Finite Group Databasebeta

← #14535808

Subgroups — #14535808

Order 2176.

592 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 592 conjugacy classes of subgroups — too many to draw. Only its 137 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×#10587633C211yesmaximal, characteristic, complemented
2#11762725C212yesmaximal, complemented
3#11762718C212yesmaximal, complemented
4#11762717C212yesmaximal, complemented
5#11762716C211yesmaximal, characteristic, complemented
6#11762709C212yesmaximal, complemented
7#11762700C212yesmaximal, complemented
8#11762697C212yesmaximal, complemented
9C2×C17×#124C211yesmaximal, characteristic, complemented
10C22×Q8×C17C2211yescharacteristic
11C22×D4×C17C2211yescharacteristic
12C22×C22⋊D17C2212yes—
13C22×C22⋊D17C2212yes—
14C22×D68C2211yescharacteristic
15C17×#124—24no—
16C2×C22⋊C4×C17C2212yes—
17C2×C22⋊C4×C17C2212yes—
18C2×C42×C17C2211yescharacteristic
19C2×C22⋊Dic17C2212yes—
20C2×C22⋊Dic17C2212yes—
21#10587633—24no—
22C2×#63227C2212yes—
23C2×#63227C2212yes—
24C2×#63227C2212yes—
25C2×#63227C2212yes—
26C2×C4⋊Dic17C2211yescharacteristic
27C2×#63225C2212yes—
28C2×C4×Dic17C2211yescharacteristic
29C2×C4×Dic17C2211yescharacteristic
30#10587502C2212yes—
31#10587502C2212yes—
32#10587502C2212yes—
33C24×C17C2312yes—
34C23×D17C2312yes—
35C2×Q8×C17—41no—
36C2×Q8×C17—41no—
37C2×Q8×C17D414yescomplemented
38C2×D4×C17—22no—
39C2×D4×C17—41no—
40C2×D4×C17—41no—
41C22×C4×C17C2312yes—
42C22×C4×C17C2311yescharacteristic
43C22×C4×C17C2311yescharacteristic
44C22×C4×C17C2311yescharacteristic
45C2×C22⋊D17—24no—
46C2×C22⋊D17—42no—
47C2×C22⋊D17—42no—
48C2×C22⋊D17—42no—
49C2×C22⋊D17—42no—
50C2×C22⋊D17—42no—
51C22×Dic17C2312yes—
52C22×Dic17C2312yes—
53C22×Dic17C2312yes—
54C2×D68—22no—
55C2×D68—41no—
56C2×D68—41no—
57C22⋊C4×C17—24no—
58C22⋊C4×C17—24no—
59C42×C17—22no—
60C22⋊Dic17—24no—
61C22⋊Dic17—42no—
62#63227—24no—
63#63227—24no—
64#63227—24no—
65#63227—42no—
66C4⋊Dic17—41no—
67#63225—42no—
68C4×Dic17—22no—
69C4×Dic17—41no—
70C23×C17—22no—
71C23×C17—22no—
72C23×C17—22no—
73C23×C17—22no—
74C23×C17—22no—
75C23×C17—22no—
76C23×C17—22no—
77C23×C17C2411yescharacteristic
78C22×D17—22no—
79C22×D17—22no—
80C22×D17—22no—
81C22×D17—22no—
82C22×D17—22no—
83C22×D17—22no—
84C22×D17—22no—
85Q8×C17—44no—
86D4×C17—44no—
87C2×C4×C17—21no—
88C2×C4×C17—21no—
89C2×C4×C17—21no—
90C2×C4×C17—21no—
91C2×C4×C17—21no—
92C2×C4×C17—21no—
93C2×C4×C17—22no—
94C2×C4×C17—22no—
95C2×C4×C17—22no—
96C2×C4×C17C2×D412yes—
97C2×C4×C17C2×D412yes—
98C2×C4×C17C2×D412yes—
99C22⋊D17—48no—
100C22⋊D17—48no—
101C2×Dic17—22no—
102C2×Dic17—22no—
103C2×Dic17—22no—
104C2×Dic17—22no—
105C2×Dic17—22no—
106C2×Dic17—22no—
107C2×Dic17—22no—
108C2×Dic17—22no—
109C2×Dic17#4114yes—
110D68—44no—
111#2733—171nomaximal, Sylow
112C22×C17—42no—
113C22×C17—42no—
114C22×C17—42no—
115C22×C17—42no—
116C22×C17—42no—
117C22×C17—42no—
118C22×C17—42no—
119C22×C17C22×D411yescharacteristic
120C22×C17C2×#4112yes—
121C22×C17H(5,𝔽2)11yescharacteristic
122C22×C17H(5,𝔽2)11yescharacteristic
123C22×C17H(5,𝔽2)12yes—
124C2×D17—42no—
125C2×D17—42no—
126C2×D17—42no—
127C2×D17—42no—
128C2×D17—42no—
129C2×D17—42no—
130C2×D17—42no—
131C4×C17—22no—
132C4×C17—22no—
133C4×C17—22no—
134C4×C17—42no—
135Dic17—24no—
136Dic17—42no—
137Dic17—42no—
138C2×#124—171no—
139C2×#124D1711yescharacteristic, complemented
140#394—172no—
141#392—172no—
142#392—172no—
143#391—171no—
144#390—172no—
145#388—172no—
146#388—172no—
147C2×C17—82no—
148C2×C17#53411yescharacteristic
149C2×C17#53812yes—
150C2×C17#55012yes—
151C2×C17#55511yescharacteristic
152C2×C17H(3,𝔽4)11yescharacteristic
153D17—82no—
154C22×Q8C2×D1711yescharacteristic
155C22×D4—172no—
156C22×D4—172no—
157C22×D4—171no—
158C22×D4C2×D1711yescharacteristic
159#124—24no—
160#124—344no—
161C2×C4⋊C4—172no—
162C2×C4⋊C4—171no—
163C2×C22⋊C4—172no—
164C2×C22⋊C4—172no—
165C2×C22⋊C4—172no—
166C2×C22⋊C4—172no—
167C2×C22⋊C4—172no—
168C2×C22⋊C4—172no—
169C2×C22⋊C4C2×D1712yes—
170C2×C22⋊C4C2×D1712yes—
171C2×C42—171no—
172C2×C42—171no—
173C2×C42C2×D1711yescharacteristic
174#95—172no—
175#95—172no—
176#95—172no—
177C17#273311yescharacteristic, complemented, Sylow
178C24—172no—
179C24C22×D1712yes—
180C2×Q8—41no—
181C2×Q8—41no—
182C2×Q8C22⋊D1714yescomplemented
183C2×D4—22no—
184C2×D4—41no—
185C2×D4—41no—
186C2×D4—344no—
187C2×D4—342no—
188C2×D4—681no—
189C2×D4—681no—
190C2×D4—682no—
191C2×D4—682no—
192C2×D4—682no—
193C2×D4—682no—
194C2×D4—682no—
195C22×C4—172no—
196C22×C4—172no—
197C22×C4—172no—
198C22×C4C22×D1712yes—
199C22×C4C22×D1711yescharacteristic
200C22×C4C22×D1711yescharacteristic
201C22×C4C22×D1711yescharacteristic
202C4⋊C4—682no—
203C4⋊C4—681no—
204C22⋊C4—24no—
205C22⋊C4—24no—
206C22⋊C4—344no—
207C22⋊C4—344no—
208C22⋊C4—344no—
209C22⋊C4—344no—
210C22⋊C4—682no—
211C22⋊C4—682no—
212C42—22no—
213C42—342no—
214C42—681no—
215C23—22no—
216C23—22no—
217C23—22no—
218C23—22no—
219C23—22no—
220C23—22no—
221C23—22no—
222C23—342no—
223C23—342no—
224C23—342no—
225C23—342no—
226C23—342no—
227C23—342no—
228C23—342no—
229C23C23×D1711yescharacteristic
230Q8—44no—
231D4—44no—
232D4—684no—
233D4—688no—
234D4—688no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—174no—
245C2×C4—342no—
246C2×C4—342no—
247C2×C4—342no—
248C2×C4—342no—
249C2×C4—342no—
250C2×C4—342no—
251C2×C4—342no—
252C2×C4—342no—
253C2×C4C2×C22⋊D1712yes—
254C2×C4C2×C22⋊D1712yes—
255C2×C4C2×C22⋊D1712yes—
256C22—42no—
257C22—42no—
258C22—42no—
259C22—42no—
260C22—42no—
261C22—42no—
262C22—42no—
263C22—682no—
264C22—682no—
265C22—682no—
266C22—682no—
267C22—682no—
268C22—682no—
269C22—682no—
270C22#1058769111yescharacteristic
271C22#1058769111yescharacteristic
272C22#41×D1712yes—
273C22#1058769612yes—
274C22C22×C22⋊D1711yescharacteristic
275C4—22no—
276C4—22no—
277C4—22no—
278C4—42no—
279C4—344no—
280C4—682no—
281C4—682no—
282C2—82no—
283C2—1362no—
284C2#1176336811yescharacteristic
285C2#1176339312yes—
286C2#1176347812yes—
287C2#1176352111yescharacteristic
288C2#1176355111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC2×C17×#124C2
hypercenterC2×#124D17
hypocenterC17#2733
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. C2×#10587633C2semidirect
2. #11762725C2semidirect
3. #11762718C2semidirect
4. #11762717C2semidirect
5. #11762716C2semidirect
6. #11762709C2semidirect
7. #11762700C2semidirect
8. #11762697C2semidirect
9. C2×C17×#124C2semidirect
37. C2×Q8×C17D4semidirect
139. C2×#124D17semidirect
177. C17#2733semidirect
182. C2×Q8C22⋊D17semidirect

Maximal subgroups

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

subgroupindex
C2×#105876332
#117627252
#117627182
#117627172
#117627162
#117627092
#117627002
#117626972
C2×C17×#1242
#273317