Finite Group Databasebeta

← #14534902

Subgroups — #14534902

Order 2176.

448 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 153 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×#10587630C211yesmaximal, characteristic
2C2×#10587575C211yesmaximal, characteristic
3#11762743C212yesmaximal
4#11762742C212yesmaximal
5#11762730C211yesmaximal, characteristic
6#11762729C211yesmaximal, characteristic
7#11762728C211yesmaximal, characteristic
8#11762678C212yesmaximal
9#11762673C212yesmaximal
10#11762654C211yesmaximal, characteristic
11C17×#395C211yesmaximal, characteristic
12C22×Q8×C17C2211yescharacteristic
13C22×Dic34C2211yescharacteristic
14C2×C4⋊C4×C17C2211yescharacteristic
15C2×C4⋊C4×C17C2212yes—
16#10587630—24no—
17C2×C4⋊Dic17C2211yescharacteristic
18C2×C4⋊Dic17C2212yes—
19C2×#63225C2211yescharacteristic
20C2×#63225C2212yes—
21C2×#63225C2211yescharacteristic
22C2×#63225C2211yescharacteristic
23C2×#63225C2211yescharacteristic
24C2×#63225C2212yes—
25C2×#63225C2212yes—
26C2×#63225C2212yes—
27C2×C4×Dic17C2211yescharacteristic
28C2×C4×Dic17C2212yes—
29#10587575—24no—
30C17×#95C2211yescharacteristic
31C17×#95C2212yes—
32#10587502C2211yescharacteristic
33#10587502C2212yes—
34#10587502C2212yes—
35#10587502C2212yes—
36#10587502C2212yes—
37C2×Q8×C17—22no—
38C2×Q8×C17—42no—
39C22×C4×C17C2312yes—
40C22×C4×C17C2311yescharacteristic
41C22×C4×C17C2311yescharacteristic
42C22×C4×C17C2311yescharacteristic
43C22×C4×C17C2312yes—
44C22×Dic17C2312yes—
45C22×Dic17C2311yescharacteristic
46C22×Dic17C2311yescharacteristic
47C22×Dic17C2311yescharacteristic
48C22×Dic17C2312yes—
49C22×Dic17C2311yescharacteristic
50C2×Dic34—22no—
51C2×Dic34—42no—
52C4⋊C4×C17—22no—
53C4⋊C4×C17—42no—
54C4⋊Dic17—22no—
55C4⋊Dic17—42no—
56#63225—22no—
57#63225—22no—
58#63225—22no—
59#63225—22no—
60#63225—24no—
61#63225—24no—
62#63225—42no—
63#63225—42no—
64C4×Dic17D414yes—
65C4×Dic17Q818yescomplemented
66C23×C17C2411yescharacteristic
67Q8×C17—44no—
68C2×C4×C17—22no—
69C2×C4×C17—22no—
70C2×C4×C17—22no—
71C2×C4×C17—21no—
72C2×C4×C17—21no—
73C2×C4×C17—22no—
74C2×C4×C17—22no—
75C2×C4×C17—22no—
76C2×C4×C17—22no—
77C2×C4×C17—21no—
78C2×C4×C17—21no—
79C2×C4×C17C2×D412yes—
80C2×C4×C17C2×Q814yes—
81C2×Dic17—22no—
82C2×Dic17—21no—
83C2×Dic17—22no—
84C2×Dic17—21no—
85C2×Dic17—22no—
86C2×Dic17—21no—
87C2×Dic17—21no—
88C2×Dic17—22no—
89C2×Dic17—22no—
90C2×Dic17—22no—
91C2×Dic17—21no—
92C2×Dic17—21no—
93C2×Dic17C2×D412yes—
94C2×Dic17C2×D412yes—
95C2×Dic17C2×Q814yes—
96C2×Dic17C2×Q814yes—
97Dic34—44no—
98#2660—171nomaximal, Sylow
99C22×C17C22×D411yescharacteristic
100C22×C17C22×Q812yes—
101C22×C17H(5,𝔽2)12yes—
102C22×C172−1+411yescharacteristic
103C22×C172−1+411yescharacteristic
104C4×C17—22no—
105C4×C17—24no—
106C4×C17—41no—
107C4×C17—42no—
108C4×C17—41no—
109Dic17—22no—
110Dic17—24no—
111Dic17—42no—
112Dic17—41no—
113Dic17—41no—
114Dic17C4⋊Q814yes—
115C2×C4⋊Q8—171no—
116C2×#125—171no—
117#398—172no—
118#398—172no—
119#395—171no—
120#395—171no—
121#395—171no—
122#395D1711yescharacteristic
123#384—172no—
124#382—172no—
125#376—171no—
126C2×C17C2×C4⋊Q811yescharacteristic
127C2×C17#53511yescharacteristic
128C2×C17#53611yescharacteristic
129C2×C17#54412yes—
130C2×C17#54412yes—
131C22×Q8—171no—
132C22×Q8C2×D1711yescharacteristic
133C4⋊Q8—344no—
134#125—344no—
135C2×C4⋊C4—171no—
136C2×C4⋊C4—171no—
137C2×C4⋊C4—171no—
138C2×C4⋊C4—171no—
139C2×C4⋊C4—171no—
140C2×C4⋊C4—172no—
141C2×C4⋊C4—172no—
142C2×C4⋊C4—172no—
143C2×C4⋊C4—172no—
144C2×C4⋊C4—172no—
145C2×C4⋊C4C2×D1711yescharacteristic
146C2×C4⋊C4C2×D1712yes—
147C2×C42—171no—
148C2×C42—172no—
149#95—171no—
150#95—172no—
151#95—172no—
152#95—172no—
153#95—172no—
154#95C2×D1711yescharacteristic
155#95C2×D1712yes—
156C17#266011yescharacteristic, complemented, Sylow
157C2×Q8—22no—
158C2×Q8—42no—
159C2×Q8—342no—
160C2×Q8—682no—
161C22×C4—171no—
162C22×C4—172no—
163C22×C4—171no—
164C22×C4—171no—
165C22×C4—172no—
166C22×C4—171no—
167C22×C4C22×D1711yescharacteristic
168C22×C4C22×D1712yes—
169C22×C4C22×D1711yescharacteristic
170C22×C4C22×D1711yescharacteristic
171C22×C4C22×D1712yes—
172C4⋊C4—22no—
173C4⋊C4—42no—
174C4⋊C4—342no—
175C4⋊C4—342no—
176C4⋊C4—342no—
177C4⋊C4—342no—
178C4⋊C4—344no—
179C4⋊C4—344no—
180C4⋊C4—342no—
181C4⋊C4—682no—
182C4⋊C4—682no—
183C4⋊C4—682no—
184C42—174no—
185C42—178no—
186C23C23×D1711yescharacteristic
187Q8—44no—
188Q8—684no—
189C2×C4—22no—
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—21no—
199C2×C4—21no—
200C2×C4—172no—
201C2×C4—174no—
202C2×C4—172no—
203C2×C4—174no—
204C2×C4—342no—
205C2×C4—341no—
206C2×C4—342no—
207C2×C4—341no—
208C2×C4—341no—
209C2×C4—341no—
210C2×C4—342no—
211C2×C4—342no—
212C2×C4—342no—
213C2×C4—342no—
214C2×C4—341no—
215C2×C4—341no—
216C2×C4D4×D1712yes—
217C2×C4Q8×D1714yes—
218C22C2×D4×D1711yescharacteristic
219C22#1058769112yes—
220C22C2×Q8×D1712yes—
221C22#1058769411yescharacteristic
222C22#1058769711yescharacteristic
223C4—22no—
224C4—24no—
225C4—42no—
226C4—41no—
227C4—41no—
228C4—174no—
229C4—342no—
230C4—344no—
231C4—682no—
232C4—681no—
233C4—681no—
234C2C4⋊Q8×D1711yescharacteristic
235C2#1176337411yescharacteristic
236C2#1176338211yescharacteristic
237C2#1176343612yes—
238C2#1176343812yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, prime radical, cosocleC23×C17C24
FittingC17×#395C2
hypercenter#395D17
hypocenterC17#2660
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
65. C4×Dic17Q8semidirect
156. C17#2660semidirect

Maximal subgroups

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

subgroupindex
C2×#105876302
C2×#105875752
#117627432
#117627422
#117627302
#117627292
#117627282
#117626782
#117626732
#117626542
C17×#3952
#266017