Finite Group Databasebeta

← #17949759

Subgroups — #17949759

Order 4352.

566 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 566 conjugacy classes of subgroups — too many to draw. Only its 132 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#14544674C211yesmaximal, characteristic
2#14544674C211yesmaximal, characteristic
3#14543602C214yesmaximal, complemented
4C2×#11762708C211yesmaximal, characteristic
5C2×C22⋊#3624—21no—
6C2×C22⋊#3624—21no—
7#11763675—21no—
8#11763675—21no—
9#11763675C2212yes—
10#11763675C2212yes—
11C22×C4⋊Dic17—21no—
12C22×C4⋊Dic17C411yescharacteristic
13C22×C4×Dic17C2211yescharacteristic
14#11762708—21no—
15#11762708—21no—
16#11762708C412yes—
17#11762708C2212yes—
18C2×#10587502—21no—
19C22×C4⋊C4×C17C411yescharacteristic
20C23×C4×C17—21no—
21C23×C4×C17C2×C411yescharacteristic
22C23×Dic17—21no—
23C23×Dic17D411yescharacteristic
24C23×Dic17D411yescharacteristic
25C22×#3624—22no—
26C22×#3624—22no—
27C2×C4⋊C4×C17—21no—
28C2×C4⋊C4×C17—22no—
29C2×C4⋊C4×C17C814yescomplemented
30C2×C4⋊C4×C17C2×C412yes—
31C2×C4⋊Dic17—21no—
32C2×C4⋊Dic17—22no—
33C2×C4⋊Dic17—22no—
34C2×C4⋊Dic17—44no—
35C2×C4⋊Dic17—42no—
36C2×C4⋊Dic17C2×C412yes—
37C2×C4×Dic17—42no—
38C2×C4×Dic17C2×C411yescharacteristic
39C2×C4×Dic17D411yescharacteristic
40C2×C4×Dic17D411yescharacteristic
41C2×C4×Dic17C2311yescharacteristic
42C22⋊#3624—42no—
43C22⋊#3624—42no—
44#10587502—22no—
45#10587502—22no—
46C24×C17C22⋊C411yescharacteristic
47C22×C4×C17—22no—
48C22×C4×C17—22no—
49C22×C4×C17—24no—
50C22×C4×C17—22no—
51C22×C4×C17—42no—
52C22×C4×C17—42no—
53C22×C4×C17C22⋊C411yescharacteristic
54C22×C4×C17C2×C812yes—
55C22×C4×C17M1612yes—
56C22×C4×C17C22×C411yescharacteristic
57C22×Dic17—22no—
58C22×Dic17—22no—
59C22×Dic17—22no—
60C22×Dic17—22no—
61C22×Dic17—22no—
62C22×Dic17—41no—
63C22×Dic17—41no—
64C22×Dic17—42no—
65C22×Dic17—42no—
66C22×Dic17—42no—
67C22×Dic17C22⋊C411yescharacteristic
68C22×Dic17C22⋊C411yescharacteristic
69C22×Dic17C2×D411yescharacteristic
70C22×Dic17C2×D411yescharacteristic
71C2×#3624—42no—
72C2×#3624—42no—
73C2×#3624—42no—
74C2×#3624—42no—
75C2×#3624—42no—
76C2×#3624—42no—
77C4⋊C4×C17—24no—
78C4⋊C4×C17—22no—
79C4⋊C4×C17C2×C814yes—
80C4⋊Dic17—44no—
81C4⋊Dic17—42no—
82C4⋊Dic17—44no—
83C4⋊Dic17—42no—
84C4×Dic17—41no—
85C4×Dic17—41no—
86C4×Dic17—41no—
87C4×Dic17—41no—
88#7593—171nomaximal, Sylow
89C23×C17—21no—
90C23×C17—21no—
91C23×C17—21no—
92C23×C17—21no—
93C23×C17C22⋊C812yes—
94C23×C17#9912yes—
95C23×C17#10112yes—
96C23×C17C2×C22⋊C411yescharacteristic
97C2×C4×C17—24no—
98C2×C4×C17—24no—
99C2×C4×C17—21no—
100C2×C4×C17—21no—
101C2×C4×C17—21no—
102C2×C4×C17—21no—
103C2×C4×C17—24no—
104C2×C4×C17—48no—
105C2×C4×C17—42no—
106C2×C4×C17—42no—
107C2×C4×C17C22⋊C812yes—
108C2×C4×C17C22×C811yescharacteristic
109C2×C4×C17C2×M1611yescharacteristic
110C2×Dic17—21no—
111C2×Dic17—21no—
112C2×Dic17—21no—
113C2×Dic17—21no—
114C2×Dic17—21no—
115C2×Dic17—21no—
116C2×Dic17—21no—
117C2×Dic17—21no—
118C2×Dic17—21no—
119C2×Dic17—21no—
120C2×Dic17—44no—
121C2×Dic17—44no—
122C2×Dic17—42no—
123C2×Dic17—42no—
124C2×Dic17—42no—
125C2×Dic17—84no—
126C2×Dic17AGL(1,ℤ/8)12yes—
127C2×Dic17#13712yes—
128#3624—82no—
129#3624—82no—
130C2×#389—171no—
131#1817—171no—
132#1817—171no—
133#1271—174no—
134C22×C17—22no—
135C22×C17—22no—
136C22×C17—22no—
137C22×C17—22no—
138C22×C17—22no—
139C22×C17—22no—
140C22×C17#32414yes—
141C22×C17C2×C22⋊C811yescharacteristic
142C22×C17C2×#9911yescharacteristic
143C22×C17C2×#10111yescharacteristic
144C22×C17#41811yescharacteristic
145C22×C17#41911yescharacteristic
146C22×C17#42111yescharacteristic
147C22×C17#42111yescharacteristic
148C4×C17—22no—
149C4×C17—21no—
150C4×C17—21no—
151C4×C17—44no—
152Dic17—22no—
153Dic17—22no—
154Dic17—41no—
155Dic17—41no—
156Dic17—82no—
157C22×C4⋊C4—171no—
158C22×C4⋊C4—341no—
159C22×C4⋊C4#59511yescharacteristic
160C22×C42—171no—
161C2×C22⋊C8—341no—
162C2×C22⋊C8—341no—
163#389—172no—
164#389—172no—
165#389—341no—
166#389—341no—
167C2×#95—341no—
168#336—172no—
169#336—172no—
170#336—341no—
171#336—341no—
172C2×C17—24no—
173C2×C17C2×#32411yescharacteristic
174C2×C17#147811yescharacteristic
175C2×C17#147911yescharacteristic
176C2×C17#151011yescharacteristic
177C2×C17#151111yescharacteristic
178C2×C17#153411yescharacteristic
179C2×C17#155611yescharacteristic
180C23×C4—21no—
181C23×C4—171no—
182C23×C4—171no—
183C23×C4—341no—
184C23×C4C2×#59511yescharacteristic
185C22×C8—342no—
186C22×C8—342no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—172no—
190C2×C4⋊C4—342no—
191C2×C4⋊C4—342no—
192C2×C4⋊C4—341no—
193C2×C4⋊C4—682no—
194C2×C4⋊C4—684no—
195C2×C4⋊C4#362414yescomplemented
196C2×C4⋊C4C2×#59512yes—
197C2×C42—171no—
198C2×C42—171no—
199C2×C42—171no—
200C2×C42—171no—
201C2×C42—682no—
202C22⋊C8—682no—
203C22⋊C8—682no—
204#95—342no—
205#95—342no—
206C17#759311yescharacteristic, complemented, Sylow
207C24C22⋊#59511yescharacteristic
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—24no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—171no—
215C22×C4—171no—
216C22×C4—171no—
217C22×C4—171no—
218C22×C4—342no—
219C22×C4—342no—
220C22×C4—342no—
221C22×C4—342no—
222C22×C4—342no—
223C22×C4—681no—
224C22×C4—681no—
225C22×C4—682no—
226C22×C4—682no—
227C22×C4—682no—
228C22×C4C2×#362412yes—
229C22×C4#6324712yes—
230C22×C4C22⋊#59511yescharacteristic
231C22×C4C22×#59511yescharacteristic
232C2×C8—682no—
233C2×C8—682no—
234C2×C8—682no—
235C2×C8—682no—
236C2×C8—682no—
237C2×C8—682no—
238C4⋊C4—22no—
239C4⋊C4—24no—
240C4⋊C4—682no—
241C4⋊C4—682no—
242C4⋊C4—684no—
243C4⋊C4—684no—
244C4⋊C4C2×#362414yes—
245C42—681no—
246C42—681no—
247C42—681no—
248C42—681no—
249C23—21no—
250C23—21no—
251C23—21no—
252C23—21no—
253C23#1058753912yes—
254C23#1058754512yes—
255C23C22⋊#362412yes—
256C23C2×C22⋊#59511yescharacteristic
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—24no—
262C2×C4—24no—
263C2×C4—24no—
264C2×C4—48no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—172no—
268C2×C4—172no—
269C2×C4—341no—
270C2×C4—341no—
271C2×C4—341no—
272C2×C4—341no—
273C2×C4—341no—
274C2×C4—341no—
275C2×C4—341no—
276C2×C4—341no—
277C2×C4—341no—
278C2×C4—341no—
279C2×C4—684no—
280C2×C4—684no—
281C2×C4—682no—
282C2×C4—682no—
283C2×C4—682no—
284C2×C4—1364no—
285C2×C4C22⋊#362412yes—
286C2×C4C22×#362411yescharacteristic
287C2×C4C2×#6324711yescharacteristic
288C8—1362no—
289C8—1362no—
290C22—22no—
291C22—22no—
292C22—22no—
293C22—22no—
294C22—22no—
295C22—22no—
296C22#1176365414yes—
297C22C2×C22⋊#362411yescharacteristic
298C22C2×#1058753911yescharacteristic
299C22C2×#1058754511yescharacteristic
300C22#1176380111yescharacteristic
301C22#1176380211yescharacteristic
302C22#1176380711yescharacteristic
303C22#1176380811yescharacteristic
304C4—21no—
305C4—21no—
306C4—22no—
307C4—44no—
308C4—342no—
309C4—342no—
310C4—681no—
311C4—681no—
312C4—1362no—
313C2—24no—
314C2C2×#1176365411yescharacteristic
315C2#1454408311yescharacteristic
316C2#1454408611yescharacteristic
317C2#1454419111yescharacteristic
318C2#1454419211yescharacteristic
319C2#1454426611yescharacteristic
320C2#1454431311yescharacteristic
subgroupG/H
centerC23C2×C22⋊#595
commutatorC2×C4×C17C22×C8
FrattiniC22×C4C22×#595
FittingC22×C4⋊C4×C17C4
socleC23×C17C2×C22⋊C4
prime radicalC24×C17C22⋊C4
cosocleC2×C4×Dic17C23
hypercenterC22×C4⋊C4#595
hypocenterC17#7593
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. #14543602C2semidirect
29. C2×C4⋊C4×C17C8semidirect
195. C2×C4⋊C4#3624semidirect
206. C17#7593semidirect

Maximal subgroups

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

subgroupindex
#145446742
#145436022
C2×#117627082
#759317