Finite Group Databasebeta

← #14544923

Subgroups — #14544923

Order 2176.

408 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 74 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×#10587551C214yesmaximal, complemented
2#11763751C211yesmaximal, characteristic, complemented
3C4×C22⋊#595C211yesmaximal, characteristic, complemented
4C2×#10587624C211yesmaximal, characteristic
5C22×D4×C17C411yescharacteristic, complemented
6C22×C22⋊D17C2212yes—
7C22×C4×D17C2211yescharacteristic
8C2×C22⋊#595C2212yes—
9C2×C22⋊#595C2212yes—
10C2×C4⋊#595—21no—
11C2×C4×#595—21no—
12C2×C22⋊Dic17C412yescomplemented
13#10587624—21no—
14#10587624—21no—
15#10587624—21no—
16#10587624—21no—
17C2×C4⋊Dic17C411yescharacteristic, complemented
18#10587551—28no—
19(C2×#595)⋊C4—21no—
20(C2×#595)⋊C4—21no—
21C24×C17C2×C412yes—
22C23×D17C2311yescharacteristic
23C22×#595—22no—
24C22×#595—22no—
25C2×D4×C17—21no—
26C2×D4×C17—21no—
27C2×D4×C17—42no—
28C22×C4×C17C2×C411yescharacteristic
29C2×C22⋊D17—24no—
30C2×C22⋊D17—22no—
31C2×C22⋊D17—22no—
32C2×C22⋊D17—24no—
33C22×Dic17C2×C412yes—
34C22×Dic17C2×C411yescharacteristic
35C2×C4×D17—42no—
36C2×C4×D17D411yescharacteristic
37C2×C4×D17D411yescharacteristic
38C2×C4×D17D411yescharacteristic
39C2×C4×D17D411yescharacteristic
40C22⋊#595—22no—
41C22⋊#595—24no—
42C22⋊#595D414yescomplemented
43C4⋊#595—42no—
44C4×#595—42no—
45C22⋊Dic17—22no—
46C22⋊Dic17—22no—
47C4⋊Dic17—21no—
48C4⋊Dic17—21no—
49C23×C17—22no—
50C23×C17—24no—
51C23×C17—42no—
52C23×C17—42no—
53C23×C17C22×C411yescharacteristic
54C22×D17—41no—
55C22×D17—41no—
56C22×D17C2×D411yescharacteristic
57C22×D17C2×D411yescharacteristic
58C22×D17C2×D412yes—
59C22×D17#4112yes—
60C2×#595—24no—
61C2×#595—42no—
62C2×#595—42no—
63C2×#595—42no—
64C2×#595—42no—
65C2×#595—42no—
66D4×C17—42no—
67D4×C17—42no—
68C2×C4×C17—22no—
69C2×C4×C17C22⋊C411yescharacteristic
70C2×C4×C17C22⋊C411yescharacteristic
71C22⋊D17—44no—
72C22⋊D17—42no—
73C22⋊D17—42no—
74C2×Dic17—22no—
75C2×Dic17—24no—
76C2×Dic17—41no—
77C2×Dic17C22⋊C411yescharacteristic
78C2×Dic17C22⋊C411yescharacteristic
79C4×D17—22no—
80C4×D17—22no—
81C4×D17—41no—
82C4×D17—41no—
83#1964—171nomaximal, Sylow
84C22×C17—21no—
85C22×C17—21no—
86C22×C17—42no—
87C22×C17—42no—
88C22×C17—42no—
89C22×C17—84no—
90C22×C17C2×C22⋊C411yescharacteristic
91C22×C17C4×D412yes—
92C2×D17—21no—
93C2×D17—21no—
94C2×D17—21no—
95C2×D17—21no—
96C2×D17—42no—
97C2×D17—42no—
98C2×D17#12112yes—
99C2×D17#12411yescharacteristic
100C2×D17#12711yescharacteristic
101#595—44no—
102#595—82no—
103C4×C17—21no—
104C4×C17#9912yescomplemented
105Dic17—21no—
106Dic17—21no—
107Dic17—42no—
108C2×#121—171no—
109C2×#99—174no—
110#380—171no—
111C4×C22⋊C4—171no—
112C2×C17—22no—
113C2×C17—82no—
114C2×C17#39011yescharacteristic
115C2×C17C2×#9911yescharacteristic
116C2×C17#41011yescharacteristic
117D17—22no—
118D17—21no—
119D17—21no—
120C22×D4—172no—
121C22×D4#59511yescharacteristic, complemented
122C23×C4—171no—
123#121—341no—
124#121—341no—
125#121—341no—
126#121—341no—
127C2×C4⋊C4—171no—
128C2×C4⋊C4—341no—
129C2×C22⋊C4—172no—
130C2×C22⋊C4—172no—
131C2×C22⋊C4—172no—
132C2×C42—341no—
133#99—348no—
134#95—341no—
135#95—341no—
136C17#196411yescharacteristic, complemented, Sylow
137C24—171no—
138C24C2×#59512yes—
139C2×D4—21no—
140C2×D4—21no—
141C2×D4—42no—
142C2×D4—344no—
143C2×D4—342no—
144C2×D4—342no—
145C2×D4—344no—
146C22×C4—171no—
147C22×C4—171no—
148C22×C4—171no—
149C22×C4—171no—
150C22×C4—171no—
151C22×C4—172no—
152C22×C4—342no—
153C22×C4—342no—
154C22×C4—682no—
155C22×C4C2×#59511yescharacteristic
156C4⋊C4—341no—
157C4⋊C4—341no—
158C4⋊C4—682no—
159C22⋊C4—174no—
160C22⋊C4—342no—
161C22⋊C4—342no—
162C22⋊C4—342no—
163C22⋊C4—344no—
164C42—682no—
165C23—22no—
166C23—24no—
167C23—42no—
168C23—42no—
169C23—171no—
170C23—171no—
171C23—172no—
172C23—172no—
173C23—681no—
174C23—681no—
175C23C22×#59511yescharacteristic
176D4—42no—
177D4—42no—
178D4—684no—
179D4—682no—
180D4—682no—
181C2×C4—22no—
182C2×C4—171no—
183C2×C4—171no—
184C2×C4—342no—
185C2×C4—342no—
186C2×C4—344no—
187C2×C4—344no—
188C2×C4—342no—
189C2×C4—681no—
190C2×C4—681no—
191C2×C4—681no—
192C2×C4—682no—
193C2×C4—682no—
194C2×C4—682no—
195C2×C4—682no—
196C2×C4—682no—
197C2×C4C22⋊#59511yescharacteristic
198C2×C4C22⋊#59511yescharacteristic
199C22—21no—
200C22—21no—
201C22—42no—
202C22—42no—
203C22—42no—
204C22—84no—
205C22—172no—
206C22—171no—
207C22—171no—
208C22—341no—
209C22—341no—
210C22—341no—
211C22—341no—
212C22—682no—
213C22—682no—
214C22D4×#59512yes—
215C22C2×C22⋊#59511yescharacteristic
216C4—21no—
217C4—341no—
218C4—341no—
219C4—684no—
220C4—682no—
221C4—1362no—
222C4#1058755112yescomplemented
223C2—22no—
224C2—82no—
225C2—342no—
226C2—341no—
227C2—341no—
228C2#1176376111yescharacteristic
229C2C2×#1058755111yescharacteristic
230C2#1176378911yescharacteristic
subgroupG/H
centerC22C2×C22⋊#595
commutatorC23×C17C22×C4
FrattiniC23C22×#595
FittingC22×D4×C17C4
socleC22×C17C2×C22⋊C4
prime radicalC2×#10587624C2
cosocleC23×D17C23
hypercenterC22×D4#595
hypocenterC17#1964
LCSG ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. C2×#10587551C2semidirect
2. #11763751C2semidirect
3. C4×C22⋊#595C2semidirect
5. C22×D4×C17C4semidirect
12. C2×C22⋊Dic17C4semidirect
17. C2×C4⋊Dic17C4semidirect
42. C22⋊#595D4semidirect
104. C4×C17#99semidirect
121. C22×D4#595semidirect
136. C17#1964semidirect
222. C4#10587551semidirect

Maximal subgroups

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

subgroupindex
C2×#105875512
#117637512
C4×C22⋊#5952
C2×#105876242
#196417