Finite Group Databasebeta

← #33004277

Subgroups — #33004277

Order 18688.

624 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 624 conjugacy classes of subgroups — too many to draw. Only its 107 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
1C2×#17988377C212yesmaximal, complemented
2C2×#17988374C212yesmaximal
3#22609919C211yesmaximal, characteristic, complemented
4C4×#14556508C211yesmaximal, characteristic, complemented
5C73×#2534C211yesmaximal, characteristic
6C2×#14556508C2212yes—
7C2×#14556506C2212yes—
8C4×#11924012—21no—
9#17988447—21no—
10#17988377—24no—
11#17988374—24no—
12#17988356—21no—
13#17988354—21no—
14C2×#123×C73C2211yescharacteristic
15C2×#120×C73C2211yescharacteristic
16C2×C4×D4×C73C2211yescharacteristic
17C22×C22⋊C4×C73C411yescharacteristic, complemented
18#396×C73—21no—
19#394×C73—21no—
20#388×C73—21no—
21#386×C73—21no—
22#381×C73C411yescharacteristic, complemented
23#380×C73C411yescharacteristic, complemented
24#379×C73C411yescharacteristic, complemented
25C2×#11924012—22no—
26C2×C4⋊Dic73—41no—
27C2×C22⋊Dic73—22no—
28#14556508—22no—
29#14556508D414yes—
30#14556506—22no—
31#14556506—22no—
32#14556502—41no—
33C4×#10596831—41no—
34#14556498—41no—
35C25×C73C2×C411yescharacteristic
36C22×D4×C73—21no—
37C22×D4×C73C2311yescharacteristic
38C23×C4×C73C2×C411yescharacteristic
39C23×C4×C73C2×C411yescharacteristic
40C23×C4×C73D411yescharacteristic
41#123×C73—22no—
42#123×C73—22no—
43#120×C73—22no—
44#120×C73—22no—
45C4×D4×C73—41no—
46C4×D4×C73—41no—
47C2×C4⋊C4×C73—21no—
48C2×C4⋊C4×C73—21no—
49C2×C4⋊C4×C73D411yescharacteristic
50C2×C22⋊C4×C73—21no—
51C2×C22⋊C4×C73—21no—
52C2×C22⋊C4×C73—21no—
53C2×C22⋊C4×C73—21no—
54C2×C22⋊C4×C73—21no—
55C2×C22⋊C4×C73—21no—
56C2×C22⋊C4×C73—41no—
57C2×C22⋊C4×C73—41no—
58C2×C22⋊C4×C73C2×C411yescharacteristic
59C2×C22⋊C4×C73C2×C411yescharacteristic
60C2×C22⋊C4×C73C2×C411yescharacteristic
61C2×C22⋊C4×C73D411yescharacteristic
62C2×C42×C73D411yescharacteristic
63#95×C73—21no—
64#95×C73—21no—
65#95×C73—21no—
66C22×Dic73—42no—
67#11924012—24no—
68#11924012—42no—
69C2×#10596831—42no—
70C4⋊Dic73—82no—
71C22⋊Dic73—42no—
72C22⋊Dic73—42no—
73C24×C73—21no—
74C24×C73—22no—
75C24×C73—41no—
76C24×C73—42no—
77C24×C73—41no—
78C24×C73—81no—
79C24×C73C22×C411yescharacteristic
80C24×C73C2×D411yescharacteristic
81C2×D4×C73—41no—
82C2×D4×C73—41no—
83C2×D4×C73—42no—
84C2×D4×C73—41no—
85C2×D4×C73—41no—
86C2×D4×C73—81no—
87C2×D4×C73C2×D412yes—
88C2×D4×C73#4112yes—
89C22×C4×C73—21no—
90C22×C4×C73—21no—
91C22×C4×C73—21no—
92C22×C4×C73—21no—
93C22×C4×C73—21no—
94C22×C4×C73—21no—
95C22×C4×C73—21no—
96C22×C4×C73—22no—
97C22×C4×C73—41no—
98C22×C4×C73—41no—
99C22×C4×C73—81no—
100C22×C4×C73—81no—
101C22×C4×C73C22⋊C411yescharacteristic
102C22×C4×C73C22⋊C411yescharacteristic
103C22×C4×C73C22⋊C411yescharacteristic
104C22×C4×C73C22⋊C411yescharacteristic
105C22×C4×C73C2×D411yescharacteristic
106C4⋊C4×C73—41no—
107C4⋊C4×C73—42no—
108C4⋊C4×C73—81no—
109C22⋊C4×C73—22no—
110C22⋊C4×C73—22no—
111C22⋊C4×C73—41no—
112C22⋊C4×C73—41no—
113C22⋊C4×C73—42no—
114C22⋊C4×C73—42no—
115C22⋊C4×C73—41no—
116C22⋊C4×C73—41no—
117C22⋊C4×C73—41no—
118C22⋊C4×C73—41no—
119C22⋊C4×C73—42no—
120C22⋊C4×C73—81no—
121C42×C73—21no—
122C42×C73—21no—
123C23×C73—21no—
124C23×C73—21no—
125C23×C73—21no—
126C23×C73—21no—
127C23×C73—22no—
128C23×C73—41no—
129C23×C73—41no—
130C23×C73—41no—
131C23×C73—41no—
132C23×C73—41no—
133C23×C73—41no—
134C23×C73—41no—
135C23×C73—42no—
136C23×C73—42no—
137C23×C73—41no—
138C23×C73—41no—
139C23×C73—41no—
140C23×C73—82no—
141C23×C73—84no—
142C23×C73—84no—
143C23×C73—82no—
144C23×C73C2×C22⋊C411yescharacteristic
145C23×C73C4×D412yes—
146C23×C73#12411yescharacteristic
147C23×C73#12711yescharacteristic
148D4×C73—42no—
149D4×C73—42no—
150D4×C73—82no—
151D4×C73—82no—
152C2×C4×C73—21no—
153C2×C4×C73—21no—
154C2×C4×C73—21no—
155C2×C4×C73—21no—
156C2×C4×C73—21no—
157C2×C4×C73—41no—
158C2×C4×C73—42no—
159C2×C4×C73—41no—
160C2×C4×C73—41no—
161C2×C4×C73—41no—
162C2×C4×C73—41no—
163C2×C4×C73—41no—
164C2×C4×C73—41no—
165C2×C4×C73—41no—
166C2×C4×C73—41no—
167C2×C4×C73—41no—
168C2×C4×C73—41no—
169C2×C4×C73—41no—
170C2×C4×C73—41no—
171C2×C4×C73—81no—
172C2×C4×C73—81no—
173C2×C4×C73—81no—
174C2×C4×C73—82no—
175C2×C4×C73#9911yescharacteristic
176C2×C4×C73#9911yescharacteristic
177C2×C4×C73#12112yes—
178C2×Dic73—82no—
179C2×Dic73—82no—
180C2×Dic73—82no—
181#10596831—44no—
182C22×C73—21no—
183C22×C73—21no—
184C22×C73—21no—
185C22×C73—22no—
186C22×C73—21no—
187C22×C73—41no—
188C22×C73—41no—
189C22×C73—42no—
190C22×C73—42no—
191C22×C73—42no—
192C22×C73—42no—
193C22×C73—41no—
194C22×C73—41no—
195C22×C73—41no—
196C22×C73—41no—
197C22×C73—42no—
198C22×C73—81no—
199C22×C73—81no—
200C22×C73—81no—
201C22×C73—84no—
202C22×C73—84no—
203C22×C73—81no—
204C22×C73—81no—
205C22×C73#39011yescharacteristic
206C22×C73C2×#9911yescharacteristic
207C22×C73#41011yescharacteristic
208C4×C73—21no—
209C4×C73—21no—
210C4×C73—22no—
211C4×C73—41no—
212C4×C73—41no—
213C4×C73—41no—
214C4×C73—81no—
215C4×C73—81no—
216C4×C73—81no—
217Dic73—162no—
218#13219—731nomaximal, Sylow
219C2×C73—21no—
220C2×C73—21no—
221C2×C73—41no—
222C2×C73—42no—
223C2×C73—41no—
224C2×C73—82no—
225C2×C73#196411yescharacteristic
226C2×C73#212011yescharacteristic
227C2×C73#212111yescharacteristic
228#2534D7311yescharacteristic
229C2×#352—732no—
230C2×AΣL(1,𝔽16)—732no—
231#1799—731no—
232C4×#100—731no—
233C73#1321911yescharacteristic, complemented, Sylow
234C2×#123C2×D7311yescharacteristic
235C2×#120C2×D7311yescharacteristic
236C2×C4×D4C2×D7311yescharacteristic
237C22×C22⋊C4Dic7311yescharacteristic, complemented
238C2×#100—732no—
239C2×#99—732no—
240C4×M16—1461no—
241#396—21no—
242#394—21no—
243#388—21no—
244#386—21no—
245#381Dic7311yescharacteristic, complemented
246#380—1461no—
247#380Dic7311yescharacteristic, complemented
248#379Dic7311yescharacteristic, complemented
249#352—1464no—
250AΣL(1,𝔽16)—1464no—
251#343—1461no—
252#342—1461no—
253C25C2×Dic7311yescharacteristic
254C22×D4—21no—
255C22×D4C22×D7311yescharacteristic
256C23×C4C2×Dic7311yescharacteristic
257C23×C4C2×Dic7311yescharacteristic
258C23×C4C22⋊D7311yescharacteristic
259C2×M16—1462no—
260#123—22no—
261#123—22no—
262#120—22no—
263#120—22no—
264C4×D4—41no—
265C4×D4—41no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—2921no—
269C2×C4⋊C4C22⋊D7311yescharacteristic
270C2×C22⋊C4—21no—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—41no—
277C2×C22⋊C4—41no—
278C2×C22⋊C4—1462no—
279C2×C22⋊C4C2×Dic7311yescharacteristic
280C2×C22⋊C4C2×Dic7311yescharacteristic
281C2×C22⋊C4C2×Dic7311yescharacteristic
282C2×C22⋊C4C22⋊D7311yescharacteristic
283C2×C42C22⋊D7311yescharacteristic
284#100—734no—
285#100—1462no—
286#99—1462no—
287#99—1462no—
288#97—2921no—
289C4×C8—2921no—
290#95—21no—
291#95—21no—
292#95—21no—
293#95—2921no—
294C24—21no—
295C24—22no—
296C24—41no—
297C24—42no—
298C24—41no—
299C24—81no—
300C24C22×Dic7311yescharacteristic
301C24C2×C22⋊D7311yescharacteristic
302C2×D4—41no—
303C2×D4—41no—
304C2×D4—42no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—81no—
308C2×D4D4×D7312yes—
309C2×D4#1192402712yes—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—22no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—81no—
321C22×C4—81no—
322C22×C4—2922no—
323C22×C4C22⋊Dic7311yescharacteristic
324C22×C4C22⋊Dic7311yescharacteristic
325C22×C4C22⋊Dic7311yescharacteristic
326C22×C4C22⋊Dic7311yescharacteristic
327C22×C4C2×C22⋊D7311yescharacteristic
328M16—2922no—
329M16—1464no—
330C2×C8—2922no—
331C4⋊C4—41no—
332C4⋊C4—42no—
333C4⋊C4—81no—
334C4⋊C4—5842no—
335C22⋊C4—22no—
336C22⋊C4—22no—
337C22⋊C4—41no—
338C22⋊C4—41no—
339C22⋊C4—42no—
340C22⋊C4—42no—
341C22⋊C4—41no—
342C22⋊C4—41no—
343C22⋊C4—41no—
344C22⋊C4—41no—
345C22⋊C4—42no—
346C22⋊C4—81no—
347C22⋊C4—2922no—
348C22⋊C4—2922no—
349C42—21no—
350C42—21no—
351C23—21no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—22no—
356C23—41no—
357C23—41no—
358C23—41no—
359C23—41no—
360C23—41no—
361C23—41no—
362C23—41no—
363C23—42no—
364C23—42no—
365C23—41no—
366C23—41no—
367C23—41no—
368C23—82no—
369C23—84no—
370C23—84no—
371C23—82no—
372C23C2×C22⋊Dic7311yescharacteristic
373C23D4×Dic7312yes—
374C23#1455657811yescharacteristic
375C23#1455658711yescharacteristic
376D4—42no—
377D4—42no—
378D4—82no—
379D4—82no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—41no—
386C2×C4—42no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—82no—
403C2×C4—5842no—
404C2×C4—5842no—
405C2×C4—5842no—
406C2×C4#1455650611yescharacteristic
407C2×C4#1455650611yescharacteristic
408C2×C4#1455656212yes—
409C8—2924no—
410C22—21no—
411C22—21no—
412C22—22no—
413C22—21no—
414C22—21no—
415C22—41no—
416C22—41no—
417C22—42no—
418C22—42no—
419C22—42no—
420C22—42no—
421C22—41no—
422C22—41no—
423C22—41no—
424C22—41no—
425C22—42no—
426C22—81no—
427C22—81no—
428C22—81no—
429C22—84no—
430C22—84no—
431C22—81no—
432C22—81no—
433C22#1798849011yescharacteristic
434C22C2×#1455650611yescharacteristic
435C22#1798856111yescharacteristic
436C4—22no—
437C4—21no—
438C4—21no—
439C4—41no—
440C4—41no—
441C4—41no—
442C4—81no—
443C4—81no—
444C4—81no—
445C4—11682no—
446C2—21no—
447C2—21no—
448C2—41no—
449C2—42no—
450C2—41no—
451C2—82no—
452C2#2261076811yescharacteristic
453C2#2261157211yescharacteristic
454C2#2261157711yescharacteristic
subgroupG/H
centerC22C2×#14556506
commutatorC24×C73C22×C4
FrattiniC22×D4C22×D73
FittingC73×#2534C2
socleC22×C73C2×#99
prime radicalC2×#120×C73C22
cosocleC22×D4×C73C23
hypercenter#2534D73
hypocenterC73#13219
LCSG ▷ C24×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2534

Complements

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

NH ≅ G/Ntype
1. C2×#17988377C2semidirect
3. #22609919C2semidirect
4. C4×#14556508C2semidirect
17. C22×C22⋊C4×C73C4semidirect
22. #381×C73C4semidirect
23. #380×C73C4semidirect
24. #379×C73C4semidirect
233. C73#13219semidirect
237. C22×C22⋊C4Dic73semidirect
245. #381Dic73semidirect
247. #380Dic73semidirect
248. #379Dic73semidirect

Maximal subgroups

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

subgroupindex
C2×#179883772
C2×#179883742
#226099192
C4×#145565082
C73×#25342
#1321973