Finite Group Databasebeta

← #14535155

Subgroups — #14535155

Order 2176.

592 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 592 conjugacy classes of subgroups — too many to draw. Only its 157 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×#10587583C212yesmaximal, complemented
2C2×#10587582C212yesmaximal, complemented
3#11762731C212yesmaximal, complemented
4#11762731C212yesmaximal, complemented
5#11762730C212yesmaximal, complemented
6#11762711C211yesmaximal, characteristic
7#11762707C211yesmaximal, characteristic, complemented
8#11762687C212yesmaximal
9C17×#389C211yesmaximal, characteristic, complemented
10C22×D68C2211yescharacteristic
11C22×C4×D17C2212yes—
12C22×Dic34C2212yes—
13C2×C4⋊C4×C17C2212yes—
14C2×C4⋊C4×C17C2211yescharacteristic
15C2×C4⋊C4×C17C2211yescharacteristic
16C2×C42×C17C2211yescharacteristic
17C2×#63227C2212yes—
18C2×#63227C2212yes—
19C2×#63227C2212yes—
20C2×#63227C2212yes—
21C2×#63227C2212yes—
22C2×C4⋊Dic17C2212yes—
23C2×C4⋊Dic17C2211yescharacteristic
24C2×C4⋊Dic17C2211yescharacteristic
25C2×C4⋊Dic17C2211yescharacteristic
26C2×#63225C2212yes—
27C2×#63225C2212yes—
28#10587583—28no—
29#10587582—28no—
30C17×#95C2212yes—
31#10587502C2212yes—
32#10587502C2212yes—
33C23×D17C2312yes—
34C22×C4×C17C2312yes—
35C22×C4×C17C2312yes—
36C22×C4×C17C2311yescharacteristic
37C22×C4×C17C2311yescharacteristic
38C22×C4×C17C2311yescharacteristic
39C22×Dic17C2312yes—
40C22×Dic17C2312yes—
41C22×Dic17C2312yes—
42C2×D68—41no—
43C2×D68—41no—
44C2×D68Q814yescomplemented
45C2×C4×D17—24no—
46C2×C4×D17—24no—
47C2×C4×D17—42no—
48C2×Dic34—24no—
49C2×Dic34—24no—
50C2×Dic34—42no—
51C4⋊C4×C17—24no—
52C4⋊C4×C17—41no—
53C4⋊C4×C17D414yescomplemented
54C42×C17—41no—
55#63227—24no—
56#63227—24no—
57#63227—24no—
58#63227—24no—
59#63227—42no—
60C4⋊Dic17—24no—
61C4⋊Dic17—41no—
62C4⋊Dic17—41no—
63C4⋊Dic17D414yescomplemented
64#63225—24no—
65#63225—24no—
66C23×C17C2411yescharacteristic
67C22×D17—22no—
68C22×D17—22no—
69C22×D17—22no—
70C22×D17—22no—
71C22×D17—22no—
72C22×D17—22no—
73C22×D17C2×Q814yes—
74C2×C4×C17—22no—
75C2×C4×C17—21no—
76C2×C4×C17—21no—
77C2×C4×C17—21no—
78C2×C4×C17—22no—
79C2×C4×C17—21no—
80C2×C4×C17—21no—
81C2×C4×C17—21no—
82C2×C4×C17—22no—
83C2×C4×C17—22no—
84C2×C4×C17—22no—
85C2×C4×C17C2×D414yes—
86C2×C4×C17C2×D412yes—
87C2×C4×C17C2×D412yes—
88C2×C4×C17C2×Q812yes—
89C2×Dic17—22no—
90C2×Dic17—22no—
91C2×Dic17—22no—
92C2×Dic17—22no—
93C2×Dic17—22no—
94C2×Dic17—22no—
95C2×Dic17—22no—
96C2×Dic17—22no—
97C2×Dic17C2×D414yes—
98D68—44no—
99C4×D17—48no—
100Dic34—48no—
101#2683—171nomaximal, Sylow
102C22×C17C22×D411yescharacteristic
103C22×C17C22×D411yescharacteristic
104C22×C17C22×Q811yescharacteristic
105C22×C17H(5,𝔽2)11yescharacteristic
106C22×C17H(5,𝔽2)11yescharacteristic
107C22×C172−1+411yescharacteristic
108C22×C172−1+411yescharacteristic
109C2×D17—24no—
110C2×D17—24no—
111C2×D17—24no—
112C2×D17—42no—
113C2×D17—42no—
114C2×D17—42no—
115C2×D17—42no—
116C4×C17—22no—
117C4×C17—22no—
118C4×C17—22no—
119C4×C17—24no—
120C4×C17—42no—
121Dic17—24no—
122Dic17—42no—
123Dic17—42no—
124C2×C22⋊Q8—172no—
125C2×C22⋊Q8—172no—
126#396—172no—
127#396—172no—
128#395—172no—
129#390—171no—
130#389—171no—
131#389D1711yescharacteristic, complemented
132#386—172no—
133C2×C17#53711yescharacteristic
134C2×C17#53711yescharacteristic
135C2×C17#54311yescharacteristic
136C2×C17D4211yescharacteristic
137C2×C17D4×Q811yescharacteristic
138C2×C17D4×Q811yescharacteristic
139C2×C17#56311yescharacteristic
140D17—44no—
141C22×Q8—172no—
142C22×D4—171no—
143C23×C4—172no—
144C22⋊Q8—348no—
145C22⋊Q8—348no—
146C2×C4⋊C4—171no—
147C2×C4⋊C4—172no—
148C2×C4⋊C4—172no—
149C2×C4⋊C4—172no—
150C2×C4⋊C4—171no—
151C2×C4⋊C4—171no—
152C2×C4⋊C4C2×D1712yes—
153C2×C4⋊C4C2×D1711yescharacteristic
154C2×C4⋊C4C2×D1711yescharacteristic
155C2×C22⋊C4—172no—
156C2×C22⋊C4—172no—
157C2×C22⋊C4—172no—
158C2×C22⋊C4—172no—
159C2×C22⋊C4—172no—
160C2×C42C2×D1711yescharacteristic
161#95—172no—
162#95—172no—
163#95C2×D1712yes—
164C17#268311yescharacteristic, complemented, Sylow
165C24—172no—
166C2×Q8—344no—
167C2×Q8—344no—
168C2×Q8—682no—
169C2×D4—174no—
170C2×D4—681no—
171C2×D4—681no—
172C22×C4—172no—
173C22×C4—172no—
174C22×C4—172no—
175C22×C4—344no—
176C22×C4—344no—
177C22×C4—682no—
178C22×C4C22×D1712yes—
179C22×C4C22×D1712yes—
180C22×C4C22×D1711yescharacteristic
181C22×C4C22×D1711yescharacteristic
182C22×C4C22×D1711yescharacteristic
183C4⋊C4—24no—
184C4⋊C4—41no—
185C4⋊C4—174no—
186C4⋊C4—344no—
187C4⋊C4—344no—
188C4⋊C4—344no—
189C4⋊C4—681no—
190C4⋊C4—681no—
191C4⋊C4D6814yescomplemented
192C22⋊C4—344no—
193C22⋊C4—344no—
194C22⋊C4—344no—
195C22⋊C4—344no—
196C22⋊C4—682no—
197C42—41no—
198C23—174no—
199C23—342no—
200C23—342no—
201C23—342no—
202C23—342no—
203C23—342no—
204C23—342no—
205C23C23×D1711yescharacteristic
206Q8—688no—
207D4—684no—
208C2×C4—22no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—22no—
215C2×C4—21no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—174no—
220C2×C4—342no—
221C2×C4—342no—
222C2×C4—342no—
223C2×C4—342no—
224C2×C4—342no—
225C2×C4—342no—
226C2×C4—342no—
227C2×C4—342no—
228C2×C4—688no—
229C2×C4C2×D6812yes—
230C2×C4C2×D6814yes—
231C2×C4D4×D1712yes—
232C2×C4Q8×D1712yes—
233C22—344no—
234C22—344no—
235C22—344no—
236C22—682no—
237C22—682no—
238C22—682no—
239C22—682no—
240C22C22×D6811yescharacteristic
241C22C2×D4×D1711yescharacteristic
242C22#1058769111yescharacteristic
243C22C2×Q8×D1711yescharacteristic
244C22#1058769411yescharacteristic
245C22#1058769611yescharacteristic
246C22#1058769711yescharacteristic
247C4—22no—
248C4—22no—
249C4—22no—
250C4—24no—
251C4—42no—
252C4—344no—
253C4—682no—
254C4—682no—
255C2—684no—
256C2#1176338611yescharacteristic
257C2#1176339011yescharacteristic
258C2#1176343411yescharacteristic
259C2D4×D6811yescharacteristic
260C2#1176347411yescharacteristic
261C2Q8×D6811yescharacteristic
262C2#1176356311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#389C2
prime radicalC22×D68C22
hypercenter#389D17
hypocenterC17#2683
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #389

Complements

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

NH ≅ G/Ntype
1. C2×#10587583C2semidirect
2. C2×#10587582C2semidirect
3. #11762731C2semidirect
4. #11762731C2semidirect
5. #11762730C2semidirect
7. #11762707C2semidirect
9. C17×#389C2semidirect
44. C2×D68Q8semidirect
53. C4⋊C4×C17D4semidirect
63. C4⋊Dic17D4semidirect
131. #389D17semidirect
164. C17#2683semidirect
191. C4⋊C4D68semidirect

Maximal subgroups

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

subgroupindex
C2×#105875832
C2×#105875822
#117627312
#117627302
#117627112
#117627072
#117626872
C17×#3892
#268317