Finite Group Databasebeta

← #33004651

Subgroups — #33004651

Order 18688.

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

normal subgroups only

This group has 488 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#22609923C211yesmaximal, characteristic
2C4×#14556506C211yesmaximal, characteristic
3#22608049C212yesmaximal
4#22608049C212yesmaximal
5C73×#2579C211yesmaximal, characteristic
6#17988611—21no—
7C2×#14556508C2212yes—
8C2×#14556506C2212yes—
9C4×C22⋊Dic73—21no—
10#17988356—21no—
11#17988354—21no—
12C2×C4×D4×C73C2211yescharacteristic
13C22×C4⋊C4×C73C411yescharacteristic, complemented
14#398×C73—21no—
15#396×C73—21no—
16#387×C73—21no—
17#386×C73C2211yescharacteristic
18#386×C73C2211yescharacteristic
19#385×C73—21no—
20#385×C73C411yescharacteristic, complemented
21#385×C73C411yescharacteristic, complemented
22#380×C73C411yescharacteristic, complemented
23C2×#11924012—22no—
24C2×C22⋊Dic73—22no—
25C2×C4×Dic73—41no—
26C4⋊#10596831—41no—
27C4⋊#10596831—41no—
28#14556508—24no—
29#14556506—22no—
30#14556506Q814yes—
31#14556498—41no—
32C22×D4×C73C2311yescharacteristic
33C23×C4×C73C2×C411yescharacteristic
34C23×C4×C73C2×C411yescharacteristic
35C23×C4×C73C2×C411yescharacteristic
36C23×C4×C73D411yescharacteristic
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×C73—41no—
45C2×C4⋊C4×C73—41no—
46C2×C4⋊C4×C73D411yescharacteristic
47C2×C22⋊C4×C73—21no—
48C2×C22⋊C4×C73—21no—
49C2×C22⋊C4×C73C2×C411yescharacteristic
50C2×C22⋊C4×C73C2×C411yescharacteristic
51C2×C22⋊C4×C73C2×C411yescharacteristic
52C2×C22⋊C4×C73D411yescharacteristic
53C2×C42×C73D411yescharacteristic
54#95×C73—21no—
55#95×C73—21no—
56#95×C73—21no—
57#95×C73—21no—
58#95×C73—21no—
59C22×Dic73—42no—
60#11924012—42no—
61#11924012—42no—
62C2×#10596831—42no—
63C22⋊Dic73—24no—
64C22⋊Dic73—42no—
65C4×Dic73—82no—
66C24×C73C22×C411yescharacteristic
67C24×C73C2×D411yescharacteristic
68C2×D4×C73—41no—
69C2×D4×C73—41no—
70C2×D4×C73C2×Q812yes—
71C2×D4×C73#4112yes—
72C22×C4×C73—21no—
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—42no—
81C22×C4×C73—41no—
82C22×C4×C73—41no—
83C22×C4×C73—81no—
84C22×C4×C73—81no—
85C22×C4×C73—81no—
86C22×C4×C73C22⋊C411yescharacteristic
87C22×C4×C73C22⋊C411yescharacteristic
88C22×C4×C73C22⋊C411yescharacteristic
89C22×C4×C73C22⋊C411yescharacteristic
90C22×C4×C73C2×D411yescharacteristic
91C4⋊C4×C73—41no—
92C4⋊C4×C73—42no—
93C4⋊C4×C73—42no—
94C4⋊C4×C73—41no—
95C4⋊C4×C73—41no—
96C4⋊C4×C73—81no—
97C4⋊C4×C73—81no—
98C22⋊C4×C73—41no—
99C22⋊C4×C73—41no—
100C22⋊C4×C73—42no—
101C22⋊C4×C73—41no—
102C22⋊C4×C73—41no—
103C22⋊C4×C73—81no—
104C42×C73—21no—
105C42×C73—21no—
106C23×C73—21no—
107C23×C73—21no—
108C23×C73—21no—
109C23×C73—21no—
110C23×C73—41no—
111C23×C73—41no—
112C23×C73—41no—
113C23×C73—41no—
114C23×C73C2×C22⋊C411yescharacteristic
115C23×C73C4×Q812yes—
116C23×C73C22⋊Q812yes—
117D4×C73—42no—
118D4×C73—42no—
119C2×C4×C73—21no—
120C2×C4×C73—21no—
121C2×C4×C73—22no—
122C2×C4×C73—21no—
123C2×C4×C73—21no—
124C2×C4×C73—21no—
125C2×C4×C73—41no—
126C2×C4×C73—42no—
127C2×C4×C73—41no—
128C2×C4×C73—42no—
129C2×C4×C73—41no—
130C2×C4×C73—41no—
131C2×C4×C73—41no—
132C2×C4×C73—41no—
133C2×C4×C73—42no—
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—81no—
144C2×C4×C73—81no—
145C2×C4×C73—81no—
146C2×C4×C73—82no—
147C2×C4×C73#9911yescharacteristic
148C2×C4×C73#9911yescharacteristic
149C2×C4×C73#12411yescharacteristic
150C2×C4×C73C4⋊Q811yescharacteristic
151C2×Dic73—44no—
152C2×Dic73—82no—
153C2×Dic73—82no—
154#10596831—82no—
155C22×C73—21no—
156C22×C73—21no—
157C22×C73—22no—
158C22×C73—22no—
159C22×C73—42no—
160C22×C73—42no—
161C22×C73—42no—
162C22×C73—41no—
163C22×C73—41no—
164C22×C73—41no—
165C22×C73—41no—
166C22×C73—81no—
167C22×C73#39111yescharacteristic
168C22×C73C2×#9911yescharacteristic
169C22×C73#41011yescharacteristic
170C4×C73—21no—
171C4×C73—21no—
172C4×C73—21no—
173C4×C73—21no—
174C4×C73—41no—
175C4×C73—42no—
176C4×C73—41no—
177C4×C73—41no—
178C4×C73—81no—
179C4×C73—81no—
180C4×C73—81no—
181Dic73—84no—
182#13297—731nomaximal, Sylow
183C2×C73—21no—
184C2×C73—21no—
185C2×C73—42no—
186C2×C73—42no—
187C2×C73#199611yescharacteristic
188C2×C73#212011yescharacteristic
189C2×C73#212111yescharacteristic
190#2579D7311yescharacteristic
191#1800—731no—
192C4×#99—731no—
193#1390—732no—
194#1390—732no—
195C73#1329711yescharacteristic, complemented, Sylow
196C2×C4×D4C2×D7311yescharacteristic
197C22×C4⋊C4Dic7311yescharacteristic, complemented
198#423—1461no—
199C2×#100—732no—
200C2×#99—732no—
201#398—21no—
202#396—21no—
203#387—21no—
204#386C2×D7311yescharacteristic
205#386C2×D7311yescharacteristic
206#385—21no—
207#385Dic7311yescharacteristic, complemented
208#385Dic7311yescharacteristic, complemented
209#380Dic7311yescharacteristic, complemented
210C4×C22⋊C4—1461no—
211#343—1461no—
212#342—1461no—
213C22×D4C22×D7311yescharacteristic
214C23×C4C2×Dic7311yescharacteristic
215C23×C4C2×Dic7311yescharacteristic
216C23×C4C2×Dic7311yescharacteristic
217C23×C4C22⋊D7311yescharacteristic
218C2×M16—1462no—
219C4×D4—41no—
220C4×D4—41no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—41no—
227C2×C4⋊C4—41no—
228C2×C4⋊C4C22⋊D7311yescharacteristic
229C2×C22⋊C4—21no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—1462no—
232C2×C22⋊C4C2×Dic7311yescharacteristic
233C2×C22⋊C4C2×Dic7311yescharacteristic
234C2×C22⋊C4C2×Dic7311yescharacteristic
235C2×C22⋊C4C22⋊D7311yescharacteristic
236C2×C42—2921no—
237C2×C42C22⋊D7311yescharacteristic
238C4⋊C8—2921no—
239C4⋊C8—2921no—
240#100—1464no—
241#99—734no—
242#99—1462no—
243#95—21no—
244#95—21no—
245#95—21no—
246#95—21no—
247#95—21no—
248#95—2921no—
249C24C22×Dic7311yescharacteristic
250C24C2×C22⋊D7311yescharacteristic
251C2×D4—41no—
252C2×D4—41no—
253C2×D4Q8×D7312yes—
254C2×D4#1192402912yes—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—42no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—81no—
267C22×C4—81no—
268C22×C4—81no—
269C22×C4—2922no—
270C22×C4C22⋊Dic7311yescharacteristic
271C22×C4C22⋊Dic7311yescharacteristic
272C22×C4C22⋊Dic7311yescharacteristic
273C22×C4C22⋊Dic7311yescharacteristic
274C22×C4C2×C22⋊D7311yescharacteristic
275M16—2922no—
276M16—2922no—
277C2×C8—2922no—
278C4⋊C4—41no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—81no—
284C4⋊C4—81no—
285C22⋊C4—41no—
286C22⋊C4—41no—
287C22⋊C4—42no—
288C22⋊C4—41no—
289C22⋊C4—41no—
290C22⋊C4—81no—
291C22⋊C4—2922no—
292C22⋊C4—1464no—
293C42—21no—
294C42—21no—
295C42—5842no—
296C23—21no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23—41no—
301C23—41no—
302C23—41no—
303C23—41no—
304C23C2×C22⋊Dic7311yescharacteristic
305C23Q8×Dic7312yes—
306C23#1455656712yes—
307D4—42no—
308D4—42no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—22no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—41no—
316C2×C4—42no—
317C2×C4—41no—
318C2×C4—42no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—42no—
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—81no—
334C2×C4—81no—
335C2×C4—81no—
336C2×C4—82no—
337C2×C4—2924no—
338C2×C4—5842no—
339C2×C4—5842no—
340C2×C4#1455650611yescharacteristic
341C2×C4#1455650611yescharacteristic
342C2×C4#1455658011yescharacteristic
343C2×C4#1455658911yescharacteristic
344C8—5842no—
345C22—21no—
346C22—21no—
347C22—22no—
348C22—22no—
349C22—42no—
350C22—42no—
351C22—42no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—81no—
357C22#1798849411yescharacteristic
358C22C2×#1455650611yescharacteristic
359C22#1798856111yescharacteristic
360C4—21no—
361C4—21no—
362C4—21no—
363C4—21no—
364C4—42no—
365C4—41no—
366C4—41no—
367C4—41no—
368C4—81no—
369C4—81no—
370C4—81no—
371C4—5844no—
372C2—21no—
373C2—21no—
374C2—42no—
375C2—42no—
376C2#2261091711yescharacteristic
377C2#2261157211yescharacteristic
378C2#2261157711yescharacteristic
subgroupG/H
centerC22C2×#14556506
commutatorC24×C73C22×C4
FrattiniC22×D4C22×D73
FittingC73×#2579C2
socleC22×C73C2×#99
prime radical, cosocleC22×D4×C73C23
hypercenter#2579D73
hypocenterC73#13297
LCSG ▷ C24×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2579

Complements

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

NH ≅ G/Ntype
13. C22×C4⋊C4×C73C4semidirect
20. #385×C73C4semidirect
21. #385×C73C4semidirect
22. #380×C73C4semidirect
195. C73#13297semidirect
197. C22×C4⋊C4Dic73semidirect
207. #385Dic73semidirect
208. #385Dic73semidirect
209. #380Dic73semidirect

Maximal subgroups

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

subgroupindex
#226099232
C4×#145565062
#226080492
C73×#25792
#1329773