Finite Group Databasebeta

← #30759485

Subgroups — #30759485

Order 17152.

408 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 101 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#22560201C211yesmaximal, characteristic
2#22558983C211yesmaximal, characteristic, complemented
3#22558894C211yesmaximal, characteristic, complemented
4#22558890C211yesmaximal, characteristic, complemented
5#22558600C211yesmaximal, characteristic
6#22558106C211yesmaximal, characteristic, complemented
7C2×#476×C67C211yesmaximal, characteristic
8#16863449C2211yescharacteristic
9#16863449C2211yescharacteristic
10#16863420C2211yescharacteristic
11#16863420C2211yescharacteristic
12C2×#14523960—21no—
13#16863165—21no—
14#16863148—22no—
15#16863145—22no—
16#16863142—22no—
17D4⋊(C67⋊C8)—22no—
18C2×C4⋊Q8×C67C2211yescharacteristic
19C2×C4×D4×C67C2211yescharacteristic
20#476×C67—22no—
21#476×C67—22no—
22C2×#106×C67—21no—
23C2×C4⋊C8×C67C2211yescharacteristic
24C2×#102×C67—21no—
25C2×#11761612—21no—
26C2×#11761612—21no—
27C4⋊(C67⋊C8)—21no—
28C4⋊(C67⋊C8)—21no—
29C4⋊(C67⋊C8)—21no—
30C4⋊(C67⋊C8)—21no—
31C4⋊(C67⋊C8)—21no—
32C4⋊(C67⋊C8)D412yes—
33#14523960—42no—
34C22⋊(C67⋊C8)—21no—
35C22⋊(C67⋊C8)—21no—
36C4×C67⋊C8—21no—
37C4×C67⋊C8—21no—
38C22×Q8×C67—21no—
39C22×D4×C67D411yescharacteristic
40C23×C4×C67—21no—
41C22×C8×C67—21no—
42C4⋊Q8×C67—21no—
43C4⋊Q8×C67—21no—
44C4⋊Q8×C67—21no—
45C4⋊Q8×C67D412yes—
46C4×D4×C67—21no—
47C4×D4×C67—21no—
48C4×D4×C67—21no—
49C4×D4×C67—21no—
50C2×C4⋊C4×C67—21no—
51C2×C4⋊C4×C67—21no—
52C2×C4⋊C4×C67D411yescharacteristic
53C2×C22⋊C4×C67—21no—
54C2×C42×C67C2311yescharacteristic
55#106×C67—24no—
56C4⋊C8×C67—21no—
57C4⋊C8×C67D412yes—
58#102×C67—24no—
59#11761612—41no—
60#11761612—41no—
61#11761612—41no—
62#11761612—41no—
63C2×C67⋊C8—22no—
64C2×C67⋊C8—41no—
65C2×C67⋊C8—41no—
66C2×C67⋊C8—41no—
67C24×C67—21no—
68C2×Q8×C67—22no—
69C2×Q8×C67—41no—
70C2×Q8×C67—82no—
71C2×D4×C67—21no—
72C2×D4×C67—21no—
73C2×D4×C67—21no—
74C2×D4×C67—21no—
75C2×D4×C67—21no—
76C2×D4×C67Q1612yes—
77C22×C4×C67—21no—
78C22×C4×C67—41no—
79C22×C4×C67—41no—
80C22×C4×C67—41no—
81C22×C4×C67—82no—
82C22×C4×C67C2×D411yescharacteristic
83C22×C4×C67#4111yescharacteristic
84C22×C4×C67#4111yescharacteristic
85C2×C8×C67—22no—
86C2×C8×C67—41no—
87C2×C8×C67—41no—
88C4⋊C4×C67—22no—
89C4⋊C4×C67—21no—
90C4⋊C4×C67—21no—
91C4⋊C4×C67—22no—
92C4⋊C4×C67—41no—
93C4⋊C4×C67—41no—
94C22⋊C4×C67—41no—
95C22⋊C4×C67—41no—
96C42×C67C2×D411yescharacteristic
97C42×C67C2×D411yescharacteristic
98C42×C67C2×D411yescharacteristic
99C42×C67#4111yescharacteristic
100C23×C67—22no—
101C23×C67—41no—
102C23×C67—41no—
103C23×C67—81no—
104C23×C67—81no—
105C23×C67#12311yescharacteristic
106Q8×C67—82no—
107Q8×C67—82no—
108D4×C67—22no—
109D4×C67—22no—
110D4×C67—41no—
111D4×C67—41no—
112C2×C4×C67—22no—
113C2×C4×C67—41no—
114C2×C4×C67—41no—
115C2×C4×C67—42no—
116C2×C4×C67—82no—
117C2×C4×C67—82no—
118C2×C4×C67—82no—
119C2×C4×C67—81no—
120C2×C4×C67#12011yescharacteristic
121C2×C4×C67#12111yescharacteristic
122C2×C4×C67#12111yescharacteristic
123C2×C4×C67#12111yescharacteristic
124C2×C4×C67#12311yescharacteristic
125C2×C4×C67#12411yescharacteristic
126C2×C4×C67C2×Q1611yescharacteristic
127C2×C4×C67#13511yescharacteristic
128C2×C4×C67AGL(1,ℤ/8)11yescharacteristic
129C2×C4×C67#13711yescharacteristic
130C8×C67—42no—
131C67⋊C8—42no—
132C67⋊C8—81no—
133C67⋊C8—81no—
134C67⋊C8—81no—
135C22×C67—21no—
136C22×C67—21no—
137C22×C67—42no—
138C22×C67—82no—
139C22×C67—82no—
140C22×C67—81no—
141C22×C67—81no—
142C22×C67#39411yescharacteristic
143C22×C67#48411yescharacteristic
144C22×C67#48511yescharacteristic
145C4×C67—21no—
146C4×C67—21no—
147C4×C67—21no—
148C4×C67—82no—
149C4×C67—81no—
150C4×C67#44911yescharacteristic
151C4×C67#45111yescharacteristic
152C4×C67#46711yescharacteristic
153C4×C67#46811yescharacteristic
154C4×C67#47112yes—
155#10453—671nomaximal, Sylow
156C2×C67—21no—
157C2×C67—21no—
158C2×C67—82no—
159C2×C67#203611yescharacteristic
160C2×C67#204611yescharacteristic
161C2×C67#204811yescharacteristic
162C2×#476D6711yescharacteristic
163#1800—671no—
164#1571—671no—
165#1549—671no—
166#1548—671no—
167#1478—671no—
168#1301—671no—
169C67#1045311yescharacteristic, complemented, Sylow
170C2×C4⋊Q8C2×D6711yescharacteristic
171C2×C4×D4C2×D6711yescharacteristic
172#476—22no—
173#476—22no—
174#431—671no—
175#431—671no—
176C2×#106—21no—
177#423—671no—
178#423—671no—
179C2×C4⋊C8C2×D6711yescharacteristic
180C2×#102—21no—
181C2×#100—1341no—
182#343—1341no—
183#335—1342no—
184#333—1342no—
185#332—1342no—
186D4⋊C8—1342no—
187C22×Q8—21no—
188C22×D4C22⋊D6711yescharacteristic
189C23×C4—21no—
190C2×M16—1341no—
191C2×M16—1341no—
192C22×C8—21no—
193C4⋊Q8—21no—
194C4⋊Q8—21no—
195C4⋊Q8—21no—
196C4⋊Q8C22⋊D6712yes—
197C4×D4—21no—
198C4×D4—21no—
199C4×D4—21no—
200C4×D4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4C22⋊D6711yescharacteristic
204C2×C22⋊C4—21no—
205C2×C42C22×D6711yescharacteristic
206#106—24no—
207C4⋊C8—21no—
208C4⋊C8—672no—
209C4⋊C8—1341no—
210C4⋊C8—1341no—
211C4⋊C8—1341no—
212C4⋊C8—1341no—
213C4⋊C8—1341no—
214C4⋊C8C22⋊D6712yes—
215#102—24no—
216#100—2682no—
217C22⋊C8—1341no—
218C22⋊C8—1341no—
219C4×C8—1341no—
220C4×C8—1341no—
221C24—21no—
222C2×Q8—22no—
223C2×Q8—41no—
224C2×Q8—82no—
225C2×D4—21no—
226C2×D4—21no—
227C2×D4—21no—
228C2×D4—21no—
229C2×D4—21no—
230C2×D4#1176161812yes—
231C22×C4—21no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—82no—
236C22×C4C2×C22⋊D6711yescharacteristic
237C22×C4#1176162711yescharacteristic
238C22×C4#1176162711yescharacteristic
239M16—2681no—
240M16—2681no—
241M16—2681no—
242M16—2681no—
243C2×C8—22no—
244C2×C8—41no—
245C2×C8—41no—
246C2×C8—2681no—
247C2×C8—2681no—
248C2×C8—2681no—
249C2×C8—1342no—
250C4⋊C4—22no—
251C4⋊C4—21no—
252C4⋊C4—21no—
253C4⋊C4—22no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C22⋊C4—41no—
257C22⋊C4—41no—
258C42D4×D6711yescharacteristic
259C42C2×C22⋊D6711yescharacteristic
260C42C2×C22⋊D6711yescharacteristic
261C42#1176162711yescharacteristic
262C23—22no—
263C23—41no—
264C23—41no—
265C23—81no—
266C23—81no—
267C23#1452402411yescharacteristic
268Q8—82no—
269Q8—82no—
270D4—22no—
271D4—22no—
272D4—41no—
273D4—41no—
274C2×C4—22no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—42no—
278C2×C4—82no—
279C2×C4—82no—
280C2×C4—82no—
281C2×C4—81no—
282C2×C4#1452401211yescharacteristic
283C2×C4#1452401411yescharacteristic
284C2×C4#1452401511yescharacteristic
285C2×C4#1452401511yescharacteristic
286C2×C4#1452402411yescharacteristic
287C2×C4#1452403011yescharacteristic
288C2×C4C2×#1176161811yescharacteristic
289C2×C4#1452406311yescharacteristic
290C2×C4#1452406811yescharacteristic
291C2×C4#1452407411yescharacteristic
292C8—42no—
293C8—2682no—
294C8—5361no—
295C8—5361no—
296C8—5361no—
297C22—21no—
298C22—21no—
299C22—42no—
300C22—82no—
301C22—82no—
302C22—81no—
303C22—81no—
304C22#1686331011yescharacteristic
305C22#1686372911yescharacteristic
306C22#1686373411yescharacteristic
307C4—21no—
308C4—21no—
309C4—21no—
310C4—81no—
311C4—82no—
312C4#1686354111yescharacteristic
313C4#1686355211yescharacteristic
314C4#1686363511yescharacteristic
315C4#1686364411yescharacteristic
316C4#1686365612yes—
317C2—21no—
318C2—21no—
319C2—82no—
320C2#2256140911yescharacteristic
321C2#2256146711yescharacteristic
322C2#2256147911yescharacteristic
subgroupG/H
centerC22#16863310
commutator, cosocleC2×C42×C67C23
FrattiniC2×C42C22×D67
FittingC2×#476×C67C2
socleC22×C67#394
prime radicalC22×D4×C67D4
hypercenterC2×#476D67
hypocenterC67#10453
LCSG ▷ C2×C42×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#476

Complements

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

NH ≅ G/Ntype
2. #22558983C2semidirect
3. #22558894C2semidirect
4. #22558890C2semidirect
6. #22558106C2semidirect
169. C67#10453semidirect

Maximal subgroups

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

subgroupindex
#225602012
#225589832
#225588942
#225588902
#225586002
#225581062
C2×#476×C672
#1045367