Finite Group Databasebeta

Subgroups — #20290299

Order 6784.

180 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 66 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	Q₈×#88324	C₂	1	1	yes	maximal, characteristic, complemented
2	#16811651	C₂	1	1	yes	maximal, characteristic
3	#16811485	C₂	1	1	yes	maximal, characteristic, complemented
4	Q₈⋊#88324	C₂	1	1	yes	maximal, characteristic, complemented
5	#16811471	C₂	1	1	yes	maximal, characteristic
6	C₈×#88324	C₂	1	1	yes	maximal, characteristic, complemented
7	#16810708	C₂	1	1	yes	maximal, characteristic
8	C₂×D₅₃⋊C₈	—	2	1	no	—
9	C₄⋊#88324	—	2	1	no	—
10	C₄⋊#88324	C₂²	1	1	yes	characteristic
11	C₄⋊#88324	C₂²	1	1	yes	characteristic
12	#13133274	—	2	1	no	—
13	C₄×#88324	—	2	1	no	—
14	C₄×#88324	C₂²	1	1	yes	characteristic
15	C₄×#88324	C₂²	1	1	yes	characteristic
16	C₂×#11719924	C₄	1	1	yes	characteristic
17	Q₈×Dic₅₃	C₂²	1	1	yes	characteristic
18	#13133179	C₂²	1	1	yes	characteristic
19	#13133148	C₄	1	1	yes	characteristic
20	#13133139	C₄	1	1	yes	characteristic
21	C₈×Dic₅₃	C₂²	1	1	yes	characteristic
22	#103×C₅₃	C₄	1	1	yes	characteristic
23	C₂×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂×D₂₁₂	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×D₅₃	—	2	1	no	—
26	C₂×#88324	—	2	1	no	—
27	C₂×#88324	—	2	1	no	—
28	C₂×#88324	—	2	1	no	—
29	C₂×#88324	D₄	1	1	yes	characteristic
30	C₂×#88324	D₄	1	1	yes	characteristic
31	D₅₃⋊C₈	—	4	2	no	—
32	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
34	#11719924	C₈	1	4	yes	complemented
35	#11719921	—	2	1	no	—
36	C₄⋊Dic₅₃	—	2	1	no	—
37	C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
38	C₄×Dic₅₃	—	2	1	no	—
39	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
40	C₂×#88322	C₂×C₄	1	1	yes	characteristic
41	C₂²×D₅₃	—	2	1	no	—
42	Q₈×C₅₃	—	2	1	no	—
43	Q₈×C₅₃	C₂×C₈	1	2	yes	complemented
44	C₂×C₄×C₅₃	—	2	1	no	—
45	C₂×C₄×C₅₃	—	2	1	no	—
46	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
47	C₂×Dic₅₃	—	2	1	no	—
48	C₂×Dic₅₃	C₂×D₄	1	1	yes	characteristic
49	C₂×Dic₅₃	#41	1	1	yes	characteristic
50	D₂₁₂	—	2	1	no	—
51	D₂₁₂	C₂×C₈	1	2	yes	—
52	C₄×D₅₃	—	4	1	no	—
53	C₄×D₅₃	—	4	1	no	—
54	#88324	—	2	2	no	—
55	#88324	—	2	1	no	—
56	#88324	—	4	1	no	—
57	#88324	—	4	1	no	—
58	#88324	SD₁₆	1	2	yes	complemented
59	C₈×C₅₃	—	2	1	no	—
60	#88322	C₂×C₈	1	2	yes	complemented
61	C₂²×C₅₃	C₄×D₄	1	1	yes	characteristic
62	C₂×D₅₃	—	2	2	no	—
63	C₂×D₅₃	—	4	1	no	—
64	C₂×D₅₃	—	4	1	no	—
65	C₄×C₅₃	—	2	2	no	—
66	C₄×C₅₃	—	4	1	no	—
67	C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
68	C₄×C₅₃	#131	1	1	yes	characteristic
69	Dic₅₃	—	2	1	no	—
70	Dic₅₃	—	4	1	no	—
71	Dic₅₃	C₂×SD₁₆	1	1	yes	characteristic
72	Dic₅₃	#135	1	1	yes	characteristic
73	C₈×SD₁₆	—	53	1	no	maximal, Sylow
74	C₂×C₅₃	C₈×D₄	1	1	yes	characteristic
75	C₂×C₅₃	C₄×SD₁₆	1	1	yes	characteristic
76	C₂×C₅₃	#443	1	1	yes	characteristic
77	D₅₃	—	4	2	no	—
78	C₈×Q₈	—	53	1	no	—
79	C₄×SD₁₆	—	53	1	no	—
80	C₈×D₄	—	53	1	no	—
81	#334	—	53	1	no	—
82	Q₈⋊C₈	—	53	1	no	—
83	D₄⋊C₈	—	53	1	no	—
84	C₈²	—	53	1	no	—
85	C₅₃	C₈×SD₁₆	1	1	yes	characteristic, complemented, Sylow
86	C₂×SD₁₆	—	53	1	no	—
87	C₂²×C₈	—	106	1	no	—
88	C₄×Q₈	—	53	1	no	—
89	C₄×D₄	—	53	1	no	—
90	#106	—	53	1	no	—
91	C₄⋊C₈	—	53	1	no	—
92	C₄⋊C₈	—	53	1	no	—
93	C₄⋊C₈	—	106	1	no	—
94	#103	#6157	1	1	yes	characteristic
95	#102	—	53	1	no	—
96	C₂²⋊C₈	—	106	1	no	—
97	C₄×C₈	—	53	1	no	—
98	C₄×C₈	—	53	1	no	—
99	C₄×C₈	—	53	1	no	—
100	C₄×C₈	—	106	1	no	—
101	C₂×Q₈	C₂×#6157	1	1	yes	characteristic
102	C₂×D₄	—	53	1	no	—
103	C₂²×C₄	—	106	1	no	—
104	SD₁₆	—	53	4	no	—
105	C₂×C₈	—	53	1	no	—
106	C₂×C₈	—	53	1	no	—
107	C₂×C₈	—	53	1	no	—
108	C₂×C₈	—	106	1	no	—
109	C₂×C₈	—	106	1	no	—
110	C₂×C₈	—	106	1	no	—
111	C₂×C₈	—	212	2	no	—
112	C₂×C₈	C₂×#6157	1	1	yes	characteristic
113	C₄⋊C₄	—	53	1	no	—
114	C₄⋊C₄	—	106	1	no	—
115	C₄⋊C₄	C₂×#6157	1	1	yes	characteristic
116	C₂²⋊C₄	—	106	1	no	—
117	C₄²	—	53	1	no	—
118	C₄²	—	106	1	no	—
119	C₂³	—	106	1	no	—
120	Q₈	—	2	1	no	—
121	Q₈	D₅₃⋊C₈	1	2	yes	complemented
122	D₄	—	53	2	no	—
123	D₄	—	106	1	no	—
124	C₂×C₄	—	2	1	no	—
125	C₂×C₄	—	2	1	no	—
126	C₂×C₄	—	53	1	no	—
127	C₂×C₄	—	53	1	no	—
128	C₂×C₄	—	106	1	no	—
129	C₂×C₄	—	212	1	no	—
130	C₂×C₄	—	212	1	no	—
131	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
132	C₈	—	2	1	no	—
133	C₈	—	53	2	no	—
134	C₈	—	53	2	no	—
135	C₈	—	106	1	no	—
136	C₈	—	106	2	no	—
137	C₈	—	212	1	no	—
138	C₈	—	212	1	no	—
139	C₂²	—	106	2	no	—
140	C₂²	—	212	1	no	—
141	C₂²	—	212	1	no	—
142	C₂²	D₄×#6157	1	1	yes	characteristic
143	C₄	—	2	2	no	—
144	C₄	—	4	1	no	—
145	C₄	—	53	1	no	—
146	C₄	—	53	1	no	—
147	C₄	—	106	1	no	—
148	C₄	—	212	1	no	—
149	C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
150	C₄	#13133304	1	1	yes	characteristic
151	C₂	—	212	2	no	—
152	C₂	#16811650	1	1	yes	characteristic
153	C₂	SD₁₆×#6157	1	1	yes	characteristic
154	C₂	#16811667	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×#6157
commutator	C₄×C₅₃	C₂²×C₈
Frattini	C₂×C₄	C₂²×#6157
Fitting	#103×C₅₃	C₄
socle	C₂²×C₅₃	C₄×D₄
hypercenter	#103	#6157
hypocenter	C₅₃	C₈×SD₁₆
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #103

Complements

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

N	H ≅ G/N	type
1. Q₈×#88324	C₂	semidirect
3. #16811485	C₂	semidirect
4. Q₈⋊#88324	C₂	semidirect
6. C₈×#88324	C₂	semidirect
34. #11719924	C₈	semidirect
43. Q₈×C₅₃	C₂×C₈	semidirect
58. #88324	SD₁₆	semidirect
60. #88322	C₂×C₈	semidirect
85. C₅₃	C₈×SD₁₆	semidirect
121. Q₈	D₅₃⋊C₈	semidirect