Finite Group Databasebeta

← #14540004

Subgroups — #14540004

Order 2176.

362 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 115 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
1C4⋊Q8×D17C211yesmaximal, characteristic, complemented
2#11763341C211yesmaximal, characteristic
3#11763216C212yesmaximal, complemented
4#11763167C212yesmaximal, complemented
5#11763141C212yesmaximal, complemented
6#11763121C212yesmaximal
7#11763105C211yesmaximal, characteristic, complemented
8#11763101C211yesmaximal, characteristic, complemented
9#11763099C211yesmaximal, characteristic, complemented
10#11762865C211yesmaximal, characteristic, complemented
11C17×#476C211yesmaximal, characteristic, complemented
12C2×Q8×D17—21no—
13C17×C4⋊Q8C2211yescharacteristic
14C4×D4×C17C2211yescharacteristic
15#10587630—21no—
16#10587624C2211yescharacteristic
17#10587622C2211yescharacteristic
18C4×C22⋊D17—21no—
19#10587612—21no—
20#10587583C2211yescharacteristic
21C4⋊C4×D17—21no—
22C4⋊C4×D17C2212yes—
23#10587576C2211yescharacteristic
24#10587574—21no—
25#10587574C2212yes—
26#10587569—21no—
27#10587565—21no—
28C42×D17C2211yescharacteristic
29#10587555C2211yescharacteristic
30C4×Dic34C2211yescharacteristic
31C17×#106C2212yes—
32C17×C4⋊C8C2211yescharacteristic
33C17×#102C2212yes—
34#10587503C2212yes—
35#10587491C2212yes—
36#10587489C2212yes—
37#10587487C2212yes—
38#10587486C2212yes—
39#10587481C2212yes—
40#10587479C2212yes—
41#10587478C2212yes—
42C4⋊#3622C2211yescharacteristic
43C2×Q8×C17—21no—
44C2×D4×C17C2311yescharacteristic
45C22×C4×C17—21no—
46C2×C22⋊D17—21no—
47Q8×D17—44no—
48C2×C4×D17—22no—
49C2×C4×D17D411yescharacteristic
50C2×C4×D17D411yescharacteristic
51C2×C4×D17C2311yescharacteristic
52C2×Dic34—21no—
53C2×Dic34—22no—
54C2×Dic34C2311yescharacteristic
55C2×C8×C17C2312yes—
56C4⋊C4×C17—21no—
57C4⋊C4×C17C2312yes—
58C4⋊C4×C17C2311yescharacteristic
59C22⋊C4×C17—21no—
60C42×C17C2311yescharacteristic
61C22⋊Dic17—21no—
62C22⋊Dic17—21no—
63C22⋊Dic17—21no—
64#63227—21no—
65C4⋊Dic17—21no—
66C4⋊Dic17—21no—
67C4⋊Dic17—21no—
68C4⋊Dic17C2311yescharacteristic
69C4⋊Dic17C2312yes—
70#63225—21no—
71#63225—21no—
72#63225—22no—
73#63225—22no—
74C4×Dic17—21no—
75C4×Dic17D411yescharacteristic
76C4×Dic17D411yescharacteristic
77C4×Dic17C2311yescharacteristic
78C2×#3622C2312yes—
79C23×C17—21no—
80C22×D17C2×D411yescharacteristic
81Q8×C17—42no—
82D4×C17—21no—
83D4×C17—21no—
84C2×C4×C17—21no—
85C2×C4×C17—22no—
86C2×C4×C17—42no—
87C2×C4×C17C2×D411yescharacteristic
88C2×C4×C17C2×D411yescharacteristic
89C2×C4×C17C2411yescharacteristic
90C22⋊D17—42no—
91C2×Dic17—21no—
92C2×Dic17—21no—
93C2×Dic17—22no—
94C2×Dic17C2×D411yescharacteristic
95C2×Dic17C2×D411yescharacteristic
96C2×Dic17C2×D411yescharacteristic
97C4×D17—22no—
98C4×D17—22no—
99C4×D17—21no—
100C4×D17—21no—
101C4×D17—44no—
102Dic34—21no—
103Dic34—21no—
104Dic34—44no—
105Dic34—42no—
106C8×C17—22no—
107#3622—22no—
108#3238—171nomaximal, Sylow
109C22×C17—41no—
110C22×C17—41no—
111C22×C17—41no—
112C22×C17C22×D411yescharacteristic
113C2×D17—21no—
114C2×D17—21no—
115C2×D17AGL(1,ℤ/8)12yes—
116C4×C17—21no—
117C4×C17—42no—
118C4×C17—41no—
119C4×C17#13712yes—
120C4×C172−1+411yescharacteristic
121C4×C172−1+411yescharacteristic
122Dic17—21no—
123Dic17—21no—
124Dic17—21no—
125Dic17—41no—
126Dic17—42no—
127Dic17—41no—
128C2×C4⋊Q8—171no—
129#529—171no—
130#501—172no—
131#489—172no—
132#483—172no—
133#480—172no—
134#477—171no—
135#477—171no—
136#476—171no—
137#476D1711yescharacteristic, complemented
138#432—171no—
139C2×C17—81no—
140C2×C17#53611yescharacteristic
141C2×C17C2×AGL(1,ℤ/8)11yescharacteristic
142C2×C17C2×#13711yescharacteristic
143D17—22no—
144C22×Q8—341no—
145C4⋊Q8—172no—
146C4⋊Q8—171no—
147C4⋊Q8—341no—
148C4⋊Q8—341no—
149C4⋊Q8C2×D1711yescharacteristic
150#126—341no—
151#125—171no—
152#124—171no—
153#123—341no—
154C22⋊Q8—171no—
155#121—171no—
156C4×Q8—171no—
157C4×D4—341no—
158C4×D4C2×D1711yescharacteristic
159#117—341no—
160C2×C4⋊C4—172no—
161C2×C4⋊C4—341no—
162C2×C42—171no—
163#107—172no—
164#107—172no—
165#106—172no—
166#106C2×D1712yes—
167C4⋊C8—171no—
168C4⋊C8C2×D1711yescharacteristic
169#103—172no—
170#103—172no—
171#102—172no—
172#102C2×D1712yes—
173C22⋊C8—172no—
174#97—172no—
175C17#323811yescharacteristic, complemented, Sylow
176C2×Q8—21no—
177C2×Q8—171no—
178C2×Q8—342no—
179C2×Q8—341no—
180C2×Q8—684no—
181C2×D4—341no—
182C2×D4C22×D1711yescharacteristic
183C22×C4—21no—
184C22×C4—171no—
185C22×C4—171no—
186C22×C4—171no—
187C22×C4—342no—
188C2×C8—172no—
189C2×C8C22×D1712yes—
190C4⋊C4—21no—
191C4⋊C4—172no—
192C4⋊C4—171no—
193C4⋊C4—342no—
194C4⋊C4—342no—
195C4⋊C4—341no—
196C4⋊C4—341no—
197C4⋊C4—341no—
198C4⋊C4—341no—
199C4⋊C4—341no—
200C4⋊C4C22×D1712yes—
201C4⋊C4C22×D1711yescharacteristic
202C22⋊C4—21no—
203C22⋊C4—341no—
204C22⋊C4—341no—
205C22⋊C4—341no—
206C22⋊C4—341no—
207C42—171no—
208C42—171no—
209C42—171no—
210C42—341no—
211C42C22×D1711yescharacteristic
212C23—21no—
213C23—171no—
214Q8—42no—
215Q8—341no—
216Q8—341no—
217Q8—682no—
218Q8—684no—
219D4—21no—
220D4—21no—
221D4—682no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—42no—
225C2×C4—171no—
226C2×C4—171no—
227C2×C4—171no—
228C2×C4—342no—
229C2×C4—341no—
230C2×C4—341no—
231C2×C4—342no—
232C2×C4—341no—
233C2×C4—341no—
234C2×C4—342no—
235C2×C4—684no—
236C2×C4D4×D1711yescharacteristic
237C2×C4D4×D1711yescharacteristic
238C2×C4C23×D1711yescharacteristic
239C8—22no—
240C8—342no—
241C22—41no—
242C22—41no—
243C22—41no—
244C22—172no—
245C22—341no—
246C22—341no—
247C22C2×D4×D1711yescharacteristic
248C4—21no—
249C4—42no—
250C4—41no—
251C4—341no—
252C4—341no—
253C4—341no—
254C4—682no—
255C4—681no—
256C4—681no—
257C4#1058760112yes—
258C4#1058769411yescharacteristic
259C4#1058769711yescharacteristic
260C2—81no—
261C2—342no—
262C2#1176338211yescharacteristic
263C2AGL(1,ℤ/8)×D1711yescharacteristic
264C2C2×#1058760111yescharacteristic
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#476C2
socleC22×C17C22×D4
prime radical#10587624C22
hypercenter#476D17
hypocenterC17#3238
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ #476

Complements

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

NH ≅ G/Ntype
1. C4⋊Q8×D17C2semidirect
3. #11763216C2semidirect
4. #11763167C2semidirect
5. #11763141C2semidirect
7. #11763105C2semidirect
8. #11763101C2semidirect
9. #11763099C2semidirect
10. #11762865C2semidirect
11. C17×#476C2semidirect
137. #476D17semidirect
175. C17#3238semidirect

Maximal subgroups

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

subgroupindex
C4⋊Q8×D172
#117633412
#117632162
#117631672
#117631412
#117631212
#117631052
#117631012
#117630992
#117628652
C17×#4762
#323817