Finite Group Databasebeta

Subgroups — #16811213

Order 3392.

250 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 95 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	#13133212	C₂	1	2	yes	maximal, complemented
2	#13133206	C₂	1	2	yes	maximal, complemented
3	#13133202	C₂	1	2	yes	maximal, complemented
4	#13133192	C₂	1	2	yes	maximal, complemented
5	#13133191	C₂	1	1	yes	maximal, characteristic, complemented
6	#13133187	C₂	1	2	yes	maximal, complemented
7	#13133177	C₂	1	2	yes	maximal, complemented
8	C₂×#11719921	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²⋊Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂³×D₅₃	C₂²	1	1	yes	characteristic
11	C₂×Q₈×C₅₃	C₂²	1	1	yes	characteristic, complemented
12	C₂²×C₄×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×C₂²⋊D₅₃	C₂²	1	2	yes	—
14	C₂²×Dic₅₃	C₂²	1	1	yes	characteristic
15	C₂×D₂₁₂	C₂²	1	2	yes	—
16	C₂×D₂₁₂	C₂²	1	2	yes	—
17	C₂×D₂₁₂	C₂²	1	1	yes	characteristic, complemented
18	C₂×C₄×D₅₃	C₂²	1	2	yes	complemented
19	C₄⋊C₄×C₅₃	C₂²	1	2	yes	complemented
20	C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic, complemented
21	C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
22	#11719921	C₂²	1	2	yes	—
23	#11719921	C₂²	1	2	yes	—
24	#11719921	C₂²	1	2	yes	—
25	#11719921	C₂²	1	2	yes	—
26	#11719921	C₂²	1	2	yes	complemented
27	#11719921	C₂²	1	2	yes	complemented
28	C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic, complemented
29	#11719919	C₂²	1	2	yes	complemented
30	C₄×Dic₅₃	C₂²	1	2	yes	complemented
31	C₂³×C₅₃	C₂³	1	1	yes	characteristic
32	C₂²×D₅₃	—	2	2	no	—
33	C₂²×D₅₃	—	4	1	no	—
34	C₂²×D₅₃	—	4	1	no	—
35	C₂²×D₅₃	C₂³	1	2	yes	—
36	C₂²×D₅₃	C₂³	1	2	yes	—
37	Q₈×C₅₃	—	4	1	no	—
38	C₂×C₄×C₅₃	—	4	1	no	—
39	C₂×C₄×C₅₃	C₂³	1	2	yes	—
40	C₂×C₄×C₅₃	C₂³	1	2	yes	—
41	C₂×C₄×C₅₃	C₂³	1	1	yes	characteristic
42	C₂×C₄×C₅₃	C₂³	1	1	yes	characteristic
43	C₂²⋊D₅₃	—	2	4	no	—
44	C₂×Dic₅₃	—	4	1	no	—
45	C₂×Dic₅₃	C₂³	1	2	yes	—
46	C₂×Dic₅₃	C₂³	1	2	yes	—
47	D₂₁₂	—	4	2	no	—
48	D₂₁₂	—	4	2	no	—
49	D₂₁₂	—	4	1	no	—
50	C₄×D₅₃	—	4	2	no	—
51	C₂²×C₅₃	—	2	1	no	—
52	C₂²×C₅₃	—	2	1	no	—
53	C₂²×C₅₃	#41	1	2	yes	—
54	C₂²×C₅₃	C₂⁴	1	1	yes	characteristic
55	C₂×D₅₃	—	2	2	no	—
56	C₂×D₅₃	—	2	2	no	—
57	C₂×D₅₃	—	2	2	no	—
58	C₂×D₅₃	—	2	2	no	—
59	C₂×D₅₃	—	2	2	no	—
60	C₂×D₅₃	—	2	2	no	—
61	C₂×D₅₃	—	4	4	no	—
62	C₄×C₅₃	—	2	2	no	—
63	C₄×C₅₃	—	2	1	no	—
64	C₄×C₅₃	—	2	2	no	—
65	C₄×C₅₃	—	2	1	no	—
66	Dic₅₃	—	2	2	no	—
67	Dic₅₃	—	2	2	no	—
68	C₂×C₅₃	—	2	2	no	—
69	C₂×C₅₃	C₂×#41	1	1	yes	characteristic
70	C₂×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
71	C₂×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
72	D₅₃	—	4	2	no	—
73	D₅₃	—	4	2	no	—
74	#538	—	53	1	no	maximal, Sylow
75	C₅₃	#538	1	1	yes	characteristic, complemented, Sylow
76	#126	—	53	2	no	—
77	#124	—	53	2	no	—
78	#123	—	53	2	no	—
79	C₂²⋊Q₈	D₅₃	1	1	yes	characteristic, complemented
80	#121	—	53	1	no	—
81	#121	—	53	2	no	—
82	#120	—	53	2	no	—
83	C₄×D₄	—	53	2	no	—
84	C₂×C₂²⋊C₄	—	53	1	no	—
85	C₂⁴	—	53	1	no	—
86	C₂×Q₈	C₂×D₅₃	1	1	yes	characteristic, complemented
87	C₂×D₄	—	53	2	no	—
88	C₂×D₄	—	53	2	no	—
89	C₂×D₄	—	53	2	no	—
90	C₂×D₄	—	53	1	no	—
91	C₂²×C₄	—	53	1	no	—
92	C₂²×C₄	—	53	2	no	—
93	C₂²×C₄	C₂×D₅₃	1	1	yes	characteristic
94	C₄⋊C₄	—	53	2	no	—
95	C₄⋊C₄	—	53	1	no	—
96	C₄⋊C₄	C₂×D₅₃	1	2	yes	complemented
97	C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic, complemented
98	C₂²⋊C₄	—	53	2	no	—
99	C₂²⋊C₄	—	53	2	no	—
100	C₂²⋊C₄	—	53	2	no	—
101	C₂²⋊C₄	—	53	2	no	—
102	C₂²⋊C₄	—	53	2	no	—
103	C₂²⋊C₄	—	53	2	no	—
104	C₂²⋊C₄	C₂×D₅₃	1	2	yes	—
105	C₄²	—	53	2	no	—
106	C₂³	—	53	2	no	—
107	C₂³	—	53	2	no	—
108	C₂³	—	106	2	no	—
109	C₂³	—	212	1	no	—
110	C₂³	—	212	1	no	—
111	C₂³	C₂²×D₅₃	1	1	yes	characteristic
112	Q₈	—	4	1	no	—
113	D₄	—	106	4	no	—
114	D₄	—	212	1	no	—
115	D₄	—	212	2	no	—
116	D₄	—	212	2	no	—
117	C₂×C₄	—	4	1	no	—
118	C₂×C₄	—	53	2	no	—
119	C₂×C₄	—	53	2	no	—
120	C₂×C₄	—	212	2	no	—
121	C₂×C₄	—	212	1	no	—
122	C₂×C₄	C₂²×D₅₃	1	2	yes	—
123	C₂×C₄	C₂²×D₅₃	1	1	yes	characteristic
124	C₂×C₄	C₂²×D₅₃	1	1	yes	characteristic
125	C₂×C₄	C₂²×D₅₃	1	2	yes	—
126	C₂²	—	2	1	no	—
127	C₂²	—	2	1	no	—
128	C₂²	—	106	2	no	—
129	C₂²	—	106	2	no	—
130	C₂²	—	106	2	no	—
131	C₂²	—	106	2	no	—
132	C₂²	—	106	2	no	—
133	C₂²	—	106	2	no	—
134	C₂²	—	212	4	no	—
135	C₂²	#11719949	1	2	yes	—
136	C₂²	C₂³×D₅₃	1	1	yes	characteristic
137	C₄	—	2	1	no	—
138	C₄	—	2	1	no	—
139	C₄	—	2	2	no	—
140	C₄	—	2	2	no	—
141	C₄	—	106	2	no	—
142	C₄	—	106	2	no	—
143	C₂	—	2	2	no	—
144	C₂	—	212	2	no	—
145	C₂	—	212	2	no	—
146	C₂	C₂×#11719949	1	1	yes	characteristic
147	C₂	#13133262	1	1	yes	characteristic
148	C₂	#13133263	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₅₃
commutator, socle, cosocle	C₂²×C₅₃	C₂⁴
Fitting	C₂²⋊Q₈×C₅₃	C₂
hypercenter	C₂²⋊Q₈	D₅₃
hypocenter	C₅₃	#538
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²⋊Q₈

Complements

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

N	H ≅ G/N	type
1. #13133212	C₂	semidirect
2. #13133206	C₂	semidirect
3. #13133202	C₂	semidirect
4. #13133192	C₂	semidirect
5. #13133191	C₂	semidirect
6. #13133187	C₂	semidirect
7. #13133177	C₂	semidirect
8. C₂×#11719921	C₂	semidirect
9. C₂²⋊Q₈×C₅₃	C₂	semidirect
11. C₂×Q₈×C₅₃	C₂²	semidirect
17. C₂×D₂₁₂	C₂²	semidirect
18. C₂×C₄×D₅₃	C₂²	semidirect
19. C₄⋊C₄×C₅₃	C₂²	semidirect
20. C₄⋊C₄×C₅₃	C₂²	semidirect
26. #11719921	C₂²	semidirect
27. #11719921	C₂²	semidirect
28. C₄⋊Dic₅₃	C₂²	semidirect
29. #11719919	C₂²	semidirect
30. C₄×Dic₅₃	C₂²	semidirect
75. C₅₃	#538	semidirect
79. C₂²⋊Q₈	D₅₃	semidirect
86. C₂×Q₈	C₂×D₅₃	semidirect
96. C₄⋊C₄	C₂×D₅₃	semidirect
97. C₄⋊C₄	C₂×D₅₃	semidirect

Maximal subgroups

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

subgroup	index
#13133212	2
#13133206	2
#13133202	2
#13133192	2
#13133191	2
#13133187	2
#13133177	2
C₂×#11719921	2
C₂²⋊Q₈×C₅₃	2
#538	53