Finite Group Databasebeta

← #34006347

Subgroups — #34006347

Order 18688.

588 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 588 conjugacy classes of subgroups — too many to draw. Only its 122 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
1C2×#17989604C211yesmaximal, characteristic
2#22625787C211yesmaximal, characteristic
3C2×#17989447C212yesmaximal, complemented
4#22624809C212yesmaximal, complemented
5#22618279C211yesmaximal, characteristic
6#17989604—24no—
7C2×#14556658C2212yes—
8C2×#14556658C2212yes—
9C2×C22⋊#10596833—21no—
10C2×C22⋊#10596833—21no—
11C2×C4×#10596833—21no—
12#17989453—21no—
13#17989447—24no—
14C2×#14556589C2211yescharacteristic
15C2×#14556570C411yescharacteristic
16C22×C4×Dic73C2211yescharacteristic
17#17988494C411yescharacteristic
18#17988492C411yescharacteristic
19#17988482—21no—
20#17988482—21no—
21#17988456C2211yescharacteristic
22#17988445—21no—
23#17988445—21no—
24C2×C22⋊Q8×C73C411yescharacteristic
25C22×#10596833—22no—
26C22×#10596833—22no—
27#14556658—22no—
28#14556658—22no—
29#14556658—22no—
30#14556658D414yes—
31C22⋊#10596833—24no—
32C22⋊#10596833—42no—
33C4×#10596833—24no—
34C22×Dic146C2×C411yescharacteristic
35C23×Dic73D411yescharacteristic
36C23×Dic73D411yescharacteristic
37#14556589—22no—
38#14556589—22no—
39#14556570—42no—
40C2×C4⋊Dic73C2×C411yescharacteristic
41C2×#11924007—21no—
42C2×#11924007—21no—
43C2×#11924007—21no—
44C2×#11924007—21no—
45C2×#11924007—21no—
46C2×C22⋊Dic73—21no—
47C2×C4×Dic73—22no—
48C2×C4×Dic73—22no—
49C2×C4×Dic73C2×C411yescharacteristic
50C2×C4×Dic73D411yescharacteristic
51C2×C4×Dic73D411yescharacteristic
52C2×C4×Dic73C2311yescharacteristic
53#14556498—21no—
54#14556498—21no—
55#14556498—21no—
56#14556498—21no—
57C22×Q8×C73C2×C411yescharacteristic
58C23×C4×C73C2×C411yescharacteristic
59C22⋊Q8×C73—42no—
60C2×C4⋊C4×C73—21no—
61C2×C4⋊C4×C73C2×C411yescharacteristic
62C2×C22⋊C4×C73—21no—
63C2×#10596833—24no—
64C2×#10596833—42no—
65C2×#10596833—42no—
66C2×#10596833—42no—
67C2×#10596833—42no—
68C2×#10596833—42no—
69C2×Dic146—22no—
70C2×Dic146—41no—
71C2×Dic146—41no—
72C22×Dic73—22no—
73C22×Dic73—22no—
74C22×Dic73—21no—
75C22×Dic73—21no—
76C22×Dic73—22no—
77C22×Dic73—22no—
78C22×Dic73—21no—
79C22×Dic73—21no—
80C22×Dic73C22⋊C411yescharacteristic
81C22×Dic73C22⋊C411yescharacteristic
82C22×Dic73SD1612yes—
83C22×Dic73Q1612yes—
84C22×Dic73C2×D411yescharacteristic
85C22×Dic73C2×D411yescharacteristic
86C4⋊Dic73—41no—
87#11924007—42no—
88#11924007—41no—
89#11924007—42no—
90#11924007—41no—
91#11924007—42no—
92#11924007—42no—
93C22⋊Dic73—42no—
94C4×Dic73—22no—
95C4×Dic73—21no—
96C4×Dic73—22no—
97C4×Dic73—21no—
98C4×Dic73C2×D412yes—
99C4×Dic73#4112yes—
100C24×C73C22⋊C411yescharacteristic
101C2×Q8×C73—22no—
102C2×Q8×C73—41no—
103C2×Q8×C73—41no—
104C22×C4×C73—22no—
105C22×C4×C73—22no—
106C22×C4×C73—21no—
107C22×C4×C73—21no—
108C22×C4×C73—41no—
109C22×C4×C73C22⋊C411yescharacteristic
110C22×C4×C73C22×C411yescharacteristic
111C4⋊C4×C73—41no—
112C4⋊C4×C73—42no—
113C22⋊C4×C73—42no—
114C23×C73—21no—
115C23×C73—21no—
116C23×C73—21no—
117C23×C73—21no—
118C23×C73—21no—
119C23×C73—21no—
120C23×C73#10312yes—
121C23×C73C2×C22⋊C411yescharacteristic
122Q8×C73—82no—
123C2×C4×C73—21no—
124C2×C4×C73—21no—
125C2×C4×C73—21no—
126C2×C4×C73—21no—
127C2×C4×C73—44no—
128C2×C4×C73—41no—
129C2×C4×C73—41no—
130C2×C4×C73—41no—
131C2×C4×C73—41no—
132C2×C4×C73—41no—
133C2×C4×C73—41no—
134C2×C4×C73#10012yes—
135C2×C4×C73C4×D412yes—
136C2×Dic73—21no—
137C2×Dic73—21no—
138C2×Dic73—24no—
139C2×Dic73—24no—
140C2×Dic73—21no—
141C2×Dic73—21no—
142C2×Dic73—21no—
143C2×Dic73—42no—
144C2×Dic73—42no—
145C2×Dic73—41no—
146C2×Dic73—41no—
147C2×Dic73—41no—
148C2×Dic73—41no—
149C2×Dic73—41no—
150C2×Dic73—41no—
151C2×Dic73#10312yes—
152C2×Dic73#12011yescharacteristic
153C2×Dic73#12112yes—
154C2×Dic73#12311yescharacteristic
155C2×Dic73C2×SD1611yescharacteristic
156C2×Dic73C2×Q1611yescharacteristic
157C2×Dic73#13512yes—
158C2×Dic73#13512yes—
159C2×Dic73#13712yes—
160Dic146—82no—
161#10596833—44no—
162#10596833—82no—
163C22×C73—22no—
164C22×C73—22no—
165C22×C73—22no—
166C22×C73—41no—
167C22×C73—41no—
168C22×C73—41no—
169C22×C73—41no—
170C22×C73#38611yescharacteristic
171C22×C73C2×#10011yescharacteristic
172C22×C73#41311yescharacteristic
173C22×C73C2×#10311yescharacteristic
174C22×C73#41611yescharacteristic
175C22×C73#41611yescharacteristic
176C22×C73#41911yescharacteristic
177C4×C73—22no—
178C4×C73—22no—
179C4×C73—81no—
180C4×C73—81no—
181Dic73—22no—
182Dic73—41no—
183Dic73—81no—
184Dic73—81no—
185Dic73#45212yes—
186Dic73#46512yes—
187Dic73#46712yes—
188Dic73#48512yes—
189#9128—731nomaximal, Sylow
190C2×C73—42no—
191C2×C73#154211yescharacteristic
192C2×C73#154311yescharacteristic
193C2×C73#154811yescharacteristic
194C2×C73#155411yescharacteristic
195C2×C73#187311yescharacteristic
196C2×C73#189211yescharacteristic
197C2×C73#190011yescharacteristic
198C2×#431—731no—
199#2857—731no—
200#1797—731no—
201C2×#333—732no—
202#1296—732no—
203C73#912811yescharacteristic, complemented, Sylow
204C2×C4⋊Q8—731no—
205C2×C22⋊Q8—731no—
206C2×C22⋊Q8#6440811yescharacteristic
207C22×C42—731no—
208#431—1464no—
209C2×C4⋊C8—732no—
210C2×C4⋊C8—732no—
211C2×C22⋊C8—1461no—
212C2×C22⋊C8—1461no—
213C2×C4×C8—1461no—
214#391—731no—
215#391—731no—
216#389—1461no—
217#389—1461no—
218#384—731no—
219#380—1461no—
220#380—1461no—
221#336—1461no—
222#333—1464no—
223C22×Q8—731no—
224C22×Q8C2×#6440811yescharacteristic
225C23×C4—731no—
226C23×C4—731no—
227C23×C4C2×#6440811yescharacteristic
228C22×C8—1462no—
229C22×C8—1462no—
230C4⋊Q8—1462no—
231C4⋊Q8—1462no—
232C22⋊Q8—42no—
233C22⋊Q8—2922no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—731no—
236C2×C4⋊C4—1461no—
237C2×C4⋊C4—1461no—
238C2×C4⋊C4—1461no—
239C2×C4⋊C4—1461no—
240C2×C4⋊C4—1461no—
241C2×C4⋊C4C2×#6440811yescharacteristic
242C2×C22⋊C4—21no—
243C2×C22⋊C4—1461no—
244C2×C42—731no—
245C2×C42—731no—
246C2×C42—731no—
247C2×C42—731no—
248C2×C42—1462no—
249C2×C42—1462no—
250C4⋊C8—734no—
251C4⋊C8—1462no—
252C4⋊C8—1462no—
253C4⋊C8—1462no—
254C22⋊C8—2922no—
255C22⋊C8—1464no—
256C4×C8—1464no—
257#95—1461no—
258#95—1461no—
259#95—1461no—
260#95—1461no—
261C24C22⋊#6440811yescharacteristic
262C2×Q8—22no—
263C2×Q8—41no—
264C2×Q8—41no—
265C2×Q8—2921no—
266C2×Q8—2921no—
267C2×Q8—1462no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—41no—
273C22×C4—731no—
274C22×C4—731no—
275C22×C4—732no—
276C22×C4—732no—
277C22×C4—731no—
278C22×C4—731no—
279C22×C4—1461no—
280C22×C4—1461no—
281C22×C4—1461no—
282C22×C4—1461no—
283C22×C4—1462no—
284C22×C4—1462no—
285C22×C4—1462no—
286C22×C4—1462no—
287C22×C4C22⋊#6440811yescharacteristic
288C22×C4C22×#6440811yescharacteristic
289C2×C8—2922no—
290C2×C8—2922no—
291C2×C8—2922no—
292C2×C8—2922no—
293C2×C8—2922no—
294C2×C8—1464no—
295C4⋊C4—41no—
296C4⋊C4—42no—
297C4⋊C4—2921no—
298C4⋊C4—2922no—
299C4⋊C4—2922no—
300C4⋊C4—2922no—
301C4⋊C4—2921no—
302C4⋊C4—2921no—
303C4⋊C4—2922no—
304C22⋊C4—42no—
305C22⋊C4—2922no—
306C42—732no—
307C42—732no—
308C42—1461no—
309C42—1462no—
310C42—1461no—
311C42—1462no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23#1455665412yes—
319C23C2×C22⋊#6440811yescharacteristic
320Q8—82no—
321Q8—5842no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—44no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—2922no—
334C2×C4—2922no—
335C2×C4—2921no—
336C2×C4—2921no—
337C2×C4—2921no—
338C2×C4—2921no—
339C2×C4—2921no—
340C2×C4—2921no—
341C2×C4—732no—
342C2×C4—731no—
343C2×C4—731no—
344C2×C4—731no—
345C2×C4—731no—
346C2×C4—732no—
347C2×C4—732no—
348C2×C4—732no—
349C2×C4—732no—
350C2×C4—1461no—
351C2×C4—1461no—
352C2×C4—1461no—
353C2×C4—1464no—
354C2×C4—1464no—
355C2×C4—1461no—
356C2×C4—1461no—
357C2×C4#1455665112yes—
358C2×C4D4×#6440812yes—
359C8—2924no—
360C8—5842no—
361C22—22no—
362C22—22no—
363C22—22no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22#1798953511yescharacteristic
369C22C2×#1455665111yescharacteristic
370C22#1798957411yescharacteristic
371C22C2×#1455665411yescharacteristic
372C22#1798957711yescharacteristic
373C22#1798957711yescharacteristic
374C22#1798958011yescharacteristic
375C4—22no—
376C4—22no—
377C4—81no—
378C4—81no—
379C4—2921no—
380C4—5841no—
381C4—5841no—
382C4—732no—
383C4—732no—
384C4—732no—
385C4—732no—
386C4—1462no—
387C2—42no—
388C2#2262543311yescharacteristic
389C2#2262543411yescharacteristic
390C2#2262544411yescharacteristic
391C2#2262545711yescharacteristic
392C2#2262591711yescharacteristic
393C2#2262594411yescharacteristic
394C2#2262595611yescharacteristic
subgroupG/H
centerC23C2×C22⋊#64408
commutatorC22×C4×C73C22×C4
FrattiniC22×C4C22×#64408
FittingC2×C22⋊Q8×C73C4
socleC23×C73C2×C22⋊C4
prime radicalC24×C73C22⋊C4
cosocleC2×C4×Dic73C23
hypercenterC2×C22⋊Q8#64408
hypocenterC73#9128
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. C2×#17989447C2semidirect
4. #22624809C2semidirect
203. C73#9128semidirect

Maximal subgroups

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

subgroupindex
C2×#179896042
#226257872
C2×#179894472
#226248092
#226182792
#912873