Finite Group Databasebeta

← #32996887

Subgroups — #32996887

Order 18688.

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

normal subgroups only

This group has 572 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#22610011C211yesmaximal, characteristic, complemented
2C2×#17988313C211yesmaximal, characteristic, complemented
3#22607961C212yesmaximal, complemented
4#22607934C212yesmaximal, complemented
5C73×#2406C211yesmaximal, characteristic
6#17988551—21no—
7C2×C22⋊#10596831—21no—
8#17988356—21no—
9#17988356C2212yes—
10#17988313—21no—
11#17988313C2212yes—
12C22×#41×C73C411yescharacteristic
13C2×#123×C73—21no—
14C22×C4⋊C4×C73C2211yescharacteristic
15#395×C73C411yescharacteristic
16#395×C73C2211yescharacteristic
17#394×C73C411yescharacteristic
18#394×C73C2211yescharacteristic
19#386×C73—21no—
20#385×C73—21no—
21#385×C73—21no—
22#381×C73C411yescharacteristic
23C2×#11924012—22no—
24C22×#10596831—41no—
25C22⋊#10596831—22no—
26C22⋊#10596831—41no—
27C22⋊#10596831—41no—
28C22⋊#10596831—41no—
29C2×#41×C73—41no—
30C2×#41×C73—41no—
31C2×#41×C73—44no—
32C22×Q8×C73C2×C411yescharacteristic
33C22×D4×C73—21no—
34C22×D4×C73C2×C411yescharacteristic
35C23×C4×C73—21no—
36C23×C4×C73C2×C411yescharacteristic
37C23×C4×C73D411yescharacteristic
38C23×C4×C73D411yescharacteristic
39#123×C73—44no—
40C2×C4⋊C4×C73—21no—
41C2×C4⋊C4×C73—21no—
42C2×C4⋊C4×C73—41no—
43C2×C4⋊C4×C73—41no—
44C2×C4⋊C4×C73C2×C411yescharacteristic
45C2×C4⋊C4×C73D411yescharacteristic
46C2×C4⋊C4×C73D411yescharacteristic
47C2×C4⋊C4×C73C2311yescharacteristic
48C2×C22⋊C4×C73—21no—
49C2×C22⋊C4×C73—21no—
50C2×C22⋊C4×C73—21no—
51C2×C22⋊C4×C73—21no—
52C2×C22⋊C4×C73—21no—
53#95×C73—21no—
54#95×C73—21no—
55#95×C73—21no—
56#95×C73C2×C411yescharacteristic
57#95×C73C2×C411yescharacteristic
58#11924012—42no—
59#11924012—42no—
60C2×#10596831—42no—
61C2×#10596831—42no—
62C2×#10596831—82no—
63C24×C73—21no—
64C24×C73C22⋊C411yescharacteristic
65#41×C73—44no—
66#41×C73—44no—
67#41×C73—44no—
68#41×C73—44no—
69C2×Q8×C73—42no—
70C2×Q8×C73—41no—
71C2×D4×C73—24no—
72C2×D4×C73—41no—
73C2×D4×C73—42no—
74C2×D4×C73—41no—
75C2×D4×C73—41no—
76C2×D4×C73—42no—
77C22×C4×C73—21no—
78C22×C4×C73—21no—
79C22×C4×C73—21no—
80C22×C4×C73—21no—
81C22×C4×C73—21no—
82C22×C4×C73—21no—
83C22×C4×C73—21no—
84C22×C4×C73—22no—
85C22×C4×C73—21no—
86C22×C4×C73—21no—
87C22×C4×C73—41no—
88C22×C4×C73—42no—
89C22×C4×C73—42no—
90C22×C4×C73—42no—
91C22×C4×C73—42no—
92C22×C4×C73—42no—
93C22×C4×C73C22⋊C411yescharacteristic
94C22×C4×C73C22⋊C411yescharacteristic
95C22×C4×C73C22⋊C411yescharacteristic
96C22×C4×C73C22×C411yescharacteristic
97C22×C4×C73C2×D411yescharacteristic
98C22×C4×C73C2×D411yescharacteristic
99C4⋊C4×C73—41no—
100C4⋊C4×C73—41no—
101C4⋊C4×C73—41no—
102C4⋊C4×C73—41no—
103C4⋊C4×C73—42no—
104C4⋊C4×C73—42no—
105C22⋊C4×C73—42no—
106C22⋊C4×C73—42no—
107C22⋊C4×C73—42no—
108C22⋊C4×C73—42no—
109C22⋊C4×C73—42no—
110C23×C73—22no—
111C23×C73—21no—
112C23×C73—21no—
113C23×C73—22no—
114C23×C73—41no—
115C23×C73—41no—
116C23×C73—41no—
117C23×C73—41no—
118C23×C73—41no—
119C23×C73—41no—
120C23×C73—41no—
121C23×C73#9911yescharacteristic
122C23×C73#9911yescharacteristic
123C23×C73C2×C22⋊C411yescharacteristic
124Q8×C73—42no—
125Q8×C73—42no—
126D4×C73—44no—
127D4×C73—42no—
128D4×C73—42no—
129D4×C73—44no—
130C2×C4×C73—21no—
131C2×C4×C73—21no—
132C2×C4×C73—22no—
133C2×C4×C73—21no—
134C2×C4×C73—21no—
135C2×C4×C73—21no—
136C2×C4×C73—21no—
137C2×C4×C73—21no—
138C2×C4×C73—21no—
139C2×C4×C73—21no—
140C2×C4×C73—42no—
141C2×C4×C73—42no—
142C2×C4×C73—41no—
143C2×C4×C73—42no—
144C2×C4×C73—48no—
145C2×C4×C73—42no—
146C2×C4×C73—42no—
147C2×C4×C73—41no—
148C2×C4×C73—41no—
149C2×C4×C73—41no—
150C2×C4×C73—41no—
151C2×C4×C73—41no—
152C2×C4×C73—41no—
153C2×C4×C73—41no—
154C2×C4×C73—41no—
155C2×C4×C73—41no—
156C2×C4×C73—41no—
157C2×C4×C73—41no—
158C2×C4×C73—81no—
159C2×C4×C73#13512yes—
160#10596831—82no—
161#10596831—82no—
162C22×C73—21no—
163C22×C73—21no—
164C22×C73—21no—
165C22×C73—21no—
166C22×C73—21no—
167C22×C73—21no—
168C22×C73—21no—
169C22×C73—42no—
170C22×C73—42no—
171C22×C73—42no—
172C22×C73—42no—
173C22×C73—42no—
174C22×C73—81no—
175C22×C73—81no—
176C22×C73—81no—
177C22×C73—81no—
178C22×C73AΣL(1,𝔽16)12yes—
179C22×C73C2×#9911yescharacteristic
180C22×C73#41611yescharacteristic
181C22×C73#42111yescharacteristic
182C4×C73—22no—
183C4×C73—41no—
184C4×C73—41no—
185C4×C73—42no—
186C4×C73—41no—
187C4×C73—41no—
188C4×C73—41no—
189C4×C73—81no—
190C4×C73—81no—
191#11848—731nomaximal, Sylow
192C2×C73—21no—
193C2×C73—21no—
194C2×C73—22no—
195C2×C73—41no—
196C2×C73—82no—
197C2×C73#151611yescharacteristic
198C2×C73C2×AΣL(1,𝔽16)11yescharacteristic
199C2×C73#213311yescharacteristic
200#2406D7311yescharacteristic
201#1821—731no—
202C2×#324—731no—
203#1354—732no—
204#1345—732no—
205C73#1184811yescharacteristic, complemented, Sylow
206C22×#41Dic7311yescharacteristic
207C2×#123—21no—
208C22×C4⋊C4C2×D7311yescharacteristic
209#407—1461no—
210C2×C22⋊C8—1461no—
211#395Dic7311yescharacteristic
212#395C2×D7311yescharacteristic
213#394Dic7311yescharacteristic
214#394C2×D7311yescharacteristic
215#386—21no—
216#385—21no—
217#385—21no—
218#381Dic7311yescharacteristic
219#343—732no—
220#343—1461no—
221#324—732no—
222#324—1461no—
223C2×#41—41no—
224C2×#41—41no—
225C2×#41—44no—
226C22×Q8C2×Dic7311yescharacteristic
227C22×D4—21no—
228C22×D4C2×Dic7311yescharacteristic
229C23×C4—21no—
230C23×C4C2×Dic7311yescharacteristic
231C23×C4C22⋊D7311yescharacteristic
232C23×C4C22⋊D7311yescharacteristic
233C2×M16—1462no—
234C22×C8—2921no—
235#123—44no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—41no—
239C2×C4⋊C4—41no—
240C2×C4⋊C4C2×Dic7311yescharacteristic
241C2×C4⋊C4C22⋊D7311yescharacteristic
242C2×C4⋊C4C22⋊D7311yescharacteristic
243C2×C4⋊C4C22×D7311yescharacteristic
244C2×C22⋊C4—21no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—21no—
249C22⋊C8—2921no—
250C22⋊C8—2921no—
251C22⋊C8—2921no—
252C22⋊C8—1462no—
253#95—21no—
254#95—21no—
255#95—21no—
256#95C2×Dic7311yescharacteristic
257#95C2×Dic7311yescharacteristic
258C24—21no—
259C24C22⋊Dic7311yescharacteristic
260#41—44no—
261#41—44no—
262#41—44no—
263#41—44no—
264C2×Q8—42no—
265C2×Q8—41no—
266C2×D4—24no—
267C2×D4—41no—
268C2×D4—42no—
269C2×D4—41no—
270C2×D4—41no—
271C2×D4—42no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—22no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—41no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4C22⋊Dic7311yescharacteristic
289C22×C4C22⋊Dic7311yescharacteristic
290C22×C4C22⋊Dic7311yescharacteristic
291C22×C4C22×Dic7311yescharacteristic
292C22×C4C2×C22⋊D7311yescharacteristic
293C22×C4C2×C22⋊D7311yescharacteristic
294M16—2922no—
295M16—2922no—
296C2×C8—2922no—
297C2×C8—2922no—
298C2×C8—5842no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C22⋊C4—42no—
306C22⋊C4—42no—
307C22⋊C4—42no—
308C22⋊C4—42no—
309C22⋊C4—42no—
310C23—22no—
311C23—21no—
312C23—21no—
313C23—22no—
314C23—41no—
315C23—41no—
316C23—41no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23#1455650611yescharacteristic
322C23#1455650611yescharacteristic
323C23C2×C22⋊Dic7311yescharacteristic
324Q8—42no—
325Q8—42no—
326D4—44no—
327D4—42no—
328D4—42no—
329D4—44no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—22no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—41no—
343C2×C4—42no—
344C2×C4—48no—
345C2×C4—42no—
346C2×C4—42no—
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—41no—
358C2×C4—81no—
359C2×C4#1455661112yes—
360C8—5842no—
361C8—5842no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—42no—
370C22—42no—
371C22—42no—
372C22—42no—
373C22—42no—
374C22—81no—
375C22—81no—
376C22—81no—
377C22—81no—
378C22#1798837412yes—
379C22C2×#1455650611yescharacteristic
380C22#1798858311yescharacteristic
381C22#1798860411yescharacteristic
382C4—22no—
383C4—41no—
384C4—41no—
385C4—41no—
386C4—42no—
387C4—41no—
388C4—41no—
389C4—81no—
390C4—81no—
391C2—21no—
392C2—21no—
393C2—22no—
394C2—41no—
395C2—82no—
396C2#2260848711yescharacteristic
397C2C2×#1798837411yescharacteristic
398C2#2261162011yescharacteristic
subgroupG/H
centerC22C2×#14556506
commutatorC22×C4×C73C22×C4
FrattiniC2×C4⋊C4C22×D73
FittingC73×#2406C2
socleC22×C73C2×#99
prime radicalC22×#41×C73C4
cosocleC2×C4⋊C4×C73C23
hypercenter#2406D73
hypocenterC73#11848
LCSG ▷ C22×C4×C73 ▷ C23×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ #2406

Complements

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

NH ≅ G/Ntype
1. #22610011C2semidirect
2. C2×#17988313C2semidirect
3. #22607961C2semidirect
4. #22607934C2semidirect
205. C73#11848semidirect

Maximal subgroups

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

subgroupindex
#226100112
C2×#179883132
#226079612
#226079342
C73×#24062
#1184873