Finite Group Databasebeta

Subgroups — #20245336

Order 6016.

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

normal subgroups only

This group has 226 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	#15689923	C₂	1	1	yes	maximal, characteristic, complemented
2	#15689561	C₂	1	2	yes	maximal, complemented
3	#15689376	C₂	1	1	yes	maximal, characteristic, complemented
4	#15688934	C₂	1	2	yes	maximal, complemented
5	C₄₇×#349	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#10625350	—	2	1	no	—
7	#13094057	—	2	2	no	—
8	#13094052	—	2	2	no	—
9	#13094046	—	2	1	no	—
10	#13094046	C₂²	1	2	yes	—
11	C₂×D₃₇₆	C₂²	1	1	yes	characteristic, complemented
12	#13093998	—	2	1	no	—
13	#13093969	—	2	2	no	—
14	D₇₅₂	—	2	2	no	—
15	#13093956	C₂²	1	1	yes	characteristic, complemented
16	#13093946	—	2	1	no	—
17	C₂×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
18	M₃₂×C₄₇	C₂²	1	2	yes	complemented
19	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
20	C₂×C₂²⋊D₄₇	—	4	1	no	—
21	C₂×C₂²⋊D₄₇	—	4	1	no	—
22	#10625350	—	2	2	no	—
23	#10625350	—	2	2	no	—
24	#10625350	—	4	1	no	—
25	#10625350	—	4	1	no	—
26	C₂×D₁₈₈	—	2	1	no	—
27	C₂×D₁₈₈	—	2	1	no	—
28	C₂×C₄×D₄₇	—	4	1	no	—
29	C₂×Dic₉₄	—	2	1	no	—
30	M₁₆×C₄₇	—	2	2	no	—
31	C₂×C₈×C₄₇	D₄	1	1	yes	characteristic
32	C₂×C₈×C₄₇	C₂³	1	1	yes	characteristic
33	(C₂×D₄₇)⋊C₄	—	4	1	no	—
34	C₄⋊Dic₄₇	—	2	1	no	—
35	Dic₁₈₈	—	2	1	no	—
36	Dic₁₈₈	D₄	1	2	yes	complemented
37	D₃₇₆	—	2	1	no	—
38	D₃₇₆	—	2	2	no	—
39	D₃₇₆	D₄	1	2	yes	complemented
40	#10625325	—	2	2	no	—
41	#10625325	—	2	2	no	—
42	C₁₆×C₄₇	—	2	2	no	—
43	C₂³×C₄₇	D₈	1	1	yes	characteristic
44	C₂²×D₄₇	—	4	1	no	—
45	C₂²×D₄₇	—	4	1	no	—
46	C₂×C₄×C₄₇	—	2	1	no	—
47	C₂×C₄×C₄₇	—	2	1	no	—
48	C₂×C₄×C₄₇	D₈	1	1	yes	characteristic
49	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
50	C₂²⋊D₄₇	—	4	2	no	—
51	C₂²⋊D₄₇	—	4	2	no	—
52	C₂²⋊D₄₇	—	8	2	no	—
53	C₂×Dic₄₇	—	4	1	no	—
54	C₂×Dic₄₇	—	4	1	no	—
55	D₁₈₈	—	2	2	no	—
56	D₁₈₈	—	4	1	no	—
57	D₁₈₈	—	4	1	no	—
58	D₁₈₈	—	4	1	no	—
59	C₄×D₄₇	—	4	2	no	—
60	C₄×D₄₇	—	4	2	no	—
61	Dic₉₄	—	2	2	no	—
62	Dic₉₄	—	4	1	no	—
63	C₈×C₄₇	—	2	1	no	—
64	C₈×C₄₇	C₂×D₄	1	2	yes	—
65	C₂²×C₄₇	—	2	1	no	—
66	C₂²×C₄₇	—	4	1	no	—
67	C₂²×C₄₇	C₂×D₈	1	1	yes	characteristic
68	C₂×D₄₇	—	4	2	no	—
69	C₂×D₄₇	—	8	2	no	—
70	C₂×D₄₇	—	8	1	no	—
71	C₂×D₄₇	—	8	1	no	—
72	C₂×D₄₇	—	8	1	no	—
73	C₄×C₄₇	—	2	1	no	—
74	C₄×C₄₇	#120	1	1	yes	characteristic
75	C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
76	Dic₄₇	—	4	2	no	—
77	Dic₄₇	—	8	1	no	—
78	#2190	—	47	1	no	maximal, Sylow
79	C₂×C₄₇	—	2	1	no	—
80	C₂×C₄₇	—	4	1	no	—
81	C₂×C₄₇	#447	1	1	yes	characteristic
82	D₄₇	—	8	2	no	—
83	D₄₇	—	16	1	no	—
84	#575	—	47	1	no	—
85	#509	—	47	2	no	—
86	#469	—	47	1	no	—
87	#360	—	47	2	no	—
88	#349	D₄₇	1	1	yes	characteristic, complemented
89	C₄₇	#2190	1	1	yes	characteristic, complemented, Sylow
90	C₂×#41	—	94	1	no	—
91	#137	—	94	2	no	—
92	AGL(1,ℤ/8)	—	94	2	no	—
93	#135	—	47	2	no	—
94	#135	—	94	1	no	—
95	C₂×D₈	—	47	1	no	—
96	C₂×M₁₆	C₂×D₄₇	1	1	yes	characteristic
97	#121	—	94	1	no	—
98	SD₃₂	—	94	2	no	—
99	D₁₆	—	94	2	no	—
100	M₃₂	C₂×D₄₇	1	2	yes	complemented
101	#106	—	47	1	no	—
102	#102	—	94	1	no	—
103	#41	—	94	2	no	—
104	#41	—	94	2	no	—
105	#41	—	188	1	no	—
106	#41	—	188	1	no	—
107	C₂×Q₈	—	94	1	no	—
108	C₂×D₄	—	94	1	no	—
109	C₂×D₄	—	94	1	no	—
110	C₂×D₄	—	188	1	no	—
111	C₂×D₄	—	188	1	no	—
112	C₂²×C₄	—	188	1	no	—
113	C₂²×C₄	D₁₈₈	1	1	yes	characteristic
114	Q₁₆	—	47	2	no	—
115	Q₁₆	—	94	1	no	—
116	SD₁₆	—	94	2	no	—
117	SD₁₆	—	94	2	no	—
118	D₈	—	47	2	no	—
119	D₈	—	94	1	no	—
120	D₈	—	94	2	no	—
121	M₁₆	—	2	2	no	—
122	C₂×C₈	D₁₈₈	1	1	yes	characteristic
123	C₂×C₈	C₂²×D₄₇	1	1	yes	characteristic
124	C₄⋊C₄	—	94	1	no	—
125	C₂²⋊C₄	—	188	1	no	—
126	C₁₆	—	2	2	no	—
127	C₂³	—	188	1	no	—
128	C₂³	—	188	1	no	—
129	C₂³	D₃₇₆	1	1	yes	characteristic
130	Q₈	—	94	2	no	—
131	Q₈	—	188	1	no	—
132	D₄	—	94	2	no	—
133	D₄	—	376	2	no	—
134	D₄	—	188	1	no	—
135	D₄	—	188	2	no	—
136	D₄	—	188	2	no	—
137	D₄	—	188	1	no	—
138	D₄	—	188	1	no	—
139	C₂×C₄	—	2	1	no	—
140	C₂×C₄	—	2	1	no	—
141	C₂×C₄	—	188	2	no	—
142	C₂×C₄	—	188	1	no	—
143	C₂×C₄	—	188	1	no	—
144	C₂×C₄	—	188	2	no	—
145	C₂×C₄	D₃₇₆	1	1	yes	characteristic
146	C₂×C₄	C₂×D₁₈₈	1	1	yes	characteristic
147	C₈	—	2	1	no	—
148	C₈	D₄×D₄₇	1	2	yes	—
149	C₂²	—	2	1	no	—
150	C₂²	—	4	1	no	—
151	C₂²	—	376	1	no	—
152	C₂²	—	376	1	no	—
153	C₂²	—	376	2	no	—
154	C₂²	—	376	1	no	—
155	C₂²	—	188	2	no	—
156	C₂²	C₂×D₃₇₆	1	1	yes	characteristic
157	C₄	—	2	1	no	—
158	C₄	—	376	1	no	—
159	C₄	—	188	2	no	—
160	C₄	#13093994	1	1	yes	characteristic
161	C₄	#13094052	1	1	yes	characteristic
162	C₂	—	2	1	no	—
163	C₂	—	4	1	no	—
164	C₂	—	376	2	no	—
165	C₂	—	752	1	no	—
166	C₂	#15689259	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#15689259
commutator	C₂×C₈×C₄₇	C₂³
Frattini	C₂×C₈	C₂²×D₄₇
Fitting	C₄₇×#349	C₂
socle	C₂×C₄₇	#447
hypercenter	#349	D₄₇
hypocenter	C₄₇	#2190
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×M₁₆ ◁ #349

Complements

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

N	H ≅ G/N	type
1. #15689923	C₂	semidirect
2. #15689561	C₂	semidirect
3. #15689376	C₂	semidirect
4. #15688934	C₂	semidirect
5. C₄₇×#349	C₂	semidirect
11. C₂×D₃₇₆	C₂²	semidirect
15. #13093956	C₂²	semidirect
18. M₃₂×C₄₇	C₂²	semidirect
36. Dic₁₈₈	D₄	semidirect
39. D₃₇₆	D₄	semidirect
88. #349	D₄₇	semidirect
89. C₄₇	#2190	semidirect
100. M₃₂	C₂×D₄₇	semidirect