Finite Group Databasebeta

← #33004459

Subgroups — #33004459

Order 18688.

514 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988972C211yesmaximal, characteristic
2C2×#17988405C211yesmaximal, characteristic, complemented
3C2×#17988391C211yesmaximal, characteristic, complemented
4C2×#17988389C211yesmaximal, characteristic
5C2×#17988387C211yesmaximal, characteristic
6#107×Dic73C211yesmaximal, characteristic, complemented
7C2×#358×C73C211yesmaximal, characteristic, complemented
8C22×#11924015—21no—
9C22×D584C2211yescharacteristic
10C2×#14556580—21no—
11C22×#11924003C2211yescharacteristic
12#17988972—22no—
13#17988972C414yescomplemented
14C2×#14556525C2211yescharacteristic
15C2×C8×Dic73C2211yescharacteristic
16C4⋊C4×Dic73—21no—
17#17988405—22no—
18#17988391—22no—
19#17988389—22no—
20#17988387—22no—
21#17988349—21no—
22C22×Q16×C73C2211yescharacteristic
23C22×C16×C73C2211yescharacteristic
24C2×#107×C73C2211yescharacteristic
25#358×C73—22no—
26C22×D292—21no—
27C2×#11924015—24no—
28C2×#11924015—42no—
29C2×#11924015—42no—
30C2×D584—21no—
31C2×D584—21no—
32C2×D584—21no—
33C2×D584—21no—
34C2×D584—21no—
35C2×D584C2×C412yes—
36C22×#10596831D411yescharacteristic
37#14556580—24no—
38#14556580—42no—
39C2×C4⋊Dic73—21no—
40C2×#11924005—41no—
41C2×#11924005—41no—
42C2×C4×Dic73—41no—
43C2×C4×Dic73D411yescharacteristic
44C2×#11924003—21no—
45C2×#11924003—21no—
46C2×#11924003—21no—
47#14556525—21no—
48#14556525D412yes—
49C8×Dic73C2×C412yes—
50C8×Dic73D412yes—
51#14556498—41no—
52C22×Q8×C73—21no—
53C2×Q16×C73—21no—
54C2×Q16×C73—21no—
55C2×Q16×C73—21no—
56C2×Q16×C73—21no—
57C2×Q16×C73—21no—
58C2×Q16×C73C2×C412yes—
59C22×C8×C73C2311yescharacteristic
60C2×C4⋊C4×C73—21no—
61C2×C16×C73—21no—
62C2×C16×C73—21no—
63C2×C16×C73—21no—
64#107×C73—21no—
65#107×C73D412yes—
66C23×D73—41no—
67C2×D292—22no—
68C2×D292—41no—
69C2×D292—41no—
70C2×D292—41no—
71C2×D292—41no—
72C2×D292—41no—
73C22×Dic73—41no—
74C22×Dic73D811yescharacteristic
75C22×Dic73D811yescharacteristic
76#11924015—48no—
77D584—22no—
78D584—22no—
79D584—41no—
80D584—41no—
81C2×#10596831—21no—
82C2×#10596831—21no—
83C2×#10596831C22⋊C412yes—
84C4⋊Dic73—22no—
85C4⋊Dic73—41no—
86#11924005—44no—
87#11924005—44no—
88C4×Dic73—82no—
89C4×Dic73C22⋊C412yes—
90C4×Dic73SD1612yes—
91#11924003—41no—
92C2×Q8×C73—22no—
93C2×Q8×C73—41no—
94C2×Q8×C73—41no—
95C2×Q8×C73—41no—
96C2×Q8×C73—41no—
97C2×Q8×C73—41no—
98C22×C4×C73—41no—
99C22×C4×C73—41no—
100C22×C4×C73C2×D411yescharacteristic
101Q16×C73—22no—
102Q16×C73—22no—
103Q16×C73—41no—
104Q16×C73—41no—
105C2×C8×C73C22×C411yescharacteristic
106C2×C8×C73C2×D411yescharacteristic
107C2×C8×C73C2×D411yescharacteristic
108C2×C8×C73C2×D411yescharacteristic
109C2×C8×C73#4111yescharacteristic
110C2×C8×C73#4111yescharacteristic
111C4⋊C4×C73—22no—
112C4⋊C4×C73—41no—
113C16×C73—41no—
114C23×C73C2×D811yescharacteristic
115C22×D73—42no—
116C22×D73—81no—
117C22×D73—81no—
118C22×D73—81no—
119C22×D73—81no—
120C22×D73—81no—
121C22×D73—81no—
122Q8×C73—42no—
123Q8×C73—42no—
124Q8×C73—81no—
125Q8×C73—81no—
126C2×C4×C73—42no—
127C2×C4×C73—42no—
128C2×C4×C73—81no—
129C2×C4×C73—81no—
130C2×C4×C73—81no—
131C2×C4×C73—81no—
132C2×C4×C73C2×C22⋊C411yescharacteristic
133C2×C4×C73#12111yescharacteristic
134C2×C4×C73#12111yescharacteristic
135C2×C4×C73C2×SD1611yescharacteristic
136C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
137C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
138C2×Dic73—21no—
139C2×Dic73—21no—
140C2×Dic73—21no—
141C2×Dic73—21no—
142C2×Dic73—42no—
143C2×Dic73—81no—
144C2×Dic73—81no—
145C2×Dic73#10212yes—
146C2×Dic73#10212yes—
147D292—42no—
148D292—42no—
149D292—81no—
150D292—81no—
151C8×C73C4×D411yescharacteristic
152C8×C73C4×D411yescharacteristic
153C8×C73#12411yescharacteristic
154C8×C73#12711yescharacteristic
155#10596831—22no—
156C22×C73C2×#10211yescharacteristic
157C22×C73#45911yescharacteristic
158C22×C73#45911yescharacteristic
159C22×C73#50811yescharacteristic
160C22×C73#50811yescharacteristic
161C22×C73#50911yescharacteristic
162C22×C73#51011yescharacteristic
163C2×D73—82no—
164C2×D73—82no—
165C2×D73—82no—
166C2×D73—161no—
167C2×D73—161no—
168C2×D73—161no—
169C2×D73—161no—
170C4×C73—82no—
171C4×C73—82no—
172C4×C73#39011yescharacteristic
173C4×C73#41811yescharacteristic
174C4×C73#46011yescharacteristic
175C4×C73#46011yescharacteristic
176Dic73—22no—
177Dic73—22no—
178Dic73—82no—
179#13254—731nomaximal, Sylow
180C2×C73#198211yescharacteristic
181C2×C73#213811yescharacteristic
182C2×C73#213911yescharacteristic
183C2×C73#221011yescharacteristic
184C2×C73#221011yescharacteristic
185C2×C73#221111yescharacteristic
186C2×C73#221111yescharacteristic
187D73—162no—
188C2×#495—731no—
189C2×C4⋊C16—731no—
190C2×#358—731no—
191C2×#358D7311yescharacteristic, complemented
192C2×#357—731no—
193C2×#357—731no—
194C4×#107—731no—
195C73#1325411yescharacteristic, complemented, Sylow
196C22×Q16C2×D7311yescharacteristic
197C22×SD16—1461no—
198C22×D8—731no—
199C2×#124—1461no—
200C22×C16—731no—
201C22×C16C2×D7311yescharacteristic
202#495—734no—
203#495—1462no—
204C2×#107—731no—
205C2×#107C2×D7311yescharacteristic
206C2×C4×C8—731no—
207C4×C4⋊C4—1461no—
208C4⋊C16—1462no—
209#358—22no—
210#358—1462no—
211#357—1462no—
212#357—1462no—
213#340—1461no—
214C22×Q8—21no—
215C22×D4—1461no—
216C2×Q16—21no—
217C2×Q16—21no—
218C2×Q16—21no—
219C2×Q16—21no—
220C2×Q16—21no—
221C2×Q16C4×D7312yes—
222C2×SD16—2922no—
223C2×SD16—2922no—
224C2×SD16—1464no—
225C2×D8—732no—
226C2×D8—1461no—
227C2×D8—1461no—
228C2×D8—1461no—
229C2×D8—1461no—
230C2×D8—1461no—
231C22×C8—731no—
232C22×C8C22×D7311yescharacteristic
233#124—2922no—
234#124—1464no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—1461no—
237C2×C22⋊C4—2921no—
238C2×C22⋊C4—2921no—
239C2×C42—2921no—
240C2×C42—731no—
241C2×C16—21no—
242C2×C16—21no—
243C2×C16—21no—
244C2×C16—1461no—
245C2×C16—1461no—
246C2×C16—1461no—
247#107—21no—
248#107—732no—
249#107—1461no—
250#107C22⋊D7312yes—
251C4×C8—732no—
252C4×C8—732no—
253#95—2921no—
254C24—2921no—
255C2×Q8—22no—
256C2×Q8—41no—
257C2×Q8—41no—
258C2×Q8—41no—
259C2×Q8—41no—
260C2×Q8—41no—
261C2×D4—2921no—
262C2×D4—2921no—
263C2×D4—2921no—
264C2×D4—2921no—
265C2×D4—2921no—
266C2×D4—1462no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—2921no—
270C22×C4—731no—
271C22×C4—731no—
272C22×C4D4×D7311yescharacteristic
273Q16—22no—
274Q16—22no—
275Q16—41no—
276Q16—41no—
277SD16—2928no—
278D8—2921no—
279D8—2921no—
280D8—1462no—
281D8—1462no—
282C2×C8—732no—
283C2×C8—1461no—
284C2×C8—1461no—
285C2×C8C2×C4×D7311yescharacteristic
286C2×C8D4×D7311yescharacteristic
287C2×C8D4×D7311yescharacteristic
288C2×C8C2×C22⋊D7311yescharacteristic
289C2×C8#1192402711yescharacteristic
290C2×C8#1192402811yescharacteristic
291C4⋊C4—22no—
292C4⋊C4—41no—
293C4⋊C4—2921no—
294C4⋊C4—1462no—
295C22⋊C4—2924no—
296C22⋊C4—2924no—
297C42—5842no—
298C42—732no—
299C42—732no—
300C16—41no—
301C16—2921no—
302C23—2922no—
303C23—5841no—
304C23—5841no—
305C23—5841no—
306C23—5841no—
307C23—5841no—
308C23—5841no—
309C23D8×D7311yescharacteristic
310Q8—42no—
311Q8—42no—
312Q8—81no—
313Q8—81no—
314D4—2922no—
315D4—2922no—
316D4—5841no—
317D4—5841no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—81no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4—81no—
324C2×C4—2922no—
325C2×C4—5841no—
326C2×C4—5841no—
327C2×C4—732no—
328C2×C4—732no—
329C2×C4—1461no—
330C2×C4—1461no—
331C2×C4—1461no—
332C2×C4—1461no—
333C2×C4C22⋊C4×D7311yescharacteristic
334C2×C4#1455656311yescharacteristic
335C2×C4#1455656411yescharacteristic
336C2×C4SD16×D7311yescharacteristic
337C2×C4#1455661511yescharacteristic
338C2×C4#1455661811yescharacteristic
339C8—1462no—
340C8#1455655111yescharacteristic
341C8C4×C22⋊D7311yescharacteristic
342C8#1455657711yescharacteristic
343C8#1455658711yescharacteristic
344C22—5842no—
345C22—5842no—
346C22—5842no—
347C22—11681no—
348C22—11681no—
349C22—11681no—
350C22—11681no—
351C22#102×D7311yescharacteristic
352C22#1798878011yescharacteristic
353C22#1798878111yescharacteristic
354C22#1798901811yescharacteristic
355C22#1798902011yescharacteristic
356C22#1798902411yescharacteristic
357C22#1798902911yescharacteristic
358C4—82no—
359C4—82no—
360C4—5842no—
361C4—1462no—
362C4—1462no—
363C4#1798848711yescharacteristic
364C4#1798859011yescharacteristic
365C4#1798878511yescharacteristic
366C4#1798878611yescharacteristic
367C2—11682no—
368C2#2261085611yescharacteristic
369C2#2261163711yescharacteristic
370C2#2261164211yescharacteristic
371C2#2261198811yescharacteristic
372C2#2261198911yescharacteristic
373C2#2261199511yescharacteristic
374C2#2261199611yescharacteristic
subgroupG/H
centerC23D8×D73
commutatorC2×C8×C73C22×C4
FrattiniC22×C8C22×D73
FittingC2×#358×C73C2
socleC23×C73C2×D8
prime radicalC22×D584C22
cosocleC22×C8×C73C23
hypercenterC2×#358D73
hypocenterC73#13254
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8 ◁ C2×#358

Complements

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

NH ≅ G/Ntype
2. C2×#17988405C2semidirect
3. C2×#17988391C2semidirect
6. #107×Dic73C2semidirect
7. C2×#358×C73C2semidirect
13. #17988972C4semidirect
191. C2×#358D73semidirect
195. C73#13254semidirect

Maximal subgroups

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

subgroupindex
C2×#179889722
C2×#179884052
C2×#179883912
C2×#179883892
C2×#179883872
#107×Dic732
C2×#358×C732
#1325473