Finite Group Databasebeta

← #14539961

Subgroups — #14539961

Order 2176.

332 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 113 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#11763364C211yesmaximal, characteristic, complemented
2#11763341C211yesmaximal, characteristic
3#11763210C212yesmaximal, complemented
4#11763162C212yesmaximal, complemented
5#11763152C212yesmaximal, complemented
6#11763131C212yesmaximal
7#11763105C211yesmaximal, characteristic, complemented
8#11763099C211yesmaximal, characteristic, complemented
9#11763091C211yesmaximal, characteristic, complemented
10#11762861C211yesmaximal, characteristic, complemented
11C17×#474C211yesmaximal, characteristic, complemented
12C17×C4⋊Q8C2211yescharacteristic
13C4×D4×C17C2211yescharacteristic
14#10587632—21no—
15Q8×Dic17—21no—
16#10587624C2211yescharacteristic
17#10587622C2211yescharacteristic
18C4×C22⋊D17—21no—
19#10587612—21no—
20#10587583—21no—
21#10587583C2211yescharacteristic
22#10587578C2212yes—
23#10587576C2211yescharacteristic
24#10587576C2212yes—
25#10587573—21no—
26#10587569—21no—
27#10587565—21no—
28C42×D17C2211yescharacteristic
29#10587555C2211yescharacteristic
30C4×Dic34C2211yescharacteristic
31C17×#107C2212yes—
32C17×C4⋊C8C2211yescharacteristic
33C17×#102C2212yes—
34#10587503C2212yes—
35#10587491C2212yes—
36#10587489C2212yes—
37#10587487C2212yes—
38C8×Dic17C2212yes—
39#10587481C2212yes—
40#10587479C2212yes—
41#10587479C2212yes—
42C4⋊#3622C2211yescharacteristic
43C2×Q8×C17—21no—
44C2×D4×C17C2311yescharacteristic
45C22×C4×C17—21no—
46C2×C22⋊D17—21no—
47C2×C4×D17D411yescharacteristic
48C2×C4×D17D411yescharacteristic
49C2×C4×D17C2311yescharacteristic
50C2×Dic34—21no—
51C2×Dic34C2311yescharacteristic
52C2×C8×C17C2312yes—
53C4⋊C4×C17—21no—
54C4⋊C4×C17C2312yes—
55C4⋊C4×C17C2311yescharacteristic
56C22⋊C4×C17—21no—
57C42×C17C2311yescharacteristic
58C22⋊Dic17—21no—
59C22⋊Dic17—21no—
60C22⋊Dic17—21no—
61#63227—21no—
62#63227—22no—
63C4⋊Dic17—21no—
64C4⋊Dic17—21no—
65C4⋊Dic17—21no—
66C4⋊Dic17—22no—
67C4⋊Dic17C2311yescharacteristic
68C4⋊Dic17C2312yes—
69#63225—21no—
70#63225—21no—
71#63225—22no—
72C4×Dic17—21no—
73C4×Dic17—22no—
74C4×Dic17D411yescharacteristic
75C4×Dic17D411yescharacteristic
76C4×Dic17C2311yescharacteristic
77C2×#3622C2312yes—
78C23×C17—21no—
79C22×D17C2×D411yescharacteristic
80Q8×C17—42no—
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—42no—
90C2×Dic17—21no—
91C2×Dic17—21no—
92C2×Dic17—22no—
93C2×Dic17C2×D411yescharacteristic
94C2×Dic17C2×D411yescharacteristic
95C2×Dic17C2×D411yescharacteristic
96C4×D17—22no—
97C4×D17—21no—
98C4×D17—21no—
99C4×D17—41no—
100Dic34—21no—
101Dic34—21no—
102Dic34—42no—
103C8×C17—22no—
104#3622—22no—
105#3233—171nomaximal, Sylow
106C22×C17—41no—
107C22×C17—41no—
108C22×C17—41no—
109C22×C17C22×D411yescharacteristic
110C2×D17—21no—
111C2×D17—21no—
112C2×D17—21no—
113C4×C17—21no—
114C4×C17—41no—
115C4×C17—42no—
116C4×C17#13512yes—
117C4×C172−1+411yescharacteristic
118C4×C172−1+411yescharacteristic
119Dic17—21no—
120Dic17—21no—
121Dic17—21no—
122Dic17—41no—
123Dic17—42no—
124Dic17—41no—
125#533—171no—
126#529—171no—
127#499—172no—
128#488—172no—
129#485—172no—
130#482—172no—
131#477—171no—
132#476—171no—
133#475—171no—
134#474D1711yescharacteristic, complemented
135#431—171no—
136C2×C17—81no—
137C2×C17#53611yescharacteristic
138C2×C17C2×#13511yescharacteristic
139C2×C17#57511yescharacteristic
140D17—41no—
141C4⋊Q8—171no—
142C4⋊Q8C2×D1711yescharacteristic
143#126—341no—
144#125—171no—
145#125—172no—
146#124—171no—
147#123—341no—
148C22⋊Q8—171no—
149C22⋊Q8—341no—
150C22⋊Q8—341no—
151#121—171no—
152C4×Q8—171no—
153C4×Q8—341no—
154C4×Q8—341no—
155C4×D4—341no—
156C4×D4C2×D1711yescharacteristic
157#117—172no—
158#117—341no—
159C2×C42—171no—
160#107—172no—
161#107C2×D1712yes—
162#106—172no—
163#106—172no—
164C4⋊C8—171no—
165C4⋊C8C2×D1711yescharacteristic
166#103—172no—
167#103—172no—
168#102—172no—
169#102C2×D1712yes—
170C22⋊C8—172no—
171C4×C8—172no—
172C17#323311yescharacteristic, complemented, Sylow
173C2×Q8—21no—
174C2×Q8—171no—
175C2×Q8—341no—
176C2×D4—341no—
177C2×D4C22×D1711yescharacteristic
178C22×C4—21no—
179C22×C4—171no—
180C22×C4—171no—
181C22×C4—171no—
182C2×C8—172no—
183C2×C8C22×D1712yes—
184C4⋊C4—21no—
185C4⋊C4—172no—
186C4⋊C4—171no—
187C4⋊C4—341no—
188C4⋊C4—341no—
189C4⋊C4—341no—
190C4⋊C4—342no—
191C4⋊C4—342no—
192C4⋊C4—341no—
193C4⋊C4—341no—
194C4⋊C4C22×D1712yes—
195C4⋊C4C22×D1711yescharacteristic
196C22⋊C4—21no—
197C22⋊C4—342no—
198C22⋊C4—341no—
199C22⋊C4—341no—
200C22⋊C4—341no—
201C22⋊C4—341no—
202C42—171no—
203C42—171no—
204C42—171no—
205C42—342no—
206C42—341no—
207C42C22×D1711yescharacteristic
208C23—21no—
209C23—171no—
210Q8—42no—
211Q8—341no—
212Q8—341no—
213Q8—682no—
214D4—21no—
215D4—21no—
216D4—682no—
217C2×C4—22no—
218C2×C4—21no—
219C2×C4—42no—
220C2×C4—171no—
221C2×C4—171no—
222C2×C4—171no—
223C2×C4—341no—
224C2×C4—341no—
225C2×C4—342no—
226C2×C4—341no—
227C2×C4—341no—
228C2×C4—342no—
229C2×C4—681no—
230C2×C4D4×D1711yescharacteristic
231C2×C4D4×D1711yescharacteristic
232C2×C4C23×D1711yescharacteristic
233C8—22no—
234C8—342no—
235C22—41no—
236C22—41no—
237C22—41no—
238C22—341no—
239C22—341no—
240C22—341no—
241C22C2×D4×D1711yescharacteristic
242C4—21no—
243C4—41no—
244C4—42no—
245C4—341no—
246C4—341no—
247C4—341no—
248C4—681no—
249C4—682no—
250C4—681no—
251C4#1058759812yes—
252C4#1058769411yescharacteristic
253C4#1058769711yescharacteristic
254C2—81no—
255C2—681no—
256C2#1176338211yescharacteristic
257C2C2×#1058759811yescharacteristic
258C2#1176360811yescharacteristic
subgroupG/H
centerC22C2×D4×D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#474C2
socleC22×C17C22×D4
prime radical#10587624C22
hypercenter#474D17
hypocenterC17#3233
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ #474

Complements

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

NH ≅ G/Ntype
1. #11763364C2semidirect
3. #11763210C2semidirect
4. #11763162C2semidirect
5. #11763152C2semidirect
7. #11763105C2semidirect
8. #11763099C2semidirect
9. #11763091C2semidirect
10. #11762861C2semidirect
11. C17×#474C2semidirect
134. #474D17semidirect
172. C17#3233semidirect

Maximal subgroups

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

subgroupindex
#117633642
#117633412
#117632102
#117631622
#117631522
#117631312
#117631052
#117630992
#117630912
#117628612
C17×#4742
#323317