Finite Group Databasebeta

← #17948478

Subgroups — #17948478

Order 4352.

368 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 78 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#14543622C212yesmaximal
2#14527052C211yesmaximal, characteristic
3C2×C22⋊#3624—22no—
4C2×#10587502—22no—
5C2×#10587502C411yescharacteristic
6C2×#10587502C2211yescharacteristic
7C22×C4⋊C4×C17C411yescharacteristic
8C23×C4×C17—21no—
9C23×C4×C17C2×C411yescharacteristic
10C23×Dic17—21no—
11C23×Dic17D412yes—
12C22×#3624—42no—
13C2×C4⋊C4×C17—21no—
14C2×C4⋊C4×C17—41no—
15C2×C4⋊C4×C17—41no—
16C2×C4⋊C4×C17C812yescomplemented
17C22⋊#3624—24no—
18C22⋊#3624—42no—
19#10587502—41no—
20#10587502—44no—
21#10587502—41no—
22C24×C17C22⋊C411yescharacteristic
23C22×C4×C17—21no—
24C22×C4×C17—21no—
25C22×C4×C17—22no—
26C22×C4×C17—42no—
27C22×C4×C17—42no—
28C22×C4×C17—42no—
29C22×C4×C17—41no—
30C22×C4×C17—41no—
31C22×C4×C17C2×C811yescharacteristic
32C22×C4×C17M1611yescharacteristic
33C22×Dic17—22no—
34C22×Dic17—22no—
35C22×Dic17—22no—
36C22×Dic17—42no—
37C22×Dic17—42no—
38C22×Dic17—42no—
39C22×Dic17—41no—
40C22×Dic17—41no—
41C22×Dic17—42no—
42C22×Dic17SD1612yes—
43C22×Dic17Q1612yes—
44C2×#3624—44no—
45C2×#3624—82no—
46C2×#3624—82no—
47C4⋊C4×C17—42no—
48C4⋊C4×C17—42no—
49#7132—171nomaximal, Sylow
50C23×C17—22no—
51C23×C17—21no—
52C23×C17—21no—
53C23×C17C22⋊C811yescharacteristic
54C23×C17#9911yescharacteristic
55C23×C17#10011yescharacteristic
56C23×C17#10312yes—
57C23×C17#10412yes—
58C2×C4×C17—21no—
59C2×C4×C17—21no—
60C2×C4×C17—42no—
61C2×C4×C17—42no—
62C2×C4×C17—41no—
63C2×C4×C17—41no—
64C2×C4×C17—44no—
65C2×C4×C17—42no—
66C2×C4×C17—82no—
67C2×C4×C17—81no—
68C2×C4×C17—81no—
69C2×Dic17—22no—
70C2×Dic17—22no—
71C2×Dic17—42no—
72C2×Dic17—42no—
73C2×Dic17—42no—
74C2×Dic17—42no—
75C2×Dic17—42no—
76C2×Dic17—44no—
77C2×Dic17—82no—
78C2×Dic17—81no—
79C2×Dic17—82no—
80C2×Dic17—81no—
81#3624—84no—
82#1445—171no—
83#1280—172no—
84C22×C17—21no—
85C22×C17—21no—
86C22×C17—21no—
87C22×C17—21no—
88C22×C17—22no—
89C22×C17—22no—
90C22×C17—22no—
91C22×C17—22no—
92C22×C17#32311yescharacteristic
93C22×C17Q8⋊C812yes—
94C22×C17#32812yes—
95C22×C17#32911yescharacteristic
96C22×C17#33311yescharacteristic
97C22×C17#35212yes—
98C22×C17#35311yescharacteristic
99C22×C17#35511yescharacteristic
100C4×C17—41no—
101C4×C17—41no—
102C4×C17—82no—
103Dic17—42no—
104Dic17—42no—
105Dic17—82no—
106C22×C4⋊C4#59511yescharacteristic
107C2×C22⋊C8—342no—
108C2×#95—171no—
109C2×#95—171no—
110C2×#95—342no—
111C2×C17—21no—
112C2×C17—22no—
113C2×C17—21no—
114C2×C17#127211yescharacteristic
115C2×C17#131711yescharacteristic
116C2×C17#131811yescharacteristic
117C2×C17#135212yes—
118C2×C17#135312yes—
119C23×C4—21no—
120C23×C4—172no—
121C23×C4—341no—
122C23×C4C2×#59511yescharacteristic
123C22×C8—682no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—41no—
126C2×C4⋊C4—41no—
127C2×C4⋊C4#362412yescomplemented
128C22⋊C8—344no—
129C22⋊C8—682no—
130#95—681no—
131#95—681no—
132#95—684no—
133C17#713211yescharacteristic, complemented, Sylow
134C24C22⋊#59511yescharacteristic
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—22no—
138C22×C4—42no—
139C22×C4—42no—
140C22×C4—42no—
141C22×C4—41no—
142C22×C4—41no—
143C22×C4—172no—
144C22×C4—172no—
145C22×C4—342no—
146C22×C4—342no—
147C22×C4—342no—
148C22×C4—682no—
149C22×C4—682no—
150C22×C4—681no—
151C22×C4—682no—
152C22×C4—682no—
153C22×C4—681no—
154C22×C4C2×#362411yescharacteristic
155C22×C4#6324711yescharacteristic
156C2×C8—684no—
157C2×C8—1362no—
158C2×C8—1362no—
159C4⋊C4—42no—
160C4⋊C4—42no—
161C23—21no—
162C23—21no—
163C23—22no—
164C23#1058754912yes—
165C23#1058755012yes—
166C23#1058755111yescharacteristic
167C23C22⋊#362411yescharacteristic
168C23#1058755311yescharacteristic
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—42no—
172C2×C4—42no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—44no—
176C2×C4—42no—
177C2×C4—82no—
178C2×C4—81no—
179C2×C4—81no—
180C2×C4—342no—
181C2×C4—342no—
182C2×C4—682no—
183C2×C4—682no—
184C2×C4—682no—
185C2×C4—682no—
186C2×C4—682no—
187C2×C4—684no—
188C2×C4—1362no—
189C2×C4—1361no—
190C2×C4—1362no—
191C2×C4—1361no—
192C8—1364no—
193C22—21no—
194C22—21no—
195C22—21no—
196C22—21no—
197C22—22no—
198C22—22no—
199C22—22no—
200C22—22no—
201C22#1176365211yescharacteristic
202C22Q8⋊#362412yes—
203C22#1176366312yes—
204C22#1176366511yescharacteristic
205C22#1176366911yescharacteristic
206C22#1176371112yes—
207C22#1176371311yescharacteristic
208C22#1176371711yescharacteristic
209C4—41no—
210C4—41no—
211C4—82no—
212C4—682no—
213C4—682no—
214C4—1362no—
215C2—21no—
216C2—22no—
217C2—21no—
218C2#1454360411yescharacteristic
219C2#1454371011yescharacteristic
220C2#1454371211yescharacteristic
221C2#1454378112yes—
222C2#1454378312yes—
subgroupG/H
centerC23#10587551
commutatorC22×C4×C17C2×C8
FrattiniC23×C4C2×#595
FittingC22×C4⋊C4×C17C4
socleC23×C17#99
prime radicalC24×C17C22⋊C4
cosocleC2×#10587502C22
hypercenterC22×C4⋊C4#595
hypocenterC17#7132
LCSG ▷ C22×C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C24 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
16. C2×C4⋊C4×C17C8semidirect
127. C2×C4⋊C4#3624semidirect
133. C17#7132semidirect

Maximal subgroups

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

subgroupindex
#145436222
#145270522
#713217