Finite Group Databasebeta

← #17981292

Subgroups — #17981292

Order 4352.

322 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 65 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#14545896C212yesmaximal
2#14544452C211yesmaximal, characteristic
3#11763922—22no—
4#11763895C414yes—
5#11763894—22no—
6C22×#63242C411yescharacteristic
7C2×#10587677C2211yescharacteristic
8#11763674—22no—
9C2×M16×D17C411yescharacteristic
10C22×C4×D17C2×C411yescharacteristic
11C2×C22⋊#595D411yescharacteristic
12C2×C22⋊#595Q811yescharacteristic
13C22×#3624—21no—
14#10587677—41no—
15#10587677—41no—
16C2×C4⋊#595C2×C412yes—
17C2×C4×#595D412yes—
18C2×#63242—21no—
19C2×#63242—41no—
20C2×#63242—41no—
21C2×#63242C812yescomplemented
22C22⋊#3633—24no—
23C4⋊#3633—42no—
24C4×#3633—42no—
25C2×M16×C17C811yescharacteristic, complemented
26C2×#63223C811yescharacteristic, complemented
27M16×D17—41no—
28M16×D17—41no—
29C2×#63218—21no—
30C2×C8×D17—21no—
31C23×D17C4⋊C411yescharacteristic
32C22×#595C22⋊C412yes—
33C22×#595C4⋊C412yes—
34C2×#3633—44no—
35C22×C4×C17C2×C811yescharacteristic
36C22×Dic17M1611yescharacteristic
37C2×C4×D17—21no—
38C2×C4×D17—21no—
39C2×C4×D17—21no—
40C2×C4×D17—21no—
41C2×C4×D17C4211yescharacteristic
42C2×C4×D17C22⋊C411yescharacteristic
43C2×C4×D17C2×C811yescharacteristic
44C2×C4×D17M1611yescharacteristic
45C22⋊#595—41no—
46C22⋊#595—41no—
47C2×#3624—22no—
48C2×#3624—42no—
49C4⋊#595—42no—
50C4×#595—22no—
51C4×#595—22no—
52#63242—41no—
53#63242—41no—
54#63242—42no—
55M16×C17—41no—
56C2×C8×C17—21no—
57#63223—41no—
58C2×#3622—21no—
59#63218—42no—
60C8×D17—41no—
61C8×D17—41no—
62#7190—171nomaximal, Sylow
63C23×C17C4⋊C811yescharacteristic
64C22×D17—21no—
65C22×D17—21no—
66C22×D17—42no—
67C22×D17#9511yescharacteristic
68C22×D17C4⋊C811yescharacteristic
69C2×#595—22no—
70C2×#595—22no—
71C2×#595—42no—
72C2×#595—42no—
73#3633—84no—
74C2×C4×C17—21no—
75C2×C4×C17—21no—
76C2×C4×C17C4×C811yescharacteristic
77C2×C4×C17C22⋊C811yescharacteristic
78C2×Dic17—21no—
79C2×Dic17—21no—
80C2×Dic17#9711yescharacteristic
81C2×Dic17C22⋊C811yescharacteristic
82C4×D17—21no—
83C4×D17—21no—
84C4×D17—21no—
85C4×D17—21no—
86C4×D17—21no—
87C4×D17—21no—
88C4×D17#9911yescharacteristic
89C4×D17#9911yescharacteristic
90C4×D17#10011yescharacteristic
91C4×D17#10111yescharacteristic
92#3624—42no—
93C8×C17—41no—
94#3622—41no—
95#1729—171no—
96#1458—172no—
97C22×C17—21no—
98C22×C17—41no—
99C22×C17#33611yescharacteristic
100C2×D17—22no—
101C2×D17—21no—
102C2×D17—42no—
103C2×D17—42no—
104C2×D17—41no—
105C2×D17#34211yescharacteristic
106C2×D17#34311yescharacteristic
107#595—42no—
108#595—42no—
109C4×C17—21no—
110C4×C17#32311yescharacteristic
111C4×C17#32411yescharacteristic
112Dic17—21no—
113Dic17#32311yescharacteristic
114Dic17#32411yescharacteristic
115C22×M16—171no—
116C23×C8—171no—
117C2×#117—171no—
118#431—342no—
119#336—342no—
120#324—174no—
121#323—342no—
122C2×C17—21no—
123C2×C17—41no—
124C2×C17#128011yescharacteristic
125D17—21no—
126D17—41no—
127D17#139512yes—
128C23×C4—171no—
129C2×M16—171no—
130C2×M16—341no—
131C2×M16—681no—
132C2×M16—681no—
133C2×M16#363311yescharacteristic, complemented
134C22×C8—172no—
135C22×C8—341no—
136C22×C8—341no—
137C22×C8—341no—
138C22×C8—681no—
139C22×C8—681no—
140#117—681no—
141#117—681no—
142C2×C4⋊C4—172no—
143C2×C22⋊C4—171no—
144C2×C22⋊C4—171no—
145C2×C42—172no—
146C4⋊C8—682no—
147C22⋊C8—344no—
148C4×C8—682no—
149C17#719011yescharacteristic, complemented, Sylow
150C24—171no—
151C22×C4—171no—
152C22×C4—171no—
153C22×C4—171no—
154C22×C4—171no—
155C22×C4—171no—
156C22×C4—172no—
157C22×C4—172no—
158C22×C4—341no—
159C22×C4—341no—
160C22×C4—341no—
161C22×C4—341no—
162C22×C4C2×#363311yescharacteristic
163M16—41no—
164M16—682no—
165M16—681no—
166C2×C8—21no—
167C2×C8—342no—
168C2×C8—341no—
169C2×C8—681no—
170C2×C8—681no—
171C2×C8—681no—
172C2×C8—681no—
173C2×C8—682no—
174C2×C8—682no—
175C2×C8—684no—
176C4⋊C4—682no—
177C22⋊C4—681no—
178C22⋊C4—681no—
179C42—342no—
180C42—342no—
181C23—171no—
182C23—171no—
183C23—341no—
184C23—341no—
185C23—682no—
186C23C4⋊#363311yescharacteristic
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—171no—
190C2×C4—171no—
191C2×C4—171no—
192C2×C4—171no—
193C2×C4—171no—
194C2×C4—171no—
195C2×C4—341no—
196C2×C4—341no—
197C2×C4—341no—
198C2×C4—341no—
199C2×C4—341no—
200C2×C4—341no—
201C2×C4—341no—
202C2×C4—341no—
203C2×C4—342no—
204C2×C4—342no—
205C2×C4—682no—
206C2×C4—682no—
207C2×C4C4×#363311yescharacteristic
208C2×C4C22⋊#363311yescharacteristic
209C8—41no—
210C8—682no—
211C8—681no—
212C8—1364no—
213C22—21no—
214C22—41no—
215C22—171no—
216C22—171no—
217C22—342no—
218C22—341no—
219C22—681no—
220C22—682no—
221C22—682no—
222C22(C2×#3633)⋊C411yescharacteristic
223C4—21no—
224C4—171no—
225C4—171no—
226C4—341no—
227C4—682no—
228C4—682no—
229C4#1176389411yescharacteristic
230C4#1176389511yescharacteristic
231C2—21no—
232C2—41no—
233C2—172no—
234C2—341no—
235C2—681no—
236C2#1454579311yescharacteristic
subgroupG/H
centerC4#11763894
commutatorC2×C4×C17C4×C8
FrattiniC22×C4C2×#3633
FittingC2×M16×C17C8
socleC2×C17#1280
prime radicalC23×D17C4⋊C4
cosocleC2×#10587677C22
hypercenterC2×M16#3633
hypocenterC17#7190
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
21. C2×#63242C8semidirect
25. C2×M16×C17C8semidirect
26. C2×#63223C8semidirect
133. C2×M16#3633semidirect
149. C17#7190semidirect

Maximal subgroups

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

subgroupindex
#145458962
#145444522
#719017