Finite Group Databasebeta

← #33239449

Subgroups — #33239449

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#22618849C211yesmaximal, characteristic
3#22611820C218yesmaximal, complemented
4#22611703C211yesmaximal, characteristic, complemented
5C2×#17988403C211yesmaximal, characteristic, complemented
6#22611666C211yesmaximal, characteristic
7#22611536C211yesmaximal, characteristic, complemented
8C73×#2950C211yesmaximal, characteristic, complemented
9C2×#14556613C2211yescharacteristic
10#17989353—21no—
11#17989028C414yescomplemented
12#17989028C2214yes—
13C2×#14556537C412yescomplemented
14C2×#14556534C412yescomplemented
15C2×#14556531C2211yescharacteristic
16C2×#14556531C2211yescharacteristic
17#17988710C2214yes—
18#17988705—22no—
19C2×#14556527C2211yescharacteristic
20#17988605—21no—
21#17988409C2214yescomplemented
22#17988403C2214yescomplemented
23#17988396C2214yes—
24#17988365C2214yescomplemented
25C22×Q16×C73C2211yescharacteristic
26#519×C73—21no—
27#441×C73C2214yescomplemented
28C4×Q16×C73—22no—
29#427×C73C2211yescharacteristic
30#419×C73—21no—
31#405×C73C2211yescharacteristic
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—42no—
41#14556597—24no—
42C2×#11924012—21no—
43M16×D73—42no—
44#14556537C2×C418yes—
45#14556534C2×C418yes—
46#14556531—21no—
47#14556531C2×C412yes—
48#14556531C2312yes—
49#14556531C2312yes—
50C2×#11924003—21no—
51C2×#11924003C2×C412yes—
52#14556527—21no—
53#14556527C2312yes—
54#14556519—24no—
55C22×Q8×C73—21no—
56C2×Q16×C73—21no—
57C2×Q16×C73—21no—
58C2×Q16×C73—21no—
59C2×Q16×C73C2×C412yes—
60C2×Q16×C73C2×C412yes—
61C2×Q16×C73C2312yes—
62C2×M16×C73D411yescharacteristic
63C22×C8×C73C2311yescharacteristic
64C4×Q8×C73—24no—
65C4×Q8×C73—42no—
66#117×C73—21no—
67#117×C73—42no—
68#117×C73D411yescharacteristic
69#107×C73—21no—
70#106×C73C2312yes—
71#103×C73—24no—
72#97×C73C2312yes—
73C4×C8×C73D412yes—
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—22no—
89#11924012—41no—
90#11924011—44no—
91C2×#10596831—41no—
92C8×D73—44no—
93#11924003—22no—
94#11924003C22×C414yes—
95C2×Q8×C73—22no—
96C2×Q8×C73—22no—
97C2×Q8×C73—22no—
98C2×Q8×C73—41no—
99C2×Q8×C73—41no—
100C2×Q8×C73—41no—
101C22×C4×C73—41no—
102C22×C4×C73C2×D411yescharacteristic
103Q16×C73—22no—
104Q16×C73—22no—
105Q16×C73—22no—
106Q16×C73C22×C414yes—
107M16×C73—21no—
108M16×C73—21no—
109C2×C8×C73C22×C411yescharacteristic
110C2×C8×C73C22×C411yescharacteristic
111C2×C8×C73C2×D412yes—
112C2×C8×C73C2×D411yescharacteristic
113C2×C8×C73#4111yescharacteristic
114C2×C8×C73#4111yescharacteristic
115C2×C8×C73C2411yescharacteristic
116C4⋊C4×C73—21no—
117C4⋊C4×C73—22no—
118C4⋊C4×C73—44no—
119C22⋊C4×C73—21no—
120C22⋊C4×C73—41no—
121C42×C73—44no—
122C42×C73—41no—
123C42×C73C2×D412yes—
124C23×C73AGL(1,ℤ/8)11yescharacteristic
125C22×D73—41no—
126Q8×C73—24no—
127Q8×C73—42no—
128Q8×C73—42no—
129Q8×C73—42no—
130C2×C4×C73—22no—
131C2×C4×C73—42no—
132C2×C4×C73—42no—
133C2×C4×C73—42no—
134C2×C4×C73—42no—
135C2×C4×C73C4×D411yescharacteristic
136C2×C4×C73C4×D411yescharacteristic
137C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
138C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
139C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
140C2×C4×C73C22×D411yescharacteristic
141C22⋊D73—42no—
142C22⋊D73—42no—
143C2×Dic73—41no—
144D292—22no—
145D292—41no—
146C4×D73—42no—
147C4×D73—42no—
148Dic146—22no—
149Dic146—41no—
150C8×C73—22no—
151C8×C73C4×D411yescharacteristic
152C8×C73C4×D411yescharacteristic
153C8×C73C23×C411yescharacteristic
154C8×C73C2×#4111yescharacteristic
155#10596831—42no—
156C22×C73—41no—
157C22×C73#44211yescharacteristic
158C22×C73#44211yescharacteristic
159C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
160C2×D73—42no—
161C2×D73—161no—
162C4×C73—42no—
163C4×C73—44no—
164C4×C73—82no—
165C4×C73#44211yescharacteristic
166C4×C73#44211yescharacteristic
167C4×C73C2×C4×D411yescharacteristic
168C4×C73C2×AGL(1,ℤ/8)11yescharacteristic
169Dic73—42no—
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#2950D7311yescharacteristic, complemented
179#2180—738no—
180#2155—731no—
181C2×#362—731no—
182#2145—731no—
183#2110—731no—
184C73#3347011yescharacteristic, complemented, Sylow
185C2×#135—731no—
186C22×Q16C2×D7311yescharacteristic
187#568—1461no—
188#519—21no—
189#510—734no—
190#510—734no—
191C2×Q32—732no—
192C2×SD32—732no—
193C2×M32—731no—
194C2×M32—731no—
195#444—734no—
196#443—1462no—
197#441C2×D7314yescomplemented
198C4×Q16—22no—
199C2×#108—731no—
200#427C2×D7311yescharacteristic
201#421—1461no—
202#419—21no—
203#405C2×D7311yescharacteristic
204#365—734no—
205#362—734no—
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—2922no—
226C2×M16—1461no—
227C2×M16C22⋊D7311yescharacteristic
228C22×C8C22×D7311yescharacteristic
229C4×Q8—24no—
230C4×Q8—42no—
231#117—21no—
232#117—42no—
233#117C22⋊D7311yescharacteristic
234Q32—738no—
235SD32—738no—
236M32—732no—
237M32—732no—
238M32—732no—
239M32—1461no—
240C2×C16—732no—
241C2×C16—1461no—
242#108—732no—
243#108—1461no—
244#107—21no—
245#106C22×D7312yes—
246#104—1464no—
247#103—24no—
248#97C22×D7312yes—
249C4×C8C22⋊D7312yes—
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×C4C2×C22⋊D7311yescharacteristic
266Q16—22no—
267Q16—22no—
268Q16—22no—
269Q16—732no—
270Q16—1461no—
271Q16C2×C4×D7314yes—
272SD16—1462no—
273SD16—1462no—
274D8—732no—
275D8—1461no—
276M16—21no—
277M16—21no—
278M16—2924no—
279M16—2921no—
280M16—1462no—
281C2×C8—2924no—
282C2×C8—2921no—
283C2×C8C2×C4×D7311yescharacteristic
284C2×C8C2×C4×D7311yescharacteristic
285C2×C8C2×C22⋊D7312yes—
286C2×C8C2×C22⋊D7311yescharacteristic
287C2×C8#1192402811yescharacteristic
288C2×C8#1192402811yescharacteristic
289C2×C8C23×D7311yescharacteristic
290C4⋊C4—21no—
291C4⋊C4—22no—
292C4⋊C4—44no—
293C22⋊C4—21no—
294C22⋊C4—41no—
295C42—44no—
296C42—41no—
297C42C2×C22⋊D7312yes—
298C16—734no—
299C16—1462no—
300C23—2921no—
301C23#1455661711yescharacteristic
302Q8—24no—
303Q8—42no—
304Q8—42no—
305Q8—42no—
306Q8—2921no—
307Q8—1462no—
308D4—2922no—
309D4—2922no—
310D4—2921no—
311D4—1462no—
312C2×C4—22no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—2921no—
318C2×C4—2922no—
319C2×C4—2922no—
320C2×C4C4×C22⋊D7311yescharacteristic
321C2×C4C4×C22⋊D7311yescharacteristic
322C2×C4#1455661611yescharacteristic
323C2×C4#1455661611yescharacteristic
324C2×C4#1455661711yescharacteristic
325C2×C4C22×C22⋊D7311yescharacteristic
326C8—22no—
327C8—2922no—
328C8C4×C22⋊D7311yescharacteristic
329C8C4×C22⋊D7311yescharacteristic
330C8C22×C4×D7311yescharacteristic
331C8C2×#1192402811yescharacteristic
332C22—41no—
333C22—2922no—
334C22—11681no—
335C22#1798869711yescharacteristic
336C22#1798869711yescharacteristic
337C22C2×#1455661711yescharacteristic
338C4—42no—
339C4—44no—
340C4—82no—
341C4—2922no—
342C4#1798869711yescharacteristic
343C4#1798869711yescharacteristic
344C4C2×C4×C22⋊D7311yescharacteristic
345C4C2×#1455661611yescharacteristic
346C2—21no—
347C2—21no—
348C2—21no—
349C2—5842no—
350C2C2×#1798869711yescharacteristic
subgroupG/H
centerC2C2×#17988697
commutatorC8×C73C23×C4
FrattiniC2×C8C23×D73
FittingC73×#2950C2
socleC2×C73C2×#442
prime radicalC2×#14556613C22
cosocleC2×C8×C73C24
hypercenter#2950D73
hypocenterC73#33470
LCSG ▷ C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2 ◁ C22×C4 ◁ #405 ◁ #2950

Complements

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

NH ≅ G/Ntype
3. #22611820C2semidirect
4. #22611703C2semidirect
5. C2×#17988403C2semidirect
7. #22611536C2semidirect
8. C73×#2950C2semidirect
11. #17989028C4semidirect
13. C2×#14556537C4semidirect
14. C2×#14556534C4semidirect
21. #17988409C22semidirect
22. #17988403C22semidirect
24. #17988365C22semidirect
27. #441×C73C22semidirect
178. #2950D73semidirect
184. C73#33470semidirect
197. #441C2×D73semidirect

Maximal subgroups

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

subgroupindex
C2×#179890282
#226188492
#226118202
#226117032
C2×#179884032
#226116662
#226115362
C73×#29502
#3347073