Finite Group Databasebeta

← C2×#22625160 #34015860

Subgroups — C2×#22625160 #34015860

Order 18688.

1806 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989555C212yesmaximal
2C2×#17989433C212yesmaximal, complemented
3#22625160C218yesmaximal, direct product factor
4C2×#17989432C212yesmaximal, complemented
5C22×C4⋊C4×D73C211yesmaximal, characteristic
6C22×#11924036—22no—
7C2×#14556648—22no—
8#17989555—24no—
9#17989555C2218yes—
10C2×C22⋊#10596833C2214yes—
11#17989433C2218yescomplemented
12#17989432C2218yescomplemented
13C23×C4×D73C411yescharacteristic
14C23×C4×D73C411yescharacteristic
15C23×C4×D73C2211yescharacteristic
16C22×C4⋊Dic73C2211yescharacteristic
17C22×#11924007C411yescharacteristic
18C22×#11924007C2211yescharacteristic
19C2×C4⋊C4×D73—22no—
20C2×C4⋊C4×D73—24no—
21C2×C4⋊C4×D73—22no—
22C2×C4⋊C4×D73C414yes—
23C2×C4⋊C4×D73C2214yes—
24C22×C4⋊C4×C73C411yescharacteristic
25C2×#11924036—28no—
26C2×#11924036—48no—
27C22×#10596833—24no—
28#14556648—28no—
29C22⋊#10596833—24no—
30C22⋊#10596833C2318yes—
31C24×D73C2×C411yescharacteristic
32C23×Dic73C2×C411yescharacteristic
33C23×Dic73C2×C411yescharacteristic
34C23×Dic73C2311yescharacteristic
35C22×C4×D73—22no—
36C22×C4×D73—21no—
37C22×C4×D73—21no—
38C22×C4×D73—24no—
39C22×C4×D73—22no—
40C22×C4×D73—21no—
41C22×C4×D73—21no—
42C22×C4×D73—44no—
43C22×C4×D73—44no—
44C22×C4×D73C818yescomplemented
45C22×C4×D73C2×C414yes—
46C22×C4×D73C2×C414yes—
47C22×C4×D73C2×C412yes—
48C22×C4×D73D412yes—
49C22×C4×D73D414yes—
50C22×C4×D73D412yes—
51C22×C4×D73C2312yes—
52C2×C4⋊Dic73—22no—
53C2×C4⋊Dic73—22no—
54C2×C4⋊Dic73C2×C412yes—
55C2×C4⋊Dic73C2312yes—
56C2×#11924007—22no—
57C2×#11924007—22no—
58C2×#11924007—22no—
59C2×#11924007—22no—
60C2×#11924007C2×C412yes—
61C2×#11924007C2×C414yes—
62C2×#11924007C2312yes—
63C4⋊C4×D73—28no—
64C4⋊C4×D73—28no—
65C4⋊C4×D73—28no—
66C4⋊C4×D73—28no—
67C23×C4×C73C2×C411yescharacteristic
68C23×C4×C73C2×C411yescharacteristic
69C23×C4×C73C2×C411yescharacteristic
70C2×C4⋊C4×C73—22no—
71C2×C4⋊C4×C73—22no—
72C2×C4⋊C4×C73C2×C414yes—
73#11924036—416no—
74C2×#10596833—28no—
75C2×#10596833—48no—
76C23×D73—24no—
77C23×D73—24no—
78C23×D73C22⋊C412yes—
79C23×D73C22⋊C411yescharacteristic
80C23×D73C22⋊C411yescharacteristic
81C23×D73C2×C814yes—
82C23×D73M1614yes—
83C23×D73C22×C412yes—
84C22×Dic73—21no—
85C22×Dic73—21no—
86C22×Dic73—22no—
87C22×Dic73—21no—
88C22×Dic73—21no—
89C22×Dic73—42no—
90C22×Dic73—42no—
91C22×Dic73C2×C814yescomplemented
92C22×Dic73C22×C412yes—
93C22×Dic73C22×C412yes—
94C22×Dic73C22×C411yescharacteristic
95C22×Dic73C2×D412yes—
96C22×Dic73C2×D412yes—
97C22×Dic73C2411yescharacteristic
98C2×C4×D73—28no—
99C2×C4×D73—216no—
100C2×C4×D73—28no—
101C2×C4×D73—28no—
102C2×C4×D73—24no—
103C2×C4×D73—24no—
104C2×C4×D73—28no—
105C2×C4×D73—24no—
106C2×C4×D73—24no—
107C2×C4×D73—44no—
108C2×C4×D73—48no—
109C2×C4×D73—44no—
110C2×C4×D73—44no—
111C2×C4×D73—44no—
112C2×C4×D73—48no—
113C2×C4×D73—44no—
114C2×C4×D73—48no—
115C2×C4×D73C22⋊C414yes—
116C2×C4×D73C22⋊C414yes—
117C2×C4×D73C2×C8116yescomplemented
118C2×C4×D73C2×D414yes—
119C2×C4×D73C2×D414yes—
120C4⋊Dic73—24no—
121C4⋊Dic73—24no—
122#11924007—28no—
123#11924007—24no—
124#11924007—24no—
125C24×C73C22×C411yescharacteristic
126C22×C4×C73—21no—
127C22×C4×C73—21no—
128C22×C4×C73—22no—
129C22×C4×C73—21no—
130C22×C4×C73—21no—
131C22×C4×C73—42no—
132C22×C4×C73—42no—
133C22×C4×C73C22⋊C412yes—
134C22×C4×C73C22⋊C412yes—
135C22×C4×C73C2×C814yescomplemented
136C22×C4×C73C22×C412yes—
137C22×C4×C73C22×C412yes—
138C22×C4×C73C22×C412yes—
139C4⋊C4×C73—28no—
140C23×C73—22no—
141C23×C73—22no—
142C23×C73C2×C22⋊C411yescharacteristic
143C23×C73C2×C22⋊C411yescharacteristic
144C23×C73C22×C812yes—
145C23×C73C2×M1612yes—
146C23×C73C23×C411yescharacteristic
147C22×D73—22no—
148C22×D73—22no—
149C22×D73—216no—
150C22×D73—216no—
151C22×D73—24no—
152C22×D73—24no—
153C22×D73—24no—
154C22×D73—24no—
155C22×D73C22⋊C814yes—
156C22×D73C22⋊C818yes—
157C22×D73C22⋊C814yes—
158C22×D73#9914yes—
159C22×D73#10114yes—
160C22×D73C2×C22⋊C412yes—
161C22×D73C2×C22⋊C412yes—
162C22×D73C22×C814yes—
163C22×D73C2×M1614yes—
164C2×C4×C73—22no—
165C2×C4×C73—22no—
166C2×C4×C73—24no—
167C2×C4×C73—24no—
168C2×C4×C73—24no—
169C2×C4×C73—22no—
170C2×C4×C73—42no—
171C2×C4×C73—42no—
172C2×C4×C73—42no—
173C2×C4×C73—42no—
174C2×C4×C73—42no—
175C2×C4×C73C2×C22⋊C414yes—
176C2×C4×C73C22×C814yescomplemented
177C2×Dic73—22no—
178C2×Dic73—22no—
179C2×Dic73—24no—
180C2×Dic73—24no—
181C2×Dic73—24no—
182C2×Dic73—22no—
183C2×Dic73—41no—
184C2×Dic73—42no—
185C2×Dic73—41no—
186C2×Dic73—42no—
187C2×Dic73—42no—
188C2×Dic73—42no—
189C2×Dic73C2×C22⋊C412yes—
190C2×Dic73C22×C814yescomplemented
191C2×Dic73C22×D412yes—
192C4×D73—216no—
193C4×D73—216no—
194C4×D73—28no—
195C4×D73—28no—
196C4×D73—44no—
197C4×D73—44no—
198C4×D73—48no—
199C4×D73—48no—
200#10596833—48no—
201C22×C73—21no—
202C22×C73—21no—
203C22×C73—21no—
204C22×C73—21no—
205C22×C73—24no—
206C22×C73—24no—
207C22×C73C2×C22⋊C812yes—
208C22×C73C2×C22⋊C812yes—
209C22×C73C2×#9912yes—
210C22×C73C2×#10112yes—
211C22×C73C22×C22⋊C411yescharacteristic
212C22×C73C23×C811yescharacteristic
213C22×C73C22×M1611yescharacteristic
214C2×D73—22no—
215C2×D73—22no—
216C2×D73—28no—
217C2×D73—216no—
218C2×D73—28no—
219C2×D73—28no—
220C2×D73#324116yescomplemented
221C2×D73C2×C22⋊C814yes—
222C2×D73C2×C22⋊C814yes—
223C2×D73C2×#9914yes—
224C2×D73C2×#10114yes—
225C4×C73—24no—
226C4×C73—24no—
227C4×C73—42no—
228Dic73—24no—
229Dic73—22no—
230Dic73—22no—
231Dic73—42no—
232C22×#324—731nomaximal, Sylow
233C2×C73—21no—
234C2×C73—21no—
235C2×C73—22no—
236C2×C73C2×#32414yescomplemented
237C2×C73C22×C22⋊C811yescharacteristic
238C2×C73C22×#9911yescharacteristic
239C2×C73C22×#10111yescharacteristic
240D73—24no—
241D73C2×#32418yescomplemented
242C23×C4⋊C4—731no—
243C22×C22⋊C8—732no—
244C2×#324—732no—
245C2×#324—738no—
246C2×#324—732no—
247C73C22×#32411yescharacteristic, complemented, Sylow
248C24×C4—731no—
249C24×C4—731no—
250C24×C4—731no—
251C23×C8—1462no—
252C22×C4⋊C4—731no—
253C22×C4⋊C4—731no—
254C22×C4⋊C4—731no—
255C22×C4⋊C4—734no—
256C22×C4⋊C4—734no—
257C22×C4⋊C4—1464no—
258C22×C4⋊C4—1462no—
259C22×C4⋊C4—1462no—
260C22×C4⋊C4#6440811yescharacteristic
261C2×C22⋊C8—734no—
262C2×C22⋊C8—738no—
263C2×C22⋊C8—1464no—
264C2×C22⋊C8—1462no—
265#324—738no—
266#324—738no—
267C25—731no—
268C23×C4—2924no—
269C23×C4—2924no—
270C23×C4—731no—
271C23×C4—732no—
272C23×C4—734no—
273C23×C4—732no—
274C23×C4—732no—
275C23×C4—731no—
276C23×C4—734no—
277C23×C4—731no—
278C23×C4—732no—
279C23×C4—738no—
280C23×C4—734no—
281C23×C4—1462no—
282C23×C4—1461no—
283C23×C4—1461no—
284C23×C4—1462no—
285C23×C4—1461no—
286C23×C4—1461no—
287C23×C4—1464no—
288C23×C4C2×#6440811yescharacteristic
289C23×C4C2×#6440811yescharacteristic
290C23×C4C2×#6440811yescharacteristic
291C22×C8—2928no—
292C22×C8—1464no—
293C22×C8—1468no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—732no—
297C2×C4⋊C4—732no—
298C2×C4⋊C4—732no—
299C2×C4⋊C4—734no—
300C2×C4⋊C4—732no—
301C2×C4⋊C4—1468no—
302C2×C4⋊C4—1468no—
303C2×C4⋊C4—1468no—
304C2×C4⋊C4—1468no—
305C2×C4⋊C4—1462no—
306C2×C4⋊C4—1462no—
307C2×C4⋊C4—1462no—
308C2×C4⋊C4—1462no—
309C2×C4⋊C4—1462no—
310C2×C4⋊C4—1462no—
311C2×C4⋊C4C2×#6440814yes—
312C22⋊C8—738no—
313C22⋊C8—1464no—
314C22⋊C8—1468no—
315C24—731no—
316C24—734no—
317C24—734no—
318C24—732no—
319C24—732no—
320C24—731no—
321C24—1464no—
322C24—1464no—
323C24C22×#6440811yescharacteristic
324C22×C4—22no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—2924no—
332C22×C4—2924no—
333C22×C4—2928no—
334C22×C4—2924no—
335C22×C4—2928no—
336C22×C4—2928no—
337C22×C4—2924no—
338C22×C4—2924no—
339C22×C4—2922no—
340C22×C4—2922no—
341C22×C4—731no—
342C22×C4—731no—
343C22×C4—732no—
344C22×C4—734no—
345C22×C4—734no—
346C22×C4—732no—
347C22×C4—734no—
348C22×C4—732no—
349C22×C4—732no—
350C22×C4—7316no—
351C22×C4—734no—
352C22×C4—734no—
353C22×C4—1464no—
354C22×C4—1464no—
355C22×C4—1464no—
356C22×C4—1464no—
357C22×C4—1468no—
358C22×C4—1468no—
359C22×C4—1468no—
360C22×C4—14616no—
361C22×C4—1468no—
362C22×C4—1461no—
363C22×C4—1461no—
364C22×C4—1462no—
365C22×C4—1461no—
366C22×C4—1461no—
367C22×C4C22⋊#6440812yes—
368C22×C4C22⋊#6440812yes—
369C22×C4#1192403614yescomplemented
370C22×C4C22×#6440812yes—
371C22×C4C22×#6440812yes—
372C22×C4C22×#6440812yes—
373C2×C8—2928no—
374C2×C8—29216no—
375C2×C8—1468no—
376C4⋊C4—28no—
377C4⋊C4—1464no—
378C4⋊C4—1464no—
379C4⋊C4—1464no—
380C4⋊C4—1464no—
381C4⋊C4—1468no—
382C23—22no—
383C23—22no—
384C23—732no—
385C23—734no—
386C23—734no—
387C23—734no—
388C23—732no—
389C23—734no—
390C23—734no—
391C23—738no—
392C23—734no—
393C23—1464no—
394C23—1464no—
395C23—14616no—
396C23—14616no—
397C23—1464no—
398C23—1464no—
399C23—1462no—
400C23—1462no—
401C23C2×C22⋊#6440811yescharacteristic
402C23C2×C22⋊#6440811yescharacteristic
403C23C2×#1192403612yes—
404C23D73⋊M1612yes—
405C23C23×#6440811yescharacteristic
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—24no—
409C2×C4—24no—
410C2×C4—24no—
411C2×C4—22no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—2924no—
418C2×C4—2924no—
419C2×C4—2928no—
420C2×C4—2928no—
421C2×C4—2922no—
422C2×C4—2922no—
423C2×C4—2922no—
424C2×C4—2921no—
425C2×C4—2922no—
426C2×C4—2921no—
427C2×C4—732no—
428C2×C4—732no—
429C2×C4—734no—
430C2×C4—1468no—
431C2×C4—1468no—
432C2×C4—14616no—
433C2×C4—14616no—
434C2×C4—1462no—
435C2×C4—1462no—
436C2×C4—1462no—
437C2×C4—1464no—
438C2×C4—1464no—
439C2×C4—1464no—
440C2×C4C2×C22⋊#6440814yes—
441C2×C4C2×#1192403614yescomplemented
442C8—2928no—
443C22—24no—
444C22—24no—
445C22—21no—
446C22—21no—
447C22—21no—
448C22—21no—
449C22—7316no—
450C22—734no—
451C22—734no—
452C22—734no—
453C22—734no—
454C22—1468no—
455C22—1468no—
456C22—14616no—
457C22—1468no—
458C22—1462no—
459C22—1462no—
460C22#1798955512yes—
461C22#1798955512yes—
462C22D73⋊#9912yes—
463C22D73⋊#10112yes—
464C22C22×C22⋊#6440811yescharacteristic
465C22C22×#1192403611yescharacteristic
466C22C2×D73⋊M1611yescharacteristic
467C4—24no—
468C4—24no—
469C4—42no—
470C4—2922no—
471C4—1464no—
472C4—1462no—
473C4—1462no—
474C2—21no—
475C2—21no—
476C2—22no—
477C2—738no—
478C2—1464no—
479C2#2262516014yesdirect product factor
480C2C2×#1798955511yescharacteristic
481C2C2×D73⋊#9911yescharacteristic
482C2C2×D73⋊#10111yescharacteristic
subgroupG/H
centerC23C2×C22⋊#64408
commutatorC22×C73C23×C8
FrattiniC23C23×#64408
FittingC22×C4⋊C4×C73C4
socleC23×C73C2×C22⋊C4
prime radicalC24×D73C2×C4
cosocleC22×Dic73C24
hypercenterC22×C4⋊C4#64408
hypocenterC73C22×#324
LCSG ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#17989433C2semidirect
3. #22625160C2direct (N × H)
4. C2×#17989432C2semidirect
11. #17989433C22semidirect
12. #17989432C22semidirect
44. C22×C4×D73C8semidirect
91. C22×Dic73C2×C8semidirect
117. C2×C4×D73C2×C8semidirect
135. C22×C4×C73C2×C8semidirect
176. C2×C4×C73C22×C8semidirect
190. C2×Dic73C22×C8semidirect
220. C2×D73#324semidirect
236. C2×C73C2×#324semidirect
241. D73C2×#324semidirect
247. C73C22×#324semidirect
369. C22×C4#11924036semidirect
441. C2×C4C2×#11924036semidirect
479. C2#22625160direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#179895552
C2×#179894332
#226251602
C2×#179894322
C22×C4⋊C4×D732
C22×#32473