Finite Group Databasebeta

← #14534555

Subgroups — #14534555

Order 2176.

528 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 149 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×#10587630C211yesmaximal, characteristic, complemented
2C2×#10587583C211yesmaximal, characteristic
3C2×#10587632C211yesmaximal, characteristic
4C2×#10587558C211yesmaximal, characteristic
5#11762743C214yesmaximal, complemented
6#11762738C214yesmaximal
7#11762715C211yesmaximal, characteristic, complemented
8#11762674C211yesmaximal, characteristic, complemented
9C17×#391C211yesmaximal, characteristic, complemented
10C22×Q8×C17C2211yescharacteristic
11C22×C4×D17C2211yescharacteristic
12C22×Dic34C2211yescharacteristic
13C2×C4⋊C4×C17C2211yescharacteristic
14C2×C42×C17C2211yescharacteristic
15#10587632—24no—
16#10587630—24no—
17C2×#63227C2212yes—
18C2×#63227C2212yes—
19C2×#63227C2212yes—
20C2×C4⋊Dic17C2211yescharacteristic
21C2×C4⋊Dic17C2212yes—
22C2×#63225C2214yes—
23C2×#63225C2212yes—
24C2×#63225C2212yes—
25C2×C4×Dic17C2211yescharacteristic
26#10587583—24no—
27#10587558—24no—
28C17×#95C2214yes—
29#10587502C2214yes—
30#10587502C2214yes—
31C23×D17C2311yescharacteristic
32C2×Q8×C17—22no—
33C2×Q8×C17—22no—
34C2×Q8×C17—22no—
35C22×C4×C17C2312yes—
36C22×C4×C17C2312yes—
37C22×C4×C17C2312yes—
38C22×C4×C17C2311yescharacteristic
39C22×Dic17C2312yes—
40C22×Dic17C2312yes—
41C22×Dic17C2311yescharacteristic
42C22×Dic17C2312yes—
43C2×C4×D17—22no—
44C2×C4×D17—22no—
45C2×C4×D17D414yes—
46C2×Dic34—22no—
47C2×Dic34—22no—
48C2×Dic34—22no—
49C4⋊C4×C17—22no—
50C42×C17—22no—
51#63227—24no—
52#63227—24no—
53#63227—24no—
54C4⋊Dic17—22no—
55C4⋊Dic17—24no—
56#63225—24no—
57#63225—28no—
58#63225—24no—
59C4×Dic17D414yes—
60C23×C17C2411yescharacteristic
61C22×D17—21no—
62C22×D17—21no—
63C22×D17—22no—
64C22×D17—22no—
65C22×D17C2×D412yes—
66Q8×C17—44no—
67C2×C4×C17—22no—
68C2×C4×C17—22no—
69C2×C4×C17—22no—
70C2×C4×C17—22no—
71C2×C4×C17—22no—
72C2×C4×C17—22no—
73C2×C4×C17—22no—
74C2×C4×C17—24no—
75C2×C4×C17C2×D412yes—
76C2×C4×C17C2×D412yes—
77C2×C4×C17#4112yes—
78C2×Dic17—21no—
79C2×Dic17—21no—
80C2×Dic17—22no—
81C2×Dic17—22no—
82C2×Dic17—22no—
83C2×Dic17—22no—
84C2×Dic17—22no—
85C2×Dic17—22no—
86C2×Dic17—24no—
87C2×Dic17C2×D412yes—
88C2×Dic17C2×D414yes—
89C4×D17—28no—
90Dic34—44no—
91#2621—171nomaximal, Sylow
92C22×C17C22×D411yescharacteristic
93C22×C17C22×D411yescharacteristic
94C22×C17C2×#4111yescharacteristic
95C22×C172−1+412yes—
96C22×C172−1+412yes—
97C2×D17—22no—
98C2×D17—22no—
99C2×D17—22no—
100C2×D17—42no—
101C2×D17—42no—
102C4×C17—42no—
103C4×C17—42no—
104C4×C17—42no—
105C4×C17#12114yes—
106Dic17—22no—
107Dic17—24no—
108Dic17—42no—
109Dic17—42no—
110C2×C4⋊Q8—171no—
111C2×C22⋊Q8—171no—
112C2×C22⋊Q8—171no—
113C2×#117—171no—
114#398—174no—
115#397—174no—
116#391—171no—
117#391D1711yescharacteristic, complemented
118#382—171no—
119C2×C17C2×#12111yescharacteristic
120C2×C17#53611yescharacteristic
121C2×C17#53612yes—
122C2×C17#53611yescharacteristic
123C2×C17#54112yes—
124D17—42no—
125C22×Q8—171no—
126C22×Q8C2×D1711yescharacteristic
127C23×C4—171no—
128C4⋊Q8—344no—
129C22⋊Q8—344no—
130C22⋊Q8—344no—
131#117—344no—
132C2×C4⋊C4—172no—
133C2×C4⋊C4—171no—
134C2×C4⋊C4—172no—
135C2×C4⋊C4—172no—
136C2×C4⋊C4—174no—
137C2×C4⋊C4C2×D1711yescharacteristic
138C2×C22⋊C4—172no—
139C2×C22⋊C4—172no—
140C2×C22⋊C4—172no—
141C2×C42—171no—
142C2×C42C2×D1711yescharacteristic
143#95—174no—
144#95—174no—
145#95C2×D1714yes—
146C17#262111yescharacteristic, complemented, Sylow
147C24—171no—
148C2×Q8—22no—
149C2×Q8—22no—
150C2×Q8—22no—
151C2×Q8—342no—
152C2×Q8—342no—
153C2×Q8—342no—
154C22×C4—172no—
155C22×C4—171no—
156C22×C4—174no—
157C22×C4—172no—
158C22×C4—172no—
159C22×C4—342no—
160C22×C4—342no—
161C22×C4C22×D1712yes—
162C22×C4C22×D1712yes—
163C22×C4C22×D1712yes—
164C22×C4C22×D1711yescharacteristic
165C4⋊C4—22no—
166C4⋊C4—344no—
167C4⋊C4—344no—
168C4⋊C4—344no—
169C4⋊C4—348no—
170C4⋊C4—342no—
171C22⋊C4—344no—
172C22⋊C4—344no—
173C22⋊C4—344no—
174C42—22no—
175C42—174no—
176C23—172no—
177C23—341no—
178C23—341no—
179C23—342no—
180C23—342no—
181C23C23×D1711yescharacteristic
182Q8—44no—
183Q8—684no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—24no—
191C2×C4—22no—
192C2×C4—174no—
193C2×C4—172no—
194C2×C4—341no—
195C2×C4—341no—
196C2×C4—342no—
197C2×C4—342no—
198C2×C4—348no—
199C2×C4—342no—
200C2×C4—342no—
201C2×C4—342no—
202C2×C4—342no—
203C2×C4—344no—
204C2×C4#6325212yes—
205C2×C4D4×D1712yes—
206C2×C4D4×D1712yes—
207C22—342no—
208C22—342no—
209C22—342no—
210C22—682no—
211C22—682no—
212C22C2×#6325211yescharacteristic
213C22C2×D4×D1711yescharacteristic
214C22C2×D4×D1711yescharacteristic
215C22#1058769412yes—
216C22#1058769712yes—
217C4—42no—
218C4—42no—
219C4—42no—
220C4—342no—
221C4—344no—
222C4—682no—
223C4—682no—
224C4#1058757014yes—
225C2—682no—
226C2C2×#1058757011yescharacteristic
227C2#1176337911yescharacteristic
228C2#1176338212yes—
229C2#1176338411yescharacteristic
230C2#1176341712yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#391C2
prime radicalC23×D17C23
hypercenter#391D17
hypocenterC17#2621
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
1. C2×#10587630C2semidirect
5. #11762743C2semidirect
7. #11762715C2semidirect
8. #11762674C2semidirect
9. C17×#391C2semidirect
117. #391D17semidirect
146. C17#2621semidirect

Maximal subgroups

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

subgroupindex
C2×#105876302
C2×#105875832
C2×#105876322
C2×#105875582
#117627432
#117627382
#117627152
#117626742
C17×#3912
#262117