Finite Group Databasebeta

← #16954660

Subgroups — #16954660

Order 4352.

398 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 79 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×#11762608C212yesmaximal, complemented
2#14526953C212yesmaximal, complemented
3#14526951C212yesmaximal, complemented
4C17×#3152C211yesmaximal, characteristic
5C2×#10587507—22no—
6#11762608—22no—
7#11762608C2214yes—
8C2×C17×#137—22no—
9C17×#532C411yescharacteristic
10C2×C17×C4⋊Q8C2211yescharacteristic
11C17×#497—22no—
12C17×#494C411yescharacteristic
13C17×#494C2211yescharacteristic
14C17×#492C411yescharacteristic
15C17×#492C2211yescharacteristic
16C4×M16×C17C411yescharacteristic
17C2×#41×C17—21no—
18C22×Q8×C17D412yes—
19C17×#137—44no—
20C17×#137—44no—
21C2×Q16×C17—22no—
22C2×Q16×C17—22no—
23C2×SD16×C17—22no—
24C2×SD16×C17—22no—
25C2×M16×C17—21no—
26C17×C4⋊Q8—41no—
27C17×C4⋊Q8C2×C411yescharacteristic
28C17×C4⋊Q8C2×C411yescharacteristic
29C17×C4⋊Q8D412yes—
30C17×C4⋊Q8C2311yescharacteristic
31C17×#127C2×C411yescharacteristic
32C17×#124—21no—
33C17×#121—41no—
34C4×D4×C17—41no—
35C2×C4⋊C4×C17—41no—
36C2×C42×C17C2×C411yescharacteristic
37C2×#63223—42no—
38C17×#97—21no—
39C4×C8×C17C2×C411yescharacteristic
40C4×C8×C17C2×C411yescharacteristic
41#10587507—24no—
42#10587507—42no—
43#41×C17—42no—
44#41×C17—42no—
45C2×Q8×C17—22no—
46C2×Q8×C17—21no—
47C2×Q8×C17—42no—
48C2×Q8×C17—42no—
49C2×Q8×C17C22⋊C412yes—
50C2×Q8×C17C2×D412yes—
51C2×D4×C17—21no—
52C2×D4×C17—41no—
53C2×D4×C17—41no—
54C22×C4×C17—22no—
55C22×C4×C17—21no—
56C22×C4×C17—41no—
57C22×C4×C17C22⋊C411yescharacteristic
58Q16×C17—44no—
59Q16×C17—44no—
60SD16×C17—44no—
61SD16×C17—44no—
62M16×C17—24no—
63C2×C8×C17—21no—
64C2×C8×C17—21no—
65C4⋊C4×C17—41no—
66C4⋊C4×C17—42no—
67C4⋊C4×C17—41no—
68C4⋊C4×C17—41no—
69C22⋊C4×C17—42no—
70C42×C17—21no—
71C42×C17C22⋊C411yescharacteristic
72C42×C17C22×C411yescharacteristic
73#63223—44no—
74#63223—82no—
75C2×#3622—82no—
76#13590—171nomaximal, Sylow
77C23×C17—41no—
78C23×C17#9911yescharacteristic
79Q8×C17—42no—
80Q8×C17—42no—
81Q8×C17—42no—
82Q8×C17—82no—
83D4×C17—42no—
84D4×C17—82no—
85D4×C17—82no—
86C2×C4×C17—22no—
87C2×C4×C17—21no—
88C2×C4×C17—21no—
89C2×C4×C17—22no—
90C2×C4×C17—22no—
91C2×C4×C17—42no—
92C2×C4×C17—42no—
93C2×C4×C17—42no—
94C2×C4×C17—41no—
95C2×C4×C17—81no—
96C2×C4×C17—81no—
97C2×C4×C17#9911yescharacteristic
98C2×C4×C17C2×C22⋊C411yescharacteristic
99C8×C17—22no—
100C8×C17—22no—
101#3622—84no—
102#3152D1711yescharacteristic
103C2×#356—172no—
104#1411—172no—
105#1410—172no—
106C22×C17—41no—
107C22×C17—82no—
108C22×C17—81no—
109C22×C17#35312yes—
110C22×C17C2×#9911yescharacteristic
111C4×C17—22no—
112C4×C17—21no—
113C4×C17—21no—
114C4×C17—41no—
115C4×C17—42no—
116C4×C17—42no—
117C4×C17—81no—
118C2×#137—22no—
119#532Dic1711yescharacteristic
120C2×C4⋊Q8C2×D1711yescharacteristic
121#497—22no—
122#494Dic1711yescharacteristic
123#494C2×D1711yescharacteristic
124#492Dic1711yescharacteristic
125#492C2×D1711yescharacteristic
126C2×#101—342no—
127C4×M16Dic1711yescharacteristic
128#356—174no—
129#356—342no—
130C2×C17—21no—
131C2×C17—22no—
132C2×C17—161no—
133C2×C17C2×#35311yescharacteristic
134C2×#41—21no—
135C22×Q8C22⋊D1712yes—
136#137—44no—
137#137—44no—
138C2×Q16—22no—
139C2×Q16—22no—
140C2×SD16—22no—
141C2×SD16—22no—
142C2×M16—21no—
143C2×M16—682no—
144C4⋊Q8—41no—
145C4⋊Q8C2×Dic1711yescharacteristic
146C4⋊Q8C2×Dic1711yescharacteristic
147C4⋊Q8C22⋊D1712yes—
148C4⋊Q8C22×D1711yescharacteristic
149#127C2×Dic1711yescharacteristic
150#124—21no—
151#121—41no—
152C4×D4—41no—
153C2×C4⋊C4—41no—
154C2×C42C2×Dic1711yescharacteristic
155#101—344no—
156#101—682no—
157#97—21no—
158C4×C8C2×Dic1711yescharacteristic
159C4×C8C2×Dic1711yescharacteristic
160C17#1359011yescharacteristic, complemented, Sylow
161#41—42no—
162#41—42no—
163C2×Q8—22no—
164C2×Q8—21no—
165C2×Q8—42no—
166C2×Q8—42no—
167C2×Q8C22⋊Dic1712yes—
168C2×Q8C2×C22⋊D1712yes—
169C2×D4—21no—
170C2×D4—41no—
171C2×D4—41no—
172C22×C4—21no—
173C22×C4—22no—
174C22×C4—41no—
175C22×C4C22⋊Dic1711yescharacteristic
176Q16—44no—
177Q16—44no—
178SD16—44no—
179SD16—44no—
180M16—24no—
181M16—684no—
182M16—1362no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—1362no—
186C4⋊C4—41no—
187C4⋊C4—42no—
188C4⋊C4—41no—
189C4⋊C4—41no—
190C22⋊C4—42no—
191C42—21no—
192C42C22⋊Dic1711yescharacteristic
193C42C22×Dic1711yescharacteristic
194C23—41no—
195C23#1058750511yescharacteristic
196Q8—42no—
197Q8—42no—
198Q8—42no—
199Q8—82no—
200D4—42no—
201D4—82no—
202D4—82no—
203C2×C4—21no—
204C2×C4—22no—
205C2×C4—21no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—41no—
212C2×C4—81no—
213C2×C4—81no—
214C2×C4#1058750511yescharacteristic
215C2×C4C2×C22⋊Dic1711yescharacteristic
216C8—22no—
217C8—22no—
218C8—1364no—
219C22—41no—
220C22—82no—
221C22—81no—
222C22#1176260212yes—
223C22C2×#1058750511yescharacteristic
224C4—22no—
225C4—21no—
226C4—21no—
227C4—41no—
228C4—42no—
229C4—42no—
230C4—81no—
231C2—21no—
232C2—22no—
233C2—161no—
234C2C2×#1176260211yescharacteristic
subgroupG/H
centerC2C2×#11762602
commutatorC42×C17C22×C4
FrattiniC4⋊Q8C22×D17
FittingC17×#3152C2
socleC2×C17C2×#353
prime radicalC17×#532C4
cosocleC17×C4⋊Q8C23
hypercenter#3152D17
hypocenterC17#13590
LCSG ▷ C42×C17 ▷ C2×C4×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C2 ◁ C23 ◁ C22×C4 ◁ C2×C4⋊Q8 ◁ #3152

Complements

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

NH ≅ G/Ntype
1. C2×#11762608C2semidirect
2. #14526953C2semidirect
3. #14526951C2semidirect
160. C17#13590semidirect

Maximal subgroups

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

subgroupindex
C2×#117626082
#145269532
#145269512
C17×#31522
#1359017