Finite Group Databasebeta

Subgroups — #21426170

Order 7552.

170 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 45 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	#16837837	C₂	1	1	yes	maximal, characteristic, complemented
2	#16837777	C₂	1	1	yes	maximal, characteristic, complemented
3	#16837519	C₂	1	1	yes	maximal, characteristic, complemented
4	#16837222	C₂	1	1	yes	maximal, characteristic, complemented
5	#16837216	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837215	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₉×#509	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×D₄₇₂	C₂²	1	1	yes	characteristic
9	C₂×#11745669	—	2	1	no	—
10	#14273734	—	2	1	no	—
11	#14273723	—	2	1	no	—
12	#14273670	—	2	1	no	—
13	#14273668	—	2	1	no	—
14	#14273667	C₂²	1	1	yes	characteristic
15	#14273663	C₂²	1	1	yes	characteristic
16	#14273654	—	2	1	no	—
17	C₈×Dic₅₉	C₂²	1	1	yes	characteristic
18	#135×C₅₉	C₂²	1	1	yes	characteristic
19	C₂×D₈×C₅₉	C₂²	1	1	yes	characteristic
20	SD₃₂×C₅₉	—	2	1	no	—
21	D₁₆×C₅₉	—	2	1	no	—
22	M₃₂×C₅₉	C₂²	1	1	yes	characteristic
23	#41×C₅₉	—	2	1	no	—
24	C₂×D₄×C₅₉	—	2	1	no	—
25	C₂×C₂²⋊D₅₉	—	4	2	no	—
26	C₂×D₂₃₆	—	2	1	no	—
27	Q₁₆×C₅₉	D₄	1	1	yes	characteristic, complemented
28	SD₁₆×C₅₉	—	2	1	no	—
29	D₈×C₅₉	—	2	1	no	—
30	D₈×C₅₉	—	2	1	no	—
31	D₈×C₅₉	D₄	1	1	yes	characteristic, complemented
32	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
33	#11745669	—	4	1	no	—
34	#11745669	—	4	1	no	—
35	C₄×Dic₅₉	D₄	1	1	yes	characteristic
36	#11745664	—	2	1	no	—
37	#11745664	—	4	1	no	—
38	C₂×C₅₉⋊C₈	—	4	1	no	—
39	C₂×C₅₉⋊C₈	D₄	1	1	yes	characteristic
40	D₄₇₂	—	2	1	no	—
41	D₄₇₂	—	2	1	no	—
42	C₁₆×C₅₉	—	2	1	no	—
43	C₅₉⋊C₁₆	—	2	1	no	—
44	C₂³×C₅₉	—	4	1	no	—
45	C₂²×D₅₉	—	4	1	no	—
46	Q₈×C₅₉	—	2	1	no	—
47	D₄×C₅₉	—	2	1	no	—
48	D₄×C₅₉	—	4	1	no	—
49	D₄×C₅₉	—	4	1	no	—
50	D₄×C₅₉	—	4	1	no	—
51	C₂×C₄×C₅₉	—	4	1	no	—
52	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
53	C₂²⋊D₅₉	—	8	4	no	—
54	C₂×Dic₅₉	—	2	1	no	—
55	D₂₃₆	—	4	1	no	—
56	D₂₃₆	—	4	1	no	—
57	C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
58	C₈×C₅₉	#41	1	1	yes	characteristic
59	C₅₉⋊C₈	—	4	1	no	—
60	C₅₉⋊C₈	D₈	1	1	yes	characteristic
61	C₅₉⋊C₈	D₈	1	1	yes	characteristic
62	C₂²×C₅₉	—	4	1	no	—
63	C₂²×C₅₉	—	8	2	no	—
64	C₂²×C₅₉	—	8	1	no	—
65	C₂²×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
66	C₂×D₅₉	—	8	2	no	—
67	C₂×D₅₉	—	8	1	no	—
68	C₄×C₅₉	—	4	1	no	—
69	C₄×C₅₉	#121	1	1	yes	characteristic
70	C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
71	Dic₅₉	—	2	2	no	—
72	#2213	—	59	1	no	maximal, Sylow
73	C₂×C₅₉	—	2	1	no	—
74	C₂×C₅₉	—	8	1	no	—
75	C₂×C₅₉	—	16	1	no	—
76	C₂×C₅₉	#459	1	1	yes	characteristic
77	D₅₉	—	16	1	no	—
78	#509	—	59	1	no	—
79	#509	D₅₉	1	1	yes	characteristic, complemented
80	#493	—	59	1	no	—
81	#443	—	59	1	no	—
82	#364	—	59	1	no	—
83	#361	—	59	1	no	—
84	#361	—	59	1	no	—
85	C₅₉	#2213	1	1	yes	characteristic, complemented, Sylow
86	#135	C₂×D₅₉	1	1	yes	characteristic
87	C₂×D₈	—	59	1	no	—
88	C₂×D₈	—	118	1	no	—
89	C₂×D₈	C₂×D₅₉	1	1	yes	characteristic
90	#131	—	118	1	no	—
91	#127	—	118	1	no	—
92	SD₃₂	—	2	1	no	—
93	SD₃₂	—	118	1	no	—
94	D₁₆	—	2	1	no	—
95	D₁₆	—	118	1	no	—
96	M₃₂	—	59	1	no	—
97	M₃₂	C₂×D₅₉	1	1	yes	characteristic
98	#108	—	59	1	no	—
99	#104	—	118	1	no	—
100	C₄×C₈	—	59	1	no	—
101	#41	—	2	1	no	—
102	C₂×D₄	—	2	1	no	—
103	C₂×D₄	—	118	1	no	—
104	C₂×D₄	—	236	2	no	—
105	Q₁₆	C₂²⋊D₅₉	1	1	yes	characteristic, complemented
106	SD₁₆	—	2	1	no	—
107	D₈	—	2	1	no	—
108	D₈	—	2	1	no	—
109	D₈	—	118	1	no	—
110	D₈	—	118	1	no	—
111	D₈	—	236	1	no	—
112	D₈	—	236	1	no	—
113	D₈	C₂²⋊D₅₉	1	1	yes	characteristic, complemented
114	M₁₆	—	118	1	no	—
115	M₁₆	—	236	1	no	—
116	C₂×C₈	—	59	1	no	—
117	C₂×C₈	—	236	1	no	—
118	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
119	C₄²	—	59	1	no	—
120	C₁₆	—	2	1	no	—
121	C₁₆	—	118	1	no	—
122	C₂³	—	4	1	no	—
123	C₂³	—	236	1	no	—
124	Q₈	—	2	1	no	—
125	D₄	—	2	1	no	—
126	D₄	—	4	1	no	—
127	D₄	—	4	1	no	—
128	D₄	—	4	1	no	—
129	D₄	—	472	4	no	—
130	D₄	—	236	1	no	—
131	D₄	—	236	1	no	—
132	C₂×C₄	—	4	1	no	—
133	C₂×C₄	—	118	1	no	—
134	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
135	C₈	—	59	1	no	—
136	C₈	—	59	1	no	—
137	C₈	—	236	1	no	—
138	C₈	#11745687	1	1	yes	characteristic
139	C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
140	C₂²	—	4	1	no	—
141	C₂²	—	8	2	no	—
142	C₂²	—	8	1	no	—
143	C₂²	—	472	2	no	—
144	C₂²	—	472	1	no	—
145	C₂²	#14273751	1	1	yes	characteristic
146	C₄	—	4	1	no	—
147	C₄	—	118	2	no	—
148	C₄	#14273699	1	1	yes	characteristic
149	C₄	D₈×D₅₉	1	1	yes	characteristic
150	C₂	—	2	1	no	—
151	C₂	—	8	1	no	—
152	C₂	—	16	1	no	—
153	C₂	—	944	1	no	—
154	C₂	#16837595	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#16837595
commutator	C₂×C₈×C₅₉	C₂³
Frattini	C₂×C₈	C₂²×D₅₉
Fitting	C₅₉×#509	C₂
socle	C₂×C₅₉	#459
hypercenter	#509	D₅₉
hypocenter	C₅₉	#2213
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #509

Complements

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

N	H ≅ G/N	type
1. #16837837	C₂	semidirect
2. #16837777	C₂	semidirect
3. #16837519	C₂	semidirect
4. #16837222	C₂	semidirect
5. #16837216	C₂	semidirect
6. #16837215	C₂	semidirect
7. C₅₉×#509	C₂	semidirect
27. Q₁₆×C₅₉	D₄	semidirect
31. D₈×C₅₉	D₄	semidirect
79. #509	D₅₉	semidirect
85. C₅₉	#2213	semidirect
105. Q₁₆	C₂²⋊D₅₉	semidirect
113. D₈	C₂²⋊D₅₉	semidirect