Finite Group Databasebeta

← #14535743

Subgroups — #14535743

Order 2176.

476 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 133 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×#10587565C212yesmaximal, complemented
2#11762745C212yesmaximal, complemented
3#11762732C212yesmaximal, complemented
4#11762723C212yesmaximal
5#11762707C211yesmaximal, characteristic, complemented
6#11762707C211yesmaximal, characteristic, complemented
7#11762704C212yesmaximal, complemented
8#11762702C212yesmaximal, complemented
9C17×#390C211yesmaximal, characteristic
10C22×D4×C17C2211yescharacteristic
11C2×C4⋊C4×C17C2211yescharacteristic
12C2×C22⋊C4×C17C2212yes—
13C2×C22⋊C4×C17C2212yes—
14C2×C42×C17C2211yescharacteristic
15C2×C22⋊Dic17C2212yes—
16C2×C22⋊Dic17C2212yes—
17C2×C22⋊Dic17C2212yes—
18C2×C22⋊Dic17C2212yes—
19C2×C4⋊Dic17C2211yescharacteristic
20C2×C4⋊Dic17C2211yescharacteristic
21C2×C4⋊Dic17C2211yescharacteristic
22C2×C4⋊Dic17C2211yescharacteristic
23C2×#63225C2212yes—
24C2×#63225C2212yes—
25C2×#63225C2212yes—
26C2×#63225C2212yes—
27C2×C4×Dic17C2212yes—
28#10587565—28no—
29#10587502C2212yes—
30#10587502C2212yes—
31#10587502C2212yes—
32C24×C17C2312yes—
33C2×D4×C17—22no—
34C2×D4×C17—41no—
35C2×D4×C17—41no—
36C22×C4×C17C2312yes—
37C22×C4×C17C2311yescharacteristic
38C22×C4×C17C2311yescharacteristic
39C22×C4×C17C2311yescharacteristic
40C22×Dic17C2312yes—
41C22×Dic17C2312yes—
42C22×Dic17C2312yes—
43C22×Dic17C2312yes—
44C4⋊C4×C17—41no—
45C22⋊C4×C17—24no—
46C22⋊C4×C17—24no—
47C42×C17—41no—
48C22⋊Dic17—24no—
49C22⋊Dic17—24no—
50C22⋊Dic17—42no—
51C22⋊Dic17—42no—
52C4⋊Dic17—22no—
53C4⋊Dic17—41no—
54C4⋊Dic17—41no—
55C4⋊Dic17D414yescomplemented
56#63225—24no—
57#63225—24no—
58#63225—24no—
59#63225—42no—
60C4×Dic17—24no—
61C23×C17—22no—
62C23×C17—22no—
63C23×C17—22no—
64C23×C17—22no—
65C23×C17—22no—
66C23×C17—22no—
67C23×C17—22no—
68C23×C17C2411yescharacteristic
69D4×C17—44no—
70C2×C4×C17—21no—
71C2×C4×C17—21no—
72C2×C4×C17—21no—
73C2×C4×C17—21no—
74C2×C4×C17—21no—
75C2×C4×C17—21no—
76C2×C4×C17—22no—
77C2×C4×C17—22no—
78C2×C4×C17—22no—
79C2×C4×C17C2×D412yes—
80C2×C4×C17#4112yes—
81C2×C4×C17#4112yes—
82C2×Dic17—22no—
83C2×Dic17—22no—
84C2×Dic17—22no—
85C2×Dic17—22no—
86C2×Dic17—22no—
87C2×Dic17—22no—
88C2×Dic17—22no—
89C2×Dic17—22no—
90C2×Dic17—22no—
91C2×Dic17—22no—
92C2×Dic17—22no—
93C2×Dic17C2×D414yes—
94#2727—171nomaximal, Sylow
95C22×C17—42no—
96C22×C17—42no—
97C22×C17—42no—
98C22×C17—42no—
99C22×C17—42no—
100C22×C17—42no—
101C22×C17—42no—
102C22×C17C22×D411yescharacteristic
103C22×C17C2×#4111yescharacteristic
104C22×C17C2×#4111yescharacteristic
105C22×C17H(5,𝔽2)11yescharacteristic
106C22×C17H(5,𝔽2)11yescharacteristic
107C22×C172−1+411yescharacteristic
108C22×C172−1+411yescharacteristic
109C4×C17—22no—
110C4×C17—22no—
111C4×C17—22no—
112C4×C17—42no—
113Dic17—24no—
114Dic17—42no—
115Dic17—42no—
116Dic17—42no—
117C2×#126—172no—
118#399—172no—
119#396—172no—
120#394—172no—
121#390D1711yescharacteristic
122#389—172no—
123#389—171no—
124#389—171no—
125#388—172no—
126C2×C17—82no—
127C2×C17#53711yescharacteristic
128C2×C17#53811yescharacteristic
129C2×C17#53911yescharacteristic
130C2×C17#54711yescharacteristic
131C2×C17#55011yescharacteristic
132C2×C17#55611yescharacteristic
133C2×C17#56311yescharacteristic
134C22×D4C2×D1711yescharacteristic
135#126—348no—
136C2×C4⋊C4—171no—
137C2×C4⋊C4—172no—
138C2×C4⋊C4—172no—
139C2×C4⋊C4—172no—
140C2×C4⋊C4—171no—
141C2×C4⋊C4—171no—
142C2×C4⋊C4—171no—
143C2×C4⋊C4—172no—
144C2×C4⋊C4C2×D1711yescharacteristic
145C2×C22⋊C4—172no—
146C2×C22⋊C4—172no—
147C2×C22⋊C4—172no—
148C2×C22⋊C4—172no—
149C2×C22⋊C4C2×D1712yes—
150C2×C22⋊C4C2×D1712yes—
151C2×C42—172no—
152C2×C42C2×D1711yescharacteristic
153#95—172no—
154#95—172no—
155#95—172no—
156C17#272711yescharacteristic, complemented, Sylow
157C24C22×D1712yes—
158C2×D4—22no—
159C2×D4—41no—
160C2×D4—41no—
161C22×C4—172no—
162C22×C4—172no—
163C22×C4—172no—
164C22×C4—172no—
165C22×C4C22×D1712yes—
166C22×C4C22×D1711yescharacteristic
167C22×C4C22×D1711yescharacteristic
168C22×C4C22×D1711yescharacteristic
169C4⋊C4—41no—
170C4⋊C4—174no—
171C4⋊C4—344no—
172C4⋊C4—344no—
173C4⋊C4—344no—
174C4⋊C4—342no—
175C4⋊C4—682no—
176C4⋊C4—681no—
177C4⋊C4—681no—
178C22⋊C4—24no—
179C22⋊C4—24no—
180C22⋊C4—344no—
181C22⋊C4—344no—
182C22⋊C4—682no—
183C22⋊C4—682no—
184C42—41no—
185C42—344no—
186C23—22no—
187C23—22no—
188C23—22no—
189C23—22no—
190C23—22no—
191C23—22no—
192C23—22no—
193C23C23×D1711yescharacteristic
194D4—44no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—174no—
205C2×C4—342no—
206C2×C4—342no—
207C2×C4—342no—
208C2×C4—342no—
209C2×C4—342no—
210C2×C4—342no—
211C2×C4—342no—
212C2×C4—342no—
213C2×C4—342no—
214C2×C4—342no—
215C2×C4—342no—
216C2×C4#6325212yes—
217C2×C4D4×D1712yes—
218C2×C4#6325412yes—
219C22—42no—
220C22—42no—
221C22—42no—
222C22—42no—
223C22—42no—
224C22—42no—
225C22—42no—
226C22C2×#6325211yescharacteristic
227C22C2×D4×D1711yescharacteristic
228C22C2×#6325411yescharacteristic
229C22#1058769111yescharacteristic
230C22#1058769111yescharacteristic
231C22#1058769711yescharacteristic
232C22#1058769711yescharacteristic
233C4—22no—
234C4—22no—
235C4—22no—
236C4—42no—
237C4—344no—
238C4—682no—
239C4—682no—
240C4—682no—
241C2—82no—
242C2#1176338611yescharacteristic
243C2#1176339711yescharacteristic
244C2#1176340411yescharacteristic
245C2#1176346411yescharacteristic
246C2#1176347911yescharacteristic
247C2#1176352911yescharacteristic
248C2#1176355911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#390C2
prime radicalC22×D4×C17C22
hypercenter#390D17
hypocenterC17#2727
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
1. C2×#10587565C2semidirect
2. #11762745C2semidirect
3. #11762732C2semidirect
5. #11762707C2semidirect
6. #11762707C2semidirect
7. #11762704C2semidirect
8. #11762702C2semidirect
55. C4⋊Dic17D4semidirect
156. C17#2727semidirect

Maximal subgroups

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

subgroupindex
C2×#105875652
#117627452
#117627322
#117627232
#117627072
#117627042
#117627022
C17×#3902
#272717