Finite Group Databasebeta

← #14529192

Subgroups — #14529192

Order 2176.

578 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 578 conjugacy classes of subgroups — too many to draw. Only its 93 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×#10587623C211yesmaximal, characteristic
2C2×#10587505C211yesmaximal, characteristic, complemented
3#11762664C212yesmaximal, complemented
4#11762576C212yesmaximal, complemented
5C17×#518C211yesmaximal, characteristic, complemented
6C24×D17C2211yescharacteristic
7C22×D4×C17C2211yescharacteristic
8C22×C22⋊D17—21no—
9C4×D4×C17—24no—
10C17×#117—21no—
11C2×C22⋊C4×C17—21no—
12C2×C22⋊C4×C17C2212yes—
13C2×C22⋊Dic17C2212yes—
14C2×C22⋊Dic17C2211yescharacteristic
15#10587623—22no—
16#10587623C414yescomplemented
17C2×#63227—21no—
18C2×#63227—24no—
19#10587505—22no—
20C24×C17D411yescharacteristic
21C24×C17C2311yescharacteristic
22C23×D17—21no—
23C23×D17—22no—
24C23×D17—21no—
25C23×D17—22no—
26C23×D17—44no—
27C23×D17C2×C412yescomplemented
28C2×D4×C17—22no—
29C2×D4×C17—22no—
30C2×D4×C17C2×C412yes—
31C2×D4×C17D412yes—
32C22×C4×C17—22no—
33C22×C4×C17D411yescharacteristic
34C22×C4×C17D414yescomplemented
35C2×C22⋊D17—24no—
36C2×C22⋊D17—44no—
37C22×Dic17—21no—
38C22×Dic17—22no—
39C4⋊C4×C17—22no—
40C22⋊C4×C17—22no—
41C22⋊C4×C17—22no—
42C22⋊C4×C17—22no—
43C42×C17—22no—
44C22⋊Dic17—21no—
45C22⋊Dic17—22no—
46C22⋊Dic17—22no—
47C22⋊Dic17C2×C412yescomplemented
48#63227—24no—
49#63227—48no—
50C23×C17—21no—
51C23×C17—21no—
52C23×C17—42no—
53C23×C17—42no—
54C23×C17C22⋊C412yescomplemented
55C23×C17C22×C411yescharacteristic
56C23×C17C2×D411yescharacteristic
57C23×C17C2×D411yescharacteristic
58C23×C17C2×D412yes—
59C23×C17#4112yes—
60C22×D17—24no—
61C22×D17—28no—
62C22×D17—22no—
63C22×D17—24no—
64C22×D17—44no—
65C22×D17—44no—
66C22×D17—48no—
67C22×D17—44no—
68C22×D17—42no—
69C22×D17—41no—
70C22×D17—42no—
71C22×D17—41no—
72D4×C17—24no—
73D4×C17—24no—
74C2×C4×C17—21no—
75C2×C4×C17—21no—
76C2×C4×C17—24no—
77C2×C4×C17—44no—
78C2×C4×C17—82no—
79C2×C4×C17C22⋊C412yescomplemented
80C22⋊D17—48no—
81C2×Dic17—22no—
82C2×Dic17—42no—
83C2×Dic17—44no—
84C2×Dic17—42no—
85#1889—171nomaximal, Sylow
86C22×C17—24no—
87C22×C17—22no—
88C22×C17—22no—
89C22×C17—41no—
90C22×C17—41no—
91C22×C17—42no—
92C22×C17—42no—
93C22×C17—42no—
94C22×C17—81no—
95C22×C17C2×C22⋊C411yescharacteristic
96C22×C17C4×D412yes—
97C22×C17#12011yescharacteristic
98C22×C17#12112yes—
99C22×C17#12311yescharacteristic
100C2×D17—24no—
101C2×D17—28no—
102C2×D17—42no—
103C2×D17—48no—
104C2×D17—44no—
105C2×D17—44no—
106C2×D17—42no—
107C2×D17—44no—
108C4×C17—22no—
109C4×C17—44no—
110Dic17—42no—
111Dic17—82no—
112C2×#120—171no—
113#518D1711yescharacteristic, complemented
114C2×#99—171no—
115#379—172no—
116#342—172no—
117C2×C17—21no—
118C2×C17—22no—
119C2×C17—22no—
120C2×C17—21no—
121C2×C17—42no—
122C2×C17#38611yescharacteristic
123C2×C17B(4,𝔽2)12yes—
124D17—44no—
125C25—171no—
126C22×D4—341no—
127C22×D4C2×D1711yescharacteristic
128#120—174no—
129#120—342no—
130C4×D4—24no—
131#117—21no—
132C2×C22⋊C4—21no—
133C2×C22⋊C4—171no—
134C2×C22⋊C4—172no—
135C2×C22⋊C4—341no—
136C2×C22⋊C4—344no—
137C2×C22⋊C4C2×D1712yes—
138#99—342no—
139C17#188911yescharacteristic, complemented, Sylow
140C24—172no—
141C24—342no—
142C24—341no—
143C24—341no—
144C24—342no—
145C24—684no—
146C24C22⋊D1711yescharacteristic
147C24C22×D1711yescharacteristic
148C2×D4—22no—
149C2×D4—22no—
150C2×D4—344no—
151C2×D4—684no—
152C2×D4C4×D1712yes—
153C2×D4D6812yes—
154C22×C4—22no—
155C22×C4—341no—
156C22×C4—342no—
157C22×C4C22⋊D1711yescharacteristic
158C22×C4C22⋊D1714yescomplemented
159C4⋊C4—22no—
160C22⋊C4—22no—
161C22⋊C4—22no—
162C22⋊C4—22no—
163C22⋊C4—172no—
164C22⋊C4—341no—
165C22⋊C4—344no—
166C22⋊C4—342no—
167C22⋊C4—342no—
168C22⋊C4—688no—
169C42—22no—
170C23—21no—
171C23—21no—
172C23—42no—
173C23—42no—
174C23—348no—
175C23—344no—
176C23—342no—
177C23—344no—
178C23—684no—
179C23—682no—
180C23—684no—
181C23—684no—
182C23—688no—
183C23—681no—
184C23—682no—
185C23—681no—
186C23#6322712yescomplemented
187C23C2×C4×D1711yescharacteristic
188C23C2×D6811yescharacteristic
189C23#6325212yes—
190C23C2×C22⋊D1711yescharacteristic
191C23C2×C22⋊D1712yes—
192D4—24no—
193D4—24no—
194D4—688no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—24no—
198C2×C4—44no—
199C2×C4—82no—
200C2×C4—342no—
201C2×C4—682no—
202C2×C4—684no—
203C2×C4—682no—
204C2×C4#6322712yescomplemented
205C22—24no—
206C22—22no—
207C22—22no—
208C22—41no—
209C22—41no—
210C22—42no—
211C22—42no—
212C22—42no—
213C22—81no—
214C22—344no—
215C22—348no—
216C22—682no—
217C22—688no—
218C22—684no—
219C22—684no—
220C22—682no—
221C22—684no—
222C22C2×#6322711yescharacteristic
223C22C4×C22⋊D1712yes—
224C22#1058761511yescharacteristic
225C22#1058761612yes—
226C22#1058763911yescharacteristic
227C4—22no—
228C4—44no—
229C4—682no—
230C4—1362no—
231C2—21no—
232C2—22no—
233C2—21no—
234C2—22no—
235C2—42no—
236C2—684no—
237C2#1176268811yescharacteristic
238C2#1176299612yes—
subgroupG/H
centerC22#10587639
commutatorC23×C17C22×C4
FrattiniC24C22×D17
FittingC17×#518C2
socleC22×C17#120
prime radicalC2×#10587623C2
cosocleC24×C17C23
hypercenter#518D17
hypocenterC17#1889
LCSG ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C24 ◁ #518

Complements

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

NH ≅ G/Ntype
2. C2×#10587505C2semidirect
3. #11762664C2semidirect
4. #11762576C2semidirect
5. C17×#518C2semidirect
16. #10587623C4semidirect
27. C23×D17C2×C4semidirect
34. C22×C4×C17D4semidirect
47. C22⋊Dic17C2×C4semidirect
54. C23×C17C22⋊C4semidirect
79. C2×C4×C17C22⋊C4semidirect
113. #518D17semidirect
139. C17#1889semidirect
158. C22×C4C22⋊D17semidirect
186. C23#63227semidirect
204. C2×C4#63227semidirect

Maximal subgroups

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

subgroupindex
C2×#105876232
C2×#105875052
#117626642
#117625762
C17×#5182
#188917