Finite Group Databasebeta

← #32995619

Subgroups — #32995619

Order 18688.

494 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 91 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#22608269C211yesmaximal, characteristic, complemented
2#22608249C211yesmaximal, characteristic, complemented
3#22607865C212yesmaximal, complemented
4#22607863C212yesmaximal, complemented
5C73×#2544C211yesmaximal, characteristic
6#17988551—21no—
7C2×C22⋊#10596831—21no—
8#17988313C2212yes—
9#17988313C2212yes—
10#17988311—21no—
11#17988311—21no—
12C2×#127×C73C411yescharacteristic
13C2×#124×C73C411yescharacteristic
14C2×#124×C73C411yescharacteristic
15C2×C4×D4×C73C2211yescharacteristic
16#394×C73—21no—
17#394×C73—21no—
18#391×C73C2211yescharacteristic
19#390×C73C2211yescharacteristic
20#386×C73—21no—
21#386×C73—21no—
22C4×C4⋊C4×C73C411yescharacteristic
23C2×#11924012—41no—
24C22×#10596831—41no—
25C22⋊#10596831—22no—
26C22⋊#10596831—22no—
27C22⋊#10596831—41no—
28C22⋊#10596831—41no—
29C22×Q8×C73C2×C411yescharacteristic
30C22×D4×C73—21no—
31C22×D4×C73—21no—
32C22×D4×C73C2×C411yescharacteristic
33C22×D4×C73C2×C411yescharacteristic
34C23×C4×C73D411yescharacteristic
35C23×C4×C73D411yescharacteristic
36#127×C73—42no—
37#124×C73—41no—
38#124×C73—41no—
39#124×C73—42no—
40C4×D4×C73—41no—
41C4×D4×C73—41no—
42C2×C4⋊C4×C73—21no—
43C2×C4⋊C4×C73C2311yescharacteristic
44C2×C22⋊C4×C73—21no—
45C2×C22⋊C4×C73—21no—
46C2×C22⋊C4×C73—21no—
47C2×C22⋊C4×C73—21no—
48C2×C22⋊C4×C73—21no—
49C2×C22⋊C4×C73—21no—
50C2×C22⋊C4×C73D411yescharacteristic
51C2×C22⋊C4×C73D411yescharacteristic
52C2×C42×C73C2×C411yescharacteristic
53C2×C42×C73C2×C411yescharacteristic
54C2×C42×C73C2×C411yescharacteristic
55#95×C73—21no—
56#95×C73—21no—
57#95×C73—21no—
58#11924012—82no—
59C2×#10596831—42no—
60C2×#10596831—42no—
61C2×#10596831—82no—
62C24×C73—21no—
63C24×C73C22⋊C411yescharacteristic
64C24×C73C22⋊C411yescharacteristic
65C2×Q8×C73—22no—
66C2×Q8×C73—41no—
67C2×Q8×C73—41no—
68C2×D4×C73—22no—
69C2×D4×C73—22no—
70C2×D4×C73—22no—
71C2×D4×C73—41no—
72C2×D4×C73—41no—
73C2×D4×C73—41no—
74C2×D4×C73—42no—
75C2×D4×C73—42no—
76C2×D4×C73—42no—
77C2×D4×C73—42no—
78C2×D4×C73—81no—
79C2×D4×C73—81no—
80C22×C4×C73—21no—
81C22×C4×C73—21no—
82C22×C4×C73—21no—
83C22×C4×C73—21no—
84C22×C4×C73—21no—
85C22×C4×C73—21no—
86C22×C4×C73—21no—
87C22×C4×C73—81no—
88C22×C4×C73—81no—
89C22×C4×C73C22⋊C411yescharacteristic
90C22×C4×C73C22⋊C411yescharacteristic
91C22×C4×C73C22×C411yescharacteristic
92C22×C4×C73C2×D411yescharacteristic
93C22×C4×C73C2×D411yescharacteristic
94C4⋊C4×C73—41no—
95C4⋊C4×C73—42no—
96C22⋊C4×C73—41no—
97C22⋊C4×C73—41no—
98C22⋊C4×C73—42no—
99C22⋊C4×C73—42no—
100C22⋊C4×C73—42no—
101C22⋊C4×C73—41no—
102C22⋊C4×C73—41no—
103C22⋊C4×C73—81no—
104C22⋊C4×C73—81no—
105C42×C73—21no—
106C42×C73—21no—
107C42×C73—41no—
108C42×C73—41no—
109C23×C73—21no—
110C23×C73—21no—
111C23×C73—21no—
112C23×C73—21no—
113C23×C73—21no—
114C23×C73—41no—
115C23×C73—41no—
116C23×C73—41no—
117C23×C73—41no—
118C23×C73—41no—
119C23×C73—41no—
120C23×C73—41no—
121C23×C73—41no—
122C23×C73—41no—
123C23×C73—41no—
124C23×C73—41no—
125C23×C73#10012yes—
126C23×C73C2×C22⋊C411yescharacteristic
127Q8×C73—82no—
128D4×C73—44no—
129D4×C73—82no—
130D4×C73—84no—
131D4×C73—84no—
132C2×C4×C73—21no—
133C2×C4×C73—21no—
134C2×C4×C73—22no—
135C2×C4×C73—21no—
136C2×C4×C73—21no—
137C2×C4×C73—21no—
138C2×C4×C73—41no—
139C2×C4×C73—42no—
140C2×C4×C73—42no—
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×C73—41no—
152C2×C4×C73—81no—
153C2×C4×C73—81no—
154C2×C4×C73#9911yescharacteristic
155C2×C4×C73#9911yescharacteristic
156C2×C4×C73#13112yes—
157#10596831—82no—
158#10596831—82no—
159C22×C73—21no—
160C22×C73—21no—
161C22×C73—22no—
162C22×C73—42no—
163C22×C73—42no—
164C22×C73—41no—
165C22×C73—41no—
166C22×C73—41no—
167C22×C73—41no—
168C22×C73—41no—
169C22×C73—81no—
170C22×C73—81no—
171C22×C73—81no—
172C22×C73—81no—
173C22×C73—81no—
174C22×C73—81no—
175C22×C73—81no—
176C22×C73—81no—
177C22×C73—81no—
178C22×C73#40811yescharacteristic
179C22×C73C2×#9911yescharacteristic
180C22×C73C2×#10011yescharacteristic
181C4×C73—21no—
182C4×C73—21no—
183C4×C73—22no—
184C4×C73—42no—
185C4×C73—41no—
186C4×C73—41no—
187C4×C73—41no—
188C4×C73—81no—
189C4×C73—81no—
190#11566—731nomaximal, Sylow
191C2×C73—21no—
192C2×C73—21no—
193C2×C73—42no—
194C2×C73—81no—
195C2×C73—161no—
196C2×C73#147111yescharacteristic
197C2×C73#212711yescharacteristic
198C2×C73#213211yescharacteristic
199#2544D7311yescharacteristic
200#1465—731no—
201#1459—731no—
202#1319—732no—
203#1318—732no—
204C73#1156611yescharacteristic, complemented, Sylow
205C2×#127Dic7311yescharacteristic
206C2×#124Dic7311yescharacteristic
207C2×#124Dic7311yescharacteristic
208C2×C4×D4C2×D7311yescharacteristic
209#407—1461no—
210C2×C22⋊C8—1461no—
211#394—21no—
212#394—21no—
213#391C2×D7311yescharacteristic
214#390C2×D7311yescharacteristic
215#386—21no—
216#386—21no—
217C4×C4⋊C4Dic7311yescharacteristic
218#324—732no—
219#324—732no—
220#323—1461no—
221#323—1461no—
222C22×Q8C2×Dic7311yescharacteristic
223C22×D4—21no—
224C22×D4—21no—
225C22×D4C2×Dic7311yescharacteristic
226C22×D4C2×Dic7311yescharacteristic
227C23×C4C22⋊D7311yescharacteristic
228C23×C4C22⋊D7311yescharacteristic
229C2×M16—2921no—
230C22×C8—2921no—
231#127—42no—
232#124—41no—
233#124—41no—
234#124—42no—
235C4×D4—41no—
236C4×D4—41no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4C22×D7311yescharacteristic
239C2×C22⋊C4—21no—
240C2×C22⋊C4—21no—
241C2×C22⋊C4—21no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—21no—
244C2×C22⋊C4—21no—
245C2×C22⋊C4C22⋊D7311yescharacteristic
246C2×C22⋊C4C22⋊D7311yescharacteristic
247C2×C42C2×Dic7311yescharacteristic
248C2×C42C2×Dic7311yescharacteristic
249C2×C42C2×Dic7311yescharacteristic
250C22⋊C8—2921no—
251C22⋊C8—2921no—
252C22⋊C8—1462no—
253C22⋊C8—1462no—
254#95—21no—
255#95—21no—
256#95—21no—
257C24—21no—
258C24C22⋊Dic7311yescharacteristic
259C24C22⋊Dic7311yescharacteristic
260C2×Q8—22no—
261C2×Q8—41no—
262C2×Q8—41no—
263C2×D4—22no—
264C2×D4—22no—
265C2×D4—22no—
266C2×D4—41no—
267C2×D4—41no—
268C2×D4—41no—
269C2×D4—42no—
270C2×D4—42no—
271C2×D4—42no—
272C2×D4—42no—
273C2×D4—81no—
274C2×D4—81no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—81no—
283C22×C4—81no—
284C22×C4C22⋊Dic7311yescharacteristic
285C22×C4C22⋊Dic7311yescharacteristic
286C22×C4C22×Dic7311yescharacteristic
287C22×C4C2×C22⋊D7311yescharacteristic
288C22×C4C2×C22⋊D7311yescharacteristic
289M16—5842no—
290C2×C8—2922no—
291C2×C8—2922no—
292C2×C8—5842no—
293C4⋊C4—41no—
294C4⋊C4—42no—
295C22⋊C4—41no—
296C22⋊C4—41no—
297C22⋊C4—42no—
298C22⋊C4—42no—
299C22⋊C4—42no—
300C22⋊C4—41no—
301C22⋊C4—41no—
302C22⋊C4—81no—
303C22⋊C4—81no—
304C42—21no—
305C42—21no—
306C42—41no—
307C42—41no—
308C23—21no—
309C23—21no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23—41no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23—41no—
322C23—41no—
323C23—41no—
324C23#1455650812yes—
325C23C2×C22⋊Dic7311yescharacteristic
326Q8—82no—
327D4—44no—
328D4—82no—
329D4—84no—
330D4—84no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—22no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—41no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
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—81no—
352C2×C4—81no—
353C2×C4#1455650611yescharacteristic
354C2×C4#1455650611yescharacteristic
355C2×C4#1455659812yes—
356C8—5842no—
357C8—5842no—
358C22—21no—
359C22—21no—
360C22—22no—
361C22—42no—
362C22—42no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22—81no—
369C22—81no—
370C22—81no—
371C22—81no—
372C22—81no—
373C22—81no—
374C22—81no—
375C22—81no—
376C22—81no—
377C22#1798855411yescharacteristic
378C22C2×#1455650611yescharacteristic
379C22C2×#1455650811yescharacteristic
380C4—21no—
381C4—21no—
382C4—22no—
383C4—42no—
384C4—41no—
385C4—41no—
386C4—41no—
387C4—81no—
388C4—81no—
389C2—21no—
390C2—21no—
391C2—42no—
392C2—81no—
393C2—161no—
394C2#2260829111yescharacteristic
395C2#2261160011yescharacteristic
396C2#2261161611yescharacteristic
subgroupG/H
centerC22C2×#14556506
commutatorC22×C4×C73C22×C4
FrattiniC2×C4⋊C4C22×D73
FittingC73×#2544C2
socleC22×C73C2×#99
prime radicalC2×#127×C73C4
cosocleC2×C4⋊C4×C73C23
hypercenter#2544D73
hypocenterC73#11566
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2544

Complements

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

NH ≅ G/Ntype
1. #22608269C2semidirect
2. #22608249C2semidirect
3. #22607865C2semidirect
4. #22607863C2semidirect
204. C73#11566semidirect

Maximal subgroups

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

subgroupindex
#226082692
#226082492
#226078652
#226078632
C73×#25442
#1156673