Finite Group Databasebeta

← #16834930

Subgroups — #16834930

Order 3712.

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

normal subgroups only

This group has 288 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
1C2×#11745219C211yesmaximal, characteristic
2#14272105C211yesmaximal, characteristic
3#14272017C212yesmaximal, complemented
4#14272016C212yesmaximal, complemented
5C22⋊Q8×D29C211yesmaximal, characteristic
6C2×Q8×D29C411yescharacteristic, complemented
7C22×C4×D29C2211yescharacteristic
8C2×C22⋊#1181—21no—
9C22×#6599—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—42no—
39C2×#6599—22no—
40C2×#6599—42no—
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—41no—
55C22×D29—41no—
56C22×D29C22⋊C411yescharacteristic
57C22×D29SD1612yes—
58C22×D29Q1612yes—
59C22×D29C2×D411yescharacteristic
60C2×#1181—42no—
61C2×#1181—42no—
62C2×#1181—42no—
63Q8×C29—41no—
64C2×C4×C29—21no—
65C2×C4×C29—21no—
66C2×C4×C29—41no—
67C2×C4×C29C22⋊C411yescharacteristic
68C2×C4×C29C22×C411yescharacteristic
69C2×Dic29—22no—
70C2×Dic29—21no—
71C2×Dic29—21no—
72C2×Dic29C22⋊C411yescharacteristic
73C2×Dic29C2×D411yescharacteristic
74C4×D29—22no—
75C4×D29—22no—
76C4×D29—41no—
77C4×D29—41no—
78C4×D29—42no—
79C4×D29—42no—
80Dic58—42no—
81Dic58—41no—
82#6599—42no—
83#1520—291nomaximal, Sylow
84C22×C29—21no—
85C22×C29—21no—
86C22×C29#10312yes—
87C22×C29C2×C22⋊C411yescharacteristic
88C2×D29—21no—
89C2×D29—21no—
90C2×D29—21no—
91C2×D29—21no—
92C2×D29—42no—
93C2×D29—42no—
94C2×D29#10312yes—
95C2×D29C2×SD1611yescharacteristic
96C2×D29C2×Q1611yescharacteristic
97#1181—82no—
98C4×C29—21no—
99C4×C29—21no—
100C4×C29—41no—
101C4×C29—41no—
102Dic29—21no—
103Dic29—41no—
104Dic29—41no—
105Dic29#9912yes—
106C2×C22⋊Q8—291no—
107C2×C22⋊C8—291no—
108#380—291no—
109#328—292no—
110#328—292no—
111C2×C29—22no—
112C2×C29C2×#9911yescharacteristic
113C2×C29C2×#10311yescharacteristic
114C2×C29#42111yescharacteristic
115D29—21no—
116D29—22no—
117D29—21no—
118C22×Q8—291no—
119C23×C4—291no—
120C22×C8—581no—
121C22⋊Q8—291no—
122C22⋊Q8—291no—
123C22⋊Q8—291no—
124C22⋊Q8—581no—
125C22⋊Q8—581no—
126C22⋊Q8#118111yescharacteristic, complemented
127C2×C4⋊C4—291no—
128C2×C4⋊C4—581no—
129C2×C4⋊C4—581no—
130C2×C22⋊C4—581no—
131C2×C22⋊C4—581no—
132C22⋊C8—292no—
133C22⋊C8—581no—
134#95—292no—
135C29#152011yescharacteristic, complemented, Sylow
136C24—291no—
137C2×Q8—291no—
138C2×Q8—581no—
139C2×Q8—1162no—
140C2×Q8C2×#118111yescharacteristic
141C22×C4—291no—
142C22×C4—291no—
143C22×C4—291no—
144C22×C4—291no—
145C22×C4—291no—
146C22×C4—581no—
147C22×C4—581no—
148C22×C4—582no—
149C22×C4—1162no—
150C22×C4C2×#118111yescharacteristic
151C2×C8—582no—
152C2×C8—1162no—
153C4⋊C4—21no—
154C4⋊C4—291no—
155C4⋊C4—581no—
156C4⋊C4—581no—
157C4⋊C4—581no—
158C4⋊C4—581no—
159C4⋊C4—1162no—
160C4⋊C4C2×#118111yescharacteristic
161C22⋊C4—21no—
162C22⋊C4—581no—
163C22⋊C4—581no—
164C22⋊C4—581no—
165C22⋊C4—1162no—
166C23—291no—
167C23—291no—
168C23—292no—
169C23—292no—
170C23—1161no—
171C23—1161no—
172C23C22⋊#118111yescharacteristic
173Q8—41no—
174Q8—1161no—
175Q8—1162no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—41no—
179C2×C4—291no—
180C2×C4—291no—
181C2×C4—582no—
182C2×C4—582no—
183C2×C4—582no—
184C2×C4—581no—
185C2×C4—581no—
186C2×C4—1161no—
187C2×C4—1161no—
188C2×C4—1162no—
189C2×C4—1162no—
190C2×C4—1162no—
191C2×C4—1162no—
192C2×C4—1162no—
193C2×C4C22⋊#118111yescharacteristic
194C2×C4C22×#118111yescharacteristic
195C8—1162no—
196C22—21no—
197C22—21no—
198C22—291no—
199C22—292no—
200C22—291no—
201C22—581no—
202C22—581no—
203C22—581no—
204C22—581no—
205C22—1162no—
206C22—1162no—
207C22#1174522512yes—
208C22C2×C22⋊#118111yescharacteristic
209C4—21no—
210C4—21no—
211C4—41no—
212C4—41no—
213C4—292no—
214C4—581no—
215C4—1161no—
216C4—1161no—
217C4—2322no—
218C2—22no—
219C2—581no—
220C2—582no—
221C2—581no—
222C2D29⋊#9911yescharacteristic
223C2C2×#1174522511yescharacteristic
224C2#1427216011yescharacteristic
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#1520
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
126. C22⋊Q8#1181semidirect
135. C29#1520semidirect

Maximal subgroups

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

subgroupindex
C2×#117452192
#142721052
#142720172
#142720162
C22⋊Q8×D292
#152029