Finite Group Databasebeta

← #14540024

Subgroups — #14540024

Order 2176.

328 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 328 conjugacy classes of subgroups — too many to draw. Only its 117 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
1#11763343C211yesmaximal, characteristic
2#11763215C211yesmaximal, characteristic, complemented
3#11763215C211yesmaximal, characteristic, complemented
4#11763171C211yesmaximal, characteristic, complemented
5#11763171C211yesmaximal, characteristic, complemented
6#11763150C212yesmaximal, complemented
7#11763133C212yesmaximal
8#11763113C212yesmaximal, complemented
9#11763106C212yesmaximal, complemented
10#11762866C211yesmaximal, characteristic, complemented
11C17×#533C211yesmaximal, characteristic, complemented
12C17×C4⋊Q8C2211yescharacteristic
13C17×C4⋊Q8C2211yescharacteristic
14C17×#125C2212yes—
15C17×C22⋊Q8—21no—
16C17×C22⋊Q8—21no—
17C4×Q8×C17—21no—
18C4×Q8×C17—21no—
19C17×#117C2212yes—
20C2×C42×C17C2211yescharacteristic
21#10587616C2211yescharacteristic
22#10587615—21no—
23C4×C22⋊D17—21no—
24#10587610C2211yescharacteristic
25#10587562—21no—
26#10587561C2211yescharacteristic
27C4×D68C2211yescharacteristic
28#10587558—21no—
29#10587556C2211yescharacteristic
30C4×Dic34C2211yescharacteristic
31C22⋊#3622C2212yes—
32#10587481C2212yes—
33#10587481C2212yes—
34#10587480C2212yes—
35#10587480C2212yes—
36#10587479C2212yes—
37#10587479C2212yes—
38#10587478C2212yes—
39#10587478C2212yes—
40C4⋊#3622C2212yes—
41#10587474C2211yescharacteristic
42#10587474C2211yescharacteristic
43C2×Q8×C17—21no—
44C2×Q8×C17—21no—
45C22×C4×C17D411yescharacteristic
46C22×C4×C17D411yescharacteristic
47C22×C4×C17C2311yescharacteristic
48C2×C22⋊D17—21no—
49C2×D68C2311yescharacteristic
50C2×C4×D17—21no—
51C2×Dic34C2311yescharacteristic
52C4⋊C4×C17—22no—
53C4⋊C4×C17—22no—
54C4⋊C4×C17—21no—
55C4⋊C4×C17—21no—
56C4⋊C4×C17C2312yes—
57C4⋊C4×C17C2312yes—
58C22⋊C4×C17—22no—
59C42×C17—22no—
60C42×C17D411yescharacteristic
61C42×C17D411yescharacteristic
62C42×C17C2311yescharacteristic
63C42×C17C2311yescharacteristic
64C22⋊Dic17—21no—
65#63227—21no—
66#63227—21no—
67#63227—21no—
68#63227—21no—
69C4⋊Dic17C2311yescharacteristic
70C4⋊Dic17C2311yescharacteristic
71#63225—21no—
72#63225—21no—
73#63225—21no—
74#63225—21no—
75C4×Dic17—21no—
76C2×#3622C2312yes—
77C2×#3622C2312yes—
78C23×C17C2×D411yescharacteristic
79C22×D17—21no—
80Q8×C17—42no—
81Q8×C17—42no—
82C2×C4×C17—22no—
83C2×C4×C17—22no—
84C2×C4×C17—22no—
85C2×C4×C17—41no—
86C2×C4×C17—41no—
87C2×C4×C17C2×D411yescharacteristic
88C2×C4×C17C2×D411yescharacteristic
89C2×C4×C17C2×D411yescharacteristic
90C2×C4×C17C2×D411yescharacteristic
91C2×C4×C17C2×D411yescharacteristic
92C2×C4×C17C2411yescharacteristic
93C22⋊D17—42no—
94C2×Dic17—21no—
95C2×Dic17—21no—
96C2×Dic17—21no—
97D68—22no—
98C4×D17—42no—
99Dic34—22no—
100#3622—22no—
101#3622—22no—
102#3240—171nomaximal, Sylow
103C22×C17—21no—
104C22×C17—21no—
105C22×C17—21no—
106C22×C17C22×D411yescharacteristic
107C2×D17—41no—
108C2×D17—41no—
109C2×D17—41no—
110C4×C17—21no—
111C4×C17—21no—
112C4×C17—21no—
113C4×C17—21no—
114C4×C17—21no—
115C4×C17—42no—
116C4×C17—42no—
117C4×C172−1+412yes—
118Dic17—41no—
119Dic17—41no—
120Dic17—41no—
121#533D1711yescharacteristic, complemented
122#529—171no—
123#501—171no—
124#501—171no—
125#489—171no—
126#489—171no—
127#485—172no—
128#482—172no—
129#479—172no—
130#478—172no—
131#432—171no—
132C2×C17—41no—
133C2×C17#53611yescharacteristic
134C2×C17#57511yescharacteristic
135C2×C17#57511yescharacteristic
136D17—81no—
137C4⋊Q8C2×D1711yescharacteristic
138C4⋊Q8C2×D1711yescharacteristic
139#126—341no—
140#125—171no—
141#125C2×D1712yes—
142#124—171no—
143#123—341no—
144C22⋊Q8—21no—
145C22⋊Q8—21no—
146C22⋊Q8—171no—
147#121—171no—
148C4×Q8—21no—
149C4×Q8—21no—
150C4×Q8—171no—
151C4×D4—171no—
152C4×D4—341no—
153#117—341no—
154#117C2×D1712yes—
155C2×C42C2×D1711yescharacteristic
156#107—172no—
157#107—172no—
158#106—172no—
159#106—172no—
160C4⋊C8—172no—
161#103—172no—
162#103—172no—
163#102—172no—
164#102—172no—
165C22⋊C8—172no—
166#97—171no—
167#97—171no—
168C17#324011yescharacteristic, complemented, Sylow
169C2×Q8—21no—
170C2×Q8—21no—
171C2×Q8—171no—
172C2×D4—171no—
173C2×D4—341no—
174C22×C4—341no—
175C22×C4C22⋊D1711yescharacteristic
176C22×C4C22⋊D1711yescharacteristic
177C22×C4C22×D1711yescharacteristic
178C2×C8—172no—
179C2×C8—172no—
180C4⋊C4—22no—
181C4⋊C4—22no—
182C4⋊C4—21no—
183C4⋊C4—21no—
184C4⋊C4—171no—
185C4⋊C4—171no—
186C4⋊C4—341no—
187C4⋊C4—341no—
188C4⋊C4—341no—
189C4⋊C4—341no—
190C4⋊C4C22×D1712yes—
191C4⋊C4C22×D1712yes—
192C22⋊C4—22no—
193C22⋊C4—341no—
194C22⋊C4—341no—
195C22⋊C4—341no—
196C22⋊C4—341no—
197C22⋊C4—341no—
198C42—22no—
199C42—341no—
200C42C22⋊D1711yescharacteristic
201C42C22⋊D1711yescharacteristic
202C42C22×D1711yescharacteristic
203C42C22×D1711yescharacteristic
204C23—341no—
205C23C2×C22⋊D1711yescharacteristic
206Q8—42no—
207Q8—42no—
208Q8—342no—
209D4—342no—
210D4—682no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—341no—
217C2×C4—341no—
218C2×C4—341no—
219C2×C4—682no—
220C2×C4C2×C22⋊D1711yescharacteristic
221C2×C4C2×C22⋊D1711yescharacteristic
222C2×C4C2×C22⋊D1711yescharacteristic
223C2×C4C2×C22⋊D1711yescharacteristic
224C2×C4C2×C22⋊D1711yescharacteristic
225C2×C4C23×D1711yescharacteristic
226C8—342no—
227C8—342no—
228C22—21no—
229C22—21no—
230C22—21no—
231C22—681no—
232C22—681no—
233C22—681no—
234C22C22×C22⋊D1711yescharacteristic
235C4—21no—
236C4—21no—
237C4—21no—
238C4—21no—
239C4—21no—
240C4—42no—
241C4—42no—
242C4—681no—
243C4—681no—
244C4—681no—
245C4#1058769412yes—
246C2—41no—
247C2—1361no—
248C2#1176338311yescharacteristic
249C2#1176360711yescharacteristic
250C2#1176360711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#533C2
socleC22×C17C22×D4
prime radical#10587616C22
hypercenter#533D17
hypocenterC17#3240
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #533

Complements

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

NH ≅ G/Ntype
2. #11763215C2semidirect
3. #11763215C2semidirect
4. #11763171C2semidirect
5. #11763171C2semidirect
6. #11763150C2semidirect
8. #11763113C2semidirect
9. #11763106C2semidirect
10. #11762866C2semidirect
11. C17×#533C2semidirect
121. #533D17semidirect
168. C17#3240semidirect

Maximal subgroups

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

subgroupindex
#117633432
#117632152
#117631712
#117631502
#117631332
#117631132
#117631062
#117628662
C17×#5332
#324017