Finite Group Databasebeta

← #17958158

Subgroups — #17958158

Order 4352.

526 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 112 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#14544015C212yesmaximal
2#14543714C214yesmaximal, complemented
3#14531370C211yesmaximal, characteristic
4#11763825—22no—
5#11763654C2214yes—
6#11763652—22no—
7C2×#10587612C411yescharacteristic
8C2×#10587612C2211yescharacteristic
9#11762716—22no—
10#11762716C412yes—
11#11762716C2212yes—
12#11762672—22no—
13C23×Q8×C17C411yescharacteristic
14C22×Q8×C17—21no—
15C22×Q8×C17—22no—
16C22×Q8×C17—42no—
17C22×Q8×C17—42no—
18C22×Q8×C17C814yescomplemented
19C22×Q8×C17C2×C412yes—
20C23×C4×C17—21no—
21C23×C4×C17C2×C411yescharacteristic
22C2×C22⋊Dic17—21no—
23C2×C22⋊Dic17D412yes—
24#10587612—42no—
25#10587612—42no—
26C2×C4⋊Dic17—21no—
27C2×C4⋊Dic17C2×C411yescharacteristic
28C2×C4⋊Dic17C2311yescharacteristic
29C2×C4×Dic17C2×C412yes—
30C2×C4×Dic17D412yes—
31C22⋊#3624—24no—
32C4⋊#3624—42no—
33C4×#3624—42no—
34#10587502—24no—
35#10587502—24no—
36C24×C17C22⋊C411yescharacteristic
37C2×Q8×C17—24no—
38C2×Q8×C17—22no—
39C2×Q8×C17—48no—
40C2×Q8×C17—48no—
41C2×Q8×C17—42no—
42C2×Q8×C17—42no—
43C2×Q8×C17—42no—
44C2×Q8×C17—42no—
45C2×Q8×C17C2×C814yes—
46C22×C4×C17—22no—
47C22×C4×C17—22no—
48C22×C4×C17—24no—
49C22×C4×C17—22no—
50C22×C4×C17—42no—
51C22×C4×C17—42no—
52C22×C4×C17C22⋊C411yescharacteristic
53C22×C4×C17C2×C812yes—
54C22×C4×C17M1612yes—
55C22×C4×C17C22×C411yescharacteristic
56C22×Dic17—22no—
57C22×Dic17C22⋊C412yes—
58C22×Dic17C2×D412yes—
59C2×#3624—44no—
60C22⋊Dic17—42no—
61C22⋊Dic17—42no—
62C4⋊Dic17—41no—
63C4⋊Dic17—41no—
64C4⋊Dic17—42no—
65C4×Dic17—22no—
66C4×Dic17—22no—
67C4×Dic17—42no—
68#11434—171nomaximal, Sylow
69C23×C17—21no—
70C23×C17—21no—
71C23×C17—41no—
72C23×C17—41no—
73C23×C17C22⋊C812yes—
74C23×C17C2×C22⋊C411yescharacteristic
75Q8×C17—44no—
76Q8×C17—44no—
77Q8×C17—48no—
78C2×C4×C17—21no—
79C2×C4×C17—21no—
80C2×C4×C17—24no—
81C2×C4×C17—24no—
82C2×C4×C17—24no—
83C2×C4×C17—48no—
84C2×C4×C17—42no—
85C2×C4×C17—42no—
86C2×C4×C17C22⋊C812yes—
87C2×C4×C17#9912yes—
88C2×C4×C17#10012yes—
89C2×C4×C17C22×C811yescharacteristic
90C2×C4×C17C2×M1611yescharacteristic
91C2×Dic17—22no—
92C2×Dic17—22no—
93C2×Dic17—42no—
94C2×Dic17—42no—
95C2×Dic17—42no—
96C2×Dic17—42no—
97C2×Dic17—42no—
98#3624—84no—
99#2310—171no—
100#1458—172no—
101#1319—174no—
102C22×C17—21no—
103C22×C17—21no—
104C22×C17—22no—
105C22×C17—42no—
106C22×C17—42no—
107C22×C17—41no—
108C22×C17—41no—
109C22×C17C2×C22⋊C811yescharacteristic
110C22×C17C2×#9911yescharacteristic
111C22×C17C2×#10011yescharacteristic
112C4×C17—21no—
113C4×C17—21no—
114C4×C17—44no—
115C4×C17#32314yes—
116Dic17—42no—
117Dic17—42no—
118Dic17—82no—
119C23×Q8#59511yescharacteristic
120C2×#117—171no—
121C2×#117—171no—
122#431—342no—
123#391—172no—
124#391—172no—
125#391—342no—
126#381—342no—
127#324—174no—
128#323—342no—
129C2×C17—21no—
130C2×C17—21no—
131C2×C17—42no—
132C2×C17C2×#32311yescharacteristic
133C2×C17#212911yescharacteristic
134C2×C17#213511yescharacteristic
135C22×Q8—22no—
136C22×Q8—21no—
137C22×Q8—42no—
138C22×Q8—42no—
139C22×Q8#362414yescomplemented
140C22×Q8C2×#59512yes—
141C23×C4—21no—
142C23×C4C2×#59511yescharacteristic
143#117—682no—
144#117—682no—
145C2×C4⋊C4—171no—
146C2×C4⋊C4—171no—
147C2×C4⋊C4—341no—
148C2×C22⋊C4—172no—
149C2×C22⋊C4—341no—
150C2×C42—172no—
151C2×C42—172no—
152C4⋊C8—682no—
153C22⋊C8—344no—
154C4×C8—682no—
155#95—344no—
156#95—344no—
157C17#1143411yescharacteristic, complemented, Sylow
158C24C22⋊#59511yescharacteristic
159C2×Q8—22no—
160C2×Q8—24no—
161C2×Q8—48no—
162C2×Q8—48no—
163C2×Q8—42no—
164C2×Q8—42no—
165C2×Q8—42no—
166C2×Q8—42no—
167C2×Q8C2×#362414yes—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—24no—
172C22×C4—42no—
173C22×C4—42no—
174C22×C4—172no—
175C22×C4—172no—
176C22×C4—342no—
177C22×C4C2×#362412yes—
178C22×C4#6324712yes—
179C22×C4C22⋊#59511yescharacteristic
180C22×C4C22×#59511yescharacteristic
181C2×C8—684no—
182C4⋊C4—681no—
183C4⋊C4—681no—
184C4⋊C4—682no—
185C22⋊C4—682no—
186C22⋊C4—682no—
187C42—342no—
188C42—342no—
189C42—682no—
190C23—21no—
191C23—21no—
192C23—41no—
193C23—41no—
194C23C22⋊#362412yes—
195C23C2×C22⋊#59511yescharacteristic
196Q8—44no—
197Q8—44no—
198Q8—48no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—24no—
202C2×C4—24no—
203C2×C4—24no—
204C2×C4—48no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—342no—
208C2×C4—342no—
209C2×C4—682no—
210C2×C4—682no—
211C2×C4—682no—
212C2×C4—682no—
213C2×C4—682no—
214C2×C4#1058755112yes—
215C2×C4C22⋊#362412yes—
216C2×C4#1058755312yes—
217C2×C4C22×#362411yescharacteristic
218C2×C4C2×#6324711yescharacteristic
219C8—1364no—
220C22—21no—
221C22—21no—
222C22—22no—
223C22—42no—
224C22—42no—
225C22—41no—
226C22—41no—
227C22C2×C22⋊#362411yescharacteristic
228C22C2×#1058755111yescharacteristic
229C22C2×#1058755311yescharacteristic
230C4—21no—
231C4—21no—
232C4—44no—
233C4—682no—
234C4—682no—
235C4—1362no—
236C4#1176365214yes—
237C2—21no—
238C2—21no—
239C2—42no—
240C2C2×#1176365211yescharacteristic
241C2#1454506611yescharacteristic
242C2#1454508611yescharacteristic
subgroupG/H
centerC22C2×#10587551
commutatorC2×C4×C17C22×C8
FrattiniC22×C4C22×#595
FittingC23×Q8×C17C4
socleC22×C17C2×#99
prime radicalC24×C17C22⋊C4
cosocleC2×C4⋊Dic17C23
hypercenterC23×Q8#595
hypocenterC17#11434
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ C23×Q8

Complements

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

NH ≅ G/Ntype
2. #14543714C2semidirect
18. C22×Q8×C17C8semidirect
139. C22×Q8#3624semidirect
157. C17#11434semidirect

Maximal subgroups

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

subgroupindex
#145440152
#145437142
#145313702
#1143417