Finite Group Databasebeta

← #16935194

Subgroups — #16935194

Order 4352.

528 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 105 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#14538823C212yesmaximal
2#14529765C211yesmaximal, characteristic
3#14527213C212yesmaximal
4#14526613C211yesmaximal, characteristic
5C17×#2399C211yesmaximal, characteristic
6#11763366C2211yescharacteristic
7#11763144—22no—
8#11763104—22no—
9#11763019—22no—
10#11762965—22no—
11C4×#63231—24no—
12#11762862C2212yes—
13#11762833C2211yescharacteristic
14#11762569—22no—
15Q8⋊#3622—24no—
16C2×C4×Q8×C17C2212yes—
17C17×#398—22no—
18C17×#395—22no—
19C4×C4⋊C4×C17—22no—
20C17×#376C2211yescharacteristic
21C22×Q8×C17D412yes—
22C4×Q8×C17—22no—
23C4×Q8×C17—22no—
24C4×Q8×C17—22no—
25C4×Q8×C17—22no—
26C2×C4⋊C4×C17—24no—
27C2×C4⋊C4×C17—22no—
28C2×C4⋊C4×C17D412yes—
29C2×C42×C17—22no—
30C2×C42×C17D412yes—
31C2×C42×C17C2311yescharacteristic
32C2×#63231—24no—
33#10587612—21no—
34#10587610—22no—
35C2×#63223—21no—
36#10587556—21no—
37#10587555D412yes—
38C4×Dic34—22no—
39C17×#95—22no—
40C17×#95—22no—
41C17×#95—22no—
42#10587506—24no—
43C22⋊#3622—22no—
44#10587481—24no—
45#10587478—24no—
46C4⋊#3622—22no—
47C4⋊#3622—22no—
48C4×#3622D414yes—
49C2×Q8×C17—22no—
50C2×Q8×C17—22no—
51C2×Q8×C17—44no—
52C22×C4×C17—22no—
53C22×C4×C17—22no—
54C22×C4×C17—22no—
55C22×C4×C17C2×D412yes—
56C22×C4×C17C2×D411yescharacteristic
57C2×Dic34—22no—
58C4⋊C4×C17—22no—
59C4⋊C4×C17—22no—
60C4⋊C4×C17—48no—
61C4⋊C4×C17—42no—
62C4⋊C4×C17—42no—
63C42×C17—22no—
64C42×C17—42no—
65C42×C17—42no—
66C42×C17Q1614yescomplemented
67C42×C17C2×D411yescharacteristic
68C42×C17C2×D412yes—
69C42×C17#4111yescharacteristic
70C22⋊Dic17—41no—
71#63231—48no—
72C4⋊Dic17—22no—
73C4⋊Dic17—22no—
74#63225—42no—
75C4×Dic17—41no—
76#63223—42no—
77C2×#3622—24no—
78C2×#3622—41no—
79#10017—171nomaximal, Sylow
80C23×C17#12011yescharacteristic
81Q8×C17—24no—
82Q8×C17—24no—
83Q8×C17—42no—
84Q8×C17—42no—
85C2×C4×C17—24no—
86C2×C4×C17—24no—
87C2×C4×C17—42no—
88C2×C4×C17—44no—
89C2×C4×C17—42no—
90C2×C4×C17—42no—
91C2×C4×C17—44no—
92C2×C4×C17—42no—
93C2×C4×C17#12012yes—
94C2×C4×C17#12111yescharacteristic
95C2×C4×C17#12112yes—
96C2×C4×C17#12711yescharacteristic
97C2×C4×C17C2×Q1612yes—
98C2×C4×C17#13712yes—
99C2×Dic17—42no—
100Dic34—44no—
101#3622—44no—
102#3622—81no—
103#3104—172no—
104#2399D1711yescharacteristic
105#1995—171no—
106#1487—172no—
107#1274—171no—
108C22×C17—22no—
109C22×C17#39211yescharacteristic
110C22×C17#45112yes—
111C4×C17—22no—
112C4×C17—22no—
113C4×C17—44no—
114C4×C17—44no—
115C4×C17—82no—
116C4×C17#46212yes—
117C4×C17#49411yescharacteristic
118C4×C17#49711yescharacteristic
119Dic17—82no—
120#533—171no—
121C2×C4×Q8C2×D1712yes—
122#484—342no—
123#477—342no—
124#462—342no—
125#451—342no—
126C4×Q16—344no—
127#431—172no—
128#423—171no—
129#398—22no—
130#395—22no—
131C4×C4⋊C4—22no—
132#376C2×D1711yescharacteristic
133#339—342no—
134Q8⋊C8—344no—
135C2×C17—22no—
136C2×C17#201012yes—
137C2×C17#201611yescharacteristic
138C22×Q8C22⋊D1712yes—
139C2×Q16—344no—
140C2×M16—341no—
141C4⋊Q8—172no—
142#125—341no—
143C22⋊Q8—342no—
144C4×Q8—22no—
145C4×Q8—22no—
146C4×Q8—22no—
147C4×Q8—22no—
148C4×Q8—342no—
149#117—341no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4C22⋊D1712yes—
153C2×C42—22no—
154C2×C42C22⋊D1712yes—
155C2×C42C22×D1711yescharacteristic
156#107—344no—
157C4⋊C8—342no—
158C4⋊C8—342no—
159#103—344no—
160#103—344no—
161C22⋊C8—342no—
162C4×C8—174no—
163#95—22no—
164#95—22no—
165#95—22no—
166C17#1001711yescharacteristic, complemented, Sylow
167C2×Q8—22no—
168C2×Q8—22no—
169C2×Q8—44no—
170C2×Q8—342no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4C2×C22⋊D1712yes—
175C22×C4C2×C22⋊D1711yescharacteristic
176Q16—688no—
177M16—682no—
178C2×C8—344no—
179C2×C8—681no—
180C4⋊C4—22no—
181C4⋊C4—22no—
182C4⋊C4—48no—
183C4⋊C4—42no—
184C4⋊C4—42no—
185C4⋊C4—342no—
186C4⋊C4—342no—
187C4⋊C4—682no—
188C22⋊C4—681no—
189C42—22no—
190C42—42no—
191C42—42no—
192C42—681no—
193C42#6323114yescomplemented
194C42D4×D1711yescharacteristic
195C42D4×D1712yes—
196C42#6325411yescharacteristic
197C23#1058763911yescharacteristic
198Q8—24no—
199Q8—24no—
200Q8—42no—
201Q8—42no—
202Q8—684no—
203C2×C4—24no—
204C2×C4—24no—
205C2×C4—42no—
206C2×C4—44no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—44no—
210C2×C4—42no—
211C2×C4—682no—
212C2×C4#1058762312yes—
213C2×C4#1058762411yescharacteristic
214C2×C4#1058762512yes—
215C2×C4#1058762611yescharacteristic
216C2×C4#1058762812yes—
217C2×C4C2×#6323112yes—
218C8—684no—
219C8—1361no—
220C22—22no—
221C22#1176271711yescharacteristic
222C22#1176296512yes—
223C4—22no—
224C4—22no—
225C4—44no—
226C4—44no—
227C4—82no—
228C4—1362no—
229C4#1176301612yes—
230C4#1176318811yescharacteristic
231C4#1176320111yescharacteristic
232C2—22no—
233C2#1452984912yes—
234C2#1452988411yescharacteristic
subgroupG/H
centerC2×C4#10587626
commutator, cosocleC2×C42×C17C23
FrattiniC2×C42C22×D17
FittingC17×#2399C2
socleC22×C17#392
prime radicalC23×C17#120
hypercenter#2399D17
hypocenterC17#10017
LCSG ▷ C2×C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2399

Complements

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

NH ≅ G/Ntype
66. C42×C17Q16semidirect
166. C17#10017semidirect
193. C42#63231semidirect

Maximal subgroups

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

subgroupindex
#145388232
#145297652
#145272132
#145266132
C17×#23992
#1001717