Finite Group Databasebeta

← #17949916

Subgroups — #17949916

Order 4352.

636 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 636 conjugacy classes of subgroups — too many to draw. Only its 102 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#14544501C212yesmaximal, complemented
2#14543600C214yesmaximal, complemented
3#127×Dic17C211yesmaximal, characteristic
4C2×#10587542—22no—
5#11763675—22no—
6#11763675C2214yes—
7C2×D4×Dic17—22no—
8C2×D4×Dic17C412yes—
9#11762712—22no—
10#11762712C2212yes—
11#11762679C411yescharacteristic
12C42×Dic17C2211yescharacteristic
13C2×C17×#127C411yescharacteristic
14C22×D4×C17—22no—
15C22×D4×C17C2×C412yes—
16C23×Dic17—22no—
17C22×#3624—24no—
18C17×#127—24no—
19C2×C42×C17C2×C411yescharacteristic
20C2×C22⋊Dic17—22no—
21C2×C22⋊Dic17—24no—
22D4×Dic17—48no—
23D4×Dic17—44no—
24C2×C4⋊Dic17—22no—
25C2×C4⋊Dic17C2×C412yes—
26C2×C4×Dic17C2×C411yescharacteristic
27C2×C4×Dic17D412yes—
28C2×C4×Dic17D412yes—
29C2×C4×Dic17C2311yescharacteristic
30#10587542—28no—
31C24×C17—22no—
32C2×D4×C17—28no—
33C2×D4×C17—22no—
34C2×D4×C17—42no—
35C2×D4×C17—42no—
36C2×D4×C17—44no—
37C2×D4×C17—44no—
38C2×D4×C17M1614yes—
39C22×C4×C17C22⋊C412yes—
40C22×C4×C17C22×C411yescharacteristic
41C22×Dic17—22no—
42C22×Dic17—48no—
43C22×Dic17—44no—
44C22×Dic17C22⋊C411yescharacteristic
45C22×Dic17C22⋊C411yescharacteristic
46C22×Dic17C2×D412yes—
47C2×#3624—28no—
48C2×#3624—44no—
49C2×#3624—44no—
50C42×C17—22no—
51C22⋊Dic17—44no—
52C22⋊Dic17—48no—
53C4⋊Dic17—44no—
54C4⋊Dic17—42no—
55C4×Dic17—24no—
56C4×Dic17—22no—
57C4×Dic17—24no—
58C4×Dic17—41no—
59#7637—171nomaximal, Sylow
60C23×C17—24no—
61C23×C17—42no—
62C23×C17—42no—
63C23×C17—44no—
64C23×C17—44no—
65C23×C17C2×C22⋊C411yescharacteristic
66D4×C17—44no—
67D4×C17—44no—
68D4×C17—416no—
69C2×C4×C17—22no—
70C2×C4×C17—21no—
71C2×C4×C17—21no—
72C2×C4×C17—22no—
73C2×C4×C17#10414yes—
74C2×C4×C17C2×M1612yes—
75C2×Dic17—22no—
76C2×Dic17—22no—
77C2×Dic17—22no—
78C2×Dic17—21no—
79C2×Dic17—22no—
80C2×Dic17—42no—
81C2×Dic17—44no—
82C2×Dic17—88no—
83C2×Dic17#10012yes—
84C2×Dic17AGL(1,ℤ/8)14yes—
85#3624—48no—
86C4×#127—171no—
87#1745—172no—
88#1270—174no—
89C22×C17—44no—
90C22×C17—48no—
91C22×C17—82no—
92C22×C17—84no—
93C22×C17—82no—
94C22×C17#40711yescharacteristic
95C22×C17#41011yescharacteristic
96C22×C17C2×#10011yescharacteristic
97C22×C17#41812yes—
98C22×C17C2×#10412yes—
99C4×C17—22no—
100C4×C17—24no—
101Dic17—22no—
102Dic17—24no—
103Dic17—41no—
104Dic17—82no—
105C2×#127#59511yescharacteristic
106C2×C4×D4—172no—
107C2×C4×D4—342no—
108C2×#97—342no—
109#390—172no—
110#390—342no—
111#384—171no—
112C43—171no—
113#336—174no—
114#336—342no—
115C2×C17—84no—
116C2×C17#146611yescharacteristic
117C2×C17#149012yes—
118C2×C17#150512yes—
119C2×C17#152511yescharacteristic
120C2×C17#155511yescharacteristic
121C22×D4—22no—
122C22×D4C2×#59512yes—
123C23×C4—342no—
124C22×C8—344no—
125#127—24no—
126C4×D4—688no—
127C4×D4—684no—
128C2×C4⋊C4—172no—
129C2×C4⋊C4—342no—
130C2×C22⋊C4—342no—
131C2×C22⋊C4—344no—
132C2×C42—172no—
133C2×C42—172no—
134C2×C42—171no—
135C2×C42—171no—
136C2×C42C2×#59511yescharacteristic
137#97—348no—
138C17#763711yescharacteristic, complemented, Sylow
139C24—22no—
140C2×D4—28no—
141C2×D4—22no—
142C2×D4—42no—
143C2×D4—42no—
144C2×D4—44no—
145C2×D4—44no—
146C2×D4#6324714yes—
147C22×C4—171no—
148C22×C4—172no—
149C22×C4—171no—
150C22×C4—342no—
151C22×C4—684no—
152C22×C4—688no—
153C22×C4C22⋊#59512yes—
154C22×C4C22×#59511yescharacteristic
155C2×C8—348no—
156C2×C8—684no—
157C2×C8—684no—
158C4⋊C4—682no—
159C4⋊C4—684no—
160C22⋊C4—688no—
161C22⋊C4—684no—
162C42—22no—
163C42—342no—
164C42—344no—
165C42—344no—
166C42—681no—
167C23—24no—
168C23—42no—
169C23—42no—
170C23—44no—
171C23—44no—
172C23C2×C22⋊#59511yescharacteristic
173D4—44no—
174D4—44no—
175D4—416no—
176C2×C4—22no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—22no—
180C2×C4—172no—
181C2×C4—174no—
182C2×C4—342no—
183C2×C4—342no—
184C2×C4—342no—
185C2×C4—341no—
186C2×C4—342no—
187C2×C4—682no—
188C2×C4—684no—
189C2×C4—1368no—
190C2×C4#1058754814yes—
191C2×C4C2×#6324712yes—
192C8—688no—
193C22—44no—
194C22—48no—
195C22—84no—
196C22—82no—
197C22—82no—
198C22#1176377911yescharacteristic
199C22#1176378911yescharacteristic
200C22D17⋊#10011yescharacteristic
201C22#1176380112yes—
202C22C2×#1058754812yes—
203C4—22no—
204C4—24no—
205C4—344no—
206C4—342no—
207C4—681no—
208C4—1362no—
209C2—84no—
210C2#1454404411yescharacteristic
211C2#1454412312yes—
212C2#1454417312yes—
213C2#1454423911yescharacteristic
214C2#1454431211yescharacteristic
subgroupG/H
centerC23C2×C22⋊#595
commutatorC22×C4×C17C22×C4
FrattiniC22×C4C22×#595
Fitting, prime radicalC2×C17×#127C4
socleC23×C17C2×C22⋊C4
cosocleC2×C4×Dic17C23
hypercenterC2×#127#595
hypocenterC17#7637
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#127

Complements

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

NH ≅ G/Ntype
1. #14544501C2semidirect
2. #14543600C2semidirect
138. C17#7637semidirect

Maximal subgroups

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

subgroupindex
#145445012
#145436002
#127×Dic172
#763717