Finite Group Databasebeta

← #16953413

Subgroups — #16953413

Order 4352.

370 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 99 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#14539308C211yesmaximal, characteristic
2#14530504C212yesmaximal
3#14526891C212yesmaximal
4#14526868C211yesmaximal, characteristic
5C17×#1971C211yesmaximal, characteristic
6#11763366—21no—
7C2×#10587483C2212yes—
8#11763218—24no—
9#11762847—21no—
10#11762847C2212yes—
11#11762613—24no—
12C22×Q16×C17C2211yescharacteristic
13C2×C17×#106C2211yescharacteristic
14C2×C17×#103—21no—
15C2×C17×#97C2211yescharacteristic
16C17×#391—21no—
17C22×Q8×C17—21no—
18C2×Q16×C17—22no—
19C2×Q16×C17—22no—
20C2×Q16×C17—21no—
21C2×Q16×C17Q812yescomplemented
22C22×C8×C17D411yescharacteristic
23C22×C8×C17C2311yescharacteristic
24C2×C4⋊C4×C17—21no—
25C2×C42×C17D411yescharacteristic
26#10587612—22no—
27#10587610—42no—
28#10587556—22no—
29#10587555—22no—
30C4×Dic34—42no—
31C17×#106—22no—
32C17×#103—42no—
33C17×#97—22no—
34C17×#95—42no—
35#10587489—22no—
36#10587489D414yes—
37#10587488—22no—
38#10587488—22no—
39#10587483—22no—
40#10587483—22no—
41C2×#63214—22no—
42C2×Q8×C17—22no—
43C2×Q8×C17—41no—
44C2×Q8×C17—41no—
45C2×Q8×C17—41no—
46C2×Q8×C17—41no—
47C2×Q8×C17—41no—
48C22×C4×C17—41no—
49C22×C4×C17—41no—
50C22×C4×C17D812yes—
51C22×C4×C17C2×D411yescharacteristic
52C2×Dic34—42no—
53Q16×C17—24no—
54Q16×C17—42no—
55C2×C8×C17—21no—
56C2×C8×C17—21no—
57C2×C8×C17SD1612yes—
58C2×C8×C17C2×D412yes—
59C2×C8×C17C2×Q811yescharacteristic
60C2×C8×C17#4111yescharacteristic
61C2×C8×C17#4112yes—
62C4⋊C4×C17—41no—
63C4⋊C4×C17—41no—
64C42×C17—21no—
65C42×C17—21no—
66C22⋊Dic17—42no—
67C4⋊Dic17—24no—
68C4⋊Dic17—42no—
69#63225—44no—
70C4×Dic17—42no—
71#63214—42no—
72#13368—171nomaximal, Sylow
73C23×C17C2×D811yescharacteristic
74Q8×C17—42no—
75Q8×C17—42no—
76Q8×C17—81no—
77Q8×C17—81no—
78C2×C4×C17—22no—
79C2×C4×C17—42no—
80C2×C4×C17—42no—
81C2×C4×C17—81no—
82C2×C4×C17—81no—
83C2×C4×C17—82no—
84C2×C4×C17—81no—
85C2×C4×C17#12011yescharacteristic
86C2×C4×C17C22⋊Q811yescharacteristic
87C2×C4×C17#12411yescharacteristic
88C2×C4×C17C2×SD1611yescharacteristic
89C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
90C2×C4×C17#13711yescharacteristic
91C2×Dic17—44no—
92Dic34—84no—
93C8×C17—22no—
94C8×C17C22⋊Q812yes—
95C8×C17#12412yes—
96#3162—171no—
97#2141—172no—
98#1971D1711yescharacteristic
99#1386—172no—
100#1377—171no—
101C22×C17—22no—
102C22×C17#44711yescharacteristic
103C22×C17#47411yescharacteristic
104C22×C17#48611yescharacteristic
105C4×C17—42no—
106C4×C17—82no—
107C4×C17—161no—
108C4×C17#39311yescharacteristic
109C4×C17#44811yescharacteristic
110C4×C17#47611yescharacteristic
111C4×C17#48911yescharacteristic
112Dic17—84no—
113C22×Q16C2×D1711yescharacteristic
114#533—341no—
115C2×M32—172no—
116#501—344no—
117#427—172no—
118#427—341no—
119C2×#106C2×D1711yescharacteristic
120C2×#103—21no—
121C2×#97C2×D1711yescharacteristic
122#391—21no—
123#358—344no—
124C2×C17—21no—
125C2×C17—21no—
126C2×C17#202311yescharacteristic
127C2×C17#219112yes—
128C22×Q8—21no—
129C2×Q16—21no—
130C2×Q16—22no—
131C2×Q16—22no—
132C2×Q16Dic3412yescomplemented
133C22×C8C22⋊D1711yescharacteristic
134C22×C8C22×D1711yescharacteristic
135C4⋊Q8—342no—
136#125—342no—
137C22⋊Q8—682no—
138C4×Q8—682no—
139#117—342no—
140C2×C4⋊C4—21no—
141C2×C42C22⋊D1711yescharacteristic
142M32—342no—
143M32—342no—
144C2×C16—342no—
145#107—174no—
146#107—342no—
147#106—22no—
148#106—342no—
149#106—342no—
150#103—42no—
151#97—22no—
152#95—42no—
153C17#1336811yescharacteristic, complemented, Sylow
154C2×Q8—22no—
155C2×Q8—41no—
156C2×Q8—41no—
157C2×Q8—41no—
158C2×Q8—41no—
159C2×Q8—41no—
160C2×Q8—682no—
161C22×C4—41no—
162C22×C4—41no—
163C22×C4#6322812yes—
164C22×C4C2×C22⋊D1711yescharacteristic
165Q16—24no—
166Q16—42no—
167C2×C8—21no—
168C2×C8—21no—
169C2×C8#6322912yes—
170C2×C8C2×Dic3411yescharacteristic
171C2×C8#6325211yescharacteristic
172C2×C8D4×D1712yes—
173C2×C8#6325412yes—
174C4⋊C4—41no—
175C4⋊C4—41no—
176C4⋊C4—344no—
177C4⋊C4—684no—
178C4⋊C4—682no—
179C22⋊C4—682no—
180C42—21no—
181C42—21no—
182C42—682no—
183C16—682no—
184C23C2×#6322811yescharacteristic
185Q8—42no—
186Q8—42no—
187Q8—81no—
188Q8—81no—
189Q8—1364no—
190C2×C4—22no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—81no—
194C2×C4—81no—
195C2×C4—82no—
196C2×C4—81no—
197C2×C4—684no—
198C2×C4#1058761011yescharacteristic
199C2×C4C2×#6322911yescharacteristic
200C2×C4#1058762211yescharacteristic
201C2×C4#1058762311yescharacteristic
202C2×C4#1058763511yescharacteristic
203C2×C4#1058763711yescharacteristic
204C8—22no—
205C8#1058756412yes—
206C8#1058757112yes—
207C22—22no—
208C22#1176294211yescharacteristic
209C22#1176308911yescharacteristic
210C22#1176315511yescharacteristic
211C4—42no—
212C4—82no—
213C4—161no—
214C4—1364no—
215C4#1176272111yescharacteristic
216C4#1176294711yescharacteristic
217C4#1176309911yescharacteristic
218C4#1176316811yescharacteristic
219C2—21no—
220C2—21no—
221C2#1452992011yescharacteristic
222C2#1453073412yes—
subgroupG/H
centerC22#11762942
commutator, cosocleC22×C8×C17C23
FrattiniC22×C8C22×D17
FittingC17×#1971C2
socleC22×C17#447
prime radicalC23×C17C2×D8
hypercenter#1971D17
hypocenterC17#13368
LCSG ▷ C22×C8×C17 ▷ C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#97 ◁ #1971

Complements

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

NH ≅ G/Ntype
21. C2×Q16×C17Q8semidirect
132. C2×Q16Dic34semidirect
153. C17#13368semidirect

Maximal subgroups

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

subgroupindex
#145393082
#145305042
#145268912
#145268682
C17×#19712
#1336817