Subgroups — #28298310

Order 13568.

242 conjugacy classes of subgroups.

The subgroup lattice has 170 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811486	C₂	1	2	yes	maximal
2	C₂×C₁₆×Dic₅₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊#88324	—	2	2	no	—
4	#16811486	—	2	2	no	—
5	#16811486	C₄	1	4	yes	—
6	C₂²×#11719908	C₄	1	1	yes	characteristic
7	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
8	C₁₆×Dic₅₃	—	2	2	no	—
9	C₂²×C₁₆×C₅₃	C₄	1	1	yes	characteristic
10	C₂²×#88324	—	4	2	no	—
11	C₄⋊#88324	—	2	4	no	—
12	C₄⋊#88324	—	4	2	no	—
13	C₂²×#88322	Q₈	1	1	yes	characteristic
14	C₂×C₄×Dic₅₃	D₄	1	1	yes	characteristic
15	C₂×#11719908	—	2	1	no	—
16	C₂×#11719908	—	2	1	no	—
17	C₂×#11719908	C₈	1	2	yes	complemented
18	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
19	C₈×Dic₅₃	D₄	1	2	yes	—
20	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₁₆×C₅₃	—	2	1	no	—
22	C₂×C₁₆×C₅₃	—	2	1	no	—
23	C₂×C₁₆×C₅₃	C₈	1	2	yes	complemented
24	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
25	C₂²×Dic₅₃	D₈	1	1	yes	characteristic
26	C₂²×Dic₅₃	Q₁₆	1	1	yes	characteristic
27	C₂×#88324	—	4	4	no	—
28	C₂×#88324	—	8	2	no	—
29	C₂×#88324	—	8	2	no	—
30	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
31	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
32	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
33	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
34	C₂×C₈×C₅₃	M₁₆	1	1	yes	characteristic
35	C₂×C₈×C₅₃	M₁₆	1	1	yes	characteristic
36	C₄×Dic₅₃	C₂²⋊C₄	1	2	yes	—
37	C₄×Dic₅₃	SD₁₆	1	2	yes	—
38	C₂×#88322	—	2	1	no	—
39	C₂×#88322	—	2	1	no	—
40	C₂×#88322	C₄⋊C₄	1	2	yes	—
41	C₁₆×C₅₃	—	2	2	no	—
42	#11719908	—	2	2	no	—
43	C₂³×C₅₃	#107	1	1	yes	characteristic
44	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
45	C₂×C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
46	C₂×C₄×C₅₃	C₄⋊C₈	1	1	yes	characteristic
47	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
48	C₂×C₄×C₅₃	#108	1	1	yes	characteristic
49	C₂×C₄×C₅₃	#108	1	1	yes	characteristic
50	C₂×Dic₅₃	—	2	1	no	—
51	C₂×Dic₅₃	—	2	1	no	—
52	C₂×Dic₅₃	#102	1	2	yes	—
53	C₂×Dic₅₃	#103	1	2	yes	—
54	C₂×Dic₅₃	D₁₆	1	1	yes	characteristic
55	C₂×Dic₅₃	SD₃₂	1	2	yes	—
56	C₂×Dic₅₃	Q₃₂	1	1	yes	characteristic
57	#88324	—	8	4	no	—
58	C₈×C₅₃	C₄×C₈	1	1	yes	characteristic
59	C₈×C₅₃	#97	1	1	yes	characteristic
60	C₈×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
61	C₈×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
62	#88322	—	2	2	no	—
63	#7154	—	53	1	no	maximal, Sylow
64	C₂²×C₅₃	#335	1	1	yes	characteristic
65	C₂²×C₅₃	#335	1	1	yes	characteristic
66	C₂²×C₅₃	#340	1	1	yes	characteristic
67	C₂²×C₅₃	#366	1	1	yes	characteristic
68	C₂²×C₅₃	#367	1	1	yes	characteristic
69	C₂²×C₅₃	#368	1	1	yes	characteristic
70	C₂²×C₅₃	#368	1	1	yes	characteristic
71	C₄×C₅₃	#334	1	1	yes	characteristic
72	C₄×C₅₃	#334	1	1	yes	characteristic
73	C₄×C₅₃	#336	1	1	yes	characteristic
74	C₄×C₅₃	#341	1	1	yes	characteristic
75	Dic₅₃	—	2	2	no	—
76	Dic₅₃	#357	1	2	yes	—
77	Dic₅₃	#358	1	2	yes	—
78	C₂×C₄×C₁₆	—	53	1	no	—
79	C₂×#335	—	53	2	no	—
80	C₂×C₅₃	#1277	1	1	yes	characteristic
81	C₂×C₅₃	#1371	1	1	yes	characteristic
82	C₂×C₅₃	#1371	1	1	yes	characteristic
83	C₂×C₅₃	#1372	1	1	yes	characteristic
84	C₂×C₅₃	#1372	1	1	yes	characteristic
85	C₂×C₅₃	#1380	1	1	yes	characteristic
86	C₂×C₅₃	#1381	1	1	yes	characteristic
87	C₂²×C₁₆	—	53	1	no	—
88	C₂²×C₁₆	#6157	1	1	yes	characteristic
89	C₂×C₄⋊C₈	—	106	2	no	—
90	C₂×C₄×C₈	—	53	1	no	—
91	C₄×C₁₆	—	106	2	no	—
92	#335	—	53	4	no	—
93	#335	—	106	2	no	—
94	C₅₃	#7154	1	1	yes	characteristic, complemented, Sylow
95	C₂²×C₈	—	53	1	no	—
96	C₂²×C₈	—	212	2	no	—
97	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
98	C₂×C₄²	—	53	1	no	—
99	C₂×C₁₆	—	2	1	no	—
100	C₂×C₁₆	—	2	1	no	—
101	C₂×C₁₆	—	53	2	no	—
102	C₂×C₁₆	—	106	1	no	—
103	C₂×C₁₆	—	106	1	no	—
104	C₂×C₁₆	#88324	1	2	yes	complemented
105	C₄⋊C₈	—	106	4	no	—
106	C₄⋊C₈	—	212	2	no	—
107	C₄×C₈	—	53	2	no	—
108	C₄×C₈	—	53	2	no	—
109	C₂²×C₄	—	53	1	no	—
110	C₂²×C₄	—	53	1	no	—
111	C₂²×C₄	C₄⋊#6157	1	1	yes	characteristic
112	C₂×C₈	—	53	2	no	—
113	C₂×C₈	—	106	1	no	—
114	C₂×C₈	—	106	1	no	—
115	C₂×C₈	—	424	2	no	—
116	C₂×C₈	—	424	2	no	—
117	C₂×C₈	—	212	4	no	—
118	C₂×C₈	D₅₃⋊C₈	1	1	yes	characteristic
119	C₂×C₈	#11719936	1	1	yes	characteristic
120	C₂×C₈	C₄×#6157	1	1	yes	characteristic
121	C₂×C₈	C₂×#88324	1	1	yes	characteristic
122	C₂×C₈	#11719940	1	1	yes	characteristic
123	C₂×C₈	C₂²⋊#6157	1	1	yes	characteristic
124	C₄²	—	53	2	no	—
125	C₄²	—	53	2	no	—
126	C₁₆	—	2	2	no	—
127	C₁₆	—	106	2	no	—
128	C₂³	#13133286	1	1	yes	characteristic
129	C₂×C₄	—	53	1	no	—
130	C₂×C₄	—	53	1	no	—
131	C₂×C₄	—	53	2	no	—
132	C₂×C₄	—	53	2	no	—
133	C₂×C₄	—	53	2	no	—
134	C₂×C₄	—	106	1	no	—
135	C₂×C₄	—	106	1	no	—
136	C₂×C₄	(C₂×#6157)⋊C₄	1	1	yes	characteristic
137	C₂×C₄	C₄⋊#88324	1	1	yes	characteristic
138	C₂×C₄	#13133284	1	1	yes	characteristic
139	C₂×C₄	#13133285	1	1	yes	characteristic
140	C₂×C₄	#13133287	1	1	yes	characteristic
141	C₂×C₄	#13133288	1	1	yes	characteristic
142	C₈	—	106	2	no	—
143	C₈	—	424	4	no	—
144	C₈	C₄×#88324	1	1	yes	characteristic
145	C₈	#13133272	1	1	yes	characteristic
146	C₈	C₂²⋊#88324	1	1	yes	characteristic
147	C₈	#13133274	1	1	yes	characteristic
148	C₂²	#16811486	1	1	yes	characteristic
149	C₂²	#16811487	1	1	yes	characteristic
150	C₂²	#16811497	1	1	yes	characteristic
151	C₂²	#16811549	1	1	yes	characteristic
152	C₂²	#16811550	1	1	yes	characteristic
153	C₂²	#16811551	1	1	yes	characteristic
154	C₂²	#16811552	1	1	yes	characteristic
155	C₄	—	53	2	no	—
156	C₄	—	53	2	no	—
157	C₄	—	106	2	no	—
158	C₄	#16811484	1	1	yes	characteristic
159	C₄	#16811485	1	1	yes	characteristic
160	C₄	#16811489	1	1	yes	characteristic
161	C₄	#16811499	1	1	yes	characteristic
162	C₂	#20289545	1	1	yes	characteristic
163	C₂	#20289746	1	1	yes	characteristic
164	C₂	#20289747	1	1	yes	characteristic
165	C₂	#20289748	1	1	yes	characteristic
166	C₂	#20289749	1	1	yes	characteristic
167	C₂	#20289773	1	1	yes	characteristic
168	C₂	#20289775	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13133286
commutator	C₈×C₅₃	C₄×C₈
Frattini	C₂²×C₈	C₂×#6157
Fitting	C₂²×C₁₆×C₅₃	C₄
socle, prime radical	C₂³×C₅₃	#107
cosocle	C₂×C₈×Dic₅₃	C₂²
hypercenter	C₂²×C₁₆	#6157
hypocenter	C₅₃	#7154
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆

Complements

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

N	H ≅ G/N	type
17. C₂×#11719908	C₈	semidirect
23. C₂×C₁₆×C₅₃	C₈	semidirect
94. C₅₃	#7154	semidirect
104. C₂×C₁₆	#88324	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16811486	2
C₂×C₁₆×Dic₅₃	2
#7154	53