Finite Group Databasebeta

Subgroups — #28298685

Order 13568.

196 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 52 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	#20290932	C₂	1	1	yes	maximal, characteristic
2	#20290641	C₂	1	1	yes	maximal, characteristic
3	#347×D₅₃	C₂	1	1	yes	maximal, characteristic
4	D₅₃⋊M₃₂	—	2	1	no	—
5	#16811628	C₄	1	1	yes	characteristic
6	#16811627	C₄	1	1	yes	characteristic
7	#16811516	—	2	1	no	—
8	#16811515	C₄	1	2	yes	—
9	#16811499	—	2	1	no	—
10	#16811499	—	2	1	no	—
11	M₃₂×D₅₃	—	2	1	no	—
12	#97×D₅₃	C₂²	1	1	yes	characteristic
13	#16810401	C₄	1	1	yes	characteristic
14	#16810400	—	2	1	no	—
15	C₅₃×#347	C₄	1	1	yes	characteristic
16	C₂×#11719936	—	2	1	no	—
17	C₂×#11719936	—	2	1	no	—
18	#13133292	—	2	2	no	—
19	#13133291	—	2	2	no	—
20	D₅₃⋊C₁₆	—	2	2	no	—
21	#13133284	—	4	1	no	—
22	C₄⋊#88324	—	4	1	no	—
23	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
24	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
25	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
26	#13133157	—	2	1	no	—
27	#13133156	—	2	2	no	—
28	C₁₆×D₅₃	—	2	2	no	—
29	#13133128	C₂×C₄	1	1	yes	characteristic
30	#13133127	D₄	1	1	yes	characteristic
31	#13133127	Q₈	1	1	yes	characteristic
32	M₃₂×C₅₃	—	2	1	no	—
33	#97×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄×D₅₃	—	2	1	no	—
35	C₂×C₄×D₅₃	C₂²⋊C₄	1	1	yes	characteristic
36	C₂×#88324	—	4	1	no	—
37	C₂×#88324	—	4	1	no	—
38	#11719936	—	4	1	no	—
39	#11719936	—	4	1	no	—
40	#11719936	—	4	1	no	—
41	#11719936	—	4	1	no	—
42	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
43	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
44	C₄²×C₅₃	C₄²	1	1	yes	characteristic
45	C₄×Dic₅₃	—	2	1	no	—
46	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
47	C₂×#88322	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×#88322	C₄⋊C₄	1	1	yes	characteristic
49	C₈×D₅₃	—	2	1	no	—
50	C₈×D₅₃	—	2	1	no	—
51	C₈×D₅₃	—	2	1	no	—
52	C₈×D₅₃	—	2	1	no	—
53	#11719910	—	2	4	no	—
54	C₁₆×C₅₃	—	2	2	no	—
55	#11719908	—	2	2	no	—
56	C₂²×D₅₃	#101	1	1	yes	characteristic
57	C₂×C₄×C₅₃	—	2	1	no	—
58	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
59	C₂×Dic₅₃	—	2	1	no	—
60	C₂×Dic₅₃	#100	1	1	yes	characteristic
61	C₄×D₅₃	—	8	1	no	—
62	C₄×D₅₃	#104	1	1	yes	characteristic
63	C₄×D₅₃	#104	1	1	yes	characteristic
64	C₄×D₅₃	#104	1	1	yes	characteristic
65	C₄×D₅₃	#104	1	1	yes	characteristic
66	#88324	—	8	1	no	—
67	#88324	—	8	1	no	—
68	C₈×C₅₃	—	2	1	no	—
69	C₈×C₅₃	—	2	1	no	—
70	#88322	—	2	1	no	—
71	#88322	—	2	1	no	—
72	#7267	—	53	1	no	maximal, Sylow
73	C₂²×C₅₃	#343	1	1	yes	characteristic
74	C₂×D₅₃	—	4	1	no	—
75	C₂×D₅₃	#330	1	1	yes	characteristic
76	C₂×D₅₃	#330	1	1	yes	characteristic
77	C₄×C₅₃	—	4	1	no	—
78	C₄×C₅₃	#339	1	1	yes	characteristic
79	C₄×C₅₃	#339	1	1	yes	characteristic
80	Dic₅₃	—	4	1	no	—
81	Dic₅₃	#329	1	1	yes	characteristic
82	Dic₅₃	#329	1	1	yes	characteristic
83	C₂×#347	—	53	1	no	—
84	C₂×#347	—	53	1	no	—
85	#1838	—	53	1	no	—
86	C₂×C₅₃	—	2	1	no	—
87	C₂×C₅₃	#1295	1	1	yes	characteristic
88	D₅₃	—	2	1	no	—
89	D₅₃	—	2	1	no	—
90	C₂×M₃₂	—	106	1	no	—
91	C₂×M₃₂	—	106	1	no	—
92	#423	—	53	1	no	—
93	#423	—	53	1	no	—
94	C₂×#97	—	53	1	no	—
95	#347	—	53	2	no	—
96	#347	—	53	1	no	—
97	#347	—	106	1	no	—
98	#347	—	106	1	no	—
99	#347	#6157	1	1	yes	characteristic
100	#341	—	106	1	no	—
101	#341	—	106	1	no	—
102	C₅₃	#7267	1	1	yes	characteristic, complemented, Sylow
103	C₂×M₁₆	—	106	1	no	—
104	C₂×M₁₆	—	106	1	no	—
105	C₂²×C₈	—	53	1	no	—
106	C₂²×C₈	—	53	1	no	—
107	C₂×C₄²	—	53	1	no	—
108	M₃₂	—	2	1	no	—
109	M₃₂	—	106	2	no	—
110	M₃₂	—	106	2	no	—
111	M₃₂	—	106	2	no	—
112	M₃₂	—	106	1	no	—
113	C₂×C₁₆	—	106	2	no	—
114	C₂×C₁₆	—	106	2	no	—
115	C₄⋊C₈	—	212	1	no	—
116	C₄⋊C₈	—	212	1	no	—
117	#97	—	53	1	no	—
118	#97	—	53	1	no	—
119	#97	—	53	1	no	—
120	#97	C₂×#6157	1	1	yes	characteristic
121	C₂²×C₄	—	53	1	no	—
122	C₂²×C₄	—	106	1	no	—
123	M₁₆	—	212	1	no	—
124	M₁₆	—	212	1	no	—
125	M₁₆	—	212	1	no	—
126	M₁₆	—	212	1	no	—
127	C₂×C₈	—	53	1	no	—
128	C₂×C₈	—	53	1	no	—
129	C₂×C₈	—	106	1	no	—
130	C₂×C₈	—	106	1	no	—
131	C₂×C₈	—	106	1	no	—
132	C₂×C₈	—	106	1	no	—
133	C₂×C₈	—	212	1	no	—
134	C₂×C₈	—	212	1	no	—
135	C₂×C₈	C₄⋊#6157	1	1	yes	characteristic
136	C₂×C₈	C₂²⋊#6157	1	1	yes	characteristic
137	C₄²	—	53	1	no	—
138	C₄²	—	106	1	no	—
139	C₄²	C₄×#6157	1	1	yes	characteristic
140	C₁₆	—	2	2	no	—
141	C₁₆	—	106	2	no	—
142	C₁₆	—	106	4	no	—
143	C₂³	—	53	1	no	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	53	1	no	—
146	C₂×C₄	—	53	1	no	—
147	C₂×C₄	—	53	1	no	—
148	C₂×C₄	—	53	1	no	—
149	C₂×C₄	—	53	1	no	—
150	C₂×C₄	—	106	1	no	—
151	C₂×C₄	—	424	1	no	—
152	C₂×C₄	(C₂×#6157)⋊C₄	1	1	yes	characteristic
153	C₈	—	2	1	no	—
154	C₈	—	2	1	no	—
155	C₈	—	106	1	no	—
156	C₈	—	106	1	no	—
157	C₈	—	424	1	no	—
158	C₈	—	424	1	no	—
159	C₂²	—	53	1	no	—
160	C₂²	—	53	1	no	—
161	C₂²	—	212	1	no	—
162	C₂²	#16811502	1	1	yes	characteristic
163	C₄	—	4	1	no	—
164	C₄	—	53	1	no	—
165	C₄	—	53	1	no	—
166	C₄	—	212	1	no	—
167	C₄	#16811494	1	1	yes	characteristic
168	C₄	#16811495	1	1	yes	characteristic
169	C₂	—	2	1	no	—
170	C₂	—	106	1	no	—
171	C₂	—	106	1	no	—
172	C₂	#20289592	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#20289592
commutator	C₄²×C₅₃	C₄²
Frattini	#97	C₂×#6157
Fitting	C₅₃×#347	C₄
socle	C₂×C₅₃	#1295
prime radical	C₂²×D₅₃	#101
cosocle	#97×D₅₃	C₂²
hypercenter	#347	#6157
hypocenter	C₅₃	#7267
LCS	G ▷ C₄²×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #97 ◁ #347

Complements

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

N	H ≅ G/N	type
102. C₅₃	#7267	semidirect

Maximal subgroups

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

subgroup	index
#20290932	2
#20290641	2
#347×D₅₃	2
#7267	53