Finite Group Databasebeta

← #32996313

Subgroups — #32996313

Order 18688.

606 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 606 conjugacy classes of subgroups — too many to draw. Only its 95 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
1C4×#14556506C211yesmaximal, characteristic, complemented
2#22608249C211yesmaximal, characteristic, complemented
3#22607955C212yesmaximal, complemented
4#22607928C212yesmaximal, complemented
5C73×#2397C211yesmaximal, characteristic
6C2×#14556506—21no—
7C2×C22⋊#10596831—21no—
8C4×C22⋊Dic73—21no—
9#17988354C2212yes—
10#17988313—21no—
11#17988311C2212yes—
12C2×C4×D4×C73—21no—
13C2×C4×D4×C73—21no—
14C2×C4×D4×C73C411yescharacteristic, complemented
15C2×C4×D4×C73C2211yescharacteristic
16#399×C73—21no—
17#399×C73C411yescharacteristic, complemented
18#399×C73C2211yescharacteristic
19#392×C73—21no—
20#392×C73C411yescharacteristic, complemented
21#392×C73C2211yescharacteristic
22#376×C73C411yescharacteristic, complemented
23C22×#10596831—41no—
24C2×C22⋊Dic73—22no—
25C2×C4×Dic73—41no—
26#14556506—24no—
27C22⋊#10596831—22no—
28C22⋊#10596831—41no—
29#14556498—41no—
30C22×D4×C73—21no—
31C22×D4×C73—21no—
32C22×D4×C73C2×C411yescharacteristic
33C22×D4×C73D411yescharacteristic
34C23×C4×C73—21no—
35C23×C4×C73D411yescharacteristic
36C23×C4×C73D411yescharacteristic
37C4×D4×C73—41no—
38C4×D4×C73—41no—
39C4×D4×C73—44no—
40C4×D4×C73—44no—
41C4×D4×C73—41no—
42C4×D4×C73—41no—
43C2×C4⋊C4×C73—21no—
44C2×C4⋊C4×C73—21no—
45C2×C4⋊C4×C73C2×C411yescharacteristic
46C2×C4⋊C4×C73D411yescharacteristic
47C2×C22⋊C4×C73—21no—
48C2×C22⋊C4×C73—21no—
49C2×C22⋊C4×C73—21no—
50C2×C22⋊C4×C73—21no—
51C2×C22⋊C4×C73—21no—
52C2×C22⋊C4×C73C2×C411yescharacteristic
53C2×C22⋊C4×C73C2311yescharacteristic
54C2×C42×C73—21no—
55C2×C42×C73C2×C411yescharacteristic
56#95×C73—21no—
57#95×C73C2×C411yescharacteristic
58#95×C73C2×C411yescharacteristic
59C22×Dic73—42no—
60C2×#10596831—42no—
61C2×#10596831—82no—
62C22⋊Dic73—24no—
63C22⋊Dic73—42no—
64C4×Dic73—82no—
65C24×C73—21no—
66C24×C73C22⋊C411yescharacteristic
67C24×C73C2×D411yescharacteristic
68C2×D4×C73—22no—
69C2×D4×C73—22no—
70C2×D4×C73—41no—
71C2×D4×C73—41no—
72C2×D4×C73—42no—
73C2×D4×C73—42no—
74C2×D4×C73—42no—
75C2×D4×C73—42no—
76C2×D4×C73—42no—
77C2×D4×C73—42no—
78C2×D4×C73—42no—
79C22×C4×C73—21no—
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—42no—
87C22×C4×C73—42no—
88C22×C4×C73—42no—
89C22×C4×C73—42no—
90C22×C4×C73—41no—
91C22×C4×C73—41no—
92C22×C4×C73C22⋊C411yescharacteristic
93C22×C4×C73C22⋊C411yescharacteristic
94C22×C4×C73C22⋊C411yescharacteristic
95C22×C4×C73C22×C411yescharacteristic
96C22×C4×C73C2×D411yescharacteristic
97C4⋊C4×C73—41no—
98C4⋊C4×C73—41no—
99C4⋊C4×C73—42no—
100C4⋊C4×C73—42no—
101C22⋊C4×C73—41no—
102C22⋊C4×C73—41no—
103C22⋊C4×C73—42no—
104C22⋊C4×C73—42no—
105C22⋊C4×C73—42no—
106C22⋊C4×C73—42no—
107C22⋊C4×C73—42no—
108C42×C73—24no—
109C42×C73—21no—
110C42×C73—21no—
111C23×C73—21no—
112C23×C73—21no—
113C23×C73—21no—
114C23×C73—21no—
115C23×C73—22no—
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—41no—
126C23×C73#10412yes—
127C23×C73C2×C22⋊C411yescharacteristic
128C23×C73#13512yes—
129D4×C73—44no—
130D4×C73—42no—
131D4×C73—48no—
132D4×C73—48no—
133D4×C73—42no—
134C2×C4×C73—22no—
135C2×C4×C73—22no—
136C2×C4×C73—22no—
137C2×C4×C73—21no—
138C2×C4×C73—21no—
139C2×C4×C73—21no—
140C2×C4×C73—21no—
141C2×C4×C73—21no—
142C2×C4×C73—21no—
143C2×C4×C73—41no—
144C2×C4×C73—41no—
145C2×C4×C73—41no—
146C2×C4×C73—41no—
147C2×C4×C73—42no—
148C2×C4×C73—42no—
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—81no—
156C2×C4×C73—81no—
157C2×C4×C73—82no—
158C2×C4×C73—82no—
159C2×C4×C73#9911yescharacteristic
160C2×C4×C73#9911yescharacteristic
161C2×Dic73—44no—
162C2×Dic73—82no—
163C2×Dic73—82no—
164#10596831—82no—
165C22×C73—21no—
166C22×C73—21no—
167C22×C73—22no—
168C22×C73—22no—
169C22×C73—42no—
170C22×C73—42no—
171C22×C73—42no—
172C22×C73—42no—
173C22×C73—41no—
174C22×C73—41no—
175C22×C73—41no—
176C22×C73—41no—
177C22×C73—42no—
178C22×C73—81no—
179C22×C73—81no—
180C22×C73—81no—
181C22×C73—81no—
182C22×C73—81no—
183C22×C73—81no—
184C22×C73C2×#9911yescharacteristic
185C22×C73#41611yescharacteristic
186C22×C73C2×#10411yescharacteristic
187C4×C73—21no—
188C4×C73—42no—
189C4×C73—42no—
190C4×C73—41no—
191C4×C73—41no—
192C4×C73—41no—
193C4×C73—41no—
194C4×C73—81no—
195C4×C73#35312yes—
196Dic73—84no—
197#11726—731nomaximal, Sylow
198C2×C73—21no—
199C2×C73—21no—
200C2×C73—42no—
201C2×C73—42no—
202C2×C73—82no—
203C2×C73#150111yescharacteristic
204C2×C73#212011yescharacteristic
205C2×C73C2×#35311yescharacteristic
206#2397D7311yescharacteristic
207C4×#99—731no—
208#1459—731no—
209#1352—732no—
210#1343—732no—
211C73#1172611yescharacteristic, complemented, Sylow
212C2×C4×D4—21no—
213C2×C4×D4—21no—
214C2×C4×D4Dic7311yescharacteristic, complemented
215C2×C4×D4C2×D7311yescharacteristic
216C2×#99—1461no—
217C2×C22⋊C8—1461no—
218#399—21no—
219#399Dic7311yescharacteristic, complemented
220#399C2×D7311yescharacteristic
221#392—21no—
222#392Dic7311yescharacteristic, complemented
223#392C2×D7311yescharacteristic
224C4×C22⋊C4—1461no—
225#376Dic7311yescharacteristic, complemented
226#342—732no—
227#324—1461no—
228#323—732no—
229C22×D4—21no—
230C22×D4—21no—
231C22×D4C2×Dic7311yescharacteristic
232C22×D4C22⋊D7311yescharacteristic
233C23×C4—21no—
234C23×C4C22⋊D7311yescharacteristic
235C23×C4C22⋊D7311yescharacteristic
236C22×C8—2921no—
237C4×D4—41no—
238C4×D4—41no—
239C4×D4—44no—
240C4×D4—44no—
241C4×D4—41no—
242C4×D4—41no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4C2×Dic7311yescharacteristic
246C2×C4⋊C4C22⋊D7311yescharacteristic
247C2×C22⋊C4—21no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—1462no—
253C2×C22⋊C4C2×Dic7311yescharacteristic
254C2×C22⋊C4C22×D7311yescharacteristic
255C2×C42—21no—
256C2×C42—2921no—
257C2×C42C2×Dic7311yescharacteristic
258#99—1464no—
259C22⋊C8—2921no—
260C22⋊C8—1462no—
261#95—21no—
262#95—2921no—
263#95C2×Dic7311yescharacteristic
264#95C2×Dic7311yescharacteristic
265C24—21no—
266C24C22⋊Dic7311yescharacteristic
267C24C2×C22⋊D7311yescharacteristic
268C2×D4—22no—
269C2×D4—22no—
270C2×D4—41no—
271C2×D4—41no—
272C2×D4—42no—
273C2×D4—42no—
274C2×D4—42no—
275C2×D4—42no—
276C2×D4—42no—
277C2×D4—42no—
278C2×D4—42no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—2922no—
293C22×C4C22⋊Dic7311yescharacteristic
294C22×C4C22⋊Dic7311yescharacteristic
295C22×C4C22⋊Dic7311yescharacteristic
296C22×C4C22×Dic7311yescharacteristic
297C22×C4C2×C22⋊D7311yescharacteristic
298C2×C8—2922no—
299C2×C8—5842no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C22⋊C4—41no—
305C22⋊C4—41no—
306C22⋊C4—42no—
307C22⋊C4—42no—
308C22⋊C4—42no—
309C22⋊C4—42no—
310C22⋊C4—42no—
311C22⋊C4—2922no—
312C22⋊C4—1464no—
313C42—24no—
314C42—21no—
315C42—21no—
316C42—5842no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23—22no—
322C23—41no—
323C23—41no—
324C23—41no—
325C23—41no—
326C23—41no—
327C23—41no—
328C23—41no—
329C23—41no—
330C23—41no—
331C23—41no—
332C23#1455651812yes—
333C23C2×C22⋊Dic7311yescharacteristic
334C23#1455661112yes—
335D4—44no—
336D4—42no—
337D4—48no—
338D4—48no—
339D4—42no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—82no—
364C2×C4—82no—
365C2×C4—2924no—
366C2×C4—5842no—
367C2×C4—5842no—
368C2×C4#1455650611yescharacteristic
369C2×C4#1455650611yescharacteristic
370C8—5842no—
371C22—21no—
372C22—21no—
373C22—22no—
374C22—22no—
375C22—42no—
376C22—42no—
377C22—42no—
378C22—42no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22—42no—
384C22—81no—
385C22—81no—
386C22—81no—
387C22—81no—
388C22—81no—
389C22—81no—
390C22C2×#1455650611yescharacteristic
391C22#1798858311yescharacteristic
392C22C2×#1455651811yescharacteristic
393C4—21no—
394C4—42no—
395C4—42no—
396C4—41no—
397C4—41no—
398C4—41no—
399C4—41no—
400C4—81no—
401C4—5844no—
402C4#1798838012yes—
403C2—21no—
404C2—21no—
405C2—42no—
406C2—42no—
407C2—82no—
408C2#2260842211yescharacteristic
409C2#2261157211yescharacteristic
410C2C2×#1798838011yescharacteristic
subgroupG/H
centerC2×C4#14556506
commutatorC22×C4×C73C22×C4
FrattiniC2×C22⋊C4C22×D73
Fitting, prime radicalC73×#2397C2
socleC22×C73C2×#99
cosocleC2×C22⋊C4×C73C23
hypercenter#2397D73
hypocenterC73#11726
LCSG ▷ C22×C4×C73 ▷ C23×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2397

Complements

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

NH ≅ G/Ntype
1. C4×#14556506C2semidirect
2. #22608249C2semidirect
3. #22607955C2semidirect
4. #22607928C2semidirect
14. C2×C4×D4×C73C4semidirect
17. #399×C73C4semidirect
20. #392×C73C4semidirect
22. #376×C73C4semidirect
211. C73#11726semidirect
214. C2×C4×D4Dic73semidirect
219. #399Dic73semidirect
222. #392Dic73semidirect
225. #376Dic73semidirect

Maximal subgroups

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

subgroupindex
C4×#145565062
#226082492
#226079552
#226079282
C73×#23972
#1172673