Finite Group Databasebeta

Subgroups — #28312597

Order 13568.

194 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 194 conjugacy classes of subgroups — too many to draw. Only its 78 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	#20291066	C₂	1	1	yes	maximal, characteristic
2	C₂×#16811549	C₂	1	1	yes	maximal, characteristic
3	#20289877	C₂	1	2	yes	maximal, complemented
4	#20289876	C₂	1	2	yes	maximal, complemented
5	M₆₄×D₅₃	C₂	1	1	yes	maximal, characteristic
6	#16811633	—	2	1	no	—
7	C₂×#13133286	—	2	1	no	—
8	#16811550	—	2	1	no	—
9	#16811549	—	2	1	no	—
10	#16811549	C₂²	1	1	yes	characteristic
11	#16811549	C₂²	1	2	yes	—
12	#16811549	C₂²	1	1	yes	characteristic
13	C₂×C₁₆×D₅₃	C₂²	1	1	yes	characteristic
14	#16810457	C₄	1	1	yes	characteristic, complemented
15	#16810456	C₄	1	2	yes	complemented
16	C₃₂×D₅₃	C₂²	1	2	yes	—
17	C₅₃×M₆₄	C₄	1	1	yes	characteristic, complemented
18	#13133296	—	4	1	no	—
19	C₂×C₄⋊#6157	—	4	1	no	—
20	#13133286	—	2	2	no	—
21	#13133286	—	2	1	no	—
22	#13133286	—	2	1	no	—
23	#13133286	—	4	1	no	—
24	#13133285	—	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	complemented
32	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₃₂×C₅₃	C₂×C₄	1	2	yes	complemented
34	C₂²×#6157	—	8	1	no	—
35	C₂×C₄×D₅₃	D₈	1	1	yes	characteristic
36	C₂²⋊#6157	—	8	1	no	—
37	C₄⋊#6157	—	4	1	no	—
38	C₄⋊#6157	—	4	1	no	—
39	C₄⋊#6157	—	4	2	no	—
40	C₄⋊#6157	—	8	1	no	—
41	C₄×#6157	—	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	D₁₆	1	1	yes	characteristic
53	C₂×#6157	—	8	1	no	—
54	C₂×#6157	—	8	1	no	—
55	C₂×#6157	—	8	2	no	—
56	C₂×#6157	—	16	2	no	—
57	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
58	C₂×Dic₅₃	Q₃₂	1	1	yes	characteristic
59	C₄×D₅₃	#107	1	1	yes	characteristic
60	C₄×D₅₃	D₁₆	1	1	yes	characteristic
61	C₄×D₅₃	Q₃₂	1	1	yes	characteristic
62	C₄×D₅₃	C₂×D₈	1	1	yes	characteristic
63	C₈×C₅₃	#107	1	1	yes	characteristic
64	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
65	#88322	#107	1	1	yes	characteristic
66	#88322	C₂×Q₁₆	1	1	yes	characteristic
67	#13634	—	53	1	no	maximal, Sylow
68	C₂²×C₅₃	#366	1	1	yes	characteristic
69	C₂×D₅₃	—	2	1	no	—
70	C₂×D₅₃	—	2	1	no	—
71	C₂×D₅₃	#366	1	1	yes	characteristic
72	C₂×D₅₃	C₂×D₁₆	1	1	yes	characteristic
73	#6157	—	16	1	no	—
74	#6157	—	16	2	no	—
75	#6157	—	16	1	no	—
76	C₄×C₅₃	#366	1	1	yes	characteristic
77	C₄×C₅₃	C₂×#107	1	1	yes	characteristic
78	Dic₅₃	#366	1	1	yes	characteristic
79	Dic₅₃	C₂×Q₃₂	1	1	yes	characteristic
80	C₂×M₆₄	—	53	1	no	—
81	#2158	—	53	1	no	—
82	C₂×#366	—	53	1	no	—
83	#1424	—	53	2	no	—
84	#1423	—	53	2	no	—
85	C₂×C₅₃	—	2	1	no	—
86	C₂×C₅₃	C₂×#366	1	1	yes	characteristic
87	D₅₃	—	2	1	no	—
88	D₅₃	#2263	1	1	yes	characteristic
89	D₅₃	#2264	1	1	yes	characteristic
90	C₂²×C₁₆	—	53	1	no	—
91	#427	—	106	1	no	—
92	C₂×#107	—	106	1	no	—
93	M₆₄	—	53	1	no	—
94	M₆₄	—	53	2	no	—
95	M₆₄	#6157	1	1	yes	characteristic, complemented
96	C₂×C₃₂	—	53	2	no	—
97	#367	—	106	1	no	—
98	#366	—	53	2	no	—
99	#366	—	53	1	no	—
100	#366	—	53	1	no	—
101	#366	—	106	1	no	—
102	C₅₃	#13634	1	1	yes	characteristic, complemented, Sylow
103	C₂²×C₈	—	53	1	no	—
104	#117	—	212	1	no	—
105	C₂×C₄⋊C₄	—	212	1	no	—
106	C₂×C₁₆	—	53	1	no	—
107	C₂×C₁₆	—	53	1	no	—
108	C₂×C₁₆	—	53	1	no	—
109	C₂×C₁₆	—	53	1	no	—
110	C₂×C₁₆	—	53	1	no	—
111	C₂×C₁₆	C₂×#6157	1	1	yes	characteristic
112	#107	—	106	2	no	—
113	#107	—	106	1	no	—
114	#107	—	106	1	no	—
115	#107	—	212	1	no	—
116	#106	—	212	1	no	—
117	C₃₂	—	53	2	no	—
118	C₃₂	C₂×#6157	1	2	yes	complemented
119	C₂²×C₄	—	53	1	no	—
120	C₂²×C₄	—	424	1	no	—
121	C₂×C₈	—	53	1	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	C₄⋊#6157	1	1	yes	characteristic
127	C₄⋊C₄	—	424	1	no	—
128	C₄⋊C₄	—	212	1	no	—
129	C₄⋊C₄	—	212	1	no	—
130	C₄⋊C₄	—	212	2	no	—
131	C₂²⋊C₄	—	424	1	no	—
132	C₄²	—	424	1	no	—
133	C₁₆	—	53	1	no	—
134	C₁₆	—	53	1	no	—
135	C₁₆	C₄⋊#6157	1	1	yes	characteristic
136	C₁₆	C₂²×#6157	1	1	yes	characteristic
137	C₂³	—	53	1	no	—
138	C₂×C₄	—	53	1	no	—
139	C₂×C₄	—	53	1	no	—
140	C₂×C₄	—	53	1	no	—
141	C₂×C₄	—	53	1	no	—
142	C₂×C₄	—	53	1	no	—
143	C₂×C₄	—	848	2	no	—
144	C₂×C₄	—	424	1	no	—
145	C₂×C₄	—	424	1	no	—
146	C₂×C₄	—	424	2	no	—
147	C₂×C₄	#13133286	1	1	yes	characteristic
148	C₈	—	53	1	no	—
149	C₈	—	53	1	no	—
150	C₈	#13133286	1	1	yes	characteristic
151	C₈	C₂×C₄⋊#6157	1	1	yes	characteristic
152	C₂²	—	53	1	no	—
153	C₂²	—	53	1	no	—
154	C₂²	—	106	1	no	—
155	C₂²	—	106	1	no	—
156	C₂²	#16811549	1	1	yes	characteristic
157	C₄	—	53	1	no	—
158	C₄	—	53	1	no	—
159	C₄	—	848	2	no	—
160	C₄	—	848	1	no	—
161	C₄	—	848	1	no	—
162	C₄	#16811549	1	1	yes	characteristic
163	C₄	C₂×#13133286	1	1	yes	characteristic
164	C₂	—	2	1	no	—
165	C₂	—	53	1	no	—
166	C₂	—	53	1	no	—
167	C₂	—	106	1	no	—
168	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	D₁₆
cosocle	C₁₆×D₅₃	C₂³
hypercenter	M₆₄	#6157
hypocenter	C₅₃	#13634
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. #20289877	C₂	semidirect
4. #20289876	C₂	semidirect
14. #16810457	C₄	semidirect
15. #16810456	C₄	semidirect
17. C₅₃×M₆₄	C₄	semidirect
31. #13133123	C₂×C₄	semidirect
33. C₃₂×C₅₃	C₂×C₄	semidirect
95. M₆₄	#6157	semidirect
102. C₅₃	#13634	semidirect
118. C₃₂	C₂×#6157	semidirect

Maximal subgroups

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

subgroup	index
#20291066	2
C₂×#16811549	2
#20289877	2
#20289876	2
M₆₄×D₅₃	2
#13634	53