Finite Group Databasebeta

Subgroups — #28312601

Order 13568.

178 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 178 conjugacy classes of subgroups — too many to draw. Only its 76 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₅₃⋊#368	C₂	1	1	yes	maximal, characteristic
2	C₂×#16811551	C₂	1	1	yes	maximal, characteristic
3	#20289879	C₂	1	2	yes	maximal, complemented
4	#20289878	C₂	1	2	yes	maximal, complemented
5	M₆₄×D₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133288	—	2	1	no	—
7	D₅₃⋊#108	—	2	1	no	—
8	#16811552	—	2	1	no	—
9	#16811551	—	2	1	no	—
10	#16811551	C₂²	1	1	yes	characteristic
11	#16811551	C₂²	1	2	yes	—
12	#16811551	C₂²	1	1	yes	characteristic
13	C₂×C₁₆×D₅₃	C₂²	1	1	yes	characteristic
14	#16810457	C₄	1	1	yes	characteristic
15	#16810456	C₄	1	2	yes	—
16	C₃₂×D₅₃	C₂²	1	2	yes	—
17	C₅₃×M₆₄	C₄	1	1	yes	characteristic
18	C₂×#11719936	—	4	1	no	—
19	D₅₃⋊M₁₆	—	4	1	no	—
20	#13133288	—	2	2	no	—
21	#13133288	—	2	1	no	—
22	#13133288	—	2	1	no	—
23	#13133288	—	4	1	no	—
24	#13133287	—	4	1	no	—
25	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
26	C₂×#11719908	Q₈	1	1	yes	characteristic
27	C₁₆×D₅₃	C₂×C₄	1	1	yes	characteristic
28	C₁₆×D₅₃	D₄	1	1	yes	characteristic
29	C₁₆×D₅₃	Q₈	1	1	yes	characteristic
30	C₁₆×D₅₃	C₂³	1	1	yes	characteristic
31	#13133123	C₂×C₄	1	2	yes	—
32	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₃₂×C₅₃	C₂×C₄	1	2	yes	—
34	C₂×C₄×D₅₃	D₈	1	1	yes	characteristic
35	#11719940	—	8	1	no	—
36	C₂×#88324	—	8	1	no	—
37	#11719936	—	4	1	no	—
38	#11719936	—	4	1	no	—
39	#11719936	—	4	2	no	—
40	#11719936	—	8	1	no	—
41	D₅₃⋊C₈	—	8	1	no	—
42	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
43	C₂×#88322	Q₁₆	1	1	yes	characteristic
44	C₈×D₅₃	C₄⋊C₄	1	1	yes	characteristic
45	C₈×D₅₃	D₈	1	1	yes	characteristic
46	C₈×D₅₃	Q₁₆	1	1	yes	characteristic
47	C₈×D₅₃	C₂×D₄	1	1	yes	characteristic
48	C₁₆×C₅₃	C₄⋊C₄	1	1	yes	characteristic
49	C₁₆×C₅₃	C₂²×C₄	1	1	yes	characteristic
50	#11719908	C₄⋊C₄	1	1	yes	characteristic
51	#11719908	C₂×Q₈	1	1	yes	characteristic
52	C₂²×D₅₃	Q₃₂	1	1	yes	characteristic
53	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
54	C₂×Dic₅₃	D₁₆	1	1	yes	characteristic
55	C₄×D₅₃	#107	1	1	yes	characteristic
56	C₄×D₅₃	D₁₆	1	1	yes	characteristic
57	C₄×D₅₃	Q₃₂	1	1	yes	characteristic
58	C₄×D₅₃	C₂×D₈	1	1	yes	characteristic
59	#88324	—	8	2	no	—
60	#88324	—	8	1	no	—
61	#88324	—	8	1	no	—
62	C₈×C₅₃	#107	1	1	yes	characteristic
63	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
64	#88322	#107	1	1	yes	characteristic
65	#88322	C₂×Q₁₆	1	1	yes	characteristic
66	#13636	—	53	1	no	maximal, Sylow
67	C₂²×C₅₃	#366	1	1	yes	characteristic
68	C₂×D₅₃	—	4	1	no	—
69	C₂×D₅₃	#366	1	1	yes	characteristic
70	C₂×D₅₃	C₂×Q₃₂	1	1	yes	characteristic
71	C₄×C₅₃	#366	1	1	yes	characteristic
72	C₄×C₅₃	C₂×#107	1	1	yes	characteristic
73	Dic₅₃	#366	1	1	yes	characteristic
74	Dic₅₃	C₂×D₁₆	1	1	yes	characteristic
75	C₂×M₆₄	—	53	1	no	—
76	C₂×#368	—	53	1	no	—
77	C₂×#368	—	53	1	no	—
78	#1425	—	53	2	no	—
79	#1425	—	53	2	no	—
80	C₂×C₅₃	—	2	1	no	—
81	C₂×C₅₃	C₂×#366	1	1	yes	characteristic
82	D₅₃	—	2	1	no	—
83	D₅₃	—	2	1	no	—
84	C₂²×C₁₆	—	53	1	no	—
85	C₂×#108	—	106	1	no	—
86	C₂×#108	—	106	1	no	—
87	M₆₄	—	53	1	no	—
88	M₆₄	—	53	2	no	—
89	M₆₄	#6157	1	1	yes	characteristic
90	C₂×C₃₂	—	53	2	no	—
91	#368	—	53	1	no	—
92	#368	—	53	2	no	—
93	#368	—	53	1	no	—
94	#368	—	106	1	no	—
95	#368	—	106	1	no	—
96	C₅₃	#13636	1	1	yes	characteristic, complemented, Sylow
97	C₂×M₁₆	—	212	1	no	—
98	C₂×M₁₆	—	212	1	no	—
99	C₂²×C₈	—	53	1	no	—
100	C₂×C₁₆	—	53	1	no	—
101	C₂×C₁₆	—	53	1	no	—
102	C₂×C₁₆	—	53	1	no	—
103	C₂×C₁₆	—	53	1	no	—
104	C₂×C₁₆	—	53	1	no	—
105	C₂×C₁₆	C₂×#6157	1	1	yes	characteristic
106	#108	—	106	2	no	—
107	#108	—	106	1	no	—
108	#108	—	106	1	no	—
109	#108	—	212	1	no	—
110	#108	—	212	1	no	—
111	C₃₂	—	53	2	no	—
112	C₃₂	C₂×#6157	1	2	yes	—
113	C₂²×C₄	—	53	1	no	—
114	M₁₆	—	424	1	no	—
115	M₁₆	—	424	1	no	—
116	M₁₆	—	212	1	no	—
117	M₁₆	—	212	1	no	—
118	M₁₆	—	212	2	no	—
119	C₂×C₈	—	53	1	no	—
120	C₂×C₈	—	53	1	no	—
121	C₂×C₈	—	53	1	no	—
122	C₂×C₈	—	53	1	no	—
123	C₂×C₈	—	53	1	no	—
124	C₂×C₈	—	424	1	no	—
125	C₂×C₈	—	424	1	no	—
126	C₂×C₈	C₄⋊#6157	1	1	yes	characteristic
127	C₁₆	—	53	1	no	—
128	C₁₆	—	53	1	no	—
129	C₁₆	C₄⋊#6157	1	1	yes	characteristic
130	C₁₆	C₂²×#6157	1	1	yes	characteristic
131	C₂³	—	53	1	no	—
132	C₂×C₄	—	53	1	no	—
133	C₂×C₄	—	53	1	no	—
134	C₂×C₄	—	53	1	no	—
135	C₂×C₄	—	53	1	no	—
136	C₂×C₄	—	53	1	no	—
137	C₂×C₄	#13133286	1	1	yes	characteristic
138	C₈	—	53	1	no	—
139	C₈	—	53	1	no	—
140	C₈	—	424	2	no	—
141	C₈	—	424	1	no	—
142	C₈	—	424	1	no	—
143	C₈	#13133286	1	1	yes	characteristic
144	C₈	C₂×C₄⋊#6157	1	1	yes	characteristic
145	C₂²	—	53	1	no	—
146	C₂²	—	53	1	no	—
147	C₂²	—	212	1	no	—
148	C₂²	#16811549	1	1	yes	characteristic
149	C₄	—	53	1	no	—
150	C₄	—	53	1	no	—
151	C₄	#16811549	1	1	yes	characteristic
152	C₄	C₂×#13133286	1	1	yes	characteristic
153	C₂	—	2	1	no	—
154	C₂	—	106	1	no	—
155	C₂	—	106	1	no	—
156	C₂	C₂×#16811549	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#16811549
commutator	C₁₆×C₅₃	C₂²×C₄
Frattini	C₁₆	C₂²×#6157
Fitting	C₅₃×M₆₄	C₄
socle	C₂×C₅₃	C₂×#366
prime radical	C₂²×D₅₃	Q₃₂
cosocle	C₁₆×D₅₃	C₂³
hypercenter	M₆₄	#6157
hypocenter	C₅₃	#13636
LCS	G ▷ C₁₆×C₅₃ ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ M₆₄

Complements

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

N	H ≅ G/N	type
3. #20289879	C₂	semidirect
4. #20289878	C₂	semidirect
96. C₅₃	#13636	semidirect

Maximal subgroups

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

subgroup	index
D₅₃⋊#368	2
C₂×#16811551	2
#20289879	2
#20289878	2
M₆₄×D₅₃	2
#13636	53