Finite Group Databasebeta

← #33004290

Subgroups — #33004290

Order 18688.

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

normal subgroups only

This group has 628 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#22609923C211yesmaximal, characteristic, complemented
4C4×#14556506C211yesmaximal, characteristic, complemented
5C73×#2534C211yesmaximal, characteristic
6#17988611—21no—
7C2×#14556508C2212yes—
8C2×#14556506C2212yes—
9C4×C22⋊Dic73—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×C22⋊Dic73—22no—
27C2×C4×Dic73—41no—
28C4⋊#10596831—41no—
29C4⋊#10596831—41no—
30#14556508—22no—
31#14556508—22no—
32#14556506—22no—
33#14556506D414yes—
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—42no—
68#11924012—42no—
69C2×#10596831—42no—
70C22⋊Dic73—24no—
71C22⋊Dic73—42no—
72C4×Dic73—82no—
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—42no—
85C2×D4×C73—81no—
86C2×D4×C73C2×D412yes—
87C2×D4×C73#4112yes—
88C22×C4×C73—21no—
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—22no—
96C22×C4×C73—41no—
97C22×C4×C73—41no—
98C22×C4×C73—81no—
99C22×C4×C73—81no—
100C22×C4×C73C22⋊C411yescharacteristic
101C22×C4×C73C22⋊C411yescharacteristic
102C22×C4×C73C22⋊C411yescharacteristic
103C22×C4×C73C22⋊C411yescharacteristic
104C22×C4×C73C2×D411yescharacteristic
105C4⋊C4×C73—41no—
106C4⋊C4×C73—41no—
107C4⋊C4×C73—41no—
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—42no—
116C22⋊C4×C73—41no—
117C22⋊C4×C73—41no—
118C22⋊C4×C73—41no—
119C22⋊C4×C73—41no—
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#12112yes—
147D4×C73—42no—
148D4×C73—42no—
149D4×C73—84no—
150C2×C4×C73—21no—
151C2×C4×C73—21no—
152C2×C4×C73—21no—
153C2×C4×C73—21no—
154C2×C4×C73—21no—
155C2×C4×C73—41no—
156C2×C4×C73—42no—
157C2×C4×C73—41no—
158C2×C4×C73—41no—
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—81no—
170C2×C4×C73—81no—
171C2×C4×C73—81no—
172C2×C4×C73—82no—
173C2×C4×C73#9911yescharacteristic
174C2×C4×C73#9911yescharacteristic
175C2×C4×C73#12411yescharacteristic
176C2×C4×C73#12711yescharacteristic
177C2×Dic73—44no—
178C2×Dic73—82no—
179C2×Dic73—82no—
180#10596831—82no—
181C22×C73—21no—
182C22×C73—21no—
183C22×C73—22no—
184C22×C73—22no—
185C22×C73—42no—
186C22×C73—42no—
187C22×C73—42no—
188C22×C73—42no—
189C22×C73—42no—
190C22×C73—41no—
191C22×C73—41no—
192C22×C73—41no—
193C22×C73—41no—
194C22×C73—42no—
195C22×C73—81no—
196C22×C73—81no—
197C22×C73—81no—
198C22×C73—84no—
199C22×C73—84no—
200C22×C73—81no—
201C22×C73—81no—
202C22×C73#39011yescharacteristic
203C22×C73C2×#9911yescharacteristic
204C22×C73#41011yescharacteristic
205C4×C73—21no—
206C4×C73—21no—
207C4×C73—21no—
208C4×C73—21no—
209C4×C73—41no—
210C4×C73—41no—
211C4×C73—41no—
212C4×C73—81no—
213C4×C73—81no—
214C4×C73—81no—
215Dic73—84no—
216#13222—731nomaximal, Sylow
217C2×C73—21no—
218C2×C73—21no—
219C2×C73—42no—
220C2×C73—42no—
221C2×C73—82no—
222C2×C73#196411yescharacteristic
223C2×C73#212011yescharacteristic
224C2×C73#212111yescharacteristic
225#2534D7311yescharacteristic
226C2×#352—732no—
227C2×AΣL(1,𝔽16)—732no—
228#1800—731no—
229C4×#99—731no—
230C73#1322211yescharacteristic, complemented, Sylow
231C2×#123C2×D7311yescharacteristic
232C2×#120C2×D7311yescharacteristic
233C2×C4×D4C2×D7311yescharacteristic
234C22×C22⋊C4Dic7311yescharacteristic, complemented
235#423—1461no—
236C2×#100—732no—
237C2×#99—732no—
238#396—21no—
239#394—21no—
240#388—21no—
241#386—21no—
242#381Dic7311yescharacteristic, complemented
243#380Dic7311yescharacteristic, complemented
244#379Dic7311yescharacteristic, complemented
245C4×C22⋊C4—1461no—
246#352—1464no—
247AΣL(1,𝔽16)—1464no—
248#343—1461no—
249#342—1461no—
250C25C2×Dic7311yescharacteristic
251C22×D4—21no—
252C22×D4C22×D7311yescharacteristic
253C23×C4C2×Dic7311yescharacteristic
254C23×C4C2×Dic7311yescharacteristic
255C23×C4C22⋊D7311yescharacteristic
256C2×M16—1462no—
257#123—22no—
258#123—22no—
259#120—22no—
260#120—22no—
261C4×D4—41no—
262C4×D4—41no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4C22⋊D7311yescharacteristic
266C2×C22⋊C4—21no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—41no—
273C2×C22⋊C4—41no—
274C2×C22⋊C4—1462no—
275C2×C22⋊C4C2×Dic7311yescharacteristic
276C2×C22⋊C4C2×Dic7311yescharacteristic
277C2×C22⋊C4C2×Dic7311yescharacteristic
278C2×C22⋊C4C22⋊D7311yescharacteristic
279C2×C42—2921no—
280C2×C42C22⋊D7311yescharacteristic
281C4⋊C8—2921no—
282C4⋊C8—2921no—
283#100—1462no—
284#100—1462no—
285#99—734no—
286#99—1462no—
287#95—21no—
288#95—21no—
289#95—21no—
290#95—2921no—
291C24—21no—
292C24—22no—
293C24—41no—
294C24—42no—
295C24—41no—
296C24—81no—
297C24C22×Dic7311yescharacteristic
298C24C2×C22⋊D7311yescharacteristic
299C2×D4—41no—
300C2×D4—41no—
301C2×D4—42no—
302C2×D4—42no—
303C2×D4—81no—
304C2×D4D4×D7312yes—
305C2×D4#1192402712yes—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—22no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—81no—
317C22×C4—81no—
318C22×C4—2922no—
319C22×C4C22⋊Dic7311yescharacteristic
320C22×C4C22⋊Dic7311yescharacteristic
321C22×C4C22⋊Dic7311yescharacteristic
322C22×C4C22⋊Dic7311yescharacteristic
323C22×C4C2×C22⋊D7311yescharacteristic
324M16—2922no—
325M16—2922no—
326C2×C8—2922no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—81no—
331C22⋊C4—22no—
332C22⋊C4—22no—
333C22⋊C4—41no—
334C22⋊C4—41no—
335C22⋊C4—42no—
336C22⋊C4—42no—
337C22⋊C4—42no—
338C22⋊C4—41no—
339C22⋊C4—41no—
340C22⋊C4—41no—
341C22⋊C4—41no—
342C22⋊C4—81no—
343C22⋊C4—2922no—
344C22⋊C4—1464no—
345C42—21no—
346C42—21no—
347C42—5842no—
348C23—21no—
349C23—21no—
350C23—21no—
351C23—21no—
352C23—22no—
353C23—41no—
354C23—41no—
355C23—41no—
356C23—41no—
357C23—41no—
358C23—41no—
359C23—41no—
360C23—42no—
361C23—42no—
362C23—41no—
363C23—41no—
364C23—41no—
365C23—82no—
366C23—84no—
367C23—84no—
368C23—82no—
369C23C2×C22⋊Dic7311yescharacteristic
370C23D4×Dic7312yes—
371C23#1455656212yes—
372D4—42no—
373D4—42no—
374D4—84no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—41no—
381C2×C4—42no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—81no—
395C2×C4—81no—
396C2×C4—81no—
397C2×C4—82no—
398C2×C4—2924no—
399C2×C4—5842no—
400C2×C4—5842no—
401C2×C4#1455650611yescharacteristic
402C2×C4#1455650611yescharacteristic
403C2×C4#1455657811yescharacteristic
404C2×C4#1455658711yescharacteristic
405C8—5842no—
406C22—21no—
407C22—21no—
408C22—22no—
409C22—22no—
410C22—42no—
411C22—42no—
412C22—42no—
413C22—42no—
414C22—42no—
415C22—41no—
416C22—41no—
417C22—41no—
418C22—41no—
419C22—42no—
420C22—81no—
421C22—81no—
422C22—81no—
423C22—84no—
424C22—84no—
425C22—81no—
426C22—81no—
427C22#1798849011yescharacteristic
428C22C2×#1455650611yescharacteristic
429C22#1798856111yescharacteristic
430C4—21no—
431C4—21no—
432C4—21no—
433C4—21no—
434C4—41no—
435C4—41no—
436C4—41no—
437C4—81no—
438C4—81no—
439C4—81no—
440C4—5844no—
441C2—21no—
442C2—21no—
443C2—42no—
444C2—42no—
445C2—82no—
446C2#2261076811yescharacteristic
447C2#2261157211yescharacteristic
448C2#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#13222
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. #22609923C2semidirect
4. C4×#14556506C2semidirect
17. C22×C22⋊C4×C73C4semidirect
22. #381×C73C4semidirect
23. #380×C73C4semidirect
24. #379×C73C4semidirect
230. C73#13222semidirect
234. C22×C22⋊C4Dic73semidirect
242. #381Dic73semidirect
243. #380Dic73semidirect
244. #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
#226099232
C4×#145565062
C73×#25342
#1322273