Finite Group Databasebeta

← #16921027

Subgroups — #16921027

Order 4352.

264 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 47 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#14526708C211yesmaximal, characteristic
2#14526661C211yesmaximal, characteristic
3C17×#2001C211yesmaximal, characteristic
4#11762773—21no—
5C22⋊#63214—21no—
6C22×Q16×C17C411yescharacteristic
7C2×C17×C22⋊Q8—21no—
8C2×C17×#103—21no—
9C2×C17×C22⋊C8C2211yescharacteristic
10C17×#340C411yescharacteristic
11C22×Q8×C17—21no—
12C23×C4×C17D411yescharacteristic
13C2×Q16×C17—41no—
14C2×Q16×C17—42no—
15C22×C8×C17C2×C411yescharacteristic
16C22×C8×C17D411yescharacteristic
17C17×C22⋊Q8—42no—
18C17×C22⋊Q8—42no—
19C2×C4⋊C4×C17—21no—
20C2×C4⋊C4×C17—41no—
21C2×C22⋊C4×C17—41no—
22C2×#63223—41no—
23C17×#103—42no—
24C17×C22⋊C8—41no—
25C22⋊#3622—22no—
26C22⋊#3622—41no—
27C2×#63214—41no—
28C24×C17SD1611yescharacteristic
29C2×Q8×C17—22no—
30C2×Q8×C17—41no—
31C2×Q8×C17—41no—
32C2×Q8×C17—41no—
33C2×Q8×C17—41no—
34C2×Q8×C17—41no—
35C22×C4×C17—21no—
36C22×C4×C17—21no—
37C22×C4×C17—41no—
38C22×C4×C17—41no—
39C22×C4×C17—41no—
40C22×C4×C17—41no—
41C22×C4×C17C22⋊C411yescharacteristic
42C22×C4×C17D811yescharacteristic
43Q16×C17—42no—
44Q16×C17—42no—
45C2×C8×C17—21no—
46C2×C8×C17—21no—
47C2×C8×C17—41no—
48C2×C8×C17—41no—
49C4⋊C4×C17—22no—
50C4⋊C4×C17—41no—
51C4⋊C4×C17—44no—
52C22⋊C4×C17—44no—
53#63223—81no—
54#63223—81no—
55C2×#3622—42no—
56#63214—81no—
57#7086—171nomaximal, Sylow
58C23×C17—21no—
59C23×C17—21no—
60C23×C17—21no—
61C23×C17—41no—
62C23×C17—41no—
63C23×C17#10211yescharacteristic
64Q8×C17—42no—
65Q8×C17—42no—
66Q8×C17—81no—
67Q8×C17—81no—
68C2×C4×C17—21no—
69C2×C4×C17—21no—
70C2×C4×C17—21no—
71C2×C4×C17—41no—
72C2×C4×C17—41no—
73C2×C4×C17—41no—
74C2×C4×C17—42no—
75C2×C4×C17—42no—
76C2×C4×C17—81no—
77C2×C4×C17—81no—
78C2×C4×C17—81no—
79C2×C4×C17—81no—
80C2×C4×C17—81no—
81C2×C4×C17#9911yescharacteristic
82C2×C4×C17#10411yescharacteristic
83C8×C17—41no—
84C8×C17—41no—
85#3622—82no—
86#2001D1711yescharacteristic
87#1314—171no—
88#1293—171no—
89C22×C17—21no—
90C22×C17—21no—
91C22×C17—42no—
92C22×C17—41no—
93C22×C17—41no—
94C22×C17—41no—
95C22×C17—81no—
96C22×C17#32711yescharacteristic
97C22×C17#35911yescharacteristic
98C22×C17#36211yescharacteristic
99C4×C17—21no—
100C4×C17—41no—
101C4×C17—82no—
102C4×C17—82no—
103C4×C17AΣL(1,𝔽16)11yescharacteristic
104C4×C17#35311yescharacteristic
105C22×Q16Dic1711yescharacteristic
106C2×C22⋊Q8—21no—
107C2×#103—21no—
108#407—341no—
109C2×C22⋊C8C2×D1711yescharacteristic
110C22⋊C16—341no—
111#340Dic1711yescharacteristic
112C2×C17—21no—
113C2×C17—21no—
114C2×C17—81no—
115C2×C17#134211yescharacteristic
116C2×C17#134311yescharacteristic
117C2×C17#135011yescharacteristic
118C22×Q8—21no—
119C23×C4C22⋊D1711yescharacteristic
120C2×Q16—41no—
121C2×Q16—42no—
122C2×M16—681no—
123C22×C8C2×Dic1711yescharacteristic
124C22×C8C22⋊D1711yescharacteristic
125C22⋊Q8—42no—
126C22⋊Q8—42no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—41no—
129C2×C22⋊C4—41no—
130C2×C16—681no—
131#103—42no—
132C22⋊C8—41no—
133C22⋊C8—342no—
134C22⋊C8—681no—
135C17#708611yescharacteristic, complemented, Sylow
136C24#6322911yescharacteristic
137C2×Q8—22no—
138C2×Q8—41no—
139C2×Q8—41no—
140C2×Q8—41no—
141C2×Q8—41no—
142C2×Q8—41no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—41no—
146C22×C4—41no—
147C22×C4—41no—
148C22×C4—41no—
149C22×C4#6322811yescharacteristic
150C22×C4C22⋊Dic1711yescharacteristic
151Q16—42no—
152Q16—42no—
153M16—1361no—
154M16—1361no—
155C2×C8—21no—
156C2×C8—21no—
157C2×C8—41no—
158C2×C8—41no—
159C2×C8—682no—
160C4⋊C4—22no—
161C4⋊C4—41no—
162C4⋊C4—44no—
163C22⋊C4—44no—
164C16—1361no—
165C23—21no—
166C23—21no—
167C23—21no—
168C23—41no—
169C23—41no—
170C23#1058750311yescharacteristic
171Q8—42no—
172Q8—42no—
173Q8—81no—
174Q8—81no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—81no—
184C2×C4—81no—
185C2×C4—81no—
186C2×C4—81no—
187C2×C4—81no—
188C2×C4#1058750511yescharacteristic
189C2×C4#1058750811yescharacteristic
190C8—41no—
191C8—41no—
192C8—1362no—
193C22—21no—
194C22—21no—
195C22—42no—
196C22—41no—
197C22—41no—
198C22—41no—
199C22—81no—
200C22#1176254311yescharacteristic
201C22#1176261611yescharacteristic
202C22#1176262511yescharacteristic
203C4—21no—
204C4—41no—
205C4—82no—
206C4—82no—
207C4#1176259611yescharacteristic
208C4#1176260211yescharacteristic
209C2—21no—
210C2—21no—
211C2—81no—
212C2#1452677811yescharacteristic
213C2#1452678111yescharacteristic
214C2#1452680211yescharacteristic
subgroupG/H
centerC22#11762543
commutatorC22×C8×C17C2×C4
FrattiniC2×C22⋊C8C2×D17
FittingC17×#2001C2
socleC22×C17#327
prime radicalC24×C17SD16
cosocleC2×C17×C22⋊C8C22
hypercenter#2001D17
hypocenterC17#7086
LCSG ▷ C22×C8×C17 ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C22⋊C8 ◁ #2001

Complements

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

NH ≅ G/Ntype
135. C17#7086semidirect

Maximal subgroups

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

subgroupindex
#145267082
#145266612
C17×#20012
#708617