Finite Group Databasebeta

← #16948052

Subgroups — #16948052

Order 4352.

296 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 69 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#14541649C211yesmaximal, characteristic
2#14531012C211yesmaximal, characteristic, complemented
3#14530829C211yesmaximal, characteristic
4#14528155C211yesmaximal, characteristic, complemented
5#14526843C211yesmaximal, characteristic, complemented
6#14526764C211yesmaximal, characteristic, complemented
7C17×#1331C211yesmaximal, characteristic
8#11763516—21no—
9C2×#10587498—21no—
10#11763214C2211yescharacteristic
11#11763210—21no—
12#11763189C2211yescharacteristic
13#11763107—22no—
14#11763099—21no—
15#11763095—21no—
16C8×Dic34C2211yescharacteristic
17D8×Dic17—21no—
18C4⋊#63214C2211yescharacteristic
19#11762614—22no—
20#11762612—21no—
21#11762554—21no—
22C4×D8×C17C2211yescharacteristic
23C4×C16×C17C2211yescharacteristic
24C17×#335C2211yescharacteristic
25C2×D8×C17D411yescharacteristic
26C4×D4×C17—21no—
27D4×Dic17—22no—
28C2×C4⋊Dic17—41no—
29#10587610—41no—
30C2×Dic68—21no—
31C2×Dic68—21no—
32#10587576—22no—
33#10587564—42no—
34#10587555—21no—
35#10587555—21no—
36C4×Dic34D411yescharacteristic
37C2×C16×C17—21no—
38C17×#107D411yescharacteristic
39C17×C4⋊C8—21no—
40C17×#102—21no—
41C4×C8×C17C2311yescharacteristic
42#10587503—22no—
43#10587498—42no—
44#10587489—21no—
45#10587489D412yes—
46#10587488—41no—
47#10587487—41no—
48#10587485—21no—
49C8×Dic17—21no—
50C2×#63214—21no—
51#10587479—22no—
52C4⋊#3622D411yescharacteristic
53C2×D4×C17—21no—
54C22×C4×C17—41no—
55C22×Dic17—42no—
56C2×Dic34—41no—
57C2×Dic34—41no—
58C2×Dic34D811yescharacteristic
59D8×C17—21no—
60D8×C17SD1612yes—
61C2×C8×C17—41no—
62C2×C8×C17C2×D411yescharacteristic
63C2×C8×C17C2×D411yescharacteristic
64C4⋊C4×C17—21no—
65C22⋊C4×C17—41no—
66C42×C17C2×D411yescharacteristic
67C22⋊Dic17—42no—
68C4⋊Dic17—22no—
69C4⋊Dic17—42no—
70C4⋊Dic17—41no—
71C4⋊Dic17—41no—
72C4⋊Dic17—41no—
73C4⋊Dic17D811yescharacteristic
74#63225—21no—
75#63225—42no—
76C4×Dic17—21no—
77C2×#3622—21no—
78Dic68—42no—
79Dic68—41no—
80Dic68—41no—
81C16×C17—41no—
82#63214—41no—
83#12359—171nomaximal, Sylow
84C23×C17—41no—
85D4×C17—22no—
86D4×C17—41no—
87C2×C4×C17—41no—
88C2×C4×C17—81no—
89C2×C4×C17—81no—
90C2×C4×C17#12011yescharacteristic
91C2×C4×C17C2×D811yescharacteristic
92C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
93C2×Dic17—21no—
94C2×Dic17—21no—
95C2×Dic17—42no—
96C2×Dic17—81no—
97C2×Dic17—162no—
98Dic34—21no—
99Dic34—21no—
100Dic34—81no—
101Dic34—81no—
102Dic34—161no—
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#1365—171no—
113Q8⋊C16—171no—
114#1331D1711yescharacteristic
115C22×C17—42no—
116C22×C17—81no—
117C22×C17—81no—
118C22×C17#44711yescharacteristic
119C4×C17—21no—
120C4×C17—81no—
121C4×C17#44811yescharacteristic
122C4×C17#45411yescharacteristic
123C4×C17#50811yescharacteristic
124C4×C17#51011yescharacteristic
125Dic17—41no—
126Dic17—41no—
127Dic17—82no—
128Dic17—161no—
129#554—341no—
130C2×SD32—341no—
131#500—171no—
132#499—341no—
133#494—171no—
134#478—342no—
135#476—341no—
136#475—341no—
137C8×Q8—171no—
138C4×D8—341no—
139C4×D8C2×D1711yescharacteristic
140C4⋊C16—171no—
141#358—342no—
142#358—341no—
143C4×C16C2×D1711yescharacteristic
144#335C2×D1711yescharacteristic
145#332—341no—
146C2×C17—82no—
147C2×C17#162711yescharacteristic
148C2×C17#218911yescharacteristic
149C2×C17#219411yescharacteristic
150C2×Q16—341no—
151C2×Q16—341no—
152C2×D8D6811yescharacteristic
153C4⋊Q8—341no—
154C4⋊Q8—341no—
155#125—342no—
156C22⋊Q8—682no—
157C22⋊Q8—681no—
158C4×Q8—171no—
159C4×D4—21no—
160C4×D4—342no—
161C2×C4⋊C4—681no—
162SD32—682no—
163C2×C16—21no—
164C2×C16—341no—
165#107—172no—
166#107—341no—
167#107D6811yescharacteristic
168#106—342no—
169#106—681no—
170C4⋊C8—21no—
171C4⋊C8—171no—
172C4⋊C8—341no—
173#103—681no—
174#102—21no—
175#102—342no—
176C4×C8—341no—
177C4×C8C22×D1711yescharacteristic
178C17#1235911yescharacteristic, complemented, Sylow
179C2×Q8—171no—
180C2×Q8—681no—
181C2×Q8—681no—
182C2×D4—21no—
183C22×C4—41no—
184C22×C4—682no—
185Q16—682no—
186Q16—681no—
187Q16—681no—
188D8—21no—
189D8#6321912yes—
190C2×C8—41no—
191C2×C8—341no—
192C2×C8C2×D6811yescharacteristic
193C2×C8D4×D1711yescharacteristic
194C4⋊C4—21no—
195C4⋊C4—171no—
196C4⋊C4—341no—
197C4⋊C4—342no—
198C4⋊C4—682no—
199C4⋊C4—682no—
200C4⋊C4—681no—
201C4⋊C4—681no—
202C4⋊C4—681no—
203C22⋊C4—41no—
204C22⋊C4—682no—
205C42—341no—
206C42D4×D1711yescharacteristic
207C16—41no—
208C16—681no—
209C23—41no—
210Q8—2721no—
211Q8—341no—
212Q8—341no—
213Q8—1361no—
214Q8—1361no—
215D4—22no—
216D4—41no—
217C2×C4—41no—
218C2×C4—81no—
219C2×C4—81no—
220C2×C4—2722no—
221C2×C4—341no—
222C2×C4—341no—
223C2×C4—682no—
224C2×C4—1361no—
225C2×C4#1058756811yescharacteristic
226C2×C4D8×D1711yescharacteristic
227C2×C4#1058760011yescharacteristic
228C8—21no—
229C8—81no—
230C8—681no—
231C8C2×#6321911yescharacteristic
232C8#1058759511yescharacteristic
233C22—42no—
234C22—81no—
235C22—81no—
236C22#1176294411yescharacteristic
237C4—21no—
238C4—81no—
239C4—2721no—
240C4—681no—
241C4—681no—
242C4—1362no—
243C4#1176295011yescharacteristic
244C4#1176298311yescharacteristic
245C4#1176323811yescharacteristic
246C4#1176324811yescharacteristic
247C2—82no—
248C2#1452789011yescharacteristic
249C2#1453072511yescharacteristic
250C2#1453075611yescharacteristic
subgroupG/H
centerC22#11762944
commutator, cosocleC4×C8×C17C23
FrattiniC4×C8C22×D17
FittingC17×#1331C2
socleC22×C17#447
prime radicalC2×D8×C17D4
hypercenter#1331D17
hypocenterC17#12359
LCSG ▷ C4×C8×C17 ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #1331

Complements

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

NH ≅ G/Ntype
2. #14531012C2semidirect
4. #14528155C2semidirect
5. #14526843C2semidirect
6. #14526764C2semidirect
178. C17#12359semidirect

Maximal subgroups

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

subgroupindex
#145416492
#145310122
#145308292
#145281552
#145268432
#145267642
C17×#13312
#1235917