Finite Group Databasebeta

← #32989860

Subgroups — #32989860

Order 18688.

490 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988868C211yesmaximal, characteristic
2#22610733C211yesmaximal, characteristic, complemented
3#22610725C211yesmaximal, characteristic
4#22609932C211yesmaximal, characteristic
5C2×#17988332C211yesmaximal, characteristic, complemented
6#22607815C211yesmaximal, characteristic, complemented
7C2×#334×C73C211yesmaximal, characteristic, complemented
8C2×#14556591C2211yescharacteristic
9C2×C4×D292C2211yescharacteristic
10#17988868—22no—
11#17988868—22no—
12C2×#14556525—21no—
13C2×C4⋊#10596831C2211yescharacteristic
14C2×#14556516—21no—
15C2×#14556516—21no—
16C2×#14556513—21no—
17C2×#14556513—21no—
18C2×#14556503—21no—
19C2×#14556503—21no—
20#17988457C2211yescharacteristic
21#17988350—21no—
22#17988332—22no—
23C2×C4⋊C8×C73C2211yescharacteristic
24C2×C4⋊C8×C73C2211yescharacteristic
25C2×C4×C8×C73C2211yescharacteristic
26#334×C73—22no—
27C22×Dic146—21no—
28C22×D292D411yescharacteristic
29C22×C4×D73—21no—
30C22×#10596831—21no—
31#14556591—21no—
32#14556591—21no—
33#14556591—21no—
34#14556591D412yes—
35C4×D292—21no—
36C4×D292—21no—
37C4×D292—21no—
38C4×D292—21no—
39C2×C4⋊Dic73—21no—
40C2×C4⋊Dic73—21no—
41C2×C4⋊Dic73—21no—
42C2×C4⋊Dic73—21no—
43C2×C4⋊Dic73—21no—
44C2×C4⋊Dic73D411yescharacteristic
45C2×#11924005—21no—
46#14556525—24no—
47C4⋊#10596831—21no—
48C4⋊#10596831—21no—
49#14556516—42no—
50#14556516—42no—
51#14556513—24no—
52#14556513—42no—
53#14556503—42no—
54#14556503—42no—
55C22×C8×C73—21no—
56C22×C8×C73—21no—
57C22×C8×C73D411yescharacteristic
58C22×C8×C73D411yescharacteristic
59C2×C42×C73C2311yescharacteristic
60C4⋊C8×C73—21no—
61C4⋊C8×C73—21no—
62C4⋊C8×C73—21no—
63C4⋊C8×C73D412yes—
64C4×C8×C73—21no—
65C4×C8×C73—21no—
66C23×D73—21no—
67C2×Dic146—22no—
68C2×Dic146—41no—
69C2×Dic146—41no—
70C2×Dic146—41no—
71C2×Dic146—41no—
72C2×Dic146—41no—
73C2×D292—21no—
74C2×D292—21no—
75C2×D292—21no—
76C2×D292—21no—
77C2×D292—21no—
78C2×D292SD1612yes—
79C22×Dic73—21no—
80C22×Dic73—41no—
81C22×Dic73—41no—
82C2×C4×D73—41no—
83C2×C4×D73—41no—
84C2×C4×D73—41no—
85C2×C4×D73—41no—
86C2×C4×D73—41no—
87C2×C4×D73—41no—
88C2×#10596831—41no—
89C2×#10596831—41no—
90C2×#10596831—41no—
91C4⋊Dic73—22no—
92C4⋊Dic73—21no—
93C4⋊Dic73—22no—
94C4⋊Dic73—21no—
95C4⋊Dic73—41no—
96C4⋊Dic73—41no—
97C4⋊Dic73—41no—
98C4⋊Dic73—41no—
99C4⋊Dic73—41no—
100C4⋊Dic73—41no—
101C4⋊Dic73—81no—
102#11924005—41no—
103#11924005—41no—
104C22×C4×C73C2×D411yescharacteristic
105C22×C4×C73C2×D411yescharacteristic
106C22×C4×C73#4111yescharacteristic
107C2×C8×C73—21no—
108C2×C8×C73—21no—
109C2×C8×C73—22no—
110C2×C8×C73—21no—
111C2×C8×C73—21no—
112C2×C8×C73—21no—
113C2×C8×C73—41no—
114C2×C8×C73—41no—
115C2×C8×C73—41no—
116C2×C8×C73—41no—
117C2×C8×C73—41no—
118C2×C8×C73D812yes—
119C42×C73C2×D411yescharacteristic
120C42×C73C2×D411yescharacteristic
121C42×C73#4111yescharacteristic
122C42×C73#4111yescharacteristic
123C23×C73#12111yescharacteristic
124C22×D73—22no—
125C22×D73—41no—
126C22×D73—41no—
127C22×D73—41no—
128C22×D73—41no—
129C22×D73—41no—
130C22×D73—41no—
131C2×C4×C73—21no—
132C2×C4×C73—21no—
133C2×C4×C73#12011yescharacteristic
134C2×C4×C73#12111yescharacteristic
135C2×C4×C73#12111yescharacteristic
136C2×C4×C73#12311yescharacteristic
137C2×C4×C73#12311yescharacteristic
138C2×C4×C73#12411yescharacteristic
139C2×C4×C73C2×D811yescharacteristic
140C2×C4×C73C2×SD1611yescharacteristic
141C2×C4×C73#13511yescharacteristic
142C2×C4×C73#13511yescharacteristic
143C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
144C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
145C2×C4×C73#13711yescharacteristic
146C2×C4×C73#13711yescharacteristic
147C2×Dic73—41no—
148C2×Dic73—42no—
149C2×Dic73—41no—
150C2×Dic73—41no—
151C2×Dic73—81no—
152C2×Dic73—81no—
153C2×Dic73—81no—
154C2×Dic73—81no—
155C2×Dic73—81no—
156D292—22no—
157D292—22no—
158D292—41no—
159D292—41no—
160C4×D73—81no—
161C4×D73—81no—
162C4×D73—81no—
163C4×D73—81no—
164Dic146—42no—
165Dic146—42no—
166Dic146—81no—
167Dic146—81no—
168C8×C73—22no—
169C8×C73—42no—
170C8×C73—41no—
171C8×C73—81no—
172#10596831—81no—
173C22×C73#39411yescharacteristic
174C22×C73#45911yescharacteristic
175C22×C73#46311yescharacteristic
176C22×C73#46511yescharacteristic
177C22×C73#47011yescharacteristic
178C22×C73#47111yescharacteristic
179C22×C73#47311yescharacteristic
180C2×D73—42no—
181C2×D73—42no—
182C2×D73—42no—
183C2×D73—81no—
184C2×D73—81no—
185C2×D73—81no—
186C2×D73—81no—
187C4×C73—21no—
188C4×C73—21no—
189C4×C73#44811yescharacteristic
190C4×C73#45211yescharacteristic
191C4×C73#45911yescharacteristic
192C4×C73#46311yescharacteristic
193C4×C73#46611yescharacteristic
194C4×C73#46911yescharacteristic
195C4×C73#48311yescharacteristic
196C4×C73#48511yescharacteristic
197Dic73—82no—
198Dic73—81no—
199Dic73—161no—
200#10484—731nomaximal, Sylow
201C2×C73#166511yescharacteristic
202C2×C73#167911yescharacteristic
203C2×C73#168911yescharacteristic
204C2×C73#170311yescharacteristic
205C2×C73#203511yescharacteristic
206C2×C73#205211yescharacteristic
207C2×C73#205411yescharacteristic
208D73—82no—
209C2×#474—731no—
210#1956—731no—
211#1954—731no—
212#1802—731no—
213C2×#334D7311yescharacteristic, complemented
214C2×#332—731no—
215#1296—731no—
216C73#1048411yescharacteristic, complemented, Sylow
217C2×C4⋊Q8—731no—
218C2×C4×D4—731no—
219#474—1462no—
220#474—1462no—
221C2×#107—1461no—
222C2×C4⋊C8—731no—
223C2×C4⋊C8C2×D7311yescharacteristic
224C2×C4⋊C8C2×D7311yescharacteristic
225C2×#103—1461no—
226C2×#103—1461no—
227C2×#102—1461no—
228C2×#102—1461no—
229C2×C22⋊C8—1461no—
230C2×C22⋊C8—1461no—
231C2×C4×C8C2×D7311yescharacteristic
232#384—731no—
233#340—1461no—
234#334—22no—
235#332—1462no—
236C22×Q8—1461no—
237C22×D4—731no—
238C23×C4—1461no—
239C22×C8—21no—
240C22×C8—21no—
241C22×C8—1461no—
242C22×C8D29211yescharacteristic
243C22×C8D29211yescharacteristic
244C4⋊Q8—732no—
245C4⋊Q8—1461no—
246C4⋊Q8—1461no—
247C4⋊Q8—1461no—
248C4×D4—1461no—
249C4×D4—1461no—
250C4×D4—1461no—
251C4×D4—1461no—
252C2×C4⋊C4—731no—
253C2×C4⋊C4—1461no—
254C2×C4⋊C4—1461no—
255C2×C4⋊C4—1461no—
256C2×C4⋊C4—1461no—
257C2×C4⋊C4—1461no—
258C2×C22⋊C4—1461no—
259C2×C42C22×D7311yescharacteristic
260#107—1464no—
261C4⋊C8—21no—
262C4⋊C8—21no—
263C4⋊C8—21no—
264C4⋊C8—1461no—
265C4⋊C8—1461no—
266C4⋊C8C22⋊D7312yes—
267#103—2922no—
268#103—2922no—
269#102—2922no—
270#102—1464no—
271C22⋊C8—2922no—
272C22⋊C8—2922no—
273C4×C8—21no—
274C4×C8—21no—
275C24—1461no—
276C2×Q8—2921no—
277C2×Q8—2921no—
278C2×Q8—2921no—
279C2×Q8—2921no—
280C2×Q8—2921no—
281C2×Q8—1462no—
282C2×D4—732no—
283C2×D4—1461no—
284C2×D4—1461no—
285C2×D4—1461no—
286C2×D4—1461no—
287C2×D4—1461no—
288C22×C4—2921no—
289C22×C4—2921no—
290C22×C4—2921no—
291C22×C4—2921no—
292C22×C4—2921no—
293C22×C4—2921no—
294C22×C4—2921no—
295C22×C4—2921no—
296C22×C4—1461no—
297C22×C4D4×D7311yescharacteristic
298C22×C4C2×D29211yescharacteristic
299C22×C4#1192402911yescharacteristic
300C2×C8—21no—
301C2×C8—21no—
302C2×C8—22no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—41no—
307C2×C8—41no—
308C2×C8—41no—
309C2×C8—41no—
310C2×C8—41no—
311C2×C8—2921no—
312C2×C8—2921no—
313C2×C8—2921no—
314C2×C8D58412yes—
315C4⋊C4—2921no—
316C4⋊C4—2921no—
317C4⋊C4—2921no—
318C4⋊C4—2921no—
319C4⋊C4—2921no—
320C4⋊C4—2921no—
321C4⋊C4—5841no—
322C4⋊C4—1462no—
323C4⋊C4—1461no—
324C4⋊C4—1461no—
325C4⋊C4—1462no—
326C22⋊C4—2921no—
327C22⋊C4—2921no—
328C42D4×D7311yescharacteristic
329C42C2×C22⋊D7311yescharacteristic
330C42#1192402811yescharacteristic
331C42#1192402911yescharacteristic
332C23—2921no—
333C23—2921no—
334C23—2921no—
335C23—2921no—
336C23—2921no—
337C23—2921no—
338C23—1462no—
339C23#1455656611yescharacteristic
340Q8—2922no—
341Q8—2922no—
342Q8—5841no—
343Q8—5841no—
344D4—2921no—
345D4—2921no—
346D4—1462no—
347D4—1462no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—2921no—
351C2×C4—2922no—
352C2×C4—2921no—
353C2×C4—2921no—
354C2×C4—5841no—
355C2×C4—5841no—
356C2×C4—5841no—
357C2×C4—5841no—
358C2×C4—5841no—
359C2×C4—5841no—
360C2×C4—5841no—
361C2×C4—5841no—
362C2×C4—5841no—
363C2×C4#1455655911yescharacteristic
364C2×C4#1455656511yescharacteristic
365C2×C4#1455656611yescharacteristic
366C2×C4#1455657611yescharacteristic
367C2×C4#1455657611yescharacteristic
368C2×C4#1455658011yescharacteristic
369C2×C4C2×D58411yescharacteristic
370C2×C4SD16×D7311yescharacteristic
371C2×C4#1455661211yescharacteristic
372C2×C4#1455661311yescharacteristic
373C2×C4#1455661911yescharacteristic
374C2×C4#1455661911yescharacteristic
375C2×C4#1455662011yescharacteristic
376C2×C4#1455662011yescharacteristic
377C8—22no—
378C8—42no—
379C8—41no—
380C8—81no—
381C8—5841no—
382C22—2922no—
383C22—2922no—
384C22—2922no—
385C22—5841no—
386C22—5841no—
387C22—5841no—
388C22—5841no—
389C22#1798850411yescharacteristic
390C22#1798878311yescharacteristic
391C22#1798880311yescharacteristic
392C22#1798881711yescharacteristic
393C22#1798884611yescharacteristic
394C22#1798885111yescharacteristic
395C22#1798886311yescharacteristic
396C4—21no—
397C4—21no—
398C4—5841no—
399C4—5842no—
400C4—11681no—
401C4#1798872411yescharacteristic
402C4#1798874611yescharacteristic
403C4#1798878311yescharacteristic
404C4#1798880311yescharacteristic
405C4#1798882311yescharacteristic
406C4#1798884011yescharacteristic
407C4#1798891911yescharacteristic
408C4#1798893111yescharacteristic
409C2—5842no—
410C2#2260927311yescharacteristic
411C2#2260935511yescharacteristic
412C2#2260942111yescharacteristic
413C2#2260950711yescharacteristic
414C2#2261112911yescharacteristic
415C2#2261123211yescharacteristic
416C2#2261124211yescharacteristic
subgroupG/H
centerC23#14556566
commutator, cosocleC2×C42×C73C23
FrattiniC2×C42C22×D73
FittingC2×#334×C73C2
socleC23×C73#121
prime radicalC22×D292D4
hypercenterC2×#334D73
hypocenterC73#10484
LCSG ▷ C2×C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#334

Complements

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

NH ≅ G/Ntype
2. #22610733C2semidirect
5. C2×#17988332C2semidirect
6. #22607815C2semidirect
7. C2×#334×C73C2semidirect
213. C2×#334D73semidirect
216. C73#10484semidirect

Maximal subgroups

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

subgroupindex
C2×#179888682
#226107332
#226107252
#226099322
C2×#179883322
#226078152
C2×#334×C732
#1048473