Finite Group Databasebeta

← #14544626

Subgroups — #14544626

Order 2176.

618 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 618 conjugacy classes of subgroups — too many to draw. Only its 88 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×#10587550C212yesmaximal, complemented
2C2×#10587548C212yesmaximal, complemented
3#11763780C211yesmaximal, characteristic, complemented
4C4×C22⋊#595C211yesmaximal, characteristic, complemented
5C2×#41×D17C211yesmaximal, characteristic
6C2×#41×C17C411yescharacteristic, complemented
7#41×D17—24no—
8#41×D17—24no—
9C2×#63256C2211yescharacteristic
10C2×Q8×D17C411yescharacteristic, complemented
11C2×#63254—21no—
12C2×#63254C2211yescharacteristic
13C2×D4×D17—21no—
14C2×D4×D17C411yescharacteristic, complemented
15C2×#63252—21no—
16C2×#63252C411yescharacteristic, complemented
17C22×C4×D17—21no—
18C22×C4×D17C2211yescharacteristic
19C2×C22⋊#595—21no—
20C2×C4×#595C2212yescomplemented
21C2×#63243C2212yescomplemented
22C22⋊#3624—21no—
23#10587550—24no—
24#10587548—24no—
25(C2×#595)⋊C4—21no—
26#10587543—21no—
27C23×D17—21no—
28C23×D17D411yescharacteristic
29C22×#595—22no—
30#41×C17—22no—
31#41×C17—22no—
32C2×Q8×C17C2×C411yescharacteristic
33C2×D4×C17—21no—
34C2×D4×C17C2×C411yescharacteristic
35C22×C4×C17—21no—
36C22×C4×C17C2×C411yescharacteristic
37C2×C22⋊D17—21no—
38C2×C22⋊D17—21no—
39C2×C22⋊D17—21no—
40C22×Dic17—21no—
41C22×Dic17D411yescharacteristic
42#63256—22no—
43#63256D414yes—
44Q8×D17—22no—
45Q8×D17—22no—
46#63254—28no—
47#63254—22no—
48#63254D414yes—
49D4×D17—28no—
50D4×D17—22no—
51D4×D17—22no—
52#63252—28no—
53#63252—22no—
54#63252—22no—
55C2×D68—21no—
56C2×D68C2×C411yescharacteristic
57C2×C4×D17—28no—
58C2×C4×D17—21no—
59C2×C4×D17—21no—
60C2×C4×D17—21no—
61C2×C4×D17—22no—
62C2×C4×D17—22no—
63C2×C4×D17C2×C411yescharacteristic
64C2×C4×D17D411yescharacteristic
65C2×C4×D17D411yescharacteristic
66C2×C4×D17C2311yescharacteristic
67C2×Dic34—21no—
68C2×Dic34C2×C411yescharacteristic
69C22⋊#595—24no—
70C2×#3624—22no—
71C4×#595—22no—
72C4×#595—22no—
73#63243—22no—
74#63243—22no—
75C23×C17—21no—
76C23×C17C22⋊C411yescharacteristic
77C22×D17—24no—
78C22×D17—21no—
79C22×D17—22no—
80C22×D17—22no—
81C22×D17—41no—
82C22×D17—42no—
83C22×D17—42no—
84C22×D17C22⋊C411yescharacteristic
85C22×D17C2×D411yescharacteristic
86C2×#595—24no—
87C2×#595—42no—
88C2×#595—42no—
89Q8×C17—21no—
90Q8×C17C22⋊C412yescomplemented
91D4×C17—24no—
92D4×C17—21no—
93D4×C17C22⋊C412yescomplemented
94C2×C4×C17—24no—
95C2×C4×C17—21no—
96C2×C4×C17—21no—
97C2×C4×C17—21no—
98C2×C4×C17C22⋊C411yescharacteristic
99C2×C4×C17C22×C411yescharacteristic
100C22⋊D17—24no—
101C22⋊D17—24no—
102C22⋊D17—24no—
103C2×Dic17—21no—
104C2×Dic17—22no—
105C2×Dic17—24no—
106C2×Dic17C22⋊C411yescharacteristic
107C2×Dic17C2×D411yescharacteristic
108D68—24no—
109D68—21no—
110D68C22⋊C412yescomplemented
111C4×D17—24no—
112C4×D17—24no—
113C4×D17—22no—
114C4×D17—21no—
115C4×D17—21no—
116C4×D17—22no—
117C4×D17—24no—
118C4×D17C2×D412yes—
119C4×D17C2×D412yes—
120Dic34—24no—
121Dic34—21no—
122Dic34C22⋊C412yescomplemented
123#3624—42no—
124#1789—171nomaximal, Sylow
125C22×C17—21no—
126C22×C17—22no—
127C22×C17—21no—
128C22×C17—21no—
129C22×C17—41no—
130C22×C17—41no—
131C22×C17C2×C22⋊C411yescharacteristic
132C2×D17—21no—
133C2×D17—21no—
134C2×D17—21no—
135C2×D17—21no—
136C2×D17—22no—
137C2×D17—44no—
138C2×D17—48no—
139C2×D17—41no—
140C2×D17—41no—
141C2×D17#10412yescomplemented
142C2×D17#12012yes—
143#595—44no—
144C4×C17—22no—
145C4×C17—21no—
146C4×C17C2×C22⋊C411yescharacteristic
147C4×C17C2×C22⋊C411yescharacteristic
148Dic17—21no—
149Dic17—22no—
150Dic17#12012yes—
151C22×#41—171no—
152C2×#104—172no—
153C2×#104—172no—
154#407—171no—
155C4×C22⋊C4—171no—
156C2×C17—41no—
157C2×C17—42no—
158C2×C17#37911yescharacteristic
159C2×C17C2×#10411yescharacteristic
160C2×C17#42111yescharacteristic
161D17—22no—
162D17—22no—
163D17—42no—
164C2×#41—171no—
165C2×#41—171no—
166C2×#41—171no—
167C2×#41—341no—
168C2×#41—341no—
169C2×#41—344no—
170C2×#41—344no—
171C2×#41#59511yescharacteristic, complemented
172C22×Q8—171no—
173C22×D4—171no—
174C22×D4—341no—
175C23×C4—171no—
176C23×C4—341no—
177C2×M16—172no—
178C2×C22⋊C4—341no—
179C2×C42—172no—
180#104—344no—
181#104—344no—
182C22⋊C8—341no—
183C22⋊C8—341no—
184#95—341no—
185C17#178911yescharacteristic, complemented, Sylow
186C24—171no—
187C24—341no—
188#41—22no—
189#41—22no—
190#41—174no—
191#41—174no—
192#41—342no—
193#41—342no—
194#41—348no—
195#41—342no—
196#41—342no—
197#41—348no—
198C2×Q8—171no—
199C2×Q8—341no—
200C2×Q8—342no—
201C2×Q8—342no—
202C2×Q8C2×#59511yescharacteristic
203C2×D4—21no—
204C2×D4—171no—
205C2×D4—341no—
206C2×D4—341no—
207C2×D4—342no—
208C2×D4—348no—
209C2×D4—341no—
210C2×D4—341no—
211C2×D4—342no—
212C2×D4C2×#59511yescharacteristic
213C22×C4—21no—
214C22×C4—171no—
215C22×C4—171no—
216C22×C4—171no—
217C22×C4—171no—
218C22×C4—171no—
219C22×C4—342no—
220C22×C4—342no—
221C22×C4—341no—
222C22×C4—341no—
223C22×C4—348no—
224C22×C4—341no—
225C22×C4—341no—
226C22×C4—342no—
227C22×C4C2×#59511yescharacteristic
228M16—342no—
229M16—342no—
230C2×C8—342no—
231C22⋊C4—344no—
232C42—342no—
233C42—342no—
234C23—21no—
235C23—171no—
236C23—171no—
237C23—342no—
238C23—342no—
239C23—344no—
240C23—341no—
241C23—681no—
242C23—682no—
243C23—682no—
244C23C22⋊#59511yescharacteristic
245Q8—21no—
246Q8—172no—
247Q8—344no—
248Q8—341no—
249Q8C22⋊#59512yescomplemented
250D4—24no—
251D4—21no—
252D4—172no—
253D4—344no—
254D4—344no—
255D4—344no—
256D4—344no—
257D4—341no—
258D4C22⋊#59512yescomplemented
259C2×C4—24no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—171no—
264C2×C4—172no—
265C2×C4—172no—
266C2×C4—171no—
267C2×C4—342no—
268C2×C4—344no—
269C2×C4—342no—
270C2×C4—344no—
271C2×C4—344no—
272C2×C4—342no—
273C2×C4—344no—
274C2×C4—341no—
275C2×C4—341no—
276C2×C4—341no—
277C2×C4—344no—
278C2×C4—682no—
279C2×C4—682no—
280C2×C4C22⋊#59511yescharacteristic
281C2×C4C22×#59511yescharacteristic
282C8—682no—
283C22—21no—
284C22—22no—
285C22—21no—
286C22—21no—
287C22—41no—
288C22—41no—
289C22—172no—
290C22—172no—
291C22—341no—
292C22—341no—
293C22—341no—
294C22—341no—
295C22—342no—
296C22—684no—
297C22—688no—
298C22—681no—
299C22—681no—
300C22C2×C22⋊#59511yescharacteristic
301C4—21no—
302C4—22no—
303C4—172no—
304C4—341no—
305C4—342no—
306C4—684no—
307C4C2×C22⋊#59511yescharacteristic
308C4C2×C22⋊#59511yescharacteristic
309C2—41no—
310C2—42no—
311C2—342no—
312C2—342no—
313C2—682no—
314C2#1176375011yescharacteristic
315C2D17⋊#10411yescharacteristic
316C2#1176380611yescharacteristic
subgroupG/H
centerC22C2×C22⋊#595
commutatorC2×C4×C17C22×C4
FrattiniC2×C4C22×#595
FittingC2×#41×C17C4
socleC22×C17C2×C22⋊C4
prime radicalC2×#41×D17C2
cosocleC2×C4×D17C23
hypercenterC2×#41#595
hypocenterC17#1789
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
1. C2×#10587550C2semidirect
2. C2×#10587548C2semidirect
3. #11763780C2semidirect
4. C4×C22⋊#595C2semidirect
6. C2×#41×C17C4semidirect
10. C2×Q8×D17C4semidirect
14. C2×D4×D17C4semidirect
16. C2×#63252C4semidirect
20. C2×C4×#595C22semidirect
21. C2×#63243C22semidirect
90. Q8×C17C22⋊C4semidirect
93. D4×C17C22⋊C4semidirect
110. D68C22⋊C4semidirect
122. Dic34C22⋊C4semidirect
141. C2×D17#104semidirect
171. C2×#41#595semidirect
185. C17#1789semidirect
249. Q8C22⋊#595semidirect
258. D4C22⋊#595semidirect

Maximal subgroups

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

subgroupindex
C2×#105875502
C2×#105875482
#117637802
C4×C22⋊#5952
C2×#41×D172
#178917