Finite Group Databasebeta

← #14535711

Subgroups — #14535711

Order 2176.

504 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 129 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×#10587580C212yesmaximal
2#11762739C211yesmaximal, characteristic, complemented
3#11762738C212yesmaximal, complemented
4#11762738C212yesmaximal, complemented
5#11762710C211yesmaximal, characteristic
6#11762704C212yesmaximal, complemented
7#11762683C212yesmaximal, complemented
8#11762680C212yesmaximal, complemented
9C17×#398C211yesmaximal, characteristic, complemented
10C22×D68C2211yescharacteristic
11C22×C4×D17C2212yes—
12C2×C4⋊C4×C17C2212yes—
13C2×C4⋊C4×C17C2211yescharacteristic
14C2×C4⋊C4×C17C2211yescharacteristic
15C2×#63227C2212yes—
16C2×#63227C2212yes—
17C2×#63227C2212yes—
18C2×#63227C2212yes—
19C2×#63227C2212yes—
20C2×C4⋊Dic17C2211yescharacteristic
21C2×#63225C2212yes—
22C2×#63225C2212yes—
23C2×#63225C2212yes—
24C2×C4×Dic17C2211yescharacteristic
25#10587580—28no—
26C17×#95C2211yescharacteristic
27C17×#95C2212yes—
28#10587502C2212yes—
29#10587502C2212yes—
30#10587502C2211yescharacteristic
31#10587502C2212yes—
32C23×D17C2312yes—
33C22×C4×C17C2312yes—
34C22×C4×C17C2311yescharacteristic
35C22×C4×C17C2312yes—
36C22×C4×C17C2311yescharacteristic
37C22×C4×C17C2311yescharacteristic
38C22×Dic17C2312yes—
39C22×Dic17C2312yes—
40C22×Dic17C2312yes—
41C2×D68—41no—
42C2×D68—41no—
43C2×D68D414yes—
44C2×C4×D17—24no—
45C2×C4×D17—42no—
46C2×C4×D17—42no—
47C4⋊C4×C17—24no—
48C4⋊C4×C17—41no—
49C4⋊C4×C17—41no—
50#63227—24no—
51#63227—24no—
52#63227—24no—
53#63227—24no—
54#63227—24no—
55C4⋊Dic17—41no—
56#63225—24no—
57#63225—24no—
58#63225—42no—
59C4×Dic17—41no—
60C23×C17C2411yescharacteristic
61C22×D17—22no—
62C22×D17—22no—
63C22×D17—22no—
64C22×D17—22no—
65C22×D17—22no—
66C22×D17—22no—
67C22×D17C2×D414yes—
68C2×C4×C17—21no—
69C2×C4×C17—21no—
70C2×C4×C17—22no—
71C2×C4×C17—22no—
72C2×C4×C17—22no—
73C2×C4×C17—22no—
74C2×C4×C17—22no—
75C2×C4×C17—21no—
76C2×C4×C17—22no—
77C2×C4×C17—21no—
78C2×C4×C17—22no—
79C2×C4×C17—22no—
80C2×C4×C17C2×D412yes—
81C2×Dic17—22no—
82C2×Dic17—22no—
83C2×Dic17—22no—
84C2×Dic17—22no—
85C2×Dic17—22no—
86C2×Dic17—22no—
87C2×Dic17—22no—
88C2×Dic17—22no—
89C2×Dic17#4114yes—
90D68—44no—
91C4×D17—48no—
92#2724—171nomaximal, Sylow
93C22×C17C22×D411yescharacteristic
94C22×C17C2×#4112yes—
95C22×C17H(5,𝔽2)11yescharacteristic
96C22×C17H(5,𝔽2)11yescharacteristic
97C22×C172−1+412yes—
98C2×D17—24no—
99C2×D17—24no—
100C2×D17—24no—
101C2×D17—42no—
102C2×D17—42no—
103C2×D17—42no—
104C2×D17—42no—
105C4×C17—22no—
106C4×C17—42no—
107C4×C17—42no—
108C4×C17—41no—
109C4×C17—41no—
110Dic17—24no—
111Dic17—42no—
112Dic17—42no—
113C2×#123—172no—
114#398—171no—
115#398D1711yescharacteristic, complemented
116#397—172no—
117#397—172no—
118#390—171no—
119#389—172no—
120#385—172no—
121#385—172no—
122C2×C17#53411yescharacteristic
123C2×C17#53912yes—
124C2×C17#54712yes—
125C2×C17#55911yescharacteristic
126C2×C17#56311yescharacteristic
127D17—44no—
128C22×D4—171no—
129C23×C4—172no—
130#123—348no—
131C2×C4⋊C4—172no—
132C2×C4⋊C4—172no—
133C2×C4⋊C4—172no—
134C2×C4⋊C4—171no—
135C2×C4⋊C4C2×D1712yes—
136C2×C4⋊C4C2×D1711yescharacteristic
137C2×C4⋊C4C2×D1711yescharacteristic
138C2×C22⋊C4—172no—
139C2×C22⋊C4—172no—
140C2×C22⋊C4—172no—
141C2×C22⋊C4—172no—
142C2×C22⋊C4—172no—
143C2×C42—171no—
144#95—172no—
145#95—172no—
146#95—172no—
147#95—171no—
148#95C2×D1712yes—
149#95C2×D1711yescharacteristic
150C17#272411yescharacteristic, complemented, Sylow
151C24—172no—
152C2×D4—174no—
153C2×D4—681no—
154C2×D4—681no—
155C22×C4—172no—
156C22×C4—172no—
157C22×C4—172no—
158C22×C4—344no—
159C22×C4—682no—
160C22×C4—682no—
161C22×C4C22×D1711yescharacteristic
162C22×C4C22×D1712yes—
163C22×C4C22×D1711yescharacteristic
164C22×C4C22×D1712yes—
165C22×C4C22×D1711yescharacteristic
166C4⋊C4—24no—
167C4⋊C4—41no—
168C4⋊C4—41no—
169C4⋊C4—344no—
170C4⋊C4—344no—
171C4⋊C4—682no—
172C4⋊C4—681no—
173C22⋊C4—344no—
174C22⋊C4—344no—
175C22⋊C4—344no—
176C22⋊C4—344no—
177C22⋊C4—344no—
178C42—681no—
179C23—174no—
180C23—342no—
181C23—342no—
182C23—342no—
183C23—342no—
184C23—342no—
185C23—342no—
186C23C23×D1711yescharacteristic
187D4—684no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—174no—
201C2×C4—342no—
202C2×C4—342no—
203C2×C4—342no—
204C2×C4—342no—
205C2×C4—342no—
206C2×C4—342no—
207C2×C4—342no—
208C2×C4—342no—
209C2×C4—688no—
210C2×C4D4×D1712yes—
211C22—344no—
212C22—344no—
213C22—344no—
214C22—682no—
215C22—682no—
216C22—682no—
217C22—682no—
218C22C2×D4×D1711yescharacteristic
219C22#1058769111yescharacteristic
220C22#1058769412yes—
221C22#41×D1712yes—
222C22#1058769611yescharacteristic
223C4—22no—
224C4—42no—
225C4—41no—
226C4—41no—
227C4—42no—
228C4—344no—
229C4—682no—
230C4—682no—
231C2—684no—
232C2#1176336711yescharacteristic
233C2#1176340712yes—
234C2#1176346112yes—
235C2#1176354311yescharacteristic
236C2#1176355811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#398C2
prime radicalC22×D68C22
hypercenter#398D17
hypocenterC17#2724
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
2. #11762739C2semidirect
3. #11762738C2semidirect
4. #11762738C2semidirect
6. #11762704C2semidirect
7. #11762683C2semidirect
8. #11762680C2semidirect
9. C17×#398C2semidirect
115. #398D17semidirect
150. C17#2724semidirect

Maximal subgroups

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

subgroupindex
C2×#105875802
#117627392
#117627382
#117627102
#117627042
#117626832
#117626802
C17×#3982
#272417