Finite Group Databasebeta

Subgroups — #16838024

Order 3776.

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	#14273726	C₂	1	1	yes	maximal, characteristic, complemented
2	#14273721	C₂	1	2	yes	maximal, complemented
3	#14273716	C₂	1	1	yes	maximal, characteristic, complemented
4	#14273715	C₂	1	2	yes	maximal, complemented
5	#14273712	C₂	1	2	yes	maximal, complemented
6	#14273705	C₂	1	2	yes	maximal, complemented
7	#14273703	C₂	1	1	yes	maximal, characteristic, complemented
8	#14273702	C₂	1	1	yes	maximal, characteristic
9	C₄×D₂₃₆	C₂	1	1	yes	maximal, characteristic, complemented
10	#14273684	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×Q₈×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₈×C₅₉	C₂²	1	1	yes	characteristic
13	C₂×D₂₃₆	C₂²	1	1	yes	characteristic
14	C₂×D₂₃₆	C₂²	1	2	yes	—
15	C₂×C₄×D₅₉	C₂²	1	1	yes	characteristic
16	C₂×C₄×D₅₉	C₂²	1	2	yes	—
17	C₂×Dic₁₁₈	C₂²	1	2	yes	—
18	C₄⋊C₄×C₅₉	C₂²	1	1	yes	characteristic
19	C₄⋊C₄×C₅₉	C₂²	1	2	yes	—
20	C₄²×C₅₉	C₂²	1	1	yes	characteristic
21	C₄²×C₅₉	C₂²	1	2	yes	—
22	(C₂×D₅₉)⋊C₄	C₂²	1	1	yes	characteristic
23	(C₂×D₅₉)⋊C₄	C₂²	1	1	yes	characteristic
24	(C₂×D₅₉)⋊C₄	C₂²	1	2	yes	—
25	(C₂×D₅₉)⋊C₄	C₂²	1	2	yes	—
26	(C₂×D₅₉)⋊C₄	C₂²	1	2	yes	—
27	(C₂×D₅₉)⋊C₄	C₂²	1	2	yes	—
28	(C₂×D₅₉)⋊C₄	C₂²	1	2	yes	—
29	C₄⋊Dic₅₉	C₂²	1	1	yes	characteristic
30	Dic₅₉⋊C₄	C₂²	1	1	yes	characteristic
31	Dic₅₉⋊C₄	C₂²	1	1	yes	characteristic
32	Dic₅₉⋊C₄	C₂²	1	2	yes	—
33	Dic₅₉⋊C₄	C₂²	1	2	yes	—
34	C₄×Dic₅₉	C₂²	1	1	yes	characteristic
35	C₂²×D₅₉	C₂³	1	1	yes	characteristic
36	C₂²×D₅₉	C₂³	1	2	yes	—
37	Q₈×C₅₉	—	2	2	no	—
38	C₂×C₄×C₅₉	C₂³	1	1	yes	characteristic
39	C₂×C₄×C₅₉	C₂³	1	2	yes	—
40	C₂×C₄×C₅₉	C₂³	1	2	yes	—
41	C₂×C₄×C₅₉	C₂³	1	1	yes	characteristic
42	C₂×C₄×C₅₉	C₂³	1	1	yes	characteristic
43	C₂×Dic₅₉	C₂³	1	1	yes	characteristic
44	C₂×Dic₅₉	C₂³	1	1	yes	characteristic
45	C₂×Dic₅₉	C₂³	1	1	yes	characteristic
46	C₂×Dic₅₉	C₂³	1	2	yes	—
47	D₂₃₆	—	2	2	no	—
48	D₂₃₆	—	4	2	no	—
49	C₄×D₅₉	—	2	2	no	—
50	C₄×D₅₉	—	4	2	no	—
51	Dic₁₁₈	—	4	2	no	—
52	C₂²×C₅₉	C₂⁴	1	1	yes	characteristic
53	C₂×D₅₉	—	2	1	no	—
54	C₂×D₅₉	—	2	2	no	—
55	C₂×D₅₉	—	2	2	no	—
56	C₂×D₅₉	—	2	1	no	—
57	C₂×D₅₉	—	2	1	no	—
58	C₂×D₅₉	—	2	2	no	—
59	C₄×C₅₉	—	2	2	no	—
60	C₄×C₅₉	—	2	2	no	—
61	C₄×C₅₉	—	2	1	no	—
62	C₄×C₅₉	—	2	1	no	—
63	C₄×C₅₉	#41	1	2	yes	—
64	Dic₅₉	—	2	1	no	—
65	Dic₅₉	—	2	1	no	—
66	Dic₅₉	—	2	2	no	—
67	Dic₅₉	—	2	1	no	—
68	C₂×C₅₉	C₂×#41	1	1	yes	characteristic
69	C₂×C₅₉	H(5,𝔽₂)	1	1	yes	characteristic
70	C₂×C₅₉	2₋¹⁺⁴	1	1	yes	characteristic
71	D₅₉	—	4	1	no	—
72	D₅₉	—	4	2	no	—
73	#542	—	59	1	no	maximal, Sylow
74	C₅₉	#542	1	1	yes	characteristic, complemented, Sylow
75	C₄⋊Q₈	—	59	1	no	—
76	#126	—	59	2	no	—
77	#124	—	59	1	no	—
78	#124	—	59	2	no	—
79	#123	—	59	2	no	—
80	C₂²⋊Q₈	—	59	1	no	—
81	C₂²⋊Q₈	—	59	2	no	—
82	#121	—	59	1	no	—
83	C₄×Q₈	D₅₉	1	1	yes	characteristic, complemented
84	C₄×D₄	—	59	1	no	—
85	#117	—	59	1	no	—
86	C₂×Q₈	—	59	2	no	—
87	C₂×Q₈	C₂×D₅₉	1	1	yes	characteristic
88	C₂×D₄	—	59	1	no	—
89	C₂×D₄	—	59	2	no	—
90	C₂²×C₄	—	59	1	no	—
91	C₂²×C₄	—	59	2	no	—
92	C₄⋊C₄	—	59	1	no	—
93	C₄⋊C₄	—	59	1	no	—
94	C₄⋊C₄	—	59	1	no	—
95	C₄⋊C₄	—	59	2	no	—
96	C₄⋊C₄	—	59	2	no	—
97	C₄⋊C₄	C₂×D₅₉	1	1	yes	characteristic
98	C₄⋊C₄	C₂×D₅₉	1	2	yes	—
99	C₂²⋊C₄	—	59	1	no	—
100	C₂²⋊C₄	—	59	1	no	—
101	C₂²⋊C₄	—	59	2	no	—
102	C₂²⋊C₄	—	59	2	no	—
103	C₂²⋊C₄	—	59	2	no	—
104	C₂²⋊C₄	—	59	2	no	—
105	C₂²⋊C₄	—	59	2	no	—
106	C₄²	—	59	1	no	—
107	C₄²	C₂×D₅₉	1	1	yes	characteristic
108	C₄²	C₂×D₅₉	1	2	yes	—
109	C₂³	—	59	1	no	—
110	C₂³	—	59	2	no	—
111	Q₈	—	2	2	no	—
112	Q₈	—	236	2	no	—
113	D₄	—	118	2	no	—
114	D₄	—	236	2	no	—
115	C₂×C₄	—	59	1	no	—
116	C₂×C₄	—	59	1	no	—
117	C₂×C₄	—	59	1	no	—
118	C₂×C₄	—	59	2	no	—
119	C₂×C₄	—	118	2	no	—
120	C₂×C₄	—	236	2	no	—
121	C₂×C₄	C₂²×D₅₉	1	1	yes	characteristic
122	C₂×C₄	C₂²×D₅₉	1	2	yes	—
123	C₂×C₄	C₂²×D₅₉	1	1	yes	characteristic
124	C₂×C₄	C₂²×D₅₉	1	2	yes	—
125	C₂×C₄	C₂²×D₅₉	1	1	yes	characteristic
126	C₂²	—	118	1	no	—
127	C₂²	—	118	2	no	—
128	C₂²	—	118	2	no	—
129	C₂²	—	118	1	no	—
130	C₂²	—	118	1	no	—
131	C₂²	—	118	2	no	—
132	C₂²	C₂³×D₅₉	1	1	yes	characteristic
133	C₄	—	2	2	no	—
134	C₄	—	2	1	no	—
135	C₄	—	2	2	no	—
136	C₄	—	2	1	no	—
137	C₄	—	118	1	no	—
138	C₄	—	118	2	no	—
139	C₄	—	118	1	no	—
140	C₄	—	118	1	no	—
141	C₄	#11745685	1	2	yes	—
142	C₂	—	236	1	no	—
143	C₂	—	236	2	no	—
144	C₂	C₂×#11745685	1	1	yes	characteristic
145	C₂	#14273773	1	1	yes	characteristic
146	C₂	#14273775	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₅₉
commutator, socle, cosocle	C₂²×C₅₉	C₂⁴
Fitting	C₄×Q₈×C₅₉	C₂
prime radical	#14273702	C₂
hypercenter	C₄×Q₈	D₅₉
hypocenter	C₅₉	#542
LCS	G ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. #14273726	C₂	semidirect
2. #14273721	C₂	semidirect
3. #14273716	C₂	semidirect
4. #14273715	C₂	semidirect
5. #14273712	C₂	semidirect
6. #14273705	C₂	semidirect
7. #14273703	C₂	semidirect
9. C₄×D₂₃₆	C₂	semidirect
10. #14273684	C₂	semidirect
11. C₄×Q₈×C₅₉	C₂	semidirect
74. C₅₉	#542	semidirect
83. C₄×Q₈	D₅₉	semidirect

Maximal subgroups

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

subgroup	index
#14273726	2
#14273721	2
#14273716	2
#14273715	2
#14273712	2
#14273705	2
#14273703	2
#14273702	2
C₄×D₂₃₆	2
#14273684	2
C₄×Q₈×C₅₉	2
#542	59