Finite Group Databasebeta

← #16834899

Subgroups — #16834899

Order 3712.

326 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 60 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#14272135C211yesmaximal, characteristic, complemented
2#14272106C211yesmaximal, characteristic, complemented
3#14272015C214yesmaximal, complemented
4C2×#11745257C211yesmaximal, characteristic
5C22×D116—21no—
6C22×D116C411yescharacteristic, complemented
7C22×C4×D29C2211yescharacteristic
8C2×C22⋊#1181—21no—
9C2×#90800—21no—
10C2×C4⋊C4×C29C411yescharacteristic, complemented
11C2×#90781—21no—
12#11745257—21no—
13#11745257—21no—
14#11745257C412yescomplemented
15#11745257C2212yes—
16(C2×#1181)⋊C4—21no—
17(C2×#1181)⋊C4C2212yescomplemented
18#11745219—21no—
19#11745219C2212yescomplemented
20C23×D29—21no—
21C23×D29D411yescharacteristic
22C22×#1181—22no—
23C22×C4×C29—21no—
24C22×C4×C29C2×C411yescharacteristic
25C22×Dic29D411yescharacteristic
26C2×D116—22no—
27C2×D116—22no—
28C2×D116—21no—
29C2×D116—42no—
30C2×D116—44no—
31C2×D116C2×C412yes—
32C2×C4×D29—42no—
33C2×C4×D29C2×C411yescharacteristic
34C2×C4×D29D411yescharacteristic
35C2×C4×D29D411yescharacteristic
36C2×C4×D29C2311yescharacteristic
37C22⋊#1181—42no—
38#90800—42no—
39C2×#6599—22no—
40C4⋊C4×C29—21no—
41C4⋊C4×C29C2×C412yes—
42#90781—22no—
43#90781—22no—
44C23×C29C22⋊C411yescharacteristic
45C22×D29—22no—
46C22×D29—22no—
47C22×D29—22no—
48C22×D29—41no—
49C22×D29—42no—
50C22×D29—42no—
51C22×D29—42no—
52C22×D29C22⋊C411yescharacteristic
53C22×D29C2×D411yescharacteristic
54C2×#1181—42no—
55C2×#1181—42no—
56C2×#1181—42no—
57C2×C4×C29—22no—
58C2×C4×C29—22no—
59C2×C4×C29—42no—
60C2×C4×C29C22⋊C411yescharacteristic
61C2×C4×C29C22×C411yescharacteristic
62C2×Dic29—22no—
63C2×Dic29C22⋊C411yescharacteristic
64C2×Dic29C2×D411yescharacteristic
65D116—42no—
66D116—42no—
67D116—44no—
68D116—44no—
69C4×D29—41no—
70C4×D29—41no—
71C4×D29—41no—
72C4×D29—41no—
73#6599—42no—
74#1510—291nomaximal, Sylow
75C22×C29—21no—
76C22×C29—21no—
77C22×C29#9912yes—
78C22×C29C2×C22⋊C411yescharacteristic
79C2×D29—21no—
80C2×D29—21no—
81C2×D29—21no—
82C2×D29—21no—
83C2×D29—21no—
84C2×D29—44no—
85C2×D29—42no—
86C2×D29—42no—
87C2×D29—42no—
88C2×D29—84no—
89C2×D29AGL(1,ℤ/8)12yes—
90#1181—82no—
91C4×C29—21no—
92C4×C29—21no—
93C4×C29—42no—
94Dic29—21no—
95Dic29—21no—
96C2×#121—291no—
97#407—291no—
98#381—291no—
99#327—294no—
100C2×C29—22no—
101C2×C29C2×#9911yescharacteristic
102C2×C29#41811yescharacteristic
103C2×C29#42111yescharacteristic
104D29—22no—
105D29—41no—
106D29—82no—
107C22×D4—291no—
108C22×D4—581no—
109C23×C4—291no—
110C2×M16—581no—
111#121—292no—
112#121—292no—
113#121—581no—
114#121—581no—
115C2×C4⋊C4#118111yescharacteristic, complemented
116C2×C22⋊C4—581no—
117C2×C22⋊C4—581no—
118C22⋊C8—292no—
119C22⋊C8—581no—
120#95—292no—
121#95—581no—
122C29#151011yescharacteristic, complemented, Sylow
123C24—291no—
124C24—581no—
125C2×D4—292no—
126C2×D4—582no—
127C2×D4—582no—
128C2×D4—581no—
129C2×D4—1162no—
130C2×D4—1164no—
131C22×C4—21no—
132C22×C4—291no—
133C22×C4—291no—
134C22×C4—291no—
135C22×C4—291no—
136C22×C4—291no—
137C22×C4—582no—
138C22×C4—1162no—
139C22×C4C2×#118111yescharacteristic
140M16—1162no—
141C2×C8—582no—
142C4⋊C4—21no—
143C4⋊C4C2×#118112yes—
144C22⋊C4—582no—
145C22⋊C4—582no—
146C22⋊C4—1162no—
147C23—291no—
148C23—291no—
149C23—582no—
150C23—582no—
151C23—582no—
152C23—1161no—
153C23—1162no—
154C23—1162no—
155C23—1162no—
156C23C22⋊#118111yescharacteristic
157D4—1162no—
158D4—1162no—
159D4—1164no—
160D4—1164no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—42no—
164C2×C4—291no—
165C2×C4—291no—
166C2×C4—582no—
167C2×C4—1161no—
168C2×C4—1161no—
169C2×C4—1161no—
170C2×C4—1161no—
171C2×C4—1162no—
172C2×C4—1162no—
173C2×C4—1162no—
174C2×C4C22⋊#118111yescharacteristic
175C2×C4C22×#118111yescharacteristic
176C8—1162no—
177C22—21no—
178C22—21no—
179C22—292no—
180C22—581no—
181C22—581no—
182C22—581no—
183C22—581no—
184C22—581no—
185C22—1164no—
186C22—1162no—
187C22—1162no—
188C22—1162no—
189C22—2324no—
190C22#1174521512yes—
191C22C2×C22⋊#118111yescharacteristic
192C4—21no—
193C4—21no—
194C4—42no—
195C4—581no—
196C4—581no—
197C4—2322no—
198C2—22no—
199C2—582no—
200C2—1161no—
201C2—2322no—
202C2C2×#1174521511yescharacteristic
203C2#1427215511yescharacteristic
204C2#1427216211yescharacteristic
subgroupG/H
centerC22C2×C22⋊#1181
commutatorC2×C4×C29C22×C4
FrattiniC2×C4C22×#1181
FittingC2×C4⋊C4×C29C4
socleC22×C29C2×C22⋊C4
prime radicalC2×#11745257C2
cosocleC2×C4×D29C23
hypercenterC2×C4⋊C4#1181
hypocenterC29#1510
LCSG ▷ C2×C4×C29 ▷ C22×C29 ▷ C29
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #14272135C2semidirect
2. #14272106C2semidirect
3. #14272015C2semidirect
6. C22×D116C4semidirect
10. C2×C4⋊C4×C29C4semidirect
14. #11745257C4semidirect
17. (C2×#1181)⋊C4C22semidirect
19. #11745219C22semidirect
115. C2×C4⋊C4#1181semidirect
122. C29#1510semidirect

Maximal subgroups

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

subgroupindex
#142721352
#142721062
#142720152
C2×#117452572
#151029