Finite Group Databasebeta

← #34001441

Subgroups — #34001441

Order 18688.

588 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 588 conjugacy classes of subgroups — too many to draw. Only its 118 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#22625821C211yesmaximal, characteristic, complemented
2#22625785C211yesmaximal, characteristic, complemented
3#22624749C212yesmaximal, complemented
4#22624747C212yesmaximal, complemented
5#121×Dic73C211yesmaximal, characteristic
6C2×C4⋊#10596833—21no—
7C2×C22⋊#10596833—21no—
8C2×C22⋊#10596833—21no—
9#17989453—21no—
10#17989453C2212yes—
11#17989453C2212yes—
12C2×D4×Dic73—21no—
13C2×D4×Dic73C411yescharacteristic
14C22×C4×Dic73C2211yescharacteristic
15#17988490C2211yescharacteristic
16#17988486C2211yescharacteristic
17#17988480—21no—
18#17988467—21no—
19#17988447C411yescharacteristic
20C4⋊C4×Dic73C411yescharacteristic
21C22⋊C4×Dic73—21no—
22C2×#121×C73C411yescharacteristic
23C22×#10596833—22no—
24C22×#10596833—22no—
25C4⋊#10596833—42no—
26C22⋊#10596833—42no—
27C22⋊#10596833—42no—
28C23×Dic73—21no—
29C23×Dic73D411yescharacteristic
30C23×Dic73D411yescharacteristic
31D4×Dic73—42no—
32D4×Dic73—44no—
33C2×C4⋊Dic73—21no—
34C2×C4⋊Dic73C2×C411yescharacteristic
35C2×C4⋊Dic73C2×C411yescharacteristic
36C2×C22⋊Dic73—21no—
37C2×C22⋊Dic73—21no—
38C2×C22⋊Dic73—21no—
39C2×C22⋊Dic73—21no—
40C2×C4×Dic73—21no—
41C2×C4×Dic73—42no—
42C2×C4×Dic73C2×C411yescharacteristic
43C2×C4×Dic73D411yescharacteristic
44C2×C4×Dic73D411yescharacteristic
45C2×C4×Dic73C2311yescharacteristic
46#14556498—21no—
47#14556498—21no—
48#14556498—21no—
49C22×D4×C73—21no—
50C22×D4×C73C2×C411yescharacteristic
51C23×C4×C73C2×C411yescharacteristic
52#121×C73—22no—
53#121×C73C814yescomplemented
54C2×C4⋊C4×C73C2×C411yescharacteristic
55C2×C22⋊C4×C73—21no—
56C2×#10596833—42no—
57C2×#10596833—42no—
58C2×#10596833—42no—
59C2×#10596833—42no—
60C2×#10596833—42no—
61C2×#10596833—42no—
62C22×Dic73—22no—
63C22×Dic73—22no—
64C22×Dic73—21no—
65C22×Dic73—21no—
66C22×Dic73—21no—
67C22×Dic73—21no—
68C22×Dic73—41no—
69C22×Dic73—41no—
70C22×Dic73—41no—
71C22×Dic73—42no—
72C22×Dic73—42no—
73C22×Dic73—42no—
74C22×Dic73C22⋊C411yescharacteristic
75C22×Dic73C22⋊C411yescharacteristic
76C22×Dic73C2×D411yescharacteristic
77C22×Dic73C2×D411yescharacteristic
78C4⋊Dic73—42no—
79C4⋊Dic73—41no—
80C4⋊Dic73—41no—
81C22⋊Dic73—42no—
82C22⋊Dic73—42no—
83C22⋊Dic73—41no—
84C22⋊Dic73—42no—
85C22⋊Dic73—41no—
86C4×Dic73—22no—
87C4×Dic73—22no—
88C4×Dic73—41no—
89C4×Dic73—41no—
90C4×Dic73—42no—
91C24×C73—21no—
92C24×C73C22⋊C411yescharacteristic
93C2×D4×C73—24no—
94C2×D4×C73—21no—
95C2×D4×C73—41no—
96C2×D4×C73—41no—
97C2×D4×C73—42no—
98C2×D4×C73—42no—
99C2×D4×C73C2×C812yes—
100C22×C4×C73—21no—
101C22×C4×C73—41no—
102C22×C4×C73—41no—
103C22×C4×C73C22⋊C411yescharacteristic
104C22×C4×C73C2×C812yes—
105C22×C4×C73M1612yes—
106C22×C4×C73C22×C411yescharacteristic
107C4⋊C4×C73—21no—
108C4⋊C4×C73C2×C812yes—
109C22⋊C4×C73—24no—
110C23×C73—21no—
111C23×C73—21no—
112C23×C73—21no—
113C23×C73—21no—
114C23×C73—21no—
115C23×C73—21no—
116C23×C73—22no—
117C23×C73—41no—
118C23×C73—41no—
119C23×C73—41no—
120C23×C73—41no—
121C23×C73—41no—
122C23×C73—41no—
123C23×C73C22⋊C812yes—
124C23×C73C2×C22⋊C411yescharacteristic
125D4×C73—42no—
126D4×C73—42no—
127D4×C73—48no—
128C2×C4×C73—21no—
129C2×C4×C73—21no—
130C2×C4×C73—21no—
131C2×C4×C73—21no—
132C2×C4×C73—22no—
133C2×C4×C73—42no—
134C2×C4×C73—42no—
135C2×C4×C73—41no—
136C2×C4×C73—41no—
137C2×C4×C73C22⋊C812yes—
138C2×C4×C73C22×C811yescharacteristic
139C2×C4×C73C2×M1611yescharacteristic
140C2×Dic73—21no—
141C2×Dic73—21no—
142C2×Dic73—21no—
143C2×Dic73—21no—
144C2×Dic73—21no—
145C2×Dic73—21no—
146C2×Dic73—21no—
147C2×Dic73—21no—
148C2×Dic73—21no—
149C2×Dic73—44no—
150C2×Dic73—42no—
151C2×Dic73—42no—
152C2×Dic73—41no—
153C2×Dic73—41no—
154C2×Dic73—41no—
155C2×Dic73—41no—
156C2×Dic73—41no—
157C2×Dic73—41no—
158C2×Dic73—84no—
159C2×Dic73#9912yes—
160C2×Dic73AGL(1,ℤ/8)11yescharacteristic
161C2×Dic73AGL(1,ℤ/8)12yes—
162C2×Dic73#13711yescharacteristic
163#10596833—82no—
164#10596833—82no—
165C22×C73—22no—
166C22×C73—22no—
167C22×C73—22no—
168C22×C73—41no—
169C22×C73—41no—
170C22×C73—41no—
171C22×C73—41no—
172C22×C73—42no—
173C22×C73—42no—
174C22×C73—42no—
175C22×C73—81no—
176C22×C73—81no—
177C22×C73—81no—
178C22×C73—81no—
179C22×C73C2×C22⋊C811yescharacteristic
180C22×C73C2×#9911yescharacteristic
181C22×C73#41311yescharacteristic
182C22×C73#41711yescharacteristic
183C22×C73#41811yescharacteristic
184C22×C73#42111yescharacteristic
185C22×C73#42111yescharacteristic
186C4×C73—22no—
187C4×C73—21no—
188C4×C73—21no—
189C4×C73—42no—
190Dic73—22no—
191Dic73—22no—
192Dic73—21no—
193Dic73—21no—
194Dic73—41no—
195Dic73—81no—
196Dic73—81no—
197#7599—731nomaximal, Sylow
198C2×C73—42no—
199C2×C73—82no—
200C2×C73#145911yescharacteristic
201C2×C73#147711yescharacteristic
202C2×C73#147811yescharacteristic
203C2×C73#151011yescharacteristic
204C2×C73#151311yescharacteristic
205C2×C73#153511yescharacteristic
206C2×C73#155711yescharacteristic
207C4×#121—731no—
208#1817—731no—
209#1797—731no—
210#1271—732no—
211#1270—732no—
212C73#759911yescharacteristic, complemented, Sylow
213C2×#121#6440811yescharacteristic
214C2×C4×D4—731no—
215C2×C4×D4—1461no—
216C22×C42—731no—
217C2×C4⋊C8—1461no—
218C2×C22⋊C8—1461no—
219C2×C22⋊C8—1461no—
220#390—731no—
221#389—731no—
222#388—1461no—
223#386—1461no—
224#380—731no—
225C4×C4⋊C4—731no—
226C4×C22⋊C4—1461no—
227#336—732no—
228#336—732no—
229#336—1461no—
230C22×D4—21no—
231C22×D4C2×#6440811yescharacteristic
232C23×C4—731no—
233C23×C4—731no—
234C23×C4—1461no—
235C23×C4C2×#6440811yescharacteristic
236C22×C8—1462no—
237C22×C8—1462no—
238#121—22no—
239#121#1059683314yescomplemented
240C4×D4—2924no—
241C4×D4—2922no—
242C2×C4⋊C4—731no—
243C2×C4⋊C4—731no—
244C2×C4⋊C4—1461no—
245C2×C4⋊C4C2×#6440811yescharacteristic
246C2×C22⋊C4—21no—
247C2×C22⋊C4—1461no—
248C2×C22⋊C4—1461no—
249C2×C22⋊C4—1461no—
250C2×C22⋊C4—1461no—
251C2×C42—2922no—
252C2×C42—731no—
253C2×C42—731no—
254C2×C42—731no—
255C2×C42—731no—
256C2×C42—1461no—
257C4⋊C8—2922no—
258C22⋊C8—2922no—
259C22⋊C8—2922no—
260#95—1461no—
261#95—1461no—
262#95—1461no—
263C24—21no—
264C24C22⋊#6440811yescharacteristic
265C2×D4—24no—
266C2×D4—21no—
267C2×D4—41no—
268C2×D4—41no—
269C2×D4—42no—
270C2×D4—42no—
271C2×D4C2×#1059683312yes—
272C22×C4—21no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—2921no—
276C22×C4—2921no—
277C22×C4—2921no—
278C22×C4—2922no—
279C22×C4—2922no—
280C22×C4—2922no—
281C22×C4—731no—
282C22×C4—731no—
283C22×C4—731no—
284C22×C4—731no—
285C22×C4—1461no—
286C22×C4—1461no—
287C22×C4—1461no—
288C22×C4—1462no—
289C22×C4—1462no—
290C22×C4—1461no—
291C22×C4C22⋊#6440811yescharacteristic
292C22×C4C2×#1059683312yes—
293C22×C4#1192403712yes—
294C22×C4C22×#6440811yescharacteristic
295C2×C8—2922no—
296C2×C8—2922no—
297C2×C8—2922no—
298C2×C8—2922no—
299C2×C8—2922no—
300C2×C8—2922no—
301C4⋊C4—21no—
302C4⋊C4—2921no—
303C4⋊C4—2921no—
304C4⋊C4—2922no—
305C4⋊C4C2×#1059683312yes—
306C22⋊C4—24no—
307C22⋊C4—2922no—
308C22⋊C4—2921no—
309C22⋊C4—2921no—
310C22⋊C4—2922no—
311C22⋊C4—2922no—
312C42—2921no—
313C42—2921no—
314C42—2922no—
315C42—1462no—
316C42—1462no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23—21no—
322C23—21no—
323C23—22no—
324C23—41no—
325C23—41no—
326C23—41no—
327C23—41no—
328C23—41no—
329C23—41no—
330C23C22⋊#1059683312yes—
331C23C2×C22⋊#6440811yescharacteristic
332D4—42no—
333D4—42no—
334D4—48no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—22no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—2922no—
345C2×C4—2922no—
346C2×C4—2921no—
347C2×C4—2921no—
348C2×C4—2921no—
349C2×C4—2921no—
350C2×C4—2921no—
351C2×C4—2921no—
352C2×C4—2924no—
353C2×C4—5844no—
354C2×C4—732no—
355C2×C4—731no—
356C2×C4—731no—
357C2×C4—732no—
358C2×C4—1461no—
359C2×C4—1461no—
360C2×C4—1461no—
361C2×C4—1461no—
362C2×C4—1461no—
363C2×C4—1461no—
364C2×C4—1461no—
365C2×C4—1461no—
366C2×C4—1461no—
367C2×C4C22⋊#1059683312yes—
368C2×C4C22×#1059683311yescharacteristic
369C2×C4C2×#1192403711yescharacteristic
370C8—5842no—
371C8—5842no—
372C22—22no—
373C22—22no—
374C22—22no—
375C22—41no—
376C22—41no—
377C22—41no—
378C22—41no—
379C22—42no—
380C22—42no—
381C22—42no—
382C22—81no—
383C22—81no—
384C22—81no—
385C22—81no—
386C22C2×C22⋊#1059683311yescharacteristic
387C22D73⋊#9911yescharacteristic
388C22#1798957411yescharacteristic
389C22#1798957811yescharacteristic
390C22#1798957911yescharacteristic
391C22#1798958411yescharacteristic
392C22#1798958511yescharacteristic
393C4—21no—
394C4—21no—
395C4—22no—
396C4—42no—
397C4—2921no—
398C4—5841no—
399C4—5841no—
400C4—1462no—
401C4—1462no—
402C4—1461no—
403C4—1461no—
404C2—42no—
405C2—82no—
406C2#2262516711yescharacteristic
407C2#2262522611yescharacteristic
408C2#2262523011yescharacteristic
409C2#2262533711yescharacteristic
410C2#2262534711yescharacteristic
411C2#2262541611yescharacteristic
412C2#2262546311yescharacteristic
subgroupG/H
centerC23C2×C22⋊#64408
commutatorC2×C4×C73C22×C8
FrattiniC22×C4C22×#64408
Fitting, prime radicalC2×#121×C73C4
socleC23×C73C2×C22⋊C4
cosocleC2×C4×Dic73C23
hypercenterC2×#121#64408
hypocenterC73#7599
LCSG ▷ C2×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. #22625821C2semidirect
2. #22625785C2semidirect
3. #22624749C2semidirect
4. #22624747C2semidirect
53. #121×C73C8semidirect
212. C73#7599semidirect
239. #121#10596833semidirect

Maximal subgroups

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

subgroupindex
#226258212
#226257852
#226247492
#226247472
#121×Dic732
#759973