Finite Group Databasebeta

Subgroups — #20291098

Order 6784.

164 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 164 conjugacy classes of subgroups — too many to draw. Only its 64 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	C₂×#13133292	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₅₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄⋊#11719910	C₂	1	1	yes	maximal, characteristic, complemented
4	#16811518	C₂	1	1	yes	maximal, characteristic, complemented
5	#16811518	C₂	1	1	yes	maximal, characteristic, complemented
6	#16811511	C₂	1	1	yes	maximal, characteristic, complemented
7	#16810688	C₂	1	1	yes	maximal, characteristic
8	#13133292	—	2	2	no	—
9	D₅₃⋊C₁₆	—	2	2	no	—
10	C₂×#11719910	C₂²	1	1	yes	characteristic
11	C₂×#11719910	C₂²	1	1	yes	characteristic
12	C₂×#11719910	C₂²	1	1	yes	characteristic
13	C₂×#11719910	C₂²	1	1	yes	characteristic
14	C₂×C₈×D₅₃	C₂²	1	1	yes	characteristic
15	C₂×C₈×D₅₃	C₂²	1	1	yes	characteristic
16	C₄×D₂₁₂	C₄	1	1	yes	characteristic
17	#13133129	C₄	1	1	yes	characteristic
18	#13133129	C₄	1	1	yes	characteristic
19	C₄×#88322	C₂²	1	1	yes	characteristic
20	C₄⋊C₈×C₅₃	C₄	1	1	yes	characteristic
21	C₂×D₂₁₂	C₈	1	1	yes	characteristic
22	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
26	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
27	#11719921	C₈	1	1	yes	characteristic
28	#11719921	C₈	1	1	yes	characteristic
29	C₄⋊Dic₅₃	C₈	1	1	yes	characteristic
30	C₂×#88322	C₂×C₄	1	1	yes	characteristic
31	C₂×#88322	C₂³	1	1	yes	characteristic
32	C₈×D₅₃	—	2	1	no	—
33	C₈×D₅₃	—	2	1	no	—
34	C₈×D₅₃	—	2	1	no	—
35	C₈×D₅₃	—	2	1	no	—
36	#11719910	—	2	1	no	—
37	#11719910	—	2	1	no	—
38	#11719910	—	2	1	no	—
39	#11719910	D₄	1	2	yes	complemented
40	C₂²×D₅₃	C₂×C₈	1	1	yes	characteristic
41	C₂²×D₅₃	C₂×C₈	1	1	yes	characteristic
42	C₂×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
43	C₂×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
44	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
45	C₂×Dic₅₃	C₂×C₈	1	1	yes	characteristic
46	C₂×Dic₅₃	C₂×C₈	1	1	yes	characteristic
47	D₂₁₂	—	2	2	no	—
48	C₄×D₅₃	—	2	1	no	—
49	C₄×D₅₃	—	2	1	no	—
50	C₄×D₅₃	—	2	1	no	—
51	C₄×D₅₃	—	2	1	no	—
52	C₈×C₅₃	—	2	1	no	—
53	C₈×C₅₃	—	2	1	no	—
54	#88322	—	2	1	no	—
55	#88322	C₂×D₄	1	1	yes	characteristic
56	#88322	#41	1	1	yes	characteristic
57	C₂²×C₅₃	C₂²×C₈	1	1	yes	characteristic
58	C₂×D₅₃	—	2	1	no	—
59	C₂×D₅₃	—	2	1	no	—
60	C₂×D₅₃	—	2	1	no	—
61	C₂×D₅₃	—	2	1	no	—
62	C₂×D₅₃	—	2	1	no	—
63	C₂×D₅₃	M₃₂	1	2	yes	complemented
64	C₄×C₅₃	—	2	1	no	—
65	C₄×C₅₃	—	2	1	no	—
66	C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
67	C₄×C₅₃	#131	1	1	yes	characteristic
68	Dic₅₃	—	2	1	no	—
69	Dic₅₃	—	2	1	no	—
70	#2168	—	53	1	no	maximal, Sylow
71	C₂×C₅₃	C₈×D₄	1	1	yes	characteristic
72	C₂×C₅₃	C₂×M₃₂	1	1	yes	characteristic
73	C₂×C₅₃	#504	1	1	yes	characteristic
74	D₅₃	—	2	2	no	—
75	D₅₃	—	4	1	no	—
76	C₂×M₃₂	—	53	1	no	—
77	C₂²×C₁₆	—	53	1	no	—
78	C₈×D₄	—	53	1	no	—
79	C₄⋊C₁₆	—	53	1	no	—
80	C₂²⋊C₁₆	—	53	1	no	—
81	C₂²⋊C₁₆	—	53	1	no	—
82	#346	—	53	1	no	—
83	C₅₃	#2168	1	1	yes	characteristic, complemented, Sylow
84	C₂²×C₈	—	53	1	no	—
85	C₂²×C₈	—	53	1	no	—
86	C₄×D₄	—	53	1	no	—
87	M₃₂	—	106	2	no	—
88	C₂×C₁₆	—	53	1	no	—
89	C₂×C₁₆	—	53	1	no	—
90	C₂×C₁₆	—	53	1	no	—
91	C₂×C₁₆	—	53	1	no	—
92	C₂×C₁₆	—	106	2	no	—
93	C₄⋊C₈	#6157	1	1	yes	characteristic
94	C₂²⋊C₈	—	53	1	no	—
95	C₂²⋊C₈	—	53	1	no	—
96	C₄×C₈	—	53	1	no	—
97	C₂×D₄	—	53	1	no	—
98	C₂²×C₄	—	53	1	no	—
99	C₂²×C₄	—	53	1	no	—
100	C₂×C₈	—	53	1	no	—
101	C₂×C₈	—	53	1	no	—
102	C₂×C₈	—	106	1	no	—
103	C₂×C₈	—	106	1	no	—
104	C₂×C₈	—	106	1	no	—
105	C₂×C₈	—	106	1	no	—
106	C₂×C₈	C₂×#6157	1	1	yes	characteristic
107	C₂×C₈	C₂×#6157	1	1	yes	characteristic
108	C₄⋊C₄	—	53	1	no	—
109	C₂²⋊C₄	—	53	1	no	—
110	C₂²⋊C₄	—	53	1	no	—
111	C₄²	C₂×#6157	1	1	yes	characteristic
112	C₁₆	—	53	2	no	—
113	C₁₆	—	106	1	no	—
114	C₁₆	—	106	1	no	—
115	C₁₆	—	106	1	no	—
116	C₂³	—	53	1	no	—
117	C₂³	—	53	1	no	—
118	D₄	—	106	2	no	—
119	C₂×C₄	—	53	1	no	—
120	C₂×C₄	—	53	1	no	—
121	C₂×C₄	—	106	1	no	—
122	C₂×C₄	—	106	1	no	—
123	C₂×C₄	—	106	1	no	—
124	C₂×C₄	—	106	1	no	—
125	C₂×C₄	D₅₃⋊C₈	1	1	yes	characteristic
126	C₂×C₄	D₅₃⋊C₈	1	1	yes	characteristic
127	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
128	C₈	—	2	1	no	—
129	C₈	—	2	1	no	—
130	C₈	—	53	1	no	—
131	C₈	—	53	1	no	—
132	C₈	—	106	1	no	—
133	C₂²	—	53	2	no	—
134	C₂²	—	106	1	no	—
135	C₂²	—	106	1	no	—
136	C₂²	—	106	1	no	—
137	C₂²	—	106	1	no	—
138	C₂²	—	106	1	no	—
139	C₂²	C₂×D₅₃⋊C₈	1	1	yes	characteristic
140	C₄	—	2	1	no	—
141	C₄	—	2	1	no	—
142	C₄	—	106	1	no	—
143	C₄	—	106	1	no	—
144	C₄	D₄×#6157	1	1	yes	characteristic
145	C₄	#13133305	1	1	yes	characteristic
146	C₂	—	106	2	no	—
147	C₂	—	212	1	no	—
148	C₂	#16811651	1	1	yes	characteristic
149	C₂	D₅₃⋊M₃₂	1	1	yes	characteristic
150	C₂	#16811682	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×#6157
commutator, socle	C₂²×C₅₃	C₂²×C₈
Fitting	C₄⋊C₈×C₅₃	C₄
hypercenter	C₄⋊C₈	#6157
hypocenter	C₅₃	#2168
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
1. C₂×#13133292	C₂	semidirect
2. C₂×D₅₃⋊C₁₆	C₂	semidirect
3. C₄⋊#11719910	C₂	semidirect
4. #16811518	C₂	semidirect
5. #16811518	C₂	semidirect
6. #16811511	C₂	semidirect
39. #11719910	D₄	semidirect
63. C₂×D₅₃	M₃₂	semidirect
83. C₅₃	#2168	semidirect