Finite Group Databasebeta

Subgroups — #16811247

Order 3392.

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

normal subgroups only

This group has 216 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	#13133205	C₂	1	1	yes	maximal, characteristic, complemented
4	#13133202	C₂	1	2	yes	maximal, complemented
5	#13133197	C₂	1	1	yes	maximal, characteristic, complemented
6	#13133193	C₂	1	2	yes	maximal, complemented
7	#13133192	C₂	1	1	yes	maximal, characteristic
8	#13133182	C₂	1	1	yes	maximal, characteristic, complemented
9	#13133179	C₂	1	1	yes	maximal, characteristic, complemented
10	#13133174	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄⋊Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₈×C₅₃	C₂²	1	2	yes	—
13	C₂×D₂₁₂	C₂²	1	1	yes	characteristic
14	C₂×D₂₁₂	C₂²	1	2	yes	—
15	C₂×C₄×D₅₃	C₂²	1	1	yes	characteristic
16	C₂×C₄×D₅₃	C₂²	1	2	yes	—
17	C₂×Dic₁₀₆	C₂²	1	1	yes	characteristic
18	C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
19	C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
20	C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
21	C₄²×C₅₃	C₂²	1	1	yes	characteristic
22	#11719921	C₂²	1	1	yes	characteristic
23	#11719921	C₂²	1	1	yes	characteristic
24	#11719921	C₂²	1	2	yes	—
25	#11719921	C₂²	1	2	yes	—
26	#11719921	C₂²	1	2	yes	—
27	#11719921	C₂²	1	2	yes	—
28	#11719921	C₂²	1	2	yes	—
29	C₄⋊Dic₅₃	C₂²	1	2	yes	—
30	#11719919	C₂²	1	1	yes	characteristic
31	#11719919	C₂²	1	1	yes	characteristic
32	#11719919	C₂²	1	2	yes	—
33	C₄×Dic₅₃	C₂²	1	1	yes	characteristic
34	C₄×Dic₅₃	C₂²	1	2	yes	—
35	C₂²×D₅₃	C₂³	1	1	yes	characteristic
36	C₂²×D₅₃	C₂³	1	2	yes	—
37	Q₈×C₅₃	—	4	2	no	—
38	C₂×C₄×C₅₃	C₂³	1	2	yes	—
39	C₂×C₄×C₅₃	C₂³	1	2	yes	—
40	C₂×C₄×C₅₃	C₂³	1	1	yes	characteristic
41	C₂×C₄×C₅₃	C₂³	1	1	yes	characteristic
42	C₂×C₄×C₅₃	C₂³	1	1	yes	characteristic
43	C₂×Dic₅₃	C₂³	1	2	yes	—
44	C₂×Dic₅₃	C₂³	1	1	yes	characteristic
45	C₂×Dic₅₃	C₂³	1	1	yes	characteristic
46	C₂×Dic₅₃	C₂³	1	1	yes	characteristic
47	D₂₁₂	—	2	2	no	—
48	D₂₁₂	—	4	2	no	—
49	C₄×D₅₃	—	2	2	no	—
50	C₄×D₅₃	—	4	2	no	—
51	Dic₁₀₆	—	2	2	no	—
52	C₂²×C₅₃	C₂⁴	1	1	yes	characteristic
53	C₂×D₅₃	—	2	1	no	—
54	C₂×D₅₃	—	2	2	no	—
55	C₂×D₅₃	—	2	2	no	—
56	C₂×D₅₃	—	2	1	no	—
57	C₂×D₅₃	—	2	1	no	—
58	C₂×D₅₃	—	2	2	no	—
59	C₄×C₅₃	—	2	2	no	—
60	C₄×C₅₃	—	2	2	no	—
61	C₄×C₅₃	—	2	1	no	—
62	C₄×C₅₃	—	2	1	no	—
63	C₄×C₅₃	#41	1	2	yes	—
64	Dic₅₃	—	2	2	no	—
65	Dic₅₃	—	2	1	no	—
66	Dic₅₃	—	2	1	no	—
67	Dic₅₃	—	2	1	no	—
68	C₂×C₅₃	C₂×#41	1	1	yes	characteristic
69	C₂×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
70	C₂×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
71	D₅₃	—	4	1	no	—
72	D₅₃	—	4	2	no	—
73	#542	—	53	1	no	maximal, Sylow
74	C₅₃	#542	1	1	yes	characteristic, complemented, Sylow
75	C₄⋊Q₈	D₅₃	1	1	yes	characteristic, complemented
76	#126	—	53	2	no	—
77	#124	—	53	2	no	—
78	#124	—	53	1	no	—
79	#123	—	53	2	no	—
80	C₂²⋊Q₈	—	53	2	no	—
81	C₂²⋊Q₈	—	53	1	no	—
82	#121	—	53	1	no	—
83	C₄×Q₈	—	53	1	no	—
84	C₄×D₄	—	53	1	no	—
85	#117	—	53	1	no	—
86	C₂×Q₈	—	53	1	no	—
87	C₂×Q₈	C₂×D₅₃	1	2	yes	—
88	C₂×D₄	—	53	1	no	—
89	C₂×D₄	—	53	2	no	—
90	C₂²×C₄	—	53	1	no	—
91	C₂²×C₄	—	53	2	no	—
92	C₄⋊C₄	—	53	1	no	—
93	C₄⋊C₄	—	53	1	no	—
94	C₄⋊C₄	—	53	2	no	—
95	C₄⋊C₄	—	53	2	no	—
96	C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
97	C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
98	C₄⋊C₄	C₂×D₅₃	1	2	yes	—
99	C₂²⋊C₄	—	53	1	no	—
100	C₂²⋊C₄	—	53	1	no	—
101	C₂²⋊C₄	—	53	2	no	—
102	C₂²⋊C₄	—	53	2	no	—
103	C₂²⋊C₄	—	53	2	no	—
104	C₂²⋊C₄	—	53	2	no	—
105	C₂²⋊C₄	—	53	2	no	—
106	C₄²	—	53	1	no	—
107	C₄²	—	53	2	no	—
108	C₄²	C₂×D₅₃	1	1	yes	characteristic
109	C₂³	—	53	1	no	—
110	C₂³	—	53	2	no	—
111	Q₈	—	4	2	no	—
112	Q₈	—	106	2	no	—
113	D₄	—	106	2	no	—
114	D₄	—	212	2	no	—
115	C₂×C₄	—	53	1	no	—
116	C₂×C₄	—	53	2	no	—
117	C₂×C₄	—	53	1	no	—
118	C₂×C₄	—	53	1	no	—
119	C₂×C₄	—	106	2	no	—
120	C₂×C₄	—	212	2	no	—
121	C₂×C₄	C₂²×D₅₃	1	2	yes	—
122	C₂×C₄	C₂²×D₅₃	1	1	yes	characteristic
123	C₂×C₄	C₂²×D₅₃	1	1	yes	characteristic
124	C₂×C₄	C₂²×D₅₃	1	2	yes	—
125	C₂×C₄	C₂²×D₅₃	1	1	yes	characteristic
126	C₂²	—	106	1	no	—
127	C₂²	—	106	2	no	—
128	C₂²	—	106	2	no	—
129	C₂²	—	106	1	no	—
130	C₂²	—	106	1	no	—
131	C₂²	—	106	2	no	—
132	C₂²	C₂³×D₅₃	1	1	yes	characteristic
133	C₄	—	2	2	no	—
134	C₄	—	2	2	no	—
135	C₄	—	2	1	no	—
136	C₄	—	2	1	no	—
137	C₄	—	106	1	no	—
138	C₄	—	106	2	no	—
139	C₄	—	106	1	no	—
140	C₄	—	106	1	no	—
141	C₄	#11719949	1	2	yes	—
142	C₂	—	212	1	no	—
143	C₂	—	212	2	no	—
144	C₂	C₂×#11719949	1	1	yes	characteristic
145	C₂	#13133262	1	1	yes	characteristic
146	C₂	#13133265	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₅₃
commutator, socle, cosocle	C₂²×C₅₃	C₂⁴
Fitting	C₄⋊Q₈×C₅₃	C₂
prime radical	#13133192	C₂
hypercenter	C₄⋊Q₈	D₅₃
hypocenter	C₅₃	#542
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄⋊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. #13133205	C₂	semidirect
4. #13133202	C₂	semidirect
5. #13133197	C₂	semidirect
6. #13133193	C₂	semidirect
8. #13133182	C₂	semidirect
9. #13133179	C₂	semidirect
10. #13133174	C₂	semidirect
11. C₄⋊Q₈×C₅₃	C₂	semidirect
74. C₅₃	#542	semidirect
75. C₄⋊Q₈	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
#13133205	2
#13133202	2
#13133197	2
#13133193	2
#13133192	2
#13133182	2
#13133179	2
#13133174	2
C₄⋊Q₈×C₅₃	2
#542	53