Finite Group Databasebeta

← #16936618

Subgroups — #16936618

Order 4352.

466 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 105 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#14539930C212yesmaximal
2#14539308C211yesmaximal, characteristic
3#14529604C211yesmaximal, characteristic
4#14527349C212yesmaximal
5C17×#1281C211yesmaximal, characteristic
6#11763366C2212yes—
7C2×#10587555C2211yescharacteristic
8#11763218—22no—
9#11763218—22no—
10#11763207—22no—
11#11763160—22no—
12#11763140—24no—
13#11763095—24no—
14#11762847—22no—
15C2×#10587487—22no—
16#11762675C2211yescharacteristic
17#11762547—24no—
18C17×#431C2212yes—
19C2×C17×#97C2211yescharacteristic
20C22×Dic34—21no—
21C22×C8×C17D412yes—
22C2×C42×C17C2311yescharacteristic
23#10587612—22no—
24C2×C4⋊Dic17—22no—
25C2×C4⋊Dic17—21no—
26#10587610—22no—
27#10587610—22no—
28#10587556—22no—
29#10587555—22no—
30#10587555—21no—
31#10587555D412yes—
32#10587555D412yes—
33#10587555D412yes—
34C4×Dic34—22no—
35C4×Dic34—22no—
36C17×C4⋊C8—22no—
37C17×C22⋊C8D414yes—
38C17×#97—22no—
39C4×C8×C17—22no—
40#10587502—22no—
41#10587489—24no—
42#10587488—28no—
43#10587488—22no—
44#10587488—22no—
45#10587487—28no—
46#10587487—44no—
47C22×C4×C17C2×D412yes—
48C22×C4×C17C2×D411yescharacteristic
49C22×Dic17—41no—
50C22×Dic17—41no—
51C2×Dic34—22no—
52C2×Dic34—22no—
53C2×Dic34—22no—
54C2×Dic34—42no—
55C2×Dic34—42no—
56C2×Dic34—41no—
57C2×C8×C17—22no—
58C2×C8×C17—22no—
59C2×C8×C17—24no—
60C2×C8×C17SD1614yes—
61C42×C17C2×D412yes—
62C42×C17C2×D411yescharacteristic
63C42×C17#4111yescharacteristic
64C22⋊Dic17—42no—
65C4⋊Dic17—24no—
66C4⋊Dic17—22no—
67C4⋊Dic17—22no—
68C4⋊Dic17—22no—
69C4⋊Dic17—22no—
70C4⋊Dic17—42no—
71C4⋊Dic17—42no—
72#63225—42no—
73#63225—42no—
74C4×Dic17—42no—
75#10227—171nomaximal, Sylow
76C23×C17#12711yescharacteristic
77C2×C4×C17—42no—
78C2×C4×C17#12012yes—
79C2×C4×C17#12011yescharacteristic
80C2×C4×C17#12112yes—
81C2×C4×C17#12111yescharacteristic
82C2×C4×C17C2×SD1612yes—
83C2×C4×C17#13712yes—
84C2×Dic17—42no—
85C2×Dic17—42no—
86C2×Dic17—42no—
87C2×Dic17—81no—
88C2×Dic17—81no—
89C2×Dic17—82no—
90C2×Dic17—81no—
91Dic34—44no—
92Dic34—44no—
93Dic34—82no—
94Dic34—82no—
95C8×C17—24no—
96C8×C17—44no—
97#3229—172no—
98#3162—171no—
99#1960—171no—
100#1518—172no—
101#1281D1711yescharacteristic
102C22×C17—22no—
103C22×C17#39211yescharacteristic
104C22×C17#49211yescharacteristic
105C22×C17#49711yescharacteristic
106C4×C17—22no—
107C4×C17—22no—
108C4×C17#44812yes—
109C4×C17#46112yes—
110Dic17—82no—
111Dic17—82no—
112Dic17—82no—
113Dic17—161no—
114#533—172no—
115C2×C4⋊Q8—171no—
116#501—342no—
117#501—342no—
118#498—342no—
119#487—342no—
120#483—344no—
121#475—344no—
122#431C2×D1712yes—
123#427—342no—
124C2×#103—342no—
125C2×#97C2×D1711yescharacteristic
126#382—171no—
127#328—344no—
128C2×C17—21no—
129C2×C17—21no—
130C2×C17#200912yes—
131C2×C17#201311yescharacteristic
132C22×Q8—341no—
133C22×C8D6812yes—
134C4⋊Q8—172no—
135C4⋊Q8—172no—
136C4⋊Q8—172no—
137C4⋊Q8—342no—
138C4⋊Q8—341no—
139#125—342no—
140C22⋊Q8—342no—
141C22⋊Q8—342no—
142C4×Q8—342no—
143C4×Q8—342no—
144#117—342no—
145C2×C4⋊C4—342no—
146C2×C4⋊C4—341no—
147C2×C42C22×D1711yescharacteristic
148#107—344no—
149#106—342no—
150#106—342no—
151#106—348no—
152C4⋊C8—22no—
153#103—348no—
154#103—684no—
155C22⋊C8D6814yes—
156#97—22no—
157C4×C8—22no—
158#95—342no—
159C17#1022711yescharacteristic, complemented, Sylow
160C2×Q8—342no—
161C2×Q8—342no—
162C2×Q8—342no—
163C2×Q8—681no—
164C2×Q8—682no—
165C2×Q8—682no—
166C22×C4—681no—
167C22×C4—681no—
168C22×C4C2×D6812yes—
169C22×C4C2×D6811yescharacteristic
170C2×C8—22no—
171C2×C8—22no—
172C2×C8—24no—
173C2×C8#6321914yes—
174C4⋊C4—344no—
175C4⋊C4—342no—
176C4⋊C4—342no—
177C4⋊C4—342no—
178C4⋊C4—342no—
179C4⋊C4—682no—
180C4⋊C4—682no—
181C4⋊C4—682no—
182C4⋊C4—682no—
183C22⋊C4—682no—
184C42—682no—
185C42D4×D1712yes—
186C42D4×D1711yescharacteristic
187C42#6325611yescharacteristic
188C23#1058756011yescharacteristic
189Q8—684no—
190Q8—684no—
191Q8—1362no—
192Q8—1362no—
193C2×C4—42no—
194C2×C4—682no—
195C2×C4—682no—
196C2×C4—682no—
197C2×C4—1362no—
198C2×C4—1361no—
199C2×C4—1361no—
200C2×C4—1361no—
201C2×C4#1058756812yes—
202C2×C4#1058756811yescharacteristic
203C2×C4#1058758112yes—
204C2×C4#1058758111yescharacteristic
205C2×C4C2×#6321912yes—
206C2×C4#1058759512yes—
207C8—24no—
208C8—44no—
209C22—22no—
210C22#1176271911yescharacteristic
211C22#1176318211yescharacteristic
212C22#1176320311yescharacteristic
213C4—22no—
214C4—22no—
215C4—2721no—
216C4—1362no—
217C4—1362no—
218C4—1362no—
219C4#1176295012yes—
220C4#1176301512yes—
221C2—21no—
222C2—21no—
223C2#1452984712yes—
224C2#1452986911yescharacteristic
subgroupG/H
centerC22#11762719
commutator, cosocleC2×C42×C17C23
FrattiniC2×C42C22×D17
FittingC17×#1281C2
socleC22×C17#392
prime radicalC23×C17#127
hypercenter#1281D17
hypocenterC17#10227
LCSG ▷ C2×C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1281

Complements

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

NH ≅ G/Ntype
159. C17#10227semidirect

Maximal subgroups

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

subgroupindex
#145399302
#145393082
#145296042
#145273492
C17×#12812
#1022717