Finite Group Databasebeta

← #14540014

Subgroups — #14540014

Order 2176.

362 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 125 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
1#11763343C211yesmaximal, characteristic
2#11763215C212yesmaximal, complemented
3#11763171C212yesmaximal, complemented
4#11763144C212yesmaximal, complemented
5#11763127C212yesmaximal
6#11763091C212yesmaximal, complemented
7#11763086C212yesmaximal, complemented
8#11762866C211yesmaximal, characteristic, complemented
9C2×C17×C4⋊Q8C211yesmaximal, characteristic, complemented
10C22×Q8×C17—21no—
11C17×C4⋊Q8—21no—
12C17×C4⋊Q8—21no—
13C17×C4⋊Q8C2212yes—
14C17×C4⋊Q8C2212yes—
15C2×C4⋊C4×C17—21no—
16C2×C4⋊C4×C17C2212yes—
17C2×C42×C17C2211yescharacteristic
18#10587616C2211yescharacteristic
19#10587615—21no—
20C4×C22⋊D17—21no—
21#10587610C2211yescharacteristic
22#10587562—21no—
23#10587561C2211yescharacteristic
24C4×D68C2211yescharacteristic
25#10587558—21no—
26#10587556C2211yescharacteristic
27C4×Dic34C2211yescharacteristic
28C22⋊#3622C2212yes—
29#10587481C2214yes—
30#10587480C2214yes—
31#10587479C2214yes—
32#10587478C2214yes—
33C4⋊#3622C2212yes—
34#10587474C2212yes—
35C2×Q8×C17—22no—
36C2×Q8×C17—22no—
37C2×Q8×C17—44no—
38C22×C4×C17—22no—
39C22×C4×C17D411yescharacteristic
40C22×C4×C17D411yescharacteristic
41C22×C4×C17C2311yescharacteristic
42C2×C22⋊D17—21no—
43C2×D68C2311yescharacteristic
44C2×C4×D17—21no—
45C2×Dic34C2311yescharacteristic
46C4⋊C4×C17—22no—
47C4⋊C4×C17—22no—
48C4⋊C4×C17—22no—
49C4⋊C4×C17C2314yes—
50C42×C17D411yescharacteristic
51C42×C17D411yescharacteristic
52C42×C17C2311yescharacteristic
53C42×C17C2311yescharacteristic
54C22⋊Dic17—21no—
55#63227—21no—
56#63227—21no—
57#63227—21no—
58#63227—21no—
59C4⋊Dic17C2311yescharacteristic
60C4⋊Dic17C2311yescharacteristic
61#63225—21no—
62#63225—21no—
63#63225—21no—
64#63225—21no—
65C4×Dic17—21no—
66C2×#3622C2314yes—
67C23×C17C2×D411yescharacteristic
68C22×D17—21no—
69Q8×C17—44no—
70Q8×C17—44no—
71C2×C4×C17—22no—
72C2×C4×C17—22no—
73C2×C4×C17—22no—
74C2×C4×C17—24no—
75C2×C4×C17—44no—
76C2×C4×C17C2×D411yescharacteristic
77C2×C4×C17C2×D411yescharacteristic
78C2×C4×C17C2×D411yescharacteristic
79C2×C4×C17C2×D411yescharacteristic
80C2×C4×C17C2×D411yescharacteristic
81C2×C4×C17C2411yescharacteristic
82C22⋊D17—42no—
83C2×Dic17—21no—
84C2×Dic17—21no—
85C2×Dic17—21no—
86D68—22no—
87C4×D17—42no—
88Dic34—22no—
89#3622—24no—
90#3239—171nomaximal, Sylow
91C22×C17—21no—
92C22×C17—21no—
93C22×C17#13712yes—
94C22×C17C22×D411yescharacteristic
95C2×D17—41no—
96C2×D17—41no—
97C2×D17—41no—
98C4×C17—21no—
99C4×C17—21no—
100C4×C17—21no—
101C4×C17—21no—
102C4×C17—44no—
103C4×C17AGL(1,ℤ/8)12yes—
104C4×C172−1+412yes—
105Dic17—41no—
106Dic17—41no—
107Dic17—41no—
108C2×C4⋊Q8D1711yescharacteristic, complemented
109#529—171no—
110#501—172no—
111#489—172no—
112#484—172no—
113#481—172no—
114#475—172no—
115#474—172no—
116#432—171no—
117C2×C17—22no—
118C2×C17#53611yescharacteristic
119C2×C17C2×AGL(1,ℤ/8)11yescharacteristic
120C2×C17C2×#13711yescharacteristic
121D17—81no—
122C22×Q8—21no—
123C4⋊Q8—21no—
124C4⋊Q8—21no—
125C4⋊Q8C2×D1712yes—
126C4⋊Q8C2×D1712yes—
127#126—341no—
128#125—171no—
129#124—171no—
130#123—341no—
131C22⋊Q8—171no—
132#121—171no—
133C4×Q8—171no—
134C4×D4—171no—
135C4×D4—341no—
136#117—341no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4C2×D1712yes—
139C2×C42C2×D1711yescharacteristic
140#107—174no—
141#106—174no—
142C4⋊C8—172no—
143#103—174no—
144#102—174no—
145C22⋊C8—172no—
146#97—172no—
147C17#323911yescharacteristic, complemented, Sylow
148C2×Q8—22no—
149C2×Q8—22no—
150C2×Q8—44no—
151C2×Q8—171no—
152C2×D4—171no—
153C2×D4—341no—
154C22×C4—22no—
155C22×C4—341no—
156C22×C4C22⋊D1711yescharacteristic
157C22×C4C22⋊D1711yescharacteristic
158C22×C4C22×D1711yescharacteristic
159C2×C8—174no—
160C4⋊C4—22no—
161C4⋊C4—22no—
162C4⋊C4—22no—
163C4⋊C4—171no—
164C4⋊C4—171no—
165C4⋊C4—341no—
166C4⋊C4—341no—
167C4⋊C4—341no—
168C4⋊C4—341no—
169C4⋊C4C22×D1714yes—
170C22⋊C4—341no—
171C22⋊C4—341no—
172C22⋊C4—341no—
173C22⋊C4—341no—
174C22⋊C4—341no—
175C42—341no—
176C42C22⋊D1711yescharacteristic
177C42C22⋊D1711yescharacteristic
178C42C22×D1711yescharacteristic
179C42C22×D1711yescharacteristic
180C23—341no—
181C23C2×C22⋊D1711yescharacteristic
182Q8—44no—
183Q8—44no—
184Q8—342no—
185D4—342no—
186D4—682no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—24no—
191C2×C4—44no—
192C2×C4—341no—
193C2×C4—341no—
194C2×C4—341no—
195C2×C4—682no—
196C2×C4C2×C22⋊D1711yescharacteristic
197C2×C4C2×C22⋊D1711yescharacteristic
198C2×C4C2×C22⋊D1711yescharacteristic
199C2×C4C2×C22⋊D1711yescharacteristic
200C2×C4C2×C22⋊D1711yescharacteristic
201C2×C4C23×D1711yescharacteristic
202C8—344no—
203C22—21no—
204C22—21no—
205C22—681no—
206C22—681no—
207C22—681no—
208C22#1058762812yes—
209C22C22×C22⋊D1711yescharacteristic
210C4—21no—
211C4—21no—
212C4—21no—
213C4—21no—
214C4—44no—
215C4—681no—
216C4—681no—
217C4—681no—
218C4#1058761812yes—
219C4#1058769412yes—
220C2—22no—
221C2—1361no—
222C2#1176338311yescharacteristic
223C2C2×#1058761811yescharacteristic
224C2C2×#1058762811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC2×C17×C4⋊Q8C2
socleC22×C17C22×D4
prime radical#10587616C22
hypercenterC2×C4⋊Q8D17
hypocenterC17#3239
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
2. #11763215C2semidirect
3. #11763171C2semidirect
4. #11763144C2semidirect
6. #11763091C2semidirect
7. #11763086C2semidirect
8. #11762866C2semidirect
9. C2×C17×C4⋊Q8C2semidirect
108. C2×C4⋊Q8D17semidirect
147. C17#3239semidirect

Maximal subgroups

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

subgroupindex
#117633432
#117632152
#117631712
#117631442
#117631272
#117630912
#117630862
#117628662
C2×C17×C4⋊Q82
#323917