Finite Group Databasebeta

← #16949864

Subgroups — #16949864

Order 4352.

986 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 986 conjugacy classes of subgroups — too many to draw. Only its 135 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#14530473C214yesmaximal, complemented
2#14528712C212yesmaximal, complemented
3C17×#3485C211yesmaximal, characteristic
4C2×#10587507C2214yescomplemented
5C2×#10587504—22no—
6#11762773—22no—
7#11762608—24no—
8#11762606—24no—
9C2×C17×H(5,𝔽2)C411yescharacteristic
10C22×#41×C17C2211yescharacteristic
11C17×#547—24no—
12C17×#546—24no—
13C17×#536C2212yes—
14C17×#534—22no—
15C17×#525C412yes—
16C2×C17×#123—22no—
17C17×#518C411yescharacteristic
18C17×H(5,𝔽2)—44no—
19C2×#41×C17—22no—
20C2×#41×C17—24no—
21C2×#41×C17—44no—
22C2×#41×C17C2×C412yes—
23C22×Q8×C17C2311yescharacteristic
24C22×D4×C17—22no—
25C22×D4×C17—22no—
26C22×D4×C17C2×C411yescharacteristic
27C22×D4×C17D412yes—
28C23×C4×C17C2×C411yescharacteristic
29C23×C4×C17D412yes—
30C17×C4⋊Q8D414yes—
31C17×#124D414yes—
32C17×#123—28no—
33C17×#123—24no—
34C17×#123—22no—
35C17×#123—24no—
36C17×C22⋊Q8—22no—
37C17×C22⋊Q8—24no—
38C17×#121—28no—
39C17×#121—22no—
40C17×#120—24no—
41C17×#120—22no—
42C4×D4×C17—24no—
43C4×D4×C17—24no—
44C17×#117C2×C412yes—
45C2×C4⋊C4×C17—24no—
46C2×C22⋊C4×C17—24no—
47C2×C22⋊C4×C17—22no—
48C2×#63223—24no—
49#10587507—28no—
50#10587504—44no—
51C22⋊#3622—44no—
52C24×C17—22no—
53C24×C17C22⋊C411yescharacteristic
54C24×C17C22⋊C412yes—
55#41×C17—44no—
56#41×C17—416no—
57#41×C17—48no—
58C2×Q8×C17—42no—
59C2×Q8×C17C22⋊C412yes—
60C2×Q8×C17C2×D414yes—
61C2×D4×C17—24no—
62C2×D4×C17—24no—
63C2×D4×C17—24no—
64C2×D4×C17—22no—
65C2×D4×C17—24no—
66C2×D4×C17—22no—
67C2×D4×C17—24no—
68C2×D4×C17—42no—
69C2×D4×C17—48no—
70C2×D4×C17—44no—
71C2×D4×C17—44no—
72C2×D4×C17C22⋊C412yes—
73C22×C4×C17—22no—
74C22×C4×C17—24no—
75C22×C4×C17—24no—
76C22×C4×C17—22no—
77C22×C4×C17—22no—
78C22×C4×C17—24no—
79C22×C4×C17—42no—
80C22×C4×C17—42no—
81C22×C4×C17C22⋊C411yescharacteristic
82C22×C4×C17C22×C411yescharacteristic
83C22×C4×C17C2×D412yes—
84C4⋊C4×C17—24no—
85C4⋊C4×C17—22no—
86C4⋊C4×C17—28no—
87C22⋊C4×C17—24no—
88C22⋊C4×C17—24no—
89C22⋊C4×C17—24no—
90C22⋊C4×C17—22no—
91C22⋊C4×C17—28no—
92C42×C17C22⋊C414yes—
93#63223—44no—
94#63223—44no—
95C2×#3622—44no—
96#12709—171nomaximal, Sylow
97C23×C17—22no—
98C23×C17—22no—
99C23×C17—22no—
100C23×C17—21no—
101C23×C17—22no—
102C23×C17—44no—
103C23×C17—82no—
104C23×C17—81no—
105C23×C17—84no—
106C23×C17#9914yes—
107C23×C17C2×C22⋊C411yescharacteristic
108Q8×C17—44no—
109Q8×C17—42no—
110D4×C17—28no—
111D4×C17—48no—
112D4×C17—42no—
113D4×C17—48no—
114D4×C17—44no—
115D4×C17—84no—
116C2×C4×C17—22no—
117C2×C4×C17—22no—
118C2×C4×C17—22no—
119C2×C4×C17—24no—
120C2×C4×C17—21no—
121C2×C4×C17—22no—
122C2×C4×C17—22no—
123C2×C4×C17—48no—
124C2×C4×C17—44no—
125C2×C4×C17—44no—
126C2×C4×C17—44no—
127C2×C4×C17—84no—
128C2×C4×C17C2×C22⋊C412yes—
129C2×C4×C17#12014yes—
130#3622—84no—
131#3485D1711yescharacteristic
132#2133—174no—
133#1785—172no—
134C22×C17—24no—
135C22×C17—41no—
136C22×C17—44no—
137C22×C17—44no—
138C22×C17—42no—
139C22×C17—44no—
140C22×C17—41no—
141C22×C17—42no—
142C22×C17—82no—
143C22×C17—82no—
144C22×C17—164no—
145C22×C17#37911yescharacteristic
146C22×C17C2×#9912yes—
147C4×C17—24no—
148C4×C17—22no—
149C4×C17—44no—
150C4×C17—41no—
151C4×C17—42no—
152C2×H(5,𝔽2)Dic1711yescharacteristic
153C22×#41C2×D1711yescharacteristic
154#547—24no—
155#546—24no—
156#536C2×D1712yes—
157#534—22no—
158#525Dic1712yes—
159C2×#123—22no—
160#518Dic1711yescharacteristic
161C2×#101—174no—
162C2×#100—342no—
163#407—342no—
164#356—344no—
165#355—344no—
166C2×C17—21no—
167C2×C17—22no—
168C2×C17—44no—
169C2×C17—81no—
170C2×C17—82no—
171C2×C17#178111yescharacteristic
172H(5,𝔽2)—44no—
173C2×#41—22no—
174C2×#41—24no—
175C2×#41—44no—
176C2×#41C2×Dic1712yes—
177C22×Q8C22×D1711yescharacteristic
178C22×D4—22no—
179C22×D4—22no—
180C22×D4C2×Dic1711yescharacteristic
181C22×D4C22⋊D1712yes—
182C23×C4C2×Dic1711yescharacteristic
183C23×C4C22⋊D1712yes—
184C2×M16—344no—
185C4⋊Q8C22⋊D1714yes—
186#124C22⋊D1714yes—
187#123—28no—
188#123—24no—
189#123—22no—
190#123—24no—
191C22⋊Q8—22no—
192C22⋊Q8—24no—
193#121—28no—
194#121—22no—
195#120—24no—
196#120—22no—
197C4×D4—24no—
198C4×D4—24no—
199#117C2×Dic1712yes—
200C2×C4⋊C4—24no—
201C2×C22⋊C4—24no—
202C2×C22⋊C4—22no—
203#101—348no—
204#100—684no—
205C22⋊C8—684no—
206C17#1270911yescharacteristic, complemented, Sylow
207C24—22no—
208C24C22⋊Dic1711yescharacteristic
209C24C22⋊Dic1712yes—
210#41—44no—
211#41—416no—
212#41—48no—
213C2×Q8—42no—
214C2×Q8C22⋊Dic1712yes—
215C2×Q8C2×C22⋊D1714yes—
216C2×D4—24no—
217C2×D4—24no—
218C2×D4—24no—
219C2×D4—22no—
220C2×D4—24no—
221C2×D4—22no—
222C2×D4—24no—
223C2×D4—42no—
224C2×D4—48no—
225C2×D4—44no—
226C2×D4—44no—
227C2×D4C22⋊Dic1712yes—
228C22×C4—22no—
229C22×C4—24no—
230C22×C4—24no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—24no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4C22⋊Dic1711yescharacteristic
237C22×C4C22×Dic1711yescharacteristic
238C22×C4C2×C22⋊D1712yes—
239M16—684no—
240M16—684no—
241C2×C8—684no—
242C4⋊C4—24no—
243C4⋊C4—22no—
244C4⋊C4—28no—
245C22⋊C4—24no—
246C22⋊C4—24no—
247C22⋊C4—24no—
248C22⋊C4—22no—
249C22⋊C4—28no—
250C42C22⋊Dic1714yes—
251C23—22no—
252C23—22no—
253C23—22no—
254C23—21no—
255C23—22no—
256C23—44no—
257C23—82no—
258C23—81no—
259C23—84no—
260C23#1058750514yes—
261C23C2×C22⋊Dic1711yescharacteristic
262Q8—44no—
263Q8—42no—
264D4—28no—
265D4—48no—
266D4—42no—
267D4—48no—
268D4—44no—
269D4—84no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—24no—
275C2×C4—21no—
276C2×C4—22no—
277C2×C4—48no—
278C2×C4—44no—
279C2×C4—44no—
280C2×C4—44no—
281C2×C4—84no—
282C2×C4C2×C22⋊Dic1712yes—
283C2×C4#1058763914yes—
284C8—1364no—
285C22—24no—
286C22—41no—
287C22—44no—
288C22—44no—
289C22—42no—
290C22—44no—
291C22—41no—
292C22—42no—
293C22—82no—
294C22—82no—
295C22—164no—
296C22#1176266511yescharacteristic
297C22C2×#1058750512yes—
298C4—24no—
299C4—22no—
300C4—44no—
301C4—41no—
302C4—42no—
303C2—21no—
304C2—22no—
305C2—44no—
306C2—81no—
307C2—82no—
308C2#1452869911yescharacteristic
subgroupG/H
centerC2#14528699
commutatorC22×C4×C17C22×C4
FrattiniC22×Q8C22×D17
Fitting, prime radicalC17×#3485C2
socleC2×C17#1781
cosocleC22×Q8×C17C23
hypercenter#3485D17
hypocenterC17#12709
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2 ◁ C23 ◁ C22×#41 ◁ #3485

Complements

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

NH ≅ G/Ntype
1. #14530473C2semidirect
2. #14528712C2semidirect
4. C2×#10587507C22semidirect
206. C17#12709semidirect

Maximal subgroups

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

subgroupindex
#145304732
#145287122
C17×#34852
#1270917