Finite Group Databasebeta

← #34005203

Subgroups — #34005203

Order 18688.

680 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22625786C211yesmaximal, characteristic
2C4×#14556648C211yesmaximal, characteristic
3#22624767C212yesmaximal, complemented
4#22624767C212yesmaximal, complemented
5C4×C4⋊C4×D73C211yesmaximal, characteristic
6C2×C4⋊#10596833C2212yes—
7C2×#14556648—21no—
8C2×#14556648—21no—
9C2×C4×#10596833C2212yes—
10#17989453—21no—
11#17989453—21no—
12C2×C4⋊C4×D73C411yescharacteristic
13C2×C4⋊C4×D73C411yescharacteristic
14C2×C42×D73—21no—
15C2×C42×D73C2211yescharacteristic
16C4⋊C4×Dic73C2211yescharacteristic
17C4⋊C4×Dic73C2211yescharacteristic
18C4×#11924007—21no—
19C4×C4⋊Dic73C411yescharacteristic
20#17988438—21no—
21#95×D73—21no—
22C4×C4⋊C4×C73C411yescharacteristic
23C22×#10596833—22no—
24C22×#10596833—22no—
25C4⋊#10596833—22no—
26C4⋊#10596833—22no—
27#14556648—24no—
28#14556648—42no—
29C4×#10596833—22no—
30C4×#10596833—22no—
31C22×C4×D73—21no—
32C22×C4×D73—21no—
33C22×C4×D73C2×C411yescharacteristic
34C22×C4×D73D411yescharacteristic
35C22×C4×D73D411yescharacteristic
36C2×C4⋊Dic73C2×C411yescharacteristic
37C2×C4⋊Dic73C2×C411yescharacteristic
38C2×#11924007—21no—
39C2×#11924007—21no—
40C4⋊C4×D73—22no—
41C4⋊C4×D73—22no—
42C4⋊C4×D73—22no—
43C4⋊C4×D73C814yescomplemented
44C2×C4×Dic73—21no—
45C2×C4×Dic73—21no—
46C2×C4×Dic73—21no—
47C2×C4×Dic73D411yescharacteristic
48C2×C4×Dic73D411yescharacteristic
49C2×C4×Dic73C2311yescharacteristic
50C42×D73—22no—
51C42×D73—22no—
52C42×D73—44no—
53#14556498—21no—
54#14556498—21no—
55#14556498—21no—
56C2×C4⋊C4×C73C2×C411yescharacteristic
57C2×C4⋊C4×C73C2×C411yescharacteristic
58C2×C42×C73—21no—
59C2×C42×C73C2×C411yescharacteristic
60#95×C73—21no—
61C2×#10596833—24no—
62C2×#10596833—42no—
63C2×#10596833—42no—
64C2×#10596833—42no—
65C2×#10596833—42no—
66C2×#10596833—42no—
67C23×D73C22⋊C411yescharacteristic
68C22×Dic73—21no—
69C22×Dic73—21no—
70C22×Dic73C22⋊C411yescharacteristic
71C22×Dic73C2×D411yescharacteristic
72C22×Dic73C2×D411yescharacteristic
73C2×C4×D73—21no—
74C2×C4×D73—21no—
75C2×C4×D73—21no—
76C2×C4×D73—21no—
77C2×C4×D73—22no—
78C2×C4×D73—22no—
79C2×C4×D73—24no—
80C2×C4×D73—24no—
81C2×C4×D73—22no—
82C2×C4×D73—22no—
83C2×C4×D73—41no—
84C2×C4×D73—42no—
85C2×C4×D73—42no—
86C2×C4×D73—42no—
87C2×C4×D73—42no—
88C2×C4×D73C2×C812yes—
89C2×C4×D73M1612yes—
90C2×C4×D73SD1612yes—
91C2×C4×D73Q1612yes—
92C4⋊Dic73—21no—
93C4⋊Dic73—21no—
94C4⋊Dic73C2×C812yes—
95C4⋊Dic73M1612yes—
96#11924007—24no—
97#11924007—24no—
98C4×Dic73—22no—
99C4×Dic73—22no—
100C4×Dic73—42no—
101C4×Dic73—42no—
102C4×Dic73—42no—
103C4×Dic73#4112yes—
104C4×Dic73#4112yes—
105C22×C4×C73—21no—
106C22×C4×C73—21no—
107C22×C4×C73C22⋊C411yescharacteristic
108C22×C4×C73C22⋊C411yescharacteristic
109C22×C4×C73C22×C411yescharacteristic
110C4⋊C4×C73—21no—
111C4⋊C4×C73—21no—
112C4⋊C4×C73C2×C812yes—
113C4⋊C4×C73M1612yes—
114C42×C73—21no—
115C42×C73—21no—
116C42×C73—42no—
117C23×C73C2×C22⋊C411yescharacteristic
118C22×D73—21no—
119C22×D73—21no—
120C22×D73—21no—
121C22×D73—21no—
122C22×D73C22⋊C812yes—
123C22×D73#10312yes—
124C22×D73#10412yes—
125C2×C4×C73—21no—
126C2×C4×C73—21no—
127C2×C4×C73—21no—
128C2×C4×C73—22no—
129C2×C4×C73—22no—
130C2×C4×C73—21no—
131C2×C4×C73—21no—
132C2×C4×C73—41no—
133C2×C4×C73—41no—
134C2×C4×C73—41no—
135C2×C4×C73—41no—
136C2×C4×C73#10312yes—
137C2×C4×C73#11711yescharacteristic
138C2×C4×C73#11711yescharacteristic
139C2×C4×C73C22×C811yescharacteristic
140C2×C4×C73C2×M1611yescharacteristic
141C2×C4×C73C2×M1611yescharacteristic
142C2×C4×C73C2×M1611yescharacteristic
143C2×Dic73—21no—
144C2×Dic73—21no—
145C2×Dic73—21no—
146C2×Dic73—21no—
147C2×Dic73—22no—
148C2×Dic73—22no—
149C2×Dic73—41no—
150C2×Dic73—41no—
151C2×Dic73—41no—
152C2×Dic73—41no—
153C2×Dic73C22⋊C812yes—
154C2×Dic73#12312yes—
155C2×Dic73#12312yes—
156C2×Dic73C2×SD1611yescharacteristic
157C2×Dic73C2×Q1611yescharacteristic
158C2×Dic73AGL(1,ℤ/8)12yes—
159C4×D73—22no—
160C4×D73—22no—
161C4×D73—22no—
162C4×D73—22no—
163C4×D73—24no—
164C4×D73—24no—
165C4×D73—44no—
166C4×D73—48no—
167C4×D73—48no—
168#10596833—44no—
169#10596833—82no—
170C22×C73#38111yescharacteristic
171C22×C73C2×C22⋊C811yescharacteristic
172C22×C73#40711yescharacteristic
173C22×C73C2×#10311yescharacteristic
174C22×C73#41811yescharacteristic
175C22×C73C2×#10411yescharacteristic
176C22×C73#42111yescharacteristic
177C2×D73—22no—
178C2×D73—22no—
179C2×D73—22no—
180C2×D73—22no—
181C2×D73—22no—
182C2×D73—22no—
183C2×D73Q8⋊C814yes—
184C4×C73—22no—
185C4×C73—41no—
186C4×C73—42no—
187C4×C73—42no—
188C4×C73#43111yescharacteristic
189C4×C73#43111yescharacteristic
190C4×C73#43211yescharacteristic
191C4×C73#43211yescharacteristic
192Dic73—22no—
193Dic73—22no—
194Dic73—42no—
195Dic73—42no—
196Dic73#48112yes—
197Dic73#48412yes—
198#8707—731nomaximal, Sylow
199C2×C73C2×Q8⋊C811yescharacteristic
200C2×C73#147811yescharacteristic
201C2×C73#148711yescharacteristic
202C2×C73#149011yescharacteristic
203C2×C73#181711yescharacteristic
204C2×C73#182411yescharacteristic
205C2×C73#182611yescharacteristic
206D73—24no—
207C2×C4×C4⋊C4—731no—
208#1797—731no—
209C4×C22⋊C8—731no—
210#1279—732no—
211#1279—732no—
212C73#870711yescharacteristic, complemented, Sylow
213C22×C4⋊C4—731no—
214C22×C4⋊C4—731no—
215C22×C42—731no—
216C22×C42—1461no—
217C2×C4⋊C8—732no—
218C2×C22⋊C8—1461no—
219C2×C22⋊C8—1461no—
220C2×C4×C8—732no—
221C4×C4⋊C4—731no—
222C4×C4⋊C4—731no—
223C4×C4⋊C4—731no—
224C4×C4⋊C4—1461no—
225C4×C4⋊C4—1461no—
226C4×C4⋊C4#6440811yescharacteristic
227C2×#95—1461no—
228#336—1461no—
229#336—1461no—
230C23×C4—731no—
231C23×C4—731no—
232C23×C4—731no—
233C23×C4—1461no—
234C23×C4—1461no—
235C22×C8—1462no—
236C22×C8—1462no—
237C2×C4⋊C4—731no—
238C2×C4⋊C4—731no—
239C2×C4⋊C4—734no—
240C2×C4⋊C4—1462no—
241C2×C4⋊C4—1462no—
242C2×C4⋊C4—1461no—
243C2×C4⋊C4—1461no—
244C2×C4⋊C4—1462no—
245C2×C4⋊C4C2×#6440811yescharacteristic
246C2×C4⋊C4C2×#6440811yescharacteristic
247C2×C42—21no—
248C2×C42—2924no—
249C2×C42—731no—
250C2×C42—731no—
251C2×C42—731no—
252C2×C42—1462no—
253C2×C42—1462no—
254C2×C42—1461no—
255C2×C42—1461no—
256C2×C42—1461no—
257C2×C42C2×#6440811yescharacteristic
258C4⋊C8—1462no—
259C4⋊C8—1462no—
260C22⋊C8—2922no—
261C22⋊C8—1464no—
262C4×C8—1462no—
263C4×C8—1462no—
264#95—21no—
265#95—1461no—
266#95—1461no—
267#95—1461no—
268C24—731no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—2921no—
272C22×C4—2922no—
273C22×C4—2922no—
274C22×C4—2922no—
275C22×C4—2922no—
276C22×C4—731no—
277C22×C4—731no—
278C22×C4—731no—
279C22×C4—732no—
280C22×C4—732no—
281C22×C4—732no—
282C22×C4—732no—
283C22×C4—1462no—
284C22×C4—1462no—
285C22×C4—1464no—
286C22×C4—1464no—
287C22×C4—1462no—
288C22×C4—1462no—
289C22×C4—1461no—
290C22×C4—1461no—
291C22×C4—1461no—
292C22×C4—1461no—
293C22×C4—1461no—
294C22×C4—1461no—
295C22×C4C22⋊#6440811yescharacteristic
296C22×C4C22⋊#6440811yescharacteristic
297C22×C4C22×#6440811yescharacteristic
298C2×C8—2922no—
299C2×C8—2922no—
300C2×C8—2922no—
301C2×C8—2922no—
302C2×C8—2922no—
303C2×C8—1464no—
304C4⋊C4—21no—
305C4⋊C4—21no—
306C4⋊C4—732no—
307C4⋊C4—732no—
308C4⋊C4—1464no—
309C4⋊C4—1464no—
310C4⋊C4—1461no—
311C4⋊C4—1461no—
312C4⋊C4#1192403612yes—
313C4⋊C4#1192403812yes—
314C42—21no—
315C42—21no—
316C42—42no—
317C42—2922no—
318C42—2922no—
319C42—2922no—
320C42—732no—
321C42—732no—
322C42—1462no—
323C42—1462no—
324C23—732no—
325C23—732no—
326C23—732no—
327C23—1461no—
328C23—1461no—
329C23—1461no—
330C23—1461no—
331C23C2×C22⋊#6440811yescharacteristic
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—2928no—
344C2×C4—2928no—
345C2×C4—2924no—
346C2×C4—2921no—
347C2×C4—2921no—
348C2×C4—2921no—
349C2×C4—2921no—
350C2×C4—732no—
351C2×C4—731no—
352C2×C4—731no—
353C2×C4—732no—
354C2×C4—732no—
355C2×C4—732no—
356C2×C4—1464no—
357C2×C4—1464no—
358C2×C4—1462no—
359C2×C4—1462no—
360C2×C4—1461no—
361C2×C4—1461no—
362C2×C4—1461no—
363C2×C4—1461no—
364C2×C4—1462no—
365C2×C4—1462no—
366C2×C4—1462no—
367C2×C4—1462no—
368C2×C4#1455665412yes—
369C2×C4#1455667011yescharacteristic
370C2×C4#1455667011yescharacteristic
371C2×C4C2×#1192403611yescharacteristic
372C2×C4D73⋊M1611yescharacteristic
373C2×C4D73⋊M1611yescharacteristic
374C2×C4C2×#1192403811yescharacteristic
375C8—2924no—
376C8—5842no—
377C22—734no—
378C22—1462no—
379C22—1462no—
380C22—1462no—
381C22—1462no—
382C22—1462no—
383C22—1462no—
384C22#1798953011yescharacteristic
385C22#1798955511yescharacteristic
386C22#1798955811yescharacteristic
387C22C2×#1455665411yescharacteristic
388C22#1798957911yescharacteristic
389C22D73⋊#10411yescharacteristic
390C22#1798958411yescharacteristic
391C4—22no—
392C4—41no—
393C4—42no—
394C4—42no—
395C4—2922no—
396C4—2922no—
397C4—732no—
398C4—732no—
399C4—1462no—
400C4—1462no—
401C4#1798960511yescharacteristic
402C4#1798960611yescharacteristic
403C4#1798960911yescharacteristic
404C4#1798961011yescharacteristic
405C2—1464no—
406C2#2262522011yescharacteristic
407C2#2262523111yescharacteristic
408C2#2262526011yescharacteristic
409C2#2262527111yescharacteristic
410C2#2262582311yescharacteristic
411C2#2262583511yescharacteristic
412C2#2262583911yescharacteristic
subgroupG/H
centerC23C2×C22⋊#64408
commutatorC2×C4×C73C22×C8
FrattiniC22×C4C22×#64408
FittingC4×C4⋊C4×C73C4
socleC23×C73C2×C22⋊C4
prime radicalC23×D73C22⋊C4
cosocleC2×C4×Dic73C23
hypercenterC4×C4⋊C4#64408
hypocenterC73#8707
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. #22624767C2semidirect
4. #22624767C2semidirect
43. C4⋊C4×D73C8semidirect
212. C73#8707semidirect

Maximal subgroups

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

subgroupindex
#226257862
C4×#145566482
#226247672
C4×C4⋊C4×D732
#870773