Finite Group Databasebeta

← #17949052

Subgroups — #17949052

Order 4352.

432 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 50 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#14544618C212yesmaximal
2#14528707C211yesmaximal, characteristic
3C2×#10587545—22no—
4C2×#10587545C414yes—
5#11763779—22no—
6C23×Dic34C2211yescharacteristic
7C2×#10587495—22no—
8#11762773C411yescharacteristic
9C17×#407C411yescharacteristic
10C23×C4×C17C2×C411yescharacteristic
11C23×Dic17D411yescharacteristic
12C23×Dic17Q811yescharacteristic
13C22×Dic34—44no—
14C22×Dic34—41no—
15C22×Dic34—41no—
16C22×Dic34C2×C412yes—
17C22×Dic34D412yes—
18C2×#63247—24no—
19C2×M16×C17—21no—
20C2×#63223—21no—
21C22⋊#3624—44no—
22#10587545—42no—
23#10587545—42no—
24#10587545—44no—
25C17×C22⋊C8—41no—
26C22⋊#3622—41no—
27#10587495—44no—
28C24×C17C4⋊C411yescharacteristic
29C22×C4×C17—21no—
30C22×C4×C17—21no—
31C22×C4×C17—42no—
32C22×C4×C17C4211yescharacteristic
33C22×C4×C17C22⋊C411yescharacteristic
34C22×Dic17—24no—
35C22×Dic17—41no—
36C22×Dic17—42no—
37C22×Dic17—41no—
38C22×Dic17C22⋊C412yes—
39C22×Dic17C4⋊C412yes—
40C2×Dic34—22no—
41C2×Dic34—22no—
42C2×Dic34—24no—
43C2×Dic34—24no—
44C2×Dic34—42no—
45C2×Dic34—42no—
46C2×Dic34—42no—
47C2×Dic34—48no—
48C2×Dic34—48no—
49#63247—44no—
50#63247—44no—
51C2×#3624—44no—
52M16×C17—42no—
53C2×C8×C17—41no—
54#63223—42no—
55C2×#3622—41no—
56#7315—171nomaximal, Sylow
57C23×C17—22no—
58C23×C17—21no—
59C23×C17—42no—
60C23×C17#9511yescharacteristic
61C2×C4×C17—22no—
62C2×C4×C17—22no—
63C2×C4×C17—21no—
64C2×C4×C17—42no—
65C2×C4×C17—42no—
66C2×C4×C17#9911yescharacteristic
67C2×C4×C17#9911yescharacteristic
68C2×Dic17—22no—
69C2×Dic17—22no—
70C2×Dic17—22no—
71C2×Dic17—22no—
72C2×Dic17—22no—
73C2×Dic17—44no—
74C2×Dic17—44no—
75C2×Dic17#9914yes—
76Dic34—44no—
77Dic34—44no—
78Dic34—48no—
79#3624—84no—
80C8×C17—81no—
81#3622—81no—
82#1784—171no—
83#1784—172no—
84C22×C17—24no—
85C22×C17—22no—
86C22×C17—41no—
87C22×C17—42no—
88C22×C17—41no—
89C22×C17—41no—
90C22×C17#34212yes—
91C22×C17#34211yescharacteristic
92C4×C17—21no—
93C4×C17—21no—
94C4×C17—41no—
95Dic17—24no—
96Dic17—42no—
97C23×Q8—171no—
98C2×#101—174no—
99C2×#101—342no—
100C2×#101—342no—
101#407—171no—
102#407—342no—
103#407#59511yescharacteristic
104C2×C17—22no—
105C2×C17—21no—
106C2×C17—42no—
107C2×C17#130411yescharacteristic
108C22×Q8—172no—
109C22×Q8—172no—
110C22×Q8—681no—
111C22×Q8—681no—
112C22×Q8—684no—
113C23×C4—171no—
114C23×C4—171no—
115C23×C4C2×#59511yescharacteristic
116C2×M16—21no—
117C2×M16—344no—
118C2×M16—341no—
119#101—684no—
120#101—682no—
121#101—682no—
122#101—684no—
123C22⋊C8—41no—
124C22⋊C8—684no—
125C22⋊C8—681no—
126C17#731511yescharacteristic, complemented, Sylow
127C24C4⋊#59511yescharacteristic
128C2×Q8—344no—
129C2×Q8—344no—
130C2×Q8—342no—
131C2×Q8—342no—
132C2×Q8—682no—
133C2×Q8—682no—
134C2×Q8—682no—
135C2×Q8—688no—
136C2×Q8—688no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—42no—
140C22×C4—172no—
141C22×C4—172no—
142C22×C4—344no—
143C22×C4—681no—
144C22×C4—682no—
145C22×C4—681no—
146C22×C4C4×#59511yescharacteristic
147C22×C4C22⋊#59511yescharacteristic
148M16—42no—
149M16—682no—
150M16—684no—
151M16—684no—
152C2×C8—41no—
153C2×C8—684no—
154C2×C8—681no—
155C23—22no—
156C23—21no—
157C23—42no—
158C23(C2×#595)⋊C411yescharacteristic
159Q8—684no—
160Q8—684no—
161Q8—688no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—21no—
165C2×C4—42no—
166C2×C4—42no—
167C2×C4—174no—
168C2×C4—342no—
169C2×C4—342no—
170C2×C4—342no—
171C2×C4—342no—
172C2×C4—342no—
173C2×C4—684no—
174C2×C4—684no—
175C2×C4#1058753911yescharacteristic
176C2×C4#1058755111yescharacteristic
177C8—81no—
178C8—1361no—
179C8—1364no—
180C22—24no—
181C22—22no—
182C22—41no—
183C22—42no—
184C22—41no—
185C22—41no—
186C22#1176368612yes—
187C22#1176368711yescharacteristic
188C4—21no—
189C4—21no—
190C4—41no—
191C4—344no—
192C4—682no—
193C2—22no—
194C2—21no—
195C2—42no—
196C2#1454368311yescharacteristic
subgroupG/H
centerC2#14543683
commutatorC22×C4×C17C42
FrattiniC23×C4C2×#595
FittingC17×#407C4
socleC2×C17#1304
prime radicalC24×C17C4⋊C4
cosocleC23×Dic34C22
hypercenter#407#595
hypocenterC17#7315
LCSG ▷ C22×C4×C17 ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2 ◁ C23 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
126. C17#7315semidirect

Maximal subgroups

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

subgroupindex
#145446182
#145287072
#731517