Finite Group Databasebeta

Subgroups — #20245346

Order 6016.

220 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 47 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	#15689925	C₂	1	1	yes	maximal, characteristic, complemented
2	#15689558	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×D₇₅₂	C₂	1	1	yes	maximal, characteristic, complemented
4	#15689394	C₂	1	1	yes	maximal, characteristic, complemented
5	#15688935	C₂	1	1	yes	maximal, characteristic, complemented
6	#15688906	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₇×#359	C₂	1	1	yes	maximal, characteristic, complemented
8	#13094070	—	2	1	no	—
9	#13094052	—	2	1	no	—
10	#13094051	—	2	2	no	—
11	#13094049	—	2	1	no	—
12	#13094047	—	2	1	no	—
13	C₂×D₃₇₆	C₂²	1	1	yes	characteristic
14	C₂×D₃₇₆	C₂²	1	1	yes	characteristic, complemented
15	D₈×D₄₇	—	2	1	no	—
16	#13094030	C₂²	1	1	yes	characteristic
17	#13093967	—	2	1	no	—
18	D₇₅₂	—	2	2	no	—
19	#13093965	—	2	1	no	—
20	#13093964	C₂²	1	1	yes	characteristic, complemented
21	#13093940	—	2	1	no	—
22	#135×C₄₇	C₂²	1	1	yes	characteristic
23	C₂×C₁₆×C₄₇	C₂²	1	1	yes	characteristic, complemented
24	#108×C₄₇	C₂²	1	1	yes	characteristic, complemented
25	#41×C₄₇	—	2	1	no	—
26	#10625354	—	2	2	no	—
27	D₄×D₄₇	—	2	2	no	—
28	D₄×D₄₇	—	4	2	no	—
29	#10625350	—	4	1	no	—
30	#10625350	D₄	1	1	yes	characteristic
31	C₂×D₁₈₈	—	2	1	no	—
32	C₂×D₁₈₈	—	2	1	no	—
33	C₂×D₁₈₈	—	4	1	no	—
34	Q₁₆×C₄₇	D₄	1	1	yes	characteristic
35	SD₁₆×C₄₇	—	2	1	no	—
36	D₈×C₄₇	D₄	1	1	yes	characteristic
37	M₁₆×C₄₇	—	2	1	no	—
38	C₂×C₈×C₄₇	C₂³	1	1	yes	characteristic
39	#10625336	—	2	2	no	—
40	#10625334	—	2	2	no	—
41	#10625329	D₄	1	1	yes	characteristic
42	D₃₇₆	—	2	1	no	—
43	D₃₇₆	—	2	1	no	—
44	D₃₇₆	—	2	1	no	—
45	D₃₇₆	—	4	1	no	—
46	D₃₇₆	D₄	1	2	yes	—
47	#10625325	—	4	1	no	—
48	#10625324	—	2	1	no	—
49	C₈×D₄₇	—	2	1	no	—
50	C₁₆×C₄₇	—	2	1	no	—
51	C₄₇⋊C₁₆	—	2	1	no	—
52	C₂²×D₄₇	—	4	2	no	—
53	C₂²×D₄₇	—	4	1	no	—
54	C₂²×D₄₇	—	4	1	no	—
55	Q₈×C₄₇	—	2	1	no	—
56	D₄×C₄₇	—	2	1	no	—
57	D₄×C₄₇	—	4	1	no	—
58	C₂×C₄×C₄₇	—	4	1	no	—
59	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
60	C₂²⋊D₄₇	—	2	1	no	—
61	C₂²⋊D₄₇	—	4	2	no	—
62	D₁₈₈	—	2	2	no	—
63	D₁₈₈	—	4	2	no	—
64	D₁₈₈	—	4	1	no	—
65	D₁₈₈	—	4	1	no	—
66	D₁₈₈	—	4	1	no	—
67	D₁₈₈	D₈	1	1	yes	characteristic
68	C₄×D₄₇	—	2	1	no	—
69	C₄×D₄₇	—	4	2	no	—
70	Dic₉₄	D₈	1	1	yes	characteristic
71	C₈×C₄₇	—	4	1	no	—
72	C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
73	C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
74	C₄₇⋊C₈	—	2	1	no	—
75	C₂²×C₄₇	—	4	1	no	—
76	C₂²×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
77	C₂×D₄₇	—	2	1	no	—
78	C₂×D₄₇	—	4	2	no	—
79	C₂×D₄₇	—	8	1	no	—
80	C₂×D₄₇	—	16	2	no	—
81	C₂×D₄₇	—	16	1	no	—
82	C₂×D₄₇	—	16	1	no	—
83	C₄×C₄₇	—	4	1	no	—
84	C₄×C₄₇	#120	1	1	yes	characteristic
85	C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
86	Dic₄₇	—	2	1	no	—
87	#2192	—	47	1	no	maximal, Sylow
88	C₂×C₄₇	—	2	1	no	—
89	C₂×C₄₇	—	8	1	no	—
90	C₂×C₄₇	#447	1	1	yes	characteristic
91	D₄₇	—	4	1	no	—
92	D₄₇	—	8	2	no	—
93	D₄₇	—	16	1	no	—
94	#576	—	47	1	no	—
95	#509	—	47	1	no	—
96	C₂×D₁₆	—	47	1	no	—
97	#472	—	47	1	no	—
98	#361	—	47	1	no	—
99	#359	D₄₇	1	1	yes	characteristic, complemented
100	#350	—	47	1	no	—
101	C₄₇	#2192	1	1	yes	characteristic, complemented, Sylow
102	H(5,𝔽₂)	—	94	1	no	—
103	AGL(1,ℤ/8)	—	94	1	no	—
104	AGL(1,ℤ/8)	—	94	1	no	—
105	AGL(1,ℤ/8)	—	94	2	no	—
106	#135	—	94	1	no	—
107	#135	C₂×D₄₇	1	1	yes	characteristic
108	C₂×D₈	—	47	1	no	—
109	C₂×D₈	—	47	1	no	—
110	C₂×D₈	—	94	1	no	—
111	#131	—	47	1	no	—
112	SD₃₂	—	94	1	no	—
113	D₁₆	—	94	2	no	—
114	D₁₆	—	94	1	no	—
115	M₃₂	—	47	1	no	—
116	C₂×C₁₆	C₂×D₄₇	1	1	yes	characteristic, complemented
117	#108	C₂×D₄₇	1	1	yes	characteristic, complemented
118	#100	—	94	1	no	—
119	#41	—	2	1	no	—
120	#41	—	47	1	no	—
121	#41	—	94	2	no	—
122	#41	—	188	1	no	—
123	C₂×D₄	—	94	2	no	—
124	C₂×D₄	—	94	1	no	—
125	C₂×D₄	—	94	1	no	—
126	C₂×D₄	—	188	1	no	—
127	C₂×D₄	—	188	2	no	—
128	Q₁₆	D₁₈₈	1	1	yes	characteristic
129	SD₁₆	—	2	1	no	—
130	SD₁₆	—	94	2	no	—
131	SD₁₆	—	188	1	no	—
132	D₈	—	47	2	no	—
133	D₈	—	94	2	no	—
134	D₈	—	94	1	no	—
135	D₈	—	94	1	no	—
136	D₈	—	94	1	no	—
137	D₈	—	188	1	no	—
138	D₈	D₁₈₈	1	1	yes	characteristic
139	M₁₆	—	2	1	no	—
140	M₁₆	—	47	1	no	—
141	M₁₆	—	94	1	no	—
142	C₂×C₈	—	94	1	no	—
143	C₂×C₈	C₂²×D₄₇	1	1	yes	characteristic
144	C₁₆	—	2	1	no	—
145	C₁₆	—	94	1	no	—
146	C₂³	—	188	2	no	—
147	C₂³	—	188	1	no	—
148	C₂³	—	188	1	no	—
149	Q₈	—	2	1	no	—
150	Q₈	—	47	1	no	—
151	D₄	—	2	1	no	—
152	D₄	—	4	1	no	—
153	D₄	—	47	1	no	—
154	D₄	—	94	1	no	—
155	D₄	—	94	2	no	—
156	D₄	—	188	2	no	—
157	D₄	—	188	2	no	—
158	D₄	—	188	1	no	—
159	D₄	—	188	1	no	—
160	D₄	—	188	1	no	—
161	C₂×C₄	—	4	1	no	—
162	C₂×C₄	—	94	1	no	—
163	C₂×C₄	—	188	2	no	—
164	C₂×C₄	D₄×D₄₇	1	1	yes	characteristic
165	C₈	—	4	1	no	—
166	C₈	—	94	1	no	—
167	C₈	C₂×D₁₈₈	1	1	yes	characteristic
168	C₈	D₄×D₄₇	1	1	yes	characteristic
169	C₂²	—	4	1	no	—
170	C₂²	—	94	1	no	—
171	C₂²	—	376	1	no	—
172	C₂²	—	188	2	no	—
173	C₂²	—	752	1	no	—
174	C₂²	—	752	1	no	—
175	C₂²	—	752	2	no	—
176	C₂²	#13094051	1	1	yes	characteristic
177	C₄	—	4	1	no	—
178	C₄	—	94	1	no	—
179	C₄	#13093994	1	1	yes	characteristic
180	C₄	D₈×D₄₇	1	1	yes	characteristic
181	C₂	—	2	1	no	—
182	C₂	—	8	1	no	—
183	C₂	—	376	2	no	—
184	C₂	—	188	1	no	—
185	C₂	—	752	1	no	—
186	C₂	#15689258	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#15689258
commutator	C₂×C₈×C₄₇	C₂³
Frattini	C₂×C₈	C₂²×D₄₇
Fitting	C₄₇×#359	C₂
socle	C₂×C₄₇	#447
hypercenter	#359	D₄₇
hypocenter	C₄₇	#2192
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #359

Complements

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

N	H ≅ G/N	type
1. #15689925	C₂	semidirect
2. #15689558	C₂	semidirect
3. C₂×D₇₅₂	C₂	semidirect
4. #15689394	C₂	semidirect
5. #15688935	C₂	semidirect
6. #15688906	C₂	semidirect
7. C₄₇×#359	C₂	semidirect
14. C₂×D₃₇₆	C₂²	semidirect
20. #13093964	C₂²	semidirect
23. C₂×C₁₆×C₄₇	C₂²	semidirect
24. #108×C₄₇	C₂²	semidirect
99. #359	D₄₇	semidirect
101. C₄₇	#2192	semidirect
116. C₂×C₁₆	C₂×D₄₇	semidirect
117. #108	C₂×D₄₇	semidirect