Subgroups — #27304645

Order 13568.

204 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	C₂×#16810457	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#13133123	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄⋊#13133123	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²⋊#13133123	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²⋊#13133123	C₂	1	1	yes	maximal, characteristic, complemented
6	#20272849	C₂	1	1	yes	maximal, characteristic, complemented
7	D₄×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
8	#16810457	—	2	2	no	—
9	C₂×#13133123	—	2	2	no	—
10	C₂×#13133123	C₂²	1	1	yes	characteristic
11	C₂×#13133123	C₂²	1	1	yes	characteristic
12	C₂×#13133123	C₂²	1	1	yes	characteristic
13	C₂×#13133123	C₂²	1	1	yes	characteristic
14	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
15	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
16	C₈×D₄×C₅₃	C₄	1	1	yes	characteristic
17	C₅₃×C₄⋊C₁₆	C₄	1	1	yes	characteristic
18	C₅₃×C₂²⋊C₁₆	C₄	1	1	yes	characteristic
19	C₅₃×C₂²⋊C₁₆	C₄	1	1	yes	characteristic
20	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
21	#13133123	—	2	1	no	—
22	#13133123	—	2	1	no	—
23	#13133123	—	2	1	no	—
24	#13133123	D₄	1	2	yes	complemented
25	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
27	C₄×D₄×C₅₃	C₈	1	1	yes	characteristic
28	C₂×C₁₆×C₅₃	—	2	1	no	—
29	C₂×C₁₆×C₅₃	—	2	1	no	—
30	C₂×C₁₆×C₅₃	—	2	1	no	—
31	C₂×C₁₆×C₅₃	—	2	1	no	—
32	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
36	C₄⋊C₈×C₅₃	C₈	1	1	yes	characteristic
37	C₂²⋊C₈×C₅₃	C₈	1	1	yes	characteristic
38	C₂²⋊C₈×C₅₃	C₈	1	1	yes	characteristic
39	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
40	C₂×D₄×C₅₃	C₁₆	1	1	yes	characteristic
41	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
42	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
43	C₂×C₈×C₅₃	—	2	1	no	—
44	C₂×C₈×C₅₃	—	2	1	no	—
45	C₂×C₈×C₅₃	—	2	1	no	—
46	C₂×C₈×C₅₃	—	2	1	no	—
47	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
48	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
49	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
50	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
51	C₄⋊C₄×C₅₃	C₁₆	1	1	yes	characteristic
52	C₂²⋊C₄×C₅₃	C₁₆	1	1	yes	characteristic
53	C₂²⋊C₄×C₅₃	C₁₆	1	1	yes	characteristic
54	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
55	C₁₆×C₅₃	—	2	1	no	—
56	C₁₆×C₅₃	—	2	1	no	—
57	C₁₆×C₅₃	—	2	1	no	—
58	C₁₆×C₅₃	C₂×D₄	1	1	yes	characteristic
59	C₁₆×C₅₃	#41	1	1	yes	characteristic
60	C₂³×C₅₃	C₂×C₁₆	1	1	yes	characteristic
61	C₂³×C₅₃	C₂×C₁₆	1	1	yes	characteristic
62	D₄×C₅₃	—	2	2	no	—
63	C₂×C₄×C₅₃	—	2	1	no	—
64	C₂×C₄×C₅₃	—	2	1	no	—
65	C₂×C₄×C₅₃	—	2	1	no	—
66	C₂×C₄×C₅₃	—	2	1	no	—
67	C₂×C₄×C₅₃	C₂×C₁₆	1	1	yes	characteristic
68	C₂×C₄×C₅₃	C₂×C₁₆	1	1	yes	characteristic
69	C₂×C₄×C₅₃	C₂×C₁₆	1	1	yes	characteristic
70	C₂×C₄×C₅₃	C₂×C₁₆	1	1	yes	characteristic
71	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
72	C₈×C₅₃	—	2	1	no	—
73	C₈×C₅₃	—	2	1	no	—
74	C₈×C₅₃	—	2	1	no	—
75	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
76	C₈×C₅₃	#131	1	1	yes	characteristic
77	#13645	—	53	1	no	maximal, Sylow
78	C₂²×C₅₃	—	2	1	no	—
79	C₂²×C₅₃	—	2	1	no	—
80	C₂²×C₅₃	—	2	1	no	—
81	C₂²×C₅₃	—	2	1	no	—
82	C₂²×C₅₃	—	2	1	no	—
83	C₂²×C₅₃	M₆₄	1	2	yes	complemented
84	C₂²×C₅₃	C₂²×C₁₆	1	1	yes	characteristic
85	C₄×C₅₃	—	2	1	no	—
86	C₄×C₅₃	—	2	1	no	—
87	C₄×C₅₃	—	2	1	no	—
88	C₄×C₅₃	—	2	1	no	—
89	C₄×C₅₃	C₈×D₄	1	1	yes	characteristic
90	C₄×C₅₃	#504	1	1	yes	characteristic
91	C₂×M₆₄	—	53	1	no	—
92	C₂²×C₃₂	—	53	1	no	—
93	D₄×C₁₆	D₅₃	1	1	yes	characteristic
94	C₄⋊C₃₂	—	53	1	no	—
95	C₂²⋊C₃₂	—	53	1	no	—
96	C₂²⋊C₃₂	—	53	1	no	—
97	#1397	—	53	1	no	—
98	C₂×C₅₃	—	2	2	no	—
99	C₂×C₅₃	—	4	1	no	—
100	C₂×C₅₃	D₄×C₁₆	1	1	yes	characteristic
101	C₂×C₅₃	C₂×M₆₄	1	1	yes	characteristic
102	C₂×C₅₃	#2258	1	1	yes	characteristic
103	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
104	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
105	C₈×D₄	Dic₅₃	1	1	yes	characteristic
106	M₆₄	—	106	2	no	—
107	C₂×C₃₂	—	53	1	no	—
108	C₂×C₃₂	—	53	1	no	—
109	C₂×C₃₂	—	53	1	no	—
110	C₂×C₃₂	—	53	1	no	—
111	C₂×C₃₂	—	106	2	no	—
112	C₄⋊C₁₆	Dic₅₃	1	1	yes	characteristic
113	C₂²⋊C₁₆	Dic₅₃	1	1	yes	characteristic
114	C₂²⋊C₁₆	Dic₅₃	1	1	yes	characteristic
115	C₄×C₁₆	C₂×D₅₃	1	1	yes	characteristic
116	C₅₃	#13645	1	1	yes	characteristic, complemented, Sylow
117	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
118	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
119	C₄×D₄	#88322	1	1	yes	characteristic
120	C₂×C₁₆	—	2	1	no	—
121	C₂×C₁₆	—	2	1	no	—
122	C₂×C₁₆	—	2	1	no	—
123	C₂×C₁₆	—	2	1	no	—
124	C₂×C₁₆	C₂×Dic₅₃	1	1	yes	characteristic
125	C₂×C₁₆	C₂×Dic₅₃	1	1	yes	characteristic
126	C₂×C₁₆	C₂×Dic₅₃	1	1	yes	characteristic
127	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
128	C₄⋊C₈	#88322	1	1	yes	characteristic
129	C₂²⋊C₈	#88322	1	1	yes	characteristic
130	C₂²⋊C₈	#88322	1	1	yes	characteristic
131	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
132	C₃₂	—	53	2	no	—
133	C₃₂	—	106	1	no	—
134	C₃₂	—	106	1	no	—
135	C₃₂	—	106	1	no	—
136	C₂×D₄	#11719908	1	1	yes	characteristic
137	C₂²×C₄	C₂×#88322	1	1	yes	characteristic
138	C₂²×C₄	C₂×#88322	1	1	yes	characteristic
139	C₂×C₈	—	2	1	no	—
140	C₂×C₈	—	2	1	no	—
141	C₂×C₈	—	2	1	no	—
142	C₂×C₈	—	2	1	no	—
143	C₂×C₈	C₂×#88322	1	1	yes	characteristic
144	C₂×C₈	C₂×#88322	1	1	yes	characteristic
145	C₂×C₈	C₂×#88322	1	1	yes	characteristic
146	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
147	C₄⋊C₄	#11719908	1	1	yes	characteristic
148	C₂²⋊C₄	#11719908	1	1	yes	characteristic
149	C₂²⋊C₄	#11719908	1	1	yes	characteristic
150	C₄²	C₂×#88322	1	1	yes	characteristic
151	C₁₆	—	2	1	no	—
152	C₁₆	—	2	1	no	—
153	C₁₆	—	2	1	no	—
154	C₁₆	D₄×D₅₃	1	1	yes	characteristic
155	C₁₆	#11719947	1	1	yes	characteristic
156	C₂³	C₂×#11719908	1	1	yes	characteristic
157	C₂³	C₂×#11719908	1	1	yes	characteristic
158	D₄	—	2	2	no	—
159	C₂×C₄	—	2	1	no	—
160	C₂×C₄	—	2	1	no	—
161	C₂×C₄	—	2	1	no	—
162	C₂×C₄	—	2	1	no	—
163	C₂×C₄	C₂×#11719908	1	1	yes	characteristic
164	C₂×C₄	C₂×#11719908	1	1	yes	characteristic
165	C₂×C₄	C₂×#11719908	1	1	yes	characteristic
166	C₂×C₄	C₂×#11719908	1	1	yes	characteristic
167	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
168	C₈	—	2	1	no	—
169	C₈	—	2	1	no	—
170	C₈	—	2	1	no	—
171	C₈	D₄×Dic₅₃	1	1	yes	characteristic
172	C₈	#13133224	1	1	yes	characteristic
173	C₂²	—	2	1	no	—
174	C₂²	—	2	1	no	—
175	C₂²	—	2	1	no	—
176	C₂²	—	2	1	no	—
177	C₂²	—	2	1	no	—
178	C₂²	#16810457	1	2	yes	complemented
179	C₂²	C₂²×#11719908	1	1	yes	characteristic
180	C₄	—	2	1	no	—
181	C₄	—	2	1	no	—
182	C₄	—	2	1	no	—
183	C₄	—	2	1	no	—
184	C₄	D₄×#88322	1	1	yes	characteristic
185	C₄	#16811040	1	1	yes	characteristic
186	C₂	—	2	2	no	—
187	C₂	—	4	1	no	—
188	C₂	D₄×#11719908	1	1	yes	characteristic
189	C₂	C₂×#16810457	1	1	yes	characteristic
190	C₂	#20277009	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₁₆	C₂²×D₅₃
commutator, socle	C₂²×C₅₃	C₂²×C₁₆
Fitting	D₄×C₁₆×C₅₃	C₂
prime radical	C₂×D₄×C₅₃	C₁₆
cosocle	C₂×C₁₆×C₅₃	C₂³
hypercenter	D₄×C₁₆	D₅₃
hypocenter	C₅₃	#13645
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₁₆ ◁ D₄×C₁₆

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. C₄⋊#13133123	C₂	semidirect
4. C₂²⋊#13133123	C₂	semidirect
5. C₂²⋊#13133123	C₂	semidirect
6. #20272849	C₂	semidirect
24. #13133123	D₄	semidirect
83. C₂²×C₅₃	M₆₄	semidirect
116. C₅₃	#13645	semidirect
178. C₂²	#16810457	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
C₄⋊#13133123	2
C₂²⋊#13133123	2
#20272849	2
D₄×C₁₆×C₅₃	2
#13645	53