Finite Group Databasebeta

← #18010601

Subgroups — #18010601

Order 4736.

360 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 68 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
1#14558600C211yesmaximal, characteristic
2#14558526C212yesmaximal, complemented
3D37⋊#100C211yesmaximal, characteristic
4#14558404C212yesmaximal, complemented
5#124×D37C211yesmaximal, characteristic
6D37⋊M16C2212yes—
7D4×#3866—24no—
8#11924615C2212yes—
9C2×C22⋊#3866—22no—
10#11924599—24no—
11#11924596—22no—
12C2×Q8×D37C2211yescharacteristic
13C2×D4×D37C2211yescharacteristic
14#11924525C411yescharacteristic, complemented
15#11924524C411yescharacteristic, complemented
16#11924523C411yescharacteristic, complemented
17#11924522—22no—
18C22⋊C4×D37—22no—
19C42×D37C2211yescharacteristic
20#124×C37C411yescharacteristic, complemented
21C23×D37D412yes—
22C22×#3866—24no—
23C2×Q8×C37C2×C411yescharacteristic, complemented
24C2×D4×C37C2×C411yescharacteristic
25C2×C22⋊D37D412yes—
26Q8×D37—22no—
27Q8×D37—22no—
28D4×D37—22no—
29D4×D37—22no—
30C2×D148C2×C411yescharacteristic
31C2×C4×D37—21no—
32C2×C4×D37—21no—
33C2×C4×D37C2311yescharacteristic
34C2×Dic74—21no—
35C2×Dic74C2×C411yescharacteristic, complemented
36C22⋊#3866—24no—
37C22⋊#3866—22no—
38#10596957—22no—
39C4⋊#3866D414yes—
40C4×#3866—22no—
41#10596953—22no—
42D37⋊C8—22no—
43C22⋊C4×C37—22no—
44C42×C37C2×C411yescharacteristic, complemented
45C22⋊Dic37—22no—
46#10596938—22no—
47#10596938—22no—
48C4×Dic37—21no—
49C4×Dic37C2×C411yescharacteristic, complemented
50C23×C37C22⋊C412yes—
51C22×D37—22no—
52C22×D37—22no—
53C22×D37—44no—
54C22×D37C22⋊C412yes—
55C22×D37C2×D411yescharacteristic
56C2×#3866—24no—
57C2×#3866—84no—
58Q8×C37—22no—
59D4×C37—22no—
60C2×C4×C37—21no—
61C2×C4×C37—21no—
62C2×C4×C37C22×C411yescharacteristic
63C22⋊D37—24no—
64C2×Dic37—21no—
65C2×Dic37—21no—
66C2×Dic37C2×D411yescharacteristic
67D148—22no—
68C4×D37—42no—
69C4×D37—41no—
70C4×D37—41no—
71C4×D37C2×D412yes—
72C4×D37#4112yes—
73Dic74—22no—
74Dic74—42no—
75#64438—42no—
76C22×C37—22no—
77C22×C37—42no—
78C22×C37C2×C22⋊C411yescharacteristic
79C2×D37—22no—
80C2×D37—22no—
81C2×D37—44no—
82C2×D37—42no—
83C2×D37—82no—
84C2×D37#12011yescharacteristic
85C2×D37#12311yescharacteristic
86#3866—44no—
87C4×C37—41no—
88C4×C37—41no—
89C4×C37C4×D412yes—
90Dic37—41no—
91Dic37—41no—
92Dic37#12112yes—
93#1885—371nomaximal, Sylow
94C2×C37—21no—
95C2×C37—42no—
96C2×C37#38611yescharacteristic
97D37—21no—
98D37—42no—
99D37#45512yes—
100C2×#124—371no—
101#518—371no—
102#419—372no—
103C2×#100—371no—
104#339—372no—
105C37#188511yescharacteristic, complemented, Sylow
106C22×Q8—371no—
107C22×D4—371no—
108C2×M16—372no—
109#124—371no—
110#124—371no—
111#124—371no—
112#124—742no—
113#124#386611yescharacteristic, complemented
114C4×D4—744no—
115#117—372no—
116C2×C22⋊C4—742no—
117C2×C22⋊C4—742no—
118C2×C42—371no—
119#103—744no—
120#100—742no—
121C24—372no—
122C2×Q8—371no—
123C2×Q8—741no—
124C2×Q8—742no—
125C2×Q8—742no—
126C2×Q8C2×#386611yescharacteristic, complemented
127C2×D4—372no—
128C2×D4—371no—
129C2×D4—742no—
130C2×D4—742no—
131C2×D4C2×#386611yescharacteristic
132C22×C4—371no—
133C22×C4—741no—
134C22×C4—741no—
135C22×C4—744no—
136M16—742no—
137M16—742no—
138C2×C8—742no—
139C4⋊C4—374no—
140C22⋊C4—22no—
141C22⋊C4—742no—
142C22⋊C4—742no—
143C22⋊C4—742no—
144C22⋊C4—744no—
145C22⋊C4—742no—
146C42—371no—
147C42—741no—
148C42—742no—
149C42C2×#386611yescharacteristic, complemented
150C23—371no—
151C23—372no—
152C23—742no—
153C23—742no—
154C23—1484no—
155C23C22⋊#386612yes—
156Q8—22no—
157Q8—742no—
158Q8—1482no—
159D4—22no—
160D4—744no—
161D4—742no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—371no—
165C2×C4—372no—
166C2×C4—372no—
167C2×C4—2964no—
168C2×C4—744no—
169C2×C4—741no—
170C2×C4—741no—
171C2×C4—1481no—
172C2×C4—1481no—
173C2×C4—1482no—
174C2×C4C22×#386611yescharacteristic
175C8—1482no—
176C22—22no—
177C22—42no—
178C22—371no—
179C22—371no—
180C22—2962no—
181C22—742no—
182C22—742no—
183C22—1482no—
184C22—1484no—
185C22C2×C22⋊#386611yescharacteristic
186C4—41no—
187C4—41no—
188C4—372no—
189C4—1481no—
190C4—1484no—
191C4—1481no—
192C4D4×#386612yes—
193C2—21no—
194C2—42no—
195C2—372no—
196C2—741no—
197C2—1482no—
198C2#1455848111yescharacteristic
subgroupG/H
centerC2#14558481
commutatorC2×C4×C37C22×C4
FrattiniC2×C4C22×#3866
Fitting#124×C37C4
socleC2×C37#386
prime radicalC2×D4×D37C22
cosocleC2×C4×D37C23
hypercenter#124#3866
hypocenterC37#1885
LCSG ▷ C2×C4×C37 ▷ C2×C37 ▷ C37
UCSC1 ◁ C2 ◁ C2×C4 ◁ #124

Complements

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

NH ≅ G/Ntype
2. #14558526C2semidirect
4. #14558404C2semidirect
14. #11924525C4semidirect
15. #11924524C4semidirect
16. #11924523C4semidirect
20. #124×C37C4semidirect
23. C2×Q8×C37C2×C4semidirect
35. C2×Dic74C2×C4semidirect
44. C42×C37C2×C4semidirect
49. C4×Dic37C2×C4semidirect
105. C37#1885semidirect
113. #124#3866semidirect
126. C2×Q8C2×#3866semidirect
149. C42C2×#3866semidirect

Maximal subgroups

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

subgroupindex
#145586002
#145585262
D37⋊#1002
#145584042
#124×D372
#188537