Finite Group Databasebeta

← #32982027

Subgroups — #32982027

Order 18688.

684 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 684 conjugacy classes of subgroups — too many to draw. Only its 131 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#22615736C211yesmaximal, characteristic
2#22608502C212yesmaximal, complemented
3C2×#17988311C211yesmaximal, characteristic
4#22607798C212yesmaximal, complemented
5C73×#2315C211yesmaximal, characteristic
6C2×#14556570C2211yescharacteristic
7C2×C22⋊#10596831C2212yes—
8#17988523C412yescomplemented
9#17988510—22no—
10#17988499C412yescomplemented
11#17988460—21no—
12#17988447C2211yescharacteristic
13#17988443—21no—
14#17988323—24no—
15#17988311—22no—
16C22×C22⋊C4×C73—21no—
17C22×C22⋊C4×C73C2211yescharacteristic
18#385×C73—24no—
19#380×C73—21no—
20#380×C73C2212yes—
21C22×Dic146C2×C411yescharacteristic, complemented
22C22×#10596831—22no—
23#14556570—22no—
24#14556570—22no—
25C2×C4⋊Dic73—21no—
26C2×C4⋊Dic73C2×C411yescharacteristic, complemented
27C2×#11924007—22no—
28C2×#11924007—21no—
29C2×C22⋊Dic73—22no—
30C2×C22⋊Dic73—22no—
31C2×C22⋊Dic73—21no—
32C2×C22⋊Dic73—21no—
33C22⋊#10596831—22no—
34C22⋊#10596831—22no—
35#14556498—22no—
36#14556498C2×C412yescomplemented
37C25×C73D411yescharacteristic
38C23×C4×C73—22no—
39C23×C4×C73D411yescharacteristic
40C23×C4×C73C2311yescharacteristic
41C2×C4⋊C4×C73—22no—
42C2×C4⋊C4×C73—22no—
43C2×C22⋊C4×C73—22no—
44C2×C22⋊C4×C73—22no—
45C2×C22⋊C4×C73—22no—
46C2×C22⋊C4×C73—22no—
47C2×C22⋊C4×C73—42no—
48C2×C22⋊C4×C73—42no—
49C2×C22⋊C4×C73C2×C412yes—
50C2×C22⋊C4×C73D412yes—
51#95×C73—22no—
52#95×C73D414yes—
53C2×Dic146—22no—
54C2×Dic146—81no—
55C22×Dic73—21no—
56C22×Dic73—22no—
57C22×Dic73—21no—
58C2×#10596831—42no—
59C2×#10596831—82no—
60C4⋊Dic73—22no—
61C4⋊Dic73—81no—
62#11924007—42no—
63#11924007—82no—
64C22⋊Dic73—41no—
65C22⋊Dic73—41no—
66C22⋊Dic73—41no—
67C22⋊Dic73—41no—
68C22⋊Dic73—82no—
69C22⋊Dic73—82no—
70C24×C73—21no—
71C24×C73—21no—
72C24×C73—21no—
73C24×C73—21no—
74C24×C73—81no—
75C24×C73—81no—
76C24×C73C22⋊C412yes—
77C24×C73SD1612yes—
78C24×C73Q1612yes—
79C24×C73C2×D411yescharacteristic
80C22×C4×C73—21no—
81C22×C4×C73—21no—
82C22×C4×C73—24no—
83C22×C4×C73—44no—
84C22×C4×C73—81no—
85C22×C4×C73—81no—
86C22×C4×C73—82no—
87C22×C4×C73—82no—
88C22×C4×C73C22⋊C412yes—
89C22×C4×C73C22×C411yescharacteristic
90C22×C4×C73C2×D412yes—
91C22×C4×C73C2×D411yescharacteristic
92C22×C4×C73#4112yes—
93C4⋊C4×C73—82no—
94C4⋊C4×C73—82no—
95C22⋊C4×C73—24no—
96C22⋊C4×C73—24no—
97C22⋊C4×C73—44no—
98C22⋊C4×C73—42no—
99C22⋊C4×C73—42no—
100C23×C73—22no—
101C23×C73—22no—
102C23×C73—22no—
103C23×C73—22no—
104C23×C73—22no—
105C23×C73—22no—
106C23×C73—41no—
107C23×C73—41no—
108C23×C73—42no—
109C23×C73—42no—
110C23×C73—42no—
111C23×C73—42no—
112C23×C73—81no—
113C23×C73—81no—
114C23×C73—82no—
115C23×C73—82no—
116C23×C73—82no—
117C23×C73—82no—
118C23×C73#10314yes—
119C23×C73C2×C22⋊C411yescharacteristic
120C23×C73#12011yescharacteristic
121C23×C73#12311yescharacteristic
122C23×C73C2×SD1611yescharacteristic
123C23×C73C2×Q1611yescharacteristic
124C23×C73AGL(1,ℤ/8)11yescharacteristic
125C23×C73#13711yescharacteristic
126C2×C4×C73—22no—
127C2×C4×C73—42no—
128C2×C4×C73—41no—
129C2×C4×C73—41no—
130C2×C4×C73—42no—
131C2×C4×C73—42no—
132C2×C4×C73—42no—
133C2×C4×C73—44no—
134C2×C4×C73—44no—
135C2×C4×C73—44no—
136C2×C4×C73—81no—
137C2×C4×C73—88no—
138C2×C4×C73C4×D412yes—
139C2×C4×C73#12112yes—
140C2×Dic73—42no—
141C2×Dic73—42no—
142C2×Dic73—42no—
143C2×Dic73—42no—
144C2×Dic73—41no—
145C2×Dic73—41no—
146C2×Dic73—42no—
147Dic146—82no—
148#10596831—82no—
149C22×C73—21no—
150C22×C73—21no—
151C22×C73—21no—
152C22×C73—21no—
153C22×C73—21no—
154C22×C73—21no—
155C22×C73—24no—
156C22×C73—41no—
157C22×C73—41no—
158C22×C73—41no—
159C22×C73—41no—
160C22×C73—44no—
161C22×C73—44no—
162C22×C73—44no—
163C22×C73—42no—
164C22×C73—42no—
165C22×C73—82no—
166C22×C73—82no—
167C22×C73—82no—
168C22×C73#38611yescharacteristic
169C22×C73C2×#10311yescharacteristic
170C22×C73#41711yescharacteristic
171C22×C73#45011yescharacteristic
172C22×C73#45111yescharacteristic
173C22×C73#48311yescharacteristic
174C22×C73#48411yescharacteristic
175C4×C73—22no—
176C4×C73—22no—
177C4×C73—42no—
178C4×C73—84no—
179Dic73—82no—
180Dic73—81no—
181Dic73—81no—
182#9199—731nomaximal, Sylow
183C2×C73—21no—
184C2×C73—21no—
185C2×C73—21no—
186C2×C73—21no—
187C2×C73—21no—
188C2×C73—21no—
189C2×C73—44no—
190C2×C73#187211yescharacteristic
191C2×C73#191111yescharacteristic
192C2×C73#191311yescharacteristic
193#2626—731no—
194#2315D7311yescharacteristic
195#1520—732no—
196C2×#323—731no—
197#1289—732no—
198C73#919911yescharacteristic, complemented, Sylow
199C2×C22⋊Q8—731no—
200C22×C22⋊C4—21no—
201C22×C22⋊C4C2×D7311yescharacteristic
202C2×C22⋊C8—732no—
203#399—732no—
204#396—1462no—
205#393—732no—
206#385—24no—
207#385—1461no—
208#380—21no—
209#380—731no—
210#380C2×D7312yes—
211#379—1461no—
212#328—1464no—
213#323—1462no—
214C25C22⋊D7311yescharacteristic
215C22×Q8—731no—
216C23×C4—22no—
217C23×C4C22⋊D7311yescharacteristic
218C23×C4C22×D7311yescharacteristic
219C22×C8—1462no—
220C22⋊Q8—1462no—
221C22⋊Q8—1462no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—731no—
225C2×C4⋊C4—1461no—
226C2×C4⋊C4—1462no—
227C2×C4⋊C4—1461no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—42no—
233C2×C22⋊C4—42no—
234C2×C22⋊C4—1461no—
235C2×C22⋊C4—1461no—
236C2×C22⋊C4—1462no—
237C2×C22⋊C4—1462no—
238C2×C22⋊C4C4×D7312yes—
239C2×C22⋊C4D29212yes—
240C22⋊C8—1462no—
241C22⋊C8—1462no—
242#95—22no—
243#95—732no—
244#95—1462no—
245#95C22⋊D7314yes—
246C24—21no—
247C24—21no—
248C24—21no—
249C24—21no—
250C24—81no—
251C24—81no—
252C24#1192400512yes—
253C24#1192401612yes—
254C24#1192401812yes—
255C24C2×C22⋊D7311yescharacteristic
256C2×Q8—5841no—
257C2×Q8—1462no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—24no—
261C22×C4—44no—
262C22×C4—81no—
263C22×C4—81no—
264C22×C4—82no—
265C22×C4—82no—
266C22×C4—1462no—
267C22×C4—1461no—
268C22×C4—1461no—
269C22×C4#1192400512yes—
270C22×C4C2×C4×D7311yescharacteristic
271C22×C4C2×C22⋊D7312yes—
272C22×C4C2×D29211yescharacteristic
273C22×C4#1192402812yes—
274C2×C8—2922no—
275C2×C8—5842no—
276C4⋊C4—82no—
277C4⋊C4—82no—
278C4⋊C4—2922no—
279C4⋊C4—5842no—
280C4⋊C4—5841no—
281C4⋊C4—1462no—
282C22⋊C4—24no—
283C22⋊C4—24no—
284C22⋊C4—44no—
285C22⋊C4—42no—
286C22⋊C4—42no—
287C22⋊C4—2921no—
288C22⋊C4—2921no—
289C22⋊C4—2921no—
290C22⋊C4—2921no—
291C22⋊C4—5842no—
292C22⋊C4—5842no—
293C23—22no—
294C23—22no—
295C23—22no—
296C23—22no—
297C23—22no—
298C23—22no—
299C23—41no—
300C23—41no—
301C23—42no—
302C23—42no—
303C23—42no—
304C23—42no—
305C23—81no—
306C23—81no—
307C23—82no—
308C23—82no—
309C23—82no—
310C23—82no—
311C23#1455651414yes—
312C23C2×#1192400511yescharacteristic
313C23#1455656011yescharacteristic
314C23#1455657411yescharacteristic
315C23C2×#1192401611yescharacteristic
316C23C2×#1192401811yescharacteristic
317C23#1455661711yescharacteristic
318C23#1455662211yescharacteristic
319Q8—5842no—
320C2×C4—22no—
321C2×C4—42no—
322C2×C4—41no—
323C2×C4—42no—
324C2×C4—41no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—44no—
328C2×C4—44no—
329C2×C4—44no—
330C2×C4—81no—
331C2×C4—88no—
332C2×C4—2922no—
333C2×C4—2922no—
334C2×C4—2922no—
335C2×C4—2922no—
336C2×C4—2921no—
337C2×C4—2921no—
338C2×C4—2922no—
339C2×C4C4×C22⋊D7312yes—
340C2×C4#1455656512yes—
341C8—5842no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—24no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—44no—
354C22—44no—
355C22—44no—
356C22—42no—
357C22—42no—
358C22—82no—
359C22—82no—
360C22—82no—
361C22#1798846611yescharacteristic
362C22C2×#1455651411yescharacteristic
363C22#1798858611yescharacteristic
364C22#1798873811yescharacteristic
365C22#1798874311yescharacteristic
366C22#1798891711yescharacteristic
367C22#1798892211yescharacteristic
368C4—22no—
369C4—22no—
370C4—42no—
371C4—84no—
372C4—5842no—
373C4—5841no—
374C4—5841no—
375C2—21no—
376C2—21no—
377C2—21no—
378C2—21no—
379C2—21no—
380C2—21no—
381C2—44no—
382C2#2261023811yescharacteristic
383C2#2261046311yescharacteristic
384C2#2261047311yescharacteristic
subgroupG/H
centerC22#17988466
commutatorC22×C4×C73C22×C4
FrattiniC23×C4C22×D73
FittingC73×#2315C2
socleC22×C73#386
prime radicalC25×C73D4
cosocleC23×C4×C73C23
hypercenter#2315D73
hypocenterC73#9199
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2315

Complements

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

NH ≅ G/Ntype
2. #22608502C2semidirect
4. #22607798C2semidirect
8. #17988523C4semidirect
10. #17988499C4semidirect
21. C22×Dic146C2×C4semidirect
26. C2×C4⋊Dic73C2×C4semidirect
36. #14556498C2×C4semidirect
198. C73#9199semidirect

Maximal subgroups

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

subgroupindex
#226157362
#226085022
C2×#179883112
#226077982
C73×#23152
#919973