Finite Group Databasebeta

← #17958210

Subgroups — #17958210

Order 4352.

434 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 104 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#14543727C214yesmaximal, complemented
3#14531408C211yesmaximal, characteristic
4#11763825—22no—
5#11763654C2214yes—
6#11763652—22no—
7C2×#10587612C411yescharacteristic
8C2×#10587612C2211yescharacteristic
9#11762705—22no—
10#11762682—22no—
11#11762678C2212yes—
12#11762674C412yes—
13C22×C4⋊C4×C17C411yescharacteristic
14C23×C4×C17—21no—
15C23×C4×C17C2×C411yescharacteristic
16C2×C4⋊C4×C17—21no—
17C2×C4⋊C4×C17—42no—
18C2×C4⋊C4×C17C2×C412yes—
19C2×C22⋊Dic17—21no—
20C2×C22⋊Dic17D412yes—
21#10587612—22no—
22#10587612—42no—
23#10587612C814yescomplemented
24C2×C4⋊Dic17—21no—
25C2×C4⋊Dic17C2×C411yescharacteristic
26C2×C4⋊Dic17C2311yescharacteristic
27C2×#63225—24no—
28C2×C4×Dic17C2×C412yes—
29C2×C4×Dic17D412yes—
30C22⋊#3624—24no—
31C4⋊#3624—42no—
32C4×#3624—42no—
33#10587502—24no—
34C24×C17C22⋊C411yescharacteristic
35C22×C4×C17—22no—
36C22×C4×C17—22no—
37C22×C4×C17—22no—
38C22×C4×C17—42no—
39C22×C4×C17—82no—
40C22×C4×C17C22⋊C411yescharacteristic
41C22×C4×C17C2×C812yes—
42C22×C4×C17M1612yes—
43C22×C4×C17C22×C411yescharacteristic
44C22×Dic17—22no—
45C22×Dic17C22⋊C412yes—
46C22×Dic17C2×D412yes—
47C2×#3624—44no—
48C4⋊C4×C17—42no—
49C4⋊C4×C17—42no—
50C22⋊Dic17—24no—
51C22⋊Dic17—42no—
52C4⋊Dic17—24no—
53C4⋊Dic17—41no—
54C4⋊Dic17—41no—
55#63225—84no—
56C4×Dic17—22no—
57C4×Dic17—22no—
58C4×Dic17—22no—
59C4×Dic17C2×C814yes—
60#11450—171nomaximal, Sylow
61C23×C17—21no—
62C23×C17—21no—
63C23×C17—41no—
64C23×C17—41no—
65C23×C17C22⋊C812yes—
66C23×C17C2×C22⋊C411yescharacteristic
67C2×C4×C17—21no—
68C2×C4×C17—21no—
69C2×C4×C17—24no—
70C2×C4×C17—24no—
71C2×C4×C17—44no—
72C2×C4×C17—42no—
73C2×C4×C17—84no—
74C2×C4×C17C22⋊C812yes—
75C2×C4×C17#9912yes—
76C2×C4×C17#10112yes—
77C2×C4×C17C22×C811yescharacteristic
78C2×C4×C17C2×M1611yescharacteristic
79C2×Dic17—24no—
80C2×Dic17—22no—
81C2×Dic17—22no—
82C2×Dic17—42no—
83C2×Dic17—42no—
84C2×Dic17—42no—
85C2×Dic17—42no—
86#3624—84no—
87#2317—171no—
88#1458—172no—
89#1325—174no—
90C22×C17—21no—
91C22×C17—21no—
92C22×C17—22no—
93C22×C17—42no—
94C22×C17—42no—
95C22×C17—41no—
96C22×C17—41no—
97C22×C17C2×C22⋊C811yescharacteristic
98C22×C17C2×#9911yescharacteristic
99C22×C17C2×#10111yescharacteristic
100C4×C17—21no—
101C4×C17—21no—
102C4×C17—82no—
103C4×C17#32414yes—
104Dic17—44no—
105Dic17—42no—
106Dic17—42no—
107C2×#117—171no—
108C2×#117—171no—
109C22×C4⋊C4#59511yescharacteristic
110#431—342no—
111#389—342no—
112#385—342no—
113#384—172no—
114#382—172no—
115#324—174no—
116#323—342no—
117C2×C17—21no—
118C2×C17—21no—
119C2×C17—42no—
120C2×C17C2×#32411yescharacteristic
121C2×C17#212811yescharacteristic
122C2×C17#213511yescharacteristic
123C23×C4—21no—
124C23×C4C2×#59511yescharacteristic
125#117—174no—
126#117—342no—
127#117—682no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—42no—
130C2×C4⋊C4—171no—
131C2×C4⋊C4—171no—
132C2×C4⋊C4—341no—
133C2×C4⋊C4—344no—
134C2×C4⋊C4C2×#59512yes—
135C2×C22⋊C4—172no—
136C2×C22⋊C4—341no—
137C2×C42—172no—
138C2×C42—172no—
139C4⋊C8—682no—
140C22⋊C8—344no—
141C4×C8—682no—
142#95—344no—
143C17#1145011yescharacteristic, complemented, Sylow
144C24C22⋊#59511yescharacteristic
145C22×C4—22no—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—42no—
149C22×C4—82no—
150C22×C4—172no—
151C22×C4—172no—
152C22×C4—342no—
153C22×C4#6324212yes—
154C22×C4#6324312yes—
155C22×C4C22⋊#59511yescharacteristic
156C22×C4C22×#59511yescharacteristic
157C2×C8—684no—
158C4⋊C4—42no—
159C4⋊C4—42no—
160C4⋊C4—344no—
161C4⋊C4—681no—
162C4⋊C4—681no—
163C4⋊C4—1364no—
164C22⋊C4—344no—
165C22⋊C4—682no—
166C42—174no—
167C42—342no—
168C42—342no—
169C42—342no—
170C23—21no—
171C23—21no—
172C23—41no—
173C23—41no—
174C23#1058754312yes—
175C23C2×C22⋊#59511yescharacteristic
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—24no—
179C2×C4—24no—
180C2×C4—44no—
181C2×C4—42no—
182C2×C4—84no—
183C2×C4—344no—
184C2×C4—342no—
185C2×C4—342no—
186C2×C4—682no—
187C2×C4—682no—
188C2×C4—682no—
189C2×C4—682no—
190C2×C4#1058754312yes—
191C2×C4#1058754512yes—
192C2×C4#1058755112yes—
193C2×C4C2×#6324211yescharacteristic
194C2×C4C2×#6324311yescharacteristic
195C8—1364no—
196C22—21no—
197C22—21no—
198C22—22no—
199C22—42no—
200C22—42no—
201C22—41no—
202C22—41no—
203C22C2×#1058754311yescharacteristic
204C22C2×#1058755111yescharacteristic
205C22C2×#1058754511yescharacteristic
206C4—21no—
207C4—21no—
208C4—82no—
209C4—684no—
210C4—682no—
211C4—682no—
212C4#1176365514yes—
213C2—21no—
214C2—21no—
215C2—42no—
216C2C2×#1176365511yescharacteristic
217C2#1454506311yescharacteristic
218C2#1454508611yescharacteristic
subgroupG/H
centerC22C2×#10587551
commutatorC2×C4×C17C22×C8
FrattiniC22×C4C22×#595
FittingC22×C4⋊C4×C17C4
socleC22×C17C2×#99
prime radicalC24×C17C22⋊C4
cosocleC2×C4⋊Dic17C23
hypercenterC22×C4⋊C4#595
hypocenterC17#11450
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #14543727C2semidirect
23. #10587612C8semidirect
143. C17#11450semidirect

Maximal subgroups

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

subgroupindex
#145440152
#145437272
#145314082
#1145017