Finite Group Databasebeta

← #14534607

Subgroups — #14534607

Order 2176.

488 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 143 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
1C2×#10587630C211yesmaximal, characteristic, complemented
2C2×#10587578C211yesmaximal, characteristic
3#11762751C212yesmaximal, complemented
4#11762744C212yesmaximal
5#11762729C211yesmaximal, characteristic, complemented
6#11762722C211yesmaximal, characteristic
7#11762720C211yesmaximal, characteristic
8#11762704C212yesmaximal, complemented
9#11762673C211yesmaximal, characteristic, complemented
10#11762668C212yesmaximal
11C17×#395C211yesmaximal, characteristic, complemented
12C22×Q8×C17C2211yescharacteristic
13C22×C4×D17C2211yescharacteristic
14C22×Dic34C2211yescharacteristic
15C2×C4⋊C4×C17C2211yescharacteristic
16C2×C4⋊C4×C17C2212yes—
17#10587630—24no—
18C2×#63227C2211yescharacteristic
19C2×#63227C2211yescharacteristic
20C2×#63227C2212yes—
21C2×#63227C2212yes—
22C2×C4⋊Dic17C2211yescharacteristic
23C2×#63225C2212yes—
24C2×#63225C2212yes—
25C2×#63225C2212yes—
26C2×#63225C2212yes—
27C2×C4×Dic17C2211yescharacteristic
28C2×C4×Dic17C2211yescharacteristic
29#10587578—24no—
30C17×#95C2211yescharacteristic
31C17×#95C2212yes—
32#10587502C2212yes—
33#10587502C2212yes—
34#10587502C2211yescharacteristic
35#10587502C2212yes—
36#10587502C2212yes—
37C23×D17C2311yescharacteristic
38C2×Q8×C17—22no—
39C2×Q8×C17—42no—
40C22×C4×C17C2312yes—
41C22×C4×C17C2312yes—
42C22×C4×C17C2311yescharacteristic
43C22×C4×C17C2311yescharacteristic
44C22×C4×C17C2311yescharacteristic
45C22×Dic17C2312yes—
46C22×Dic17C2311yescharacteristic
47C22×Dic17C2312yes—
48C22×Dic17C2311yescharacteristic
49C22×Dic17C2311yescharacteristic
50C2×C4×D17—22no—
51C2×C4×D17—22no—
52C2×C4×D17Q814yescomplemented
53C2×Dic34—22no—
54C2×Dic34—42no—
55C4⋊C4×C17—22no—
56C4⋊C4×C17—42no—
57#63227—22no—
58#63227—22no—
59#63227—42no—
60#63227—42no—
61C4⋊Dic17—22no—
62#63225—24no—
63#63225—24no—
64#63225—24no—
65#63225—42no—
66C4×Dic17—22no—
67C4×Dic17D414yes—
68C23×C17C2411yescharacteristic
69C22×D17—21no—
70C22×D17—21no—
71C22×D17—22no—
72C22×D17—22no—
73C22×D17C2×Q812yes—
74Q8×C17—44no—
75C2×C4×C17—21no—
76C2×C4×C17—21no—
77C2×C4×C17—22no—
78C2×C4×C17—21no—
79C2×C4×C17—22no—
80C2×C4×C17—22no—
81C2×C4×C17—22no—
82C2×C4×C17—22no—
83C2×C4×C17—22no—
84C2×C4×C17—22no—
85C2×C4×C17—22no—
86C2×C4×C17C2×D412yes—
87C2×C4×C17C2×Q812yes—
88C2×Dic17—22no—
89C2×Dic17—22no—
90C2×Dic17—21no—
91C2×Dic17—22no—
92C2×Dic17—22no—
93C2×Dic17—22no—
94C2×Dic17—22no—
95C2×Dic17—22no—
96C2×Dic17—22no—
97C2×Dic17C2×D412yes—
98C2×Dic17C2×D412yes—
99C2×Dic17C2×Q812yes—
100C2×Dic17#4112yes—
101C4×D17—28no—
102Dic34—44no—
103#2628—171nomaximal, Sylow
104C22×C17C22×D411yescharacteristic
105C22×C17C22×Q811yescharacteristic
106C22×C17C2×#4111yescharacteristic
107C22×C17H(5,𝔽2)12yes—
108C22×C172−1+412yes—
109C2×D17—22no—
110C2×D17—24no—
111C2×D17—41no—
112C2×D17—41no—
113C2×D17—42no—
114C4×C17—22no—
115C4×C17—22no—
116C4×C17—41no—
117C4×C17—42no—
118C4×C17—42no—
119Dic17—22no—
120Dic17—42no—
121Dic17—41no—
122Dic17—42no—
123Dic17C22⋊Q814yes—
124C2×C4⋊Q8—171no—
125C2×#117—171no—
126#400—172no—
127#399—172no—
128#395—171no—
129#395D1711yescharacteristic, complemented
130#393—171no—
131#393—171no—
132#389—172no—
133#382—171no—
134#380—172no—
135C2×C17C2×C22⋊Q811yescharacteristic
136C2×C17#53511yescharacteristic
137C2×C17#53611yescharacteristic
138C2×C17#54212yes—
139C2×C17#54412yes—
140D17—42no—
141C22×Q8—171no—
142C22×Q8C2×D1711yescharacteristic
143C23×C4—171no—
144C4⋊Q8—344no—
145#117—344no—
146C2×C4⋊C4—171no—
147C2×C4⋊C4—172no—
148C2×C4⋊C4—172no—
149C2×C4⋊C4—172no—
150C2×C4⋊C4—172no—
151C2×C4⋊C4C2×D1711yescharacteristic
152C2×C4⋊C4C2×D1712yes—
153C2×C22⋊C4—171no—
154C2×C22⋊C4—171no—
155C2×C22⋊C4—172no—
156C2×C22⋊C4—172no—
157C2×C42—171no—
158C2×C42—171no—
159#95—172no—
160#95—171no—
161#95—172no—
162#95—172no—
163#95—172no—
164#95C2×D1712yes—
165#95C2×D1711yescharacteristic
166C17#262811yescharacteristic, complemented, Sylow
167C24—171no—
168C2×Q8—22no—
169C2×Q8—42no—
170C2×Q8—342no—
171C2×Q8—682no—
172C22×C4—171no—
173C22×C4—171no—
174C22×C4—174no—
175C22×C4—171no—
176C22×C4—172no—
177C22×C4—172no—
178C22×C4—342no—
179C22×C4—342no—
180C22×C4C22×D1712yes—
181C22×C4C22×D1712yes—
182C22×C4C22×D1711yescharacteristic
183C22×C4C22×D1711yescharacteristic
184C22×C4C22×D1711yescharacteristic
185C4⋊C4—22no—
186C4⋊C4—42no—
187C4⋊C4—342no—
188C4⋊C4—344no—
189C4⋊C4—344no—
190C4⋊C4—344no—
191C4⋊C4—682no—
192C22⋊C4—342no—
193C22⋊C4—342no—
194C22⋊C4—682no—
195C22⋊C4—682no—
196C42—174no—
197C42—342no—
198C23—172no—
199C23—341no—
200C23—342no—
201C23—342no—
202C23—341no—
203C23C23×D1711yescharacteristic
204Q8—44no—
205Q8—684no—
206C2×C4—21no—
207C2×C4—22no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—172no—
218C2×C4—172no—
219C2×C4—172no—
220C2×C4—172no—
221C2×C4—348no—
222C2×C4—342no—
223C2×C4—341no—
224C2×C4—342no—
225C2×C4—342no—
226C2×C4—342no—
227C2×C4—342no—
228C2×C4—342no—
229C2×C4—342no—
230C2×C4—342no—
231C2×C4D4×D1712yes—
232C2×C4Q8×D1712yes—
233C22—342no—
234C22—344no—
235C22—681no—
236C22—681no—
237C22—682no—
238C22C2×D4×D1711yescharacteristic
239C22#1058769112yes—
240C22C2×Q8×D1711yescharacteristic
241C22#1058769412yes—
242C22#41×D1711yescharacteristic
243C4—22no—
244C4—22no—
245C4—42no—
246C4—41no—
247C4—42no—
248C4—174no—
249C4—342no—
250C4—682no—
251C4—681no—
252C4—682no—
253C2—682no—
254C2C22⋊Q8×D1711yescharacteristic
255C2#1176337411yescharacteristic
256C2#1176338411yescharacteristic
257C2#1176342912yes—
258C2#1176343612yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#395C2
prime radicalC23×D17C23
hypercenter#395D17
hypocenterC17#2628
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
1. C2×#10587630C2semidirect
3. #11762751C2semidirect
5. #11762729C2semidirect
8. #11762704C2semidirect
9. #11762673C2semidirect
11. C17×#395C2semidirect
52. C2×C4×D17Q8semidirect
129. #395D17semidirect
166. C17#2628semidirect

Maximal subgroups

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

subgroupindex
C2×#105876302
C2×#105875782
#117627512
#117627442
#117627292
#117627222
#117627202
#117627042
#117626732
#117626682
C17×#3952
#262817