Finite Group Databasebeta

← #33001960

Subgroups — #33001960

Order 18688.

512 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 101 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#22612603C211yesmaximal, characteristic
2C2×#17988377C211yesmaximal, characteristic
3#22609921C211yesmaximal, characteristic
4#22609917C211yesmaximal, characteristic
5#22608050C211yesmaximal, characteristic
6#22608048C211yesmaximal, characteristic
7C2×#352×C73C211yesmaximal, characteristic
8C2×D4×Dic73C2211yescharacteristic
9C22×C4⋊Dic73C411yescharacteristic, complemented
10#17988610—21no—
11C2×#14556508C2211yescharacteristic
12C2×#14556506C2211yescharacteristic
13#17988486—21no—
14#17988480—21no—
15#17988473—21no—
16#17988467C2211yescharacteristic
17#17988462—21no—
18#17988462C411yescharacteristic, complemented
19#17988450C411yescharacteristic, complemented
20#17988445—21no—
21C22⋊C4×Dic73C411yescharacteristic, complemented
22#17988377—22no—
23#17988355—21no—
24#17988353—21no—
25C2×#123×C73C2211yescharacteristic
26C2×#100×C73C2211yescharacteristic
27C2×#99×C73C2211yescharacteristic
28#352×C73—22no—
29C23×Dic73C2×C411yescharacteristic
30C23×Dic73C2×C411yescharacteristic
31C23×Dic73D411yescharacteristic
32C2×#11924012—21no—
33D4×Dic73—41no—
34D4×Dic73—41no—
35C2×C4⋊Dic73—21no—
36C2×C4⋊Dic73—21no—
37C2×C4⋊Dic73—21no—
38C2×C4⋊Dic73—41no—
39C2×C4⋊Dic73—41no—
40C2×C4⋊Dic73D411yescharacteristic
41C2×#11924007—41no—
42C2×C22⋊Dic73—21no—
43C2×C22⋊Dic73—21no—
44C2×C22⋊Dic73C2×C411yescharacteristic
45C2×C22⋊Dic73C2×C411yescharacteristic
46C2×C22⋊Dic73D411yescharacteristic
47C2×C4×Dic73—21no—
48C2×C4×Dic73D411yescharacteristic
49#14556520—41no—
50#14556520—41no—
51#14556508—21no—
52#14556508—21no—
53#14556506—21no—
54#14556506—21no—
55#14556498—21no—
56#14556498—21no—
57#14556498—21no—
58#14556498—21no—
59#14556498—21no—
60#14556498—41no—
61C22×D4×C73C2311yescharacteristic
62C23×C4×C73C2×C411yescharacteristic
63C2×M16×C73—21no—
64#123×C73—21no—
65#123×C73—21no—
66#123×C73D412yes—
67#123×C73Q812yes—
68C2×C4⋊C4×C73—21no—
69C2×C22⋊C4×C73—21no—
70C2×C22⋊C4×C73—21no—
71C2×C22⋊C4×C73C2×C411yescharacteristic
72#100×C73—21no—
73#100×C73—21no—
74#99×C73—21no—
75#99×C73—21no—
76C22×Dic73—21no—
77C22×Dic73—21no—
78C22×Dic73—21no—
79C22×Dic73—21no—
80C22×Dic73—21no—
81C22×Dic73—21no—
82C22×Dic73—41no—
83C22×Dic73—41no—
84C22×Dic73—81no—
85C22×Dic73—81no—
86C22×Dic73—81no—
87C22×Dic73C22⋊C411yescharacteristic
88C22×Dic73C22⋊C411yescharacteristic
89C22×Dic73C22⋊C411yescharacteristic
90C22×Dic73C22⋊C411yescharacteristic
91#11924012—41no—
92#11924012—41no—
93C2×#10596831—41no—
94C4⋊Dic73—41no—
95C4⋊Dic73—42no—
96C4⋊Dic73—41no—
97C4⋊Dic73—41no—
98C4⋊Dic73—81no—
99#11924007—82no—
100C22⋊Dic73—41no—
101C22⋊Dic73—41no—
102C22⋊Dic73—41no—
103C22⋊Dic73—41no—
104C22⋊Dic73—41no—
105C22⋊Dic73—81no—
106C4×Dic73—21no—
107C4×Dic73—21no—
108C4×Dic73—81no—
109C24×C73C22×C411yescharacteristic
110C24×C73C2×D411yescharacteristic
111C2×D4×C73—41no—
112C2×D4×C73—41no—
113C2×D4×C73C2×D411yescharacteristic
114C2×D4×C73C2×Q811yescharacteristic
115C2×D4×C73#4111yescharacteristic
116C2×D4×C73#4111yescharacteristic
117C22×C4×C73—21no—
118C22×C4×C73—21no—
119C22×C4×C73—21no—
120C22×C4×C73—41no—
121C22×C4×C73—41no—
122C22×C4×C73—41no—
123C22×C4×C73C4⋊C412yescomplemented
124C22×C4×C73C2×D411yescharacteristic
125M16×C73—41no—
126M16×C73—41no—
127C2×C8×C73—41no—
128C4⋊C4×C73—22no—
129C4⋊C4×C73—22no—
130C22⋊C4×C73—21no—
131C22⋊C4×C73—22no—
132C22⋊C4×C73—22no—
133C22⋊C4×C73—41no—
134C22⋊C4×C73—41no—
135C22⋊C4×C73C4⋊C412yescomplemented
136C23×C73—21no—
137C23×C73—21no—
138C23×C73—21no—
139C23×C73—21no—
140C23×C73—41no—
141C23×C73—41no—
142C23×C73—41no—
143C23×C73—41no—
144C23×C73C2×C22⋊C411yescharacteristic
145C23×C73C2×C4⋊C411yescharacteristic
146C23×C73#11711yescharacteristic
147C23×C73#12111yescharacteristic
148C23×C73C22⋊Q811yescharacteristic
149D4×C73—41no—
150D4×C73—41no—
151D4×C73—41no—
152D4×C73—41no—
153C2×C4×C73—21no—
154C2×C4×C73—21no—
155C2×C4×C73—22no—
156C2×C4×C73—22no—
157C2×C4×C73—41no—
158C2×C4×C73—42no—
159C2×C4×C73—41no—
160C2×C4×C73—41no—
161C2×C4×C73—41no—
162C2×C4×C73—42no—
163C2×C4×C73—81no—
164C2×C4×C73—81no—
165C2×C4×C73—81no—
166C2×C4×C73#12111yescharacteristic
167C2×C4×C73C22⋊Q811yescharacteristic
168C2×Dic73—21no—
169C2×Dic73—21no—
170C2×Dic73—21no—
171C2×Dic73—41no—
172C2×Dic73—42no—
173C2×Dic73—41no—
174C2×Dic73—41no—
175C2×Dic73—41no—
176C2×Dic73—41no—
177C2×Dic73—41no—
178C2×Dic73—41no—
179C2×Dic73—41no—
180C2×Dic73—41no—
181C2×Dic73—41no—
182C2×Dic73—81no—
183C2×Dic73—81no—
184C2×Dic73—81no—
185C2×Dic73—82no—
186C2×Dic73—81no—
187C2×Dic73—81no—
188C2×Dic73—81no—
189C2×Dic73#9911yescharacteristic
190C2×Dic73#9911yescharacteristic
191C8×C73—81no—
192#10596831—81no—
193C22×C73—21no—
194C22×C73—21no—
195C22×C73—21no—
196C22×C73—21no—
197C22×C73—21no—
198C22×C73—21no—
199C22×C73—41no—
200C22×C73—41no—
201C22×C73—41no—
202C22×C73—41no—
203C22×C73—42no—
204C22×C73—41no—
205C22×C73—41no—
206C22×C73—41no—
207C22×C73—41no—
208C22×C73—81no—
209C22×C73#38011yescharacteristic
210C22×C73C2×#9911yescharacteristic
211C22×C73#41011yescharacteristic
212C4×C73—21no—
213C4×C73—21no—
214C4×C73—42no—
215C4×C73—42no—
216C4×C73—161no—
217Dic73—21no—
218Dic73—21no—
219Dic73—41no—
220Dic73—41no—
221Dic73—41no—
222Dic73—81no—
223Dic73—81no—
224Dic73—161no—
225#12796—731nomaximal, Sylow
226C2×C73—21no—
227C2×C73—21no—
228C2×C73—41no—
229C2×C73—41no—
230C2×C73—41no—
231C2×C73—41no—
232C2×C73#179911yescharacteristic
233C2×C73#212111yescharacteristic
234C2×C73#212311yescharacteristic
235#2322—731no—
236C2×#352—731no—
237C2×#352D7311yescharacteristic
238#1800—731no—
239#1799—731no—
240#1390—731no—
241#1390—731no—
242C73#1279611yescharacteristic, complemented, Sylow
243C2×#123C2×D7311yescharacteristic
244C2×C4×D4—731no—
245C22×C4⋊C4—731no—
246#423—1461no—
247C2×#100—731no—
248C2×#100C2×D7311yescharacteristic
249C2×#99—731no—
250C2×#99C2×D7311yescharacteristic
251#389—1461no—
252#388—1461no—
253#387—1461no—
254#386—731no—
255#385—731no—
256#385—1461no—
257#381—731no—
258#380—1461no—
259C4×C22⋊C4—731no—
260#352—22no—
261#352—1462no—
262#343—1461no—
263#342—1461no—
264C22×D4C22×D7311yescharacteristic
265C23×C4—731no—
266C23×C4—731no—
267C23×C4—731no—
268C23×C4C4×D7311yescharacteristic
269C2×M16—21no—
270C2×M16—1461no—
271#123—21no—
272#123—21no—
273#123Dic14612yes—
274#123C22⋊D7312yes—
275C4×D4—2921no—
276C4×D4—2921no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—2921no—
279C2×C4⋊C4—2921no—
280C2×C4⋊C4—2921no—
281C2×C4⋊C4—731no—
282C2×C4⋊C4—1461no—
283C2×C4⋊C4—1461no—
284C2×C4⋊C4—1461no—
285C2×C22⋊C4—21no—
286C2×C22⋊C4—21no—
287C2×C22⋊C4—731no—
288C2×C22⋊C4—731no—
289C2×C22⋊C4—731no—
290C2×C22⋊C4—1461no—
291C2×C22⋊C4—1461no—
292C2×C22⋊C4C4×D7311yescharacteristic
293C2×C42—731no—
294C2×C42—1461no—
295C4⋊C8—2921no—
296C4⋊C8—2921no—
297#100—21no—
298#100—21no—
299#100—1461no—
300#100—1461no—
301#99—21no—
302#99—21no—
303#99—1461no—
304#99—1461no—
305#95—2921no—
306#95—1461no—
307#95—1461no—
308#95—1461no—
309#95—1461no—
310#95—1461no—
311C24C2×C4×D7311yescharacteristic
312C24D4×D7311yescharacteristic
313C2×D4—41no—
314C2×D4—41no—
315C2×D4C2×C22⋊D7311yescharacteristic
316C2×D4C2×Dic14611yescharacteristic
317C2×D4#1192402711yescharacteristic
318C2×D4#1192402711yescharacteristic
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—2921no—
326C22×C4—2921no—
327C22×C4—5841no—
328C22×C4—5841no—
329C22×C4—5841no—
330C22×C4—731no—
331C22×C4—731no—
332C22×C4—731no—
333C22×C4—731no—
334C22×C4—1461no—
335C22×C4—1461no—
336C22×C4—1461no—
337C22×C4—1461no—
338C22×C4—1461no—
339C22×C4—1461no—
340C22×C4#1192400712yescomplemented
341C22×C4D4×D7311yescharacteristic
342M16—41no—
343M16—41no—
344M16—2921no—
345M16—2921no—
346C2×C8—41no—
347C2×C8—2921no—
348C4⋊C4—22no—
349C4⋊C4—22no—
350C4⋊C4—2921no—
351C4⋊C4—2921no—
352C4⋊C4—2921no—
353C4⋊C4—2922no—
354C4⋊C4—5841no—
355C4⋊C4—5842no—
356C22⋊C4—21no—
357C22⋊C4—22no—
358C22⋊C4—22no—
359C22⋊C4—41no—
360C22⋊C4—41no—
361C22⋊C4—2921no—
362C22⋊C4—2921no—
363C22⋊C4—2921no—
364C22⋊C4—2921no—
365C22⋊C4—2921no—
366C22⋊C4—5841no—
367C22⋊C4#1192400712yescomplemented
368C42—5841no—
369C42—1461no—
370C42—1461no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23—41no—
376C23—41no—
377C23—41no—
378C23—41no—
379C23C22⋊C4×D7311yescharacteristic
380C23C2×#1192400711yescharacteristic
381C23#1455654711yescharacteristic
382C23#1455656311yescharacteristic
383C23#1455656811yescharacteristic
384D4—41no—
385D4—41no—
386D4—41no—
387D4—41no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—41no—
393C2×C4—42no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—42no—
398C2×C4—81no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—2922no—
402C2×C4—2921no—
403C2×C4—2921no—
404C2×C4—2921no—
405C2×C4—2921no—
406C2×C4—2921no—
407C2×C4—2921no—
408C2×C4—2921no—
409C2×C4—2921no—
410C2×C4—2921no—
411C2×C4—2921no—
412C2×C4—5841no—
413C2×C4—5841no—
414C2×C4—5841no—
415C2×C4—5842no—
416C2×C4—5841no—
417C2×C4—5841no—
418C2×C4—5841no—
419C2×C4—731no—
420C2×C4—731no—
421C2×C4—1461no—
422C2×C4—1461no—
423C2×C4—1461no—
424C2×C4#1455656311yescharacteristic
425C2×C4#1455656811yescharacteristic
426C8—81no—
427C8—5841no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—21no—
434C22—41no—
435C22—41no—
436C22—41no—
437C22—41no—
438C22—42no—
439C22—41no—
440C22—41no—
441C22—41no—
442C22—41no—
443C22—81no—
444C22#1798844511yescharacteristic
445C22#99×D7311yescharacteristic
446C22#1798855911yescharacteristic
447C4—21no—
448C4—21no—
449C4—42no—
450C4—42no—
451C4—161no—
452C4—2921no—
453C4—2921no—
454C4—2921no—
455C4—5841no—
456C4—5841no—
457C4—11681no—
458C4—1461no—
459C4—1461no—
460C2—21no—
461C2—21no—
462C2—41no—
463C2—41no—
464C2—41no—
465C2—41no—
466C2#2260991711yescharacteristic
467C2#2261157611yescharacteristic
468C2#2261158311yescharacteristic
subgroupG/H
centerC22#99×D73
commutatorC24×C73C22×C4
FrattiniC22×D4C22×D73
FittingC2×#352×C73C2
socleC22×C73C2×#99
prime radical, cosocleC22×D4×C73C23
hypercenterC2×#352D73
hypocenterC73#12796
LCSG ▷ C24×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ C2×#352

Complements

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

NH ≅ G/Ntype
9. C22×C4⋊Dic73C4semidirect
18. #17988462C4semidirect
19. #17988450C4semidirect
21. C22⋊C4×Dic73C4semidirect
123. C22×C4×C73C4⋊C4semidirect
135. C22⋊C4×C73C4⋊C4semidirect
242. C73#12796semidirect
340. C22×C4#11924007semidirect
367. C22⋊C4#11924007semidirect

Maximal subgroups

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

subgroupindex
#226126032
C2×#179883772
#226099212
#226099172
#226080502
#226080482
C2×#352×C732
#1279673