Finite Group Databasebeta

Subgroups — #20245357

Order 6016.

216 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 49 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	#15689928	C₂	1	1	yes	maximal, characteristic
2	#15689564	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#13093967	C₂	1	1	yes	maximal, characteristic, complemented
4	#15688935	C₂	1	1	yes	maximal, characteristic, complemented
5	#15688929	C₂	1	1	yes	maximal, characteristic, complemented
6	#15688907	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₇×#473	C₂	1	1	yes	maximal, characteristic, complemented
8	#13094070	—	2	1	no	—
9	#13094052	—	2	1	no	—
10	#13094052	—	2	2	no	—
11	#13094046	—	2	1	no	—
12	#13094046	C₂²	1	1	yes	characteristic
13	C₂×D₃₇₆	—	2	1	no	—
14	C₂×D₃₇₆	C₂²	1	1	yes	characteristic
15	#13093970	—	2	1	no	—
16	#13093967	—	2	2	no	—
17	#13093967	—	2	1	no	—
18	#13093964	C₂²	1	1	yes	characteristic, complemented
19	C₂×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic, complemented
20	#137×C₄₇	—	2	1	no	—
21	C₂×Q₁₆×C₄₇	C₂²	1	1	yes	characteristic, complemented
22	#131×C₄₇	C₂²	1	1	yes	characteristic
23	#108×C₄₇	C₂²	1	1	yes	characteristic, complemented
24	#101×C₄₇	—	2	1	no	—
25	#41×C₄₇	D₄	1	1	yes	characteristic
26	C₂×Q₈×C₄₇	—	2	1	no	—
27	#10625354	—	4	1	no	—
28	D₄×D₄₇	—	4	2	no	—
29	D₄×D₄₇	—	4	1	no	—
30	#10625350	—	2	2	no	—
31	#10625350	—	2	1	no	—
32	C₂×D₁₈₈	—	2	2	no	—
33	C₂×D₁₈₈	—	2	1	no	—
34	Q₁₆×C₄₇	—	2	1	no	—
35	Q₁₆×C₄₇	—	2	1	no	—
36	Q₁₆×C₄₇	—	4	1	no	—
37	SD₁₆×C₄₇	—	4	1	no	—
38	M₁₆×C₄₇	—	2	1	no	—
39	M₁₆×C₄₇	—	2	1	no	—
40	M₁₆×C₄₇	D₄	1	1	yes	characteristic
41	C₂×C₈×C₄₇	—	2	1	no	—
42	C₂×C₈×C₄₇	C₂³	1	1	yes	characteristic
43	Dic₁₈₈	D₄	1	1	yes	characteristic
44	D₃₇₆	—	2	2	no	—
45	D₃₇₆	—	2	1	no	—
46	D₃₇₆	D₄	1	1	yes	characteristic
47	D₃₇₆	D₄	1	2	yes	—
48	#10625325	—	2	2	no	—
49	#10625325	—	2	1	no	—
50	C₄₇⋊C₁₆	—	2	1	no	—
51	C₄₇⋊C₁₆	—	2	1	no	—
52	C₂²×D₄₇	—	4	2	no	—
53	C₂²×D₄₇	—	4	1	no	—
54	Q₈×C₄₇	—	4	1	no	—
55	Q₈×C₄₇	—	4	1	no	—
56	Q₈×C₄₇	D₈	1	1	yes	characteristic
57	D₄×C₄₇	—	2	1	no	—
58	D₄×C₄₇	D₈	1	1	yes	characteristic
59	C₂×C₄×C₄₇	—	2	1	no	—
60	C₂×C₄×C₄₇	—	4	1	no	—
61	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
62	C₂²⋊D₄₇	—	4	2	no	—
63	C₂²⋊D₄₇	—	4	1	no	—
64	D₁₈₈	—	2	2	no	—
65	D₁₈₈	—	2	1	no	—
66	D₁₈₈	—	4	2	no	—
67	D₁₈₈	—	4	1	no	—
68	C₄×D₄₇	—	4	2	no	—
69	C₄×D₄₇	—	4	1	no	—
70	Dic₉₄	—	2	1	no	—
71	C₈×C₄₇	—	2	1	no	—
72	C₈×C₄₇	—	4	1	no	—
73	C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
74	C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄₇	—	2	1	no	—
76	C₂²×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
77	C₂×D₄₇	—	4	2	no	—
78	C₂×D₄₇	—	4	1	no	—
79	C₂×D₄₇	—	16	1	no	—
80	C₂×D₄₇	—	16	2	no	—
81	C₄×C₄₇	—	2	1	no	—
82	C₄×C₄₇	—	8	1	no	—
83	C₄×C₄₇	#120	1	1	yes	characteristic
84	C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
85	Dic₄₇	—	4	1	no	—
86	#2194	—	47	1	no	maximal, Sylow
87	C₂×C₄₇	—	2	1	no	—
88	C₂×C₄₇	—	4	1	no	—
89	C₂×C₄₇	#447	1	1	yes	characteristic
90	D₄₇	—	8	2	no	—
91	D₄₇	—	8	1	no	—
92	#576	—	47	1	no	—
93	#510	—	47	1	no	—
94	C₂×SD₃₂	—	47	1	no	—
95	#473	D₄₇	1	1	yes	characteristic, complemented
96	#361	—	47	1	no	—
97	#359	—	47	1	no	—
98	#350	—	47	1	no	—
99	C₄₇	#2194	1	1	yes	characteristic, complemented, Sylow
100	H(5,𝔽₂)	—	94	1	no	—
101	#137	—	2	1	no	—
102	AGL(1,ℤ/8)	—	94	2	no	—
103	AGL(1,ℤ/8)	—	94	1	no	—
104	#135	—	47	1	no	—
105	#135	—	94	1	no	—
106	C₂×Q₁₆	C₂×D₄₇	1	1	yes	characteristic, complemented
107	C₂×D₈	—	47	1	no	—
108	C₂×D₈	—	94	1	no	—
109	#131	C₂×D₄₇	1	1	yes	characteristic
110	Q₃₂	—	94	1	no	—
111	SD₃₂	—	94	2	no	—
112	SD₃₂	—	94	1	no	—
113	M₃₂	—	47	1	no	—
114	C₂×C₁₆	—	47	1	no	—
115	#108	C₂×D₄₇	1	1	yes	characteristic, complemented
116	#101	—	2	1	no	—
117	#41	—	94	2	no	—
118	#41	—	94	1	no	—
119	#41	—	188	1	no	—
120	#41	C₂²⋊D₄₇	1	1	yes	characteristic
121	C₂×Q₈	—	2	1	no	—
122	C₂×D₄	—	94	2	no	—
123	C₂×D₄	—	94	1	no	—
124	C₂×D₄	—	188	2	no	—
125	C₂×D₄	—	188	1	no	—
126	Q₁₆	—	2	1	no	—
127	Q₁₆	—	2	1	no	—
128	Q₁₆	—	4	1	no	—
129	Q₁₆	—	47	1	no	—
130	SD₁₆	—	4	1	no	—
131	SD₁₆	—	94	2	no	—
132	SD₁₆	—	94	1	no	—
133	D₈	—	47	2	no	—
134	D₈	—	47	1	no	—
135	D₈	—	94	2	no	—
136	D₈	—	94	1	no	—
137	M₁₆	—	2	1	no	—
138	M₁₆	—	2	1	no	—
139	M₁₆	C₂²⋊D₄₇	1	1	yes	characteristic
140	C₂×C₈	—	2	1	no	—
141	C₂×C₈	C₂²×D₄₇	1	1	yes	characteristic
142	C₁₆	—	94	1	no	—
143	C₁₆	—	94	1	no	—
144	C₂³	—	188	2	no	—
145	C₂³	—	188	1	no	—
146	Q₈	—	4	1	no	—
147	Q₈	—	4	1	no	—
148	Q₈	—	94	1	no	—
149	Q₈	#10625334	1	1	yes	characteristic
150	D₄	—	2	1	no	—
151	D₄	—	94	2	no	—
152	D₄	—	94	1	no	—
153	D₄	—	188	2	no	—
154	D₄	—	188	2	no	—
155	D₄	—	188	1	no	—
156	D₄	—	188	1	no	—
157	D₄	#10625334	1	1	yes	characteristic
158	C₂×C₄	—	2	1	no	—
159	C₂×C₄	—	4	1	no	—
160	C₂×C₄	—	188	2	no	—
161	C₂×C₄	—	188	1	no	—
162	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
163	C₈	—	2	1	no	—
164	C₈	—	4	1	no	—
165	C₈	D₄×D₄₇	1	1	yes	characteristic
166	C₈	D₄×D₄₇	1	1	yes	characteristic
167	C₂²	—	2	1	no	—
168	C₂²	—	188	2	no	—
169	C₂²	—	188	1	no	—
170	C₂²	—	752	1	no	—
171	C₂²	—	752	2	no	—
172	C₂²	#13094049	1	1	yes	characteristic
173	C₄	—	2	1	no	—
174	C₄	—	8	1	no	—
175	C₄	—	188	1	no	—
176	C₄	#13093992	1	1	yes	characteristic
177	C₄	C₂×#10625334	1	1	yes	characteristic
178	C₂	—	2	1	no	—
179	C₂	—	4	1	no	—
180	C₂	—	376	2	no	—
181	C₂	—	376	1	no	—
182	C₂	#15689256	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#15689256
commutator	C₂×C₈×C₄₇	C₂³
Frattini	C₂×C₈	C₂²×D₄₇
Fitting	C₄₇×#473	C₂
socle	C₂×C₄₇	#447
hypercenter	#473	D₄₇
hypocenter	C₄₇	#2194
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #131 ◁ #473

Complements

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

N	H ≅ G/N	type
2. #15689564	C₂	semidirect
3. C₂×#13093967	C₂	semidirect
4. #15688935	C₂	semidirect
5. #15688929	C₂	semidirect
6. #15688907	C₂	semidirect
7. C₄₇×#473	C₂	semidirect
18. #13093964	C₂²	semidirect
19. C₂×C₄₇⋊C₁₆	C₂²	semidirect
21. C₂×Q₁₆×C₄₇	C₂²	semidirect
23. #108×C₄₇	C₂²	semidirect
95. #473	D₄₇	semidirect
99. C₄₇	#2194	semidirect
106. C₂×Q₁₆	C₂×D₄₇	semidirect
115. #108	C₂×D₄₇	semidirect