Finite Group Databasebeta

← #14533086

Subgroups — #14533086

Order 2176.

624 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 624 conjugacy classes of subgroups — too many to draw. Only its 173 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
1C2×#10587615C212yesmaximal
2C2×C4×C22⋊D17C211yesmaximal, characteristic
3#11762740C212yesmaximal, complemented
4#11762735C212yesmaximal, complemented
5#11762705C212yesmaximal, complemented
6#11762672C212yesmaximal, complemented
7#11762669C211yesmaximal, characteristic, complemented
8#11762666C212yesmaximal, complemented
9C22×C4⋊C4×C17C211yesmaximal, characteristic, complemented
10C23×C4×C17C2212yes—
11C23×C4×C17C2211yescharacteristic
12C22×C22⋊D17C2211yescharacteristic
13C22×C4×D17C2211yescharacteristic
14C2×C4⋊C4×C17—24no—
15C2×C4⋊C4×C17C2212yes—
16C2×C4⋊C4×C17C2212yes—
17C2×C22⋊Dic17C2211yescharacteristic
18C2×C22⋊Dic17C2212yes—
19#10587615—28no—
20C4×C22⋊D17—24no—
21C2×#63227C2211yescharacteristic
22C2×#63227C2214yes—
23C2×C4⋊Dic17C2212yes—
24C2×#63225C2214yes—
25C2×#63225C2211yescharacteristic
26C2×C4×Dic17C2211yescharacteristic
27#10587502C2214yes—
28#10587502C2212yes—
29#10587502C2214yes—
30C24×C17C2311yescharacteristic
31C23×D17C2311yescharacteristic
32C22×C4×C17—24no—
33C22×C4×C17—24no—
34C22×C4×C17—24no—
35C22×C4×C17—24no—
36C22×C4×C17D414yes—
37C22×C4×C17C2314yes—
38C22×C4×C17C2311yescharacteristic
39C22×C4×C17C2311yescharacteristic
40C2×C22⋊D17—22no—
41C2×C22⋊D17—22no—
42C2×C22⋊D17D414yes—
43C22×Dic17C2311yescharacteristic
44C22×Dic17C2311yescharacteristic
45C22×Dic17C2314yes—
46C22×Dic17C2311yescharacteristic
47C2×C4×D17—22no—
48C2×C4×D17—24no—
49C4⋊C4×C17—24no—
50C4⋊C4×C17—24no—
51C22⋊Dic17—22no—
52C22⋊Dic17—24no—
53#63227—28no—
54#63227—22no—
55C4⋊Dic17—24no—
56#63225—22no—
57#63225—28no—
58C4×Dic17—22no—
59C23×C17—22no—
60C23×C17—22no—
61C23×C17C2×D412yes—
62C23×C17C2×D412yes—
63C23×C17#4112yes—
64C23×C17C2411yescharacteristic
65C22×D17—21no—
66C22×D17—22no—
67C22×D17—21no—
68C22×D17—22no—
69C22×D17C2×D412yes—
70C2×C4×C17—22no—
71C2×C4×C17—28no—
72C2×C4×C17—24no—
73C2×C4×C17—24no—
74C2×C4×C17—24no—
75C2×C4×C17—48no—
76C2×C4×C17C2×D412yes—
77C2×C4×C17C2×D412yes—
78C2×C4×C17#4114yes—
79C22⋊D17—28no—
80C2×Dic17—21no—
81C2×Dic17—21no—
82C2×Dic17—21no—
83C2×Dic17—21no—
84C2×Dic17—24no—
85C2×Dic17—22no—
86C2×Dic17—24no—
87C2×Dic17—22no—
88C2×Dic17—24no—
89C2×Dic17C2×D412yes—
90C4×D17—44no—
91#2486—171nomaximal, Sylow
92C22×C17—22no—
93C22×C17—22no—
94C22×C17—24no—
95C22×C17—24no—
96C22×C17#12114yes—
97C22×C17C22×D411yescharacteristic
98C22×C17C22×D411yescharacteristic
99C22×C17C2×#4112yes—
100C22×C17C2×#4111yescharacteristic
101C22×C172−1+412yes—
102C2×D17—22no—
103C2×D17—24no—
104C2×D17—41no—
105C2×D17—41no—
106C2×D17—42no—
107C4×C17—22no—
108C4×C17—44no—
109C4×C17#12314yescomplemented
110Dic17—22no—
111Dic17—41no—
112Dic17—41no—
113Dic17—44no—
114C2×#123—172no—
115C2×C4×D4—171no—
116C22×C4⋊C4D1711yescharacteristic, complemented
117#398—172no—
118#397—172no—
119#389—172no—
120#381—172no—
121#380—171no—
122#380—172no—
123C2×C17—24no—
124C2×C17C2×#12111yescharacteristic
125C2×C17C2×#12311yescharacteristic
126C2×C17#53611yescharacteristic
127C2×C17#54712yes—
128C2×C17#55212yes—
129D17—42no—
130C22×D4—171no—
131C23×C4—171no—
132C23×C4C2×D1712yes—
133C23×C4C2×D1711yescharacteristic
134#123—348no—
135C4×D4—344no—
136C2×C4⋊C4—24no—
137C2×C4⋊C4—171no—
138C2×C4⋊C4—174no—
139C2×C4⋊C4—172no—
140C2×C4⋊C4C2×D1712yes—
141C2×C4⋊C4C2×D1712yes—
142C2×C22⋊C4—171no—
143C2×C22⋊C4—172no—
144C2×C22⋊C4—171no—
145C2×C22⋊C4—174no—
146C2×C42—171no—
147#95—174no—
148#95—172no—
149#95—174no—
150C17#248611yescharacteristic, complemented, Sylow
151C24—171no—
152C24C22×D1711yescharacteristic
153C2×D4—174no—
154C2×D4—342no—
155C2×D4—342no—
156C22×C4—24no—
157C22×C4—24no—
158C22×C4—24no—
159C22×C4—24no—
160C22×C4—171no—
161C22×C4—171no—
162C22×C4—171no—
163C22×C4—174no—
164C22×C4—344no—
165C22×C4—342no—
166C22×C4C22⋊D1714yes—
167C22×C4C22×D1714yes—
168C22×C4C22×D1711yescharacteristic
169C22×C4C22×D1711yescharacteristic
170C4⋊C4—24no—
171C4⋊C4—24no—
172C4⋊C4—348no—
173C4⋊C4—344no—
174C4⋊C4—342no—
175C22⋊C4—348no—
176C22⋊C4—344no—
177C22⋊C4—342no—
178C22⋊C4—342no—
179C42—342no—
180C23—22no—
181C23—22no—
182C23—172no—
183C23—342no—
184C23—341no—
185C23—342no—
186C23—341no—
187C23D4×D1712yes—
188C23#6325412yes—
189C23C2×C22⋊D1712yes—
190C23C23×D1711yescharacteristic
191D4—348no—
192C2×C4—22no—
193C2×C4—28no—
194C2×C4—24no—
195C2×C4—24no—
196C2×C4—24no—
197C2×C4—48no—
198C2×C4—172no—
199C2×C4—341no—
200C2×C4—341no—
201C2×C4—344no—
202C2×C4—342no—
203C2×C4—344no—
204C2×C4—344no—
205C2×C4—342no—
206C2×C4—341no—
207C2×C4—341no—
208C2×C4—684no—
209C2×C4#6325214yes—
210C2×C4C2×C22⋊D1712yes—
211C2×C4C2×C22⋊D1712yes—
212C22—24no—
213C22—24no—
214C22—22no—
215C22—22no—
216C22—344no—
217C22—342no—
218C22—682no—
219C22—681no—
220C22—681no—
221C22#1058762414yes—
222C22C2×#6325212yes—
223C22C2×D4×D1711yescharacteristic
224C22C2×#6325411yescharacteristic
225C22#1058769412yes—
226C22C22×C22⋊D1711yescharacteristic
227C4—22no—
228C4—44no—
229C4—342no—
230C4—684no—
231C4—681no—
232C4—681no—
233C4#1058761514yescomplemented
234C2—24no—
235C2—682no—
236C2C2×#1058762411yescharacteristic
237C2C2×#1058761511yescharacteristic
238C2#1176338311yescharacteristic
239C2#1176346012yes—
240C2#1176349412yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC22×C4⋊C4×C17C2
prime radicalC22×C22⋊D17C22
hypercenterC22×C4⋊C4D17
hypocenterC17#2486
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. #11762740C2semidirect
4. #11762735C2semidirect
5. #11762705C2semidirect
6. #11762672C2semidirect
7. #11762669C2semidirect
8. #11762666C2semidirect
9. C22×C4⋊C4×C17C2semidirect
109. C4×C17#123semidirect
116. C22×C4⋊C4D17semidirect
150. C17#2486semidirect
233. C4#10587615semidirect

Maximal subgroups

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

subgroupindex
C2×#105876152
C2×C4×C22⋊D172
#117627402
#117627352
#117627052
#117626722
#117626692
#117626662
C22×C4⋊C4×C172
#248617