Finite Group Databasebeta

← #14528771

Subgroups — #14528771

Order 2176.

380 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 87 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×#10587567C211yesmaximal, characteristic
2C2×#10587477C212yesmaximal
3#11762667C212yesmaximal, complemented
4#11762576C211yesmaximal, characteristic, complemented
5C17×#342C211yesmaximal, characteristic, complemented
6C22×C22⋊D17C2211yescharacteristic
7C23×Dic17C2211yescharacteristic
8C22×C4×D17C411yescharacteristic, complemented
9C2×C22⋊C4×C17C2212yes—
10C2×C22⋊C4×C17C2211yescharacteristic
11C2×C22⋊Dic17C2212yes—
12C2×#63227C411yescharacteristic, complemented
13C2×#63225—22no—
14C2×#63225C411yescharacteristic, complemented
15C2×C4×Dic17C411yescharacteristic, complemented
16#10587567—22no—
17#10587567—22no—
18#10587502—22no—
19#10587477—24no—
20C24×C17C2311yescharacteristic
21C23×D17C2×C411yescharacteristic
22C22×C4×C17—22no—
23C22×C4×C17C2×C411yescharacteristic
24C22×C4×C17C2×C411yescharacteristic
25C2×C22⋊D17—22no—
26C2×C22⋊D17—22no—
27C2×C22⋊D17D412yes—
28C2×C22⋊D17Q812yes—
29C22×Dic17—22no—
30C22×Dic17—42no—
31C22×Dic17C2×C411yescharacteristic
32C22×Dic17C2×C411yescharacteristic
33C22×Dic17C2×C411yescharacteristic
34C22×Dic17D412yes—
35C22×Dic17D412yes—
36C2×C4×D17—21no—
37C2×C4×D17—21no—
38C2×C4×D17—42no—
39C22⋊C4×C17—22no—
40C22⋊C4×C17—22no—
41C22⋊C4×C17—22no—
42C22⋊Dic17—22no—
43C22⋊Dic17—22no—
44#63227—22no—
45#63225—22no—
46#63225—44no—
47C4×Dic17—21no—
48C4×Dic17—21no—
49C23×C17—41no—
50C23×C17—41no—
51C23×C17C22×C411yescharacteristic
52C23×C17C2×D412yes—
53C23×C17C2×D411yescharacteristic
54C23×C17C2×Q811yescharacteristic
55C23×C17#4112yes—
56C22×D17—21no—
57C22×D17—21no—
58C22×D17—41no—
59C22×D17—41no—
60C22×D17C4⋊C412yescomplemented
61C2×C4×C17—21no—
62C2×C4×C17—21no—
63C2×C4×C17—41no—
64C2×C4×C17—41no—
65C2×C4×C17—42no—
66C2×C4×C17—42no—
67C2×C4×C17—42no—
68C22⋊D17—24no—
69C22⋊D17—24no—
70C2×Dic17—21no—
71C2×Dic17—21no—
72C2×Dic17—22no—
73C2×Dic17—24no—
74C2×Dic17—41no—
75C2×Dic17—42no—
76C2×Dic17—42no—
77C2×Dic17—42no—
78C2×Dic17—42no—
79C2×Dic17C22⋊C411yescharacteristic
80C2×Dic17C22⋊C412yes—
81C2×Dic17C22⋊C411yescharacteristic
82C2×Dic17C4⋊C412yescomplemented
83C4×D17—42no—
84C4×D17—42no—
85#1799—171nomaximal, Sylow
86C22×C17—22no—
87C22×C17—21no—
88C22×C17—22no—
89C22×C17—21no—
90C22×C17—41no—
91C22×C17—41no—
92C22×C17—42no—
93C22×C17C2×C22⋊C411yescharacteristic
94C22×C17C2×C4⋊C411yescharacteristic
95C22×C17#11711yescharacteristic
96C22×C17#12112yes—
97C22×C17C22⋊Q812yes—
98C2×D17—22no—
99C2×D17—42no—
100C2×D17—41no—
101C2×D17—41no—
102C2×D17—42no—
103C4×C17—41no—
104C4×C17—41no—
105C4×C17—82no—
106Dic17—22no—
107Dic17—21no—
108Dic17—22no—
109Dic17—82no—
110Dic17#9912yescomplemented
111C2×C4×D4—171no—
112C2×#99—172no—
113#380—172no—
114#342—171no—
115#342D1711yescharacteristic, complemented
116C2×C17—22no—
117C2×C17—21no—
118C2×C17—21no—
119C2×C17—22no—
120C2×C17#38011yescharacteristic
121C2×C17C2×#9911yescharacteristic
122C2×C17#41011yescharacteristic
123D17—42no—
124C22×D4—171no—
125C23×C4—171no—
126C23×C4—171no—
127C4×D4—342no—
128C4×D4—342no—
129C2×C4⋊C4—171no—
130C2×C4⋊C4—342no—
131C2×C22⋊C4—171no—
132C2×C22⋊C4—172no—
133C2×C22⋊C4C2×D1712yes—
134C2×C22⋊C4C2×D1711yescharacteristic
135C2×C42—171no—
136#99—344no—
137#95—342no—
138C17#179911yescharacteristic, complemented, Sylow
139C24—171no—
140C24C22×D1711yescharacteristic
141C2×D4—172no—
142C2×D4—172no—
143C2×D4—342no—
144C2×D4—342no—
145C22×C4—22no—
146C22×C4—171no—
147C22×C4—171no—
148C22×C4—172no—
149C22×C4—171no—
150C22×C4—172no—
151C22×C4—341no—
152C22×C4—341no—
153C22×C4—342no—
154C22×C4—682no—
155C22×C4—682no—
156C22×C4C4×D1711yescharacteristic
157C22×C4C4×D1711yescharacteristic
158C4⋊C4—342no—
159C4⋊C4—684no—
160C22⋊C4—22no—
161C22⋊C4—22no—
162C22⋊C4—22no—
163C22⋊C4—342no—
164C22⋊C4—342no—
165C22⋊C4—342no—
166C42—341no—
167C42—341no—
168C23—41no—
169C23—41no—
170C23—172no—
171C23—341no—
172C23—341no—
173C23—681no—
174C23—681no—
175C23C2×C4×D1711yescharacteristic
176C23#6325212yes—
177C23D4×D1712yes—
178C23D4×D1711yescharacteristic
179C23Q8×D1711yescharacteristic
180D4—344no—
181D4—344no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—171no—
190C2×C4—171no—
191C2×C4—172no—
192C2×C4—172no—
193C2×C4—344no—
194C2×C4—342no—
195C2×C4—341no—
196C2×C4—341no—
197C2×C4—682no—
198C2×C4—682no—
199C2×C4—682no—
200C2×C4—681no—
201C2×C4—682no—
202C2×C4—682no—
203C2×C4—682no—
204C22—22no—
205C22—21no—
206C22—22no—
207C22—21no—
208C22—42no—
209C22—41no—
210C22—41no—
211C22—342no—
212C22—682no—
213C22—682no—
214C22—681no—
215C22—681no—
216C22#1058755811yescharacteristic
217C22C22⋊C4×D1711yescharacteristic
218C22#1058757012yes—
219C22C4⋊C4×D1711yescharacteristic
220C22#1058758212yes—
221C4—41no—
222C4—41no—
223C4—82no—
224C4—172no—
225C4—341no—
226C4—342no—
227C4—342no—
228C4—1362no—
229C2—22no—
230C2—21no—
231C2—21no—
232C2—22no—
233C2—682no—
234C2#1176266811yescharacteristic
235C2#99×D1711yescharacteristic
236C2#1176278111yescharacteristic
subgroupG/H
centerC22C22⋊C4×D17
commutatorC23×C17C22×C4
FrattiniC24C22×D17
FittingC17×#342C2
socleC22×C17C2×C22⋊C4
prime radicalC22×C22⋊D17C22
cosocleC24×C17C23
hypercenter#342D17
hypocenterC17#1799
LCSG ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C24 ◁ #342

Complements

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

NH ≅ G/Ntype
3. #11762667C2semidirect
4. #11762576C2semidirect
5. C17×#342C2semidirect
8. C22×C4×D17C4semidirect
12. C2×#63227C4semidirect
14. C2×#63225C4semidirect
15. C2×C4×Dic17C4semidirect
60. C22×D17C4⋊C4semidirect
82. C2×Dic17C4⋊C4semidirect
110. Dic17#99semidirect
115. #342D17semidirect
138. C17#1799semidirect

Maximal subgroups

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

subgroupindex
C2×#105875672
C2×#105874772
#117626672
#117625762
C17×#3422
#179917