Finite Group Databasebeta

← #14532415

Subgroups — #14532415

Order 2176.

836 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 836 conjugacy classes of subgroups — too many to draw. Only its 169 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
1C22×#63254C211yesmaximal, characteristic
2C2×#10587630C211yesmaximal, characteristic, complemented
3C2×#10587632C211yesmaximal, characteristic, complemented
4#11762724C214yesmaximal, complemented
5#11762715C211yesmaximal, characteristic, complemented
6#11762712C211yesmaximal, characteristic, complemented
7#11762681C214yesmaximal, complemented
8C4×#63227C211yesmaximal, characteristic, complemented
9C2×C17×#124C211yesmaximal, characteristic, complemented
10C22×Q8×C17C2211yescharacteristic
11C22×D4×C17C2211yescharacteristic
12C22×C22⋊D17C2212yes—
13C23×Dic17C2212yes—
14C2×#63254—24no—
15C2×#63254—24no—
16C2×#63254—24no—
17C22×C4×D17C2211yescharacteristic
18C22×Dic34C2211yescharacteristic
19C17×#124—24no—
20C2×C22⋊C4×C17C2214yes—
21C2×C42×C17C2211yescharacteristic
22C2×C22⋊Dic17C2214yes—
23#10587632—24no—
24#10587630—24no—
25C2×#63227C2212yes—
26C2×#63227C2212yes—
27C2×C4⋊Dic17C2211yescharacteristic
28C2×#63225C2214yes—
29C2×#63225C2212yes—
30C2×C4×Dic17C2211yescharacteristic
31#10587502C2214yes—
32#10587502C2212yes—
33C24×C17C2312yes—
34C23×D17C2311yescharacteristic
35C2×Q8×C17—22no—
36C2×Q8×C17—22no—
37C2×Q8×C17—22no—
38C2×D4×C17—22no—
39C2×D4×C17—22no—
40C2×D4×C17D414yes—
41C22×C4×C17C2312yes—
42C22×C4×C17C2312yes—
43C22×C4×C17C2311yescharacteristic
44C2×C22⋊D17—24no—
45C2×C22⋊D17—24no—
46C2×C22⋊D17—24no—
47C22×Dic17—24no—
48C22×Dic17—24no—
49C22×Dic17—24no—
50C22×Dic17C2312yes—
51C22×Dic17C2312yes—
52C22×Dic17C2311yescharacteristic
53C22×Dic17C2312yes—
54#63254—232no—
55C2×C4×D17—22no—
56C2×C4×D17—22no—
57C2×C4×D17D414yes—
58C2×Dic34—22no—
59C2×Dic34—22no—
60C2×Dic34D414yes—
61C22⋊C4×C17—28no—
62C42×C17—22no—
63C22⋊Dic17—28no—
64#63227—24no—
65#63227—24no—
66C4⋊Dic17—22no—
67#63225—28no—
68#63225—24no—
69C4×Dic17—22no—
70C23×C17—22no—
71C23×C17—22no—
72C23×C17—24no—
73C23×C17—24no—
74C23×C17C2×D414yes—
75C23×C17C2411yescharacteristic
76C22×D17—21no—
77C22×D17—21no—
78C22×D17—22no—
79C22×D17—22no—
80C22×D17C2×D412yes—
81Q8×C17—44no—
82D4×C17—28no—
83C2×C4×C17—22no—
84C2×C4×C17—22no—
85C2×C4×C17—22no—
86C2×C4×C17—22no—
87C2×C4×C17—22no—
88C2×C4×C17—22no—
89C2×C4×C17C2×D412yes—
90C2×C4×C17C2×D412yes—
91C2×C4×C17C2×D412yes—
92C22⋊D17—216no—
93C2×Dic17—216no—
94C2×Dic17—22no—
95C2×Dic17—22no—
96C2×Dic17—21no—
97C2×Dic17—21no—
98C2×Dic17—22no—
99C2×Dic17—22no—
100C2×Dic17—22no—
101C2×Dic17—24no—
102C2×Dic17C2×D414yes—
103C2×Dic17C2×D412yes—
104C2×Dic17#4114yescomplemented
105C4×D17—28no—
106Dic34—28no—
107#2432—171nomaximal, Sylow
108C22×C17—24no—
109C22×C17—28no—
110C22×C17—42no—
111C22×C17—42no—
112C22×C17—44no—
113C22×C17C22×D411yescharacteristic
114C22×C17C22×D411yescharacteristic
115C22×C17C22×D411yescharacteristic
116C22×C17C2×#4112yes—
117C22×C172−1+412yes—
118C2×D17—22no—
119C2×D17—22no—
120C2×D17—22no—
121C2×D17—42no—
122C2×D17—42no—
123C4×C17—42no—
124C4×C17—42no—
125C4×C17#12014yes—
126Dic17—24no—
127Dic17—22no—
128Dic17—24no—
129Dic17—42no—
130C22×#41—171no—
131C2×C4⋊Q8—171no—
132C2×#124D1711yescharacteristic, complemented
133C2×C22⋊Q8—171no—
134#394—174no—
135#391—171no—
136#390—171no—
137#385—174no—
138C4×C22⋊C4—171no—
139C2×C17—44no—
140C2×C17C2×#12011yescharacteristic
141C2×C17#53211yescharacteristic
142C2×C17#53611yescharacteristic
143C2×C17#54712yes—
144C2×C17#54812yes—
145D17—42no—
146C2×#41—344no—
147C2×#41—344no—
148C2×#41—344no—
149C22×Q8—171no—
150C22×Q8C2×D1711yescharacteristic
151C22×D4—172no—
152C22×D4C2×D1711yescharacteristic
153C23×C4—171no—
154C23×C4—172no—
155C4⋊Q8—344no—
156#124—24no—
157C22⋊Q8—344no—
158C2×C4⋊C4—172no—
159C2×C4⋊C4—171no—
160C2×C4⋊C4—174no—
161C2×C22⋊C4—174no—
162C2×C22⋊C4—172no—
163C2×C22⋊C4—172no—
164C2×C22⋊C4C2×D1714yes—
165C2×C42—171no—
166C2×C42C2×D1711yescharacteristic
167#95—172no—
168#95—174no—
169C17#243211yescharacteristic, complemented, Sylow
170C24—171no—
171C24C22×D1712yes—
172#41—3432no—
173C2×Q8—22no—
174C2×Q8—22no—
175C2×Q8—22no—
176C2×Q8—174no—
177C2×Q8—342no—
178C2×Q8—342no—
179C2×D4—22no—
180C2×D4—22no—
181C2×D4—344no—
182C2×D4—344no—
183C2×D4—344no—
184C2×D4C22⋊D1714yes—
185C22×C4—171no—
186C22×C4—172no—
187C22×C4—172no—
188C22×C4—174no—
189C22×C4—172no—
190C22×C4—344no—
191C22×C4—344no—
192C22×C4—344no—
193C22×C4—342no—
194C22×C4—342no—
195C22×C4C22×D1712yes—
196C22×C4C22×D1712yes—
197C22×C4C22×D1711yescharacteristic
198C4⋊C4—344no—
199C4⋊C4—348no—
200C4⋊C4—342no—
201C22⋊C4—28no—
202C22⋊C4—344no—
203C22⋊C4—344no—
204C22⋊C4—348no—
205C42—22no—
206C42—342no—
207C23—22no—
208C23—22no—
209C23—24no—
210C23—24no—
211C23—172no—
212C23—341no—
213C23—342no—
214C23—342no—
215C23—341no—
216C23C2×C22⋊D1714yes—
217C23C23×D1711yescharacteristic
218Q8—44no—
219Q8—348no—
220D4—28no—
221D4—3416no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—172no—
229C2×C4—174no—
230C2×C4—174no—
231C2×C4—3416no—
232C2×C4—342no—
233C2×C4—341no—
234C2×C4—342no—
235C2×C4—342no—
236C2×C4—341no—
237C2×C4—342no—
238C2×C4—348no—
239C2×C4—342no—
240C2×C4—344no—
241C2×C4D4×D1712yes—
242C2×C4D4×D1712yes—
243C2×C4C2×C22⋊D1712yes—
244C22—28no—
245C22—24no—
246C22—44no—
247C22—42no—
248C22—42no—
249C22—342no—
250C22—342no—
251C22—342no—
252C22—682no—
253C22—682no—
254C22C2×D4×D1711yescharacteristic
255C22C2×D4×D1711yescharacteristic
256C22#41×D1712yes—
257C22#1058769712yes—
258C22C22×C22⋊D1711yescharacteristic
259C4—42no—
260C4—42no—
261C4—344no—
262C4—342no—
263C4—344no—
264C4—682no—
265C4#1058762314yes—
266C2—44no—
267C2—682no—
268C2C2×#1058762311yescharacteristic
269C2#1176335811yescharacteristic
270C2#1176337911yescharacteristic
271C2#1176345812yes—
272C2#1176346712yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC2×C17×#124C2
prime radicalC22×#63254C2
hypercenterC2×#124D17
hypocenterC17#2432
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
2. C2×#10587630C2semidirect
3. C2×#10587632C2semidirect
4. #11762724C2semidirect
5. #11762715C2semidirect
6. #11762712C2semidirect
7. #11762681C2semidirect
8. C4×#63227C2semidirect
9. C2×C17×#124C2semidirect
104. C2×Dic17#41semidirect
132. C2×#124D17semidirect
169. C17#2432semidirect

Maximal subgroups

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

subgroupindex
C22×#632542
C2×#105876302
C2×#105876322
#117627242
#117627152
#117627122
#117626812
C4×#632272
C2×C17×#1242
#243217