Finite Group Databasebeta

Subgroups — #20291037

Order 6784.

224 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 62 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	D₅₃⋊M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
2	#16811633	C₂	1	1	yes	maximal, characteristic, complemented
3	#16811540	C₂	1	4	yes	maximal, complemented
4	C₂×#13133240	C₂	1	1	yes	maximal, characteristic
5	#13133296	—	2	1	no	—
6	#13133292	C₂²	1	2	yes	complemented
7	#13133291	—	2	1	no	—
8	D₅₃⋊C₁₆	—	2	1	no	—
9	#13133286	—	2	1	no	—
10	#13133285	C₂²	1	2	yes	complemented
11	C₂×#11719949	—	2	1	no	—
12	#13133240	—	2	1	no	—
13	#13133240	—	2	1	no	—
14	#13133240	C₄	1	2	yes	complemented
15	#13133240	C₂²	1	2	yes	—
16	C₂×#11719924	—	2	1	no	—
17	C₂×D₄₂₄	C₄	1	1	yes	characteristic, complemented
18	C₂×C₈×D₅₃	C₂²	1	1	yes	characteristic
19	C₂×Q₁₆×C₅₃	C₄	1	1	yes	characteristic, complemented
20	C₂×Q₈×C₅₃	—	2	1	no	—
21	#11719949	—	2	2	no	—
22	#11719949	—	2	2	no	—
23	#11719949	—	4	1	no	—
24	#11719949	—	4	1	no	—
25	C₂×D₂₁₂	—	2	1	no	—
26	C₂×D₂₁₂	—	4	1	no	—
27	C₂×C₄×D₅₃	—	4	1	no	—
28	C₂×C₄×D₅₃	D₄	1	1	yes	characteristic
29	C₂²⋊#6157	—	4	1	no	—
30	C₄⋊#6157	—	2	2	no	—
31	C₄×#6157	—	4	1	no	—
32	Q₁₆×C₅₃	—	2	1	no	—
33	Q₁₆×C₅₃	C₂×C₄	1	2	yes	—
34	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
35	#11719924	—	2	2	no	—
36	#11719924	—	2	2	no	—
37	C₂×#88322	D₄	1	1	yes	characteristic
38	D₄₂₄	—	2	1	no	—
39	D₄₂₄	C₂×C₄	1	2	yes	—
40	C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
41	C₈×D₅₃	D₄	1	1	yes	characteristic
42	C₈×D₅₃	D₄	1	1	yes	characteristic
43	C₈×D₅₃	C₂³	1	1	yes	characteristic
44	#11719910	—	2	2	no	—
45	C₂²×D₅₃	—	4	1	no	—
46	C₂²×D₅₃	D₈	1	1	yes	characteristic
47	C₂×#6157	—	4	2	no	—
48	Q₈×C₅₃	—	2	2	no	—
49	Q₈×C₅₃	—	4	1	no	—
50	C₂×C₄×C₅₃	—	4	1	no	—
51	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
52	C₂×Dic₅₃	SD₁₆	1	1	yes	characteristic
53	D₂₁₂	—	2	2	no	—
54	D₂₁₂	—	4	2	no	—
55	D₂₁₂	—	4	2	no	—
56	D₂₁₂	—	4	1	no	—
57	C₄×D₅₃	—	4	2	no	—
58	C₄×D₅₃	—	4	2	no	—
59	C₄×D₅₃	C₂²⋊C₄	1	1	yes	characteristic
60	C₄×D₅₃	D₈	1	1	yes	characteristic
61	C₄×D₅₃	SD₁₆	1	1	yes	characteristic
62	C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
63	C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
64	C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
65	#88322	C₂²⋊C₄	1	1	yes	characteristic
66	#88322	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₅₃	#102	1	1	yes	characteristic
68	C₂×D₅₃	—	4	1	no	—
69	C₂×D₅₃	—	4	2	no	—
70	C₂×D₅₃	—	8	2	no	—
71	C₂×D₅₃	#102	1	1	yes	characteristic
72	C₂×D₅₃	C₂×D₈	1	1	yes	characteristic
73	#6157	—	8	2	no	—
74	C₄×C₅₃	—	4	2	no	—
75	C₄×C₅₃	#102	1	1	yes	characteristic
76	C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	Dic₅₃	#102	1	1	yes	characteristic
78	Dic₅₃	C₂×SD₁₆	1	1	yes	characteristic
79	#2145	—	53	1	no	maximal, Sylow
80	C₂×C₅₃	—	2	1	no	—
81	C₂×C₅₃	C₂×#102	1	1	yes	characteristic
82	D₅₃	—	2	1	no	—
83	D₅₃	—	2	1	no	—
84	D₅₃	—	8	2	no	—
85	C₂×#135	—	53	1	no	—
86	C₂×M₃₂	—	53	1	no	—
87	#427	—	53	1	no	—
88	#360	—	53	4	no	—
89	C₅₃	#2145	1	1	yes	characteristic, complemented, Sylow
90	C₂×#41	—	106	1	no	—
91	#135	—	53	2	no	—
92	#135	—	53	2	no	—
93	#135	—	106	1	no	—
94	#135	—	106	1	no	—
95	C₂×Q₁₆	#6157	1	1	yes	characteristic, complemented
96	C₂×SD₁₆	—	106	1	no	—
97	C₂×D₈	—	53	1	no	—
98	C₂²×C₈	—	53	1	no	—
99	#117	—	106	1	no	—
100	M₃₂	—	53	2	no	—
101	M₃₂	—	106	1	no	—
102	C₂×C₁₆	—	106	1	no	—
103	#107	—	106	1	no	—
104	#106	—	53	2	no	—
105	#41	—	106	2	no	—
106	#41	—	106	2	no	—
107	#41	—	212	1	no	—
108	#41	—	212	1	no	—
109	C₂×Q₈	—	2	1	no	—
110	C₂×D₄	—	106	1	no	—
111	C₂×D₄	—	212	1	no	—
112	C₂²×C₄	—	53	1	no	—
113	C₂²×C₄	—	212	1	no	—
114	Q₁₆	—	2	1	no	—
115	Q₁₆	C₂×#6157	1	2	yes	—
116	SD₁₆	—	106	2	no	—
117	SD₁₆	—	106	2	no	—
118	D₈	—	53	2	no	—
119	D₈	—	106	1	no	—
120	C₂×C₈	—	53	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	C₂×#6157	1	1	yes	characteristic
126	C₄⋊C₄	—	106	2	no	—
127	C₂²⋊C₄	—	212	1	no	—
128	C₄²	—	212	1	no	—
129	C₁₆	—	106	2	no	—
130	C₂³	—	53	1	no	—
131	C₂³	—	212	1	no	—
132	Q₈	—	2	2	no	—
133	Q₈	—	4	1	no	—
134	D₄	—	106	2	no	—
135	D₄	—	212	2	no	—
136	D₄	—	212	2	no	—
137	D₄	—	212	1	no	—
138	C₂×C₄	—	4	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	—	53	1	no	—
144	C₂×C₄	—	212	2	no	—
145	C₂×C₄	—	212	2	no	—
146	C₂×C₄	—	212	2	no	—
147	C₂×C₄	C₂²⋊#6157	1	1	yes	characteristic
148	C₈	—	53	1	no	—
149	C₈	—	53	1	no	—
150	C₈	C₂²⋊#6157	1	1	yes	characteristic
151	C₈	C₂²×#6157	1	1	yes	characteristic
152	C₂²	—	53	1	no	—
153	C₂²	—	53	1	no	—
154	C₂²	—	424	2	no	—
155	C₂²	—	212	1	no	—
156	C₂²	—	212	2	no	—
157	C₂²	#13133279	1	1	yes	characteristic
158	C₄	—	4	2	no	—
159	C₄	—	53	1	no	—
160	C₄	—	53	1	no	—
161	C₄	—	424	2	no	—
162	C₄	#13133279	1	1	yes	characteristic
163	C₄	C₂×C₂²⋊#6157	1	1	yes	characteristic
164	C₂	—	2	1	no	—
165	C₂	—	106	1	no	—
166	C₂	—	106	1	no	—
167	C₂	—	424	2	no	—
168	C₂	C₂×#13133279	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#13133279
commutator	C₈×C₅₃	C₂²×C₄
Frattini	C₈	C₂²×#6157
Fitting	C₂×Q₁₆×C₅₃	C₄
socle	C₂×C₅₃	C₂×#102
hypercenter	C₂×Q₁₆	#6157
hypocenter	C₅₃	#2145
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₁₆

Complements

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

N	H ≅ G/N	type
1. D₅₃⋊M₃₂	C₂	semidirect
2. #16811633	C₂	semidirect
3. #16811540	C₂	semidirect
6. #13133292	C₂²	semidirect
10. #13133285	C₂²	semidirect
14. #13133240	C₄	semidirect
17. C₂×D₄₂₄	C₄	semidirect
19. C₂×Q₁₆×C₅₃	C₄	semidirect
89. C₅₃	#2145	semidirect
95. C₂×Q₁₆	#6157	semidirect