Finite Group Databasebeta

← #16929645

Subgroups — #16929645

Order 4352.

530 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 530 conjugacy classes of subgroups — too many to draw. Only its 107 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#14539210C211yesmaximal, characteristic
2#14527376C211yesmaximal, characteristic, complemented
3#14527200C212yesmaximal, complemented
4#14526620C212yesmaximal, complemented
5C17×#2979C211yesmaximal, characteristic, complemented
6C2×#10587636—22no—
7#11763359C2211yescharacteristic
8#11763191—22no—
9#11763188C411yescharacteristic
10#11763185C411yescharacteristic
11#11763183C411yescharacteristic
12#11763181C411yescharacteristic
13C4×#63223C2212yes—
14C2×C4×#3622C2211yescharacteristic
15#11762549—22no—
16#11762536—22no—
17#11762536—22no—
18C17×#532C2211yescharacteristic
19C17×#436—22no—
20C8×D4×C17—22no—
21C17×#431C2212yes—
22C17×#408—22no—
23C2×#41×C17D412yes—
24C2×#63252—21no—
25C2×M16×C17—22no—
26C22×C8×C17—22no—
27C17×C4⋊Q8C2×C411yescharacteristic
28C17×#127C2×C411yescharacteristic
29C17×#124D412yes—
30C17×#121—22no—
31C4×D4×C17—22no—
32C2×C42×C17C2311yescharacteristic
33#10587636—44no—
34#10587636—44no—
35C2×#63231—22no—
36C2×#63230—22no—
37C2×#63229—22no—
38C2×#63228—22no—
39#10587616—22no—
40C2×#63223—22no—
41C22×#3622—21no—
42#10587561—21no—
43#10587560C2×C411yescharacteristic
44C4×D68—22no—
45#10587555C2×C411yescharacteristic
46C17×C4⋊C8D412yes—
47C17×C4⋊C8D412yes—
48C17×C22⋊C8—24no—
49C17×C22⋊C8—22no—
50C4×C8×C17—22no—
51#10587474—22no—
52C4×#3622—21no—
53C4×#3622—22no—
54C4×#3622C2×C411yescharacteristic
55C4×#3622C2×C411yescharacteristic
56#41×C17—42no—
57#41×C17—42no—
58C2×Q8×C17C22⋊C412yes—
59C2×D4×C17—22no—
60C2×D4×C17—22no—
61C2×D4×C17C22⋊C412yes—
62C22×C4×C17—22no—
63C22×C4×C17C2×D411yescharacteristic
64C22×C4×C17#4112yes—
65C2×C22⋊D17—41no—
66#63252—42no—
67#63252—42no—
68C2×D68—21no—
69C2×D68—22no—
70C2×C4×D17—41no—
71C2×Dic34—21no—
72M16×C17—82no—
73C2×C8×C17—22no—
74C2×C8×C17—22no—
75C2×C8×C17—82no—
76C4⋊C4×C17—22no—
77C22⋊C4×C17—24no—
78C42×C17C22×C411yescharacteristic
79C42×C17C2×D411yescharacteristic
80C42×C17C2×D412yes—
81#63231—44no—
82#63230—44no—
83#63229—44no—
84#63228—44no—
85#63227—42no—
86C4⋊Dic17—22no—
87#63223—24no—
88#63223—42no—
89C2×#3622—22no—
90C2×#3622—22no—
91C2×#3622—21no—
92C2×#3622—21no—
93C2×#3622—42no—
94#9069—171nomaximal, Sylow
95C23×C17—22no—
96C23×C17#12311yescharacteristic
97C22×D17—41no—
98Q8×C17—42no—
99D4×C17—42no—
100D4×C17—42no—
101D4×C17—42no—
102D4×C17—82no—
103C2×C4×C17—22no—
104C2×C4×C17—22no—
105C2×C4×C17—21no—
106C2×C4×C17—21no—
107C2×C4×C17—22no—
108C2×C4×C17—42no—
109C2×C4×C17—42no—
110C2×C4×C17C2×C22⋊C411yescharacteristic
111C2×C4×C17C4×D412yes—
112C2×C4×C17#12011yescharacteristic
113C2×C4×C17#12112yes—
114C22⋊D17—82no—
115C2×Dic17—41no—
116D68—42no—
117D68—42no—
118D68—42no—
119C4×D17—81no—
120C4×D17—81no—
121Dic34—42no—
122C8×C17—42no—
123C8×C17—42no—
124#3622—22no—
125#3622—22no—
126#3622—44no—
127#3147—171no—
128#2979D1711yescharacteristic, complemented
129#1525—171no—
130#1484—172no—
131#1278—172no—
132C22×C17—22no—
133C22×C17—21no—
134C22×C17—44no—
135C22×C17—42no—
136C22×C17#38611yescharacteristic
137C2×D17—82no—
138C2×D17—81no—
139C4×C17—22no—
140C4×C17—21no—
141C4×C17—42no—
142C4×C17#44312yes—
143C4×C17#44412yes—
144C4×C17#45311yescharacteristic
145C4×C17#45311yescharacteristic
146Dic17—81no—
147C2×#135—342no—
148#532—171no—
149#532C2×D1711yescharacteristic
150#495—342no—
151#494—171no—
152#493—171no—
153#492—171no—
154#492—171no—
155#436—22no—
156C8×D4—22no—
157#431C2×D1712yes—
158#408—22no—
159C4×M16—172no—
160C2×C4×C8—171no—
161#329—342no—
162D4⋊C8—342no—
163D4⋊C8—342no—
164C2×C17—41no—
165C2×C17—82no—
166C2×C17#187612yes—
167C2×C17#188711yescharacteristic
168D17—161no—
169C2×#41—341no—
170C2×#41C22⋊D1712yes—
171#135—684no—
172#135—684no—
173C2×Q16—342no—
174C2×SD16—342no—
175C2×SD16—342no—
176C2×D8—342no—
177C2×M16—22no—
178C2×M16—342no—
179C22×C8—22no—
180C22×C8—341no—
181C4⋊Q8—171no—
182C4⋊Q8C4×D1711yescharacteristic
183#127—171no—
184#127C4×D1711yescharacteristic
185#124—341no—
186#124D6812yes—
187#121—22no—
188#121—342no—
189C4×D4—22no—
190C4×D4—342no—
191C2×C42C22×D1711yescharacteristic
192C4⋊C8C22⋊D1712yes—
193C4⋊C8C22⋊D1712yes—
194C22⋊C8—24no—
195C22⋊C8—22no—
196#97—342no—
197C4×C8—22no—
198C4×C8—171no—
199C4×C8—171no—
200C4×C8—341no—
201C4×C8—342no—
202C17#906911yescharacteristic, complemented, Sylow
203#41—42no—
204#41—42no—
205#41—682no—
206#41—682no—
207C2×Q8—341no—
208C2×Q8#6322712yes—
209C2×D4—22no—
210C2×D4—22no—
211C2×D4—341no—
212C2×D4—342no—
213C2×D4—681no—
214C2×D4#6322712yes—
215C22×C4—22no—
216C22×C4—681no—
217C22×C4#6325212yes—
218C22×C4C2×C22⋊D1711yescharacteristic
219Q16—684no—
220SD16—684no—
221SD16—684no—
222D8—684no—
223M16—82no—
224M16—344no—
225M16—682no—
226C2×C8—22no—
227C2×C8—22no—
228C2×C8—82no—
229C2×C8—342no—
230C2×C8—342no—
231C2×C8—341no—
232C2×C8—341no—
233C2×C8—682no—
234C4⋊C4—22no—
235C4⋊C4—342no—
236C22⋊C4—24no—
237C22⋊C4—682no—
238C42C2×C4×D1711yescharacteristic
239C42C2×D6811yescharacteristic
240C42C2×C22⋊D1712yes—
241C23—22no—
242C23—681no—
243C23#1058761511yescharacteristic
244Q8—42no—
245Q8—682no—
246D4—42no—
247D4—42no—
248D4—42no—
249D4—82no—
250D4—682no—
251D4—682no—
252D4—682no—
253D4—1362no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—681no—
262C2×C4—1361no—
263C2×C4—1361no—
264C2×C4C2×#6322711yescharacteristic
265C2×C4C4×C22⋊D1712yes—
266C2×C4#1058761612yes—
267C2×C4#1058763911yescharacteristic
268C8—42no—
269C8—42no—
270C8—342no—
271C8—342no—
272C8—684no—
273C22—22no—
274C22—21no—
275C22—44no—
276C22—42no—
277C22—1362no—
278C22—1361no—
279C22#1176268811yescharacteristic
280C4—22no—
281C4—21no—
282C4—42no—
283C4—1361no—
284C4#1176292412yes—
285C4#1176292912yes—
286C4#1176297711yescharacteristic
287C4#1176297711yescharacteristic
288C2—41no—
289C2—82no—
290C2—2721no—
291C2#1452911112yes—
292C2#1452918211yescharacteristic
subgroupG/H
centerC2×C4#10587639
commutatorC42×C17C22×C4
FrattiniC2×C42C22×D17
FittingC17×#2979C2
socleC22×C17#386
prime radical#14539210C2
cosocleC2×C42×C17C23
hypercenter#2979D17
hypocenterC17#9069
LCSG ▷ C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2979

Complements

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

NH ≅ G/Ntype
2. #14527376C2semidirect
3. #14527200C2semidirect
4. #14526620C2semidirect
5. C17×#2979C2semidirect
128. #2979D17semidirect
202. C17#9069semidirect

Maximal subgroups

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

subgroupindex
#145392102
#145273762
#145272002
#145266202
C17×#29792
#906917