Finite Group Databasebeta

Subgroups — #20245099

Order 6016.

232 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 69 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	C₂×#13094047	C₂	1	1	yes	maximal, characteristic
2	C₂×#13093963	C₂	1	1	yes	maximal, characteristic, complemented
3	#107×D₄₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#15688927	C₂	1	1	yes	maximal, characteristic, complemented
5	#15688925	C₂	1	1	yes	maximal, characteristic, complemented
6	#15688923	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₇×#358	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#10625354	—	2	1	no	—
9	#13094047	—	2	2	no	—
10	#13094047	C₄	1	4	yes	complemented
11	C₂×#10625336	—	2	1	no	—
12	C₂×D₃₇₆	C₂²	1	1	yes	characteristic
13	C₂×C₈×D₄₇	C₂²	1	1	yes	characteristic
14	C₄⋊C₄×D₄₇	—	2	1	no	—
15	#13093963	—	2	2	no	—
16	C₂×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
17	#13093958	C₂²	1	1	yes	characteristic
18	#13093957	—	2	1	no	—
19	C₂×Q₁₆×C₄₇	C₂²	1	1	yes	characteristic
20	C₂×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
21	#107×C₄₇	C₂²	1	1	yes	characteristic
22	C₂×Q₈×C₄₇	—	2	1	no	—
23	#10625354	—	2	4	no	—
24	#10625354	—	4	2	no	—
25	C₂×D₁₈₈	—	2	1	no	—
26	C₂×D₁₈₈	—	4	1	no	—
27	C₂×C₄×D₄₇	—	4	1	no	—
28	C₂×C₄×D₄₇	—	4	1	no	—
29	C₂×C₄×D₄₇	D₄	1	1	yes	characteristic
30	Q₁₆×C₄₇	—	2	1	no	—
31	Q₁₆×C₄₇	C₂×C₄	1	2	yes	complemented
32	C₂×C₈×C₄₇	C₂³	1	1	yes	characteristic
33	C₄⋊C₄×C₄₇	—	2	1	no	—
34	#10625336	—	2	4	no	—
35	C₄⋊Dic₄₇	—	2	1	no	—
36	Dic₄₇⋊C₄	—	4	1	no	—
37	C₂×C₄₇⋊C₈	D₄	1	1	yes	characteristic
38	D₃₇₆	—	2	1	no	—
39	D₃₇₆	C₂×C₄	1	2	yes	complemented
40	C₈×D₄₇	C₂×C₄	1	2	yes	—
41	C₈×D₄₇	D₄	1	2	yes	—
42	C₁₆×C₄₇	—	2	1	no	—
43	C₄₇⋊C₁₆	—	2	1	no	—
44	C₂²×D₄₇	—	4	1	no	—
45	C₂²×D₄₇	D₈	1	1	yes	characteristic
46	Q₈×C₄₇	—	2	2	no	—
47	Q₈×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×Dic₄₇	—	4	1	no	—
52	C₂×Dic₄₇	D₈	1	1	yes	characteristic
53	D₁₈₈	—	2	2	no	—
54	D₁₈₈	—	4	4	no	—
55	D₁₈₈	—	4	1	no	—
56	C₄×D₄₇	—	4	4	no	—
57	C₄×D₄₇	—	8	2	no	—
58	C₄×D₄₇	C₂²⋊C₄	1	2	yes	—
59	C₄×D₄₇	SD₁₆	1	2	yes	—
60	C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
61	C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
62	C₄₇⋊C₈	C₂²⋊C₄	1	2	yes	—
63	C₂²×C₄₇	C₂×D₈	1	1	yes	characteristic
64	C₂×D₄₇	—	2	1	no	—
65	C₂×D₄₇	—	2	1	no	—
66	C₂×D₄₇	—	4	2	no	—
67	C₂×D₄₇	—	8	1	no	—
68	C₂×D₄₇	—	8	1	no	—
69	C₂×D₄₇	#102	1	2	yes	—
70	C₄×C₄₇	—	4	2	no	—
71	C₄×C₄₇	—	8	1	no	—
72	C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
73	C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
74	Dic₄₇	—	8	1	no	—
75	Dic₄₇	#102	1	2	yes	—
76	#2139	—	47	1	no	maximal, Sylow
77	C₂×C₄₇	C₂×#102	1	1	yes	characteristic
78	C₂×C₄₇	#509	1	1	yes	characteristic
79	C₂×C₄₇	#510	1	1	yes	characteristic
80	D₄₇	—	2	2	no	—
81	D₄₇	—	8	2	no	—
82	C₂×#135	—	47	1	no	—
83	C₂×M₃₂	—	47	1	no	—
84	C₂×#107	—	47	1	no	—
85	#358	—	47	1	no	—
86	#358	D₄₇	1	1	yes	characteristic, complemented
87	#357	—	47	1	no	—
88	#357	—	47	1	no	—
89	C₄₇	#2139	1	1	yes	characteristic, complemented, Sylow
90	C₂×#41	—	94	1	no	—
91	#135	—	47	4	no	—
92	#135	—	94	2	no	—
93	C₂×Q₁₆	C₂×D₄₇	1	1	yes	characteristic
94	C₂×SD₁₆	—	94	1	no	—
95	C₂×D₈	—	47	1	no	—
96	C₂²×C₈	—	47	1	no	—
97	C₂×C₄⋊C₄	—	94	1	no	—
98	M₃₂	—	94	2	no	—
99	C₂×C₁₆	—	47	1	no	—
100	C₂×C₁₆	C₂×D₄₇	1	1	yes	characteristic
101	#107	—	47	1	no	—
102	#107	—	94	1	no	—
103	#107	C₂×D₄₇	1	1	yes	characteristic
104	#41	—	94	4	no	—
105	#41	—	188	2	no	—
106	C₂×Q₈	—	2	1	no	—
107	C₂×D₄	—	94	1	no	—
108	C₂×D₄	—	188	1	no	—
109	C₂²×C₄	—	47	1	no	—
110	C₂²×C₄	—	188	1	no	—
111	C₂²×C₄	—	188	1	no	—
112	Q₁₆	—	2	1	no	—
113	Q₁₆	C₄×D₄₇	1	2	yes	complemented
114	SD₁₆	—	94	4	no	—
115	D₈	—	47	2	no	—
116	D₈	—	94	1	no	—
117	C₂×C₈	—	47	1	no	—
118	C₂×C₈	—	47	2	no	—
119	C₂×C₈	—	47	2	no	—
120	C₂×C₈	C₂²×D₄₇	1	1	yes	characteristic
121	C₄⋊C₄	—	2	1	no	—
122	C₄⋊C₄	—	94	1	no	—
123	C₄⋊C₄	—	188	1	no	—
124	C₁₆	—	2	1	no	—
125	C₁₆	—	94	1	no	—
126	C₂³	—	47	1	no	—
127	C₂³	—	188	1	no	—
128	Q₈	—	2	2	no	—
129	Q₈	—	4	1	no	—
130	D₄	—	94	2	no	—
131	D₄	—	188	4	no	—
132	D₄	—	188	1	no	—
133	C₂×C₄	—	4	1	no	—
134	C₂×C₄	—	4	1	no	—
135	C₂×C₄	—	47	1	no	—
136	C₂×C₄	—	47	2	no	—
137	C₂×C₄	—	47	2	no	—
138	C₂×C₄	—	376	2	no	—
139	C₂×C₄	—	188	4	no	—
140	C₂×C₄	—	188	1	no	—
141	C₂×C₄	D₄×D₄₇	1	1	yes	characteristic
142	C₈	—	47	2	no	—
143	C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
144	C₈	D₄×D₄₇	1	1	yes	characteristic
145	C₂²	—	47	2	no	—
146	C₂²	—	94	1	no	—
147	C₂²	—	94	1	no	—
148	C₂²	—	376	1	no	—
149	C₂²	—	376	1	no	—
150	C₂²	—	188	2	no	—
151	C₂²	D₈×D₄₇	1	1	yes	characteristic
152	C₄	—	4	2	no	—
153	C₄	—	8	1	no	—
154	C₄	—	47	2	no	—
155	C₄	—	376	1	no	—
156	C₄	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
157	C₄	SD₁₆×D₄₇	1	1	yes	characteristic
158	C₂	—	94	2	no	—
159	C₂	—	376	2	no	—
160	C₂	#102×D₄₇	1	1	yes	characteristic
161	C₂	#15689560	1	1	yes	characteristic
162	C₂	#15689565	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₈×D₄₇
commutator	C₈×C₄₇	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₄₇
Fitting	C₄₇×#358	C₂
socle	C₂²×C₄₇	C₂×D₈
hypercenter	#358	D₄₇
hypocenter	C₄₇	#2139
LCS	G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #358

Complements

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

N	H ≅ G/N	type
2. C₂×#13093963	C₂	semidirect
3. #107×D₄₇	C₂	semidirect
4. #15688927	C₂	semidirect
5. #15688925	C₂	semidirect
6. #15688923	C₂	semidirect
7. C₄₇×#358	C₂	semidirect
10. #13094047	C₄	semidirect
31. Q₁₆×C₄₇	C₂×C₄	semidirect
39. D₃₇₆	C₂×C₄	semidirect
86. #358	D₄₇	semidirect
89. C₄₇	#2139	semidirect
113. Q₁₆	C₄×D₄₇	semidirect