Finite Group Databasebeta

← #16927906

Subgroups — #16927906

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—44no—
27C2×C4⋊Dic17—22no—
28C2×C4⋊Dic17—22no—
29C2×C4⋊Dic17—21no—
30C2×C4⋊Dic17—21no—
31C2×C4×Dic17—22no—
32C2×C4×Dic17—21no—
33C2×#63223—22no—
34C17×C4⋊C8—22no—
35C17×C22⋊C8—22no—
36C4×C8×C17—22no—
37#10587507—42no—
38#10587502—22no—
39#10587502—21no—
40#10587502C2×C411yescharacteristic
41#10587502C2×C412yes—
42#10587502C2×C411yescharacteristic
43C4⋊#3622—22no—
44C4⋊#3622—22no—
45C2×Q8×C17—22no—
46C2×Q8×C17—22no—
47C2×Q8×C17SD1612yes—
48C2×Q8×C17Q1612yes—
49C22×C4×C17C22⋊C412yes—
50C22×C4×C17C22⋊C412yes—
51C22×C4×C17C22×C411yescharacteristic
52C22×C4×C17C2×D411yescharacteristic
53C22×C4×C17C2×D411yescharacteristic
54C22×Dic17—21no—
55C22×Dic17—21no—
56C22×Dic17—22no—
57C2×C8×C17—42no—
58C4⋊C4×C17—22no—
59C4⋊C4×C17—24no—
60C42×C17—22no—
61C42×C17—22no—
62C42×C17#4112yes—
63C42×C17#4112yes—
64C4⋊Dic17—42no—
65C4⋊Dic17—44no—
66C4⋊Dic17—44no—
67C4⋊Dic17—42no—
68C4×Dic17—44no—
69C4×Dic17—42no—
70#63223—42no—
71#63223—42no—
72C2×#3622—42no—
73#8767—171nomaximal, Sylow
74C23×C17C2×C22⋊C411yescharacteristic
75Q8×C17—24no—
76Q8×C17—24no—
77C2×C4×C17—22no—
78C2×C4×C17—22no—
79C2×C4×C17—22no—
80C2×C4×C17—41no—
81C2×C4×C17—41no—
82C2×C4×C17—42no—
83C2×C4×C17#10314yes—
84C2×C4×C17#11711yescharacteristic
85C2×C4×C17#11711yescharacteristic
86C2×C4×C17#12312yes—
87C2×C4×C17#12312yes—
88C2×C4×C17C2×SD1611yescharacteristic
89C2×C4×C17C2×Q1611yescharacteristic
90C2×C4×C17AGL(1,ℤ/8)12yes—
91C2×Dic17—22no—
92C2×Dic17—41no—
93C2×Dic17—41no—
94C2×Dic17—41no—
95C2×Dic17—42no—
96C2×Dic17—41no—
97C2×Dic17—42no—
98C2×Dic17—42no—
99C2×Dic17—41no—
100C8×C17—82no—
101#3622—82no—
102#2351—171no—
103#1801—171no—
104#1461D1711yescharacteristic
105#1303—172no—
106#1286—172no—
107C22×C17—21no—
108C22×C17—21no—
109C22×C17#38111yescharacteristic
110C22×C17C2×#10311yescharacteristic
111C22×C17#41811yescharacteristic
112C4×C17—24no—
113C4×C17—22no—
114C4×C17—21no—
115C4×C17—21no—
116C4×C17—42no—
117C4×C17#48112yes—
118C4×C17#48412yes—
119Dic17—42no—
120Dic17—82no—
121Dic17—81no—
122C2×C4×Q8—341no—
123C2×C4×Q8C2×D1711yescharacteristic
124#431C2×D1712yes—
125#423—172no—
126C2×#101—341no—
127#389—341no—
128#389—342no—
129#382—172no—
130#382—171no—
131C4×C4⋊C4—172no—
132C4×C4⋊C4—171no—
133#343—341no—
134#324—22no—
135C2×C17—21no—
136C2×C17—21no—
137C2×C17#182411yescharacteristic
138C2×C17#182711yescharacteristic
139C2×C17#183011yescharacteristic
140C22×Q8D6811yescharacteristic
141C2×M16—342no—
142C4×Q8—22no—
143C4×Q8—22no—
144C4×Q8—684no—
145C2×C4⋊C4—341no—
146C2×C4⋊C4—341no—
147C2×C4⋊C4—342no—
148C2×C4⋊C4—342no—
149C2×C4⋊C4D6811yescharacteristic
150C2×C4⋊C4C22⋊D1712yes—
151C2×C42—341no—
152C2×C42—342no—
153C2×C42C4×D1712yes—
154C2×C42C22×D1711yescharacteristic
155C4⋊C8—22no—
156C4⋊C8—342no—
157C4⋊C8—342no—
158#101—682no—
159C22⋊C8—22no—
160C4×C8—22no—
161#95—171no—
162#95—172no—
163#95—171no—
164#95—341no—
165#95—342no—
166C17#876711yescharacteristic, complemented, Sylow
167C2×Q8—22no—
168C2×Q8—22no—
169C2×Q8#6321912yes—
170C2×Q8Dic6812yes—
171C22×C4—341no—
172C22×C4—342no—
173C22×C4—341no—
174C22×C4#6322712yes—
175C22×C4#6322712yes—
176C22×C4C2×C4×D1711yescharacteristic
177C22×C4C2×D6811yescharacteristic
178C22×C4C2×C22⋊D1711yescharacteristic
179M16—682no—
180M16—682no—
181C2×C8—42no—
182C2×C8—682no—
183C4⋊C4—22no—
184C4⋊C4—24no—
185C4⋊C4—684no—
186C4⋊C4—682no—
187C4⋊C4—684no—
188C4⋊C4—682no—
189C42—22no—
190C42—22no—
191C42—684no—
192C42—682no—
193C42#6325412yes—
194C42#6325412yes—
195C23C2×#6322711yescharacteristic
196Q8—24no—
197Q8—24no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—42no—
204C2×C4—342no—
205C2×C4—681no—
206C2×C4—681no—
207C2×C4—681no—
208C2×C4—682no—
209C2×C4—682no—
210C2×C4—681no—
211C2×C4—682no—
212C2×C4—681no—
213C2×C4#1058748714yes—
214C2×C4#1058756311yescharacteristic
215C2×C4#1058756311yescharacteristic
216C2×C4#1058757212yes—
217C2×C4C2×#6321911yescharacteristic
218C2×C4C2×Dic6811yescharacteristic
219C2×C4#1058759412yes—
220C2×C4#1058762112yes—
221C8—82no—
222C8—1362no—
223C22—21no—
224C22—21no—
225C22#1176267011yescharacteristic
226C22C2×#1058748711yescharacteristic
227C22#1176281511yescharacteristic
228C4—24no—
229C4—22no—
230C4—21no—
231C4—21no—
232C4—42no—
233C4—682no—
234C4—1362no—
235C4—1361no—
236C4#1176312812yes—
237C4#1176314512yes—
238C2—21no—
239C2—21no—
240C2#1452887511yescharacteristic
241C2#1452888511yescharacteristic
242C2#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#8767
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
166. C17#8767semidirect

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
#876717