Finite Group Databasebeta

← #14536839

Subgroups — #14536839

Order 2176.

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

normal subgroups only

This group has 464 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×#10587575C211yesmaximal, characteristic, complemented
2C2×#10587633C211yesmaximal, characteristic
3#11762744C212yesmaximal, complemented
4#11762738C212yesmaximal, complemented
5#11762733C212yesmaximal, complemented
6#11762731C212yesmaximal, complemented
7#11762728C211yesmaximal, characteristic, complemented
8#11762727C211yesmaximal, characteristic
9#11762726C211yesmaximal, characteristic
10#11762673C211yesmaximal, characteristic, complemented
11C17×#395C211yesmaximal, characteristic, complemented
12C22×Q8×C17C2211yescharacteristic
13C22×D68C2211yescharacteristic
14C2×C4⋊C4×C17C2211yescharacteristic
15C2×C4⋊C4×C17C2212yes—
16#10587633—24no—
17C2×#63227C2212yes—
18C2×#63227C2212yes—
19C2×#63227C2212yes—
20C2×#63227C2212yes—
21C2×#63227C2212yes—
22C2×#63227C2212yes—
23C2×C4⋊Dic17C2211yescharacteristic
24C2×#63225C2211yescharacteristic
25C2×#63225C2211yescharacteristic
26C2×#63225C2211yescharacteristic
27C2×#63225C2211yescharacteristic
28C2×#63225C2212yes—
29C2×#63225C2212yes—
30C2×C4×Dic17C2211yescharacteristic
31#10587575—24no—
32C17×#95C2211yescharacteristic
33C17×#95C2212yes—
34#10587502C2211yescharacteristic
35#10587502C2212yes—
36#10587502C2212yes—
37C23×D17C2312yes—
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×Dic17C2311yescharacteristic
48C22×Dic17C2311yescharacteristic
49C22×Dic17C2311yescharacteristic
50C2×D68—22no—
51C2×D68—42no—
52C4⋊C4×C17—22no—
53C4⋊C4×C17—42no—
54#63227—24no—
55#63227—24no—
56#63227—42no—
57#63227—42no—
58#63227—42no—
59#63227—42no—
60C4⋊Dic17—22no—
61#63225—22no—
62#63225—22no—
63#63225—22no—
64#63225—22no—
65#63225—42no—
66#63225—42no—
67C4×Dic17D414yes—
68C23×C17C2411yescharacteristic
69C22×D17—22no—
70C22×D17—22no—
71C22×D17—22no—
72C22×D17—22no—
73C22×D17—22no—
74C22×D17—22no—
75C22×D17—22no—
76Q8×C17—44no—
77C2×C4×C17—22no—
78C2×C4×C17—21no—
79C2×C4×C17—21no—
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×C17—21no—
87C2×C4×C17—21no—
88C2×C4×C17—22no—
89C2×C4×C17C2×D412yes—
90C2×Dic17—21no—
91C2×Dic17—22no—
92C2×Dic17—21no—
93C2×Dic17—21no—
94C2×Dic17—21no—
95C2×Dic17—22no—
96C2×Dic17—22no—
97C2×Dic17—22no—
98C2×Dic17—21no—
99C2×Dic17—21no—
100C2×Dic17—22no—
101C2×Dic17C2×D412yes—
102C2×Dic17C2×D412yes—
103D68—44no—
104#2820—171nomaximal, Sylow
105C22×C17C22×D411yescharacteristic
106C22×C17H(5,𝔽2)12yes—
107C22×C17H(5,𝔽2)11yescharacteristic
108C22×C172−1+412yes—
109C22×C172−1+411yescharacteristic
110C2×D17—42no—
111C2×D17—42no—
112C2×D17—42no—
113C2×D17—42no—
114C2×D17—42no—
115C2×D17—42no—
116C2×D17—42no—
117C4×C17—22no—
118C4×C17—41no—
119C4×C17—42no—
120C4×C17—41no—
121C4×C17—42no—
122Dic17—22no—
123Dic17—22no—
124Dic17—41no—
125Dic17—41no—
126Dic17—42no—
127C2×#125—171no—
128C2×#124—171no—
129#399—172no—
130#397—172no—
131#396—172no—
132#396—172no—
133#395—171no—
134#395D1711yescharacteristic, complemented
135#394—171no—
136#394—171no—
137#382—171no—
138C2×C17#53511yescharacteristic
139C2×C17#53611yescharacteristic
140C2×C17#53711yescharacteristic
141C2×C17#56212yes—
142C2×C17#56312yes—
143D17—82no—
144C22×Q8C2×D1711yescharacteristic
145C22×D4—171no—
146#125—344no—
147#124—344no—
148C2×C4⋊C4—171no—
149C2×C4⋊C4—171no—
150C2×C4⋊C4—171no—
151C2×C4⋊C4—171no—
152C2×C4⋊C4—171no—
153C2×C4⋊C4—172no—
154C2×C4⋊C4—172no—
155C2×C4⋊C4C2×D1711yescharacteristic
156C2×C4⋊C4C2×D1712yes—
157C2×C22⋊C4—172no—
158C2×C22⋊C4—172no—
159C2×C22⋊C4—172no—
160C2×C22⋊C4—172no—
161C2×C22⋊C4—172no—
162C2×C22⋊C4—172no—
163C2×C42—171no—
164#95—171no—
165#95—172no—
166#95—172no—
167#95C2×D1711yescharacteristic
168#95C2×D1712yes—
169C17#282011yescharacteristic, complemented, Sylow
170C24—172no—
171C2×Q8—22no—
172C2×Q8—42no—
173C2×D4—342no—
174C2×D4—682no—
175C22×C4—171no—
176C22×C4—171no—
177C22×C4—172no—
178C22×C4—171no—
179C22×C4—171no—
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—342no—
189C4⋊C4—342no—
190C4⋊C4—342no—
191C4⋊C4—342no—
192C4⋊C4—682no—
193C4⋊C4—682no—
194C22⋊C4—344no—
195C22⋊C4—344no—
196C22⋊C4—682no—
197C22⋊C4—682no—
198C22⋊C4—682no—
199C22⋊C4—682no—
200C42—174no—
201C23—342no—
202C23—342no—
203C23—342no—
204C23—342no—
205C23—342no—
206C23—342no—
207C23—342no—
208C23C23×D1711yescharacteristic
209Q8—44no—
210D4—684no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—22no—
223C2×C4—172no—
224C2×C4—172no—
225C2×C4—342no—
226C2×C4—341no—
227C2×C4—341no—
228C2×C4—341no—
229C2×C4—341no—
230C2×C4—342no—
231C2×C4—342no—
232C2×C4—342no—
233C2×C4—341no—
234C2×C4—341no—
235C2×C4—342no—
236C2×C4D4×D1712yes—
237C22—682no—
238C22—682no—
239C22—682no—
240C22—682no—
241C22—682no—
242C22—682no—
243C22—682no—
244C22C2×D4×D1711yescharacteristic
245C22#1058769112yes—
246C22#1058769412yes—
247C22#1058769411yescharacteristic
248C22#1058769611yescharacteristic
249C4—22no—
250C4—42no—
251C4—42no—
252C4—41no—
253C4—41no—
254C4—342no—
255C4—342no—
256C4—682no—
257C4—681no—
258C4—681no—
259C2—1362no—
260C2#1176337411yescharacteristic
261C2#1176338411yescharacteristic
262C2#1176338911yescharacteristic
263C2#1176355512yes—
264C2#1176355812yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#395C2
prime radicalC22×D68C22
hypercenter#395D17
hypocenterC17#2820
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
1. C2×#10587575C2semidirect
3. #11762744C2semidirect
4. #11762738C2semidirect
5. #11762733C2semidirect
6. #11762731C2semidirect
7. #11762728C2semidirect
10. #11762673C2semidirect
11. C17×#395C2semidirect
134. #395D17semidirect
169. C17#2820semidirect

Maximal subgroups

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

subgroupindex
C2×#105875752
C2×#105876332
#117627442
#117627382
#117627332
#117627312
#117627282
#117627272
#117627262
#117626732
C17×#3952
#282017