Finite Group Databasebeta

← #16948050

Subgroups — #16948050

Order 4352.

296 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 67 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#14541653C211yesmaximal, characteristic
2#14531013C211yesmaximal, characteristic, complemented
3#14530830C211yesmaximal, characteristic
4#14528155C211yesmaximal, characteristic, complemented
5#14526762C211yesmaximal, characteristic, complemented
6#14526744C211yesmaximal, characteristic
7C17×#1365C211yesmaximal, characteristic, complemented
8#11763520—21no—
9C2×#10587471—21no—
10#11763210—21no—
11#11763189C2211yescharacteristic
12#11763112—22no—
13#11763096—21no—
14#11763095—21no—
15C8×Dic34C2211yescharacteristic
16C4×D136C2211yescharacteristic
17#11762891—21no—
18#11762614—21no—
19#11762612—22no—
20C4×#63214C2211yescharacteristic
21#11762554—21no—
22C17×#500C2211yescharacteristic
23C17×C4⋊C16C2211yescharacteristic
24C17×#335C2211yescharacteristic
25C17×C4⋊Q8—21no—
26#10587632—42no—
27C2×Dic68—21no—
28C2×Dic68—21no—
29C2×D136D411yescharacteristic
30#10587583—41no—
31#10587579—22no—
32C4⋊C4×D17—41no—
33#10587576—22no—
34C4×D68—21no—
35#10587555—21no—
36C4×Dic34D411yescharacteristic
37C2×C16×C17—21no—
38C17×#107—21no—
39C17×#107D412yes—
40C17×C4⋊C8—21no—
41C4×C8×C17C2311yescharacteristic
42#10587492—21no—
43#10587489D411yescharacteristic
44#10587488—41no—
45#10587487—41no—
46#10587485—21no—
47C8×Dic17—21no—
48C2×#63214—21no—
49#10587480—22no—
50#10587479—22no—
51C4⋊#3622D411yescharacteristic
52#10587471—42no—
53C2×Q8×C17—41no—
54C2×D68—21no—
55C2×C4×D17—41no—
56C2×C4×D17—42no—
57C2×Dic34—41no—
58C2×Dic34D811yescharacteristic
59C2×C8×C17—41no—
60C2×C8×C17C2×D411yescharacteristic
61C2×C8×C17C2×D411yescharacteristic
62C4⋊C4×C17—22no—
63C4⋊C4×C17—41no—
64C42×C17C2×D411yescharacteristic
65#63227—42no—
66#63227—41no—
67C4⋊Dic17—21no—
68C4⋊Dic17—42no—
69C4⋊Dic17—41no—
70C4⋊Dic17—41no—
71C4⋊Dic17D811yescharacteristic
72#63225—21no—
73#63225—42no—
74C4×Dic17—21no—
75C2×#3622—21no—
76Dic68—42no—
77Dic68—41no—
78Dic68—41no—
79D136—21no—
80D136SD1612yes—
81C16×C17—41no—
82#63214—41no—
83#12359—171nomaximal, Sylow
84C22×D17—41no—
85Q8×C17—161no—
86C2×C4×C17—42no—
87C2×C4×C17#12011yescharacteristic
88C2×C4×C17C2×D811yescharacteristic
89C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
90C2×Dic17—21no—
91C2×Dic17—21no—
92C2×Dic17—41no—
93C2×Dic17—81no—
94D68—22no—
95D68—41no—
96C4×D17—81no—
97C4×D17—81no—
98C4×D17—162no—
99Dic34—21no—
100Dic34—21no—
101Dic34—81no—
102Dic34—81no—
103C8×C17—21no—
104C8×C17—81no—
105C8×C17C2×SD1611yescharacteristic
106C8×C17#13711yescharacteristic
107#3622—41no—
108#3380—171no—
109#2241—171no—
110#2208—171no—
111#1670—171no—
112#1365D1711yescharacteristic, complemented
113Q8⋊C16—171no—
114#1331—171no—
115C22×C17#44711yescharacteristic
116C2×D17—42no—
117C2×D17—81no—
118C2×D17—81no—
119C4×C17—21no—
120C4×C17—82no—
121C4×C17#44811yescharacteristic
122C4×C17#45411yescharacteristic
123C4×C17#50811yescharacteristic
124C4×C17#51011yescharacteristic
125Dic17—41no—
126Dic17—41no—
127Dic17—81no—
128Dic17—161no—
129#554—341no—
130C2×SD32—341no—
131#500C2×D1711yescharacteristic
132#499—341no—
133#494—171no—
134#478—342no—
135#476—341no—
136#475—341no—
137C8×Q8—171no—
138C4×D8—171no—
139C4×D8—341no—
140C4⋊C16C2×D1711yescharacteristic
141#358—341no—
142#358—342no—
143C4×C16—171no—
144#335C2×D1711yescharacteristic
145#332—341no—
146C2×C17#162711yescharacteristic
147C2×C17#218911yescharacteristic
148C2×C17#219411yescharacteristic
149D17—82no—
150C2×Q16—341no—
151C2×Q16—341no—
152C2×D8—171no—
153C4⋊Q8—21no—
154C4⋊Q8—341no—
155#125—342no—
156C22⋊Q8—681no—
157C22⋊Q8—682no—
158C4×Q8—171no—
159C4×D4—342no—
160C4×D4—341no—
161C2×C4⋊C4—681no—
162SD32—682no—
163C2×C16—21no—
164C2×C16—341no—
165#107—21no—
166#107—171no—
167#107D6812yes—
168#106—342no—
169#106—681no—
170C4⋊C8—21no—
171C4⋊C8—171no—
172C4⋊C8—341no—
173#103—681no—
174#102—342no—
175#102—341no—
176C4×C8—341no—
177C4×C8C22×D1711yescharacteristic
178C17#1235911yescharacteristic, complemented, Sylow
179C2×Q8—41no—
180C2×Q8—171no—
181C2×Q8—681no—
182C2×D4—341no—
183C22×C4—682no—
184C22×C4—681no—
185Q16—682no—
186Q16—681no—
187Q16—681no—
188D8—172no—
189D8—341no—
190C2×C8—41no—
191C2×C8—341no—
192C2×C8C2×D6811yescharacteristic
193C2×C8D4×D1711yescharacteristic
194C4⋊C4—22no—
195C4⋊C4—41no—
196C4⋊C4—171no—
197C4⋊C4—341no—
198C4⋊C4—341no—
199C4⋊C4—682no—
200C4⋊C4—682no—
201C4⋊C4—681no—
202C4⋊C4—681no—
203C22⋊C4—682no—
204C22⋊C4—681no—
205C42—341no—
206C42D4×D1711yescharacteristic
207C16—41no—
208C16—681no—
209C23—681no—
210Q8—161no—
211Q8—341no—
212Q8—341no—
213Q8—1361no—
214Q8—1361no—
215D4—342no—
216D4—681no—
217C2×C4—42no—
218C2×C4—2722no—
219C2×C4—341no—
220C2×C4—341no—
221C2×C4—681no—
222C2×C4—1361no—
223C2×C4—1361no—
224C2×C4—1361no—
225C2×C4#1058756811yescharacteristic
226C2×C4D8×D1711yescharacteristic
227C2×C4#1058760011yescharacteristic
228C8—21no—
229C8—81no—
230C8—681no—
231C8SD16×D1711yescharacteristic
232C8#1058760411yescharacteristic
233C22—682no—
234C22—1361no—
235C22—1361no—
236C22#1176294411yescharacteristic
237C4—21no—
238C4—82no—
239C4—2721no—
240C4—681no—
241C4—681no—
242C4—1361no—
243C4#1176294911yescharacteristic
244C4#1176298411yescharacteristic
245C4#1176324011yescharacteristic
246C4#1176325111yescharacteristic
247C2—1362no—
248C2#1452788811yescharacteristic
249C2#1453072711yescharacteristic
250C2#1453075811yescharacteristic
subgroupG/H
centerC22#11762944
commutator, cosocleC4×C8×C17C23
FrattiniC4×C8C22×D17
FittingC17×#1365C2
socleC22×C17#447
prime radicalC2×D136D4
hypercenter#1365D17
hypocenterC17#12359
LCSG ▷ C4×C8×C17 ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #1365

Complements

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

NH ≅ G/Ntype
2. #14531013C2semidirect
4. #14528155C2semidirect
5. #14526762C2semidirect
7. C17×#1365C2semidirect
112. #1365D17semidirect
178. C17#12359semidirect

Maximal subgroups

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

subgroupindex
#145416532
#145310132
#145308302
#145281552
#145267622
#145267442
C17×#13652
#1235917