Finite Group Databasebeta

← #33239454

Subgroups — #33239454

Order 18688.

632 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989028C211yesmaximal, characteristic
2#22618847C211yesmaximal, characteristic
3#22618799C211yesmaximal, characteristic, complemented
4#22611825C218yesmaximal, complemented
5#22611705C211yesmaximal, characteristic, complemented
6#22611668C211yesmaximal, characteristic
7#22611535C211yesmaximal, characteristic, complemented
8C2×#362×C73C211yesmaximal, characteristic, complemented
9C2×#14556613C2211yescharacteristic
10#17989353—21no—
11#17989065—21no—
12#17989028C414yescomplemented
13#17989028C2214yes—
14C2×#14556537C412yescomplemented
15C2×#14556534C412yescomplemented
16C2×#14556531C2211yescharacteristic
17#17988708C2214yes—
18#17988703—22no—
19#17988694C2214yescomplemented
20Q16×Dic73—22no—
21#17988625C2211yescharacteristic
22#17988603—21no—
23#17988596—21no—
24#17988542C2211yescharacteristic
25#17988411C2214yescomplemented
26#17988398C2214yes—
27#17988364C2214yescomplemented
28C22×Q16×C73C2211yescharacteristic
29C2×M32×C73C2211yescharacteristic
30C2×#108×C73C2211yescharacteristic
31#362×C73C2214yescomplemented
32C2×#11924028—21no—
33#14556613—21no—
34#14556613—21no—
35#14556613C2×C412yes—
36#14556613C2312yes—
37C2×Dic292C2×C411yescharacteristic
38C2×#11924017—21no—
39C2×D584C2×C411yescharacteristic
40#14556599—24no—
41#14556597—42no—
42C2×#11924012D411yescharacteristic
43M16×D73—42no—
44Q8×Dic73—24no—
45Q8×Dic73—42no—
46#14556550—21no—
47#14556550—42no—
48#14556550D411yescharacteristic
49#14556537C2×C418yes—
50#14556534C2×C418yes—
51#14556531C2×C412yes—
52#14556531C2312yes—
53C2×#11924003C2×C412yes—
54#14556525—21no—
55#14556523C2312yes—
56#14556517—24no—
57#14556515—24no—
58#14556501C2312yes—
59C8×Dic73D412yes—
60C22×Q8×C73—21no—
61C2×Q16×C73—21no—
62C2×Q16×C73—21no—
63C2×Q16×C73—21no—
64C2×Q16×C73C2×C412yes—
65C2×Q16×C73C2×C412yes—
66C2×Q16×C73C2312yes—
67C2×M16×C73—21no—
68C22×C8×C73C2311yescharacteristic
69M32×C73—21no—
70M32×C73C2312yes—
71C2×C16×C73—21no—
72#108×C73—21no—
73#108×C73C2312yes—
74#11924028—22no—
75#11924028—22no—
76#11924028—41no—
77#11924028—41no—
78C2×Dic146—21no—
79C2×D292—21no—
80C2×C22⋊D73—41no—
81C2×C4×D73—41no—
82Dic292—21no—
83Dic292C22×C412yes—
84#11924017—22no—
85#11924017—22no—
86D584—21no—
87D584C22×C412yes—
88#11924012—21no—
89#11924012—21no—
90#11924011—44no—
91C2×#10596831C2×D412yes—
92C8×D73—44no—
93C4⋊Dic73—21no—
94C4⋊Dic73—22no—
95C4⋊Dic73—44no—
96C22⋊Dic73—21no—
97C22⋊Dic73—41no—
98C4×Dic73—44no—
99C4×Dic73—41no—
100C4×Dic73C2×D412yes—
101#11924003C22×C414yes—
102C2×Q8×C73—22no—
103C2×Q8×C73—22no—
104C2×Q8×C73—22no—
105C2×Q8×C73—41no—
106C2×Q8×C73—41no—
107C2×Q8×C73—41no—
108C22×C4×C73—41no—
109C22×C4×C73C2×D411yescharacteristic
110Q16×C73—22no—
111Q16×C73—22no—
112Q16×C73—22no—
113Q16×C73C22×C414yes—
114M16×C73—22no—
115M16×C73—41no—
116C2×C8×C73—41no—
117C2×C8×C73C22×C411yescharacteristic
118C2×C8×C73C22×C411yescharacteristic
119C2×C8×C73C2×D411yescharacteristic
120C2×C8×C73#4111yescharacteristic
121C2×C8×C73#4111yescharacteristic
122C2×C8×C73C2411yescharacteristic
123C16×C73—22no—
124C23×C73AGL(1,ℤ/8)11yescharacteristic
125C22×D73—41no—
126Q8×C73—24no—
127Q8×C73—42no—
128Q8×C73—42no—
129Q8×C73—42no—
130C2×C4×C73—42no—
131C2×C4×C73—42no—
132C2×C4×C73—42no—
133C2×C4×C73C4×D411yescharacteristic
134C2×C4×C73C4×D411yescharacteristic
135C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
136C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
137C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
138C2×C4×C73C22×D411yescharacteristic
139C22⋊D73—42no—
140C22⋊D73—42no—
141C2×Dic73—22no—
142C2×Dic73—41no—
143C2×Dic73—42no—
144D292—22no—
145D292—41no—
146C4×D73—42no—
147C4×D73—42no—
148Dic146—22no—
149Dic146—41no—
150C8×C73—42no—
151C8×C73C4×D411yescharacteristic
152C8×C73C4×D411yescharacteristic
153C8×C73C23×C411yescharacteristic
154C8×C73C2×#4111yescharacteristic
155#10596831—22no—
156C22×C73—41no—
157C22×C73#44211yescharacteristic
158C22×C73#44211yescharacteristic
159C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
160C2×D73—42no—
161C2×D73—161no—
162C4×C73—44no—
163C4×C73#44211yescharacteristic
164C4×C73#44211yescharacteristic
165C4×C73C2×C4×D411yescharacteristic
166C4×C73C2×AGL(1,ℤ/8)11yescharacteristic
167Dic73—42no—
168Dic73—42no—
169Dic73—82no—
170#33470—731nomaximal, Sylow
171C2×C73—21no—
172C2×C73—21no—
173C2×C73—21no—
174C2×C73C2×#44211yescharacteristic
175D73—82no—
176C2×#510—731no—
177#2957—731no—
178#2950—731no—
179#2180—738no—
180#2155—731no—
181C2×#362D7311yescharacteristic, complemented
182#2145—731no—
183#2110—731no—
184C73#3347011yescharacteristic, complemented, Sylow
185C2×#135—731no—
186C22×Q16C2×D7311yescharacteristic
187#568—1461no—
188#519—1461no—
189#510—734no—
190#510—734no—
191C2×Q32—732no—
192C2×SD32—732no—
193C2×M32—731no—
194C2×M32C2×D7311yescharacteristic
195#444—734no—
196#443—1462no—
197#441—734no—
198C4×Q16—1462no—
199C2×#108C2×D7311yescharacteristic
200#427—731no—
201#421—1461no—
202#419—1461no—
203#405—731no—
204#365—734no—
205#362C2×D7314yescomplemented
206#360—734no—
207#347—734no—
208C2×#41—1461no—
209C22×Q8—21no—
210#135—732no—
211#135—732no—
212#135—1461no—
213#135—1461no—
214C2×Q16—21no—
215C2×Q16—21no—
216C2×Q16—21no—
217C2×Q16—731no—
218C2×Q16C4×D7312yes—
219C2×Q16C4×D7312yes—
220C2×Q16C22×D7312yes—
221C2×SD16—1461no—
222C2×D8—731no—
223#131—2922no—
224#131—1464no—
225C2×M16—21no—
226C2×M16—2922no—
227C2×M16—731no—
228C22×C8C22×D7311yescharacteristic
229C4×Q8—2922no—
230C4×Q8—1464no—
231#117—2922no—
232#117—731no—
233#117—1461no—
234Q32—738no—
235SD32—738no—
236M32—21no—
237M32—732no—
238M32—732no—
239M32C22×D7312yes—
240C2×C16—21no—
241C2×C16—732no—
242#108—21no—
243#108C22×D7312yes—
244#107—1461no—
245#106—732no—
246#104—1464no—
247#103—1464no—
248#97—732no—
249C4×C8—732no—
250#41—2921no—
251#41—2921no—
252#41—1462no—
253#41—1462no—
254C2×Q8—22no—
255C2×Q8—22no—
256C2×Q8—22no—
257C2×Q8—41no—
258C2×Q8—41no—
259C2×Q8—41no—
260C2×Q8—1461no—
261C2×D4—2921no—
262C2×D4—1461no—
263C22×C4—41no—
264C22×C4—2921no—
265C22×C4D4×D7311yescharacteristic
266Q16—22no—
267Q16—22no—
268Q16—22no—
269Q16—732no—
270Q16—1461no—
271Q16C2×C4×D7314yes—
272SD16—1462no—
273SD16—1462no—
274D8—732no—
275D8—1461no—
276M16—22no—
277M16—41no—
278M16—2924no—
279M16—1461no—
280M16—1461no—
281C2×C8—41no—
282C2×C8—2924no—
283C2×C8—732no—
284C2×C8C2×C4×D7311yescharacteristic
285C2×C8C2×C4×D7311yescharacteristic
286C2×C8D4×D7311yescharacteristic
287C2×C8#1192402711yescharacteristic
288C2×C8#1192402711yescharacteristic
289C2×C8C23×D7311yescharacteristic
290C4⋊C4—2924no—
291C4⋊C4—1462no—
292C4⋊C4—1461no—
293C22⋊C4—2921no—
294C22⋊C4—1461no—
295C42—2924no—
296C42—2921no—
297C42—732no—
298C16—22no—
299C16—734no—
300C23—2921no—
301C23#1455661811yescharacteristic
302Q8—24no—
303Q8—42no—
304Q8—42no—
305Q8—42no—
306Q8—2921no—
307Q8—1462no—
308D4—2922no—
309D4—2922no—
310D4—2921no—
311D4—1462no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—2921no—
316C2×C4—2922no—
317C2×C4—2922no—
318C2×C4—2922no—
319C2×C4—1462no—
320C2×C4#1455655111yescharacteristic
321C2×C4#1455655111yescharacteristic
322C2×C4#1455661511yescharacteristic
323C2×C4#1455661511yescharacteristic
324C2×C4#1455661811yescharacteristic
325C2×C4C2×D4×D7311yescharacteristic
326C8—42no—
327C8—1462no—
328C8#1455655111yescharacteristic
329C8#1455655111yescharacteristic
330C8C22×C4×D7311yescharacteristic
331C8C2×#1192402711yescharacteristic
332C22—41no—
333C22—2922no—
334C22—11681no—
335C22#1798869611yescharacteristic
336C22#1798869611yescharacteristic
337C22C2×#1455661811yescharacteristic
338C4—44no—
339C4—2922no—
340C4—2922no—
341C4—5842no—
342C4#1798869611yescharacteristic
343C4#1798869611yescharacteristic
344C4C2×#1455655111yescharacteristic
345C4C2×#1455661511yescharacteristic
346C2—21no—
347C2—21no—
348C2—21no—
349C2—5842no—
350C2C2×#1798869611yescharacteristic
subgroupG/H
centerC2C2×#17988696
commutatorC8×C73C23×C4
FrattiniC2×C8C23×D73
FittingC2×#362×C73C2
socleC2×C73C2×#442
prime radicalC2×#14556613C22
cosocleC2×C8×C73C24
hypercenterC2×#362D73
hypocenterC73#33470
LCSG ▷ C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2 ◁ C22×C4 ◁ C22×C8 ◁ C2×#362

Complements

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

NH ≅ G/Ntype
3. #22618799C2semidirect
4. #22611825C2semidirect
5. #22611705C2semidirect
7. #22611535C2semidirect
8. C2×#362×C73C2semidirect
12. #17989028C4semidirect
14. C2×#14556537C4semidirect
15. C2×#14556534C4semidirect
19. #17988694C22semidirect
25. #17988411C22semidirect
27. #17988364C22semidirect
31. #362×C73C22semidirect
181. C2×#362D73semidirect
184. C73#33470semidirect
205. #362C2×D73semidirect

Maximal subgroups

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

subgroupindex
C2×#179890282
#226188472
#226187992
#226118252
#226117052
#226116682
#226115352
C2×#362×C732
#3347073