Finite Group Databasebeta

← #14540652

Subgroups — #14540652

Order 2176.

420 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 123 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#11763474C211yesmaximal, characteristic, complemented
2#11763464C211yesmaximal, characteristic
3#11763159C211yesmaximal, characteristic, complemented
4#11763138C212yesmaximal, complemented
5#11763095C211yesmaximal, characteristic, complemented
6#11763035C212yesmaximal, complemented
7#11762946C212yesmaximal, complemented
8C4×#63229C211yesmaximal, characteristic, complemented
9#11762874C211yesmaximal, characteristic, complemented
10#103×D17C212yesmaximal, complemented
11C17×#461C211yesmaximal, characteristic, complemented
12C2×Q8×D17C2212yes—
13C2×#63252—21no—
14C2×SD16×C17C2212yes—
15C17×C4⋊Q8C2211yescharacteristic
16C4×D4×C17C2211yescharacteristic
17#10587632C2212yes—
18#10587624C2212yes—
19C2×#63229C2211yescharacteristic
20#10587610—21no—
21C2×C8×D17C2212yes—
22#10587582—22no—
23#10587579—21no—
24C4⋊C4×D17C2212yes—
25#10587574—21no—
26#10587569—22no—
27C4×D68C2211yescharacteristic
28#10587555C2211yescharacteristic
29C4×Dic34C2211yescharacteristic
30C17×C4⋊C8C2211yescharacteristic
31C17×#103C2212yes—
32#10587506C2212yes—
33#10587506C2212yes—
34#10587503C2211yescharacteristic
35#10587491C2212yes—
36#10587488C2212yes—
37#10587487C2212yes—
38#10587481C2211yescharacteristic
39#10587479C2211yescharacteristic
40C4×#3622C2211yescharacteristic
41C2×Q8×C17C2312yes—
42C2×D4×C17C2311yescharacteristic
43C22×C4×C17—21no—
44C2×C22⋊D17—22no—
45Q8×D17—24no—
46Q8×D17—42no—
47#63252—44no—
48C2×D68D411yescharacteristic
49C2×C4×D17—22no—
50C2×C4×D17—22no—
51C2×C4×D17C2312yes—
52C2×Dic34—21no—
53C2×Dic34—22no—
54C2×Dic34C2311yescharacteristic
55SD16×C17—42no—
56C2×C8×C17C2312yes—
57C4⋊C4×C17—21no—
58C4⋊C4×C17—21no—
59C4⋊C4×C17C2311yescharacteristic
60C22⋊C4×C17—21no—
61C42×C17C2311yescharacteristic
62C22⋊Dic17—22no—
63#63229—22no—
64#63227—22no—
65#63227D412yes—
66C4⋊Dic17—21no—
67C4⋊Dic17D411yescharacteristic
68C4⋊Dic17C2311yescharacteristic
69C4⋊Dic17C2312yes—
70#63225—21no—
71#63225—22no—
72#63225—22no—
73C4×Dic17—21no—
74C2×#3622C2311yescharacteristic
75C2×#3622C2311yescharacteristic
76C8×D17—42no—
77C23×C17—21no—
78C22×D17C2×D412yes—
79Q8×C17—22no—
80Q8×C17—22no—
81D4×C17—21no—
82D4×C17—21no—
83C2×C4×C17—21no—
84C2×C4×C17—22no—
85C2×C4×C17—42no—
86C2×C4×C17C2×D411yescharacteristic
87C2×C4×C17C2×D411yescharacteristic
88C2×C4×C17C2411yescharacteristic
89C22⋊D17—44no—
90C2×Dic17—21no—
91C2×Dic17—22no—
92C2×Dic17—21no—
93C2×Dic17C2×D412yes—
94D68SD1614yescomplemented
95C4×D17—22no—
96C4×D17—22no—
97C4×D17—44no—
98C4×D17—44no—
99Dic34—21no—
100Dic34—44no—
101Dic34—42no—
102Dic34#4112yescomplemented
103C8×C17—22no—
104#3622—21no—
105#3622—21no—
106#3298—171nomaximal, Sylow
107C22×C17—41no—
108C22×C17—41no—
109C22×C17—41no—
110C22×C17C22×D411yescharacteristic
111C2×D17—22no—
112C2×D17—22no—
113C2×D17C2×SD1614yes—
114C4×C17—21no—
115C4×C17—42no—
116C4×C17—41no—
117C4×C17C2×SD1612yes—
118C4×C17C2×#4111yescharacteristic
119C4×C17H(5,𝔽2)11yescharacteristic
120Dic17—22no—
121Dic17—22no—
122Dic17—42no—
123Dic17—41no—
124D4×Q8—171no—
125#547—171no—
126#487—171no—
127#483—172no—
128#475—171no—
129#465—172no—
130#461D1711yescharacteristic, complemented
131#448—172no—
132C4×SD16—171no—
133C8×D4—171no—
134C2×#103—172no—
135C2×C17—81no—
136C2×C17#54611yescharacteristic
137C2×C17C22×SD1611yescharacteristic
138C2×C17#57811yescharacteristic
139D17—24no—
140C2×#41—341no—
141C22×Q8—172no—
142C2×SD16—171no—
143C2×SD16C2×D1712yes—
144C22×C8—172no—
145C4⋊Q8—171no—
146C4⋊Q8—341no—
147C4⋊Q8C2×D1711yescharacteristic
148#123—342no—
149C22⋊Q8—172no—
150C22⋊Q8—342no—
151C22⋊Q8—341no—
152#121—172no—
153C4×Q8—171no—
154C4×D4—171no—
155C4×D4—341no—
156C4×D4C2×D1711yescharacteristic
157C2×C4⋊C4—172no—
158#106—172no—
159#106—171no—
160C4⋊C8C2×D1711yescharacteristic
161#103—172no—
162#103—172no—
163#103—172no—
164#103—171no—
165#103C2×D1712yes—
166#102—171no—
167C22⋊C8—172no—
168C4×C8—171no—
169C17#329811yescharacteristic, complemented, Sylow
170#41—684no—
171C2×Q8—171no—
172C2×Q8—342no—
173C2×Q8—341no—
174C2×Q8—344no—
175C2×Q8—682no—
176C2×Q8C22×D1712yes—
177C2×D4—171no—
178C2×D4—342no—
179C2×D4C22×D1711yescharacteristic
180C22×C4—21no—
181C22×C4—172no—
182C22×C4—342no—
183C22×C4—342no—
184SD16—42no—
185SD16—342no—
186C2×C8—171no—
187C2×C8—171no—
188C2×C8—682no—
189C2×C8C22×D1712yes—
190C4⋊C4—21no—
191C4⋊C4—21no—
192C4⋊C4—171no—
193C4⋊C4—171no—
194C4⋊C4—172no—
195C4⋊C4—342no—
196C4⋊C4—342no—
197C4⋊C4—341no—
198C4⋊C4—341no—
199C4⋊C4C22×D1711yescharacteristic
200C22⋊C4—21no—
201C22⋊C4—172no—
202C22⋊C4—342no—
203C22⋊C4—342no—
204C42—341no—
205C42C22×D1711yescharacteristic
206C23—21no—
207C23—172no—
208Q8—22no—
209Q8—22no—
210Q8—172no—
211Q8—341no—
212Q8—684no—
213Q8—682no—
214D4—21no—
215D4—21no—
216D4—174no—
217D4—684no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—42no—
221C2×C4—172no—
222C2×C4—341no—
223C2×C4—342no—
224C2×C4—342no—
225C2×C4—341no—
226C2×C4—342no—
227C2×C4—684no—
228C2×C4—684no—
229C2×C4D4×D1711yescharacteristic
230C2×C4D4×D1711yescharacteristic
231C2×C4C23×D1711yescharacteristic
232C8—22no—
233C8—341no—
234C8—341no—
235C22—41no—
236C22—41no—
237C22—41no—
238C22—174no—
239C22—342no—
240C22—342no—
241C22C2×D4×D1711yescharacteristic
242C4—21no—
243C4—42no—
244C4—41no—
245C4—342no—
246C4—342no—
247C4—682no—
248C4—681no—
249C4SD16×D1712yes—
250C4#1058769111yescharacteristic
251C4#41×D1711yescharacteristic
252C2—81no—
253C2—344no—
254C2#1176345611yescharacteristic
255C2C2×SD16×D1711yescharacteristic
256C2#1176362511yescharacteristic
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#461C2
socleC22×C17C22×D4
prime radical#11763464C2
hypercenter#461D17
hypocenterC17#3298
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. #11763474C2semidirect
3. #11763159C2semidirect
4. #11763138C2semidirect
5. #11763095C2semidirect
6. #11763035C2semidirect
7. #11762946C2semidirect
8. C4×#63229C2semidirect
9. #11762874C2semidirect
10. #103×D17C2semidirect
11. C17×#461C2semidirect
94. D68SD16semidirect
102. Dic34#41semidirect
130. #461D17semidirect
169. C17#3298semidirect

Maximal subgroups

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

subgroupindex
#117634742
#117634642
#117631592
#117631382
#117630952
#117630352
#117629462
C4×#632292
#117628742
#103×D172
C17×#4612
#329817