Finite Group Databasebeta

← #32974966

Subgroups — #32974966

Order 18688.

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

normal subgroups only

This group has 478 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#22611264C211yesmaximal, characteristic, complemented
2#22611259C211yesmaximal, characteristic, complemented
3#22611013C211yesmaximal, characteristic
4#22607754C211yesmaximal, characteristic, complemented
5#22607754C211yesmaximal, characteristic, complemented
6#22607752C211yesmaximal, characteristic
7C73×#2727C211yesmaximal, characteristic
8C22×#11924016—21no—
9C2×#14556591C2211yescharacteristic
10C2×#14556524—21no—
11C2×#14556522—21no—
12C2×#14556514—21no—
13#17988349—21no—
14#17988349—21no—
15#17988342C2211yescharacteristic
16#17988342C2211yescharacteristic
17#17988342C2211yescharacteristic
18C2×#126×C73—21no—
19#399×C73—21no—
20#396×C73—21no—
21#394×C73—21no—
22#390×C73C2211yescharacteristic
23#389×C73—21no—
24#389×C73C2211yescharacteristic
25#389×C73C2211yescharacteristic
26#388×C73—21no—
27C22×Dic146—21no—
28C2×#11924016—42no—
29C2×#11924016—42no—
30C2×#11924016—42no—
31C22×#10596831—21no—
32C22×#10596831—21no—
33C22×#10596831—21no—
34#14556591—42no—
35C2×C4⋊Dic73—21no—
36C2×C4⋊Dic73—21no—
37#14556524—42no—
38#14556522—42no—
39#14556514—42no—
40C22×D4×C73D411yescharacteristic
41#126×C73—42no—
42#126×C73—42no—
43C2×C4⋊C4×C73—21no—
44C2×C4⋊C4×C73—21no—
45C2×C4⋊C4×C73—21no—
46C2×C4⋊C4×C73—21no—
47C2×C4⋊C4×C73D411yescharacteristic
48C2×C4⋊C4×C73D411yescharacteristic
49C2×C4⋊C4×C73D411yescharacteristic
50C2×C4⋊C4×C73D411yescharacteristic
51C2×C4⋊C4×C73D411yescharacteristic
52C2×C22⋊C4×C73—21no—
53C2×C22⋊C4×C73—21no—
54C2×C22⋊C4×C73—21no—
55C2×C22⋊C4×C73—21no—
56C2×C22⋊C4×C73—21no—
57C2×C22⋊C4×C73—21no—
58C2×C42×C73—21no—
59C2×C42×C73C2311yescharacteristic
60#95×C73—21no—
61#95×C73—21no—
62#95×C73—21no—
63C2×Dic146—22no—
64C2×Dic146—41no—
65C2×Dic146—41no—
66C2×Dic146—41no—
67C2×Dic146—41no—
68C2×Dic146—41no—
69C22×Dic73—41no—
70#11924016—48no—
71C2×#10596831—22no—
72C2×#10596831—22no—
73C2×#10596831—22no—
74C2×#10596831—41no—
75C2×#10596831—41no—
76C2×#10596831—41no—
77C2×#10596831—41no—
78C2×#10596831—41no—
79C2×#10596831—41no—
80C4⋊Dic73—22no—
81C4⋊Dic73—22no—
82C4⋊Dic73—41no—
83C4⋊Dic73—41no—
84C24×C73—21no—
85C2×D4×C73—21no—
86C2×D4×C73—21no—
87C2×D4×C73—41no—
88C2×D4×C73—41no—
89C22×C4×C73—21no—
90C22×C4×C73—21no—
91C22×C4×C73—21no—
92C22×C4×C73—21no—
93C22×C4×C73—21no—
94C22×C4×C73C2×D411yescharacteristic
95C22×C4×C73C2×D411yescharacteristic
96C22×C4×C73C2×D411yescharacteristic
97C4⋊C4×C73—21no—
98C4⋊C4×C73—21no—
99C4⋊C4×C73—21no—
100C4⋊C4×C73—41no—
101C4⋊C4×C73—41no—
102C4⋊C4×C73—41no—
103C4⋊C4×C73—42no—
104C4⋊C4×C73—41no—
105C4⋊C4×C73—41no—
106C4⋊C4×C73—41no—
107C4⋊C4×C73—41no—
108C4⋊C4×C73—81no—
109C4⋊C4×C73SD1612yescomplemented
110C22⋊C4×C73—42no—
111C22⋊C4×C73—42no—
112C22⋊C4×C73—41no—
113C22⋊C4×C73—41no—
114C22⋊C4×C73—41no—
115C22⋊C4×C73—41no—
116C22⋊C4×C73—81no—
117C22⋊C4×C73—81no—
118C42×C73—41no—
119C42×C73—42no—
120C23×C73—41no—
121C23×C73—41no—
122C23×C73—41no—
123C23×C73—41no—
124C23×C73—41no—
125C23×C73—41no—
126C23×C73—41no—
127C23×C73#12011yescharacteristic
128D4×C73—42no—
129D4×C73—81no—
130C2×C4×C73—21no—
131C2×C4×C73—21no—
132C2×C4×C73—21no—
133C2×C4×C73—21no—
134C2×C4×C73—21no—
135C2×C4×C73—21no—
136C2×C4×C73—22no—
137C2×C4×C73—41no—
138C2×C4×C73—41no—
139C2×C4×C73—41no—
140C2×C4×C73—41no—
141C2×C4×C73—41no—
142C2×C4×C73—41no—
143C2×C4×C73—41no—
144C2×C4×C73—41no—
145C2×C4×C73—41no—
146C2×C4×C73—41no—
147C2×C4×C73—41no—
148C2×C4×C73—41no—
149C2×C4×C73—41no—
150C2×C4×C73—41no—
151C2×C4×C73C2×SD1611yescharacteristic
152C2×C4×C73#13511yescharacteristic
153C2×C4×C73#13511yescharacteristic
154C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
155C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
156C2×C4×C73#13711yescharacteristic
157C2×Dic73—42no—
158C2×Dic73—81no—
159C2×Dic73—81no—
160Dic146—42no—
161Dic146—42no—
162Dic146—81no—
163Dic146—81no—
164#10596831—42no—
165#10596831—42no—
166#10596831—42no—
167C22×C73—81no—
168C22×C73—81no—
169C22×C73—81no—
170C22×C73—81no—
171C22×C73—81no—
172C22×C73—81no—
173C22×C73—81no—
174C22×C73#44911yescharacteristic
175C22×C73#45011yescharacteristic
176C22×C73#45211yescharacteristic
177C22×C73#45311yescharacteristic
178C22×C73#45611yescharacteristic
179C22×C73#45611yescharacteristic
180C22×C73B(4,𝔽2)11yescharacteristic
181C4×C73—21no—
182C4×C73—21no—
183C4×C73—21no—
184C4×C73—21no—
185C4×C73—21no—
186C4×C73—21no—
187C4×C73—42no—
188C4×C73—81no—
189C4×C73—81no—
190C4×C73—81no—
191C4×C73—81no—
192Dic73—82no—
193#7780—731nomaximal, Sylow
194C2×C73—161no—
195C2×C73#160011yescharacteristic
196C2×C73#160311yescharacteristic
197C2×C73#160411yescharacteristic
198C2×C73#163111yescharacteristic
199C2×C73#165011yescharacteristic
200C2×C73#165411yescharacteristic
201C2×C73#166011yescharacteristic
202#2727D7311yescharacteristic
203#2058—731no—
204#2057—731no—
205#2013—731no—
206#1271—731no—
207#1271—731no—
208#1270—731no—
209C73#778011yescharacteristic, complemented, Sylow
210C22×SD16—1461no—
211C2×C4⋊Q8—731no—
212C2×#126—21no—
213C2×#107—1461no—
214C2×#106—1461no—
215C2×#103—1461no—
216#399—21no—
217#396—21no—
218#394—21no—
219#390C2×D7311yescharacteristic
220#389—21no—
221#389C2×D7311yescharacteristic
222#389C2×D7311yescharacteristic
223#388—21no—
224#340—1461no—
225#340—1461no—
226#336—731no—
227#336—731no—
228#336—731no—
229C22×Q8—1461no—
230C22×D4C22⋊D7311yescharacteristic
231C2×SD16—2922no—
232C2×SD16—2922no—
233C2×SD16—2922no—
234C22×C8—1461no—
235C22×C8—1461no—
236C22×C8—1461no—
237C4⋊Q8—2922no—
238#126—42no—
239#126—42no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—1461no—
245C2×C4⋊C4—1461no—
246C2×C4⋊C4C22⋊D7311yescharacteristic
247C2×C4⋊C4C22⋊D7311yescharacteristic
248C2×C4⋊C4C22⋊D7311yescharacteristic
249C2×C4⋊C4C22⋊D7311yescharacteristic
250C2×C4⋊C4C22⋊D7311yescharacteristic
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—21no—
257C2×C42—21no—
258C2×C42C22×D7311yescharacteristic
259#107—2922no—
260#106—2922no—
261#103—2922no—
262#95—21no—
263#95—21no—
264#95—21no—
265C24—21no—
266C2×Q8—2921no—
267C2×Q8—2921no—
268C2×Q8—2921no—
269C2×Q8—2921no—
270C2×Q8—2921no—
271C2×Q8—1462no—
272C2×D4—21no—
273C2×D4—21no—
274C2×D4—41no—
275C2×D4—41no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—2921no—
282C22×C4C2×C22⋊D7311yescharacteristic
283C22×C4C2×C22⋊D7311yescharacteristic
284C22×C4C2×C22⋊D7311yescharacteristic
285SD16—2928no—
286C2×C8—2921no—
287C2×C8—2921no—
288C2×C8—2921no—
289C2×C8—2921no—
290C2×C8—2921no—
291C2×C8—2921no—
292C2×C8—1462no—
293C2×C8—1462no—
294C2×C8—1462no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—21no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—42no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—81no—
307C4⋊C4—2921no—
308C4⋊C4—2921no—
309C4⋊C4—1462no—
310C4⋊C4—1462no—
311C4⋊C4#1192401612yescomplemented
312C22⋊C4—42no—
313C22⋊C4—42no—
314C22⋊C4—41no—
315C22⋊C4—41no—
316C22⋊C4—41no—
317C22⋊C4—41no—
318C22⋊C4—81no—
319C22⋊C4—81no—
320C42—41no—
321C42—42no—
322C23—41no—
323C23—41no—
324C23—41no—
325C23—41no—
326C23—41no—
327C23—41no—
328C23—41no—
329C23#1455656011yescharacteristic
330Q8—2922no—
331Q8—2922no—
332Q8—5841no—
333Q8—5841no—
334D4—42no—
335D4—81no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—22no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—2922no—
358C2×C4—5841no—
359C2×C4—5841no—
360C2×C4C2×#1192401611yescharacteristic
361C2×C4#1455661111yescharacteristic
362C2×C4#1455661111yescharacteristic
363C2×C4#1455661611yescharacteristic
364C2×C4#1455661711yescharacteristic
365C2×C4#1455662111yescharacteristic
366C8—2922no—
367C8—2922no—
368C8—2922no—
369C22—81no—
370C22—81no—
371C22—81no—
372C22—81no—
373C22—81no—
374C22—81no—
375C22—81no—
376C22#1798873211yescharacteristic
377C22#1798873811yescharacteristic
378C22#1798874911yescharacteristic
379C22#1798875511yescharacteristic
380C22#1798877011yescharacteristic
381C22#1798877011yescharacteristic
382C22#1798877411yescharacteristic
383C4—21no—
384C4—21no—
385C4—21no—
386C4—21no—
387C4—21no—
388C4—21no—
389C4—42no—
390C4—81no—
391C4—81no—
392C4—81no—
393C4—81no—
394C4—5842no—
395C2—161no—
396C2#2260887711yescharacteristic
397C2#2260889411yescharacteristic
398C2#2260890011yescharacteristic
399C2#2260905811yescharacteristic
400C2#2260918711yescharacteristic
401C2#2260921511yescharacteristic
402C2#2260924211yescharacteristic
subgroupG/H
centerC23#14556560
commutator, cosocleC2×C42×C73C23
FrattiniC2×C42C22×D73
FittingC73×#2727C2
socleC23×C73#120
prime radicalC22×D4×C73D4
hypercenter#2727D73
hypocenterC73#7780
LCSG ▷ C2×C42×C73 ▷ C23×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2727

Complements

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

NH ≅ G/Ntype
1. #22611264C2semidirect
2. #22611259C2semidirect
4. #22607754C2semidirect
5. #22607754C2semidirect
109. C4⋊C4×C73SD16semidirect
209. C73#7780semidirect
311. C4⋊C4#11924016semidirect

Maximal subgroups

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

subgroupindex
#226112642
#226112592
#226110132
#226077542
#226077522
C73×#27272
#778073