Finite Group Databasebeta

← #14540026

Subgroups — #14540026

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#11763366C211yesmaximal, characteristic, complemented
2#11763343C211yesmaximal, characteristic
3#11763218C211yesmaximal, characteristic, complemented
4#11763218C211yesmaximal, characteristic, complemented
5#11763169C211yesmaximal, characteristic, complemented
6#11763169C211yesmaximal, characteristic, complemented
7#11763151C212yesmaximal, complemented
8#11763134C212yesmaximal
9#11763117C212yesmaximal, complemented
10#11763110C212yesmaximal, complemented
11C17×#432C211yesmaximal, characteristic, complemented
12C2×C42×C17C2211yescharacteristic
13#10587616C2211yescharacteristic
14#10587615—21no—
15C4×C22⋊D17—21no—
16#10587612C2212yes—
17#10587610—21no—
18#10587610—21no—
19#10587610C2211yescharacteristic
20#10587562—21no—
21#10587561C2211yescharacteristic
22C4×D68C2211yescharacteristic
23#10587558—21no—
24#10587556C2211yescharacteristic
25#10587556C2212yes—
26#10587555C2211yescharacteristic
27#10587555C2211yescharacteristic
28C4×Dic34—21no—
29C4×Dic34—21no—
30C4×Dic34C2211yescharacteristic
31C17×C4⋊C8C2212yes—
32C17×C22⋊C8C2212yes—
33C17×#97C2211yescharacteristic
34C17×#97C2211yescharacteristic
35#10587492C2212yes—
36#10587492C2212yes—
37#10587489C2212yes—
38#10587489C2212yes—
39#10587488C2212yes—
40#10587488C2212yes—
41#10587487C2212yes—
42#10587487C2212yes—
43C22×C4×C17D411yescharacteristic
44C22×C4×C17D411yescharacteristic
45C22×C4×C17C2311yescharacteristic
46C2×C22⋊D17—21no—
47C2×D68C2311yescharacteristic
48C2×C4×D17—21no—
49C2×Dic34—21no—
50C2×Dic34—21no—
51C2×Dic34C2311yescharacteristic
52C2×C8×C17C2312yes—
53C2×C8×C17C2312yes—
54C42×C17D411yescharacteristic
55C42×C17D411yescharacteristic
56C42×C17C2311yescharacteristic
57C42×C17C2311yescharacteristic
58C22⋊Dic17—22no—
59C22⋊Dic17—21no—
60#63227—21no—
61#63227—21no—
62#63227—21no—
63#63227—21no—
64C4⋊Dic17—21no—
65C4⋊Dic17—21no—
66C4⋊Dic17C2312yes—
67C4⋊Dic17C2312yes—
68C4⋊Dic17C2311yescharacteristic
69C4⋊Dic17C2311yescharacteristic
70#63225—22no—
71#63225—22no—
72#63225—21no—
73#63225—21no—
74#63225—21no—
75#63225—21no—
76C4×Dic17—22no—
77C4×Dic17—21no—
78C23×C17C2×D411yescharacteristic
79C22×D17—21no—
80C2×C4×C17—22no—
81C2×C4×C17—41no—
82C2×C4×C17—41no—
83C2×C4×C17C2×D411yescharacteristic
84C2×C4×C17C2×D411yescharacteristic
85C2×C4×C17C2×D411yescharacteristic
86C2×C4×C17C2×D411yescharacteristic
87C2×C4×C17C2×D411yescharacteristic
88C2×C4×C17C2411yescharacteristic
89C22⋊D17—42no—
90C2×Dic17—22no—
91C2×Dic17—22no—
92C2×Dic17—21no—
93C2×Dic17—21no—
94C2×Dic17—21no—
95D68—22no—
96C4×D17—42no—
97Dic34—22no—
98Dic34—42no—
99Dic34—42no—
100C8×C17—22no—
101C8×C17—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×C172−1+412yes—
116Dic17—41no—
117Dic17—42no—
118Dic17—42no—
119Dic17—41no—
120Dic17—41no—
121#533—171no—
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#432D1711yescharacteristic, complemented
132C2×C17—41no—
133C2×C17#53611yescharacteristic
134C2×C17#57511yescharacteristic
135C2×C17#57511yescharacteristic
136D17—81no—
137C4⋊Q8—171no—
138C4⋊Q8—171no—
139#126—341no—
140#125—171no—
141#125—172no—
142#124—171no—
143#123—341no—
144C22⋊Q8—171no—
145C22⋊Q8—341no—
146C22⋊Q8—341no—
147#121—171no—
148C4×Q8—171no—
149C4×Q8—341no—
150C4×Q8—341no—
151C4×D4—171no—
152C4×D4—341no—
153#117—172no—
154#117—341no—
155C2×C42C2×D1711yescharacteristic
156#107—172no—
157#107—172no—
158#106—172no—
159#106—172no—
160C4⋊C8C2×D1712yes—
161#103—172no—
162#103—172no—
163#102—172no—
164#102—172no—
165C22⋊C8C2×D1712yes—
166#97C2×D1711yescharacteristic
167#97C2×D1711yescharacteristic
168C17#324011yescharacteristic, complemented, Sylow
169C2×Q8—171no—
170C2×Q8—341no—
171C2×Q8—341no—
172C2×D4—171no—
173C2×D4—341no—
174C22×C4—341no—
175C22×C4D6811yescharacteristic
176C22×C4D6811yescharacteristic
177C22×C4C22×D1711yescharacteristic
178C2×C8C22×D1712yes—
179C2×C8C22×D1712yes—
180C4⋊C4—172no—
181C4⋊C4—172no—
182C4⋊C4—171no—
183C4⋊C4—171no—
184C4⋊C4—341no—
185C4⋊C4—341no—
186C4⋊C4—341no—
187C4⋊C4—341no—
188C4⋊C4—342no—
189C4⋊C4—342no—
190C4⋊C4—341no—
191C4⋊C4—341no—
192C22⋊C4—342no—
193C22⋊C4—341no—
194C22⋊C4—341no—
195C22⋊C4—341no—
196C22⋊C4—341no—
197C22⋊C4—341no—
198C42—342no—
199C42—341no—
200C42D6811yescharacteristic
201C42D6811yescharacteristic
202C42C22×D1711yescharacteristic
203C42C22×D1711yescharacteristic
204C23—341no—
205C23C2×D6811yescharacteristic
206Q8—342no—
207Q8—682no—
208Q8—682no—
209D4—342no—
210D4—682no—
211C2×C4—22no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—341no—
215C2×C4—341no—
216C2×C4—342no—
217C2×C4—342no—
218C2×C4—341no—
219C2×C4—682no—
220C2×C4C2×D6811yescharacteristic
221C2×C4C2×D6811yescharacteristic
222C2×C4C2×D6811yescharacteristic
223C2×C4C2×D6811yescharacteristic
224C2×C4C2×D6811yescharacteristic
225C2×C4C23×D1711yescharacteristic
226C8—22no—
227C8—22no—
228C22—21no—
229C22—21no—
230C22—21no—
231C22—681no—
232C22—681no—
233C22—681no—
234C22C22×D6811yescharacteristic
235C4—21no—
236C4—21no—
237C4—21no—
238C4—21no—
239C4—21no—
240C4—681no—
241C4—681no—
242C4—682no—
243C4—682no—
244C4—681no—
245C4#1058769712yes—
246C2—41no—
247C2—1361no—
248C2#1176338511yescharacteristic
249C2#1176360911yescharacteristic
250C2#1176360911yescharacteristic
subgroupG/H
centerC22C22×D68
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#432C2
socleC22×C17C22×D4
prime radical#10587616C22
hypercenter#432D17
hypocenterC17#3240
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #432

Complements

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

NH ≅ G/Ntype
1. #11763366C2semidirect
3. #11763218C2semidirect
4. #11763218C2semidirect
5. #11763169C2semidirect
6. #11763169C2semidirect
7. #11763151C2semidirect
9. #11763117C2semidirect
10. #11763110C2semidirect
11. C17×#432C2semidirect
131. #432D17semidirect
168. C17#3240semidirect

Maximal subgroups

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

subgroupindex
#117633662
#117633432
#117632182
#117631692
#117631512
#117631342
#117631172
#117631102
C17×#4322
#324017