Finite Group Databasebeta

Subgroups — #15689939

Order 3008.

248 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 99 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	#13094072	C₂	1	1	yes	maximal, characteristic, complemented
2	#13094071	C₂	1	1	yes	maximal, characteristic, complemented
3	#13094057	C₂	1	1	yes	maximal, characteristic, complemented
4	#13094056	C₂	1	2	yes	maximal, complemented
5	#13094053	C₂	1	2	yes	maximal, complemented
6	#13094045	C₂	1	2	yes	maximal, complemented
7	#13094044	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#10625337	C₂	1	1	yes	maximal, characteristic, complemented
9	Q₁₆×D₄₇	C₂	1	2	yes	maximal, complemented
10	#13094032	C₂	1	1	yes	maximal, characteristic, complemented
11	#137×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
12	#41×C₄₇	C₂²	1	1	yes	characteristic
13	C₂×Q₈×C₄₇	C₂²	1	1	yes	characteristic
14	#10625354	—	2	1	no	—
15	#10625354	C₂²	1	2	yes	—
16	Q₈×D₄₇	—	2	1	no	—
17	Q₈×D₄₇	C₂²	1	2	yes	—
18	Q₈×D₄₇	C₂²	1	2	yes	—
19	#10625352	—	2	2	no	—
20	#10625352	C₂²	1	2	yes	—
21	#10625350	—	2	2	no	—
22	#10625350	—	2	1	no	—
23	#10625350	C₂²	1	1	yes	characteristic
24	#10625350	C₂²	1	1	yes	characteristic
25	C₂×Dic₉₄	—	2	1	no	—
26	C₂×Dic₉₄	C₂²	1	1	yes	characteristic
27	Q₁₆×C₄₇	C₂²	1	2	yes	—
28	SD₁₆×C₄₇	C₂²	1	2	yes	—
29	M₁₆×C₄₇	C₂²	1	1	yes	characteristic
30	#10625337	C₂²	1	1	yes	characteristic
31	#10625337	C₂²	1	2	yes	—
32	#10625337	C₂²	1	2	yes	—
33	#10625336	C₂²	1	1	yes	characteristic
34	#10625335	C₂²	1	1	yes	characteristic
35	#10625334	C₂²	1	1	yes	characteristic
36	C₂×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
37	Dic₁₈₈	C₂²	1	2	yes	—
38	#10625325	C₂²	1	2	yes	—
39	#10625324	C₂²	1	2	yes	—
40	C₈×D₄₇	C₂²	1	2	yes	—
41	Q₈×C₄₇	—	2	1	no	—
42	Q₈×C₄₇	C₂³	1	1	yes	characteristic
43	Q₈×C₄₇	C₂³	1	2	yes	—
44	D₄×C₄₇	—	2	1	no	—
45	D₄×C₄₇	C₂³	1	1	yes	characteristic
46	C₂×C₄×C₄₇	—	2	1	no	—
47	C₂×C₄×C₄₇	—	2	1	no	—
48	C₂×C₄×C₄₇	C₂³	1	1	yes	characteristic
49	C₂²⋊D₄₇	—	2	1	no	—
50	C₂²⋊D₄₇	—	2	2	no	—
51	C₂²⋊D₄₇	D₄	1	2	yes	—
52	C₂×Dic₄₇	—	2	1	no	—
53	C₂×Dic₄₇	—	2	2	no	—
54	D₁₈₈	—	2	2	no	—
55	D₁₈₈	D₄	1	1	yes	characteristic
56	D₁₈₈	C₂³	1	1	yes	characteristic
57	C₄×D₄₇	—	2	2	no	—
58	C₄×D₄₇	—	2	2	no	—
59	C₄×D₄₇	—	2	1	no	—
60	C₄×D₄₇	—	2	2	no	—
61	C₄×D₄₇	C₂³	1	2	yes	—
62	Dic₉₄	—	2	1	no	—
63	Dic₉₄	—	2	2	no	—
64	Dic₉₄	—	2	2	no	—
65	Dic₉₄	D₄	1	1	yes	characteristic
66	Dic₉₄	C₂³	1	2	yes	—
67	Dic₉₄	C₂³	1	1	yes	characteristic
68	C₈×C₄₇	C₂³	1	2	yes	—
69	C₄₇⋊C₈	C₂³	1	1	yes	characteristic
70	C₄₇⋊C₈	C₂³	1	1	yes	characteristic
71	C₂²×C₄₇	—	2	1	no	—
72	C₂²×C₄₇	C₂×D₄	1	1	yes	characteristic
73	C₂×D₄₇	—	2	1	no	—
74	C₂×D₄₇	C₂×D₄	1	2	yes	—
75	C₄×C₄₇	—	2	1	no	—
76	C₄×C₄₇	—	2	2	no	—
77	C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
78	C₄×C₄₇	C₂⁴	1	1	yes	characteristic
79	Dic₄₇	—	2	1	no	—
80	Dic₄₇	—	2	2	no	—
81	Dic₄₇	C₂×D₄	1	2	yes	—
82	C₂×C₄₇	—	2	1	no	—
83	C₂×C₄₇	—	4	1	no	—
84	C₂×C₄₇	C₂²×D₄	1	1	yes	characteristic
85	D₄₇	—	2	2	no	—
86	D₄₇	—	4	1	no	—
87	#578	—	47	1	no	maximal, Sylow
88	C₄₇	#578	1	1	yes	characteristic, complemented, Sylow
89	2₋¹⁺⁴	—	47	1	no	—
90	2₋¹⁺⁴	—	47	1	no	—
91	#137	—	47	2	no	—
92	#137	—	47	2	no	—
93	#137	—	47	1	no	—
94	#137	D₄₇	1	1	yes	characteristic, complemented
95	#135	—	47	1	no	—
96	#135	—	47	2	no	—
97	C₂×Q₁₆	—	47	1	no	—
98	C₂×Q₁₆	—	47	2	no	—
99	#131	—	47	1	no	—
100	#41	—	47	1	no	—
101	#41	—	47	2	no	—
102	#41	—	47	2	no	—
103	#41	—	47	1	no	—
104	#41	—	94	2	no	—
105	#41	—	94	1	no	—
106	#41	—	94	2	no	—
107	#41	—	94	1	no	—
108	#41	C₂×D₄₇	1	1	yes	characteristic
109	C₂×Q₈	—	47	2	no	—
110	C₂×Q₈	—	47	2	no	—
111	C₂×Q₈	—	47	1	no	—
112	C₂×Q₈	—	94	1	no	—
113	C₂×Q₈	—	94	1	no	—
114	C₂×Q₈	C₂×D₄₇	1	1	yes	characteristic
115	Q₁₆	—	47	1	no	—
116	Q₁₆	—	47	2	no	—
117	Q₁₆	—	47	2	no	—
118	Q₁₆	—	47	2	no	—
119	Q₁₆	C₂×D₄₇	1	2	yes	—
120	SD₁₆	—	47	1	no	—
121	SD₁₆	—	47	1	no	—
122	SD₁₆	—	47	2	no	—
123	SD₁₆	C₂×D₄₇	1	2	yes	—
124	D₈	—	47	1	no	—
125	M₁₆	—	47	2	no	—
126	M₁₆	C₂×D₄₇	1	1	yes	characteristic
127	C₂×C₈	—	47	1	no	—
128	C₂×C₈	—	47	2	no	—
129	Q₈	—	2	1	no	—
130	Q₈	—	47	1	no	—
131	Q₈	—	47	2	no	—
132	Q₈	—	47	1	no	—
133	Q₈	—	94	2	no	—
134	Q₈	—	94	2	no	—
135	Q₈	—	94	1	no	—
136	Q₈	C₂²×D₄₇	1	1	yes	characteristic
137	Q₈	C₂²×D₄₇	1	2	yes	—
138	D₄	—	2	1	no	—
139	D₄	—	47	2	no	—
140	D₄	—	47	1	no	—
141	D₄	—	47	1	no	—
142	D₄	—	94	2	no	—
143	D₄	—	94	2	no	—
144	D₄	—	94	1	no	—
145	D₄	C₂²×D₄₇	1	1	yes	characteristic
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	47	2	no	—
149	C₂×C₄	—	94	2	no	—
150	C₂×C₄	—	94	2	no	—
151	C₂×C₄	—	94	2	no	—
152	C₂×C₄	—	94	2	no	—
153	C₂×C₄	—	94	1	no	—
154	C₂×C₄	—	94	1	no	—
155	C₂×C₄	C₂²×D₄₇	1	1	yes	characteristic
156	C₈	—	47	1	no	—
157	C₈	—	47	1	no	—
158	C₈	C₂²×D₄₇	1	2	yes	—
159	C₂²	—	2	1	no	—
160	C₂²	—	47	2	no	—
161	C₂²	—	94	1	no	—
162	C₂²	D₄×D₄₇	1	1	yes	characteristic
163	C₄	—	2	1	no	—
164	C₄	—	2	2	no	—
165	C₄	—	47	2	no	—
166	C₄	—	94	1	no	—
167	C₄	—	94	2	no	—
168	C₄	D₄×D₄₇	1	1	yes	characteristic
169	C₄	C₂³×D₄₇	1	1	yes	characteristic
170	C₂	—	2	1	no	—
171	C₂	—	4	1	no	—
172	C₂	—	94	2	no	—
173	C₂	—	188	1	no	—
174	C₂	C₂×D₄×D₄₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₄×D₄₇
commutator, cosocle	C₄×C₄₇	C₂⁴
Frattini	C₄	C₂³×D₄₇
Fitting	#137×C₄₇	C₂
socle	C₂×C₄₇	C₂²×D₄
hypercenter	#137	D₄₇
hypocenter	C₄₇	#578
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #137

Complements

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

N	H ≅ G/N	type
1. #13094072	C₂	semidirect
2. #13094071	C₂	semidirect
3. #13094057	C₂	semidirect
4. #13094056	C₂	semidirect
5. #13094053	C₂	semidirect
6. #13094045	C₂	semidirect
7. #13094044	C₂	semidirect
8. C₂×#10625337	C₂	semidirect
9. Q₁₆×D₄₇	C₂	semidirect
10. #13094032	C₂	semidirect
11. #137×C₄₇	C₂	semidirect
88. C₄₇	#578	semidirect
94. #137	D₄₇	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#13094072	2
#13094071	2
#13094057	2
#13094056	2
#13094053	2
#13094045	2
#13094044	2
C₂×#10625337	2
Q₁₆×D₄₇	2
#13094032	2
#137×C₄₇	2
#578	47