Finite Group Databasebeta

← #16927901

Subgroups — #16927901

Order 4352.

422 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 123 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#14531705C211yesmaximal, characteristic
2#14528780C211yesmaximal, characteristic
3#14526686C212yesmaximal
4#14526643C212yesmaximal
5C17×#1461C211yesmaximal, characteristic
6C2×Q8×Dic17—21no—
7#11762833C2212yes—
8C2×#10587507—21no—
9#11762708—21no—
10#11762708—22no—
11#11762675C412yes—
12#11762675C2211yescharacteristic
13C4×C4⋊Dic17C412yes—
14C4×C4⋊Dic17C2211yescharacteristic
15#11762578—21no—
16C2×C4×Q8×C17C2211yescharacteristic
17C17×#431C2212yes—
18C17×#324—22no—
19C22×Q8×C17D411yescharacteristic
20C4×Q8×C17—22no—
21C4×Q8×C17—22no—
22C2×C4⋊C4×C17D411yescharacteristic
23C2×C4⋊C4×C17D412yes—
24C2×C42×C17C2×C412yes—
25C2×C42×C17C2311yescharacteristic
26Q8×Dic17—42no—
27Q8×Dic17—42no—
28C2×C4⋊Dic17—22no—
29C2×C4⋊Dic17—22no—
30C2×C4⋊Dic17—21no—
31C2×C4⋊Dic17—21no—
32C2×C4×Dic17—22no—
33C2×C4×Dic17—21no—
34C2×#63223—22no—
35C17×C4⋊C8—22no—
36C17×C22⋊C8—22no—
37C4×C8×C17—22no—
38#10587507—42no—
39#10587502—22no—
40#10587502—21no—
41#10587502C2×C411yescharacteristic
42#10587502C2×C412yes—
43#10587502C2×C411yescharacteristic
44C4⋊#3622—22no—
45C4⋊#3622—22no—
46C2×Q8×C17—22no—
47C2×Q8×C17—22no—
48C2×Q8×C17D812yes—
49C2×Q8×C17SD1612yes—
50C22×C4×C17C22⋊C412yes—
51C22×C4×C17C22⋊C412yes—
52C22×C4×C17C22×C411yescharacteristic
53C22×C4×C17C2×D411yescharacteristic
54C22×C4×C17C2×D411yescharacteristic
55C22×Dic17—21no—
56C22×Dic17—21no—
57C22×Dic17—22no—
58C2×C8×C17—42no—
59C4⋊C4×C17—22no—
60C4⋊C4×C17—24no—
61C42×C17—22no—
62C42×C17—22no—
63C42×C17#4112yes—
64C42×C17#4112yes—
65C4⋊Dic17—42no—
66C4⋊Dic17—44no—
67C4⋊Dic17—44no—
68C4⋊Dic17—41no—
69C4⋊Dic17—41no—
70C4×Dic17—44no—
71C4×Dic17—41no—
72C4×Dic17—41no—
73#63223—42no—
74#63223—42no—
75C2×#3622—42no—
76#8766—171nomaximal, Sylow
77C23×C17C2×C22⋊C411yescharacteristic
78Q8×C17—24no—
79Q8×C17—24no—
80C2×C4×C17—22no—
81C2×C4×C17—22no—
82C2×C4×C17—22no—
83C2×C4×C17—41no—
84C2×C4×C17—41no—
85C2×C4×C17—42no—
86C2×C4×C17#10214yes—
87C2×C4×C17#11711yescharacteristic
88C2×C4×C17#11711yescharacteristic
89C2×C4×C17#12312yes—
90C2×C4×C17#12312yes—
91C2×C4×C17C2×D811yescharacteristic
92C2×C4×C17C2×SD1611yescharacteristic
93C2×C4×C17#13712yes—
94C2×Dic17—22no—
95C2×Dic17—41no—
96C2×Dic17—41no—
97C2×Dic17—41no—
98C2×Dic17—42no—
99C2×Dic17—41no—
100C2×Dic17—42no—
101C2×Dic17—42no—
102C2×Dic17—41no—
103C8×C17—82no—
104#3622—82no—
105#2351—171no—
106#1801—171no—
107#1461D1711yescharacteristic
108#1303—172no—
109#1286—172no—
110C22×C17—21no—
111C22×C17—21no—
112C22×C17#38111yescharacteristic
113C22×C17C2×#10211yescharacteristic
114C22×C17#41911yescharacteristic
115C4×C17—24no—
116C4×C17—22no—
117C4×C17—21no—
118C4×C17—21no—
119C4×C17—42no—
120C4×C17#48012yes—
121C4×C17#48312yes—
122Dic17—42no—
123Dic17—82no—
124Dic17—81no—
125C2×C4×Q8—341no—
126C2×C4×Q8C2×D1711yescharacteristic
127#431C2×D1712yes—
128#423—172no—
129C2×#101—341no—
130#389—341no—
131#389—342no—
132#382—172no—
133#382—171no—
134C4×C4⋊C4—172no—
135C4×C4⋊C4—171no—
136#343—341no—
137#324—22no—
138C2×C17—21no—
139C2×C17—21no—
140C2×C17#182311yescharacteristic
141C2×C17#182711yescharacteristic
142C2×C17#183011yescharacteristic
143C22×Q8D6811yescharacteristic
144C2×M16—342no—
145C4×Q8—22no—
146C4×Q8—22no—
147C4×Q8—682no—
148C4×Q8—682no—
149C2×C4⋊C4—341no—
150C2×C4⋊C4—341no—
151C2×C4⋊C4—342no—
152C2×C4⋊C4—342no—
153C2×C4⋊C4D6811yescharacteristic
154C2×C4⋊C4C22⋊D1712yes—
155C2×C42—341no—
156C2×C42—342no—
157C2×C42C4×D1712yes—
158C2×C42C22×D1711yescharacteristic
159C4⋊C8—22no—
160C4⋊C8—342no—
161C4⋊C8—342no—
162#101—682no—
163C22⋊C8—22no—
164C4×C8—22no—
165#95—171no—
166#95—172no—
167#95—171no—
168#95—341no—
169#95—342no—
170C17#876611yescharacteristic, complemented, Sylow
171C2×Q8—22no—
172C2×Q8—22no—
173C2×Q8#6321912yes—
174C2×Q8D13612yes—
175C22×C4—341no—
176C22×C4—342no—
177C22×C4—341no—
178C22×C4#6322712yes—
179C22×C4#6322712yes—
180C22×C4C2×C4×D1711yescharacteristic
181C22×C4C2×D6811yescharacteristic
182C22×C4C2×C22⋊D1711yescharacteristic
183M16—682no—
184M16—682no—
185C2×C8—42no—
186C2×C8—682no—
187C4⋊C4—22no—
188C4⋊C4—24no—
189C4⋊C4—682no—
190C4⋊C4—684no—
191C4⋊C4—684no—
192C4⋊C4—681no—
193C4⋊C4—681no—
194C42—22no—
195C42—22no—
196C42—684no—
197C42—681no—
198C42—681no—
199C42#6325412yes—
200C42#6325412yes—
201C23C2×#6322711yescharacteristic
202Q8—24no—
203Q8—24no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—42no—
210C2×C4—342no—
211C2×C4—681no—
212C2×C4—681no—
213C2×C4—681no—
214C2×C4—682no—
215C2×C4—682no—
216C2×C4—681no—
217C2×C4—682no—
218C2×C4—681no—
219C2×C4#1058749214yes—
220C2×C4#1058756311yescharacteristic
221C2×C4#1058756311yescharacteristic
222C2×C4#1058757212yes—
223C2×C4C2×#6321911yescharacteristic
224C2×C4C2×D13611yescharacteristic
225C2×C4#1058759512yes—
226C2×C4#1058762112yes—
227C8—82no—
228C8—1362no—
229C22—21no—
230C22—21no—
231C22#1176267011yescharacteristic
232C22C2×#1058749211yescharacteristic
233C22#1176281911yescharacteristic
234C4—24no—
235C4—22no—
236C4—21no—
237C4—21no—
238C4—42no—
239C4—682no—
240C4—1362no—
241C4—1361no—
242C4#1176312312yes—
243C4#1176314012yes—
244C2—21no—
245C2—21no—
246C2#1452887111yescharacteristic
247C2#1452888511yescharacteristic
248C2#1452889611yescharacteristic
subgroupG/H
centerC22#11762670
commutatorC22×C4×C17C22×C4
FrattiniC2×C42C22×D17
FittingC17×#1461C2
socleC22×C17#381
prime radicalC23×C17C2×C22⋊C4
cosocleC2×C42×C17C23
hypercenter#1461D17
hypocenterC17#8766
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1461

Complements

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

NH ≅ G/Ntype
170. C17#8766semidirect

Maximal subgroups

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

subgroupindex
#145317052
#145287802
#145266862
#145266432
C17×#14612
#876617