Finite Group Databasebeta

← #17958300

Subgroups — #17958300

Order 4352.

616 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 616 conjugacy classes of subgroups — too many to draw. Only its 100 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
1#14544020C212yesmaximal, complemented
2#14543726C212yesmaximal, complemented
3#14543712C212yesmaximal, complemented
4#127×Dic17C211yesmaximal, characteristic
5C2×C22⋊#3624—22no—
6#11763654C2214yes—
7#11763652—22no—
8C2×D4×Dic17—22no—
9C2×D4×Dic17C411yescharacteristic
10C2×D4×Dic17C2211yescharacteristic
11#11762712—22no—
12#11762712C411yescharacteristic
13#11762712C2211yescharacteristic
14#11762679C2211yescharacteristic
15C42×Dic17C411yescharacteristic
16C2×C17×#127C411yescharacteristic
17C22×D4×C17—22no—
18C22×D4×C17C2×C411yescharacteristic
19C22×D4×C17C2×C411yescharacteristic
20C23×Dic17—21no—
21C23×Dic17D412yes—
22C22×#3624—42no—
23C17×#127—22no—
24C17×#127C814yescomplemented
25C2×C42×C17C2×C411yescharacteristic
26C2×C22⋊Dic17—22no—
27C2×C22⋊Dic17—21no—
28C2×C22⋊Dic17—22no—
29C2×C22⋊Dic17D412yes—
30D4×Dic17—41no—
31D4×Dic17—41no—
32D4×Dic17—48no—
33D4×Dic17—41no—
34D4×Dic17—41no—
35C2×C4⋊Dic17—22no—
36C2×C4⋊Dic17C2×C411yescharacteristic
37C2×C4⋊Dic17C2311yescharacteristic
38C2×C4×Dic17—21no—
39C2×C4×Dic17—21no—
40C2×C4×Dic17C2×C411yescharacteristic
41C2×C4×Dic17C2×C411yescharacteristic
42C22⋊#3624—24no—
43C22⋊#3624—42no—
44C24×C17—21no—
45C24×C17C22⋊C412yes—
46C2×D4×C17—28no—
47C2×D4×C17—21no—
48C2×D4×C17—41no—
49C2×D4×C17—41no—
50C2×D4×C17—41no—
51C2×D4×C17—41no—
52C2×D4×C17—44no—
53C2×D4×C17—44no—
54C2×D4×C17C2×C812yes—
55C2×D4×C17C2×C812yescomplemented
56C2×D4×C17M1612yes—
57C22×C4×C17—21no—
58C22×C4×C17C22×C411yescharacteristic
59C22×Dic17—21no—
60C22×Dic17—22no—
61C22×Dic17—22no—
62C22×Dic17—21no—
63C22×Dic17—44no—
64C22×Dic17—44no—
65C22×Dic17—41no—
66C22×Dic17—41no—
67C22×Dic17C22⋊C411yescharacteristic
68C22×Dic17C22⋊C411yescharacteristic
69C22×Dic17C2×D412yes—
70C2×#3624—44no—
71C2×#3624—84no—
72C42×C17—21no—
73C42×C17C2×C812yescomplemented
74C22⋊Dic17—42no—
75C22⋊Dic17—44no—
76C22⋊Dic17—44no—
77C22⋊Dic17—42no—
78C4⋊Dic17—41no—
79C4⋊Dic17—44no—
80C4⋊Dic17—41no—
81C4×Dic17—24no—
82C4×Dic17—21no—
83C4×Dic17—21no—
84C4×Dic17—42no—
85C4×Dic17—41no—
86#11475—171nomaximal, Sylow
87C23×C17—22no—
88C23×C17—22no—
89C23×C17—22no—
90C23×C17—42no—
91C23×C17—42no—
92C23×C17—42no—
93C23×C17—42no—
94C23×C17—41no—
95C23×C17—41no—
96C23×C17C22⋊C814yes—
97C23×C17C2×C22⋊C411yescharacteristic
98D4×C17—42no—
99D4×C17—42no—
100D4×C17—42no—
101D4×C17—416no—
102D4×C17—42no—
103C2×C4×C17—22no—
104C2×C4×C17—22no—
105C2×C4×C17—22no—
106C2×C4×C17—21no—
107C2×C4×C17—21no—
108C2×C4×C17C22×C811yescharacteristic
109C2×C4×C17C2×M1611yescharacteristic
110C2×Dic17—22no—
111C2×Dic17—22no—
112C2×Dic17—21no—
113C2×Dic17—22no—
114C2×Dic17—41no—
115C2×Dic17—41no—
116C2×Dic17—44no—
117C2×Dic17—42no—
118C2×Dic17—41no—
119C2×Dic17—42no—
120C2×Dic17—41no—
121C2×Dic17—82no—
122C2×Dic17—82no—
123C2×Dic17—82no—
124C2×Dic17#9911yescharacteristic
125C2×Dic17#9911yescharacteristic
126#3624—84no—
127C4×#127—171no—
128#1459—172no—
129#1325—172no—
130#1318—172no—
131C22×C17—21no—
132C22×C17—21no—
133C22×C17—24no—
134C22×C17—44no—
135C22×C17—44no—
136C22×C17—42no—
137C22×C17—42no—
138C22×C17—42no—
139C22×C17—82no—
140C22×C17—82no—
141C22×C17—81no—
142C22×C17—81no—
143C22×C17C2×C22⋊C811yescharacteristic
144C22×C17C2×#9911yescharacteristic
145C22×C17#41311yescharacteristic
146C4×C17—24no—
147C4×C17—21no—
148C4×C17—21no—
149Dic17—21no—
150Dic17—42no—
151Dic17—42no—
152Dic17—41no—
153Dic17—81no—
154Dic17#35312yes—
155C2×#127#59511yescharacteristic
156C2×C4×D4—171no—
157C2×C4×D4—171no—
158C2×C4×D4—342no—
159C2×C22⋊C8—342no—
160#390—171no—
161#390—171no—
162#390—342no—
163#384—171no—
164C43—171no—
165#324—174no—
166#323—342no—
167C2×C17—21no—
168C2×C17—21no—
169C2×C17—44no—
170C2×C17—82no—
171C2×C17#145911yescharacteristic
172C2×C17C2×#35311yescharacteristic
173C2×C17#213211yescharacteristic
174C22×D4—22no—
175C22×D4C2×#59511yescharacteristic
176C22×D4C2×#59511yescharacteristic
177C23×C4—172no—
178C23×C4—341no—
179C22×C8—682no—
180#127—22no—
181#127#362414yescomplemented
182C4×D4—681no—
183C4×D4—681no—
184C4×D4—688no—
185C4×D4—681no—
186C4×D4—681no—
187C2×C4⋊C4—171no—
188C2×C4⋊C4—171no—
189C2×C4⋊C4—342no—
190C2×C22⋊C4—172no—
191C2×C22⋊C4—341no—
192C2×C22⋊C4—342no—
193C2×C22⋊C4—342no—
194C2×C42—171no—
195C2×C42—171no—
196C2×C42—341no—
197C2×C42—341no—
198C2×C42C2×#59511yescharacteristic
199C22⋊C8—344no—
200C22⋊C8—682no—
201C17#1147511yescharacteristic, complemented, Sylow
202C24—21no—
203C24C22⋊#59512yes—
204C2×D4—28no—
205C2×D4—21no—
206C2×D4—41no—
207C2×D4—41no—
208C2×D4—41no—
209C2×D4—41no—
210C2×D4—44no—
211C2×D4—44no—
212C2×D4C2×#362412yes—
213C2×D4C2×#362412yescomplemented
214C2×D4#6324712yes—
215C22×C4—21no—
216C22×C4—171no—
217C22×C4—171no—
218C22×C4—172no—
219C22×C4—342no—
220C22×C4—342no—
221C22×C4—341no—
222C22×C4—341no—
223C22×C4—681no—
224C22×C4—681no—
225C22×C4—684no—
226C22×C4—684no—
227C22×C4C22×#59511yescharacteristic
228C2×C8—684no—
229C2×C8—1364no—
230C4⋊C4—681no—
231C4⋊C4—681no—
232C4⋊C4—684no—
233C22⋊C4—682no—
234C22⋊C4—684no—
235C22⋊C4—684no—
236C22⋊C4—682no—
237C42—21no—
238C42—341no—
239C42—341no—
240C42—344no—
241C42—681no—
242C42—682no—
243C42C2×#362412yescomplemented
244C23—22no—
245C23—22no—
246C23—22no—
247C23—42no—
248C23—42no—
249C23—42no—
250C23—42no—
251C23—41no—
252C23—41no—
253C23C22⋊#362414yes—
254C23C2×C22⋊#59511yescharacteristic
255D4—42no—
256D4—42no—
257D4—42no—
258D4—416no—
259D4—42no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—171no—
266C2×C4—171no—
267C2×C4—342no—
268C2×C4—342no—
269C2×C4—341no—
270C2×C4—342no—
271C2×C4—684no—
272C2×C4—681no—
273C2×C4—681no—
274C2×C4—681no—
275C2×C4—682no—
276C2×C4—682no—
277C2×C4—681no—
278C2×C4—1362no—
279C2×C4—1362no—
280C2×C4—1362no—
281C2×C4C22×#362411yescharacteristic
282C2×C4C2×#6324711yescharacteristic
283C8—1364no—
284C22—21no—
285C22—21no—
286C22—24no—
287C22—44no—
288C22—44no—
289C22—42no—
290C22—42no—
291C22—42no—
292C22—82no—
293C22—82no—
294C22—81no—
295C22—81no—
296C22C2×C22⋊#362411yescharacteristic
297C22D17⋊#9911yescharacteristic
298C22#1176379611yescharacteristic
299C4—24no—
300C4—21no—
301C4—21no—
302C4—172no—
303C4—341no—
304C4—682no—
305C4—682no—
306C4—681no—
307C4—1361no—
308C2—21no—
309C2—21no—
310C2—44no—
311C2—82no—
312C2#1454402011yescharacteristic
313C2D17⋊#35311yescharacteristic
314C2#1454507611yescharacteristic
subgroupG/H
centerC22D17⋊#99
commutatorC2×C4×C17C22×C8
FrattiniC22×C4C22×#595
Fitting, prime radicalC2×C17×#127C4
socleC22×C17C2×#99
cosocleC2×C4⋊Dic17C23
hypercenterC2×#127#595
hypocenterC17#11475
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ C2×#127

Complements

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

NH ≅ G/Ntype
1. #14544020C2semidirect
2. #14543726C2semidirect
3. #14543712C2semidirect
24. C17×#127C8semidirect
55. C2×D4×C17C2×C8semidirect
73. C42×C17C2×C8semidirect
181. #127#3624semidirect
201. C17#11475semidirect
213. C2×D4C2×#3624semidirect
243. C42C2×#3624semidirect

Maximal subgroups

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

subgroupindex
#145440202
#145437262
#145437122
#127×Dic172
#1147517