Finite Group Databasebeta

← #34003227

Subgroups — #34003227

Order 18688.

606 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22626607C211yesmaximal, characteristic
2#22626607C211yesmaximal, characteristic
3#22624832C214yesmaximal
4#22612334C211yesmaximal, characteristic
5#17989608—21no—
6#17989608C412yescomplemented
7#17989607—21no—
8#17989607—21no—
9#17989604—21no—
10#17989604C412yescomplemented
11#17989603—21no—
12#17989603—21no—
13#17989590—21no—
14#17989590C412yescomplemented
15#17989557C2212yes—
16#17989557C2212yes—
17C4×#11924037—21no—
18C4×#11924037C412yescomplemented
19C2×#14556550C411yescharacteristic
20C22×C4×Dic73C2211yescharacteristic
21#17988456—21no—
22#17988452—21no—
23#17988445C412yes—
24#17988445C2212yes—
25C4×#11924007—21no—
26C4×#11924007—21no—
27C2×#117×C73C411yescharacteristic
28C2×#11924037—22no—
29C2×#11924037C2×C414yes—
30#14556658—22no—
31#14556658—22no—
32C4⋊#10596833—22no—
33C4⋊#10596833—22no—
34C22⋊#10596833—22no—
35C22⋊#10596833—22no—
36C22⋊#10596833—22no—
37C22⋊#10596833C2×C414yes—
38#14556646—22no—
39#14556646—22no—
40C4×#10596833—22no—
41C4×#10596833—22no—
42C23×Dic73C2×C411yescharacteristic
43C23×Dic73C2311yescharacteristic
44#14556550—42no—
45C2×C4⋊Dic73—21no—
46C2×#11924007—22no—
47C2×#11924007—22no—
48C2×C22⋊Dic73C2×C412yes—
49C2×C4×Dic73—21no—
50C2×C4×Dic73C2×C411yescharacteristic
51C2×C4×Dic73C2×C411yescharacteristic
52C2×C4×Dic73C2×C411yescharacteristic
53C2×C4×Dic73C2×C411yescharacteristic
54C2×C4×Dic73D411yescharacteristic
55C2×C4×Dic73D411yescharacteristic
56C2×C4×Dic73D411yescharacteristic
57C2×C4×Dic73D411yescharacteristic
58C2×C4×Dic73D411yescharacteristic
59C2×C4×Dic73D411yescharacteristic
60C2×C4×Dic73Q811yescharacteristic
61C2×C4×Dic73Q811yescharacteristic
62#14556498—22no—
63#14556498—22no—
64C23×C4×C73C2×C411yescharacteristic
65#117×C73—42no—
66C2×C4⋊C4×C73—21no—
67C2×C22⋊C4×C73C2×C412yes—
68C2×C42×C73—21no—
69#11924037—44no—
70#11924037—44no—
71C2×#10596833—24no—
72C2×#10596833—24no—
73C22×Dic73—22no—
74C22×Dic73—42no—
75C22×Dic73—42no—
76C22×Dic73C4211yescharacteristic
77C22×Dic73C4211yescharacteristic
78C22×Dic73C22⋊C411yescharacteristic
79C22×Dic73C22⋊C411yescharacteristic
80C22×Dic73C4⋊C411yescharacteristic
81C22×Dic73C4⋊C411yescharacteristic
82C22×Dic73C22×C411yescharacteristic
83C22×Dic73C22×C411yescharacteristic
84C22×Dic73C2×D411yescharacteristic
85C22×Dic73C2×D411yescharacteristic
86C22×Dic73C2×D411yescharacteristic
87C22×Dic73C2×Q811yescharacteristic
88C4⋊Dic73—42no—
89#11924007—44no—
90#11924007—44no—
91C22⋊Dic73—42no—
92C4×Dic73—42no—
93C4×Dic73C22⋊C411yescharacteristic
94C4×Dic73C22⋊C411yescharacteristic
95C4×Dic73C22⋊C411yescharacteristic
96C4×Dic73C22⋊C411yescharacteristic
97C4×Dic73C22⋊C411yescharacteristic
98C4×Dic73C22⋊C411yescharacteristic
99C4×Dic73C22⋊C411yescharacteristic
100C4×Dic73C22⋊C411yescharacteristic
101C4×Dic73C4⋊C411yescharacteristic
102C4×Dic73C4⋊C411yescharacteristic
103C4×Dic73C4⋊C411yescharacteristic
104C4×Dic73C4⋊C411yescharacteristic
105C4×Dic73C4⋊C411yescharacteristic
106C4×Dic73C4⋊C411yescharacteristic
107C4×Dic73C4⋊C411yescharacteristic
108C4×Dic73C4⋊C411yescharacteristic
109C24×C73C22×C411yescharacteristic
110C22×C4×C73—22no—
111C22×C4×C73—22no—
112C22×C4×C73—22no—
113C22×C4×C73C4211yescharacteristic
114C22×C4×C73C4211yescharacteristic
115C22×C4×C73C22⋊C411yescharacteristic
116C22×C4×C73C22⋊C411yescharacteristic
117C22×C4×C73C4⋊C411yescharacteristic
118C22×C4×C73C4⋊C411yescharacteristic
119C4⋊C4×C73—42no—
120C22⋊C4×C73—42no—
121C42×C73—42no—
122C23×C73—21no—
123C23×C73—21no—
124C23×C73—41no—
125C23×C73—41no—
126C23×C73C2×C4211yescharacteristic
127C23×C73C2×C22⋊C411yescharacteristic
128C23×C73C2×C4⋊C411yescharacteristic
129C2×C4×C73—21no—
130C2×C4×C73—21no—
131C2×C4×C73—22no—
132C2×C4×C73—22no—
133C2×C4×C73—22no—
134C2×C4×C73—22no—
135C2×C4×C73—42no—
136C2×C4×C73—42no—
137C2×C4×C73—42no—
138C2×C4×C73#9511yescharacteristic
139C2×C4×C73#9511yescharacteristic
140C2×C4×C73#9511yescharacteristic
141C2×C4×C73#9511yescharacteristic
142C2×C4×C73#10012yes—
143C2×C4×C73#10112yes—
144C2×Dic73—22no—
145C2×Dic73—22no—
146C2×Dic73—22no—
147C2×Dic73—22no—
148C2×Dic73—22no—
149C2×Dic73—22no—
150C2×Dic73—41no—
151C2×Dic73—41no—
152C2×Dic73—41no—
153C2×Dic73—41no—
154C2×Dic73—41no—
155C2×Dic73—41no—
156C2×Dic73—42no—
157C2×Dic73—42no—
158C2×Dic73—42no—
159C2×Dic73#9511yescharacteristic
160C2×Dic73#9511yescharacteristic
161C2×Dic73#9511yescharacteristic
162C2×Dic73#9511yescharacteristic
163C2×Dic73C2×C22⋊C411yescharacteristic
164C2×Dic73C2×C22⋊C411yescharacteristic
165C2×Dic73C2×C4⋊C411yescharacteristic
166C2×Dic73C2×C4⋊C411yescharacteristic
167#10596833—44no—
168#10596833—44no—
169C22×C73—21no—
170C22×C73—21no—
171C22×C73—21no—
172C22×C73—21no—
173C22×C73—21no—
174C22×C73—21no—
175C22×C73—21no—
176C22×C73—21no—
177C22×C73—42no—
178C22×C73—42no—
179C22×C73C2×#9511yescharacteristic
180C22×C73C2×#10011yescharacteristic
181C22×C73C2×#10111yescharacteristic
182C4×C73—21no—
183C4×C73—21no—
184C4×C73—82no—
185C4×C73#34312yes—
186C4×C73#34312yes—
187Dic73—21no—
188Dic73—21no—
189Dic73—21no—
190Dic73—21no—
191Dic73—21no—
192Dic73—21no—
193Dic73—82no—
194Dic73#34412yes—
195Dic73#41012yes—
196#8105—731nomaximal, Sylow
197C2×C73—21no—
198C2×C73—21no—
199C2×C73—41no—
200C2×C73—41no—
201C2×C73#174011yescharacteristic
202C2×C73C2×#34311yescharacteristic
203C2×C73C2×#34411yescharacteristic
204#2920—731no—
205#2920—731no—
206#2285—731no—
207#1305—734no—
208C73#810511yescharacteristic, complemented, Sylow
209C2×#117—731no—
210C2×#117#6440811yescharacteristic
211C22×C42—731no—
212#432—732no—
213#432—1461no—
214#432—1461no—
215#432—1461no—
216#431—732no—
217#431—1461no—
218#431—1461no—
219#431—1461no—
220#423—732no—
221#423—1461no—
222#407—732no—
223#407—732no—
224C4×M16—732no—
225C4×M16—1461no—
226#384—1461no—
227#382—1461no—
228#380—732no—
229#380—732no—
230C4×C4⋊C4—1461no—
231C4×C4⋊C4—1461no—
232C23×C4—731no—
233C23×C4—731no—
234C23×C4C2×#6440811yescharacteristic
235C2×M16—734no—
236C2×M16—1462no—
237#117—42no—
238#117—2922no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—1462no—
241C2×C4⋊C4—1462no—
242C2×C4⋊C4—1461no—
243C2×C22⋊C4—732no—
244C2×C22⋊C4C2×#6440812yes—
245C2×C42—21no—
246C2×C42—731no—
247C2×C42—731no—
248C2×C42—731no—
249C2×C42—731no—
250C2×C42—731no—
251C2×C42—731no—
252C2×C42—731no—
253C2×C42—731no—
254C2×C42—731no—
255C2×C42—731no—
256C2×C42—731no—
257C2×C42—731no—
258C2×C42—1461no—
259C4⋊C8—1462no—
260C4⋊C8—1462no—
261C4⋊C8—1462no—
262C4⋊C8—1462no—
263C22⋊C8—734no—
264C22⋊C8—1462no—
265C22⋊C8—1462no—
266C22⋊C8—1462no—
267#97—1462no—
268#97—1462no—
269C4×C8—1462no—
270C4×C8—1462no—
271#95—1462no—
272#95—1462no—
273C24C22×#6440811yescharacteristic
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—2922no—
278C22×C4—2922no—
279C22×C4—731no—
280C22×C4—731no—
281C22×C4—731no—
282C22×C4—731no—
283C22×C4—731no—
284C22×C4—731no—
285C22×C4—731no—
286C22×C4—731no—
287C22×C4—731no—
288C22×C4—731no—
289C22×C4—731no—
290C22×C4—731no—
291C22×C4—1462no—
292C22×C4C4×#6440811yescharacteristic
293C22×C4C4×#6440811yescharacteristic
294C22×C4C22⋊#6440811yescharacteristic
295C22×C4C22⋊#6440811yescharacteristic
296C22×C4C4⋊#6440811yescharacteristic
297C22×C4C4⋊#6440811yescharacteristic
298M16—2924no—
299M16—2924no—
300C2×C8—1464no—
301C2×C8—1464no—
302C4⋊C4—42no—
303C4⋊C4—2922no—
304C4⋊C4—2924no—
305C4⋊C4—2924no—
306C22⋊C4—42no—
307C22⋊C4—2922no—
308C42—42no—
309C42—2922no—
310C42—731no—
311C42—731no—
312C42—731no—
313C42—731no—
314C42—731no—
315C42—731no—
316C42—731no—
317C42—731no—
318C42—731no—
319C42—731no—
320C42—731no—
321C42—731no—
322C42—731no—
323C42—731no—
324C42—731no—
325C42—731no—
326C23—21no—
327C23—21no—
328C23—41no—
329C23—41no—
330C23C2×C4×#6440811yescharacteristic
331C23C2×C22⋊#6440811yescharacteristic
332C23C2×C4⋊#6440811yescharacteristic
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—2922no—
343C2×C4—2922no—
344C2×C4—2922no—
345C2×C4—2921no—
346C2×C4—2921no—
347C2×C4—2921no—
348C2×C4—2921no—
349C2×C4—2921no—
350C2×C4—2921no—
351C2×C4—731no—
352C2×C4—731no—
353C2×C4—731no—
354C2×C4—731no—
355C2×C4—731no—
356C2×C4—731no—
357C2×C4—731no—
358C2×C4—731no—
359C2×C4—1462no—
360C2×C4—1462no—
361C2×C4—1462no—
362C2×C4—1462no—
363C2×C4—1462no—
364C2×C4—1462no—
365C2×C4(C2×#64408)⋊C411yescharacteristic
366C2×C4(C2×#64408)⋊C411yescharacteristic
367C2×C4(C2×#64408)⋊C411yescharacteristic
368C2×C4(C2×#64408)⋊C411yescharacteristic
369C2×C4#1455665112yes—
370C2×C4#1455665212yes—
371C8—2924no—
372C8—2924no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—42no—
382C22—42no—
383C22C2×(C2×#64408)⋊C411yescharacteristic
384C22C2×#1455665111yescharacteristic
385C22C2×#1455665211yescharacteristic
386C4—21no—
387C4—21no—
388C4—82no—
389C4—5842no—
390C4—732no—
391C4—732no—
392C4—1461no—
393C4—1461no—
394C4—1461no—
395C4—1461no—
396C4—1461no—
397C4—1461no—
398C4#1798946712yes—
399C4#1798946712yes—
400C2—21no—
401C2—21no—
402C2—41no—
403C2—41no—
404C2#2262562911yescharacteristic
405C2C2×#1798946711yescharacteristic
406C2#2262563911yescharacteristic
subgroupG/H
centerC2×C4(C2×#64408)⋊C4
commutatorC23×C73C2×C42
FrattiniC24C22×#64408
FittingC2×#117×C73C4
socleC22×C73C2×#95
prime radicalC24×C73C22×C4
cosocleC23×Dic73C23
hypercenterC2×#117#64408
hypocenterC73#8105
LCSG ▷ C23×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
6. #17989608C4semidirect
10. #17989604C4semidirect
14. #17989590C4semidirect
18. C4×#11924037C4semidirect
208. C73#8105semidirect

Maximal subgroups

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

subgroupindex
#226266072
#226248322
#226123342
#810573