Finite Group Databasebeta

Subgroups — #21423627

Order 7552.

170 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 51 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	#16837748	C₂	1	1	yes	maximal, characteristic, complemented
2	#16837692	C₂	1	1	yes	maximal, characteristic, complemented
3	#16837480	C₂	1	1	yes	maximal, characteristic, complemented
4	#16837154	C₂	1	1	yes	maximal, characteristic, complemented
5	#16837150	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837146	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₉×#459	C₂	1	1	yes	maximal, characteristic
8	C₂×#11745664	—	2	1	no	—
9	#14273727	C₂²	1	1	yes	characteristic
10	#14273658	—	2	1	no	—
11	C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
12	C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
13	#14273648	—	2	1	no	—
14	#14273648	—	2	1	no	—
15	#14273648	—	2	1	no	—
16	C₂²⋊(C₅₉⋊C₈)	—	2	1	no	—
17	C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×D₈×C₅₉	—	2	1	no	—
19	#127×C₅₉	C₂²	1	1	yes	characteristic
20	C₄×D₄×C₅₉	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
22	#102×C₅₉	—	2	1	no	—
23	C₂×D₄×C₅₉	—	2	1	no	—
24	C₂×D₄×C₅₉	—	2	1	no	—
25	C₂×D₄×C₅₉	—	2	1	no	—
26	C₂×D₄×C₅₉	D₄	1	1	yes	characteristic
27	C₂²×C₄×C₅₉	—	2	1	no	—
28	C₂×Dic₁₁₈	—	2	1	no	—
29	D₈×C₅₉	—	4	2	no	—
30	C₂×C₈×C₅₉	—	2	1	no	—
31	C₄⋊C₄×C₅₉	D₄	1	1	yes	characteristic
32	C₂²⋊C₄×C₅₉	—	2	1	no	—
33	C₄²×C₅₉	C₂³	1	1	yes	characteristic
34	C₄⋊Dic₅₉	—	2	1	no	—
35	C₄⋊Dic₅₉	—	2	1	no	—
36	#11745664	—	4	2	no	—
37	C₂×C₅₉⋊C₈	—	2	1	no	—
38	C₂×C₅₉⋊C₈	—	2	1	no	—
39	C₂×C₅₉⋊C₈	D₄	1	1	yes	characteristic
40	C₂×C₅₉⋊C₈	D₄	1	1	yes	characteristic
41	C₂³×C₅₉	—	2	1	no	—
42	C₂³×C₅₉	—	4	1	no	—
43	D₄×C₅₉	—	2	1	no	—
44	D₄×C₅₉	—	2	1	no	—
45	D₄×C₅₉	—	4	1	no	—
46	D₄×C₅₉	—	4	1	no	—
47	D₄×C₅₉	—	4	1	no	—
48	D₄×C₅₉	—	4	1	no	—
49	D₄×C₅₉	—	8	1	no	—
50	C₂×C₄×C₅₉	—	2	1	no	—
51	C₂×C₄×C₅₉	—	4	1	no	—
52	C₂×C₄×C₅₉	—	4	1	no	—
53	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
56	C₂×Dic₅₉	—	4	1	no	—
57	Dic₁₁₈	—	8	1	no	—
58	C₈×C₅₉	—	4	1	no	—
59	C₅₉⋊C₈	—	2	1	no	—
60	C₅₉⋊C₈	—	2	1	no	—
61	C₅₉⋊C₈	—	4	1	no	—
62	C₅₉⋊C₈	—	4	1	no	—
63	C₂²×C₅₉	—	4	1	no	—
64	C₂²×C₅₉	—	4	1	no	—
65	C₂²×C₅₉	—	4	1	no	—
66	C₂²×C₅₉	—	8	1	no	—
67	C₂²×C₅₉	—	8	1	no	—
68	C₂²×C₅₉	—	8	1	no	—
69	C₂²×C₅₉	#121	1	1	yes	characteristic
70	C₄×C₅₉	—	2	1	no	—
71	C₄×C₅₉	—	4	1	no	—
72	C₄×C₅₉	#135	1	1	yes	characteristic
73	C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
74	C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
75	C₄×C₅₉	#137	1	1	yes	characteristic
76	Dic₅₉	—	8	1	no	—
77	#1697	—	59	1	no	maximal, Sylow
78	C₂×C₅₉	—	8	1	no	—
79	C₂×C₅₉	—	16	1	no	—
80	C₂×C₅₉	#463	1	1	yes	characteristic
81	C₂×C₅₉	#468	1	1	yes	characteristic
82	C₂×C₅₉	#472	1	1	yes	characteristic
83	#486	—	59	1	no	—
84	#476	—	59	1	no	—
85	#459	D₅₉	1	1	yes	characteristic
86	#436	—	59	1	no	—
87	#335	—	59	1	no	—
88	#333	—	59	1	no	—
89	#331	—	59	1	no	—
90	C₅₉	#1697	1	1	yes	characteristic, complemented, Sylow
91	C₂×D₈	—	2	1	no	—
92	C₂×M₁₆	—	118	1	no	—
93	C₄⋊Q₈	—	59	1	no	—
94	#127	C₂×D₅₉	1	1	yes	characteristic
95	C₄×D₄	C₂×D₅₉	1	1	yes	characteristic
96	#106	—	118	1	no	—
97	C₄⋊C₈	—	59	1	no	—
98	C₄⋊C₈	—	59	1	no	—
99	C₄⋊C₈	C₂×D₅₉	1	1	yes	characteristic
100	#102	—	2	1	no	—
101	#102	—	118	1	no	—
102	#102	—	118	1	no	—
103	#102	—	118	1	no	—
104	C₂²⋊C₈	—	118	1	no	—
105	C₄×C₈	—	59	1	no	—
106	C₂×Q₈	—	118	1	no	—
107	C₂×D₄	—	2	1	no	—
108	C₂×D₄	—	2	1	no	—
109	C₂×D₄	—	2	1	no	—
110	C₂×D₄	C₂²⋊D₅₉	1	1	yes	characteristic
111	C₂²×C₄	—	2	1	no	—
112	D₈	—	4	2	no	—
113	M₁₆	—	236	2	no	—
114	C₂×C₈	—	2	1	no	—
115	C₂×C₈	—	59	1	no	—
116	C₂×C₈	—	59	1	no	—
117	C₂×C₈	—	118	1	no	—
118	C₂×C₈	—	118	1	no	—
119	C₄⋊C₄	—	118	1	no	—
120	C₄⋊C₄	—	118	1	no	—
121	C₄⋊C₄	C₂²⋊D₅₉	1	1	yes	characteristic
122	C₂²⋊C₄	—	2	1	no	—
123	C₄²	C₂²×D₅₉	1	1	yes	characteristic
124	C₂³	—	2	1	no	—
125	C₂³	—	4	1	no	—
126	Q₈	—	472	1	no	—
127	D₄	—	2	1	no	—
128	D₄	—	2	1	no	—
129	D₄	—	4	1	no	—
130	D₄	—	4	1	no	—
131	D₄	—	4	1	no	—
132	D₄	—	4	1	no	—
133	D₄	—	8	1	no	—
134	C₂×C₄	—	2	1	no	—
135	C₂×C₄	—	4	1	no	—
136	C₂×C₄	—	4	1	no	—
137	C₂×C₄	—	236	1	no	—
138	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
139	C₂×C₄	#11745687	1	1	yes	characteristic
140	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
141	C₈	—	4	1	no	—
142	C₈	—	118	1	no	—
143	C₈	—	118	1	no	—
144	C₈	—	236	1	no	—
145	C₈	—	236	1	no	—
146	C₂²	—	4	1	no	—
147	C₂²	—	4	1	no	—
148	C₂²	—	4	1	no	—
149	C₂²	—	8	1	no	—
150	C₂²	—	8	1	no	—
151	C₂²	—	8	1	no	—
152	C₂²	#14273699	1	1	yes	characteristic
153	C₄	—	2	1	no	—
154	C₄	—	4	1	no	—
155	C₄	—	472	1	no	—
156	C₄	#14273746	1	1	yes	characteristic
157	C₄	#14273751	1	1	yes	characteristic
158	C₄	#14273753	1	1	yes	characteristic
159	C₄	#14273757	1	1	yes	characteristic
160	C₂	—	8	1	no	—
161	C₂	—	16	1	no	—
162	C₂	#16837615	1	1	yes	characteristic
163	C₂	#16837646	1	1	yes	characteristic
164	C₂	#16837670	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#14273699
commutator	C₄²×C₅₉	C₂³
Frattini	C₄²	C₂²×D₅₉
Fitting	C₅₉×#459	C₂
socle	C₂²×C₅₉	#121
hypercenter	#459	D₅₉
hypocenter	C₅₉	#1697
LCS	G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #459

Complements

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

N	H ≅ G/N	type
1. #16837748	C₂	semidirect
2. #16837692	C₂	semidirect
3. #16837480	C₂	semidirect
4. #16837154	C₂	semidirect
5. #16837150	C₂	semidirect
6. #16837146	C₂	semidirect
90. C₅₉	#1697	semidirect