Finite Group Databasebeta

← #14538902

Subgroups — #14538902

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
1C2×C4×Dic34C211yesmaximal, characteristic, complemented
2#11763120C212yesmaximal, complemented
3#11763101C212yesmaximal, complemented
4#11763006C212yesmaximal, complemented
5#11762947C212yesmaximal, complemented
6#11762909C214yesmaximal, complemented
7#11762866C211yesmaximal, characteristic, complemented
8C17×#532C211yesmaximal, characteristic
9C2×#41×C17—21no—
10C22×Dic34C2211yescharacteristic
11C17×C4⋊Q8C2211yescharacteristic
12C17×#127C2211yescharacteristic
13C17×#124—21no—
14C17×#121—21no—
15C17×#121C2212yes—
16C4×D4×C17—21no—
17C4×D4×C17C2212yes—
18C2×C42×C17C2211yescharacteristic
19C2×#63229C2214yes—
20C2×C4⋊Dic17C2211yescharacteristic
21C2×#63225—21no—
22C2×C4×Dic17—21no—
23C4×Dic34—21no—
24C4×Dic34—21no—
25C4×Dic34C2212yes—
26C4×Dic34C2212yes—
27#10587503C2214yes—
28C22⋊#3622C2212yes—
29#10587481C2214yes—
30#10587478C2214yes—
31C4⋊#3622C2212yes—
32#10587474C2212yes—
33#41×C17—44no—
34C2×Q8×C17—21no—
35C2×D4×C17—22no—
36C2×D4×C17—21no—
37C2×D4×C17—21no—
38C2×D4×C17C2312yes—
39C22×C4×C17—22no—
40C22×C4×C17D411yescharacteristic
41C22×C4×C17D411yescharacteristic
42C22×C4×C17C2311yescharacteristic
43C22×Dic17—21no—
44C22×Dic17—21no—
45C2×Dic34—21no—
46C2×Dic34—24no—
47C2×Dic34C2312yes—
48C4⋊C4×C17—21no—
49C4⋊C4×C17C2312yes—
50C22⋊C4×C17—22no—
51C22⋊C4×C17—22no—
52C42×C17D411yescharacteristic
53C42×C17D411yescharacteristic
54C42×C17C2311yescharacteristic
55C42×C17C2311yescharacteristic
56#63229—28no—
57C4⋊Dic17—21no—
58C4⋊Dic17C2312yes—
59#63225—22no—
60#63225—22no—
61C4×Dic17—22no—
62C4×Dic17—22no—
63C2×#3622C2314yes—
64C23×C17—22no—
65C23×C17C2×D411yescharacteristic
66Q8×C17—42no—
67D4×C17—24no—
68D4×C17—42no—
69D4×C17—44no—
70D4×C17—42no—
71C2×C4×C17—22no—
72C2×C4×C17—22no—
73C2×C4×C17—22no—
74C2×C4×C17—22no—
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
82C2×Dic17—24no—
83C2×Dic17—22no—
84C2×Dic17—22no—
85C2×Dic17—42no—
86Dic34—24no—
87Dic34—22no—
88Dic34#4114yescomplemented
89#3622—24no—
90#3112—171nomaximal, Sylow
91C22×C17—21no—
92C22×C17—21no—
93C22×C17—42no—
94C22×C17—42no—
95C22×C17—42no—
96C22×C17AGL(1,ℤ/8)12yes—
97C22×C17C22×D411yescharacteristic
98C4×C17—21no—
99C4×C17—21no—
100C4×C17—21no—
101C4×C17—21no—
102C4×C17—42no—
103C4×C17#13712yes—
104C4×C17C2×#4112yes—
105Dic17—24no—
106Dic17—42no—
107#532D1711yescharacteristic
108C2×C4×Q8—171no—
109#480—172no—
110#477—172no—
111#460—172no—
112#448—172no—
113#440—174no—
114#432—171no—
115C2×C17—22no—
116C2×C17—82no—
117C2×C17#52511yescharacteristic
118C2×C17C2×AGL(1,ℤ/8)11yescharacteristic
119C2×C17C2×#13711yescharacteristic
120C2×#41—21no—
121C22×Q8—171no—
122C2×SD16—174no—
123C4⋊Q8C2×D1711yescharacteristic
124#127C2×D1711yescharacteristic
125#124—21no—
126#121—21no—
127#121C2×D1712yes—
128C4×Q8—172no—
129C4×Q8—172no—
130C4×Q8—341no—
131C4×Q8—341no—
132C4×D4—21no—
133C4×D4C2×D1712yes—
134C2×C4⋊C4—171no—
135C2×C4⋊C4—341no—
136C2×C42—341no—
137C2×C42C2×D1711yescharacteristic
138#107—174no—
139C4⋊C8—172no—
140#103—174no—
141#102—174no—
142C22⋊C8—172no—
143#97—172no—
144C17#311211yescharacteristic, complemented, Sylow
145#41—44no—
146C2×Q8—21no—
147C2×Q8—172no—
148C2×Q8—344no—
149C2×Q8—341no—
150C2×D4—22no—
151C2×D4—21no—
152C2×D4—21no—
153C2×D4C22×D1712yes—
154C22×C4—22no—
155C22×C4—341no—
156C22×C4—341no—
157C22×C4C22⋊D1711yescharacteristic
158C22×C4C22⋊D1711yescharacteristic
159C22×C4C22×D1711yescharacteristic
160SD16—348no—
161C2×C8—174no—
162C4⋊C4—21no—
163C4⋊C4—172no—
164C4⋊C4—341no—
165C4⋊C4—342no—
166C4⋊C4—342no—
167C4⋊C4C22×D1712yes—
168C22⋊C4—22no—
169C22⋊C4—22no—
170C42—342no—
171C42—342no—
172C42C22⋊D1711yescharacteristic
173C42C22⋊D1711yescharacteristic
174C42C22×D1711yescharacteristic
175C42C22×D1711yescharacteristic
176C23—22no—
177C23C2×C22⋊D1711yescharacteristic
178Q8—42no—
179Q8—174no—
180Q8—344no—
181Q8—342no—
182D4—24no—
183D4—42no—
184D4—44no—
185D4—42no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—44no—
191C2×C4—344no—
192C2×C4—342no—
193C2×C4—342no—
194C2×C4—682no—
195C2×C4C2×C22⋊D1711yescharacteristic
196C2×C4C2×C22⋊D1711yescharacteristic
197C2×C4C2×C22⋊D1711yescharacteristic
198C2×C4C2×C22⋊D1711yescharacteristic
199C2×C4C2×C22⋊D1711yescharacteristic
200C2×C4C23×D1711yescharacteristic
201C8—344no—
202C22—21no—
203C22—21no—
204C22—42no—
205C22—42no—
206C22—42no—
207C22#1058761812yes—
208C22C22×C22⋊D1711yescharacteristic
209C4—21no—
210C4—21no—
211C4—21no—
212C4—21no—
213C4—42no—
214C4—344no—
215C4—682no—
216C4#1058763712yes—
217C4#41×D1712yes—
218C2—22no—
219C2—82no—
220C2#1176331611yescharacteristic
221C2C2×#1058761811yescharacteristic
222C2C2×#1058763711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D17
commutator, cosocleC2×C4×C17C24
FrattiniC2×C4C23×D17
Fitting, prime radicalC17×#532C2
socleC22×C17C22×D4
hypercenter#532D17
hypocenterC17#3112
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #532

Complements

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

NH ≅ G/Ntype
1. C2×C4×Dic34C2semidirect
2. #11763120C2semidirect
3. #11763101C2semidirect
4. #11763006C2semidirect
5. #11762947C2semidirect
6. #11762909C2semidirect
7. #11762866C2semidirect
88. Dic34#41semidirect
144. C17#3112semidirect

Maximal subgroups

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

subgroupindex
C2×C4×Dic342
#117631202
#117631012
#117630062
#117629472
#117629092
#117628662
C17×#5322
#311217