Finite Group Databasebeta

← #16935254

Subgroups — #16935254

Order 4352.

564 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 85 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#14538871C212yesmaximal, complemented
2#14529581C211yesmaximal, characteristic, complemented
3#14527395C212yesmaximal, complemented
4#14526654C211yesmaximal, characteristic, complemented
5C17×#3245C211yesmaximal, characteristic, complemented
6C2×C4×D68C2211yescharacteristic
7#11763125—22no—
8#11763096—22no—
9#11763001—22no—
10#11762942—22no—
11#11762919—24no—
12#11762866C2212yes—
13C2×#10587491—22no—
14C2×#10587474C2211yescharacteristic
15#11762549—24no—
16C17×#532C2212yes—
17C17×#474—22no—
18C17×#466—22no—
19C17×#459—22no—
20C2×C17×C4⋊C8C2211yescharacteristic
21C2×#41×C17—22no—
22C22×D68D411yescharacteristic
23C22×C4×D17—21no—
24C2×D8×C17—22no—
25C22×C8×C17D412yes—
26C17×C4⋊Q8D412yes—
27C17×#127D412yes—
28C17×#124—22no—
29C17×#121—22no—
30C17×#121—22no—
31C4×D4×C17—22no—
32C4×D4×C17—22no—
33C2×C42×C17C2311yescharacteristic
34C2×#63228—24no—
35C2×#63227—21no—
36C2×C4⋊Dic17D411yescharacteristic
37C22×#3622—21no—
38C4×D68—22no—
39C4×D68—22no—
40C17×#107—22no—
41C17×C4⋊C8—22no—
42C17×#102—24no—
43#10587503—24no—
44C22⋊#3622—22no—
45#10587491—44no—
46#10587480—24no—
47#10587479—24no—
48C4⋊#3622—22no—
49#10587474—22no—
50#10587474D414yes—
51C23×D17—21no—
52#41×C17—84no—
53C2×Q8×C17—22no—
54C2×D4×C17—22no—
55C2×D4×C17—22no—
56C2×D4×C17—22no—
57C2×D4×C17—42no—
58C22×C4×C17—42no—
59C22×C4×C17C2×D411yescharacteristic
60C22×C4×C17C2×D411yescharacteristic
61C22×C4×C17#4111yescharacteristic
62C22×Dic17—21no—
63C2×D68—22no—
64C2×D68—42no—
65C2×C4×D17—42no—
66C2×C4×D17—42no—
67C2×C4×D17—41no—
68C2×C4×D17—41no—
69D8×C17—82no—
70C2×C8×C17—22no—
71C2×C8×C17—42no—
72C4⋊C4×C17—22no—
73C4⋊C4×C17—22no—
74C22⋊C4×C17—42no—
75C22⋊C4×C17—42no—
76C42×C17C2×D412yes—
77C42×C17C2×D412yes—
78#63228—48no—
79#63227—42no—
80C4⋊Dic17—22no—
81C2×#3622—22no—
82C2×#3622—24no—
83C2×#3622—41no—
84C2×#3622—41no—
85#10026—171nomaximal, Sylow
86C23×C17—42no—
87C23×C17#12111yescharacteristic
88C22×D17—22no—
89C22×D17—42no—
90C22×D17—41no—
91C22×D17—41no—
92C22×D17—42no—
93Q8×C17—82no—
94D4×C17—44no—
95D4×C17—42no—
96D4×C17—42no—
97D4×C17—82no—
98D4×C17—84no—
99C2×C4×C17—22no—
100C2×C4×C17—21no—
101C2×C4×C17—21no—
102C2×C4×C17—41no—
103C2×C4×C17—42no—
104C2×C4×C17—84no—
105C2×C4×C17#12012yes—
106C2×C4×C17#12011yescharacteristic
107C2×C4×C17#12112yes—
108C2×C4×C17#12711yescharacteristic
109C2×Dic17—42no—
110C2×Dic17—41no—
111D68—24no—
112D68—42no—
113C4×D17—82no—
114C4×D17—81no—
115C4×D17—81no—
116C8×C17—42no—
117#3622—42no—
118#3622—44no—
119#3245D1711yescharacteristic, complemented
120#3109—172no—
121#1955—171no—
122#1530—172no—
123#1290—171no—
124C22×C17—21no—
125C22×C17—21no—
126C22×C17—82no—
127C22×C17—82no—
128C22×C17—82no—
129C22×C17#39211yescharacteristic
130C22×C17#47212yes—
131C2×D17—44no—
132C2×D17—42no—
133C2×D17—82no—
134C2×D17—81no—
135C2×D17—81no—
136C4×C17—21no—
137C4×C17—22no—
138C4×C17—22no—
139C4×C17—82no—
140C4×C17#45311yescharacteristic
141C4×C17#45411yescharacteristic
142Dic17—81no—
143#532C2×D1712yes—
144C2×C4×D4—171no—
145#481—342no—
146#476—342no—
147#474—22no—
148#466—22no—
149#459—22no—
150#459—342no—
151#447—342no—
152#442—344no—
153#432—172no—
154C2×C4⋊C8C2×D1711yescharacteristic
155C2×C22⋊C8—342no—
156C2×#97—171no—
157#329—344no—
158C2×C17—22no—
159C2×C17—162no—
160C2×C17#200611yescharacteristic
161C2×C17#201011yescharacteristic
162C2×C17#201811yescharacteristic
163D17—82no—
164C2×#41—22no—
165C22×D4—171no—
166C23×C4—341no—
167C2×D8—22no—
168C2×D8—344no—
169C22×C8—341no—
170C22×C8C22⋊D1712yes—
171C4⋊Q8C22⋊D1712yes—
172#127C22⋊D1712yes—
173#124—22no—
174#121—22no—
175#121—22no—
176C4×D4—22no—
177C4×D4—22no—
178C4×D4—342no—
179C4×D4—342no—
180C2×C4⋊C4—171no—
181C2×C22⋊C4—341no—
182C2×C42C22×D1711yescharacteristic
183#107—22no—
184#106—344no—
185C4⋊C8—22no—
186C4⋊C8—342no—
187#102—24no—
188#102—344no—
189#102—344no—
190C22⋊C8—342no—
191C22⋊C8—684no—
192#97—174no—
193#97—342no—
194C17#1002611yescharacteristic, complemented, Sylow
195C24—341no—
196#41—84no—
197C2×Q8—22no—
198C2×D4—22no—
199C2×D4—22no—
200C2×D4—22no—
201C2×D4—42no—
202C2×D4—342no—
203C2×D4—682no—
204C22×C4—42no—
205C22×C4—341no—
206C22×C4—682no—
207C22×C4—682no—
208C22×C4—681no—
209C22×C4—681no—
210C22×C4D4×D1711yescharacteristic
211C22×C4#6325411yescharacteristic
212C22×C4C2×C22⋊D1711yescharacteristic
213D8—82no—
214D8—688no—
215C2×C8—22no—
216C2×C8—42no—
217C2×C8—342no—
218C2×C8—344no—
219C2×C8—681no—
220C2×C8—681no—
221C4⋊C4—22no—
222C4⋊C4—22no—
223C4⋊C4—342no—
224C22⋊C4—42no—
225C22⋊C4—42no—
226C22⋊C4—682no—
227C42D4×D1712yes—
228C42C2×C22⋊D1712yes—
229C23—42no—
230C23—342no—
231C23—682no—
232C23—681no—
233C23—681no—
234C23—682no—
235C23#1058762411yescharacteristic
236Q8—82no—
237D4—44no—
238D4—42no—
239D4—42no—
240D4—82no—
241D4—84no—
242D4—344no—
243D4—682no—
244C2×C4—22no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—41no—
248C2×C4—42no—
249C2×C4—84no—
250C2×C4—682no—
251C2×C4—681no—
252C2×C4—1362no—
253C2×C4—1361no—
254C2×C4—1361no—
255C2×C4#1058762312yes—
256C2×C4#1058762512yes—
257C2×C4#1058762611yescharacteristic
258C2×C4#1058763911yescharacteristic
259C8—42no—
260C8—682no—
261C8—684no—
262C22—21no—
263C22—21no—
264C22—82no—
265C22—82no—
266C22—82no—
267C22—684no—
268C22—682no—
269C22—1362no—
270C22—1361no—
271C22—1361no—
272C22#1176271711yescharacteristic
273C22#1176307612yes—
274C4—21no—
275C4—22no—
276C4—22no—
277C4—82no—
278C4—1361no—
279C4#1176297711yescharacteristic
280C4#1176298211yescharacteristic
281C2—22no—
282C2—162no—
283C2—1362no—
284C2#1452982611yescharacteristic
285C2#1452984811yescharacteristic
286C2#1452989111yescharacteristic
subgroupG/H
centerC22#11762717
commutator, cosocleC2×C42×C17C23
FrattiniC2×C42C22×D17
FittingC17×#3245C2
socleC22×C17#392
hypercenter#3245D17
hypocenterC17#10026
LCSG ▷ C2×C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3245

Complements

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

NH ≅ G/Ntype
1. #14538871C2semidirect
2. #14529581C2semidirect
3. #14527395C2semidirect
4. #14526654C2semidirect
5. C17×#3245C2semidirect
119. #3245D17semidirect
194. C17#10026semidirect

Maximal subgroups

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

subgroupindex
#145388712
#145295812
#145273952
#145266542
C17×#32452
#1002617