Finite Group Databasebeta

Subgroups — #20242923

Order 6016.

170 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 51 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	#15689471	C₂	1	1	yes	maximal, characteristic, complemented
2	#15689318	C₂	1	1	yes	maximal, characteristic
3	#15689201	C₂	1	1	yes	maximal, characteristic, complemented
4	#15688875	C₂	1	1	yes	maximal, characteristic, complemented
5	#15688872	C₂	1	1	yes	maximal, characteristic, complemented
6	#15688866	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₇×#476	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#10625334	—	2	1	no	—
9	C₂×#10625329	—	2	1	no	—
10	#13094021	C₂²	1	1	yes	characteristic
11	C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
12	C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
13	#13093944	—	2	1	no	—
14	#13093944	—	2	1	no	—
15	#13093944	—	2	1	no	—
16	C₂²⋊(C₄₇⋊C₈)	—	2	1	no	—
17	C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
18	C₄⋊Q₈×C₄₇	C₂²	1	1	yes	characteristic
19	C₄×D₄×C₄₇	C₂²	1	1	yes	characteristic
20	#106×C₄₇	—	2	1	no	—
21	C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
22	#102×C₄₇	—	2	1	no	—
23	C₂×Q₈×C₄₇	—	2	1	no	—
24	C₂×D₄×C₄₇	D₄	1	1	yes	characteristic
25	C₂²×C₄×C₄₇	—	2	1	no	—
26	C₂×D₁₈₈	—	2	1	no	—
27	C₂×D₁₈₈	—	2	1	no	—
28	C₂×D₁₈₈	—	2	1	no	—
29	C₂×C₈×C₄₇	—	2	1	no	—
30	C₄⋊C₄×C₄₇	—	2	1	no	—
31	C₄⋊C₄×C₄₇	—	2	1	no	—
32	C₄⋊C₄×C₄₇	D₄	1	1	yes	characteristic
33	C₂²⋊C₄×C₄₇	—	2	1	no	—
34	C₄²×C₄₇	C₂³	1	1	yes	characteristic
35	#10625334	—	4	2	no	—
36	#10625329	—	4	2	no	—
37	C₂×C₄₇⋊C₈	—	2	1	no	—
38	C₂×C₄₇⋊C₈	—	2	1	no	—
39	C₂×C₄₇⋊C₈	D₄	1	1	yes	characteristic
40	C₂×C₄₇⋊C₈	D₄	1	1	yes	characteristic
41	C₂³×C₄₇	—	2	1	no	—
42	C₂²×D₄₇	—	4	1	no	—
43	Q₈×C₄₇	—	8	1	no	—
44	D₄×C₄₇	—	2	1	no	—
45	D₄×C₄₇	—	2	1	no	—
46	C₂×C₄×C₄₇	—	2	1	no	—
47	C₂×C₄×C₄₇	—	4	1	no	—
48	C₂×C₄×C₄₇	—	4	1	no	—
49	C₂×C₄×C₄₇	—	4	1	no	—
50	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₄₇	#41	1	1	yes	characteristic
53	D₁₈₈	—	4	1	no	—
54	D₁₈₈	—	4	1	no	—
55	D₁₈₈	—	4	1	no	—
56	D₁₈₈	—	4	1	no	—
57	D₁₈₈	—	8	1	no	—
58	C₈×C₄₇	—	4	1	no	—
59	C₄₇⋊C₈	—	2	1	no	—
60	C₄₇⋊C₈	—	2	1	no	—
61	C₄₇⋊C₈	—	4	1	no	—
62	C₄₇⋊C₈	—	4	1	no	—
63	C₂²×C₄₇	—	4	1	no	—
64	C₂²×C₄₇	—	4	1	no	—
65	C₂²×C₄₇	—	4	1	no	—
66	C₂²×C₄₇	#121	1	1	yes	characteristic
67	C₂×D₄₇	—	8	1	no	—
68	C₂×D₄₇	—	8	1	no	—
69	C₂×D₄₇	—	8	1	no	—
70	C₄×C₄₇	—	2	1	no	—
71	C₄×C₄₇	—	4	1	no	—
72	C₄×C₄₇	—	8	1	no	—
73	C₄×C₄₇	#135	1	1	yes	characteristic
74	C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
75	C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
76	C₄×C₄₇	#137	1	1	yes	characteristic
77	#1697	—	47	1	no	maximal, Sylow
78	C₂×C₄₇	—	8	1	no	—
79	C₂×C₄₇	#463	1	1	yes	characteristic
80	C₂×C₄₇	#468	1	1	yes	characteristic
81	C₂×C₄₇	#472	1	1	yes	characteristic
82	D₄₇	—	16	1	no	—
83	#486	—	47	1	no	—
84	#476	D₄₇	1	1	yes	characteristic, complemented
85	#459	—	47	1	no	—
86	#436	—	47	1	no	—
87	#335	—	47	1	no	—
88	#333	—	47	1	no	—
89	#331	—	47	1	no	—
90	C₄₇	#1697	1	1	yes	characteristic, complemented, Sylow
91	C₂×D₈	—	94	1	no	—
92	C₂×M₁₆	—	94	1	no	—
93	C₄⋊Q₈	C₂×D₄₇	1	1	yes	characteristic
94	#127	—	47	1	no	—
95	C₄×D₄	C₂×D₄₇	1	1	yes	characteristic
96	#106	—	2	1	no	—
97	C₄⋊C₈	—	47	1	no	—
98	C₄⋊C₈	—	47	1	no	—
99	C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
100	#102	—	2	1	no	—
101	#102	—	94	1	no	—
102	#102	—	94	1	no	—
103	#102	—	94	1	no	—
104	C₂²⋊C₈	—	94	1	no	—
105	C₄×C₈	—	47	1	no	—
106	C₂×Q₈	—	2	1	no	—
107	C₂×D₄	—	94	1	no	—
108	C₂×D₄	—	94	1	no	—
109	C₂×D₄	—	94	1	no	—
110	C₂×D₄	C₂²⋊D₄₇	1	1	yes	characteristic
111	C₂²×C₄	—	2	1	no	—
112	D₈	—	188	2	no	—
113	M₁₆	—	188	2	no	—
114	C₂×C₈	—	2	1	no	—
115	C₂×C₈	—	47	1	no	—
116	C₂×C₈	—	47	1	no	—
117	C₂×C₈	—	94	1	no	—
118	C₂×C₈	—	94	1	no	—
119	C₄⋊C₄	—	2	1	no	—
120	C₄⋊C₄	—	2	1	no	—
121	C₄⋊C₄	C₂²⋊D₄₇	1	1	yes	characteristic
122	C₂²⋊C₄	—	2	1	no	—
123	C₄²	C₂²×D₄₇	1	1	yes	characteristic
124	C₂³	—	2	1	no	—
125	C₂³	—	188	1	no	—
126	Q₈	—	8	1	no	—
127	D₄	—	2	1	no	—
128	D₄	—	2	1	no	—
129	D₄	—	376	1	no	—
130	D₄	—	188	1	no	—
131	D₄	—	188	1	no	—
132	D₄	—	188	1	no	—
133	D₄	—	188	1	no	—
134	C₂×C₄	—	2	1	no	—
135	C₂×C₄	—	4	1	no	—
136	C₂×C₄	—	4	1	no	—
137	C₂×C₄	—	4	1	no	—
138	C₂×C₄	D₄×D₄₇	1	1	yes	characteristic
139	C₂×C₄	#10625352	1	1	yes	characteristic
140	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
141	C₈	—	4	1	no	—
142	C₈	—	94	1	no	—
143	C₈	—	94	1	no	—
144	C₈	—	188	1	no	—
145	C₈	—	188	1	no	—
146	C₂²	—	4	1	no	—
147	C₂²	—	4	1	no	—
148	C₂²	—	4	1	no	—
149	C₂²	—	376	1	no	—
150	C₂²	—	376	1	no	—
151	C₂²	—	376	1	no	—
152	C₂²	#13093996	1	1	yes	characteristic
153	C₄	—	2	1	no	—
154	C₄	—	4	1	no	—
155	C₄	—	8	1	no	—
156	C₄	#13094045	1	1	yes	characteristic
157	C₄	#13094049	1	1	yes	characteristic
158	C₄	#13094051	1	1	yes	characteristic
159	C₄	#13094055	1	1	yes	characteristic
160	C₂	—	8	1	no	—
161	C₂	—	752	1	no	—
162	C₂	#15689340	1	1	yes	characteristic
163	C₂	#15689371	1	1	yes	characteristic
164	C₂	#15689391	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#13093996
commutator	C₄²×C₄₇	C₂³
Frattini	C₄²	C₂²×D₄₇
Fitting	C₄₇×#476	C₂
socle	C₂²×C₄₇	#121
hypercenter	#476	D₄₇
hypocenter	C₄₇	#1697
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #476

Complements

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

N	H ≅ G/N	type
1. #15689471	C₂	semidirect
3. #15689201	C₂	semidirect
4. #15688875	C₂	semidirect
5. #15688872	C₂	semidirect
6. #15688866	C₂	semidirect
7. C₄₇×#476	C₂	semidirect
84. #476	D₄₇	semidirect
90. C₄₇	#1697	semidirect