Finite Group Databasebeta

← #16948019

Subgroups — #16948019

Order 4352.

312 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 69 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#14541652C211yesmaximal, characteristic, complemented
2#14531007C211yesmaximal, characteristic, complemented
3#14530816C211yesmaximal, characteristic
4#14526843C211yesmaximal, characteristic, complemented
5Q8⋊#63214C211yesmaximal, characteristic, complemented
6#14526744C211yesmaximal, characteristic, complemented
7C17×#1668C211yesmaximal, characteristic, complemented
8#11763519—21no—
9C2×#10587497—21no—
10#11763214C2211yescharacteristic
11#11763211—21no—
12#11763110—22no—
13#11763100—21no—
14C4×D136—21no—
15C4×D136C2211yescharacteristic
16C4⋊#63214C2211yescharacteristic
17#11762610—21no—
18#11762610—22no—
19C4×#63214C2211yescharacteristic
20C17×#493C2211yescharacteristic
21C17×#460—21no—
22C8×Q8×C17C2211yescharacteristic
23C17×#335C2211yescharacteristic
24C17×#331—21no—
25C2×SD16×C17—41no—
26C2×D8×C17—21no—
27C2×D8×C17—21no—
28C17×#127—21no—
29C4×Q8×C17D411yescharacteristic
30#10587632—41no—
31C2×D136D411yescharacteristic
32#10587582—42no—
33C4⋊C4×D17—41no—
34C4×D68—22no—
35C4×D68—21no—
36#10587556—22no—
37#10587555—21no—
38C17×C4⋊C8—21no—
39C17×C4⋊C8—21no—
40C17×C4⋊C8D411yescharacteristic
41C17×#102—41no—
42C4×C8×C17—21no—
43C4×C8×C17C2311yescharacteristic
44#10587497—42no—
45#10587492—22no—
46#10587492—21no—
47#10587489—21no—
48#10587489D411yescharacteristic
49#10587489D412yes—
50#10587488—22no—
51C2×#63214—21no—
52C2×#63214—21no—
53C2×Q8×C17D811yescharacteristic
54C2×D4×C17—41no—
55C2×D4×C17—41no—
56C2×D4×C17—41no—
57C2×D68—21no—
58C2×C4×D17—41no—
59C2×C4×D17—42no—
60C2×Dic34—41no—
61SD16×C17—82no—
62D8×C17—42no—
63D8×C17—41no—
64D8×C17—41no—
65C2×C8×C17—21no—
66C2×C8×C17—41no—
67C2×C8×C17C2×D411yescharacteristic
68C2×C8×C17C2×D411yescharacteristic
69C4⋊C4×C17—21no—
70C4⋊C4×C17D811yescharacteristic
71C42×C17—21no—
72C42×C17C2×D411yescharacteristic
73#63227—42no—
74#63227—41no—
75C4⋊Dic17—21no—
76C4⋊Dic17—22no—
77C4⋊Dic17—41no—
78#63225—42no—
79#63225—42no—
80D136—21no—
81D136SD1612yescomplemented
82#63214—41no—
83#63214—41no—
84#12355—171nomaximal, Sylow
85C23×C17—81no—
86C22×D17—41no—
87Q8×C17—21no—
88Q8×C17—21no—
89D4×C17—81no—
90D4×C17—81no—
91D4×C17—81no—
92D4×C17—81no—
93D4×C17—161no—
94C2×C4×C17—21no—
95C2×C4×C17—21no—
96C2×C4×C17#12011yescharacteristic
97C2×C4×C17C2×D811yescharacteristic
98C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
99C2×Dic17—42no—
100C2×Dic17—41no—
101D68—22no—
102D68—41no—
103C4×D17—81no—
104C4×D17—81no—
105C4×D17—162no—
106Dic34—161no—
107C8×C17—21no—
108C8×C17—41no—
109C8×C17—81no—
110C8×C17C2×SD1611yescharacteristic
111C8×C17#13711yescharacteristic
112#3380—171no—
113#2240—171no—
114#2206—171no—
115#1668D1711yescharacteristic, complemented
116#1365—171no—
117Q8⋊C16—171no—
118#1331—171no—
119C22×C17—161no—
120C22×C17—161no—
121C22×C17—161no—
122C22×C17#44711yescharacteristic
123C2×D17—42no—
124C2×D17—81no—
125C2×D17—81no—
126C4×C17—21no—
127C4×C17—41no—
128C4×C17—41no—
129C4×C17#44811yescharacteristic
130C4×C17#45411yescharacteristic
131C4×C17#50811yescharacteristic
132C4×C17#50911yescharacteristic
133Dic17—81no—
134Dic17—82no—
135#554—341no—
136C2×D16—341no—
137#500—171no—
138#499—341no—
139#493C2×D1711yescharacteristic
140#478—342no—
141#476—341no—
142#460—21no—
143C8×Q8C2×D1711yescharacteristic
144C4×D8—171no—
145C4×D8—341no—
146C4⋊C16—171no—
147#357—341no—
148#357—342no—
149C4×C16—171no—
150#335C2×D1711yescharacteristic
151#331—21no—
152C2×C17—321no—
153C2×C17#162711yescharacteristic
154C2×C17#218611yescharacteristic
155C2×C17#219211yescharacteristic
156D17—82no—
157C2×SD16—41no—
158C2×D8—21no—
159C2×D8—21no—
160C2×D8—171no—
161C4⋊Q8—341no—
162#127—21no—
163#125—342no—
164C22⋊Q8—682no—
165C22⋊Q8—681no—
166C4×Q8C22⋊D1711yescharacteristic
167C4×D4—342no—
168C4×D4—341no—
169C2×C4⋊C4—681no—
170D16—682no—
171C2×C16—341no—
172C2×C16—341no—
173#107—171no—
174#107—172no—
175#107—341no—
176#106—342no—
177C4⋊C8—21no—
178C4⋊C8—21no—
179C4⋊C8C22⋊D1711yescharacteristic
180#102—41no—
181#102—342no—
182#102—341no—
183C4×C8—21no—
184C4×C8C22×D1711yescharacteristic
185C17#1235511yescharacteristic, complemented, Sylow
186C2×Q8—681no—
187C2×Q8#6322811yescharacteristic
188C2×D4—41no—
189C2×D4—41no—
190C2×D4—41no—
191C2×D4—341no—
192C22×C4—682no—
193C22×C4—681no—
194SD16—82no—
195D8—42no—
196D8—41no—
197D8—41no—
198D8—172no—
199D8—341no—
200C2×C8—21no—
201C2×C8—41no—
202C2×C8D4×D1711yescharacteristic
203C2×C8D4×D1711yescharacteristic
204C4⋊C4—21no—
205C4⋊C4—341no—
206C4⋊C4—342no—
207C4⋊C4—682no—
208C4⋊C4—682no—
209C4⋊C4—681no—
210C4⋊C4#6322811yescharacteristic
211C22⋊C4—682no—
212C22⋊C4—681no—
213C42—21no—
214C42C2×C22⋊D1711yescharacteristic
215C16—681no—
216C16—681no—
217C23—81no—
218C23—681no—
219Q8—21no—
220Q8—21no—
221Q8—2721no—
222D4—81no—
223D4—81no—
224D4—81no—
225D4—81no—
226D4—161no—
227D4—342no—
228D4—681no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—2722no—
232C2×C4—681no—
233C2×C4—682no—
234C2×C4—1361no—
235C2×C4—1361no—
236C2×C4C2×#6322811yescharacteristic
237C2×C4#1058762311yescharacteristic
238C2×C4#1058763511yescharacteristic
239C8—21no—
240C8—41no—
241C8—81no—
242C8SD16×D1711yescharacteristic
243C8#1058760111yescharacteristic
244C22—161no—
245C22—161no—
246C22—161no—
247C22—682no—
248C22—1361no—
249C22—1361no—
250C22#1176294211yescharacteristic
251C4—21no—
252C4—41no—
253C4—41no—
254C4—1361no—
255C4—1362no—
256C4#1176294611yescharacteristic
257C4#1176298011yescharacteristic
258C4#1176323911yescharacteristic
259C4#1176324511yescharacteristic
260C2—321no—
261C2—1362no—
262C2#1452788511yescharacteristic
263C2#1453070511yescharacteristic
264C2#1453073911yescharacteristic
subgroupG/H
centerC22#11762942
commutator, cosocleC4×C8×C17C23
FrattiniC4×C8C22×D17
FittingC17×#1668C2
socleC22×C17#447
prime radical#14530816C2
hypercenter#1668D17
hypocenterC17#12355
LCSG ▷ C4×C8×C17 ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ C8×Q8 ◁ #1668

Complements

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

NH ≅ G/Ntype
1. #14541652C2semidirect
2. #14531007C2semidirect
4. #14526843C2semidirect
5. Q8⋊#63214C2semidirect
6. #14526744C2semidirect
7. C17×#1668C2semidirect
81. D136SD16semidirect
115. #1668D17semidirect
185. C17#12355semidirect

Maximal subgroups

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

subgroupindex
#145416522
#145310072
#145308162
#145268432
Q8⋊#632142
#145267442
C17×#16682
#1235517