Finite Group Databasebeta

Subgroups — #20244839

Order 6016.

190 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 190 conjugacy classes of subgroups — too many to draw. Only its 59 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	#15689141	C₂	1	1	yes	maximal, characteristic
2	#15689120	C₂	1	1	yes	maximal, characteristic
3	#15689105	C₂	1	1	yes	maximal, characteristic
4	#15688895	C₂	1	1	yes	maximal, characteristic, complemented
5	#15688888	C₂	1	1	yes	maximal, characteristic, complemented
6	#15688879	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₇×#424	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#10625350	C₂²	1	1	yes	characteristic
9	C₂×#10625329	C₂²	1	1	yes	characteristic
10	C₂×#10625329	C₂²	1	1	yes	characteristic
11	#13093983	C₂²	1	1	yes	characteristic
12	#13093952	—	2	2	no	—
13	#13093949	—	2	1	no	—
14	#13093946	—	2	1	no	—
15	#13093942	—	2	1	no	—
16	#13093940	—	2	1	no	—
17	C₂×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
18	C₂²×C₈×C₄₇	C₂²	1	1	yes	characteristic
19	#117×C₄₇	C₂²	1	1	yes	characteristic
20	C₄⋊C₈×C₄₇	—	2	2	no	—
21	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
22	C₂×C₂²⋊D₄₇	—	2	1	no	—
23	#10625350	—	2	1	no	—
24	#10625350	—	2	1	no	—
25	#10625350	—	2	1	no	—
26	#10625350	—	2	1	no	—
27	C₂×D₁₈₈	D₄	1	1	yes	characteristic
28	C₂×C₄×D₄₇	—	2	1	no	—
29	C₂×Dic₉₄	D₄	1	1	yes	characteristic
30	M₁₆×C₄₇	—	2	1	no	—
31	M₁₆×C₄₇	—	2	1	no	—
32	C₂×C₈×C₄₇	—	2	1	no	—
33	C₂×C₈×C₄₇	—	2	1	no	—
34	C₂×C₈×C₄₇	—	2	1	no	—
35	C₂×C₈×C₄₇	—	2	1	no	—
36	C₄⋊C₄×C₄₇	—	2	1	no	—
37	C₂²⋊C₄×C₄₇	—	2	1	no	—
38	C₄²×C₄₇	D₄	1	2	yes	—
39	C₂²⋊Dic₄₇	—	2	1	no	—
40	C₄⋊Dic₄₇	—	2	1	no	—
41	C₄×Dic₄₇	—	2	1	no	—
42	#10625329	—	2	1	no	—
43	#10625329	—	2	1	no	—
44	#10625329	—	2	1	no	—
45	#10625329	—	2	1	no	—
46	C₂×C₄₇⋊C₈	—	2	1	no	—
47	C₂×C₄₇⋊C₈	—	2	1	no	—
48	C₂³×C₄₇	#41	1	1	yes	characteristic
49	C₂²×D₄₇	—	2	1	no	—
50	C₂×C₄×C₄₇	—	2	2	no	—
51	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₄₇	#41	1	1	yes	characteristic
54	C₂×C₄×C₄₇	#41	1	1	yes	characteristic
55	C₂×C₄×C₄₇	#41	1	1	yes	characteristic
56	C₂×C₄×C₄₇	#41	1	1	yes	characteristic
57	C₂²⋊D₄₇	—	4	1	no	—
58	C₂²⋊D₄₇	—	4	1	no	—
59	C₂×Dic₄₇	—	2	1	no	—
60	C₂×Dic₄₇	—	4	1	no	—
61	D₁₈₈	—	2	1	no	—
62	D₁₈₈	—	2	1	no	—
63	C₄×D₄₇	—	4	1	no	—
64	C₄×D₄₇	—	4	1	no	—
65	Dic₉₄	—	2	1	no	—
66	Dic₉₄	—	2	1	no	—
67	C₈×C₄₇	—	4	1	no	—
68	C₈×C₄₇	—	4	1	no	—
69	C₄₇⋊C₈	—	4	1	no	—
70	C₄₇⋊C₈	—	4	1	no	—
71	C₂²×C₄₇	—	4	1	no	—
72	C₂²×C₄₇	#123	1	1	yes	characteristic
73	C₂²×C₄₇	#123	1	1	yes	characteristic
74	C₂²×C₄₇	#126	1	1	yes	characteristic
75	C₂×D₄₇	—	4	1	no	—
76	C₂×D₄₇	—	8	1	no	—
77	C₄×C₄₇	—	4	2	no	—
78	C₄×C₄₇	#121	1	1	yes	characteristic
79	C₄×C₄₇	#123	1	1	yes	characteristic
80	C₄×C₄₇	#124	1	1	yes	characteristic
81	C₄×C₄₇	#126	1	1	yes	characteristic
82	Dic₄₇	—	4	1	no	—
83	Dic₄₇	—	8	1	no	—
84	#2080	—	47	1	no	maximal, Sylow
85	C₂×C₄₇	—	2	1	no	—
86	C₂×C₄₇	—	2	1	no	—
87	C₂×C₄₇	—	2	1	no	—
88	C₂×C₄₇	#397	1	1	yes	characteristic
89	D₄₇	—	8	1	no	—
90	#424	D₄₇	1	1	yes	characteristic, complemented
91	#421	—	47	1	no	—
92	#416	—	47	1	no	—
93	#413	—	47	1	no	—
94	#344	—	47	1	no	—
95	#341	—	47	1	no	—
96	H(3,ℤ/4)	—	47	1	no	—
97	C₄₇	#2080	1	1	yes	characteristic, complemented, Sylow
98	C₂×#41	—	47	1	no	—
99	C₂×M₁₆	—	47	1	no	—
100	C₂×M₁₆	—	47	1	no	—
101	C₂×M₁₆	C₂×D₄₇	1	1	yes	characteristic
102	C₂²×C₈	C₂×D₄₇	1	1	yes	characteristic
103	#117	—	47	1	no	—
104	#117	C₂×D₄₇	1	1	yes	characteristic
105	C₄⋊C₈	—	2	2	no	—
106	#104	—	94	2	no	—
107	#103	—	94	1	no	—
108	#102	—	94	1	no	—
109	#101	—	94	1	no	—
110	#100	—	94	1	no	—
111	#41	—	94	1	no	—
112	#41	—	94	1	no	—
113	#41	—	94	1	no	—
114	#41	—	94	1	no	—
115	C₂×Q₈	—	47	1	no	—
116	C₂×D₄	—	47	1	no	—
117	C₂×D₄	—	94	1	no	—
118	C₂²×C₄	—	94	1	no	—
119	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
120	M₁₆	—	2	1	no	—
121	M₁₆	—	2	1	no	—
122	M₁₆	—	94	1	no	—
123	M₁₆	—	94	1	no	—
124	M₁₆	—	94	1	no	—
125	M₁₆	—	94	1	no	—
126	C₂×C₈	—	2	1	no	—
127	C₂×C₈	—	2	1	no	—
128	C₂×C₈	—	2	1	no	—
129	C₂×C₈	—	2	1	no	—
130	C₂×C₈	—	94	1	no	—
131	C₂×C₈	—	94	1	no	—
132	C₄⋊C₄	—	2	1	no	—
133	C₄⋊C₄	—	94	1	no	—
134	C₂²⋊C₄	—	2	1	no	—
135	C₂²⋊C₄	—	94	1	no	—
136	C₄²	—	94	1	no	—
137	C₄²	C₂²⋊D₄₇	1	2	yes	—
138	C₂³	—	94	1	no	—
139	C₂³	#10625350	1	1	yes	characteristic
140	Q₈	—	94	1	no	—
141	Q₈	—	94	1	no	—
142	D₄	—	94	1	no	—
143	D₄	—	94	1	no	—
144	D₄	—	188	1	no	—
145	D₄	—	188	1	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	94	1	no	—
148	C₂×C₄	—	188	1	no	—
149	C₂×C₄	—	188	1	no	—
150	C₂×C₄	—	188	1	no	—
151	C₂×C₄	#10625350	1	1	yes	characteristic
152	C₂×C₄	D₄×D₄₇	1	1	yes	characteristic
153	C₂×C₄	#10625352	1	1	yes	characteristic
154	C₂×C₄	#10625354	1	1	yes	characteristic
155	C₂×C₄	#10625354	1	1	yes	characteristic
156	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
157	C₈	—	4	1	no	—
158	C₈	—	4	1	no	—
159	C₈	—	188	1	no	—
160	C₈	—	188	1	no	—
161	C₂²	—	4	1	no	—
162	C₂²	—	376	1	no	—
163	C₂²	—	188	1	no	—
164	C₂²	#13094007	1	1	yes	characteristic
165	C₂²	#13094009	1	1	yes	characteristic
166	C₂²	#13094019	1	1	yes	characteristic
167	C₄	—	4	2	no	—
168	C₄	—	376	1	no	—
169	C₄	—	188	1	no	—
170	C₄	#13093995	1	1	yes	characteristic
171	C₄	#13094009	1	1	yes	characteristic
172	C₄	#13094013	1	1	yes	characteristic
173	C₄	#13094019	1	1	yes	characteristic
174	C₂	—	2	1	no	—
175	C₂	—	2	1	no	—
176	C₂	—	2	1	no	—
177	C₂	—	376	1	no	—
178	C₂	#15689049	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#13094013
commutator	C₂²×C₄×C₄₇	C₂³
Frattini	C₂²×C₄	C₂²×D₄₇
Fitting	C₄₇×#424	C₂
socle	C₂×C₄₇	#397
hypercenter	#424	D₄₇
hypocenter	C₄₇	#2080
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #424

Complements

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

N	H ≅ G/N	type
4. #15688895	C₂	semidirect
5. #15688888	C₂	semidirect
6. #15688879	C₂	semidirect
7. C₄₇×#424	C₂	semidirect
90. #424	D₄₇	semidirect
97. C₄₇	#2080	semidirect