Finite Group Databasebeta

← #14535802

Subgroups — #14535802

Order 2176.

528 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 149 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×#10587632C212yesmaximal, complemented
2#11762746C212yesmaximal, complemented
3#11762741C212yesmaximal, complemented
4#11762731C212yesmaximal, complemented
5#11762711C211yesmaximal, characteristic
6#11762707C211yesmaximal, characteristic, complemented
7#11762687C212yesmaximal
8#11762683C212yesmaximal, complemented
9C17×#391C211yesmaximal, characteristic, complemented
10C22×Q8×C17C2211yescharacteristic
11C22×D68C2211yescharacteristic
12C22×C4×D17C2212yes—
13C2×C4⋊C4×C17C2211yescharacteristic
14C2×C42×C17C2211yescharacteristic
15#10587632—28no—
16C2×#63227C2212yes—
17C2×#63227C2212yes—
18C2×#63227C2212yes—
19C2×#63227C2212yes—
20C2×#63227C2212yes—
21C2×C4⋊Dic17C2211yescharacteristic
22C2×C4⋊Dic17C2212yes—
23C2×C4⋊Dic17C2211yescharacteristic
24C2×C4⋊Dic17C2211yescharacteristic
25C2×C4⋊Dic17C2212yes—
26C2×#63225C2212yes—
27C2×#63225C2212yes—
28C17×#95C2212yes—
29C17×#95C2212yes—
30#10587502C2212yes—
31#10587502C2212yes—
32C23×D17C2312yes—
33C2×Q8×C17—41no—
34C2×Q8×C17—41no—
35C2×Q8×C17D414yescomplemented
36C22×C4×C17C2312yes—
37C22×C4×C17C2312yes—
38C22×C4×C17C2311yescharacteristic
39C22×C4×C17C2311yescharacteristic
40C22×C4×C17C2311yescharacteristic
41C22×Dic17C2312yes—
42C22×Dic17C2312yes—
43C22×Dic17C2312yes—
44C2×D68—41no—
45C2×D68—41no—
46C2×D68Q814yescomplemented
47C2×C4×D17—24no—
48C2×C4×D17—42no—
49C2×C4×D17—42no—
50C4⋊C4×C17—41no—
51C42×C17—41no—
52#63227—24no—
53#63227—24no—
54#63227—24no—
55#63227—42no—
56#63227—42no—
57C4⋊Dic17—24no—
58C4⋊Dic17—22no—
59C4⋊Dic17—41no—
60C4⋊Dic17—41no—
61C4⋊Dic17—42no—
62#63225—24no—
63#63225—24no—
64C23×C17C2411yescharacteristic
65C22×D17—22no—
66C22×D17—22no—
67C22×D17—22no—
68C22×D17—22no—
69C22×D17—22no—
70C22×D17—22no—
71C22×D17C2×Q814yes—
72Q8×C17—44no—
73C2×C4×C17—21no—
74C2×C4×C17—21no—
75C2×C4×C17—21no—
76C2×C4×C17—21no—
77C2×C4×C17—22no—
78C2×C4×C17—22no—
79C2×C4×C17—21no—
80C2×C4×C17—21no—
81C2×C4×C17—22no—
82C2×C4×C17—22no—
83C2×C4×C17—22no—
84C2×C4×C17C2×D414yes—
85C2×C4×C17C2×D412yes—
86C2×C4×C17C2×Q812yes—
87C2×C4×C17#4112yes—
88C2×Dic17—22no—
89C2×Dic17—22no—
90C2×Dic17—22no—
91C2×Dic17—22no—
92C2×Dic17—22no—
93C2×Dic17—22no—
94C2×Dic17—22no—
95C2×Dic17—22no—
96C2×Dic17—22no—
97D68—44no—
98C4×D17—48no—
99#2732—171nomaximal, Sylow
100C22×C17C22×D411yescharacteristic
101C22×C17C22×Q811yescharacteristic
102C22×C17C2×#4111yescharacteristic
103C22×C17H(5,𝔽2)11yescharacteristic
104C22×C17H(5,𝔽2)11yescharacteristic
105C22×C17H(5,𝔽2)11yescharacteristic
106C22×C172−1+411yescharacteristic
107C2×D17—24no—
108C2×D17—24no—
109C2×D17—24no—
110C2×D17—42no—
111C2×D17—42no—
112C2×D17—42no—
113C2×D17—42no—
114C4×C17—24no—
115C4×C17—22no—
116C4×C17—22no—
117C4×C17—22no—
118C4×C17—42no—
119Dic17—42no—
120Dic17—42no—
121Dic17—42no—
122C2×C22⋊Q8—172no—
123#399—172no—
124#398—172no—
125#396—172no—
126#391D1711yescharacteristic, complemented
127#390—171no—
128#389—171no—
129#386—172no—
130#385—172no—
131C2×C17#53411yescharacteristic
132C2×C17#53911yescharacteristic
133C2×C17#54311yescharacteristic
134C2×C17D4×Q811yescharacteristic
135C2×C17#55011yescharacteristic
136C2×C17#55411yescharacteristic
137C2×C17#56211yescharacteristic
138D17—44no—
139C22×Q8C2×D1711yescharacteristic
140C22×D4—171no—
141C23×C4—172no—
142C22⋊Q8—348no—
143C2×C4⋊C4—172no—
144C2×C4⋊C4—172no—
145C2×C4⋊C4—172no—
146C2×C4⋊C4—171no—
147C2×C4⋊C4—171no—
148C2×C4⋊C4—171no—
149C2×C4⋊C4—172no—
150C2×C4⋊C4C2×D1711yescharacteristic
151C2×C22⋊C4—172no—
152C2×C22⋊C4—172no—
153C2×C22⋊C4—172no—
154C2×C22⋊C4—172no—
155C2×C22⋊C4—172no—
156C2×C42C2×D1711yescharacteristic
157#95—172no—
158#95—172no—
159#95C2×D1712yes—
160#95C2×D1712yes—
161C17#273211yescharacteristic, complemented, Sylow
162C24—172no—
163C2×Q8—41no—
164C2×Q8—41no—
165C2×Q8D6814yescomplemented
166C2×D4—174no—
167C2×D4—681no—
168C2×D4—681no—
169C22×C4—172no—
170C22×C4—172no—
171C22×C4—172no—
172C22×C4—344no—
173C22×C4—682no—
174C22×C4—682no—
175C22×C4C22×D1712yes—
176C22×C4C22×D1712yes—
177C22×C4C22×D1711yescharacteristic
178C22×C4C22×D1711yescharacteristic
179C22×C4C22×D1711yescharacteristic
180C4⋊C4—41no—
181C4⋊C4—344no—
182C4⋊C4—344no—
183C4⋊C4—344no—
184C4⋊C4—342no—
185C4⋊C4—681no—
186C4⋊C4—681no—
187C4⋊C4—682no—
188C22⋊C4—344no—
189C22⋊C4—344no—
190C22⋊C4—344no—
191C22⋊C4—682no—
192C22⋊C4—682no—
193C42—41no—
194C23—174no—
195C23—342no—
196C23—342no—
197C23—342no—
198C23—342no—
199C23—342no—
200C23—342no—
201C23C23×D1711yescharacteristic
202Q8—44no—
203D4—684no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—342no—
216C2×C4—342no—
217C2×C4—342no—
218C2×C4—342no—
219C2×C4—342no—
220C2×C4—342no—
221C2×C4—342no—
222C2×C4—342no—
223C2×C4—342no—
224C2×C4—688no—
225C2×C4C2×D6812yes—
226C2×C4C2×D6814yes—
227C2×C4Q8×D1712yes—
228C2×C4#6325612yes—
229C22—344no—
230C22—344no—
231C22—344no—
232C22—682no—
233C22—682no—
234C22—682no—
235C22—682no—
236C22C22×D6811yescharacteristic
237C22#1058769111yescharacteristic
238C22#1058769111yescharacteristic
239C22C2×Q8×D1711yescharacteristic
240C22C2×#6325611yescharacteristic
241C22#1058769611yescharacteristic
242C22#1058769711yescharacteristic
243C4—24no—
244C4—22no—
245C4—22no—
246C4—22no—
247C4—42no—
248C4—682no—
249C4—682no—
250C4—682no—
251C2—684no—
252C2#1176337111yescharacteristic
253C2#1176340811yescharacteristic
254C2#1176343411yescharacteristic
255C2Q8×D6811yescharacteristic
256C2#1176348111yescharacteristic
257C2#1176352011yescharacteristic
258C2#1176355311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#391C2
prime radicalC22×D68C22
hypercenter#391D17
hypocenterC17#2732
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
1. C2×#10587632C2semidirect
2. #11762746C2semidirect
3. #11762741C2semidirect
4. #11762731C2semidirect
6. #11762707C2semidirect
8. #11762683C2semidirect
9. C17×#391C2semidirect
35. C2×Q8×C17D4semidirect
46. C2×D68Q8semidirect
126. #391D17semidirect
161. C17#2732semidirect
165. C2×Q8D68semidirect

Maximal subgroups

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

subgroupindex
C2×#105876322
#117627462
#117627412
#117627312
#117627112
#117627072
#117626872
#117626832
C17×#3912
#273217