Finite Group Databasebeta

← #17993847

Subgroups — #17993847

Order 4736.

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

normal subgroups only

This group has 234 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

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×D148C211yesmaximal, characteristic
2#14557938C211yesmaximal, characteristic, complemented
3#14557814C211yesmaximal, characteristic
4#14557743C211yesmaximal, characteristic, complemented
5#14557278C211yesmaximal, characteristic, complemented
6#14557262C211yesmaximal, characteristic
7C37×Q8⋊C8C211yesmaximal, characteristic, complemented
8C2×Q8×D37—21no—
9C2×#10596942—21no—
10#11924536—21no—
11#11924536C2211yescharacteristic
12#11924536C2211yescharacteristic
13#11924516—21no—
14#11924516—22no—
15C4×D148—21no—
16C4×D148C2211yescharacteristic
17#11924470—21no—
18#11924470—22no—
19C4⋊#64436C2211yescharacteristic
20#11924459—21no—
21#11924458—21no—
22C4×Q8×C37C2211yescharacteristic
23C4⋊C8×C37C2211yescharacteristic
24C4×C8×C37C2211yescharacteristic
25C2×Q8×C37D411yescharacteristic
26Q8×D37—42no—
27Q8×D37—42no—
28C2×D148D411yescharacteristic
29C2×C4×D37—22no—
30C2×C4×D37—21no—
31C2×Dic74—22no—
32C2×Dic74—21no—
33C2×Dic74—21no—
34C2×C8×C37—21no—
35C2×C8×C37—21no—
36C4⋊C4×C37—21no—
37C4⋊C4×C37D411yescharacteristic
38C42×C37—21no—
39C42×C37C2311yescharacteristic
40#10596942—42no—
41#10596938—22no—
42#10596938—21no—
43C4⋊Dic37—22no—
44C4⋊Dic37—21no—
45C4⋊Dic37—21no—
46C4⋊Dic37—21no—
47C4⋊Dic37D411yescharacteristic
48C4⋊Dic37D412yes—
49C2×#64436—21no—
50C22×D37—21no—
51Q8×C37—21no—
52Q8×C37D812yes—
53C2×C4×C37—21no—
54C2×C4×C37—21no—
55C2×C4×C37C2×D411yescharacteristic
56C2×C4×C37C2×D411yescharacteristic
57C2×C4×C37C2×D411yescharacteristic
58C2×Dic37—21no—
59C2×Dic37—22no—
60C2×Dic37—41no—
61D148—21no—
62D148Q1612yescomplemented
63C4×D37—44no—
64C4×D37—41no—
65C4×D37—41no—
66Dic74—44no—
67Dic74—41no—
68Dic74—41no—
69Dic74—41no—
70Dic74—41no—
71C8×C37—22no—
72C8×C37—41no—
73#64436—41no—
74C22×C37#12011yescharacteristic
75C2×D37—22no—
76C2×D37—41no—
77C2×D37—41no—
78C4×C37—22no—
79C4×C37—21no—
80C4×C37—41no—
81C4×C37C2×D811yescharacteristic
82C4×C37C2×Q1611yescharacteristic
83C4×C37AGL(1,ℤ/8)11yescharacteristic
84C4×C37#13711yescharacteristic
85Dic37—41no—
86Dic37—42no—
87Dic37—81no—
88#1649—371nomaximal, Sylow
89C2×C37#44711yescharacteristic
90C2×C37#45111yescharacteristic
91C2×C37#45611yescharacteristic
92D37—42no—
93D4×Q8—371no—
94#500—371no—
95#474—371no—
96#462—371no—
97#332—371no—
98Q8⋊C8D3711yescharacteristic, complemented
99D4⋊C8—371no—
100C37#164911yescharacteristic, complemented, Sylow
101C22×Q8—741no—
102C2×Q16—741no—
103C4⋊Q8—371no—
104C4⋊Q8—371no—
105C4⋊Q8—741no—
106C22⋊Q8—742no—
107C22⋊Q8—741no—
108C4×Q8C2×D3711yescharacteristic
109C4×D4—371no—
110C4×D4—741no—
111#107—741no—
112#107—742no—
113C4⋊C8—371no—
114C4⋊C8C2×D3711yescharacteristic
115#103—741no—
116#102—741no—
117C4×C8C2×D3711yescharacteristic
118C2×Q8—742no—
119C2×Q8—741no—
120C2×Q8—741no—
121C2×Q8—1482no—
122C2×Q8—1482no—
123C2×Q8D14811yescharacteristic
124C2×D4—371no—
125C22×C4—742no—
126C22×C4—741no—
127Q16—1482no—
128C2×C8—21no—
129C2×C8—21no—
130C2×C8—741no—
131C4⋊C4—21no—
132C4⋊C4—371no—
133C4⋊C4—372no—
134C4⋊C4—742no—
135C4⋊C4—741no—
136C4⋊C4—741no—
137C4⋊C4—741no—
138C4⋊C4D14811yescharacteristic
139C22⋊C4—742no—
140C22⋊C4—741no—
141C42—21no—
142C42C22×D3711yescharacteristic
143C23—741no—
144Q8—21no—
145Q8—1484no—
146Q8—1481no—
147Q8—1481no—
148Q8—1481no—
149Q8—1481no—
150Q8D29612yes—
151D4—372no—
152D4—741no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—742no—
156C2×C4—741no—
157C2×C4—1484no—
158C2×C4—1481no—
159C2×C4—1481no—
160C2×C4—1481no—
161C2×C4C2×D14811yescharacteristic
162C2×C4D4×D3711yescharacteristic
163C2×C4D4×D3711yescharacteristic
164C8—22no—
165C8—41no—
166C8—1481no—
167C22—742no—
168C22—1481no—
169C22—1481no—
170C22#1192450611yescharacteristic
171C4—21no—
172C4—22no—
173C4—41no—
174C4—2961no—
175C4—1482no—
176C4—1481no—
177C4C2×D29611yescharacteristic
178C4Q16×D3711yescharacteristic
179C4#1192456411yescharacteristic
180C4#1192456511yescharacteristic
181C2—1482no—
182C2#1455766911yescharacteristic
183C2#1455769011yescharacteristic
184C2#1455771711yescharacteristic
subgroupG/H
centerC22#11924506
commutator, cosocleC42×C37C23
FrattiniC42C22×D37
FittingC37×Q8⋊C8C2
socleC22×C37#120
prime radicalC2×D148D4
hypercenterQ8⋊C8D37
hypocenterC37#1649
LCSG ▷ C42×C37 ▷ C22×C37 ▷ C37
UCSC1 ◁ C22 ◁ C42 ◁ Q8⋊C8

Complements

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

NH ≅ G/Ntype
2. #14557938C2semidirect
4. #14557743C2semidirect
5. #14557278C2semidirect
7. C37×Q8⋊C8C2semidirect
62. D148Q16semidirect
98. Q8⋊C8D37semidirect
100. C37#1649semidirect

Maximal subgroups

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

subgroupindex
Q8×D1482
#145579382
#145578142
#145577432
#145572782
#145572622
C37×Q8⋊C82
#164937