Finite Group Databasebeta

← #14534166

Subgroups — #14534166

Order 2176.

468 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 161 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×#10587575C211yesmaximal, characteristic
2C2×Q8×Dic17C211yesmaximal, characteristic
3#11762741C211yesmaximal, characteristic
4#11762739C211yesmaximal, characteristic
5#11762728C211yesmaximal, characteristic
6#11762708C212yesmaximal
7#11762706C212yesmaximal
8#11762706C212yesmaximal
9#11762704C212yesmaximal
10#11762660C211yesmaximal, characteristic
11C17×#395C211yesmaximal, characteristic
12C22×Q8×C17C2211yescharacteristic
13C2×C4⋊C4×C17C2212yes—
14C2×C4⋊C4×C17C2211yescharacteristic
15Q8×Dic17—24no—
16C2×C4⋊Dic17C2212yes—
17C2×C4⋊Dic17C2211yescharacteristic
18C2×C4⋊Dic17C2211yescharacteristic
19C2×C4⋊Dic17C2212yes—
20C2×#63225C2211yescharacteristic
21C2×#63225C2212yes—
22C2×#63225C2211yescharacteristic
23C2×#63225C2211yescharacteristic
24C2×#63225C2211yescharacteristic
25C2×#63225C2212yes—
26C2×#63225C2212yes—
27C2×C4×Dic17C2212yes—
28C2×C4×Dic17C2212yes—
29C2×C4×Dic17C2211yescharacteristic
30#10587575—24no—
31C17×#95C2212yes—
32C17×#95C2211yescharacteristic
33#10587502C2212yes—
34#10587502C2212yes—
35#10587502C2212yes—
36#10587502C2211yescharacteristic
37C2×Q8×C17—22no—
38C2×Q8×C17—24no—
39C22×C4×C17C2311yescharacteristic
40C22×C4×C17C2311yescharacteristic
41C22×C4×C17C2311yescharacteristic
42C22×C4×C17C2312yes—
43C22×C4×C17C2312yes—
44C22×Dic17C2312yes—
45C22×Dic17C2311yescharacteristic
46C22×Dic17C2311yescharacteristic
47C22×Dic17C2311yescharacteristic
48C22×Dic17C2312yes—
49C22×Dic17C2311yescharacteristic
50C4⋊C4×C17—22no—
51C4⋊C4×C17—24no—
52C4⋊Dic17—22no—
53C4⋊Dic17—22no—
54C4⋊Dic17—24no—
55C4⋊Dic17—42no—
56#63225—22no—
57#63225—24no—
58#63225—22no—
59#63225—22no—
60#63225—22no—
61#63225—42no—
62#63225Q818yescomplemented
63C4×Dic17—24no—
64C4×Dic17—42no—
65C4×Dic17D414yes—
66C23×C17C2411yescharacteristic
67Q8×C17—28no—
68C2×C4×C17—22no—
69C2×C4×C17—22no—
70C2×C4×C17—22no—
71C2×C4×C17—22no—
72C2×C4×C17—22no—
73C2×C4×C17—21no—
74C2×C4×C17—21no—
75C2×C4×C17—22no—
76C2×C4×C17—21no—
77C2×C4×C17—21no—
78C2×C4×C17C2×D412yes—
79C2×C4×C17C2×Q814yes—
80C2×C4×C17#4114yes—
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—21no—
91C2×Dic17—22no—
92C2×Dic17—21no—
93C2×Dic17C2×D412yes—
94C2×Dic17C2×D412yes—
95C2×Dic17C2×Q814yes—
96C2×Dic17C2×Q814yes—
97#2584—171nomaximal, Sylow
98C22×C17C22×D411yescharacteristic
99C22×C17C22×Q812yes—
100C22×C17C2×#4112yes—
101C22×C17H(5,𝔽2)11yescharacteristic
102C22×C172−1+411yescharacteristic
103C4×C17—22no—
104C4×C17—24no—
105C4×C17—24no—
106C4×C17—41no—
107C4×C17—41no—
108Dic17—22no—
109Dic17—24no—
110Dic17—41no—
111Dic17—42no—
112Dic17—41no—
113Dic17C4⋊Q814yes—
114C2×#125—171no—
115C2×C4×Q8—171no—
116#398—171no—
117#398—171no—
118#395—171no—
119#395D1711yescharacteristic
120#389—172no—
121#389—172no—
122#389—172no—
123#389—172no—
124C4×C4⋊C4—171no—
125C2×C17#52511yescharacteristic
126C2×C17C2×C4⋊Q811yescharacteristic
127C2×C17#53711yescharacteristic
128C2×C17#55412yes—
129C2×C17#55712yes—
130C22×Q8C2×D1711yescharacteristic
131#125—344no—
132C4×Q8—344no—
133C2×C4⋊C4—172no—
134C2×C4⋊C4—172no—
135C2×C4⋊C4—171no—
136C2×C4⋊C4—171no—
137C2×C4⋊C4—171no—
138C2×C4⋊C4—171no—
139C2×C4⋊C4—171no—
140C2×C4⋊C4—171no—
141C2×C4⋊C4—172no—
142C2×C4⋊C4—172no—
143C2×C4⋊C4—172no—
144C2×C4⋊C4C2×D1711yescharacteristic
145C2×C4⋊C4C2×D1712yes—
146C2×C42—171no—
147C2×C42—172no—
148C2×C42—172no—
149#95—171no—
150#95—172no—
151#95—172no—
152#95—172no—
153#95C2×D1711yescharacteristic
154#95C2×D1712yes—
155C17#258411yescharacteristic, complemented, Sylow
156C2×Q8—22no—
157C2×Q8—24no—
158C22×C4—171no—
159C22×C4—172no—
160C22×C4—171no—
161C22×C4—172no—
162C22×C4—171no—
163C22×C4—171no—
164C22×C4C22×D1711yescharacteristic
165C22×C4C22×D1711yescharacteristic
166C22×C4C22×D1711yescharacteristic
167C22×C4C22×D1712yes—
168C22×C4C22×D1712yes—
169C4⋊C4—22no—
170C4⋊C4—24no—
171C4⋊C4—178no—
172C4⋊C4—344no—
173C4⋊C4—342no—
174C4⋊C4—342no—
175C4⋊C4—342no—
176C4⋊C4—342no—
177C4⋊C4—342no—
178C4⋊C4—342no—
179C4⋊C4—344no—
180C4⋊C4—682no—
181C4⋊C4—682no—
182C42—174no—
183C42—344no—
184C42—682no—
185C23C23×D1711yescharacteristic
186Q8—28no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—22no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—172no—
198C2×C4—174no—
199C2×C4—172no—
200C2×C4—174no—
201C2×C4—342no—
202C2×C4—341no—
203C2×C4—341no—
204C2×C4—342no—
205C2×C4—342no—
206C2×C4—341no—
207C2×C4—341no—
208C2×C4—342no—
209C2×C4—342no—
210C2×C4—341no—
211C2×C4—342no—
212C2×C4—341no—
213C2×C4D4×D1712yes—
214C2×C4#6325414yes—
215C2×C4Q8×D1714yes—
216C22C2×D4×D1711yescharacteristic
217C22C2×#6325412yes—
218C22C2×Q8×D1712yes—
219C22#1058769411yescharacteristic
220C22#1058769611yescharacteristic
221C4—24no—
222C4—22no—
223C4—24no—
224C4—41no—
225C4—41no—
226C4—174no—
227C4—344no—
228C4—342no—
229C4—681no—
230C4—682no—
231C4—681no—
232C2#1176331511yescharacteristic
233C2C4⋊Q8×D1711yescharacteristic
234C2#1176338911yescharacteristic
235C2#1176351712yes—
236C2#1176353412yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, prime radical, cosocleC23×C17C24
FittingC17×#395C2
hypercenter#395D17
hypocenterC17#2584
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
62. #63225Q8semidirect
155. C17#2584semidirect

Maximal subgroups

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

subgroupindex
C2×#105875752
C2×Q8×Dic172
#117627412
#117627392
#117627282
#117627082
#117627062
#117627042
#117626602
C17×#3952
#258417