Finite Group Databasebeta

← #14541437

Subgroups — #14541437

Order 2176.

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

normal subgroups only

This group has 350 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#11763533C211yesmaximal, characteristic, complemented
2#11763516C211yesmaximal, characteristic, complemented
3#11763165C212yesmaximal, complemented
4#11763159C211yesmaximal, characteristic, complemented
5#11763114C212yesmaximal, complemented
6#11763100C211yesmaximal, characteristic, complemented
7#11763026C212yesmaximal
8#11762933C211yesmaximal, characteristic, complemented
9SD16×Dic17C212yesmaximal, complemented
10C4×#63228C211yesmaximal, characteristic
11C17×#461C211yesmaximal, characteristic, complemented
12C2×SD16×C17C2212yes—
13C17×C4⋊Q8C2211yescharacteristic
14C4×D4×C17C2211yescharacteristic
15#10587633C2212yes—
16Q8×Dic17C2212yes—
17D4×Dic17C2212yes—
18C2×#63228C2211yescharacteristic
19C2×C4⋊Dic17—21no—
20#10587610—21no—
21#10587584—22no—
22#10587583—21no—
23#10587578—21no—
24#10587576C2212yes—
25#10587564—22no—
26C4×D68C2211yescharacteristic
27#10587555C2211yescharacteristic
28C4×Dic34C2211yescharacteristic
29C17×C4⋊C8C2211yescharacteristic
30C17×#103C2212yes—
31#10587506C2212yes—
32#10587506C2212yes—
33#10587503C2211yescharacteristic
34#10587492C2212yes—
35#10587488C2212yes—
36#10587485C2212yes—
37C8×Dic17C2212yes—
38#10587480C2211yescharacteristic
39#10587478C2211yescharacteristic
40C4×#3622C2211yescharacteristic
41C2×Q8×C17C2312yes—
42C2×D4×C17C2311yescharacteristic
43C22×C4×C17—21no—
44C22×Dic17—22no—
45C2×D68C2311yescharacteristic
46C2×C4×D17—21no—
47C2×Dic34—21no—
48C2×Dic34D411yescharacteristic
49SD16×C17—24no—
50C2×C8×C17C2312yes—
51C4⋊C4×C17—21no—
52C4⋊C4×C17—21no—
53C4⋊C4×C17C2311yescharacteristic
54C22⋊C4×C17—21no—
55C42×C17C2311yescharacteristic
56C22⋊Dic17—22no—
57#63228—22no—
58#63227—21no—
59#63227—22no—
60#63227—22no—
61C4⋊Dic17—21no—
62C4⋊Dic17—22no—
63C4⋊Dic17—22no—
64C4⋊Dic17D411yescharacteristic
65C4⋊Dic17C2311yescharacteristic
66C4⋊Dic17C2312yes—
67#63225—22no—
68#63225D412yes—
69C4×Dic17—22no—
70C4×Dic17C2312yes—
71C2×#3622C2311yescharacteristic
72C2×#3622C2311yescharacteristic
73C23×C17—21no—
74C22×D17—21no—
75Q8×C17—22no—
76Q8×C17—22no—
77D4×C17—21no—
78D4×C17#4112yes—
79C2×C4×C17—21no—
80C2×C4×C17—22no—
81C2×C4×C17—42no—
82C2×C4×C17C2×D411yescharacteristic
83C2×C4×C17C2×D411yescharacteristic
84C2×C4×C17C2411yescharacteristic
85C2×Dic17—22no—
86C2×Dic17—21no—
87C2×Dic17—82no—
88C2×Dic17C2×D412yes—
89C2×Dic17C2×D412yes—
90D68—21no—
91D68—21no—
92C4×D17—42no—
93Dic34—22no—
94Dic34—81no—
95C8×C17—22no—
96#3622—21no—
97#3622—21no—
98#3360—171nomaximal, Sylow
99C22×C17—22no—
100C22×C17—41no—
101C22×C17—41no—
102C22×C17C22×D411yescharacteristic
103C2×D17—41no—
104C2×D17—41no—
105C2×D17—41no—
106C4×C17—21no—
107C4×C17—42no—
108C4×C17—41no—
109C4×C17#13512yes—
110C4×C17C2×#4111yescharacteristic
111C4×C172−1+411yescharacteristic
112Dic17—22no—
113Dic17—22no—
114Dic17—41no—
115Dic17—42no—
116#556—171no—
117#554—171no—
118#488—172no—
119#487—171no—
120#479—172no—
121#476—171no—
122#463—172no—
123#461D1711yescharacteristic, complemented
124C8×Q8—171no—
125C4×SD16—172no—
126C4×D8—171no—
127C2×C17—42no—
128C2×C17#54811yescharacteristic
129C2×C17C2×#13511yescharacteristic
130C2×C17#57711yescharacteristic
131D17—81no—
132C2×SD16C2×D1712yes—
133C2×D8—171no—
134C4⋊Q8—171no—
135C4⋊Q8C2×D1711yescharacteristic
136#126—342no—
137#125—172no—
138#124—172no—
139C22⋊Q8—342no—
140C22⋊Q8—341no—
141C22⋊Q8—341no—
142C4×Q8—171no—
143C4×Q8—172no—
144C4×D4—172no—
145C4×D4—171no—
146C4×D4C2×D1711yescharacteristic
147#117—341no—
148C2×C4⋊C4—341no—
149#107—171no—
150#106—172no—
151C4⋊C8—172no—
152C4⋊C8C2×D1711yescharacteristic
153#103—172no—
154#103—172no—
155#103C2×D1712yes—
156#102—171no—
157#102—172no—
158#102—171no—
159C4×C8—172no—
160C4×C8—171no—
161C17#336011yescharacteristic, complemented, Sylow
162C2×Q8—171no—
163C2×Q8—341no—
164C2×Q8C22×D1712yes—
165C2×D4—171no—
166C2×D4C22×D1711yescharacteristic
167C22×C4—21no—
168C22×C4—342no—
169C22×C4—341no—
170SD16—24no—
171D8—342no—
172C2×C8—171no—
173C2×C8—171no—
174C2×C8C22×D1712yes—
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—171no—
178C4⋊C4—172no—
179C4⋊C4—172no—
180C4⋊C4—171no—
181C4⋊C4—342no—
182C4⋊C4—342no—
183C4⋊C4—342no—
184C4⋊C4—341no—
185C4⋊C4C22×D1711yescharacteristic
186C22⋊C4—21no—
187C22⋊C4—342no—
188C22⋊C4—342no—
189C22⋊C4—342no—
190C22⋊C4—341no—
191C42—172no—
192C42—342no—
193C42C22×D1711yescharacteristic
194C23—21no—
195C23—341no—
196Q8—22no—
197Q8—22no—
198Q8—342no—
199Q8—1361no—
200D4—21no—
201D4—341no—
202D4—341no—
203D4#6325412yes—
204C2×C4—22no—
205C2×C4—21no—
206C2×C4—42no—
207C2×C4—172no—
208C2×C4—172no—
209C2×C4—341no—
210C2×C4—342no—
211C2×C4—682no—
212C2×C4—1362no—
213C2×C4D4×D1711yescharacteristic
214C2×C4D4×D1711yescharacteristic
215C2×C4C23×D1711yescharacteristic
216C8—22no—
217C8—341no—
218C8—341no—
219C22—22no—
220C22—41no—
221C22—41no—
222C22—681no—
223C22—681no—
224C22—681no—
225C22C2×D4×D1711yescharacteristic
226C4—21no—
227C4—42no—
228C4—41no—
229C4—342no—
230C4—342no—
231C4—681no—
232C4—682no—
233C4#1058760212yes—
234C4C2×#6325411yescharacteristic
235C4#1058769711yescharacteristic
236C2—42no—
237C2—1361no—
238C2#1176346511yescharacteristic
239C2C2×#1058760211yescharacteristic
240C2#1176361811yescharacteristic
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#461C2
socleC22×C17C22×D4
prime radicalC2×#63228C22
hypercenter#461D17
hypocenterC17#3360
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. #11763533C2semidirect
2. #11763516C2semidirect
3. #11763165C2semidirect
4. #11763159C2semidirect
5. #11763114C2semidirect
6. #11763100C2semidirect
8. #11762933C2semidirect
9. SD16×Dic17C2semidirect
11. C17×#461C2semidirect
123. #461D17semidirect
161. C17#3360semidirect

Maximal subgroups

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

subgroupindex
#117635332
#117635162
#117631652
#117631592
#117631142
#117631002
#117630262
#117629332
SD16×Dic172
C4×#632282
C17×#4612
#336017