Finite Group Databasebeta

← #14538907

Subgroups — #14538907

Order 2176.

424 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 129 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#11763359C211yesmaximal, characteristic
2C2×C4×Dic34C211yesmaximal, characteristic, complemented
3#11763123C212yesmaximal, complemented
4#11763105C212yesmaximal, complemented
5#11763010C212yesmaximal, complemented
6#11762950C212yesmaximal, complemented
7#11762911C214yesmaximal, complemented
8C17×#432C211yesmaximal, characteristic, complemented
9C2×#63252—21no—
10C22×Dic34C2211yescharacteristic
11C2×C42×C17C2211yescharacteristic
12#10587616—21no—
13#10587616C2212yes—
14C2×C4⋊Dic17C2211yescharacteristic
15C2×#63225—21no—
16C2×C4×Dic17—21no—
17C2×#63219C2214yes—
18#10587561—21no—
19#10587560C2211yescharacteristic
20C4×D68—21no—
21C4×D68C2212yes—
22#10587555C2211yescharacteristic
23C4×Dic34—21no—
24C4×Dic34—21no—
25C4×Dic34C2212yes—
26C4×Dic34C2212yes—
27C17×C4⋊C8C2212yes—
28C17×C22⋊C8C2212yes—
29C17×#97C2212yes—
30#10587492C2214yes—
31#10587489C2214yes—
32#10587487C2214yes—
33C22×C4×C17D411yescharacteristic
34C22×C4×C17D411yescharacteristic
35C22×C4×C17C2311yescharacteristic
36C2×C22⋊D17—22no—
37C22×Dic17—21no—
38C22×Dic17—21no—
39#63252—44no—
40C2×D68—21no—
41C2×D68—21no—
42C2×D68C2312yes—
43C2×C4×D17—22no—
44C2×Dic34—21no—
45C2×Dic34—24no—
46C2×Dic34—21no—
47C2×Dic34C2312yes—
48C2×C8×C17C2314yes—
49C42×C17D411yescharacteristic
50C42×C17D411yescharacteristic
51C42×C17C2311yescharacteristic
52C42×C17C2311yescharacteristic
53#63227—22no—
54#63227—22no—
55C4⋊Dic17—21no—
56C4⋊Dic17—21no—
57C4⋊Dic17C2312yes—
58C4⋊Dic17C2312yes—
59#63225—22no—
60#63225—22no—
61C4×Dic17—22no—
62C4×Dic17—22no—
63#63219—28no—
64C23×C17C2×D411yescharacteristic
65C22×D17—22no—
66C2×C4×C17—22no—
67C2×C4×C17—22no—
68C2×C4×C17—22no—
69C2×C4×C17C2×D411yescharacteristic
70C2×C4×C17C2×D411yescharacteristic
71C2×C4×C17C2×D411yescharacteristic
72C2×C4×C17C2×D411yescharacteristic
73C2×C4×C17C2×D411yescharacteristic
74C2×C4×C17C2411yescharacteristic
75C22⋊D17—44no—
76C2×Dic17—24no—
77C2×Dic17—22no—
78C2×Dic17—22no—
79C2×Dic17—22no—
80C2×Dic17—42no—
81D68—24no—
82D68—42no—
83D68—42no—
84C4×D17—44no—
85Dic34—24no—
86Dic34—22no—
87Dic34—42no—
88Dic34#4114yescomplemented
89C8×C17—24no—
90#3112—171nomaximal, Sylow
91C22×C17—21no—
92C22×C17—21no—
93C22×C17AGL(1,ℤ/8)12yes—
94C22×C17C22×D411yescharacteristic
95C2×D17—42no—
96C2×D17—42no—
97C2×D17—42no—
98C4×C17—21no—
99C4×C17—21no—
100C4×C17—21no—
101C4×C17—21no—
102C4×C17#13712yes—
103C4×C17C2×#4112yes—
104Dic17—24no—
105Dic17—42no—
106Dic17—42no—
107#532—171no—
108C2×C4×Q8—171no—
109#480—172no—
110#477—172no—
111#460—172no—
112#448—172no—
113#440—174no—
114#432D1711yescharacteristic, complemented
115C2×C17—22no—
116C2×C17#52511yescharacteristic
117C2×C17C2×AGL(1,ℤ/8)11yescharacteristic
118C2×C17C2×#13711yescharacteristic
119D17—82no—
120C2×#41—341no—
121C22×Q8—171no—
122C2×SD16—174no—
123C4⋊Q8—171no—
124#127—171no—
125#124—341no—
126#121—172no—
127#121—341no—
128C4×Q8—172no—
129C4×Q8—172no—
130C4×Q8—341no—
131C4×Q8—341no—
132C4×D4—172no—
133C4×D4—341no—
134C2×C4⋊C4—171no—
135C2×C4⋊C4—341no—
136C2×C42—341no—
137C2×C42C2×D1711yescharacteristic
138#107—174no—
139C4⋊C8C2×D1712yes—
140#103—174no—
141#102—174no—
142C22⋊C8C2×D1712yes—
143#97C2×D1712yes—
144C17#311211yescharacteristic, complemented, Sylow
145#41—684no—
146C2×Q8—172no—
147C2×Q8—344no—
148C2×Q8—341no—
149C2×Q8—341no—
150C2×D4—172no—
151C2×D4—342no—
152C2×D4—341no—
153C2×D4—341no—
154C22×C4—341no—
155C22×C4—341no—
156C22×C4—342no—
157C22×C4D6811yescharacteristic
158C22×C4D6811yescharacteristic
159C22×C4C22×D1711yescharacteristic
160SD16—348no—
161C2×C8C22×D1714yes—
162C4⋊C4—172no—
163C4⋊C4—172no—
164C4⋊C4—341no—
165C4⋊C4—342no—
166C4⋊C4—342no—
167C4⋊C4—341no—
168C22⋊C4—342no—
169C22⋊C4—342no—
170C42—342no—
171C42—342no—
172C42D6811yescharacteristic
173C42D6811yescharacteristic
174C42C22×D1711yescharacteristic
175C42C22×D1711yescharacteristic
176C23—342no—
177C23C2×D6811yescharacteristic
178Q8—174no—
179Q8—344no—
180Q8—342no—
181Q8—682no—
182D4—344no—
183D4—682no—
184D4—684no—
185D4—682no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—344no—
190C2×C4—342no—
191C2×C4—342no—
192C2×C4—342no—
193C2×C4—684no—
194C2×C4—682no—
195C2×C4C2×D6811yescharacteristic
196C2×C4C2×D6811yescharacteristic
197C2×C4C2×D6811yescharacteristic
198C2×C4C2×D6811yescharacteristic
199C2×C4C2×D6811yescharacteristic
200C2×C4C23×D1711yescharacteristic
201C8—24no—
202C22—21no—
203C22—21no—
204C22—682no—
205C22—682no—
206C22—682no—
207C22#1058759412yes—
208C22C22×D6811yescharacteristic
209C4—21no—
210C4—21no—
211C4—21no—
212C4—21no—
213C4—344no—
214C4—682no—
215C4—682no—
216C4#1058759512yes—
217C4#41×D1712yes—
218C2—22no—
219C2—1362no—
220C2#1176332111yescharacteristic
221C2C2×#1058759411yescharacteristic
222C2C2×#1058759511yescharacteristic
subgroupG/H
centerC22C22×D68
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
FittingC17×#432C2
socleC22×C17C22×D4
prime radical#11763359C2
hypercenter#432D17
hypocenterC17#3112
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #432

Complements

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

NH ≅ G/Ntype
2. C2×C4×Dic34C2semidirect
3. #11763123C2semidirect
4. #11763105C2semidirect
5. #11763010C2semidirect
6. #11762950C2semidirect
7. #11762911C2semidirect
8. C17×#432C2semidirect
88. Dic34#41semidirect
114. #432D17semidirect
144. C17#3112semidirect

Maximal subgroups

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

subgroupindex
#117633592
C2×C4×Dic342
#117631232
#117631052
#117630102
#117629502
#117629112
C17×#4322
#311217