Finite Group Databasebeta

Subgroups — #28328554

Order 13568.

444 conjugacy classes of subgroups; only the 250 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 250 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	#20291130	C₂	1	8	yes	maximal, complemented
2	#20291080	C₂	1	4	yes	maximal
3	#20291076	C₂	1	2	yes	maximal
4	#20283650	C₂	1	1	yes	maximal, characteristic
5	C₄⋊#11719910	C₂²	1	4	yes	—
6	C₄⋊#11719910	C₂²	1	8	yes	—
7	C₂²⋊#11719910	C₂²	1	4	yes	—
8	#16811511	C₂²	1	4	yes	—
9	#16811511	C₂²	1	4	yes	—
10	C₄×#11719910	C₂²	1	4	yes	—
11	#16810754	C₄	1	4	yes	—
12	Q₈×#88322	C₂²	1	4	yes	—
13	#16810650	C₄	1	2	yes	—
14	#16810647	C₄	1	1	yes	characteristic
15	#16810579	C₂²	1	2	yes	—
16	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
17	C₅₃×#533	C₄	1	1	yes	characteristic
18	C₂×#11719910	C₂³	1	8	yes	—
19	C₂×#11719917	C₂×C₄	1	2	yes	—
20	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
21	C₂²×#88322	C₂³	1	1	yes	characteristic
22	C₄⋊#88322	C₂×C₄	1	4	yes	—
23	C₄⋊#88322	C₂×C₄	1	8	yes	—
24	#13133128	C₂×C₄	1	4	yes	—
25	C₄×#88322	C₂×C₄	1	2	yes	—
26	C₄×#88322	C₂³	1	2	yes	—
27	C₄×#88322	C₂³	1	4	yes	—
28	C₄⋊Q₈×C₅₃	C₈	1	2	yes	—
29	#125×C₅₃	C₈	1	2	yes	—
30	C₂²⋊Q₈×C₅₃	C₈	1	4	yes	—
31	C₄×Q₈×C₅₃	C₂×C₄	1	4	yes	—
32	#117×C₅₃	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂×Q₈×C₅₃	C₂×C₈	1	4	yes	—
35	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
36	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
37	C₄⋊C₄×C₅₃	C₂×C₈	1	4	yes	—
38	C₄⋊C₄×C₅₃	C₂×C₈	1	8	yes	—
39	C₄⋊C₄×C₅₃	C₂×C₈	1	4	yes	—
40	C₂²⋊C₄×C₅₃	C₂×C₈	1	4	yes	—
41	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
42	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
43	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
44	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
45	#11719917	—	4	2	no	—
46	C₂×#88322	—	2	2	no	—
47	C₂×#88322	—	4	1	no	—
48	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
49	C₂×#88322	C₂²×C₄	1	2	yes	—
50	C₂×#88322	C₂²×C₄	1	4	yes	—
51	C₂×#88322	C₂⁴	1	1	yes	characteristic
52	#11719910	—	2	8	no	—
53	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
54	Q₈×C₅₃	—	4	4	no	—
55	C₂×C₄×C₅₃	—	2	1	no	—
56	C₂×C₄×C₅₃	—	2	1	no	—
57	C₂×C₄×C₅₃	—	4	2	no	—
58	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
59	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
60	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
61	C₂×C₄×C₅₃	C₂²×C₈	1	8	yes	—
62	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
63	#88322	—	2	1	no	—
64	#88322	—	2	2	no	—
65	#88322	—	2	4	no	—
66	#88322	2₋¹⁺⁴	1	2	yes	—
67	#33501	—	53	1	no	maximal, Sylow
68	C₂²×C₅₃	—	2	2	no	—
69	C₂²×C₅₃	—	2	1	no	—
70	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
71	C₄×C₅₃	—	2	2	no	—
72	C₄×C₅₃	—	2	2	no	—
73	C₄×C₅₃	—	2	8	no	—
74	C₄×C₅₃	—	2	1	no	—
75	C₄×C₅₃	#519	1	1	yes	characteristic
76	C₄×C₅₃	#568	1	1	yes	characteristic
77	#2960	—	53	1	no	—
78	#2183	—	53	8	no	—
79	#2164	—	53	4	no	—
80	#2163	—	53	2	no	—
81	C₂×C₅₃	—	4	1	no	—
82	C₂×C₅₃	#2982	1	1	yes	characteristic
83	C₂×C₅₃	#3407	1	2	yes	—
84	#533	#6157	1	1	yes	characteristic
85	#446	—	53	4	no	—
86	C₈×Q₈	—	53	4	no	—
87	#424	—	53	2	no	—
88	#423	—	53	1	no	—
89	#405	—	53	2	no	—
90	C₂×C₄×C₈	—	53	1	no	—
91	C₄⋊C₁₆	—	53	4	no	—
92	C₄⋊C₁₆	—	53	8	no	—
93	C₂²⋊C₁₆	—	53	4	no	—
94	#346	—	53	4	no	—
95	#346	—	53	4	no	—
96	C₄×C₁₆	—	53	4	no	—
97	C₅₃	#33501	1	1	yes	characteristic, complemented, Sylow
98	C₂×M₁₆	—	53	2	no	—
99	C₂²×C₈	—	53	1	no	—
100	C₂²×C₈	—	53	1	no	—
101	C₄⋊Q₈	#88324	1	2	yes	—
102	#125	#88324	1	2	yes	—
103	C₂²⋊Q₈	#88324	1	4	yes	—
104	C₄×Q₈	C₂×#6157	1	4	yes	—
105	#117	C₂×#6157	1	2	yes	—
106	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
107	C₂×C₁₆	—	53	8	no	—
108	C₄⋊C₈	—	53	4	no	—
109	C₄⋊C₈	—	53	8	no	—
110	#97	—	53	4	no	—
111	C₄×C₈	—	53	2	no	—
112	C₄×C₈	—	53	4	no	—
113	C₄×C₈	—	53	2	no	—
114	C₂×Q₈	C₂×#88324	1	4	yes	—
115	C₂²×C₄	C₂×#88324	1	2	yes	—
116	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
117	M₁₆	—	212	2	no	—
118	C₂×C₈	—	53	1	no	—
119	C₂×C₈	—	53	1	no	—
120	C₂×C₈	—	53	2	no	—
121	C₂×C₈	—	53	4	no	—
122	C₂×C₈	—	106	2	no	—
123	C₂×C₈	—	212	1	no	—
124	C₄⋊C₄	C₂×#88324	1	4	yes	—
125	C₄⋊C₄	C₂×#88324	1	8	yes	—
126	C₄⋊C₄	C₂×#88324	1	4	yes	—
127	C₂²⋊C₄	C₂×#88324	1	4	yes	—
128	C₄²	C₂×#88324	1	1	yes	characteristic
129	C₄²	C₂×#88324	1	1	yes	characteristic
130	C₄²	C₂²×#6157	1	2	yes	—
131	C₄²	C₂²×#6157	1	4	yes	—
132	C₁₆	—	106	8	no	—
133	C₂³	C₂²×#88324	1	1	yes	characteristic
134	Q₈	—	4	4	no	—
135	C₂×C₄	—	2	1	no	—
136	C₂×C₄	—	2	1	no	—
137	C₂×C₄	—	4	2	no	—
138	C₂×C₄	C₂²×#88324	1	1	yes	characteristic
139	C₂×C₄	C₂²×#88324	1	2	yes	—
140	C₂×C₄	C₂²×#88324	1	2	yes	—
141	C₂×C₄	C₂²×#88324	1	8	yes	—
142	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
143	C₈	—	53	2	no	—
144	C₈	—	106	1	no	—
145	C₈	—	106	2	no	—
146	C₈	—	106	4	no	—
147	C₂²	—	2	1	no	—
148	C₂²	—	2	2	no	—
149	C₂²	C₂³×#88324	1	1	yes	characteristic
150	C₄	—	2	1	no	—
151	C₄	—	2	2	no	—
152	C₄	—	2	2	no	—
153	C₄	—	2	8	no	—
154	C₄	#16811691	1	1	yes	characteristic
155	C₄	#16811703	1	1	yes	characteristic
156	C₂	—	4	1	no	—
157	C₂	#20291608	1	1	yes	characteristic
158	C₂	#20291682	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#6157
commutator, socle	C₂²×C₅₃	C₂³×C₈
Fitting	C₅₃×#533	C₄
prime radical	C₂³×C₅₃	C₂²×C₈
cosocle	C₂×#88322	C₂⁴
hypercenter	#533	#6157
hypocenter	C₅₃	#33501
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ #533

Complements

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

N	H ≅ G/N	type
1. #20291130	C₂	semidirect
97. C₅₃	#33501	semidirect

Maximal subgroups

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

subgroup	index
#20291130	2
#20291080	2
#20291076	2
#20283650	2
#33501	53