Finite Group Databasebeta

← #14536848

Subgroups — #14536848

Order 2176.

430 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 125 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×#10587584C211yesmaximal, characteristic
3#11762750C212yesmaximal, complemented
4#11762744C212yesmaximal
5#11762742C212yesmaximal, complemented
6#11762738C212yesmaximal
7#11762729C211yesmaximal, characteristic, complemented
8#11762722C211yesmaximal, characteristic
9#11762720C211yesmaximal, characteristic
10#11762676C211yesmaximal, characteristic, complemented
11C17×#395C211yesmaximal, characteristic, complemented
12C22×Q8×C17C2211yescharacteristic
13C22×Dic34C2211yescharacteristic
14C2×C4⋊C4×C17C2211yescharacteristic
15C2×C4⋊C4×C17C2212yes—
16#10587630—24no—
17C2×#63227C2211yescharacteristic
18C2×#63227C2211yescharacteristic
19C2×#63227C2211yescharacteristic
20C2×#63227C2212yes—
21C2×#63227C2212yes—
22C2×C4⋊Dic17C2211yescharacteristic
23C2×#63225C2212yes—
24C2×#63225C2212yes—
25C2×#63225C2211yescharacteristic
26C2×#63225C2212yes—
27C2×C4×Dic17C2211yescharacteristic
28#10587584—24no—
29C17×#95C2211yescharacteristic
30C17×#95C2212yes—
31#10587502C2211yescharacteristic
32#10587502C2212yes—
33#10587502C2212yes—
34#10587502C2212yes—
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×Dic17C2312yes—
47C22×Dic17C2311yescharacteristic
48C22×Dic17C2311yescharacteristic
49C22×Dic17C2311yescharacteristic
50C2×Dic34—22no—
51C2×Dic34—42no—
52C4⋊C4×C17—22no—
53C4⋊C4×C17—42no—
54#63227—22no—
55#63227—22no—
56#63227—22no—
57#63227—42no—
58#63227—42no—
59C4⋊Dic17—22no—
60#63225—22no—
61#63225—24no—
62#63225—24no—
63#63225—42no—
64C4×Dic17D414yes—
65C23×C17C2411yescharacteristic
66C22×D17—21no—
67C22×D17—22no—
68C22×D17—21no—
69C22×D17—22no—
70C22×D17—21no—
71Q8×C17—44no—
72C2×C4×C17—22no—
73C2×C4×C17—21no—
74C2×C4×C17—21no—
75C2×C4×C17—22no—
76C2×C4×C17—22no—
77C2×C4×C17—22no—
78C2×C4×C17—22no—
79C2×C4×C17—22no—
80C2×C4×C17—21no—
81C2×C4×C17—22no—
82C2×C4×C17—21no—
83C2×C4×C17—22no—
84C2×C4×C17C2×D412yes—
85C2×Dic17—21no—
86C2×Dic17—22no—
87C2×Dic17—21no—
88C2×Dic17—22no—
89C2×Dic17—21no—
90C2×Dic17—22no—
91C2×Dic17—22no—
92C2×Dic17—22no—
93C2×Dic17—22no—
94C2×Dic17—22no—
95C2×Dic17—22no—
96C2×Dic17C2×D412yes—
97C2×Dic17C2×D412yes—
98Dic34—44no—
99#2821—171nomaximal, Sylow
100C22×C17C22×D411yescharacteristic
101C22×C17H(5,𝔽2)12yes—
102C22×C172−1+411yescharacteristic
103C22×C172−1+412yes—
104C22×C172−1+411yescharacteristic
105C2×D17—41no—
106C2×D17—41no—
107C2×D17—41no—
108C2×D17—42no—
109C2×D17—42no—
110C4×C17—22no—
111C4×C17—41no—
112C4×C17—42no—
113C4×C17—41no—
114C4×C17—42no—
115Dic17—22no—
116Dic17—22no—
117Dic17—42no—
118Dic17—41no—
119Dic17—42no—
120C2×C4⋊Q8—171no—
121C2×#126—171no—
122#400—172no—
123#399—172no—
124#398—172no—
125#397—172no—
126#395—171no—
127#395D1711yescharacteristic, complemented
128#393—171no—
129#393—171no—
130#383—171no—
131C2×C17#53511yescharacteristic
132C2×C17#53611yescharacteristic
133C2×C17#53611yescharacteristic
134C2×C17#56312yes—
135C2×C17#56312yes—
136D17—81no—
137C22×Q8—171no—
138C22×Q8C2×D1711yescharacteristic
139C4⋊Q8—344no—
140#126—344no—
141C2×C4⋊C4—171no—
142C2×C4⋊C4—171no—
143C2×C4⋊C4—172no—
144C2×C4⋊C4—172no—
145C2×C4⋊C4—172no—
146C2×C4⋊C4C2×D1711yescharacteristic
147C2×C4⋊C4C2×D1712yes—
148C2×C22⋊C4—171no—
149C2×C22⋊C4—171no—
150C2×C22⋊C4—171no—
151C2×C22⋊C4—172no—
152C2×C22⋊C4—172no—
153C2×C42—171no—
154#95—171no—
155#95—172no—
156#95—172no—
157#95—172no—
158#95—172no—
159#95—172no—
160#95C2×D1711yescharacteristic
161#95C2×D1712yes—
162C17#282111yescharacteristic, complemented, Sylow
163C24—171no—
164C2×Q8—22no—
165C2×Q8—42no—
166C2×Q8—342no—
167C2×Q8—682no—
168C22×C4—171no—
169C22×C4—171no—
170C22×C4—172no—
171C22×C4—171no—
172C22×C4—172no—
173C22×C4C22×D1712yes—
174C22×C4C22×D1711yescharacteristic
175C22×C4C22×D1712yes—
176C22×C4C22×D1711yescharacteristic
177C22×C4C22×D1711yescharacteristic
178C4⋊C4—22no—
179C4⋊C4—42no—
180C4⋊C4—342no—
181C4⋊C4—342no—
182C4⋊C4—344no—
183C4⋊C4—344no—
184C4⋊C4—682no—
185C22⋊C4—342no—
186C22⋊C4—342no—
187C22⋊C4—342no—
188C22⋊C4—682no—
189C22⋊C4—682no—
190C42—174no—
191C23—341no—
192C23—342no—
193C23—341no—
194C23—342no—
195C23—341no—
196C23C23×D1711yescharacteristic
197Q8—44no—
198Q8—684no—
199C2×C4—22no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—21no—
209C2×C4—22no—
210C2×C4—21no—
211C2×C4—172no—
212C2×C4—172no—
213C2×C4—341no—
214C2×C4—342no—
215C2×C4—341no—
216C2×C4—341no—
217C2×C4—342no—
218C2×C4—342no—
219C2×C4—342no—
220C2×C4—342no—
221C2×C4—342no—
222C2×C4—342no—
223C2×C4—342no—
224C2×C4D4×D1712yes—
225C22—682no—
226C22—682no—
227C22—681no—
228C22—681no—
229C22—681no—
230C22C2×D4×D1711yescharacteristic
231C22#1058769112yes—
232C22#1058769411yescharacteristic
233C22#1058769412yes—
234C22#1058769711yescharacteristic
235C4—22no—
236C4—42no—
237C4—42no—
238C4—41no—
239C4—41no—
240C4—342no—
241C4—342no—
242C4—682no—
243C4—682no—
244C4—681no—
245C2—1361no—
246C2#1176337411yescharacteristic
247C2#1176338211yescharacteristic
248C2#1176338411yescharacteristic
249C2#1176355812yes—
250C2#1176356012yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#395C2
prime radicalC23×D17C23
hypercenter#395D17
hypocenterC17#2821
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. #11762750C2semidirect
5. #11762742C2semidirect
7. #11762729C2semidirect
10. #11762676C2semidirect
11. C17×#395C2semidirect
127. #395D17semidirect
162. C17#2821semidirect

Maximal subgroups

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

subgroupindex
C2×#105876302
C2×#105875842
#117627502
#117627442
#117627422
#117627382
#117627292
#117627222
#117627202
#117626762
C17×#3952
#282117