Finite Group Databasebeta

← #14544766

Subgroups — #14544766

Order 2176.

376 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 72 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×D17⋊M16C211yesmaximal, characteristic, complemented
2C2×#10587550C212yesmaximal, complemented
3C4×C22⋊#595C211yesmaximal, characteristic, complemented
4#11763679C212yesmaximal, complemented
5#11763322C211yesmaximal, characteristic
6C2×#63256C2211yescharacteristic
7C22×C4×D17C2211yescharacteristic
8C2×C22⋊#595—21no—
9C2×#63247—21no—
10C2×C4×#595C2212yes—
11D17⋊M16—24no—
12C2×#63243C2212yes—
13C2×#63242—21no—
14C17×C22⋊Q8C411yescharacteristic, complemented
15#10587632C411yescharacteristic, complemented
16#10587616C411yescharacteristic, complemented
17#10587581C411yescharacteristic, complemented
18#10587580—21no—
19#10587579—21no—
20#10587578C2211yescharacteristic
21#10587568—21no—
22#10587567—21no—
23#10587550—24no—
24(C2×#595)⋊C4—21no—
25C23×D17D411yescharacteristic
26C22×#595—22no—
27C2×Q8×C17C2×C411yescharacteristic
28C22×C4×C17C2×C411yescharacteristic
29C2×C22⋊D17—21no—
30C22×Dic17D411yescharacteristic
31#63256—22no—
32#63256—22no—
33C2×D68—21no—
34C2×D68—21no—
35C2×D68C2×C411yescharacteristic
36C2×C4×D17—22no—
37C2×C4×D17—22no—
38C2×C4×D17—21no—
39C2×C4×D17C2×C411yescharacteristic
40C2×C4×D17D411yescharacteristic
41C2×C4×D17D411yescharacteristic
42C2×C4×D17C2311yescharacteristic
43C22⋊#595—24no—
44#63247—24no—
45C2×#3624—22no—
46C4×#595—22no—
47C4×#595—22no—
48#63243—22no—
49#63243D414yescomplemented
50#63242—24no—
51C4⋊C4×C17—21no—
52C4⋊C4×C17C2×C411yescharacteristic
53C22⋊C4×C17—21no—
54#63227—21no—
55#63227—21no—
56#63227—21no—
57#63227—21no—
58C4⋊Dic17C2×C411yescharacteristic
59#63225—21no—
60C4×Dic17—21no—
61C23×C17C22⋊C411yescharacteristic
62C22×D17—22no—
63C22×D17—22no—
64C22×D17—21no—
65C22×D17—41no—
66C22×D17C22⋊C411yescharacteristic
67C22×D17C2×D411yescharacteristic
68C2×#595—24no—
69C2×#595—42no—
70C2×#595—42no—
71Q8×C17—22no—
72C2×C4×C17—22no—
73C2×C4×C17—21no—
74C2×C4×C17—21no—
75C2×C4×C17C22⋊C411yescharacteristic
76C2×C4×C17C22×C411yescharacteristic
77C22⋊D17—42no—
78C2×Dic17—21no—
79C2×Dic17—21no—
80C2×Dic17—21no—
81C2×Dic17C22⋊C411yescharacteristic
82C2×Dic17C2×D411yescharacteristic
83D68—22no—
84D68—42no—
85D68—42no—
86C4×D17—22no—
87C4×D17—22no—
88C4×D17—21no—
89C4×D17—21no—
90C4×D17—41no—
91C4×D17—41no—
92C4×D17C2×D412yes—
93C4×D17#4112yes—
94#3624—24no—
95#1871—171nomaximal, Sylow
96C22×C17—21no—
97C22×C17—21no—
98C22×C17—21no—
99C22×C17C2×C22⋊C411yescharacteristic
100C2×D17—21no—
101C2×D17—21no—
102C2×D17—21no—
103C2×D17—21no—
104C2×D17—44no—
105C2×D17—41no—
106C2×D17—41no—
107C2×D17—41no—
108C2×D17#10412yescomplemented
109C2×D17#12112yes—
110#595—44no—
111C4×C17—21no—
112C4×C17—41no—
113C4×C17—41no—
114C4×C17C4×D412yes—
115Dic17—21no—
116Dic17—41no—
117Dic17#12011yescharacteristic
118Dic17#12311yescharacteristic
119C22×M16—171no—
120#525—171no—
121C2×#104—172no—
122C4×C22⋊C4—171no—
123#339—172no—
124C2×C17—41no—
125C2×C17#38611yescharacteristic
126C2×C17C2×#10411yescharacteristic
127C2×C17#42111yescharacteristic
128D17—22no—
129D17—22no—
130D17—81no—
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#59511yescharacteristic, complemented
140#121—171no—
141#121—171no—
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×Q8C2×#59511yescharacteristic
155C2×D4—171no—
156C2×D4—341no—
157C2×D4—341no—
158C2×D4—341no—
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—21no—
175C4⋊C4—171no—
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—171no—
188C23—171no—
189C23—342no—
190C23—342no—
191C23—341no—
192C23—681no—
193C23C22⋊#59511yescharacteristic
194Q8—22no—
195D4—342no—
196D4—682no—
197D4—682no—
198D4—682no—
199C2×C4—22no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—171no—
203C2×C4—172no—
204C2×C4—172no—
205C2×C4—171no—
206C2×C4—341no—
207C2×C4—341no—
208C2×C4—342no—
209C2×C4—341no—
210C2×C4—341no—
211C2×C4—342no—
212C2×C4—344no—
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—21no—
224C22—172no—
225C22—172no—
226C22—341no—
227C22—341no—
228C22—341no—
229C22—341no—
230C22—684no—
231C22—681no—
232C22—681no—
233C22—681no—
234C22C2×C22⋊#59511yescharacteristic
235C4—21no—
236C4—41no—
237C4—41no—
238C4—171no—
239C4—171no—
240C4—341no—
241C4—684no—
242C4—681no—
243C4D4×#59512yes—
244C2—41no—
245C2—342no—
246C2—342no—
247C2—1361no—
248C2#1176375711yescharacteristic
249C2D17⋊#10411yescharacteristic
250C2#1176380811yescharacteristic
subgroupG/H
centerC22C2×C22⋊#595
commutatorC2×C4×C17C22×C4
FrattiniC2×C4C22×#595
FittingC17×C22⋊Q8C4
socleC22×C17C2×C22⋊C4
prime radical#11763322C2
cosocleC2×C4×D17C23
hypercenterC22⋊Q8#595
hypocenterC17#1871
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×D17⋊M16C2semidirect
2. C2×#10587550C2semidirect
3. C4×C22⋊#595C2semidirect
4. #11763679C2semidirect
14. C17×C22⋊Q8C4semidirect
15. #10587632C4semidirect
16. #10587616C4semidirect
17. #10587581C4semidirect
49. #63243D4semidirect
108. C2×D17#104semidirect
139. C22⋊Q8#595semidirect
150. C17#1871semidirect

Maximal subgroups

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

subgroupindex
C2×D17⋊M162
C2×#105875502
C4×C22⋊#5952
#117636792
#117633222
#187117