Finite Group Databasebeta

Subgroups — #28307799

Order 13568.

282 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 144 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	C₄×#13133291	C₂	1	1	yes	maximal, characteristic
2	C₂×#16811511	C₂	1	1	yes	maximal, characteristic
3	#20289633	C₂	1	4	yes	maximal, complemented
4	#20272999	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133291	C₄	1	2	yes	—
6	C₂²×#11719910	C₄	1	2	yes	—
7	#16811511	C₄	1	2	yes	—
8	#16811511	C₂²	1	2	yes	—
9	#16811511	C₂²	1	2	yes	—
10	C₄×#11719910	C₄	1	2	yes	—
11	#16810676	C₄	1	1	yes	characteristic
12	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
13	#16810345	C₂²	1	2	yes	—
14	C₈×#88322	C₄	1	2	yes	—
15	C₅₃×#431	C₄	1	1	yes	characteristic
16	#13133291	C₈	1	8	yes	complemented
17	C₂×#11719910	—	2	4	no	—
18	C₂×#11719910	C₂×C₄	1	4	yes	—
19	C₂×#11719910	C₂×C₄	1	4	yes	—
20	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
21	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
22	C₄⋊#88322	—	2	1	no	—
23	C₂²⋊#88322	—	2	1	no	—
24	C₄×#88322	C₂×C₄	1	2	yes	—
25	C₄×#88322	C₂×C₄	1	1	yes	characteristic
26	C₄×#88322	C₂×C₄	1	1	yes	characteristic
27	C₄×#88322	C₂×C₄	1	1	yes	characteristic
28	C₄×#88322	C₂³	1	1	yes	characteristic
29	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₄⋊C₈×C₅₃	—	2	1	no	—
31	C₂²⋊C₈×C₅₃	—	2	1	no	—
32	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
33	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
34	C₂²×C₄×C₅₃	M₁₆	1	1	yes	characteristic
35	C₂²×C₄×C₅₃	M₁₆	1	1	yes	characteristic
36	C₂×C₈×C₅₃	—	2	2	no	—
37	C₄²×C₅₃	C₄²	1	1	yes	characteristic
38	C₄²×C₅₃	M₁₆	1	1	yes	characteristic
39	C₄²×C₅₃	M₁₆	1	1	yes	characteristic
40	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
41	C₂×#88322	—	2	2	no	—
42	C₂×#88322	C₄²	1	1	yes	characteristic
43	C₂×#88322	C₄²	1	1	yes	characteristic
44	C₂×#88322	C₂×C₈	1	4	yes	—
45	C₂×#88322	M₁₆	1	4	yes	—
46	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
47	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
48	#11719910	—	2	4	no	—
49	#11719910	C₂×C₈	1	8	yes	—
50	C₂³×C₅₃	#97	1	1	yes	characteristic
51	C₂×C₄×C₅₃	—	4	1	no	—
52	C₂×C₄×C₅₃	—	4	1	no	—
53	C₂×C₄×C₅₃	C₄×C₈	1	2	yes	—
54	C₂×C₄×C₅₃	#97	1	2	yes	—
55	C₂×C₄×C₅₃	#97	1	1	yes	characteristic
56	C₂×C₄×C₅₃	#97	1	1	yes	characteristic
57	C₂×C₄×C₅₃	#97	1	1	yes	characteristic
58	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
59	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
60	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
61	C₈×C₅₃	—	4	2	no	—
62	#88322	—	4	2	no	—
63	#88322	C₄×C₈	1	2	yes	—
64	#88322	#97	1	2	yes	—
65	#88322	C₂²×C₈	1	2	yes	—
66	#88322	C₂×M₁₆	1	2	yes	—
67	#11406	—	53	1	no	maximal, Sylow
68	C₂²×C₅₃	—	2	1	no	—
69	C₂²×C₅₃	—	2	1	no	—
70	C₂²×C₅₃	#322	1	2	yes	—
71	C₂²×C₅₃	C₂×#97	1	1	yes	characteristic
72	C₄×C₅₃	—	2	1	no	—
73	C₄×C₅₃	—	2	1	no	—
74	C₄×C₅₃	—	2	1	no	—
75	C₄×C₅₃	—	2	1	no	—
76	C₄×C₅₃	#322	1	2	yes	—
77	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
78	C₄×C₅₃	C₂×#97	1	1	yes	characteristic
79	C₄×M₃₂	—	53	1	no	—
80	C₂×#346	—	53	1	no	—
81	#1450	—	53	1	no	—
82	#1313	—	53	4	no	—
83	C₂×C₅₃	—	2	2	no	—
84	C₂×C₅₃	C₂×#322	1	1	yes	characteristic
85	C₂×C₅₃	#2110	1	1	yes	characteristic
86	C₂×C₅₃	#2110	1	1	yes	characteristic
87	C₂×M₃₂	—	53	2	no	—
88	C₂²×C₁₆	—	53	2	no	—
89	#431	—	53	1	no	—
90	#431	#6157	1	1	yes	characteristic
91	C₂×C₄×C₈	—	53	1	no	—
92	#346	—	53	2	no	—
93	#346	—	53	2	no	—
94	#346	—	53	2	no	—
95	C₄×C₁₆	—	53	2	no	—
96	#322	—	53	2	no	—
97	C₈²	—	53	2	no	—
98	C₅₃	#11406	1	1	yes	characteristic, complemented, Sylow
99	C₂²×C₈	—	53	1	no	—
100	C₂²×C₈	—	53	1	no	—
101	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
102	M₃₂	—	53	8	no	—
103	C₂×C₁₆	—	53	4	no	—
104	C₂×C₁₆	—	53	4	no	—
105	C₂×C₁₆	—	106	4	no	—
106	C₄⋊C₈	—	2	1	no	—
107	C₄⋊C₈	—	106	1	no	—
108	C₂²⋊C₈	—	2	1	no	—
109	C₂²⋊C₈	—	106	1	no	—
110	C₄×C₈	—	53	1	no	—
111	C₄×C₈	—	53	1	no	—
112	C₄×C₈	—	53	1	no	—
113	C₄×C₈	—	53	1	no	—
114	C₄×C₈	—	53	2	no	—
115	C₄×C₈	C₂×#6157	1	2	yes	—
116	C₂²×C₄	#11719936	1	1	yes	characteristic
117	C₂²×C₄	C₄×#6157	1	1	yes	characteristic
118	C₂²×C₄	#11719940	1	1	yes	characteristic
119	C₂×C₈	—	2	2	no	—
120	C₂×C₈	—	53	4	no	—
121	C₂×C₈	—	53	4	no	—
122	C₂×C₈	—	53	1	no	—
123	C₂×C₈	—	53	1	no	—
124	C₂×C₈	—	53	1	no	—
125	C₂×C₈	—	53	1	no	—
126	C₂×C₈	—	106	2	no	—
127	C₄²	#11719936	1	1	yes	characteristic
128	C₄²	C₄×#6157	1	1	yes	characteristic
129	C₄²	#11719940	1	1	yes	characteristic
130	C₄²	C₂²×#6157	1	1	yes	characteristic
131	C₁₆	—	53	8	no	—
132	C₁₆	—	106	4	no	—
133	C₂³	#13133272	1	1	yes	characteristic
134	C₂×C₄	—	4	1	no	—
135	C₂×C₄	—	4	1	no	—
136	C₂×C₄	C₈×#6157	1	2	yes	—
137	C₂×C₄	#13133271	1	2	yes	—
138	C₂×C₄	#13133272	1	1	yes	characteristic
139	C₂×C₄	#13133272	1	1	yes	characteristic
140	C₂×C₄	#13133272	1	1	yes	characteristic
141	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
142	C₂×C₄	C₂×#11719940	1	1	yes	characteristic
143	C₂×C₄	C₂×#11719936	1	1	yes	characteristic
144	C₈	—	4	2	no	—
145	C₈	—	53	2	no	—
146	C₈	—	53	2	no	—
147	C₈	—	53	2	no	—
148	C₈	—	53	2	no	—
149	C₈	—	212	2	no	—
150	C₂²	—	2	1	no	—
151	C₂²	—	2	1	no	—
152	C₂²	#88324⋊C₈	1	2	yes	—
153	C₂²	C₂×#13133272	1	1	yes	characteristic
154	C₄	—	2	1	no	—
155	C₄	—	2	1	no	—
156	C₄	—	2	1	no	—
157	C₄	—	2	1	no	—
158	C₄	#88324⋊C₈	1	2	yes	—
159	C₄	C₂×C₈×#6157	1	1	yes	characteristic
160	C₄	C₂×#13133271	1	1	yes	characteristic
161	C₂	—	2	2	no	—
162	C₂	C₂×#88324⋊C₈	1	1	yes	characteristic
163	C₂	#20290936	1	1	yes	characteristic
164	C₂	#20290938	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₄×#6157
commutator	C₄×C₅₃	C₂×C₄×C₈
Frattini	C₄²	C₂²×#6157
Fitting	C₅₃×#431	C₄
socle	C₂²×C₅₃	C₂×#97
prime radical	C₂³×C₅₃	#97
cosocle	C₄×#88322	C₂³
hypercenter	#431	#6157
hypocenter	C₅₃	#11406
LCS	G ▷ C₄×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ #431

Complements

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

N	H ≅ G/N	type
3. #20289633	C₂	semidirect
16. #13133291	C₈	semidirect
98. C₅₃	#11406	semidirect

Maximal subgroups

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

subgroup	index
C₄×#13133291	2
C₂×#16811511	2
#20289633	2
#20272999	2
#11406	53