Finite Group Databasebeta

← #16927907

Subgroups — #16927907

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#14527108C211yesmaximal, characteristic
3#14526685C212yesmaximal
4#14526644C212yesmaximal
5C17×#1801C211yesmaximal, characteristic
6#11762862C2212yes—
7#11762716—21no—
8#11762695—22no—
9#11762677C412yes—
10#11762675C2211yescharacteristic
11C4⋊C4×Dic17—21no—
12C4×C4⋊Dic17C2211yescharacteristic
13#11762655C412yes—
14#11762535—22no—
15C2×C4×Q8×C17C2211yescharacteristic
16C17×#423C2212yes—
17C2×C17×#101—21no—
18C17×#343—21no—
19C22×Q8×C17D411yescharacteristic
20C2×M16×C17—22no—
21C4×Q8×C17—22no—
22C4×Q8×C17—22no—
23C2×C4⋊C4×C17D412yes—
24C2×C4⋊C4×C17D411yescharacteristic
25C2×C42×C17C2×C412yes—
26C2×C42×C17C2311yescharacteristic
27C2×C4⋊Dic17—21no—
28C2×C4⋊Dic17—21no—
29C2×C4×Dic17—22no—
30C2×C4×Dic17—21no—
31C17×C4⋊C8—22no—
32C17×C4⋊C8—22no—
33C17×#101—42no—
34#10587502—22no—
35#10587502—22no—
36#10587502—22no—
37#10587502—21no—
38#10587502C2×C411yescharacteristic
39#10587502C2×C412yes—
40#10587502C2×C411yescharacteristic
41C22⋊#3622—22no—
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—
56M16×C17—42no—
57M16×C17—42no—
58C2×C8×C17—42no—
59C4⋊C4×C17—24no—
60C4⋊C4×C17—22no—
61C42×C17—22no—
62C42×C17—22no—
63C42×C17#4112yes—
64C42×C17#4112yes—
65C4⋊Dic17—42no—
66C4⋊Dic17—42no—
67C4×Dic17—44no—
68C4×Dic17—41no—
69C4×Dic17—41no—
70C2×#3622—42no—
71#8768—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—42no—
97C2×Dic17—41no—
98C8×C17—82no—
99#3622—82no—
100#2356—171no—
101#1801D1711yescharacteristic
102#1461—171no—
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#431—172no—
122#423C2×D1712yes—
123C2×#101—21no—
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—21no—
132#324—342no—
133C2×C17—21no—
134C2×C17—21no—
135C2×C17#182511yescharacteristic
136C2×C17#182711yescharacteristic
137C2×C17#183011yescharacteristic
138C22×Q8C22⋊D1711yescharacteristic
139C2×M16—22no—
140C4×Q8—22no—
141C4×Q8—22no—
142C2×C4⋊C4—341no—
143C2×C4⋊C4—341no—
144C2×C4⋊C4D6812yes—
145C2×C4⋊C4C22⋊D1711yescharacteristic
146C2×C42—341no—
147C2×C42—342no—
148C2×C42C4×D1712yes—
149C2×C42C22×D1711yescharacteristic
150C4⋊C8—22no—
151C4⋊C8—22no—
152C4⋊C8—342no—
153#101—42no—
154C22⋊C8—342no—
155C4×C8—342no—
156#95—171no—
157#95—172no—
158#95—171no—
159#95—341no—
160#95—342no—
161#95—342no—
162#95—342no—
163C17#876811yescharacteristic, complemented, Sylow
164C2×Q8—21no—
165C2×Q8—22no—
166C2×Q8—21no—
167C2×Q8—22no—
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—42no—
177M16—42no—
178C2×C8—42no—
179C2×C8—682no—
180C4⋊C4—24no—
181C4⋊C4—22no—
182C4⋊C4—682no—
183C4⋊C4—682no—
184C42—22no—
185C42—22no—
186C42—684no—
187C42—681no—
188C42—681no—
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—682no—
209C2×C4—681no—
210C2×C4#1058756311yescharacteristic
211C2×C4#1058756311yescharacteristic
212C2×C4#1058757212yes—
213C2×C4#1058761811yescharacteristic
214C2×C4#1058762112yes—
215C2×C4#1058762811yescharacteristic
216C2×C4#1058763612yes—
217C8—82no—
218C8—1362no—
219C22—21no—
220C22—21no—
221C22#1176267011yescharacteristic
222C22#1176280311yescharacteristic
223C22#1176280811yescharacteristic
224C4—22no—
225C4—21no—
226C4—21no—
227C4—42no—
228C4—42no—
229C4—682no—
230C4—1361no—
231C4—1362no—
232C4#1176313012yes—
233C4#1176314712yes—
234C2—21no—
235C2—21no—
236C2#1452887611yescharacteristic
237C2#1452888911yescharacteristic
238C2#1452889611yescharacteristic
subgroupG/H
centerC22#11762670
commutatorC22×C4×C17C22×C4
FrattiniC2×C42C22×D17
FittingC17×#1801C2
socleC22×C17#381
prime radicalC23×C17C2×C22⋊C4
cosocleC2×C42×C17C23
hypercenter#1801D17
hypocenterC17#8768
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1801

Complements

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

NH ≅ G/Ntype
163. C17#8768semidirect

Maximal subgroups

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

subgroupindex
#145317492
#145271082
#145266852
#145266442
C17×#18012
#876817