Finite Group Databasebeta

Subgroups — #27274530

Order 13568.

354 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 155 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	#20274402	C₂	1	1	yes	maximal, characteristic
2	#20272999	C₂	1	2	yes	maximal
3	#20272545	C₂	1	2	yes	maximal
4	#20272544	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1274	C₂	1	1	yes	maximal, characteristic
6	#16810676	C₂²	1	2	yes	—
7	C₂×#13133128	C₄	1	2	yes	—
8	C₂×#13133128	C₂²	1	1	yes	characteristic
9	C₂×C₄×#88322	C₄	1	2	yes	—
10	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
11	#16810345	C₄	1	4	yes	complemented
12	C₈×#88322	C₄	1	4	yes	complemented
13	C₅₃×#431	C₂²	1	2	yes	—
14	C₅₃×#376	C₂²	1	1	yes	characteristic
15	C₂²×#88322	C₂×C₄	1	2	yes	—
16	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
17	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
18	C₄⋊#88322	—	2	2	no	—
19	C₂²⋊#88322	—	2	2	no	—
20	#13133128	—	2	1	no	—
21	#13133128	—	2	1	no	—
22	#13133128	—	2	4	no	—
23	C₄×#88322	C₈	1	8	yes	complemented
24	C₄×#88322	C₂×C₄	1	4	yes	—
25	C₄×#88322	C₂×C₄	1	2	yes	—
26	C₄×#88322	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
29	C₄⋊C₈×C₅₃	—	2	2	no	—
30	C₂²⋊C₈×C₅₃	—	2	2	no	—
31	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
32	#95×C₅₃	—	2	2	no	—
33	C₂²×C₄×C₅₃	—	2	2	no	—
34	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
35	C₂²×C₄×C₅₃	#41	1	2	yes	—
36	C₂×C₈×C₅₃	—	2	4	no	—
37	C₄²×C₅₃	—	2	2	no	—
38	C₄²×C₅₃	C₂×C₈	1	4	yes	—
39	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
40	C₄²×C₅₃	#41	1	1	yes	characteristic
41	C₄²×C₅₃	#41	1	1	yes	characteristic
42	C₂×#88322	—	2	2	no	—
43	C₂×#88322	—	2	2	no	—
44	C₂×#88322	—	2	4	no	—
45	C₂×#88322	C₄²	1	1	yes	characteristic
46	C₂×#88322	C₄²	1	1	yes	characteristic
47	C₂×#88322	C₄²	1	2	yes	—
48	C₂×#88322	C₂×C₈	1	4	yes	—
49	C₂×#88322	C₂×C₈	1	4	yes	—
50	C₂×#88322	M₁₆	1	4	yes	—
51	C₂³×C₅₃	#117	1	1	yes	characteristic
52	C₂×C₄×C₅₃	—	2	4	no	—
53	C₂×C₄×C₅₃	—	4	2	no	—
54	C₂×C₄×C₅₃	—	4	4	no	—
55	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
56	C₂×C₄×C₅₃	#117	1	2	yes	—
57	C₂×C₄×C₅₃	#117	1	2	yes	—
58	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
59	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
60	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
61	C₈×C₅₃	—	4	4	no	—
62	#88322	—	2	4	no	—
63	#88322	—	4	4	no	—
64	#88322	C₄×C₈	1	2	yes	—
65	#88322	C₄×C₈	1	2	yes	—
66	#88322	#97	1	4	yes	—
67	#8137	—	53	1	no	maximal, Sylow
68	C₂²×C₅₃	—	2	2	no	—
69	C₂²×C₅₃	#376	1	1	yes	characteristic
70	C₂²×C₅₃	#431	1	2	yes	—
71	C₄×C₅₃	—	2	2	no	—
72	C₄×C₅₃	—	2	2	no	—
73	C₄×C₅₃	—	4	4	no	—
74	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
75	C₄×C₅₃	C₂×#97	1	1	yes	characteristic
76	C₄×C₅₃	#431	1	2	yes	—
77	#1726	—	53	1	no	—
78	#1450	—	53	2	no	—
79	#1275	—	53	2	no	—
80	#1274	—	53	1	no	—
81	#1274	D₅₃	1	1	yes	characteristic
82	C₂×C₅₃	—	2	2	no	—
83	C₂×C₅₃	#1744	1	1	yes	characteristic
84	C₂×C₅₃	#1747	1	2	yes	—
85	#431	—	53	2	no	—
86	#431	C₂×D₅₃	1	2	yes	—
87	C₂×#97	—	53	2	no	—
88	C₂×#97	—	53	1	no	—
89	C₂×C₄×C₈	—	53	2	no	—
90	C₂×C₄×C₈	—	53	1	no	—
91	#376	C₂×D₅₃	1	1	yes	characteristic
92	#322	—	53	4	no	—
93	C₈²	—	53	4	no	—
94	C₅₃	#8137	1	1	yes	characteristic, complemented, Sylow
95	C₂²×C₈	—	53	1	no	—
96	C₂²×C₈	—	53	1	no	—
97	C₂²×C₈	—	53	2	no	—
98	C₂×C₄²	C₄×D₅₃	1	2	yes	—
99	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
100	C₄⋊C₈	—	2	2	no	—
101	C₄⋊C₈	—	106	2	no	—
102	C₂²⋊C₈	—	2	2	no	—
103	C₂²⋊C₈	—	106	2	no	—
104	#97	—	106	4	no	—
105	#97	—	106	1	no	—
106	#97	—	106	1	no	—
107	C₄×C₈	—	53	2	no	—
108	C₄×C₈	—	53	2	no	—
109	C₄×C₈	—	53	4	no	—
110	C₄×C₈	—	53	8	no	—
111	C₄×C₈	C₄×D₅₃	1	4	yes	—
112	#95	—	2	2	no	—
113	C₂²×C₄	—	2	2	no	—
114	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
115	C₂²×C₄	#11719947	1	2	yes	—
116	C₂×C₈	—	2	4	no	—
117	C₂×C₈	—	53	1	no	—
118	C₂×C₈	—	53	1	no	—
119	C₂×C₈	—	53	2	no	—
120	C₂×C₈	—	53	4	no	—
121	C₂×C₈	—	53	4	no	—
122	C₂×C₈	—	53	4	no	—
123	C₂×C₈	—	106	2	no	—
124	C₂×C₈	—	106	2	no	—
125	C₂×C₈	—	106	4	no	—
126	C₄²	—	2	2	no	—
127	C₄²	C₈×D₅₃	1	4	yes	—
128	C₄²	C₂×C₄×D₅₃	1	2	yes	—
129	C₄²	#11719947	1	1	yes	characteristic
130	C₄²	#11719949	1	1	yes	characteristic
131	C₂³	#13133173	1	1	yes	characteristic
132	C₂×C₄	—	2	4	no	—
133	C₂×C₄	—	4	2	no	—
134	C₂×C₄	—	4	4	no	—
135	C₂×C₄	C₄²×D₅₃	1	1	yes	characteristic
136	C₂×C₄	#13133173	1	2	yes	—
137	C₂×C₄	#13133175	1	2	yes	—
138	C₂×C₄	#13133175	1	1	yes	characteristic
139	C₂×C₄	C₂×C₈×D₅₃	1	2	yes	—
140	C₂×C₄	M₁₆×D₅₃	1	2	yes	—
141	C₈	—	4	4	no	—
142	C₈	—	53	2	no	—
143	C₈	—	53	2	no	—
144	C₈	—	53	4	no	—
145	C₈	—	106	4	no	—
146	C₈	—	212	4	no	—
147	C₂²	—	2	2	no	—
148	C₂²	#16810468	1	1	yes	characteristic
149	C₂²	#16810673	1	2	yes	—
150	C₄	—	2	2	no	—
151	C₄	—	2	2	no	—
152	C₄	—	4	4	no	—
153	C₄	C₄×C₈×D₅₃	1	1	yes	characteristic
154	C₄	#97×D₅₃	1	1	yes	characteristic
155	C₄	#16810675	1	2	yes	—
156	C₂	—	2	2	no	—
157	C₂	#20274459	1	1	yes	characteristic
158	C₂	#20274469	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₄²×D₅₃
commutator	C₄×C₅₃	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1274	C₂
socle	C₂²×C₅₃	#376
prime radical	C₂³×C₅₃	#117
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1274	D₅₃
hypocenter	C₅₃	#8137
LCS	G ▷ C₄×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ #1274

Complements

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

N	H ≅ G/N	type
11. #16810345	C₄	semidirect
12. C₈×#88322	C₄	semidirect
23. C₄×#88322	C₈	semidirect
94. C₅₃	#8137	semidirect

Maximal subgroups

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

subgroup	index
#20274402	2
#20272999	2
#20272545	2
#20272544	2
C₅₃×#1274	2
#8137	53