Finite Group Databasebeta

← #14536837

Subgroups — #14536837

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×#10587622C211yesmaximal, characteristic
3#11762745C212yesmaximal, complemented
4#11762737C212yesmaximal, complemented
5#11762732C212yesmaximal, complemented
6#11762732C212yesmaximal, complemented
7#11762730C211yesmaximal, characteristic, complemented
8#11762729C211yesmaximal, characteristic, complemented
9#11762723C211yesmaximal, characteristic
10#11762673C211yesmaximal, characteristic, complemented
11C17×#394C211yesmaximal, characteristic
12C22×D4×C17C2211yescharacteristic
13C22×Dic34C2211yescharacteristic
14C2×C4⋊C4×C17C2211yescharacteristic
15C2×C22⋊C4×C17C2212yes—
16C2×C22⋊C4×C17C2212yes—
17C2×C22⋊Dic17C2212yes—
18C2×C22⋊Dic17C2212yes—
19C2×C22⋊Dic17C2212yes—
20C2×C22⋊Dic17C2212yes—
21#10587622—24no—
22C2×C4⋊Dic17C2211yescharacteristic
23C2×C4⋊Dic17C2212yes—
24C2×#63225C2211yescharacteristic
25C2×#63225C2211yescharacteristic
26C2×#63225C2211yescharacteristic
27C2×#63225C2211yescharacteristic
28C2×#63225C2212yes—
29C2×#63225C2212yes—
30C2×C4×Dic17C2211yescharacteristic
31#10587575—24no—
32C17×#95C2211yescharacteristic
33#10587502C2211yescharacteristic
34#10587502C2212yes—
35#10587502C2212yes—
36#10587502C2212yes—
37C24×C17C2312yes—
38C2×D4×C17—22no—
39C2×D4×C17—42no—
40C22×C4×C17C2312yes—
41C22×C4×C17C2311yescharacteristic
42C22×C4×C17C2311yescharacteristic
43C22×C4×C17C2311yescharacteristic
44C22×Dic17C2312yes—
45C22×Dic17C2312yes—
46C22×Dic17C2311yescharacteristic
47C22×Dic17C2311yescharacteristic
48C22×Dic17C2311yescharacteristic
49C22×Dic17C2311yescharacteristic
50C2×Dic34—22no—
51C2×Dic34—42no—
52C4⋊C4×C17—22no—
53C22⋊C4×C17—42no—
54C22⋊C4×C17—42no—
55C22⋊Dic17—24no—
56C22⋊Dic17—24no—
57C22⋊Dic17—42no—
58C22⋊Dic17—42no—
59C4⋊Dic17—22no—
60C4⋊Dic17—42no—
61#63225—22no—
62#63225—22no—
63#63225—22no—
64#63225—22no—
65#63225—42no—
66#63225—42no—
67C4×Dic17D414yes—
68C23×C17—22no—
69C23×C17—22no—
70C23×C17—22no—
71C23×C17—22no—
72C23×C17—22no—
73C23×C17—22no—
74C23×C17—22no—
75C23×C17C2411yescharacteristic
76D4×C17—44no—
77C2×C4×C17—22no—
78C2×C4×C17—21no—
79C2×C4×C17—21no—
80C2×C4×C17—21no—
81C2×C4×C17—22no—
82C2×C4×C17—22no—
83C2×C4×C17—22no—
84C2×C4×C17—21no—
85C2×C4×C17—22no—
86C2×C4×C17C2×D412yes—
87C2×Dic17—21no—
88C2×Dic17—22no—
89C2×Dic17—21no—
90C2×Dic17—22no—
91C2×Dic17—21no—
92C2×Dic17—21no—
93C2×Dic17—21no—
94C2×Dic17—22no—
95C2×Dic17—22no—
96C2×Dic17—22no—
97C2×Dic17—22no—
98C2×Dic17—22no—
99C2×Dic17—21no—
100C2×Dic17—22no—
101C2×Dic17C2×D412yes—
102C2×Dic17C2×D412yes—
103Dic34—44no—
104#2820—171nomaximal, Sylow
105C22×C17—42no—
106C22×C17—42no—
107C22×C17—42no—
108C22×C17—42no—
109C22×C17—42no—
110C22×C17—42no—
111C22×C17—42no—
112C22×C17C22×D411yescharacteristic
113C22×C17H(5,𝔽2)12yes—
114C22×C17H(5,𝔽2)11yescharacteristic
115C22×C172−1+412yes—
116C22×C172−1+411yescharacteristic
117C4×C17—22no—
118C4×C17—41no—
119C4×C17—41no—
120C4×C17—42no—
121Dic17—22no—
122Dic17—22no—
123Dic17—42no—
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#395—171no—
135#394—171no—
136#394D1711yescharacteristic
137#382—171no—
138C2×C17—82no—
139C2×C17#53511yescharacteristic
140C2×C17#53611yescharacteristic
141C2×C17#53711yescharacteristic
142C2×C17#56212yes—
143C2×C17#56312yes—
144C22×Q8—171no—
145C22×D4C2×D1711yescharacteristic
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⋊C4—172no—
156C2×C4⋊C4C2×D1711yescharacteristic
157C2×C22⋊C4—172no—
158C2×C22⋊C4—172no—
159C2×C22⋊C4—172no—
160C2×C22⋊C4—172no—
161C2×C22⋊C4C2×D1712yes—
162C2×C22⋊C4C2×D1712yes—
163C2×C42—171no—
164#95—171no—
165#95—172no—
166#95—172no—
167#95—172no—
168#95C2×D1711yescharacteristic
169C17#282011yescharacteristic, complemented, Sylow
170C24C22×D1712yes—
171C2×Q8—342no—
172C2×Q8—682no—
173C2×D4—22no—
174C2×D4—42no—
175C22×C4—171no—
176C22×C4—171no—
177C22×C4—172no—
178C22×C4—171no—
179C22×C4—172no—
180C22×C4—171no—
181C22×C4C22×D1712yes—
182C22×C4C22×D1711yescharacteristic
183C22×C4C22×D1711yescharacteristic
184C22×C4C22×D1711yescharacteristic
185C4⋊C4—22no—
186C4⋊C4—342no—
187C4⋊C4—342no—
188C4⋊C4—342no—
189C4⋊C4—342no—
190C4⋊C4—342no—
191C4⋊C4—682no—
192C4⋊C4—682no—
193C4⋊C4—682no—
194C22⋊C4—42no—
195C22⋊C4—42no—
196C22⋊C4—344no—
197C22⋊C4—344no—
198C22⋊C4—682no—
199C22⋊C4—682no—
200C42—174no—
201C23—22no—
202C23—22no—
203C23—22no—
204C23—22no—
205C23—22no—
206C23—22no—
207C23—22no—
208C23C23×D1711yescharacteristic
209Q8—684no—
210D4—44no—
211C2×C4—22no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—21no—
220C2×C4—172no—
221C2×C4—172no—
222C2×C4—341no—
223C2×C4—342no—
224C2×C4—341no—
225C2×C4—342no—
226C2×C4—341no—
227C2×C4—341no—
228C2×C4—341no—
229C2×C4—342no—
230C2×C4—342no—
231C2×C4—342no—
232C2×C4—342no—
233C2×C4—342no—
234C2×C4—342no—
235C2×C4—341no—
236C2×C4D4×D1712yes—
237C22—42no—
238C22—42no—
239C22—42no—
240C22—42no—
241C22—42no—
242C22—42no—
243C22—42no—
244C22C2×D4×D1711yescharacteristic
245C22#1058769111yescharacteristic
246C22#1058769112yes—
247C22#1058769712yes—
248C22#1058769711yescharacteristic
249C4—22no—
250C4—42no—
251C4—41no—
252C4—41no—
253C4—342no—
254C4—342no—
255C4—682no—
256C4—682no—
257C4—681no—
258C4—681no—
259C2—82no—
260C2#1176337411yescharacteristic
261C2#1176337911yescharacteristic
262C2#1176338811yescharacteristic
263C2#1176355412yes—
264C2#1176356212yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#394C2
prime radicalC22×D4×C17C22
hypercenter#394D17
hypocenterC17#2820
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C2×#10587575C2semidirect
3. #11762745C2semidirect
4. #11762737C2semidirect
5. #11762732C2semidirect
6. #11762732C2semidirect
7. #11762730C2semidirect
8. #11762729C2semidirect
10. #11762673C2semidirect
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×#105876222
#117627452
#117627372
#117627322
#117627302
#117627292
#117627232
#117626732
C17×#3942
#282017