Finite Group Databasebeta

← #17961574

Subgroups — #17961574

Order 4352.

566 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 566 conjugacy classes of subgroups — too many to draw. Only its 80 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#14544618C212yesmaximal
2#14543881C214yesmaximal, complemented
3#14531368C211yesmaximal, characteristic
4C2×#10587545—22no—
5C2×#10587545C2214yes—
6#11763779—22no—
7C23×Dic34C2211yescharacteristic
8C2×#10587612C411yescharacteristic
9#11762715C412yes—
10#11762715C2212yes—
11#11762714—22no—
12#11762670—22no—
13C2×C17×#117C411yescharacteristic
14C23×C4×C17C2×C411yescharacteristic
15C23×Dic17D412yes—
16C22×Dic34—24no—
17C22×Dic34—42no—
18C22×Dic34—42no—
19C22×Dic34C2×C411yescharacteristic
20C22×Dic34D412yes—
21C22×Dic34C2311yescharacteristic
22C2×#63247—24no—
23C17×#117—42no—
24C2×C4⋊C4×C17—21no—
25C2×C22⋊C4×C17—21no—
26C2×C42×C17C2×C412yes—
27C2×C22⋊Dic17—21no—
28#10587612—42no—
29C2×C4⋊Dic17C2×C412yes—
30C2×C4×Dic17—21no—
31C22⋊#3624—44no—
32#10587545—28no—
33#10587545—44no—
34#10587502—24no—
35#10587502—24no—
36C24×C17C22⋊C411yescharacteristic
37C22×C4×C17—22no—
38C22×C4×C17—22no—
39C22×C4×C17—22no—
40C22×C4×C17—22no—
41C22×C4×C17C22⋊C411yescharacteristic
42C22×C4×C17C22×C411yescharacteristic
43C22×Dic17—24no—
44C22×Dic17—24no—
45C22×Dic17—22no—
46C22×Dic17—42no—
47C22×Dic17C22⋊C412yes—
48C22×Dic17C2×D412yes—
49C2×Dic34—24no—
50C2×Dic34—22no—
51C2×Dic34—24no—
52C2×Dic34—41no—
53C2×Dic34—41no—
54C2×Dic34—42no—
55C2×Dic34—41no—
56C2×Dic34—41no—
57C2×Dic34—416no—
58C2×Dic34#4114yes—
59#63247—44no—
60#63247—44no—
61C2×#3624—44no—
62C4⋊C4×C17—42no—
63C22⋊C4×C17—42no—
64C42×C17—42no—
65C22⋊Dic17—42no—
66C4⋊Dic17—42no—
67C4×Dic17—42no—
68#12862—171nomaximal, Sylow
69C23×C17—21no—
70C23×C17—21no—
71C23×C17—42no—
72C23×C17#9912yes—
73C23×C17C2×C22⋊C411yescharacteristic
74C2×C4×C17—28no—
75C2×C4×C17—21no—
76C2×C4×C17—21no—
77C2×C4×C17—21no—
78C2×C4×C17—21no—
79C2×C4×C17—44no—
80C2×C4×C17—42no—
81C2×C4×C17#9912yes—
82C2×C4×C17#11712yes—
83C2×Dic17—22no—
84C2×Dic17—22no—
85C2×Dic17—22no—
86C2×Dic17—22no—
87C2×Dic17—22no—
88C2×Dic17—48no—
89C2×Dic17—44no—
90C2×Dic17—42no—
91C2×Dic17#12314yes—
92Dic34—44no—
93Dic34—48no—
94Dic34—44no—
95#3624—84no—
96#2310—171no—
97#1784—172no—
98#1394—174no—
99C22×C17—22no—
100C22×C17—22no—
101C22×C17—24no—
102C22×C17—42no—
103C22×C17—42no—
104C22×C17#38111yescharacteristic
105C22×C17C2×#9911yescharacteristic
106C22×C17C2×#9911yescharacteristic
107C4×C17—22no—
108C4×C17—22no—
109C4×C17—82no—
110Dic17—24no—
111Dic17—42no—
112Dic17—82no—
113C23×Q8—171no—
114C2×#117—171no—
115C2×#117#59511yescharacteristic
116C2×#101—174no—
117C2×#101—342no—
118#407—342no—
119#391—172no—
120#391—172no—
121#391—342no—
122#381—342no—
123C2×C17—21no—
124C2×C17—21no—
125C2×C17—42no—
126C2×C17#181911yescharacteristic
127C2×C17#213312yes—
128C22×Q8—172no—
129C22×Q8—171no—
130C22×Q8—171no—
131C22×Q8—344no—
132C22×Q8—682no—
133C22×Q8—682no—
134C23×C4—172no—
135C23×C4C2×#59511yescharacteristic
136C2×M16—344no—
137#117—42no—
138#117—682no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—172no—
141C2×C22⋊C4—21no—
142C2×C22⋊C4—341no—
143C2×C42—341no—
144C2×C42C2×#59512yes—
145#101—348no—
146#101—684no—
147C22⋊C8—684no—
148#95—344no—
149#95—344no—
150C17#1286211yescharacteristic, complemented, Sylow
151C24C22⋊#59511yescharacteristic
152C2×Q8—174no—
153C2×Q8—344no—
154C2×Q8—342no—
155C2×Q8—344no—
156C2×Q8—681no—
157C2×Q8—681no—
158C2×Q8—682no—
159C2×Q8—681no—
160C2×Q8—681no—
161C2×Q8—6816no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—22no—
165C22×C4—22no—
166C22×C4—172no—
167C22×C4—172no—
168C22×C4—344no—
169C22×C4—344no—
170C22×C4—342no—
171C22×C4—682no—
172C22×C4C22⋊#59511yescharacteristic
173C22×C4C22×#59511yescharacteristic
174M16—684no—
175M16—684no—
176C2×C8—684no—
177C4⋊C4—42no—
178C4⋊C4—682no—
179C22⋊C4—42no—
180C22⋊C4—682no—
181C42—42no—
182C42—682no—
183C23—21no—
184C23—21no—
185C23—42no—
186C23#1058753912yes—
187C23C2×C22⋊#59511yescharacteristic
188Q8—684no—
189Q8—688no—
190Q8—684no—
191C2×C4—28no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—44no—
197C2×C4—42no—
198C2×C4—174no—
199C2×C4—342no—
200C2×C4—342no—
201C2×C4—342no—
202C2×C4—342no—
203C2×C4—342no—
204C2×C4—688no—
205C2×C4—684no—
206C2×C4—682no—
207C2×C4#1058755112yes—
208C2×C4#1058767712yes—
209C8—1364no—
210C22—22no—
211C22—22no—
212C22—24no—
213C22—42no—
214C22—42no—
215C22#1176375211yescharacteristic
216C22C2×#1058755111yescharacteristic
217C22C2×#1058753911yescharacteristic
218C4—22no—
219C4—22no—
220C4—82no—
221C4—344no—
222C4—682no—
223C4—1362no—
224C2—21no—
225C2—21no—
226C2—42no—
227C2#1454467911yescharacteristic
228C2#1454508112yes—
subgroupG/H
centerC22C2×#10587539
commutatorC22×C4×C17C22×C4
FrattiniC22×C4C22×#595
FittingC2×C17×#117C4
socleC22×C17C2×#99
prime radicalC24×C17C22⋊C4
cosocleC22×Dic34C23
hypercenterC2×#117#595
hypocenterC17#12862
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
2. #14543881C2semidirect
150. C17#12862semidirect

Maximal subgroups

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

subgroupindex
#145446182
#145438812
#145313682
#1286217