	yes	—
125	#125	—	53	32	no	—
126	#125	—	106	48	no	—
127	C₂×C₄⋊C₄	—	2	6	no	—
128	C₂×C₄⋊C₄	—	53	12	no	—
129	C₂×C₄⋊C₄	—	106	24	no	—
130	C₂×C₄⋊C₄	—	106	24	no	—
131	C₂×C₄⋊C₄	—	106	6	no	—
132	C₂×C₄⋊C₄	C₂²×D₅₃	1	12	yes	—
133	C₂×C₄²	—	53	6	no	—
134	C₂×C₄²	—	53	6	no	—
135	#107	—	53	32	no	—
136	#106	—	53	16	no	—
137	#106	C₂²×D₅₃	1	16	yes	—
138	C₄×C₈	—	53	16	no	—
139	C₂⁴	D₄×D₅₃	1	1	yes	characteristic
140	C₂²×C₄	—	2	12	no	—
141	C₂²×C₄	—	4	8	no	—
142	C₂²×C₄	—	53	12	no	—
143	C₂²×C₄	—	106	8	no	—
144	C₂²×C₄	—	106	12	no	—
145	C₂²×C₄	—	212	8	no	—
146	C₂²×C₄	C₂×Dic₁₀₆	1	4	yes	—
147	C₂²×C₄	D₄×D₅₃	1	3	yes	—
148	C₂²×C₄	Q₈×D₅₃	1	4	yes	—
149	C₂²×C₄	C₂³×D₅₃	1	3	yes	—
150	C₂×C₈	—	53	24	no	—
151	C₂×C₈	—	53	4	no	—
152	C₂×C₈	C₂×Dic₁₀₆	1	24	yes	—
153	C₂×C₈	C₂³×D₅₃	1	4	yes	—
154	C₄⋊C₄	—	2	12	no	—
155	C₄⋊C₄	—	53	8	no	—
156	C₄⋊C₄	—	106	32	no	—
157	C₄⋊C₄	—	106	32	no	—
158	C₄⋊C₄	—	106	12	no	—
159	C₄⋊C₄	C₂³×D₅₃	1	8	yes	—
160	C₄²	—	53	12	no	—
161	C₄²	—	53	4	no	—
162	C₂³	#13133216	1	4	yes	—
163	C₂³	#13133238	1	8	yes	—
164	C₂³	C₂×D₄×D₅₃	1	3	yes	—
165	C₂×C₄	—	2	8	no	—
166	C₂×C₄	—	4	24	no	—
167	C₂×C₄	—	53	8	no	—
168	C₂×C₄	—	106	24	no	—
169	C₂×C₄	—	106	8	no	—
170	C₂×C₄	—	212	24	no	—
171	C₂×C₄	#13133216	1	12	yes	—
172	C₂×C₄	C₂×D₄×D₅₃	1	3	yes	—
173	C₂×C₄	C₂×Q₈×D₅₃	1	6	yes	—
174	C₂×C₄	C₂²×Dic₁₀₆	1	6	yes	—
175	C₂×C₄	C₂⁴×D₅₃	1	1	yes	characteristic
176	C₈	—	53	8	no	—
177	C₈	C₂²×Dic₁₀₆	1	8	yes	—
178	C₂²	#16811023	1	16	yes	direct product factor
179	C₂²	C₂×#13133216	1	6	yes	—
180	C₂²	C₂×#13133238	1	12	yes	—
181	C₂²	C₂²×D₄×D₅₃	1	1	yes	characteristic
182	C₄	—	4	8	no	—
183	C₄	—	106	8	no	—
184	C₄	—	212	8	no	—
185	C₄	C₂×#13133216	1	6	yes	—
186	C₄	C₂²×Q₈×D₅₃	1	1	yes	characteristic
187	C₄	C₂³×Dic₁₀₆	1	1	yes	characteristic
188	C₂	C₂×#16811023	1	12	yes	direct product factor
189	C₂	C₂²×#13133216	1	1	yes	characteristic
190	C₂	C₂²×#13133238	1	2	yes	—

	subgroup	G/H
center	C₂⁴	D₄×D₅₃
commutator, cosocle	C₂×C₄×C₅₃	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₅₃
Fitting	C₂²×#106×C₅₃	C₂
socle, prime radical	C₂⁴×C₅₃	C₂×D₄
hypercenter	C₂²×#106	D₅₃
hypocenter	C₅₃	C₂²×#499
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂²×#106

Complements

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

N	H ≅ G/N	type
2. C₂×#16811023	C₂	direct (N × H)
14. #16811023	C₂²	direct (N × H)
83. C₂²×C₅₃	#499	semidirect
99. C₂×C₅₃	C₂×#499	semidirect
116. C₅₃	C₂²×#499	semidirect
178. C₂²	#16811023	direct (N × H)
188. C₂	C₂×#16811023	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×#13133208	2
C₂×#16811023	2
C₂²×#13133150	2
C₂²×#13133149	2
C₂²×C₈×Dic₅₃	2
C₂²×#106×C₅₃	2
C₂²×#499	53

Subgroups — C22×#16811023 #28253684

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16811023 #28253684