Finite Group Databasebeta

Subgroups — #21425497

Order 7552.

196 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 65 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	#16837451	C₂	1	1	yes	maximal, characteristic
2	#16837450	C₂	1	1	yes	maximal, characteristic
3	#16837450	C₂	1	1	yes	maximal, characteristic
4	C₂×#14273646	C₂	1	1	yes	maximal, characteristic
5	C₂×#14273646	C₂	1	1	yes	maximal, characteristic
6	C₅₉⋊#338	C₂	1	1	yes	maximal, characteristic
7	C₂×#101×C₅₉	C₂	1	1	yes	maximal, characteristic
8	C₂×#11745664	C₂²	1	1	yes	characteristic
9	C₂×#11745664	C₂²	1	1	yes	characteristic
10	C₂×#11745664	C₂²	1	1	yes	characteristic
11	C₂×#11745664	C₂²	1	1	yes	characteristic
12	#14273663	—	2	2	no	—
13	#14273662	—	2	2	no	—
14	#14273662	—	2	2	no	—
15	#14273646	—	2	2	no	—
16	#14273646	—	2	2	no	—
17	C₂²×Q₈×C₅₉	C₂²	1	1	yes	characteristic
18	C₂×M₁₆×C₅₉	C₂²	1	1	yes	characteristic
19	C₂×M₁₆×C₅₉	C₂²	1	1	yes	characteristic
20	#101×C₅₉	—	2	2	no	—
21	C₂×Q₈×C₅₉	—	2	1	no	—
22	C₂×Q₈×C₅₉	—	2	1	no	—
23	C₂×Q₈×C₅₉	—	2	1	no	—
24	C₂×Q₈×C₅₉	—	2	1	no	—
25	C₂×Q₈×C₅₉	—	2	1	no	—
26	C₂×Q₈×C₅₉	—	2	1	no	—
27	C₂²×C₄×C₅₉	Q₈	1	2	yes	—
28	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
29	M₁₆×C₅₉	—	2	1	no	—
30	M₁₆×C₅₉	—	2	1	no	—
31	M₁₆×C₅₉	—	2	1	no	—
32	M₁₆×C₅₉	D₄	1	2	yes	—
33	C₂×C₈×C₅₉	—	2	1	no	—
34	C₂×C₈×C₅₉	—	2	1	no	—
35	#11745664	—	2	1	no	—
36	#11745664	—	2	1	no	—
37	#11745664	—	2	1	no	—
38	#11745664	—	2	1	no	—
39	#11745664	—	2	1	no	—
40	#11745664	—	2	1	no	—
41	#11745664	D₄	1	2	yes	—
42	#11745664	D₄	1	2	yes	—
43	C₂×C₅₉⋊C₈	—	2	1	no	—
44	C₂×C₅₉⋊C₈	—	2	1	no	—
45	C₂×C₅₉⋊C₈	—	2	1	no	—
46	C₂×C₅₉⋊C₈	—	2	1	no	—
47	C₂³×C₅₉	C₂×Q₈	1	1	yes	characteristic
48	Q₈×C₅₉	—	4	1	no	—
49	Q₈×C₅₉	—	4	1	no	—
50	Q₈×C₅₉	—	4	1	no	—
51	Q₈×C₅₉	—	4	1	no	—
52	C₂×C₄×C₅₉	—	2	2	no	—
53	C₂×C₄×C₅₉	—	2	2	no	—
54	C₂×C₄×C₅₉	—	2	2	no	—
55	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
59	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
60	C₂×C₄×C₅₉	#41	1	1	yes	characteristic
61	C₈×C₅₉	—	2	2	no	—
62	C₈×C₅₉	—	4	1	no	—
63	C₅₉⋊C₈	—	2	2	no	—
64	C₅₉⋊C₈	—	2	2	no	—
65	C₅₉⋊C₈	—	4	1	no	—
66	C₅₉⋊C₈	—	4	1	no	—
67	C₂²×C₅₉	—	4	1	no	—
68	C₂²×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
69	C₂²×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
70	C₂²×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
71	C₄×C₅₉	—	4	2	no	—
72	C₄×C₅₉	#121	1	1	yes	characteristic
73	C₄×C₅₉	#121	1	1	yes	characteristic
74	C₄×C₅₉	#121	1	1	yes	characteristic
75	C₄×C₅₉	#126	1	1	yes	characteristic
76	#2070	—	59	1	no	maximal, Sylow
77	C₂×C₅₉	—	2	1	no	—
78	C₂×C₅₉	—	2	1	no	—
79	C₂×C₅₉	—	2	1	no	—
80	C₂×C₅₉	#396	1	1	yes	characteristic
81	#430	—	59	1	no	—
82	#430	—	59	1	no	—
83	#430	—	59	1	no	—
84	C₂×#101	—	59	1	no	—
85	C₂×#101	—	59	1	no	—
86	C₂×#101	D₅₉	1	1	yes	characteristic
87	#338	—	59	1	no	—
88	C₅₉	#2070	1	1	yes	characteristic, complemented, Sylow
89	C₂²×Q₈	C₂×D₅₉	1	1	yes	characteristic
90	C₂×M₁₆	—	59	1	no	—
91	C₂×M₁₆	—	59	1	no	—
92	C₂×M₁₆	—	59	1	no	—
93	C₂×M₁₆	—	59	1	no	—
94	C₂×M₁₆	C₂×D₅₉	1	1	yes	characteristic
95	C₂×M₁₆	C₂×D₅₉	1	1	yes	characteristic
96	#108	—	118	2	no	—
97	#108	—	118	2	no	—
98	#108	—	118	2	no	—
99	#101	—	2	2	no	—
100	#101	—	118	2	no	—
101	#101	—	118	2	no	—
102	C₂×Q₈	—	2	1	no	—
103	C₂×Q₈	—	2	1	no	—
104	C₂×Q₈	—	2	1	no	—
105	C₂×Q₈	—	2	1	no	—
106	C₂×Q₈	—	2	1	no	—
107	C₂×Q₈	—	2	1	no	—
108	C₂²×C₄	Dic₁₁₈	1	2	yes	—
109	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
110	M₁₆	—	2	1	no	—
111	M₁₆	—	2	1	no	—
112	M₁₆	—	2	1	no	—
113	M₁₆	—	59	2	no	—
114	M₁₆	—	59	2	no	—
115	M₁₆	—	118	1	no	—
116	M₁₆	—	118	1	no	—
117	M₁₆	—	118	1	no	—
118	M₁₆	—	118	1	no	—
119	M₁₆	—	118	1	no	—
120	M₁₆	—	118	1	no	—
121	M₁₆	C₂²⋊D₅₉	1	2	yes	—
122	C₂×C₈	—	2	1	no	—
123	C₂×C₈	—	2	1	no	—
124	C₂×C₈	—	118	1	no	—
125	C₂×C₈	—	118	1	no	—
126	C₂×C₈	—	118	1	no	—
127	C₂×C₈	—	118	1	no	—
128	C₂³	C₂×Dic₁₁₈	1	1	yes	characteristic
129	Q₈	—	4	1	no	—
130	Q₈	—	4	1	no	—
131	Q₈	—	4	1	no	—
132	Q₈	—	4	1	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	#11745685	1	1	yes	characteristic
137	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
138	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
139	C₂×C₄	#11745687	1	1	yes	characteristic
140	C₂×C₄	#11745687	1	1	yes	characteristic
141	C₂×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
142	C₈	—	2	2	no	—
143	C₈	—	4	1	no	—
144	C₈	—	118	2	no	—
145	C₈	—	118	2	no	—
146	C₈	—	236	1	no	—
147	C₈	—	236	1	no	—
148	C₂²	—	4	1	no	—
149	C₂²	#14273704	1	1	yes	characteristic
150	C₂²	#14273704	1	1	yes	characteristic
151	C₂²	#14273706	1	1	yes	characteristic
152	C₄	—	4	2	no	—
153	C₄	#14273698	1	1	yes	characteristic
154	C₄	#14273699	1	1	yes	characteristic
155	C₄	#14273700	1	1	yes	characteristic
156	C₄	#14273720	1	1	yes	characteristic
157	C₂	—	2	1	no	—
158	C₂	—	2	1	no	—
159	C₂	—	2	1	no	—
160	C₂	#16837325	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#16837325
commutator	C₂²×C₄×C₅₉	C₂³
Frattini	C₂²×C₄	C₂²×D₅₉
Fitting	C₂×#101×C₅₉	C₂
socle	C₂×C₅₉	#396
hypercenter	C₂×#101	D₅₉
hypocenter	C₅₉	#2070
LCS	G ▷ C₂²×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#101

Complements

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

N	H ≅ G/N	type
88. C₅₉	#2070	semidirect