Finite Group Databasebeta

← #16834890

Subgroups — #16834890

Order 3712.

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

normal subgroups only

This group has 294 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
2C4×C22⋊#1181C211yesmaximal, characteristic
3#14272017C212yesmaximal, complemented
4#14272016C212yesmaximal, complemented
5C22⋊Q8×D29C211yesmaximal, characteristic
6C2×Q8×D29C411yescharacteristic, complemented
7C22×C4×D29C2211yescharacteristic
8C2×C22⋊#1181—21no—
9C2×#90800—21no—
10C2×C4×#1181—21no—
11C29×C22⋊Q8C411yescharacteristic, complemented
12#11745308C2211yescharacteristic
13#11745286C411yescharacteristic, complemented
14#11745259C2211yescharacteristic
15#11745258—21no—
16C4⋊C4×D29—21no—
17C4⋊C4×D29C411yescharacteristic, complemented
18C22⋊C4×D29—21no—
19#11745240—21no—
20(C2×#1181)⋊C4C2212yes—
21#11745219—21no—
22#11745219C2212yes—
23C23×D29D411yescharacteristic
24C22×#1181—22no—
25C2×Q8×C29C2×C411yescharacteristic
26C22×C4×C29C2×C411yescharacteristic
27C22×Dic29D411yescharacteristic
28Q8×D29—42no—
29C2×C4×D29—21no—
30C2×C4×D29—21no—
31C2×C4×D29—42no—
32C2×C4×D29C2×C411yescharacteristic
33C2×C4×D29D411yescharacteristic
34C2×C4×D29D411yescharacteristic
35C2×C4×D29C2311yescharacteristic
36C2×Dic58—21no—
37C2×Dic58C2×C411yescharacteristic
38C22⋊#1181—24no—
39#90800—42no—
40C2×#6599—22no—
41C4×#1181—42no—
42C4⋊C4×C29—21no—
43C4⋊C4×C29C2×C411yescharacteristic
44C22⋊C4×C29—21no—
45C22⋊Dic29—21no—
46#90781—21no—
47#90781—21no—
48C4⋊Dic29—21no—
49C4⋊Dic29C2×C411yescharacteristic
50#90779—21no—
51#90779—21no—
52#90779—21no—
53C23×C29C22⋊C411yescharacteristic
54C22×D29—22no—
55C22×D29—22no—
56C22×D29—41no—
57C22×D29C22⋊C411yescharacteristic
58C22×D29C2×D411yescharacteristic
59C2×#1181—24no—
60C2×#1181—42no—
61C2×#1181—42no—
62Q8×C29—41no—
63C2×C4×C29—21no—
64C2×C4×C29—21no—
65C2×C4×C29—41no—
66C2×C4×C29C22⋊C411yescharacteristic
67C2×C4×C29C22×C411yescharacteristic
68C2×Dic29—22no—
69C2×Dic29—21no—
70C2×Dic29—21no—
71C2×Dic29C22⋊C411yescharacteristic
72C2×Dic29C2×D411yescharacteristic
73C4×D29—22no—
74C4×D29—22no—
75C4×D29—41no—
76C4×D29—41no—
77C4×D29—42no—
78C4×D29—42no—
79Dic58—42no—
80Dic58—41no—
81#6599—42no—
82#1508—291nomaximal, Sylow
83C22×C29—21no—
84C22×C29—21no—
85C22×C29—21no—
86C22×C29C2×C22⋊C411yescharacteristic
87C2×D29—21no—
88C2×D29—21no—
89C2×D29—21no—
90C2×D29—21no—
91C2×D29—44no—
92C2×D29#10412yes—
93C2×D29#13712yes—
94#1181—44no—
95C4×C29—21no—
96C4×C29—21no—
97C4×C29—41no—
98C4×C29—41no—
99Dic29—21no—
100Dic29—41no—
101Dic29—41no—
102Dic29#9912yes—
103C2×C22⋊Q8—291no—
104#407—291no—
105C4×C22⋊C4—291no—
106#328—292no—
107#328—292no—
108C2×C29—41no—
109C2×C29C2×#9911yescharacteristic
110C2×C29#41911yescharacteristic
111C2×C29C2×#10411yescharacteristic
112D29—22no—
113D29—22no—
114C22×Q8—291no—
115C23×C4—291no—
116C2×M16—581no—
117C22⋊Q8—291no—
118C22⋊Q8—291no—
119C22⋊Q8—291no—
120C22⋊Q8—581no—
121C22⋊Q8—581no—
122C22⋊Q8#118111yescharacteristic, complemented
123C2×C4⋊C4—291no—
124C2×C4⋊C4—581no—
125C2×C22⋊C4—581no—
126C2×C22⋊C4—581no—
127C2×C42—581no—
128C22⋊C8—292no—
129C22⋊C8—581no—
130#95—292no—
131C29#150811yescharacteristic, complemented, Sylow
132C24—291no—
133C2×Q8—291no—
134C2×Q8—581no—
135C2×Q8—1162no—
136C2×Q8C2×#118111yescharacteristic
137C22×C4—291no—
138C22×C4—291no—
139C22×C4—291no—
140C22×C4—291no—
141C22×C4—291no—
142C22×C4—581no—
143C22×C4—581no—
144C22×C4—582no—
145C22×C4—1162no—
146C22×C4C2×#118111yescharacteristic
147M16—1162no—
148C2×C8—582no—
149C4⋊C4—21no—
150C4⋊C4—291no—
151C4⋊C4—581no—
152C4⋊C4—581no—
153C4⋊C4—581no—
154C4⋊C4—581no—
155C4⋊C4C2×#118111yescharacteristic
156C22⋊C4—21no—
157C22⋊C4—581no—
158C22⋊C4—581no—
159C22⋊C4—584no—
160C22⋊C4—581no—
161C42—1162no—
162C23—291no—
163C23—291no—
164C23—582no—
165C23—582no—
166C23—1161no—
167C23C22⋊#118111yescharacteristic
168Q8—41no—
169Q8—1161no—
170Q8—1162no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—41no—
174C2×C4—291no—
175C2×C4—291no—
176C2×C4—582no—
177C2×C4—582no—
178C2×C4—582no—
179C2×C4—584no—
180C2×C4—581no—
181C2×C4—581no—
182C2×C4—1161no—
183C2×C4—1161no—
184C2×C4—1162no—
185C2×C4—1162no—
186C2×C4—1162no—
187C2×C4—1162no—
188C2×C4C22⋊#118111yescharacteristic
189C2×C4C22×#118111yescharacteristic
190C8—1162no—
191C22—21no—
192C22—21no—
193C22—21no—
194C22—292no—
195C22—292no—
196C22—581no—
197C22—581no—
198C22—581no—
199C22—581no—
200C22—1164no—
201C22C2×C22⋊#118111yescharacteristic
202C4—21no—
203C4—21no—
204C4—41no—
205C4—41no—
206C4—292no—
207C4—581no—
208C4—1164no—
209C4—1161no—
210C4—1161no—
211C2—41no—
212C2—582no—
213C2—582no—
214C2D29⋊#9911yescharacteristic
215C2#1427215611yescharacteristic
216C2D29⋊#10411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#1181
commutatorC2×C4×C29C22×C4
FrattiniC2×C4C22×#1181
FittingC29×C22⋊Q8C4
socleC22×C29C2×C22⋊C4
prime radicalC23×D29D4
cosocleC2×C4×D29C23
hypercenterC22⋊Q8#1181
hypocenterC29#1508
LCSG ▷ C2×C4×C29 ▷ C22×C29 ▷ C29
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. #14272017C2semidirect
4. #14272016C2semidirect
6. C2×Q8×D29C4semidirect
11. C29×C22⋊Q8C4semidirect
13. #11745286C4semidirect
17. C4⋊C4×D29C4semidirect
122. C22⋊Q8#1181semidirect
131. C29#1508semidirect

Maximal subgroups

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

subgroupindex
#142721352
C4×C22⋊#11812
#142720172
#142720162
C22⋊Q8×D292
#150829