Finite Group Databasebeta

Subgroups — #28309966

Order 13568.

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

normal subgroups only

This group has 258 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	#20290296	C₂	1	2	yes	maximal, complemented
2	#20289746	C₂	1	1	yes	maximal, characteristic, complemented
3	#20289670	C₂	1	2	yes	maximal, complemented
4	C₁₆×#88324	C₂	1	1	yes	maximal, characteristic, complemented
5	#20276561	C₂	1	1	yes	maximal, characteristic
6	#16811651	—	2	2	no	—
7	#16811509	C₂²	1	1	yes	characteristic
8	#16811486	C₂²	1	2	yes	—
9	#16811471	—	2	2	no	—
10	C₈×#88324	C₂²	1	1	yes	characteristic
11	C₂×D₈₄₈	C₄	1	1	yes	characteristic
12	#16810705	C₂²	1	2	yes	—
13	#16810423	C₄	1	2	yes	—
14	C₁₆×Dic₅₃	C₂²	1	1	yes	characteristic
15	C₅₃×#366	C₄	1	1	yes	characteristic
16	C₂×D₅₃⋊C₈	—	4	2	no	—
17	C₂×#11719910	—	2	1	no	—
18	C₄⋊#88324	—	2	2	no	—
19	#13133274	—	4	2	no	—
20	C₄×#88324	D₄	1	1	yes	characteristic
21	C₄×#88324	D₄	1	1	yes	characteristic
22	C₂×D₄₂₄	C₂×C₄	1	2	yes	—
23	#13133179	—	2	2	no	—
24	D₈₄₈	C₈	1	4	yes	complemented
25	C₂×#11719908	C₂×C₄	1	1	yes	characteristic
26	#13133137	—	2	2	no	—
27	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
28	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
29	#107×C₅₃	C₂×C₄	1	2	yes	—
30	C₂×D₂₁₂	—	2	2	no	—
31	C₂×C₄×D₅₃	—	4	2	no	—
32	C₂×#88324	—	2	1	no	—
33	C₂×#88324	—	4	2	no	—
34	C₂×#88324	D₈	1	1	yes	characteristic
35	C₂×#88324	D₈	1	1	yes	characteristic
36	D₅₃⋊C₈	—	8	4	no	—
37	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
38	C₄⋊C₄×C₅₃	—	2	2	no	—
39	#11719921	—	4	2	no	—
40	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
41	C₂×#88322	#41	1	1	yes	characteristic
42	D₄₂₄	—	2	2	no	—
43	D₄₂₄	C₂×C₈	1	4	yes	complemented
44	#11719910	—	2	2	no	—
45	C₁₆×C₅₃	C₂×C₈	1	2	yes	complemented
46	#11719908	—	2	1	no	—
47	C₂²×D₅₃	—	4	2	no	—
48	C₂×C₄×C₅₃	—	4	2	no	—
49	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
50	C₂×Dic₅₃	C₂×D₈	1	1	yes	characteristic
51	C₂×Dic₅₃	#135	1	1	yes	characteristic
52	D₂₁₂	—	2	4	no	—
53	D₂₁₂	—	4	2	no	—
54	C₄×D₅₃	—	8	2	no	—
55	C₄×D₅₃	—	8	2	no	—
56	#88324	—	2	2	no	—
57	#88324	—	2	1	no	—
58	#88324	—	8	2	no	—
59	#88324	D₁₆	1	2	yes	complemented
60	C₈×C₅₃	C₂²×C₈	1	1	yes	characteristic
61	C₈×C₅₃	#131	1	1	yes	characteristic
62	#88322	—	2	1	no	—
63	C₈×D₁₆	—	53	1	no	maximal, Sylow
64	C₂²×C₅₃	C₄×D₈	1	1	yes	characteristic
65	C₂×D₅₃	—	4	4	no	—
66	C₂×D₅₃	—	8	2	no	—
67	C₂×D₅₃	—	8	2	no	—
68	C₄×C₅₃	—	8	2	no	—
69	C₄×C₅₃	C₈×D₄	1	1	yes	characteristic
70	C₄×C₅₃	#443	1	1	yes	characteristic
71	Dic₅₃	—	2	1	no	—
72	Dic₅₃	C₂×D₁₆	1	1	yes	characteristic
73	Dic₅₃	#508	1	1	yes	characteristic
74	C₄×D₁₆	—	53	1	no	—
75	C₈×D₈	—	53	2	no	—
76	#1371	—	53	1	no	—
77	#1331	—	53	2	no	—
78	C₈×C₁₆	—	53	1	no	—
79	C₂×C₅₃	C₈×D₈	1	1	yes	characteristic
80	C₂×C₅₃	C₄×D₁₆	1	1	yes	characteristic
81	C₂×C₅₃	#2178	1	1	yes	characteristic
82	D₅₃	—	8	4	no	—
83	C₂×D₁₆	—	53	1	no	—
84	C₄×D₈	—	53	2	no	—
85	C₈×D₄	—	106	2	no	—
86	#366	#6157	1	1	yes	characteristic
87	#357	—	53	2	no	—
88	C₄×C₁₆	—	53	1	no	—
89	C₄×C₁₆	—	53	1	no	—
90	#335	—	53	2	no	—
91	D₄⋊C₈	—	106	2	no	—
92	C₈²	—	53	1	no	—
93	C₅₃	C₈×D₁₆	1	1	yes	characteristic, complemented, Sylow
94	C₂×D₈	—	53	2	no	—
95	C₂²×C₈	—	212	2	no	—
96	C₄×D₄	—	106	2	no	—
97	D₁₆	—	53	4	no	—
98	C₂×C₁₆	—	53	1	no	—
99	C₂×C₁₆	—	106	1	no	—
100	C₂×C₁₆	C₂×#6157	1	1	yes	characteristic
101	#107	C₂×#6157	1	2	yes	—
102	C₄⋊C₈	—	106	2	no	—
103	#102	—	106	2	no	—
104	C₂²⋊C₈	—	212	2	no	—
105	C₄×C₈	—	53	1	no	—
106	C₄×C₈	—	53	1	no	—
107	C₄×C₈	—	53	1	no	—
108	C₂×D₄	—	106	2	no	—
109	C₂²×C₄	—	212	2	no	—
110	D₈	—	53	4	no	—
111	D₈	—	106	2	no	—
112	C₂×C₈	—	53	1	no	—
113	C₂×C₈	—	53	1	no	—
114	C₂×C₈	—	53	1	no	—
115	C₂×C₈	—	106	1	no	—
116	C₂×C₈	—	424	4	no	—
117	C₂×C₈	—	212	2	no	—
118	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
119	C₄⋊C₄	—	2	2	no	—
120	C₂²⋊C₄	—	212	2	no	—
121	C₄²	—	53	1	no	—
122	C₁₆	—	106	2	no	—
123	C₁₆	—	106	1	no	—
124	C₁₆	D₅₃⋊C₈	1	2	yes	complemented
125	C₂³	—	212	2	no	—
126	D₄	—	106	4	no	—
127	D₄	—	212	2	no	—
128	C₂×C₄	—	4	2	no	—
129	C₂×C₄	—	53	1	no	—
130	C₂×C₄	—	53	1	no	—
131	C₂×C₄	—	424	2	no	—
132	C₂×C₄	—	424	2	no	—
133	C₂×C₄	D₄×#6157	1	1	yes	characteristic
134	C₈	—	53	2	no	—
135	C₈	—	106	1	no	—
136	C₈	—	106	2	no	—
137	C₈	—	106	1	no	—
138	C₈	—	424	2	no	—
139	C₈	C₂×D₅₃⋊C₈	1	1	yes	characteristic
140	C₈	#13133305	1	1	yes	characteristic
141	C₂²	—	424	2	no	—
142	C₂²	—	424	2	no	—
143	C₂²	—	212	4	no	—
144	C₂²	D₈×#6157	1	1	yes	characteristic
145	C₄	—	8	2	no	—
146	C₄	—	53	1	no	—
147	C₄	—	53	1	no	—
148	C₄	—	106	1	no	—
149	C₄	#16811651	1	1	yes	characteristic
150	C₄	#16811667	1	1	yes	characteristic
151	C₂	—	424	4	no	—
152	C₂	#20290296	1	1	yes	characteristic
153	C₂	D₁₆×#6157	1	1	yes	characteristic
154	C₂	#20291116	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₈×#6157
commutator	C₈×C₅₃	C₂²×C₈
Frattini	C₂×C₈	C₂²×#6157
Fitting	C₅₃×#366	C₄
socle	C₂²×C₅₃	C₄×D₈
prime radical	C₂×D₈₄₈	C₄
cosocle	C₈×Dic₅₃	C₂³
hypercenter	#366	#6157
hypocenter	C₅₃	C₈×D₁₆
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #366

Complements

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

N	H ≅ G/N	type
1. #20290296	C₂	semidirect
2. #20289746	C₂	semidirect
3. #20289670	C₂	semidirect
4. C₁₆×#88324	C₂	semidirect
24. D₈₄₈	C₈	semidirect
43. D₄₂₄	C₂×C₈	semidirect
45. C₁₆×C₅₃	C₂×C₈	semidirect
59. #88324	D₁₆	semidirect
93. C₅₃	C₈×D₁₆	semidirect
124. C₁₆	D₅₃⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#20290296	2
#20289746	2
#20289670	2
C₁₆×#88324	2
#20276561	2
C₈×D₁₆	53