Finite Group Databasebeta

← #14544807

Subgroups — #14544807

Order 2176.

378 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 70 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×#10587677C211yesmaximal, characteristic, complemented
2C2×#10587551C212yesmaximal
3#11763752C211yesmaximal, characteristic, complemented
4#11763687C212yesmaximal, complemented
5C2×#10587569C211yesmaximal, characteristic
6C22×C22⋊D17C2211yescharacteristic
7C22×C4×D17C2211yescharacteristic
8C2×C22⋊#595C2212yes—
9C2×C22⋊#595C2212yes—
10#10587677—24no—
11C2×C4⋊#595—21no—
12C2×C4×#595—21no—
13C2×C22⋊C4×C17C411yescharacteristic, complemented
14C2×C22⋊Dic17C411yescharacteristic, complemented
15C2×#63227C2211yescharacteristic
16C2×C4⋊Dic17C411yescharacteristic, complemented
17C2×#63225C411yescharacteristic, complemented
18#10587569—21no—
19#10587569—21no—
20#10587569—21no—
21#10587569—21no—
22#10587551—24no—
23(C2×#595)⋊C4—21no—
24(C2×#595)⋊C4—21no—
25C24×C17C2×C411yescharacteristic
26C23×D17C2311yescharacteristic
27C22×#595—22no—
28C22×#595—22no—
29C22×C4×C17C2×C411yescharacteristic
30C22×C4×C17C2×C411yescharacteristic
31C2×C22⋊D17—22no—
32C2×C22⋊D17—21no—
33C2×C22⋊D17—21no—
34C2×C22⋊D17—22no—
35C22×Dic17C2×C411yescharacteristic
36C22×Dic17C2×C411yescharacteristic
37C22×Dic17C2×C411yescharacteristic
38C2×C4×D17—22no—
39C2×C4×D17—22no—
40C2×C4×D17D411yescharacteristic
41C2×C4×D17D411yescharacteristic
42C2×C4×D17D411yescharacteristic
43C2×C4×D17D411yescharacteristic
44C22⋊#595—22no—
45C22⋊#595—22no—
46C22⋊#595—22no—
47C22⋊#595D414yes—
48C4⋊#595—24no—
49C4×#595—24no—
50C22⋊C4×C17—21no—
51C22⋊C4×C17—21no—
52C22⋊Dic17—21no—
53C22⋊Dic17—21no—
54#63227—21no—
55#63227—21no—
56C4⋊Dic17—21no—
57C4⋊Dic17—21no—
58#63225—21no—
59#63225—21no—
60C23×C17—21no—
61C23×C17—22no—
62C23×C17—41no—
63C23×C17—41no—
64C23×C17C22×C411yescharacteristic
65C22×D17—41no—
66C22×D17—41no—
67C22×D17C2×D411yescharacteristic
68C22×D17C2×D411yescharacteristic
69C22×D17C2×D412yes—
70C22×D17#4112yes—
71C2×#595—24no—
72C2×#595—42no—
73C2×#595—42no—
74C2×#595—42no—
75C2×#595—42no—
76C2×#595—42no—
77C2×C4×C17—22no—
78C2×C4×C17—21no—
79C2×C4×C17—41no—
80C2×C4×C17C22⋊C411yescharacteristic
81C2×C4×C17C22⋊C411yescharacteristic
82C22⋊D17—41no—
83C22⋊D17—42no—
84C22⋊D17—41no—
85C2×Dic17—22no—
86C2×Dic17—21no—
87C2×Dic17—21no—
88C2×Dic17—22no—
89C2×Dic17—41no—
90C2×Dic17C22⋊C411yescharacteristic
91C2×Dic17C22⋊C411yescharacteristic
92C4×D17—21no—
93C4×D17—21no—
94C4×D17—22no—
95C4×D17—22no—
96C4×D17—21no—
97C4×D17—21no—
98#1899—171nomaximal, Sylow
99C22×C17—21no—
100C22×C17—21no—
101C22×C17—41no—
102C22×C17—41no—
103C22×C17—41no—
104C22×C17—82no—
105C22×C17C2×C22⋊C411yescharacteristic
106C22×C17C4×D412yes—
107C2×D17—21no—
108C2×D17—21no—
109C2×D17—21no—
110C2×D17—21no—
111C2×D17—42no—
112C2×D17—42no—
113C2×D17#12011yescharacteristic
114C2×D17#12112yes—
115C2×D17#12311yescharacteristic
116#595—44no—
117#595—82no—
118C4×C17—21no—
119C4×C17—21no—
120C4×C17—41no—
121Dic17—21no—
122Dic17—21no—
123Dic17—41no—
124Dic17—41no—
125C2×#123—171no—
126C2×#117—171no—
127C2×#99—172no—
128#381—171no—
129#342—172no—
130C2×C17—22no—
131C2×C17—81no—
132C2×C17#38611yescharacteristic
133C2×C17#41011yescharacteristic
134C2×C17#41011yescharacteristic
135D17—21no—
136D17—22no—
137D17—21no—
138C22×D4—171no—
139C23×C4—171no—
140#123—341no—
141#123—341no—
142#123—341no—
143#123—341no—
144#117—344no—
145C2×C4⋊C4—171no—
146C2×C4⋊C4—171no—
147C2×C4⋊C4—341no—
148C2×C22⋊C4—172no—
149C2×C22⋊C4—171no—
150C2×C22⋊C4—171no—
151C2×C22⋊C4—172no—
152C2×C22⋊C4#59511yescharacteristic, complemented
153C2×C42—341no—
154#99—344no—
155#95—341no—
156#95—341no—
157C17#189911yescharacteristic, complemented, Sylow
158C24—171no—
159C24C2×#59511yescharacteristic
160C2×D4—342no—
161C2×D4—341no—
162C2×D4—341no—
163C2×D4—342no—
164C22×C4—171no—
165C22×C4—171no—
166C22×C4—171no—
167C22×C4—171no—
168C22×C4—171no—
169C22×C4—171no—
170C22×C4—171no—
171C22×C4—342no—
172C22×C4—342no—
173C22×C4—342no—
174C22×C4—342no—
175C22×C4C2×#59511yescharacteristic
176C22×C4C2×#59511yescharacteristic
177C4⋊C4—341no—
178C4⋊C4—341no—
179C4⋊C4—344no—
180C4⋊C4—341no—
181C4⋊C4—341no—
182C22⋊C4—21no—
183C22⋊C4—21no—
184C22⋊C4—174no—
185C22⋊C4—341no—
186C22⋊C4—341no—
187C22⋊C4—341no—
188C22⋊C4—341no—
189C22⋊C4—342no—
190C22⋊C4—342no—
191C22⋊C4—342no—
192C42—344no—
193C23—21no—
194C23—22no—
195C23—41no—
196C23—41no—
197C23—171no—
198C23—171no—
199C23—172no—
200C23—172no—
201C23—681no—
202C23—681no—
203C23C22×#59511yescharacteristic
204D4—681no—
205D4—682no—
206D4—681no—
207C2×C4—22no—
208C2×C4—21no—
209C2×C4—41no—
210C2×C4—171no—
211C2×C4—171no—
212C2×C4—341no—
213C2×C4—341no—
214C2×C4—342no—
215C2×C4—342no—
216C2×C4—341no—
217C2×C4—341no—
218C2×C4—342no—
219C2×C4—344no—
220C2×C4—342no—
221C2×C4—341no—
222C2×C4—341no—
223C2×C4—682no—
224C2×C4—682no—
225C2×C4—681no—
226C2×C4—682no—
227C2×C4—682no—
228C2×C4—682no—
229C2×C4C22⋊#59511yescharacteristic
230C2×C4C22⋊#59511yescharacteristic
231C22—21no—
232C22—21no—
233C22—41no—
234C22—41no—
235C22—41no—
236C22—82no—
237C22—171no—
238C22—172no—
239C22—171no—
240C22—341no—
241C22—341no—
242C22—341no—
243C22—341no—
244C22—682no—
245C22—682no—
246C22D4×#59512yes—
247C22C2×C22⋊#59511yescharacteristic
248C4—21no—
249C4—21no—
250C4—41no—
251C4—341no—
252C4—341no—
253C4—684no—
254C4—681no—
255C4—681no—
256C4—1362no—
257C2—22no—
258C2—81no—
259C2—341no—
260C2—342no—
261C2—341no—
262C2#1176375711yescharacteristic
263C2#1176378811yescharacteristic
264C2#1176378911yescharacteristic
subgroupG/H
centerC22C2×C22⋊#595
commutatorC23×C17C22×C4
FrattiniC23C22×#595
FittingC2×C22⋊C4×C17C4
socleC22×C17C2×C22⋊C4
prime radicalC22×C22⋊D17C22
cosocleC23×D17C23
hypercenterC2×C22⋊C4#595
hypocenterC17#1899
LCSG ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#10587677C2semidirect
3. #11763752C2semidirect
4. #11763687C2semidirect
13. C2×C22⋊C4×C17C4semidirect
14. C2×C22⋊Dic17C4semidirect
16. C2×C4⋊Dic17C4semidirect
17. C2×#63225C4semidirect
152. C2×C22⋊C4#595semidirect
157. C17#1899semidirect

Maximal subgroups

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

subgroupindex
C2×#105876772
C2×#105875512
#117637522
#117636872
C2×#105875692
#189917