Finite Group Databasebeta

← #16927916

Subgroups — #16927916

Order 4352.

378 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 107 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#14531749C211yesmaximal, characteristic
2#14528780C211yesmaximal, characteristic
3#14526686C212yesmaximal
4#14526643C212yesmaximal
5C17×#1461C211yesmaximal, characteristic
6#11762833C2212yes—
7C2×#10587507—21no—
8#11762716—21no—
9#11762695—22no—
10#11762677C412yes—
11#11762675C2211yescharacteristic
12C4⋊C4×Dic17—21no—
13C4×C4⋊Dic17C2211yescharacteristic
14#11762655C412yes—
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
26C2×C4⋊Dic17—21no—
27C2×C4⋊Dic17—21no—
28C2×C4×Dic17—22no—
29C2×C4×Dic17—21no—
30C2×#63223—22no—
31C17×C4⋊C8—22no—
32C17×C22⋊C8—22no—
33C4×C8×C17—22no—
34#10587507—42no—
35#10587502—22no—
36#10587502—22no—
37#10587502—22no—
38#10587502—21no—
39#10587502C2×C411yescharacteristic
40#10587502C2×C412yes—
41#10587502C2×C411yescharacteristic
42C4⋊#3622—22no—
43C4⋊#3622—22no—
44C2×Q8×C17—22no—
45C2×Q8×C17—22no—
46C2×Q8×C17—21no—
47C2×Q8×C17—21no—
48C22×C4×C17C22⋊C412yes—
49C22×C4×C17C22⋊C412yes—
50C22×C4×C17C22×C411yescharacteristic
51C22×C4×C17C2×D411yescharacteristic
52C22×C4×C17C2×D411yescharacteristic
53C22×Dic17—21no—
54C22×Dic17—22no—
55C22×Dic17—21no—
56C2×C8×C17—42no—
57C4⋊C4×C17—22no—
58C4⋊C4×C17—24no—
59C42×C17—22no—
60C42×C17—22no—
61C42×C17#4112yes—
62C42×C17#4112yes—
63C4⋊Dic17—42no—
64C4⋊Dic17—41no—
65C4⋊Dic17—41no—
66C4×Dic17—44no—
67C4×Dic17—42no—
68#63223—42no—
69#63223—42no—
70C2×#3622—42no—
71#8769—171nomaximal, Sylow
72C23×C17C2×C22⋊C411yescharacteristic
73Q8×C17—42no—
74Q8×C17—42no—
75C2×C4×C17—22no—
76C2×C4×C17—22no—
77C2×C4×C17—22no—
78C2×C4×C17—22no—
79C2×C4×C17—41no—
80C2×C4×C17—41no—
81C2×C4×C17—42no—
82C2×C4×C17#11711yescharacteristic
83C2×C4×C17#11711yescharacteristic
84C2×C4×C17#12312yes—
85C2×C4×C17#12312yes—
86C2×C4×C17#13512yes—
87C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
88C2×C4×C17#13711yescharacteristic
89C2×Dic17—22no—
90C2×Dic17—41no—
91C2×Dic17—41no—
92C2×Dic17—41no—
93C2×Dic17—42no—
94C2×Dic17—42no—
95C2×Dic17—41no—
96C2×Dic17—41no—
97C2×Dic17—42no—
98C8×C17—82no—
99#3622—82no—
100#2356—171no—
101#1801—171no—
102#1461D1711yescharacteristic
103#1303—172no—
104#1286—172no—
105C22×C17—21no—
106C22×C17—21no—
107C22×C17#38111yescharacteristic
108C22×C17#41611yescharacteristic
109C22×C17#41711yescharacteristic
110C4×C17—22no—
111C4×C17—21no—
112C4×C17—21no—
113C4×C17—42no—
114C4×C17—42no—
115C4×C17#48212yes—
116C4×C17#48512yes—
117Dic17—42no—
118Dic17—81no—
119Dic17—82no—
120C2×C4×Q8C2×D1711yescharacteristic
121#431C2×D1712yes—
122#423—172no—
123C2×#101—341no—
124#391—341no—
125#387—342no—
126#383—172no—
127#382—171no—
128C4×C4⋊C4—171no—
129C4×C4⋊C4—341no—
130#376—172no—
131#343—341no—
132#324—22no—
133C2×C17—21no—
134C2×C17—21no—
135C2×C17#182511yescharacteristic
136C2×C17#182711yescharacteristic
137C2×C17#183011yescharacteristic
138C22×Q8D6811yescharacteristic
139C2×M16—342no—
140C4×Q8—22no—
141C4×Q8—22no—
142C2×C4⋊C4—341no—
143C2×C4⋊C4—341no—
144C2×C4⋊C4D6811yescharacteristic
145C2×C4⋊C4C22⋊D1712yes—
146C2×C42—341no—
147C2×C42—342no—
148C2×C42C4×D1712yes—
149C2×C42C22×D1711yescharacteristic
150C4⋊C8—22no—
151C4⋊C8—342no—
152C4⋊C8—342no—
153#101—682no—
154C22⋊C8—22no—
155C4×C8—22no—
156#95—171no—
157#95—172no—
158#95—171no—
159#95—341no—
160#95—342no—
161#95—342no—
162#95—342no—
163C17#876911yescharacteristic, complemented, Sylow
164C2×Q8—22no—
165C2×Q8—22no—
166C2×Q8—21no—
167C2×Q8—21no—
168C22×C4—341no—
169C22×C4—342no—
170C22×C4—341no—
171C22×C4#6322712yes—
172C22×C4#6322712yes—
173C22×C4C2×C4×D1711yescharacteristic
174C22×C4C2×D6811yescharacteristic
175C22×C4C2×C22⋊D1711yescharacteristic
176M16—682no—
177M16—682no—
178C2×C8—42no—
179C2×C8—682no—
180C4⋊C4—22no—
181C4⋊C4—24no—
182C4⋊C4—682no—
183C4⋊C4—681no—
184C4⋊C4—681no—
185C42—22no—
186C42—22no—
187C42—684no—
188C42—682no—
189C42#6325412yes—
190C42#6325412yes—
191C23C2×#6322711yescharacteristic
192Q8—42no—
193Q8—42no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—42no—
201C2×C4—342no—
202C2×C4—681no—
203C2×C4—681no—
204C2×C4—682no—
205C2×C4—681no—
206C2×C4—681no—
207C2×C4—682no—
208C2×C4—681no—
209C2×C4—682no—
210C2×C4#1058756311yescharacteristic
211C2×C4#1058756311yescharacteristic
212C2×C4#1058757212yes—
213C2×C4#1058759012yes—
214C2×C4#1058759411yescharacteristic
215C2×C4#1058759511yescharacteristic
216C2×C4#1058762112yes—
217C8—82no—
218C8—1362no—
219C22—21no—
220C22—21no—
221C22#1176267011yescharacteristic
222C22#1176280611yescharacteristic
223C22#1176281111yescharacteristic
224C4—22no—
225C4—21no—
226C4—21no—
227C4—42no—
228C4—42no—
229C4—682no—
230C4—1361no—
231C4—1362no—
232C4#1176313412yes—
233C4#1176315112yes—
234C2—21no—
235C2—21no—
236C2#1452888011yescharacteristic
237C2#1452888511yescharacteristic
238C2#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#8769
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
163. C17#8769semidirect

Maximal subgroups

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

subgroupindex
#145317492
#145287802
#145266862
#145266432
C17×#14612
#876917