Finite Group Databasebeta

← #33004664

Subgroups — #33004664

Order 18688.

492 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 107 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#22609919C211yesmaximal, characteristic
2C4×#14556506C211yesmaximal, characteristic
3#22608056C212yesmaximal
4#22608056C212yesmaximal
5C73×#2566C211yesmaximal, characteristic
6C2×#14556506C2212yes—
7C2×#14556506C2212yes—
8#17988447—21no—
9C4×C22⋊Dic73—21no—
10#17988354—21no—
11#17988354—21no—
12C2×#125×C73C411yescharacteristic, complemented
13C2×C4×D4×C73C2211yescharacteristic
14#396×C73—21no—
15#396×C73—21no—
16#390×C73C2211yescharacteristic
17#390×C73C2211yescharacteristic
18#385×C73—21no—
19#385×C73—21no—
20#384×C73C411yescharacteristic, complemented
21#382×C73C411yescharacteristic, complemented
22C4×C4⋊C4×C73C411yescharacteristic, complemented
23C2×C4⋊Dic73—41no—
24C2×C22⋊Dic73—22no—
25C2×C22⋊Dic73—22no—
26C2×C4×Dic73—41no—
27#14556506—22no—
28#14556506—24no—
29#14556506Q814yes—
30#14556498—41no—
31#14556498—41no—
32C22×D4×C73C2311yescharacteristic
33C23×C4×C73D411yescharacteristic
34C23×C4×C73D411yescharacteristic
35#125×C73—41no—
36#125×C73—41no—
37C4×D4×C73—41no—
38C4×D4×C73—41no—
39C2×C4⋊C4×C73—21no—
40C2×C4⋊C4×C73—21no—
41C2×C4⋊C4×C73—21no—
42C2×C4⋊C4×C73—21no—
43C2×C4⋊C4×C73—21no—
44C2×C4⋊C4×C73C2×C411yescharacteristic
45C2×C4⋊C4×C73C2×C411yescharacteristic
46C2×C4⋊C4×C73C2×C411yescharacteristic
47C2×C22⋊C4×C73—21no—
48C2×C22⋊C4×C73—21no—
49C2×C22⋊C4×C73—21no—
50C2×C22⋊C4×C73—21no—
51C2×C22⋊C4×C73D411yescharacteristic
52C2×C22⋊C4×C73D411yescharacteristic
53C2×C42×C73C2×C411yescharacteristic
54C2×C42×C73C2×C411yescharacteristic
55C2×C42×C73C2×C411yescharacteristic
56#95×C73—21no—
57#95×C73—21no—
58#95×C73—21no—
59C22×Dic73—42no—
60C22×Dic73—42no—
61C4⋊Dic73—82no—
62C22⋊Dic73—24no—
63C22⋊Dic73—42no—
64C22⋊Dic73—42no—
65C22⋊Dic73—42no—
66C4×Dic73—82no—
67C24×C73C2×D411yescharacteristic
68C24×C73C2×D411yescharacteristic
69C2×D4×C73—41no—
70C2×D4×C73—41no—
71C2×D4×C73C2×Q812yes—
72C2×D4×C73#4112yes—
73C22×C4×C73—21no—
74C22×C4×C73—21no—
75C22×C4×C73—21no—
76C22×C4×C73—21no—
77C22×C4×C73—21no—
78C22×C4×C73—21no—
79C22×C4×C73—21no—
80C22×C4×C73—21no—
81C22×C4×C73—81no—
82C22×C4×C73—81no—
83C22×C4×C73C22⋊C411yescharacteristic
84C22×C4×C73C22⋊C411yescharacteristic
85C22×C4×C73C22⋊C411yescharacteristic
86C22×C4×C73C22⋊C411yescharacteristic
87C22×C4×C73C22×C411yescharacteristic
88C4⋊C4×C73—41no—
89C4⋊C4×C73—41no—
90C4⋊C4×C73—41no—
91C4⋊C4×C73—42no—
92C4⋊C4×C73—42no—
93C4⋊C4×C73—41no—
94C4⋊C4×C73—41no—
95C4⋊C4×C73—81no—
96C4⋊C4×C73—81no—
97C22⋊C4×C73—41no—
98C22⋊C4×C73—41no—
99C22⋊C4×C73—42no—
100C22⋊C4×C73—42no—
101C22⋊C4×C73—81no—
102C22⋊C4×C73—81no—
103C42×C73—21no—
104C42×C73—21no—
105C42×C73—41no—
106C42×C73—41no—
107C23×C73—21no—
108C23×C73—21no—
109C23×C73—21no—
110C23×C73—21no—
111C23×C73—41no—
112C23×C73—41no—
113C23×C73—41no—
114C23×C73—41no—
115C23×C73C2×C22⋊C411yescharacteristic
116C23×C73C22⋊Q812yes—
117C23×C73#12411yescharacteristic
118C23×C73C4⋊Q811yescharacteristic
119D4×C73—42no—
120D4×C73—42no—
121C2×C4×C73—21no—
122C2×C4×C73—21no—
123C2×C4×C73—22no—
124C2×C4×C73—21no—
125C2×C4×C73—21no—
126C2×C4×C73—21no—
127C2×C4×C73—41no—
128C2×C4×C73—42no—
129C2×C4×C73—42no—
130C2×C4×C73—41no—
131C2×C4×C73—41no—
132C2×C4×C73—41no—
133C2×C4×C73—41no—
134C2×C4×C73—41no—
135C2×C4×C73—41no—
136C2×C4×C73—41no—
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—81no—
145C2×C4×C73—81no—
146C2×C4×C73#9911yescharacteristic
147C2×C4×C73#9911yescharacteristic
148C2×C4×C73C4×Q812yes—
149C2×Dic73—44no—
150C2×Dic73—82no—
151C2×Dic73—82no—
152C2×Dic73—82no—
153C2×Dic73—82no—
154C2×Dic73—82no—
155C22×C73—21no—
156C22×C73—21no—
157C22×C73—22no—
158C22×C73—21no—
159C22×C73—21no—
160C22×C73—42no—
161C22×C73—41no—
162C22×C73—41no—
163C22×C73—41no—
164C22×C73—41no—
165C22×C73—41no—
166C22×C73—41no—
167C22×C73—81no—
168C22×C73—81no—
169C22×C73#39111yescharacteristic
170C22×C73C2×#9911yescharacteristic
171C22×C73#41011yescharacteristic
172C4×C73—21no—
173C4×C73—21no—
174C4×C73—22no—
175C4×C73—42no—
176C4×C73—41no—
177C4×C73—41no—
178C4×C73—41no—
179C4×C73—81no—
180C4×C73—81no—
181C4×C73—81no—
182Dic73—84no—
183Dic73—162no—
184#13300—731nomaximal, Sylow
185C2×C73—21no—
186C2×C73—21no—
187C2×C73—42no—
188C2×C73—41no—
189C2×C73—41no—
190C2×C73#199611yescharacteristic
191C2×C73#212611yescharacteristic
192C2×C73#212711yescharacteristic
193#2566D7311yescharacteristic
194#1799—731no—
195C4×#99—731no—
196#1393—732no—
197#1393—732no—
198C73#1330011yescharacteristic, complemented, Sylow
199C2×#125Dic7311yescharacteristic, complemented
200C2×C4×D4C2×D7311yescharacteristic
201C2×#99—732no—
202C2×#99—732no—
203#396—21no—
204#396—21no—
205#390C2×D7311yescharacteristic
206#390C2×D7311yescharacteristic
207#385—21no—
208#385—21no—
209#384Dic7311yescharacteristic, complemented
210#382Dic7311yescharacteristic, complemented
211#380—1461no—
212C4×C4⋊C4Dic7311yescharacteristic, complemented
213C4×C22⋊C4—1461no—
214#342—1461no—
215#342—1461no—
216C22×D4C22×D7311yescharacteristic
217C23×C4C22⋊D7311yescharacteristic
218C23×C4C22⋊D7311yescharacteristic
219#125—41no—
220#125—41no—
221C4×D4—41no—
222C4×D4—41no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—2921no—
229C2×C4⋊C4C2×Dic7311yescharacteristic
230C2×C4⋊C4C2×Dic7311yescharacteristic
231C2×C4⋊C4C2×Dic7311yescharacteristic
232C2×C22⋊C4—21no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—1462no—
237C2×C22⋊C4—1462no—
238C2×C22⋊C4C22⋊D7311yescharacteristic
239C2×C22⋊C4C22⋊D7311yescharacteristic
240C2×C42—2921no—
241C2×C42C2×Dic7311yescharacteristic
242C2×C42C2×Dic7311yescharacteristic
243C2×C42C2×Dic7311yescharacteristic
244#99—734no—
245#99—1464no—
246#99—1462no—
247#95—21no—
248#95—21no—
249#95—21no—
250#95—2921no—
251#95—2921no—
252C24C2×C22⋊D7311yescharacteristic
253C24C2×C22⋊D7311yescharacteristic
254C2×D4—41no—
255C2×D4—41no—
256C2×D4Q8×D7312yes—
257C2×D4#1192402912yes—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—81no—
267C22×C4—81no—
268C22×C4—2922no—
269C22×C4—2922no—
270C22×C4C22⋊Dic7311yescharacteristic
271C22×C4C22⋊Dic7311yescharacteristic
272C22×C4C22⋊Dic7311yescharacteristic
273C22×C4C22⋊Dic7311yescharacteristic
274C22×C4C22×Dic7311yescharacteristic
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—42no—
279C4⋊C4—42no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—81no—
283C4⋊C4—81no—
284C4⋊C4—5842no—
285C22⋊C4—41no—
286C22⋊C4—41no—
287C22⋊C4—42no—
288C22⋊C4—42no—
289C22⋊C4—81no—
290C22⋊C4—81no—
291C22⋊C4—2922no—
292C22⋊C4—2922no—
293C22⋊C4—2922no—
294C22⋊C4—1464no—
295C42—21no—
296C42—21no—
297C42—41no—
298C42—41no—
299C42—5842no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—41no—
305C23—41no—
306C23—41no—
307C23—41no—
308C23C2×C22⋊Dic7311yescharacteristic
309C23#1455656712yes—
310C23#1455658011yescharacteristic
311C23#1455658911yescharacteristic
312D4—42no—
313D4—42no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—22no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—41no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4—2924no—
340C2×C4—5842no—
341C2×C4—5842no—
342C2×C4—5842no—
343C2×C4—5842no—
344C2×C4—5842no—
345C2×C4#1455650611yescharacteristic
346C2×C4#1455650611yescharacteristic
347C2×C4Q8×Dic7312yes—
348C22—21no—
349C22—21no—
350C22—22no—
351C22—21no—
352C22—21no—
353C22—42no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—81no—
361C22—81no—
362C22#1798849411yescharacteristic
363C22C2×#1455650611yescharacteristic
364C22#1798856111yescharacteristic
365C4—21no—
366C4—21no—
367C4—22no—
368C4—42no—
369C4—41no—
370C4—41no—
371C4—41no—
372C4—81no—
373C4—81no—
374C4—81no—
375C4—5844no—
376C4—11682no—
377C2—21no—
378C2—21no—
379C2—42no—
380C2—41no—
381C2—41no—
382C2#2261091711yescharacteristic
383C2#2261159611yescharacteristic
384C2#2261160011yescharacteristic
subgroupG/H
centerC22C2×#14556506
commutatorC22×C4×C73C22×C4
FrattiniC22×D4C22×D73
FittingC73×#2566C2
socleC22×C73C2×#99
prime radical, cosocleC22×D4×C73C23
hypercenter#2566D73
hypocenterC73#13300
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2566

Complements

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

NH ≅ G/Ntype
12. C2×#125×C73C4semidirect
20. #384×C73C4semidirect
21. #382×C73C4semidirect
22. C4×C4⋊C4×C73C4semidirect
198. C73#13300semidirect
199. C2×#125Dic73semidirect
209. #384Dic73semidirect
210. #382Dic73semidirect
212. C4×C4⋊C4Dic73semidirect

Maximal subgroups

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

subgroupindex
#226099192
C4×#145565062
#226080562
C73×#25662
#1330073