Finite Group Databasebeta

← #33003049

Subgroups — #33003049

Order 18688.

526 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
1C22×#14556536C211yesmaximal, characteristic
2C2×#17988681C211yesmaximal, characteristic, complemented
3C2×#17988390C211yesmaximal, characteristic, complemented
4C2×#17988387C211yesmaximal, characteristic
5#22610795C211yesmaximal, characteristic
6#22608008C211yesmaximal, characteristic, complemented
7C73×#1380C211yesmaximal, characteristic, complemented
8C22×Dic292C2211yescharacteristic
9C22×D584C2211yescharacteristic
10C2×#14556556—21no—
11C2×#14556536—22no—
12C2×#14556536—22no—
13C2×#14556536C414yescomplemented
14C22×#11924003C2211yescharacteristic
15#17988681—22no—
16#17988681—22no—
17C2×#14556514—21no—
18C2×#14556511—21no—
19C2×C8×Dic73C2211yescharacteristic
20#17988488—21no—
21#17988390—22no—
22#17988387—22no—
23C22×C16×C73C2211yescharacteristic
24C2×#107×C73C2211yescharacteristic
25C2×#107×C73C2211yescharacteristic
26C22×Dic146—21no—
27C22×D292—21no—
28C2×Dic292—21no—
29C2×Dic292—21no—
30C2×Dic292—21no—
31C2×Dic292C2×C412yes—
32C2×Dic292D412yes—
33C2×Dic292D412yes—
34C2×D584—21no—
35C2×D584—21no—
36C2×D584—21no—
37C2×D584—21no—
38C2×D584—21no—
39C2×D584C2×C412yes—
40C22×#10596831D411yescharacteristic
41#14556556—24no—
42#14556556—42no—
43C2×#11924007—41no—
44C2×#11924005—41no—
45C2×#11924005—41no—
46C2×C4×Dic73—41no—
47C2×C4×Dic73D411yescharacteristic
48#14556536—28no—
49C2×#11924003—21no—
50C2×#11924003—21no—
51C2×#11924003—21no—
52#14556514—24no—
53#14556511—42no—
54C8×Dic73—21no—
55C8×Dic73—21no—
56C22×C8×C73C2311yescharacteristic
57C2×C4⋊C4×C73—21no—
58C2×C4⋊C4×C73—21no—
59C2×C16×C73—21no—
60C2×C16×C73—21no—
61C2×C16×C73C2×C412yes—
62#107×C73—21no—
63#107×C73—21no—
64#107×C73—21no—
65#107×C73D412yes—
66C23×D73—41no—
67C2×Dic146—22no—
68C2×Dic146—22no—
69C2×Dic146—22no—
70C2×Dic146—41no—
71C2×Dic146—41no—
72C2×Dic146—41no—
73C2×D292—22no—
74C2×D292—41no—
75C2×D292—41no—
76C2×D292—41no—
77C2×D292—41no—
78C2×D292—41no—
79C22×Dic73—41no—
80C22×Dic73D811yescharacteristic
81C22×Dic73D811yescharacteristic
82Dic292—22no—
83Dic292—22no—
84Dic292—22no—
85Dic292C22⋊C414yescomplemented
86D584—22no—
87D584—22no—
88D584—41no—
89D584—41no—
90C2×#10596831—21no—
91C2×#10596831—21no—
92C2×#10596831—21no—
93#11924007—44no—
94#11924005—82no—
95#11924005—82no—
96C4×Dic73—21no—
97C4×Dic73—21no—
98C4×Dic73—44no—
99#11924003—41no—
100C22×C4×C73—41no—
101C22×C4×C73—41no—
102C22×C4×C73C2×D411yescharacteristic
103C2×C8×C73C22×C411yescharacteristic
104C2×C8×C73C2×D411yescharacteristic
105C2×C8×C73C2×D411yescharacteristic
106C2×C8×C73C2×D411yescharacteristic
107C2×C8×C73#4111yescharacteristic
108C2×C8×C73#4111yescharacteristic
109C4⋊C4×C73—22no—
110C4⋊C4×C73—41no—
111C4⋊C4×C73—41no—
112C4⋊C4×C73—41no—
113C16×C73—22no—
114C23×C73C2×D811yescharacteristic
115C22×D73—42no—
116C22×D73—81no—
117C22×D73—81no—
118C22×D73—81no—
119C22×D73—81no—
120C22×D73—81no—
121C22×D73—81no—
122C2×C4×C73—42no—
123C2×C4×C73—81no—
124C2×C4×C73—81no—
125C2×C4×C73—81no—
126C2×C4×C73—81no—
127C2×C4×C73—81no—
128C2×C4×C73C4×D411yescharacteristic
129C2×C4×C73#12011yescharacteristic
130C2×C4×C73#12111yescharacteristic
131C2×C4×C73#13511yescharacteristic
132C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
133C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
134C2×Dic73—21no—
135C2×Dic73—21no—
136C2×Dic73—21no—
137C2×Dic73—21no—
138C2×Dic73—21no—
139C2×Dic73—42no—
140C2×Dic73—42no—
141C2×Dic73—42no—
142C2×Dic73SD3212yescomplemented
143D292—42no—
144D292—42no—
145D292—81no—
146D292—81no—
147Dic146—24no—
148Dic146—42no—
149Dic146—42no—
150Dic146—42no—
151C8×C73C2×C22⋊C411yescharacteristic
152C8×C73C4×D411yescharacteristic
153C8×C73#12111yescharacteristic
154C8×C73#12311yescharacteristic
155#10596831—41no—
156C22×C73C4×D811yescharacteristic
157C22×C73#44711yescharacteristic
158C22×C73#45911yescharacteristic
159C22×C73C2×SD3211yescharacteristic
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—161no—
172C4×C73#38611yescharacteristic
173C4×C73#44211yescharacteristic
174C4×C73#44911yescharacteristic
175C4×C73#46311yescharacteristic
176Dic73—22no—
177Dic73—44no—
178Dic73—41no—
179#13013—731nomaximal, Sylow
180C2×C73#187911yescharacteristic
181C2×C73C4×SD3211yescharacteristic
182C2×C73#217511yescharacteristic
183C2×C73#218511yescharacteristic
184C2×C73#218611yescharacteristic
185C2×C73#220711yescharacteristic
186C2×C73#221111yescharacteristic
187D73—162no—
188C22×SD32—731no—
189C2×C4×Q16—731no—
190C2×#358—731no—
191C2×#357—731no—
192#1970—731no—
193#1380D7311yescharacteristic, complemented
194#1374—731no—
195C73#1301311yescharacteristic, complemented, Sylow
196C22×Q16—731no—
197C22×D8—731no—
198C2×C4×Q8—1461no—
199C2×SD32—734no—
200C2×SD32—1462no—
201C2×SD32—1462no—
202C22×C16—731no—
203C22×C16C2×D7311yescharacteristic
204C4×Q16—1462no—
205C4×Q16—1462no—
206C2×#107C2×D7311yescharacteristic
207C2×#107C2×D7311yescharacteristic
208C2×#103—1461no—
209C2×#102—1461no—
210C2×C4×C8—731no—
211#390—1461no—
212#358—1462no—
213#357—1462no—
214C22×Q8—1461no—
215C22×D4—1461no—
216C2×Q16—732no—
217C2×Q16—732no—
218C2×Q16—732no—
219C2×Q16—1461no—
220C2×Q16—1461no—
221C2×Q16—1461no—
222C2×D8—732no—
223C2×D8—1461no—
224C2×D8—1461no—
225C2×D8—1461no—
226C2×D8—1461no—
227C2×D8—1461no—
228C22×C8—731no—
229C22×C8C22×D7311yescharacteristic
230C4×Q8—2922no—
231C4×Q8—1464no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—2921no—
235C2×C22⋊C4—2921no—
236C2×C22⋊C4—2921no—
237C2×C42—2921no—
238C2×C42—731no—
239SD32—1468no—
240C2×C16—21no—
241C2×C16—21no—
242C2×C16—1461no—
243C2×C16—1461no—
244C2×C16—1461no—
245C2×C16C4×D7312yes—
246#107—21no—
247#107—21no—
248#107—21no—
249#107D29212yes—
250#103—1464no—
251#102—2922no—
252C4×C8—1461no—
253C4×C8—1461no—
254C24—2921no—
255C2×Q8—2921no—
256C2×Q8—2921no—
257C2×Q8—2921no—
258C2×Q8—1462no—
259C2×Q8—1462no—
260C2×Q8—1462no—
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—734no—
274Q16—1462no—
275Q16—1462no—
276Q16—1462no—
277D8—2921no—
278D8—2921no—
279D8—1462no—
280D8—1462no—
281C2×C8—1461no—
282C2×C8—1461no—
283C2×C8—1461no—
284C2×C8C2×C4×D7311yescharacteristic
285C2×C8D4×D7311yescharacteristic
286C2×C8D4×D7311yescharacteristic
287C2×C8C2×D29211yescharacteristic
288C2×C8#1192402811yescharacteristic
289C2×C8#1192402911yescharacteristic
290C4⋊C4—22no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—2924no—
295C22⋊C4—5842no—
296C22⋊C4—5842no—
297C42—2924no—
298C42—1461no—
299C42—1461no—
300C16—22no—
301C16—2921no—
302C23—2922no—
303C23—5841no—
304C23—5841no—
305C23—5841no—
306C23—5841no—
307C23—5841no—
308C23—5841no—
309C23D8×D7311yescharacteristic
310Q8—2922no—
311Q8—2922no—
312Q8—2922no—
313Q8—1464no—
314D4—2922no—
315D4—2922no—
316D4—5841no—
317D4—5841no—
318C2×C4—42no—
319C2×C4—81no—
320C2×C4—81no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4—81no—
324C2×C4—2922no—
325C2×C4—2922no—
326C2×C4—2922no—
327C2×C4—732no—
328C2×C4—1461no—
329C2×C4—1461no—
330C2×C4—1461no—
331C2×C4—1461no—
332C2×C4—1461no—
333C2×C4#1455655311yescharacteristic
334C2×C4#1455656111yescharacteristic
335C2×C4#1455656411yescharacteristic
336C2×C4#1455661411yescharacteristic
337C2×C4#1455661811yescharacteristic
338C2×C4#1455661811yescharacteristic
339C8—2921no—
340C8C22⋊C4×D7311yescharacteristic
341C8C4×D29211yescharacteristic
342C8#1455656611yescharacteristic
343C8#1455657611yescharacteristic
344C22—5842no—
345C22—5842no—
346C22—5842no—
347C22—11681no—
348C22—11681no—
349C22—11681no—
350C22—11681no—
351C22#1798866911yescharacteristic
352C22#1798872211yescharacteristic
353C22#1798878111yescharacteristic
354C22SD32×D7311yescharacteristic
355C22#1798901811yescharacteristic
356C22#1798902111yescharacteristic
357C22#1798902911yescharacteristic
358C4—82no—
359C4—161no—
360C4—2924no—
361C4—2921no—
362C4—1462no—
363C4#1798846511yescharacteristic
364C4#1798869811yescharacteristic
365C4#1798873511yescharacteristic
366C4#1798880511yescharacteristic
367C2—11682no—
368C2#2261028011yescharacteristic
369C2#2261178211yescharacteristic
370C2#2261179411yescharacteristic
371C2#2261184911yescharacteristic
372C2#2261185511yescharacteristic
373C2#2261197011yescharacteristic
374C2#2261199611yescharacteristic
subgroupG/H
centerC23D8×D73
commutatorC2×C8×C73C22×C4
FrattiniC22×C8C22×D73
FittingC73×#1380C2
socleC23×C73C2×D8
prime radicalC22×D584C22
cosocleC22×C8×C73C23
hypercenter#1380D73
hypocenterC73#13013
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8 ◁ #1380

Complements

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

NH ≅ G/Ntype
2. C2×#17988681C2semidirect
3. C2×#17988390C2semidirect
6. #22608008C2semidirect
7. C73×#1380C2semidirect
13. C2×#14556536C4semidirect
85. Dic292C22⋊C4semidirect
142. C2×Dic73SD32semidirect
193. #1380D73semidirect
195. C73#13013semidirect

Maximal subgroups

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

subgroupindex
C22×#145565362
C2×#179886812
C2×#179883902
C2×#179883872
#226107952
#226080082
C73×#13802
#1301373