Finite Group Databasebeta

← #14533732

Subgroups — #14533732

Order 2176.

676 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 676 conjugacy classes of subgroups — too many to draw. Only its 171 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×#10587574C211yesmaximal, characteristic, complemented
2C2×#10587582C212yesmaximal, complemented
3C2×#10587564C212yesmaximal, complemented
4C2×#10587567C211yesmaximal, characteristic
5#11762746C211yesmaximal, characteristic, complemented
6#11762745C211yesmaximal, characteristic, complemented
7#11762744C211yesmaximal, characteristic, complemented
8#11762709C212yesmaximal
9#11762678C212yesmaximal, complemented
10#11762660C211yesmaximal, characteristic, complemented
11C17×#399C211yesmaximal, characteristic, complemented
12C22×C22⋊D17C2211yescharacteristic
13C23×Dic17C2211yescharacteristic
14C22×C4×D17C2211yescharacteristic
15C22×Dic34C2212yes—
16C2×C4⋊C4×C17C2211yescharacteristic
17C2×C4⋊C4×C17C2212yes—
18C2×C22⋊C4×C17C2212yes—
19C2×C22⋊C4×C17C2211yescharacteristic
20C2×C22⋊Dic17C2212yes—
21C2×#63227C2212yes—
22C2×#63227C2212yes—
23C2×#63227C2211yescharacteristic
24C2×C4⋊Dic17C2211yescharacteristic
25C2×C4⋊Dic17C2212yes—
26C2×#63225C2212yes—
27C2×#63225C2212yes—
28C2×#63225C2212yes—
29C2×#63225C2211yescharacteristic
30C2×#63225C2212yes—
31C2×C4×Dic17C2212yes—
32C2×C4×Dic17C2211yescharacteristic
33#10587582—28no—
34#10587574—24no—
35#10587567—24no—
36#10587564—28no—
37C17×#95C2211yescharacteristic
38#10587502C2211yescharacteristic
39C24×C17C2311yescharacteristic
40C23×D17C2311yescharacteristic
41C22×C4×C17C2312yes—
42C22×C4×C17C2311yescharacteristic
43C22×C4×C17C2311yescharacteristic
44C22×C4×C17C2312yes—
45C2×C22⋊D17—22no—
46C2×C22⋊D17—22no—
47C2×C22⋊D17Q814yescomplemented
48C22×Dic17—24no—
49C22×Dic17—22no—
50C22×Dic17C2312yes—
51C22×Dic17C2311yescharacteristic
52C22×Dic17C2312yes—
53C22×Dic17C2311yescharacteristic
54C22×Dic17C2311yescharacteristic
55C2×C4×D17—22no—
56C2×C4×D17—24no—
57C2×Dic34—24no—
58C2×Dic34—24no—
59C2×Dic34—24no—
60C4⋊C4×C17—22no—
61C4⋊C4×C17—24no—
62C22⋊C4×C17—24no—
63C22⋊C4×C17—22no—
64C22⋊Dic17—24no—
65#63227—22no—
66#63227—24no—
67#63227—24no—
68C4⋊Dic17—24no—
69C4⋊Dic17—22no—
70#63225—24no—
71#63225—22no—
72#63225—24no—
73#63225—24no—
74#63225D418yescomplemented
75C4×Dic17—24no—
76C4×Dic17D414yes—
77C23×C17—22no—
78C23×C17—21no—
79C23×C17—21no—
80C23×C17—22no—
81C23×C17C2×Q812yes—
82C23×C17C2411yescharacteristic
83C22×D17—21no—
84C22×D17—22no—
85C22×D17—21no—
86C22×D17—22no—
87C22×D17C2×Q812yes—
88C2×C4×C17—22no—
89C2×C4×C17—22no—
90C2×C4×C17—22no—
91C2×C4×C17—22no—
92C2×C4×C17—21no—
93C2×C4×C17—22no—
94C2×C4×C17—22no—
95C2×C4×C17—22no—
96C2×C4×C17C2×D412yes—
97C2×C4×C17C2×D414yes—
98C22⋊D17—28no—
99C2×Dic17—22no—
100C2×Dic17—22no—
101C2×Dic17—22no—
102C2×Dic17—22no—
103C2×Dic17—22no—
104C2×Dic17—22no—
105C2×Dic17—21no—
106C2×Dic17—44no—
107C2×Dic17C2×D414yes—
108C2×Dic17C2×D412yes—
109C2×Dic17C2×D412yes—
110C2×Dic17C2×D414yes—
111C2×Dic17C2×Q812yes—
112C2×Dic17#4112yes—
113C4×D17—44no—
114Dic34—48no—
115#2542—171nomaximal, Sylow
116C22×C17—22no—
117C22×C17—24no—
118C22×C17—41no—
119C22×C17—41no—
120C22×C17—42no—
121C22×C17C22×D411yescharacteristic
122C22×C17C22×D412yes—
123C22×C17C22×Q811yescharacteristic
124C22×C17C2×#4111yescharacteristic
125C22×C172−1+412yes—
126C2×D17—22no—
127C2×D17—24no—
128C2×D17—41no—
129C2×D17—41no—
130C2×D17—42no—
131C4×C17—22no—
132C4×C17—24no—
133C4×C17—42no—
134C4×C17—41no—
135Dic17—24no—
136Dic17—42no—
137Dic17—41no—
138Dic17C22⋊Q814yescomplemented
139Dic17#12714yes—
140C2×C4⋊Q8—171no—
141C2×C22⋊Q8—172no—
142C2×C22⋊Q8—172no—
143C2×C4×D4—171no—
144#399—171no—
145#399—171no—
146#399—171no—
147#399D1711yescharacteristic, complemented
148#390—172no—
149#384—172no—
150C4×C4⋊C4—171no—
151C2×C17—42no—
152C2×C17C2×C22⋊Q811yescharacteristic
153C2×C17C2×#12711yescharacteristic
154C2×C17#53611yescharacteristic
155C2×C17#54712yes—
156C2×C17D4×Q812yes—
157D17—42no—
158C22×Q8—172no—
159C22×D4—171no—
160C23×C4—171no—
161C23×C4—171no—
162C4⋊Q8—344no—
163C22⋊Q8—348no—
164C22⋊Q8—348no—
165C4×D4—344no—
166C2×C4⋊C4—172no—
167C2×C4⋊C4—171no—
168C2×C4⋊C4—172no—
169C2×C4⋊C4—172no—
170C2×C4⋊C4—172no—
171C2×C4⋊C4—172no—
172C2×C4⋊C4—171no—
173C2×C4⋊C4C2×D1712yes—
174C2×C4⋊C4C2×D1711yescharacteristic
175C2×C22⋊C4—172no—
176C2×C22⋊C4—171no—
177C2×C22⋊C4—172no—
178C2×C22⋊C4—172no—
179C2×C22⋊C4C2×D1711yescharacteristic
180C2×C22⋊C4C2×D1712yes—
181C2×C42—171no—
182C2×C42—172no—
183#95—171no—
184#95C2×D1711yescharacteristic
185C17#254211yescharacteristic, complemented, Sylow
186C24—171no—
187C24C22×D1711yescharacteristic
188C2×Q8—344no—
189C2×Q8—344no—
190C2×Q8—344no—
191C2×D4—174no—
192C2×D4—342no—
193C2×D4—342no—
194C22×C4—171no—
195C22×C4—171no—
196C22×C4—172no—
197C22×C4—172no—
198C22×C4—171no—
199C22×C4—342no—
200C22×C4—344no—
201C22×C4—342no—
202C22×C4—344no—
203C22×C4C22×D1712yes—
204C22×C4C22×D1711yescharacteristic
205C22×C4C22×D1711yescharacteristic
206C22×C4C22×D1712yes—
207C4⋊C4—24no—
208C4⋊C4—22no—
209C4⋊C4—178no—
210C4⋊C4—342no—
211C4⋊C4—344no—
212C4⋊C4—344no—
213C4⋊C4—344no—
214C4⋊C4—344no—
215C4⋊C4—342no—
216C22⋊C4—22no—
217C22⋊C4—24no—
218C22⋊C4—344no—
219C22⋊C4—344no—
220C22⋊C4—342no—
221C22⋊C4—344no—
222C42—174no—
223C42—344no—
224C23—22no—
225C23—21no—
226C23—21no—
227C23—22no—
228C23—172no—
229C23—342no—
230C23—341no—
231C23—341no—
232C23—342no—
233C23Q8×D1712yes—
234C23C23×D1711yescharacteristic
235Q8—688no—
236D4—348no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—172no—
246C2×C4—172no—
247C2×C4—174no—
248C2×C4—172no—
249C2×C4—172no—
250C2×C4—174no—
251C2×C4—342no—
252C2×C4—342no—
253C2×C4—341no—
254C2×C4—342no—
255C2×C4—342no—
256C2×C4—342no—
257C2×C4—342no—
258C2×C4—684no—
259C2×C4—684no—
260C2×C4D4×D1712yes—
261C2×C4D4×D1714yes—
262C22—24no—
263C22—22no—
264C22—42no—
265C22—41no—
266C22—41no—
267C22—344no—
268C22—342no—
269C22—682no—
270C22—681no—
271C22—681no—
272C22C2×D4×D1712yes—
273C22C2×D4×D1711yescharacteristic
274C22C2×Q8×D1711yescharacteristic
275C22#41×D1711yescharacteristic
276C22#1058769712yes—
277C4—24no—
278C4—22no—
279C4—41no—
280C4—42no—
281C4—174no—
282C4—174no—
283C4—344no—
284C4—681no—
285C4—682no—
286C2—42no—
287C2—682no—
288C2C22⋊Q8×D1711yescharacteristic
289C2#127×D1711yescharacteristic
290C2#1176338111yescharacteristic
291C2#1176345712yes—
292C2#1176347212yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#399C2
prime radicalC22×C22⋊D17C22
hypercenter#399D17
hypocenterC17#2542
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. C2×#10587574C2semidirect
2. C2×#10587582C2semidirect
3. C2×#10587564C2semidirect
5. #11762746C2semidirect
6. #11762745C2semidirect
7. #11762744C2semidirect
9. #11762678C2semidirect
10. #11762660C2semidirect
11. C17×#399C2semidirect
47. C2×C22⋊D17Q8semidirect
74. #63225D4semidirect
138. Dic17C22⋊Q8semidirect
147. #399D17semidirect
185. C17#2542semidirect

Maximal subgroups

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

subgroupindex
C2×#105875742
C2×#105875822
C2×#105875642
C2×#105875672
#117627462
#117627452
#117627442
#117627092
#117626782
#117626602
C17×#3992
#254217