Finite Group Databasebeta

← #17949657

Subgroups — #17949657

Order 4352.

550 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 550 conjugacy classes of subgroups — too many to draw. Only its 136 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
1C4×C22⋊#3624C212yesmaximal
2#14543602C214yesmaximal, complemented
3#14531268C211yesmaximal, characteristic
4C2×C22⋊#3624—22no—
5C2×C4×#3624—22no—
6#11763675C2214yes—
7C2×#10587612C411yescharacteristic
8C22×C4×Dic17C2211yescharacteristic
9#11762708C412yes—
10#11762708C2212yes—
11#11762684—22no—
12C4×C4⋊Dic17—22no—
13C2×C17×#117C411yescharacteristic
14C23×C4×C17C2×C411yescharacteristic
15C23×Dic17D412yes—
16C22×#3624—24no—
17C17×#117—22no—
18C17×#117C814yescomplemented
19C2×C4⋊C4×C17C2×C412yes—
20C2×C22⋊C4×C17—21no—
21C2×C42×C17—21no—
22C2×C22⋊Dic17—21no—
23#10587612—42no—
24C2×C4⋊Dic17—22no—
25C2×C4⋊Dic17—22no—
26C2×C4⋊Dic17C2×C412yes—
27C2×C4×Dic17—21no—
28C2×C4×Dic17—42no—
29C2×C4×Dic17C2×C411yescharacteristic
30C2×C4×Dic17D412yes—
31C2×C4×Dic17C2311yescharacteristic
32C22⋊#3624—28no—
33C4×#3624—44no—
34#10587502—22no—
35#10587502—22no—
36C24×C17C22⋊C411yescharacteristic
37C22×C4×C17—22no—
38C22×C4×C17—22no—
39C22×C4×C17—22no—
40C22×C4×C17C22⋊C411yescharacteristic
41C22×C4×C17C2×C812yes—
42C22×C4×C17M1612yes—
43C22×C4×C17C22×C411yescharacteristic
44C22×Dic17—24no—
45C22×Dic17—24no—
46C22×Dic17—22no—
47C22×Dic17—42no—
48C22×Dic17C22⋊C412yes—
49C22×Dic17C2×D412yes—
50C2×#3624—28no—
51C2×#3624—44no—
52C2×#3624—44no—
53C4⋊C4×C17—22no—
54C4⋊C4×C17C2×C814yes—
55C22⋊C4×C17—24no—
56C42×C17—24no—
57C22⋊Dic17—42no—
58C4⋊Dic17—44no—
59C4⋊Dic17—44no—
60C4⋊Dic17—42no—
61C4×Dic17—24no—
62C4×Dic17—41no—
63C4×Dic17—41no—
64C4×Dic17—42no—
65#7564—171nomaximal, Sylow
66C23×C17—21no—
67C23×C17—21no—
68C23×C17—22no—
69C23×C17—22no—
70C23×C17C22⋊C812yes—
71C23×C17C2×C22⋊C411yescharacteristic
72C2×C4×C17—21no—
73C2×C4×C17—21no—
74C2×C4×C17—24no—
75C2×C4×C17—24no—
76C2×C4×C17—24no—
77C2×C4×C17—42no—
78C2×C4×C17—42no—
79C2×C4×C17C22⋊C812yes—
80C2×C4×C17#9912yes—
81C2×C4×C17#10112yes—
82C2×C4×C17C22×C811yescharacteristic
83C2×C4×C17C2×M1611yescharacteristic
84C2×Dic17—22no—
85C2×Dic17—22no—
86C2×Dic17—22no—
87C2×Dic17—22no—
88C2×Dic17—48no—
89C2×Dic17—42no—
90C2×Dic17—42no—
91C2×Dic17—42no—
92C2×Dic17#10414yes—
93C2×Dic17#13514yes—
94#3624—48no—
95#2296—171no—
96C4×C22⋊C8—172no—
97#1271—174no—
98C22×C17—22no—
99C22×C17—24no—
100C22×C17—41no—
101C22×C17—42no—
102C22×C17—41no—
103C22×C17C2×C22⋊C811yescharacteristic
104C22×C17C2×#9911yescharacteristic
105C22×C17C2×#10111yescharacteristic
106C22×C17#41612yes—
107C22×C17C2×#10412yes—
108C4×C17—21no—
109C4×C17—21no—
110C4×C17—44no—
111C4×C17#32414yes—
112Dic17—24no—
113Dic17—24no—
114Dic17—82no—
115C2×#117—171no—
116C2×#117#59511yescharacteristic
117C22×C42—171no—
118C2×C22⋊C8—342no—
119C2×C4×C8—342no—
120#389—172no—
121#389—172no—
122#385—342no—
123C4×C4⋊C4—342no—
124#336—174no—
125C2×C17—42no—
126C2×C17C2×#32411yescharacteristic
127C2×C17#147612yes—
128C2×C17#150112yes—
129C2×C17#152611yescharacteristic
130C2×C17#154411yescharacteristic
131C23×C4—172no—
132C23×C4C2×#59511yescharacteristic
133C22×C8—344no—
134#117—22no—
135#117—682no—
136#117#362414yescomplemented
137C2×C4⋊C4—172no—
138C2×C4⋊C4—342no—
139C2×C4⋊C4—342no—
140C2×C4⋊C4C2×#59512yes—
141C2×C22⋊C4—21no—
142C2×C22⋊C4—341no—
143C2×C42—21no—
144C2×C42—172no—
145C2×C42—171no—
146C2×C42—171no—
147C2×C42—341no—
148C2×C42—682no—
149C22⋊C8—348no—
150C4×C8—684no—
151#95—342no—
152#95—342no—
153C17#756411yescharacteristic, complemented, Sylow
154C24C22⋊#59511yescharacteristic
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—22no—
158C22×C4—172no—
159C22×C4—172no—
160C22×C4—344no—
161C22×C4—344no—
162C22×C4—342no—
163C22×C4—682no—
164C22×C4C2×#362412yes—
165C22×C4#6324712yes—
166C22×C4C22⋊#59511yescharacteristic
167C22×C4C22×#59511yescharacteristic
168C2×C8—348no—
169C2×C8—684no—
170C2×C8—684no—
171C4⋊C4—22no—
172C4⋊C4—682no—
173C4⋊C4—684no—
174C4⋊C4—684no—
175C4⋊C4C2×#362414yes—
176C22⋊C4—24no—
177C22⋊C4—682no—
178C42—24no—
179C42—344no—
180C42—681no—
181C42—681no—
182C42—682no—
183C23—21no—
184C23—21no—
185C23—22no—
186C23—22no—
187C23C22⋊#362412yes—
188C23C2×C22⋊#59511yescharacteristic
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—24no—
192C2×C4—24no—
193C2×C4—24no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—174no—
197C2×C4—174no—
198C2×C4—342no—
199C2×C4—342no—
200C2×C4—342no—
201C2×C4—342no—
202C2×C4—688no—
203C2×C4—682no—
204C2×C4—682no—
205C2×C4—682no—
206C2×C4#1058753912yes—
207C2×C4#1058754512yes—
208C2×C4C22⋊#362412yes—
209C2×C4C22×#362411yescharacteristic
210C2×C4C2×#6324711yescharacteristic
211C8—688no—
212C22—22no—
213C22—24no—
214C22—42no—
215C22—41no—
216C22—41no—
217C22C2×C22⋊#362411yescharacteristic
218C22C2×#1058753911yescharacteristic
219C22C2×#1058754511yescharacteristic
220C22#1176379912yes—
221C22D17⋊#10412yes—
222C4—21no—
223C4—21no—
224C4—44no—
225C4—344no—
226C4—344no—
227C4—1362no—
228C4#1176365414yes—
229C2—42no—
230C2C2×#1176365411yescharacteristic
231C2#1454407512yes—
232C2#1454416112yes—
233C2#1454424111yescharacteristic
234C2#1454428911yescharacteristic
subgroupG/H
centerC22×C4C22⋊#595
commutatorC2×C4×C17C22×C8
FrattiniC22×C4C22×#595
FittingC2×C17×#117C4
socleC23×C17C2×C22⋊C4
prime radicalC24×C17C22⋊C4
cosocleC2×C4×Dic17C23
hypercenterC2×#117#595
hypocenterC17#7564
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22×C4 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
2. #14543602C2semidirect
18. C17×#117C8semidirect
136. #117#3624semidirect
153. C17#7564semidirect

Maximal subgroups

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

subgroupindex
C4×C22⋊#36242
#145436022
#145312682
#756417