Finite Group Databasebeta

← #16948002

Subgroups — #16948002

Order 4352.

338 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 73 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
1Q8×D136C211yesmaximal, characteristic, complemented
2#14531008C211yesmaximal, characteristic, complemented
3#14530820C211yesmaximal, characteristic
4#14528144C211yesmaximal, characteristic, complemented
5#14526838C211yesmaximal, characteristic, complemented
6#14526746C211yesmaximal, characteristic, complemented
7C17×Q8⋊C16C211yesmaximal, characteristic, complemented
8Q8×D68—21no—
9C2×D272—21no—
10#11763214—21no—
11#11763214C2211yescharacteristic
12#11763186C2211yescharacteristic
13#11763090—21no—
14#11763090—22no—
15#11763009—21no—
16C4×D136—21no—
17C4×D136C2211yescharacteristic
18#11762611—21no—
19#11762611—22no—
20#11762562C2211yescharacteristic
21#11762552—21no—
22C8×Q8×C17C2211yescharacteristic
23C17×C4⋊C16C2211yescharacteristic
24C4×C16×C17C2211yescharacteristic
25C2×Q8×D17—41no—
26C4×Q8×C17D411yescharacteristic
27C2×#63230—41no—
28C2×D136—21no—
29C2×D136—21no—
30C2×D136D411yescharacteristic
31#10587583—42no—
32#10587583—41no—
33#10587560—21no—
34C4×D68—22no—
35C4×D68—21no—
36#10587555—22no—
37#10587555—21no—
38C2×C16×C17—21no—
39C2×C16×C17—21no—
40C17×C4⋊C8—21no—
41C17×C4⋊C8D411yescharacteristic
42C4×C8×C17—21no—
43C4×C8×C17C2311yescharacteristic
44#10587492—22no—
45#10587492—21no—
46#10587489—22no—
47#10587489—21no—
48#10587489D411yescharacteristic
49#10587489D412yes—
50#10587480—41no—
51C4⋊#3622—21no—
52D272—42no—
53C2×Q8×C17D811yescharacteristic
54Q8×D17—82no—
55Q8×D17—82no—
56C2×D68—21no—
57C2×D68—41no—
58C2×D68—41no—
59C2×D68—41no—
60C2×C4×D17—42no—
61C2×C4×D17—41no—
62C2×Dic34—42no—
63C2×Dic34—41no—
64C2×C8×C17—21no—
65C2×C8×C17C2×D411yescharacteristic
66C2×C8×C17C2×D411yescharacteristic
67C4⋊C4×C17—21no—
68C4⋊C4×C17D811yescharacteristic
69C42×C17—21no—
70C42×C17C2×D411yescharacteristic
71#63230—82no—
72#63227—42no—
73#63227—41no—
74C4⋊Dic17—21no—
75C4⋊Dic17—22no—
76C4⋊Dic17—42no—
77C4⋊Dic17—41no—
78C2×#3622—41no—
79D136—21no—
80D136—42no—
81D136—41no—
82D136—41no—
83D136SD1612yescomplemented
84C16×C17—41no—
85C16×C17—41no—
86#12352—171nomaximal, Sylow
87C22×D17—41no—
88C22×D17—81no—
89Q8×C17—21no—
90Q8×C17D1612yescomplemented
91C2×C4×C17—21no—
92C2×C4×C17—21no—
93C2×C4×C17#12011yescharacteristic
94C2×C4×C17C2×D811yescharacteristic
95C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
96C2×Dic17—41no—
97C2×Dic17—42no—
98D68—22no—
99D68—41no—
100D68—81no—
101D68—81no—
102D68—81no—
103D68—81no—
104D68—161no—
105C4×D17—84no—
106C4×D17—81no—
107C4×D17—81no—
108Dic34—84no—
109Dic34—81no—
110Dic34—81no—
111C8×C17—21no—
112C8×C17—41no—
113C8×C17C2×SD1611yescharacteristic
114C8×C17#13711yescharacteristic
115#3622—81no—
116Q8×D8—171no—
117#2240—171no—
118#2206—171no—
119#1668—171no—
120#1364—171no—
121Q8⋊C16D1711yescharacteristic, complemented
122#1331—171no—
123C22×C17#44711yescharacteristic
124C2×D17—42no—
125C2×D17—81no—
126C2×D17—81no—
127C2×D17—161no—
128C2×D17—161no—
129C2×D17—161no—
130C4×C17—22no—
131C4×C17—21no—
132C4×C17—41no—
133C4×C17#44811yescharacteristic
134C4×C17#45411yescharacteristic
135C4×C17C2×D1611yescharacteristic
136C4×C17#50911yescharacteristic
137Dic17—81no—
138Dic17—82no—
139D4×Q8—341no—
140C2×D16—341no—
141#500—171no—
142#500—341no—
143#493—171no—
144#474—342no—
145#474—341no—
146#460—341no—
147C8×Q8C2×D1711yescharacteristic
148C4×D8—171no—
149C4×D8—341no—
150C4⋊C16C2×D1711yescharacteristic
151#357—341no—
152#357—342no—
153C4×C16C2×D1711yescharacteristic
154#335—171no—
155#331—341no—
156C2×C17#162711yescharacteristic
157C2×C17#218411yescharacteristic
158C2×C17#219311yescharacteristic
159D17—82no—
160D17—321no—
161C22×Q8—681no—
162C2×SD16—681no—
163C2×D8—171no—
164C2×D8—341no—
165C2×D8—341no—
166C4⋊Q8—342no—
167C4⋊Q8—341no—
168#127—341no—
169C22⋊Q8—682no—
170C22⋊Q8—681no—
171C4×Q8D6811yescharacteristic
172C4×D4—342no—
173C4×D4—341no—
174D16—682no—
175C2×C16—21no—
176C2×C16—21no—
177#107—171no—
178#107—172no—
179#107—342no—
180#107—341no—
181C4⋊C8—21no—
182C4⋊C8—341no—
183C4⋊C8D6811yescharacteristic
184#102—342no—
185#102—341no—
186#102—681no—
187C4×C8—21no—
188C4×C8C22×D1711yescharacteristic
189C17#1235211yescharacteristic, complemented, Sylow
190C2×Q8—682no—
191C2×Q8—681no—
192C2×Q8—1362no—
193C2×Q8—1362no—
194C2×Q8D13611yescharacteristic
195C2×D4—341no—
196C2×D4—681no—
197C2×D4—681no—
198C2×D4—681no—
199C22×C4—682no—
200C22×C4—681no—
201SD16—1362no—
202D8—172no—
203D8—341no—
204D8—682no—
205D8—681no—
206D8—681no—
207C2×C8—21no—
208C2×C8—681no—
209C2×C8D4×D1711yescharacteristic
210C2×C8D4×D1711yescharacteristic
211C4⋊C4—21no—
212C4⋊C4—342no—
213C4⋊C4—341no—
214C4⋊C4—682no—
215C4⋊C4—681no—
216C4⋊C4D13611yescharacteristic
217C22⋊C4—682no—
218C22⋊C4—681no—
219C42—21no—
220C42C2×D6811yescharacteristic
221C16—41no—
222C16—41no—
223C23—681no—
224C23—1361no—
225Q8—21no—
226Q8—1364no—
227Q8—1361no—
228Q8—1361no—
229Q8D27212yescomplemented
230D4—2721no—
231D4—342no—
232D4—681no—
233D4—1361no—
234D4—1361no—
235D4—1361no—
236D4—1361no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—682no—
240C2×C4—681no—
241C2×C4—1364no—
242C2×C4—1361no—
243C2×C4—1361no—
244C2×C4#1058756811yescharacteristic
245C2×C4C2×D13611yescharacteristic
246C2×C4#1058759411yescharacteristic
247C8—21no—
248C8—41no—
249C8—1361no—
250C8SD16×D1711yescharacteristic
251C8#1058760411yescharacteristic
252C22—2721no—
253C22—2721no—
254C22—2721no—
255C22—682no—
256C22—1361no—
257C22—1361no—
258C22#1176294511yescharacteristic
259C4—22no—
260C4—21no—
261C4—41no—
262C4—1362no—
263C4—1361no—
264C4#1176294911yescharacteristic
265C4#1176298311yescharacteristic
266C4C2×D27211yescharacteristic
267C4#1176324711yescharacteristic
268C2—5441no—
269C2—1362no—
270C2#1452788911yescharacteristic
271C2#1453069811yescharacteristic
272C2#1453075011yescharacteristic
subgroupG/H
centerC22#11762945
commutator, cosocleC4×C8×C17C23
FrattiniC4×C8C22×D17
FittingC17×Q8⋊C16C2
socleC22×C17#447
prime radical#14530820C2
hypercenterQ8⋊C16D17
hypocenterC17#12352
LCSG ▷ C4×C8×C17 ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C42 ◁ C8×Q8 ◁ Q8⋊C16

Complements

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

NH ≅ G/Ntype
1. Q8×D136C2semidirect
2. #14531008C2semidirect
4. #14528144C2semidirect
5. #14526838C2semidirect
6. #14526746C2semidirect
7. C17×Q8⋊C16C2semidirect
83. D136SD16semidirect
90. Q8×C17D16semidirect
121. Q8⋊C16D17semidirect
189. C17#12352semidirect
229. Q8D272semidirect

Maximal subgroups

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

subgroupindex
Q8×D1362
#145310082
#145308202
#145281442
#145268382
#145267462
C17×Q8⋊C162
#1235217