Finite Group Databasebeta

← #14533768

Subgroups — #14533768

Order 2176.

540 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 171 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×#10587574C211yesmaximal, characteristic
2C2×#10587574C212yesmaximal
3C2×#10587573C211yesmaximal, characteristic
4#11762741C211yesmaximal, characteristic
5#11762740C211yesmaximal, characteristic
6#11762739C211yesmaximal, characteristic
7#11762713C212yesmaximal
8#11762706C212yesmaximal
9#11762704C212yesmaximal
10#11762660C211yesmaximal, characteristic
11C17×#398C211yesmaximal, characteristic
12C22×Dic34C2212yes—
13C22×Dic34C2211yescharacteristic
14C2×C4⋊C4×C17C2212yes—
15C2×C4⋊C4×C17C2211yescharacteristic
16C2×C4⋊C4×C17C2211yescharacteristic
17C2×C4⋊Dic17C2212yes—
18C2×C4⋊Dic17C2211yescharacteristic
19C2×#63225C2212yes—
20C2×#63225C2212yes—
21C2×#63225C2212yes—
22C2×#63225C2211yescharacteristic
23C2×#63225C2211yescharacteristic
24C2×#63225C2212yes—
25C2×C4×Dic17C2211yescharacteristic
26C2×C4×Dic17C2211yescharacteristic
27C2×C4×Dic17C2212yes—
28C2×C4×Dic17C2211yescharacteristic
29#10587574—24no—
30#10587574—28no—
31#10587573—24no—
32C17×#95C2212yes—
33C17×#95C2211yescharacteristic
34#10587502C2212yes—
35#10587502C2212yes—
36#10587502C2212yes—
37#10587502C2211yescharacteristic
38C22×C4×C17C2312yes—
39C22×C4×C17C2311yescharacteristic
40C22×C4×C17C2311yescharacteristic
41C22×C4×C17C2311yescharacteristic
42C22×C4×C17C2312yes—
43C22×Dic17C2312yes—
44C22×Dic17C2311yescharacteristic
45C22×Dic17C2311yescharacteristic
46C22×Dic17C2312yes—
47C22×Dic17C2311yescharacteristic
48C22×Dic17C2311yescharacteristic
49C2×Dic34—22no—
50C2×Dic34—22no—
51C2×Dic34—24no—
52C2×Dic34—24no—
53C2×Dic34—24no—
54C2×Dic34D414yes—
55C4⋊C4×C17—22no—
56C4⋊C4×C17—24no—
57C4⋊C4×C17—22no—
58C4⋊Dic17—22no—
59C4⋊Dic17—24no—
60#63225—22no—
61#63225—24no—
62#63225—22no—
63#63225—24no—
64#63225—24no—
65#63225Q818yescomplemented
66C4×Dic17—22no—
67C4×Dic17—22no—
68C4×Dic17—24no—
69C4×Dic17D414yes—
70C23×C17C2411yescharacteristic
71C2×C4×C17—21no—
72C2×C4×C17—22no—
73C2×C4×C17—21no—
74C2×C4×C17—22no—
75C2×C4×C17—22no—
76C2×C4×C17—22no—
77C2×C4×C17—22no—
78C2×C4×C17—21no—
79C2×C4×C17—22no—
80C2×C4×C17—22no—
81C2×C4×C17C2×D412yes—
82C2×C4×C17C2×D412yes—
83C2×C4×C17C2×Q814yes—
84C2×Dic17—21no—
85C2×Dic17—22no—
86C2×Dic17—21no—
87C2×Dic17—22no—
88C2×Dic17—22no—
89C2×Dic17—22no—
90C2×Dic17—21no—
91C2×Dic17—22no—
92C2×Dic17—22no—
93C2×Dic17C2×D412yes—
94C2×Dic17C2×D412yes—
95C2×Dic17C2×D412yes—
96C2×Dic17C2×D412yes—
97C2×Dic17C2×Q814yes—
98C2×Dic17C2×Q814yes—
99C2×Dic17#4112yes—
100Dic34—28no—
101Dic34—48no—
102#2546—171nomaximal, Sylow
103C22×C17C22×D411yescharacteristic
104C22×C17C22×D411yescharacteristic
105C22×C17C22×Q812yes—
106C22×C17C2×#4111yescharacteristic
107C22×C172−1+412yes—
108C4×C17—22no—
109C4×C17—22no—
110C4×C17—24no—
111C4×C17—42no—
112C4×C17—41no—
113Dic17—22no—
114Dic17—24no—
115Dic17—42no—
116Dic17—41no—
117Dic17#12114yes—
118Dic17C4⋊Q814yescomplemented
119C2×C4⋊Q8—171no—
120C2×C4⋊Q8—172no—
121C2×C4×Q8—171no—
122#398—171no—
123#398—171no—
124#398—171no—
125#398D1711yescharacteristic
126#391—172no—
127#389—172no—
128#389—172no—
129C4×C4⋊C4—171no—
130C2×C17C2×#12111yescharacteristic
131C2×C17C2×C4⋊Q811yescharacteristic
132C2×C17#53611yescharacteristic
133C2×C17D4×Q812yes—
134C2×C17#55712yes—
135C22×Q8—171no—
136C22×Q8—172no—
137C4⋊Q8—344no—
138C4⋊Q8—348no—
139C4×Q8—344no—
140C2×C4⋊C4—172no—
141C2×C4⋊C4—171no—
142C2×C4⋊C4—171no—
143C2×C4⋊C4—172no—
144C2×C4⋊C4—172no—
145C2×C4⋊C4—172no—
146C2×C4⋊C4—172no—
147C2×C4⋊C4—171no—
148C2×C4⋊C4C2×D1711yescharacteristic
149C2×C4⋊C4C2×D1712yes—
150C2×C4⋊C4C2×D1711yescharacteristic
151C2×C42—171no—
152C2×C42—171no—
153C2×C42—171no—
154C2×C42—172no—
155#95—171no—
156#95—172no—
157#95—172no—
158#95—172no—
159#95C2×D1711yescharacteristic
160#95C2×D1712yes—
161C17#254611yescharacteristic, complemented, Sylow
162C2×Q8—174no—
163C2×Q8—342no—
164C2×Q8—342no—
165C2×Q8—344no—
166C2×Q8—344no—
167C2×Q8—344no—
168C22×C4—171no—
169C22×C4—171no—
170C22×C4—172no—
171C22×C4—171no—
172C22×C4—171no—
173C22×C4—172no—
174C22×C4C22×D1711yescharacteristic
175C22×C4C22×D1712yes—
176C22×C4C22×D1711yescharacteristic
177C22×C4C22×D1711yescharacteristic
178C22×C4C22×D1712yes—
179C4⋊C4—22no—
180C4⋊C4—24no—
181C4⋊C4—22no—
182C4⋊C4—178no—
183C4⋊C4—342no—
184C4⋊C4—344no—
185C4⋊C4—344no—
186C4⋊C4—344no—
187C4⋊C4—342no—
188C4⋊C4—344no—
189C4⋊C4—342no—
190C42—174no—
191C42—342no—
192C42—342no—
193C42—344no—
194C23C23×D1711yescharacteristic
195Q8—348no—
196Q8—688no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—21no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—172no—
208C2×C4—172no—
209C2×C4—172no—
210C2×C4—174no—
211C2×C4—172no—
212C2×C4—172no—
213C2×C4—174no—
214C2×C4—342no—
215C2×C4—341no—
216C2×C4—341no—
217C2×C4—342no—
218C2×C4—341no—
219C2×C4—342no—
220C2×C4—342no—
221C2×C4—342no—
222C2×C4—342no—
223C2×C4D4×D1712yes—
224C2×C4D4×D1712yes—
225C2×C4Q8×D1714yes—
226C22C2×D4×D1711yescharacteristic
227C22C2×D4×D1711yescharacteristic
228C22C2×Q8×D1712yes—
229C22#41×D1711yescharacteristic
230C22#1058769712yes—
231C4—24no—
232C4—22no—
233C4—22no—
234C4—42no—
235C4—41no—
236C4—174no—
237C4—174no—
238C4—344no—
239C4—342no—
240C4—682no—
241C4—681no—
242C2#121×D1711yescharacteristic
243C2C4⋊Q8×D1711yescharacteristic
244C2#1176338111yescharacteristic
245C2#1176347212yes—
246C2#1176353612yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, prime radical, cosocleC23×C17C24
FittingC17×#398C2
hypercenter#398D17
hypocenterC17#2546
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
65. #63225Q8semidirect
118. Dic17C4⋊Q8semidirect
161. C17#2546semidirect

Maximal subgroups

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

subgroupindex
C2×#105875742
C2×#105875732
#117627412
#117627402
#117627392
#117627132
#117627062
#117627042
#117626602
C17×#3982
#254617