Finite Group Databasebeta

← #32996468

Subgroups — #32996468

Order 18688.

586 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 586 conjugacy classes of subgroups — too many to draw. Only its 87 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22615089C211yesmaximal, characteristic
2#22608258C211yesmaximal, characteristic
3#22607951C212yesmaximal, complemented
4#22607930C212yesmaximal, complemented
5C2×#342×C73C211yesmaximal, characteristic
6C2×#14556570C411yescharacteristic, complemented
7#17988551—21no—
8#17988512C411yescharacteristic, complemented
9#17988512C2211yescharacteristic
10#17988499C411yescharacteristic, complemented
11#17988499C2211yescharacteristic
12#17988481—21no—
13#17988481—21no—
14#17988450—21no—
15#17988443—21no—
16C4×C22⋊Dic73C411yescharacteristic, complemented
17#17988311—21no—
18#17988311C2212yes—
19C22×C22⋊C4×C73—21no—
20C22×C22⋊C4×C73C2211yescharacteristic
21#342×C73—21no—
22#342×C73C2212yes—
23C22×Dic146C2×C411yescharacteristic, complemented
24C2×#11924012—41no—
25#14556570—41no—
26#14556570—41no—
27C2×C4⋊Dic73C2×C411yescharacteristic, complemented
28C2×#11924007—21no—
29C2×C22⋊Dic73—21no—
30C2×C22⋊Dic73—21no—
31C2×C22⋊Dic73—21no—
32C2×C22⋊Dic73—21no—
33C2×C22⋊Dic73—21no—
34C2×C22⋊Dic73—21no—
35C2×C4×Dic73—21no—
36C22⋊#10596831—22no—
37C22⋊#10596831—41no—
38#14556498—21no—
39#14556498—21no—
40#14556498—21no—
41#14556498—21no—
42#14556498C2×C411yescharacteristic, complemented
43#14556498C2×C411yescharacteristic, complemented
44C25×C73D411yescharacteristic
45C23×C4×C73—41no—
46C23×C4×C73C2×C411yescharacteristic
47C23×C4×C73D411yescharacteristic
48C2×C22⋊C4×C73—24no—
49C2×C22⋊C4×C73—22no—
50C2×C22⋊C4×C73—41no—
51C2×C22⋊C4×C73—41no—
52C2×C22⋊C4×C73—41no—
53C2×C22⋊C4×C73—42no—
54C2×C22⋊C4×C73C2×C411yescharacteristic
55C2×C22⋊C4×C73D411yescharacteristic
56C2×C22⋊C4×C73D411yescharacteristic
57C2×C22⋊C4×C73C2311yescharacteristic
58C2×Dic146—41no—
59C2×Dic146—81no—
60C22×Dic73—21no—
61C22×Dic73—21no—
62C22×Dic73—21no—
63C22×Dic73—21no—
64#11924012—82no—
65C2×#10596831—42no—
66C4⋊Dic73—41no—
67#11924007—42no—
68C22⋊Dic73—42no—
69C22⋊Dic73—41no—
70C22⋊Dic73—41no—
71C22⋊Dic73—42no—
72C22⋊Dic73—41no—
73C22⋊Dic73—41no—
74C22⋊Dic73—81no—
75C22⋊Dic73—81no—
76C4×Dic73—81no—
77C24×C73—21no—
78C24×C73—21no—
79C24×C73—22no—
80C24×C73—21no—
81C24×C73—21no—
82C24×C73—42no—
83C24×C73—81no—
84C24×C73C22⋊C411yescharacteristic
85C24×C73C22⋊C411yescharacteristic
86C24×C73C2×D411yescharacteristic
87C22×C4×C73—21no—
88C22×C4×C73—21no—
89C22×C4×C73—21no—
90C22×C4×C73—21no—
91C22×C4×C73—42no—
92C22×C4×C73—44no—
93C22×C4×C73—42no—
94C22×C4×C73—81no—
95C22×C4×C73—82no—
96C22×C4×C73—82no—
97C22×C4×C73C22⋊C411yescharacteristic
98C22×C4×C73C22⋊C411yescharacteristic
99C22×C4×C73C22×C411yescharacteristic
100C22×C4×C73C2×D411yescharacteristic
101C22⋊C4×C73—24no—
102C22⋊C4×C73—41no—
103C22⋊C4×C73—41no—
104C22⋊C4×C73—41no—
105C22⋊C4×C73—41no—
106C22⋊C4×C73—44no—
107C22⋊C4×C73—42no—
108C23×C73—22no—
109C23×C73—22no—
110C23×C73—22no—
111C23×C73—22no—
112C23×C73—22no—
113C23×C73—21no—
114C23×C73—21no—
115C23×C73—21no—
116C23×C73—21no—
117C23×C73—21no—
118C23×C73—21no—
119C23×C73—41no—
120C23×C73—42no—
121C23×C73—41no—
122C23×C73—41no—
123C23×C73—41no—
124C23×C73—41no—
125C23×C73—42no—
126C23×C73—81no—
127C23×C73—81no—
128C23×C73—84no—
129C23×C73—82no—
130C23×C73—82no—
131C23×C73#9911yescharacteristic
132C23×C73#9911yescharacteristic
133C23×C73#10412yes—
134C23×C73C2×C22⋊C411yescharacteristic
135C23×C73AGL(1,ℤ/8)11yescharacteristic
136C23×C73#13711yescharacteristic
137C2×C4×C73—21no—
138C2×C4×C73—21no—
139C2×C4×C73—21no—
140C2×C4×C73—21no—
141C2×C4×C73—42no—
142C2×C4×C73—41no—
143C2×C4×C73—41no—
144C2×C4×C73—41no—
145C2×C4×C73—41no—
146C2×C4×C73—44no—
147C2×C4×C73—81no—
148C2×C4×C73—81no—
149C2×C4×C73—81no—
150C2×C4×C73—88no—
151C2×C4×C73—82no—
152C2×C4×C73—82no—
153C2×Dic73—41no—
154C2×Dic73—41no—
155C2×Dic73—41no—
156C2×Dic73—41no—
157C2×Dic73—41no—
158C2×Dic73—41no—
159C2×Dic73—41no—
160C2×Dic73—41no—
161C2×Dic73—41no—
162C2×Dic73—41no—
163C2×Dic73—41no—
164C2×Dic73—41no—
165Dic146—82no—
166#10596831—82no—
167C22×C73—21no—
168C22×C73—21no—
169C22×C73—21no—
170C22×C73—21no—
171C22×C73—21no—
172C22×C73—21no—
173C22×C73—21no—
174C22×C73—21no—
175C22×C73—22no—
176C22×C73—21no—
177C22×C73—42no—
178C22×C73—42no—
179C22×C73—42no—
180C22×C73—41no—
181C22×C73—41no—
182C22×C73—44no—
183C22×C73—44no—
184C22×C73—41no—
185C22×C73—41no—
186C22×C73—41no—
187C22×C73—41no—
188C22×C73—81no—
189C22×C73—81no—
190C22×C73—82no—
191C22×C73—82no—
192C22×C73#35412yes—
193C22×C73C2×#9911yescharacteristic
194C22×C73#41711yescharacteristic
195C22×C73C2×#10411yescharacteristic
196C4×C73—41no—
197C4×C73—41no—
198C4×C73—41no—
199C4×C73—41no—
200C4×C73—84no—
201Dic73—81no—
202Dic73—81no—
203Dic73—81no—
204Dic73—81no—
205#11759—731nomaximal, Sylow
206C2×C73—21no—
207C2×C73—21no—
208C2×C73—22no—
209C2×C73—41no—
210C2×C73—41no—
211C2×C73—42no—
212C2×C73—41no—
213C2×C73#150411yescharacteristic
214C2×C73#212111yescharacteristic
215C2×C73C2×#35411yescharacteristic
216#2561—731no—
217C2×#342D7311yescharacteristic
218#1462—731no—
219#1351—732no—
220#1344—732no—
221C73#1175911yescharacteristic, complemented, Sylow
222C2×C22⋊Q8—731no—
223C22×C22⋊C4—21no—
224C22×C22⋊C4C2×D7311yescharacteristic
225#407—1461no—
226#397—731no—
227#397—731no—
228#393—731no—
229#393—731no—
230#388—1461no—
231#388—1461no—
232#381—1461no—
233#379—1461no—
234C4×C22⋊C4—731no—
235#342—21no—
236#342C2×D7312yes—
237#323—732no—
238#323—1461no—
239C25C22⋊D7311yescharacteristic
240C22×Q8—731no—
241C23×C4—41no—
242C23×C4C4×D7311yescharacteristic
243C23×C4D29211yescharacteristic
244C2×M16—2921no—
245C22⋊Q8—2921no—
246C22⋊Q8—2921no—
247C2×C4⋊C4—731no—
248C2×C4⋊C4—1461no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—41no—
252C2×C22⋊C4—41no—
253C2×C22⋊C4—41no—
254C2×C22⋊C4—42no—
255C2×C22⋊C4—1461no—
256C2×C22⋊C4—1461no—
257C2×C22⋊C4—1461no—
258C2×C22⋊C4—1461no—
259C2×C22⋊C4—1461no—
260C2×C22⋊C4—1461no—
261C2×C22⋊C4C4×D7311yescharacteristic
262C2×C22⋊C4D29211yescharacteristic
263C2×C22⋊C4C22⋊D7311yescharacteristic
264C2×C22⋊C4C22×D7311yescharacteristic
265C2×C42—1461no—
266C22⋊C8—2921no—
267C22⋊C8—1462no—
268#95—731no—
269#95—731no—
270#95—1461no—
271#95—1461no—
272#95—1461no—
273#95—1461no—
274C24—21no—
275C24—21no—
276C24—22no—
277C24—21no—
278C24—21no—
279C24—42no—
280C24—81no—
281C24#1192400511yescharacteristic
282C24#1192400511yescharacteristic
283C24C2×C22⋊D7311yescharacteristic
284C2×Q8—2921no—
285C2×Q8—5841no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—42no—
291C22×C4—44no—
292C22×C4—42no—
293C22×C4—81no—
294C22×C4—82no—
295C22×C4—82no—
296C22×C4—1461no—
297C22×C4—1461no—
298C22×C4—1461no—
299C22×C4—1461no—
300C22×C4#1192400511yescharacteristic
301C22×C4#1192400511yescharacteristic
302C22×C4C2×C4×D7311yescharacteristic
303C22×C4C2×D29211yescharacteristic
304M16—5842no—
305C2×C8—2922no—
306C4⋊C4—2921no—
307C4⋊C4—2922no—
308C22⋊C4—24no—
309C22⋊C4—41no—
310C22⋊C4—41no—
311C22⋊C4—41no—
312C22⋊C4—41no—
313C22⋊C4—44no—
314C22⋊C4—42no—
315C22⋊C4—2922no—
316C22⋊C4—2921no—
317C22⋊C4—2921no—
318C22⋊C4—2922no—
319C22⋊C4—2921no—
320C22⋊C4—2921no—
321C22⋊C4—5841no—
322C22⋊C4—5841no—
323C42—5841no—
324C23—22no—
325C23—22no—
326C23—22no—
327C23—22no—
328C23—22no—
329C23—21no—
330C23—21no—
331C23—21no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23—41no—
336C23—42no—
337C23—41no—
338C23—41no—
339C23—41no—
340C23—41no—
341C23—42no—
342C23—81no—
343C23—81no—
344C23—84no—
345C23—82no—
346C23—82no—
347C23#1455650511yescharacteristic
348C23#1455650511yescharacteristic
349C23#1455651912yes—
350C23C2×#1192400511yescharacteristic
351C23#1455661711yescharacteristic
352C23#1455662211yescharacteristic
353Q8—5842no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—42no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—44no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—88no—
368C2×C4—82no—
369C2×C4—82no—
370C2×C4—2921no—
371C2×C4—2921no—
372C2×C4—2921no—
373C2×C4—2921no—
374C2×C4—2921no—
375C2×C4—2921no—
376C2×C4—2921no—
377C2×C4—2921no—
378C2×C4—2921no—
379C2×C4—2921no—
380C2×C4—2921no—
381C2×C4—2921no—
382C8—5842no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—22no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—42no—
394C22—42no—
395C22—42no—
396C22—41no—
397C22—41no—
398C22—44no—
399C22—44no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—81no—
405C22—81no—
406C22—82no—
407C22—82no—
408C22#1798838112yes—
409C22C2×#1455650511yescharacteristic
410C22#1798858611yescharacteristic
411C22C2×#1455651911yescharacteristic
412C4—41no—
413C4—41no—
414C4—41no—
415C4—41no—
416C4—84no—
417C4—5841no—
418C4—5841no—
419C4—5841no—
420C4—5841no—
421C2—21no—
422C2—21no—
423C2—22no—
424C2—41no—
425C2—41no—
426C2—42no—
427C2—41no—
428C2#2260843311yescharacteristic
429C2#2261157511yescharacteristic
430C2C2×#1798838111yescharacteristic
subgroupG/H
centerC22C2×#14556505
commutatorC22×C4×C73C22×C4
FrattiniC2×C22⋊C4C22×D73
FittingC2×#342×C73C2
socleC22×C73C2×#99
prime radicalC25×C73D4
cosocleC2×C22⋊C4×C73C23
hypercenterC2×#342D73
hypocenterC73#11759
LCSG ▷ C22×C4×C73 ▷ C23×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ C2×#342

Complements

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

NH ≅ G/Ntype
3. #22607951C2semidirect
4. #22607930C2semidirect
6. C2×#14556570C4semidirect
8. #17988512C4semidirect
10. #17988499C4semidirect
16. C4×C22⋊Dic73C4semidirect
23. C22×Dic146C2×C4semidirect
27. C2×C4⋊Dic73C2×C4semidirect
42. #14556498C2×C4semidirect
43. #14556498C2×C4semidirect
221. C73#11759semidirect

Maximal subgroups

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

subgroupindex
#226150892
#226082582
#226079512
#226079302
C2×#342×C732
#1175973