Finite Group Databasebeta

← #14534145

Subgroups — #14534145

Order 2176.

468 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 151 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×Q8×Dic17C211yesmaximal, characteristic
2C2×#10587573C211yesmaximal, characteristic
3#11762749C211yesmaximal, characteristic
4#11762748C211yesmaximal, characteristic
5#11762728C211yesmaximal, characteristic
6#11762713C212yesmaximal
7#11762704C212yesmaximal
8#11762696C212yesmaximal
9#11762695C212yesmaximal
10#11762673C211yesmaximal, characteristic
11C17×#395C211yesmaximal, characteristic
12C22×Q8×C17C2211yescharacteristic
13C22×Dic34C2211yescharacteristic
14C2×C4⋊C4×C17C2212yes—
15C2×C4⋊C4×C17C2211yescharacteristic
16Q8×Dic17—24no—
17C2×C4⋊Dic17C2212yes—
18C2×C4⋊Dic17C2211yescharacteristic
19C2×#63225C2212yes—
20C2×#63225C2211yescharacteristic
21C2×#63225C2212yes—
22C2×#63225C2211yescharacteristic
23C2×C4×Dic17C2211yescharacteristic
24C2×C4×Dic17C2211yescharacteristic
25C2×C4×Dic17C2212yes—
26C2×C4×Dic17C2211yescharacteristic
27#10587573—24no—
28C17×#95C2212yes—
29C17×#95C2211yescharacteristic
30#10587502C2212yes—
31#10587502C2212yes—
32#10587502C2212yes—
33#10587502C2212yes—
34#10587502C2212yes—
35#10587502C2212yes—
36#10587502C2211yescharacteristic
37C2×Q8×C17—22no—
38C2×Q8×C17—24no—
39C22×C4×C17C2312yes—
40C22×C4×C17C2311yescharacteristic
41C22×C4×C17C2311yescharacteristic
42C22×C4×C17C2311yescharacteristic
43C22×C4×C17C2312yes—
44C22×Dic17C2312yes—
45C22×Dic17C2312yes—
46C22×Dic17C2311yescharacteristic
47C22×Dic17C2311yescharacteristic
48C22×Dic17C2311yescharacteristic
49C22×Dic17C2311yescharacteristic
50C2×Dic34—22no—
51C2×Dic34—22no—
52C2×Dic34Q814yescomplemented
53C4⋊C4×C17—22no—
54C4⋊C4×C17—24no—
55C4⋊Dic17—22no—
56C4⋊Dic17—24no—
57#63225—22no—
58#63225—24no—
59#63225—22no—
60#63225—24no—
61C4×Dic17—22no—
62C4×Dic17—22no—
63C4×Dic17—24no—
64C4×Dic17D414yes—
65C23×C17C2411yescharacteristic
66Q8×C17—28no—
67C2×C4×C17—21no—
68C2×C4×C17—22no—
69C2×C4×C17—21no—
70C2×C4×C17—22no—
71C2×C4×C17—22no—
72C2×C4×C17—21no—
73C2×C4×C17—22no—
74C2×C4×C17—22no—
75C2×C4×C17—22no—
76C2×C4×C17—22no—
77C2×C4×C17C2×D412yes—
78C2×C4×C17C2×Q812yes—
79C2×C4×C17#4114yes—
80C2×Dic17—21no—
81C2×Dic17—21no—
82C2×Dic17—22no—
83C2×Dic17—21no—
84C2×Dic17—22no—
85C2×Dic17—22no—
86C2×Dic17—22no—
87C2×Dic17—22no—
88C2×Dic17—24no—
89C2×Dic17—22no—
90C2×Dic17C2×D412yes—
91C2×Dic17C2×D412yes—
92C2×Dic17C2×Q812yes—
93C2×Dic17C2×Q812yes—
94C2×Dic17#4112yes—
95Dic34—28no—
96#2581—171nomaximal, Sylow
97C22×C17C22×D411yescharacteristic
98C22×C17C22×Q811yescharacteristic
99C22×C17C2×#4112yes—
100C22×C17C2×#4111yescharacteristic
101C22×C172−1+412yes—
102C4×C17—22no—
103C4×C17—22no—
104C4×C17—24no—
105C4×C17—41no—
106C4×C17—42no—
107Dic17—22no—
108Dic17—22no—
109Dic17—41no—
110Dic17—42no—
111Dic17—42no—
112Dic17C22⋊Q814yes—
113C2×C4×Q8—171no—
114C2×C4×Q8—171no—
115#400—171no—
116#400—171no—
117#395—171no—
118#395D1711yescharacteristic
119#391—172no—
120#389—172no—
121#387—172no—
122#387—172no—
123#382—171no—
124C2×C17C2×C22⋊Q811yescharacteristic
125C2×C17#52511yescharacteristic
126C2×C17#53611yescharacteristic
127C2×C17#55612yes—
128C2×C17#55712yes—
129C22×Q8—171no—
130C22×Q8C2×D1711yescharacteristic
131C4×Q8—344no—
132C4×Q8—344no—
133C2×C4⋊C4—172no—
134C2×C4⋊C4—171no—
135C2×C4⋊C4—171no—
136C2×C4⋊C4—171no—
137C2×C4⋊C4—172no—
138C2×C4⋊C4—172no—
139C2×C4⋊C4C2×D1711yescharacteristic
140C2×C4⋊C4C2×D1712yes—
141C2×C42—171no—
142C2×C42—171no—
143C2×C42—171no—
144C2×C42—172no—
145#95—172no—
146#95—171no—
147#95—172no—
148#95—172no—
149#95—172no—
150#95—172no—
151#95—172no—
152#95C2×D1711yescharacteristic
153#95C2×D1712yes—
154C17#258111yescharacteristic, complemented, Sylow
155C2×Q8—22no—
156C2×Q8—24no—
157C2×Q8—174no—
158C2×Q8—342no—
159C2×Q8—342no—
160C22×C4—171no—
161C22×C4—171no—
162C22×C4—171no—
163C22×C4—172no—
164C22×C4—172no—
165C22×C4—171no—
166C22×C4C22×D1712yes—
167C22×C4C22×D1711yescharacteristic
168C22×C4C22×D1711yescharacteristic
169C22×C4C22×D1712yes—
170C22×C4C22×D1711yescharacteristic
171C4⋊C4—22no—
172C4⋊C4—24no—
173C4⋊C4—342no—
174C4⋊C4—344no—
175C4⋊C4—344no—
176C4⋊C4—342no—
177C4⋊C4—342no—
178C4⋊C4—344no—
179C42—174no—
180C42—342no—
181C42—344no—
182C42—342no—
183C23C23×D1711yescharacteristic
184Q8—28no—
185Q8—348no—
186C2×C4—21no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—172no—
197C2×C4—172no—
198C2×C4—172no—
199C2×C4—172no—
200C2×C4—172no—
201C2×C4—341no—
202C2×C4—342no—
203C2×C4—341no—
204C2×C4—341no—
205C2×C4—342no—
206C2×C4—342no—
207C2×C4—342no—
208C2×C4—342no—
209C2×C4—344no—
210C2×C4—342no—
211C2×C4D4×D1712yes—
212C2×C4#6325414yes—
213C2×C4Q8×D1712yes—
214C22C2×D4×D1711yescharacteristic
215C22C2×#6325412yes—
216C22C2×Q8×D1711yescharacteristic
217C22#1058769412yes—
218C22#41×D1711yescharacteristic
219C4—22no—
220C4—22no—
221C4—24no—
222C4—42no—
223C4—41no—
224C4—174no—
225C4—342no—
226C4—342no—
227C4—682no—
228C4—681no—
229C4—682no—
230C2C22⋊Q8×D1711yescharacteristic
231C2#1176331511yescharacteristic
232C2#1176338411yescharacteristic
233C2#1176353012yes—
234C2#1176353412yes—
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, prime radical, cosocleC23×C17C24
FittingC17×#395C2
hypercenter#395D17
hypocenterC17#2581
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
52. C2×Dic34Q8semidirect
154. C17#2581semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×Dic172
C2×#105875732
#117627492
#117627482
#117627282
#117627132
#117627042
#117626962
#117626952
#117626732
C17×#3952
#258117