Finite Group Databasebeta

← #14544767

Subgroups — #14544767

Order 2176.

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

normal subgroups only

This group has 376 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
1C22×#63243C211yesmaximal, characteristic, complemented
2C2×#10587548C212yesmaximal, complemented
3C4×C22⋊#595C211yesmaximal, characteristic, complemented
4#11763679C212yesmaximal, complemented
5#11763319C211yesmaximal, characteristic
6C2×#63254C2211yescharacteristic
7C22×C4×D17C2211yescharacteristic
8C2×C22⋊#595—21no—
9C22×#3624—21no—
10C2×C4×#595C2212yes—
11C2×#63243—21no—
12C2×#63243—24no—
13C2×#63243C2212yes—
14C17×#121C411yescharacteristic, complemented
15#10587624C411yescharacteristic, complemented
16#10587623—21no—
17D4×Dic17—21no—
18#10587610C411yescharacteristic, complemented
19#10587583C411yescharacteristic, complemented
20#10587578C2211yescharacteristic
21#10587569—21no—
22#10587567—21no—
23#10587548—24no—
24(C2×#595)⋊C4—21no—
25C23×D17D411yescharacteristic
26C22×#595—22no—
27C2×D4×C17—21no—
28C2×D4×C17C2×C411yescharacteristic
29C22×C4×C17C2×C411yescharacteristic
30C2×C22⋊D17—21no—
31C2×C22⋊D17—21no—
32C22×Dic17—21no—
33C22×Dic17D411yescharacteristic
34#63254—22no—
35#63254—22no—
36C2×C4×D17—22no—
37C2×C4×D17—22no—
38C2×C4×D17C2×C411yescharacteristic
39C2×C4×D17D411yescharacteristic
40C2×C4×D17D411yescharacteristic
41C2×C4×D17C2311yescharacteristic
42C2×Dic34C2×C411yescharacteristic
43C22⋊#595—24no—
44C2×#3624—24no—
45C2×#3624—22no—
46C4×#595—22no—
47C4×#595—22no—
48#63243—24no—
49#63243—22no—
50#63243D414yescomplemented
51C4⋊C4×C17C2×C411yescharacteristic
52C22⋊C4×C17—21no—
53C22⋊Dic17—21no—
54C22⋊Dic17—21no—
55#63227—21no—
56#63227—21no—
57C4⋊Dic17—21no—
58C4⋊Dic17C2×C411yescharacteristic
59#63225—21no—
60C4×Dic17—21no—
61C23×C17—21no—
62C23×C17C22⋊C411yescharacteristic
63C22×D17—22no—
64C22×D17—22no—
65C22×D17—41no—
66C22×D17C22⋊C411yescharacteristic
67C22×D17C2×D411yescharacteristic
68C2×#595—24no—
69C2×#595—42no—
70C2×#595—42no—
71D4×C17—22no—
72D4×C17—42no—
73C2×C4×C17—22no—
74C2×C4×C17—21no—
75C2×C4×C17C22⋊C411yescharacteristic
76C2×C4×C17C22×C411yescharacteristic
77C22⋊D17—42no—
78C22⋊D17—42no—
79C2×Dic17—21no—
80C2×Dic17—21no—
81C2×Dic17—21no—
82C2×Dic17—21no—
83C2×Dic17—41no—
84C2×Dic17—41no—
85C2×Dic17C22⋊C411yescharacteristic
86C2×Dic17C2×D411yescharacteristic
87C4×D17—22no—
88C4×D17—22no—
89C4×D17—21no—
90C4×D17—21no—
91C4×D17C2×D412yes—
92C4×D17#4112yes—
93Dic34—22no—
94#3624—24no—
95#1871—171nomaximal, Sylow
96C22×C17—21no—
97C22×C17—21no—
98C22×C17—41no—
99C22×C17—41no—
100C22×C17—41no—
101C22×C17#10412yescomplemented
102C22×C17C2×C22⋊C411yescharacteristic
103C2×D17—21no—
104C2×D17—21no—
105C2×D17—21no—
106C2×D17—21no—
107C2×D17—21no—
108C2×D17—44no—
109C2×D17#12112yes—
110#595—44no—
111C4×C17—21no—
112C4×C17—41no—
113C4×C17C4×D412yes—
114Dic17—21no—
115Dic17—41no—
116Dic17—41no—
117Dic17#12011yescharacteristic
118Dic17#12311yescharacteristic
119C22×M16—171no—
120#525—171no—
121C2×#104—172no—
122C4×C22⋊C4—171no—
123#339—172no—
124C2×C17—22no—
125C2×C17—81no—
126C2×C17#38611yescharacteristic
127C2×C17C2×#10411yescharacteristic
128C2×C17#42111yescharacteristic
129D17—22no—
130D17—41no—
131C2×#41—171no—
132C23×C4—171no—
133C2×M16—172no—
134C2×M16—344no—
135C2×M16—341no—
136C22×C8—341no—
137#123—341no—
138C22⋊Q8—171no—
139C22⋊Q8—171no—
140#121—171no—
141#121#59511yescharacteristic, complemented
142#120—341no—
143C4×D4—341no—
144C4×D4—341no—
145#117—171no—
146C2×C22⋊C4—341no—
147C2×C42—172no—
148#104—344no—
149#95—341no—
150C17#187111yescharacteristic, complemented, Sylow
151C24—171no—
152#41—342no—
153#41—342no—
154C2×Q8—171no—
155C2×D4—21no—
156C2×D4—341no—
157C2×D4—341no—
158C2×D4C2×#59511yescharacteristic
159C22×C4—171no—
160C22×C4—171no—
161C22×C4—171no—
162C22×C4—171no—
163C22×C4—171no—
164C22×C4—342no—
165C22×C4—342no—
166C22×C4—341no—
167C22×C4—342no—
168C22×C4C2×#59511yescharacteristic
169M16—174no—
170M16—342no—
171M16—344no—
172C2×C8—342no—
173C2×C8—344no—
174C4⋊C4—171no—
175C4⋊C4—341no—
176C4⋊C4—341no—
177C4⋊C4C2×#59511yescharacteristic
178C22⋊C4—21no—
179C22⋊C4—341no—
180C22⋊C4—341no—
181C22⋊C4—341no—
182C22⋊C4—344no—
183C22⋊C4—341no—
184C42—341no—
185C42—342no—
186C42—342no—
187C23—21no—
188C23—171no—
189C23—171no—
190C23—342no—
191C23—342no—
192C23—681no—
193C23C22⋊#59511yescharacteristic
194Q8—342no—
195D4—22no—
196D4—42no—
197D4—682no—
198D4—682no—
199C2×C4—22no—
200C2×C4—21no—
201C2×C4—171no—
202C2×C4—172no—
203C2×C4—172no—
204C2×C4—171no—
205C2×C4—341no—
206C2×C4—341no—
207C2×C4—342no—
208C2×C4—341no—
209C2×C4—341no—
210C2×C4—342no—
211C2×C4—344no—
212C2×C4—341no—
213C2×C4—341no—
214C2×C4—681no—
215C2×C4—681no—
216C2×C4—682no—
217C2×C4—682no—
218C2×C4C22⋊#59511yescharacteristic
219C2×C4C22×#59511yescharacteristic
220C8—344no—
221C22—21no—
222C22—21no—
223C22—41no—
224C22—41no—
225C22—41no—
226C22—172no—
227C22—341no—
228C22—341no—
229C22—341no—
230C22—341no—
231C22—341no—
232C22—684no—
233C22#1058754812yescomplemented
234C22C2×C22⋊#59511yescharacteristic
235C4—21no—
236C4—41no—
237C4—171no—
238C4—171no—
239C4—341no—
240C4—684no—
241C4—681no—
242C4—681no—
243C4D4×#59512yes—
244C2—22no—
245C2—81no—
246C2—342no—
247C2—681no—
248C2#1176375711yescharacteristic
249C2C2×#1058754811yescharacteristic
250C2#1176380611yescharacteristic
subgroupG/H
centerC22C2×C22⋊#595
commutatorC2×C4×C17C22×C4
FrattiniC2×C4C22×#595
FittingC17×#121C4
socleC22×C17C2×C22⋊C4
prime radical#11763319C2
cosocleC2×C4×D17C23
hypercenter#121#595
hypocenterC17#1871
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C22×#63243C2semidirect
2. C2×#10587548C2semidirect
3. C4×C22⋊#595C2semidirect
4. #11763679C2semidirect
14. C17×#121C4semidirect
15. #10587624C4semidirect
18. #10587610C4semidirect
19. #10587583C4semidirect
50. #63243D4semidirect
101. C22×C17#104semidirect
141. #121#595semidirect
150. C17#1871semidirect
233. C22#10587548semidirect

Maximal subgroups

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

subgroupindex
C22×#632432
C2×#105875482
C4×C22⋊#5952
#117636792
#117633192
#187117