Finite Group Databasebeta

← #34034769

Subgroups — #34034769

Order 18688.

582 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22627137C212yesmaximal, complemented
2#22627136C212yesmaximal, complemented
3#22627135C211yesmaximal, characteristic, complemented
4#22627134C211yesmaximal, characteristic, complemented
5#22627133C211yesmaximal, characteristic, complemented
6#22627132C211yesmaximal, characteristic, complemented
7#22627122C212yesmaximal, complemented
8#22627101C212yesmaximal
9C4×#14556687C211yesmaximal, characteristic, complemented
10C2×#17989681C211yesmaximal, characteristic, complemented
11C4×#14556678C211yesmaximal, characteristic
12C2×#14556687C2212yes—
13C22×#11924040C2212yes—
14#17989690C2212yes—
15#17989690C2212yes—
16C4⋊#11924040C2212yes—
17C4⋊#11924040C2212yes—
18#17989684C2212yes—
19#17989684C2212yes—
20C22⋊#11924040C2212yes—
21C22⋊#11924040C2212yes—
22#17989681C2211yescharacteristic
23#17989681C2211yescharacteristic
24#17989681C2212yes—
25#17989681C2211yescharacteristic
26#17989681C2211yescharacteristic
27C4×#11924040C2212yes—
28C2×#14556678C411yescharacteristic
29C2×#14556678C2211yescharacteristic
30#17989552C411yescharacteristic
31#17989552C2211yescharacteristic
32#17989551C411yescharacteristic
33#17989551C411yescharacteristic
34#17989551C2211yescharacteristic
35#17989551C2211yescharacteristic
36C4×#11924038C411yescharacteristic
37C4×#11924037C411yescharacteristic
38C4×#11924037C411yescharacteristic
39C4×#11924036C2211yescharacteristic
40C2×C4×#10596833C2211yescharacteristic
41C2×C4×#10596833C2211yescharacteristic
42C4×#11924027C411yescharacteristic
43#14556687—24no—
44C2×#11924040—24no—
45C2×#11924040C2312yes—
46C2×#11924040C2312yes—
47C2×#11924040C2312yes—
48C2×#11924040C2312yes—
49#14556678—24no—
50#14556678—24no—
51C2×#11924038C2×C411yescharacteristic
52C2×#11924038C2×C411yescharacteristic
53C2×#11924037C2×C411yescharacteristic
54C2×#11924037C2×C411yescharacteristic
55C2×#11924037C2×C411yescharacteristic
56C2×#11924037C2×C411yescharacteristic
57C2×#11924036C2×C411yescharacteristic
58C2×#11924036C2311yescharacteristic
59C22×#10596833C2×C411yescharacteristic
60C22×#10596833C2×C411yescharacteristic
61C22×#10596833C2311yescharacteristic
62C22×#10596833C2311yescharacteristic
63#14556646C2×C411yescharacteristic
64#14556646C2×C411yescharacteristic
65#14556646C2×C411yescharacteristic
66#14556646C2×C411yescharacteristic
67#14556646C2×C411yescharacteristic
68#14556646C2×C411yescharacteristic
69C4×#10596833C2×C411yescharacteristic
70C4×#10596833C2×C411yescharacteristic
71C4×#10596833C2×C411yescharacteristic
72C4×#10596833C2×C411yescharacteristic
73C4×#10596833C2×C411yescharacteristic
74C4×#10596833C2×C411yescharacteristic
75C4×#10596833C2311yescharacteristic
76C4×#10596833C2311yescharacteristic
77C4×#10596833C2311yescharacteristic
78C4×#10596833C2311yescharacteristic
79C2×#11924027C2×C411yescharacteristic
80C4×Dic146C811yescharacteristic
81#14556556C811yescharacteristic
82D4×Dic73C811yescharacteristic
83C4×C22⋊D73C811yescharacteristic
84C4×C22⋊D73C811yescharacteristic
85#14556551C811yescharacteristic
86#14556551C811yescharacteristic
87#14556549C2×C411yescharacteristic
88#14556547C2×C411yescharacteristic
89#14556547C2×C411yescharacteristic
90C2×C4×Dic73C2×C411yescharacteristic
91C2×C4×Dic73C2×C411yescharacteristic
92C42×D73C2×C411yescharacteristic
93C4×D4×C73C811yescharacteristic
94#11924040—22no—
95#11924040—22no—
96#11924040—22no—
97#11924040—22no—
98#11924038—22no—
99#11924038—22no—
100#11924037—22no—
101#11924037—22no—
102#11924037—22no—
103#11924037—22no—
104#11924036—22no—
105#11924036—22no—
106C2×#10596833—22no—
107C2×#10596833—22no—
108C2×#10596833M1614yes—
109C2×#10596833C22×C411yescharacteristic
110C2×#10596833C22×C411yescharacteristic
111C2×#10596833C22×C411yescharacteristic
112C2×#10596833C22×C411yescharacteristic
113C2×#10596833C22×C411yescharacteristic
114C2×#10596833C22×C411yescharacteristic
115C2×#10596833C22×C411yescharacteristic
116C2×#10596833#4114yes—
117C2×#10596833C2411yescharacteristic
118#11924027—22no—
119#11924027—22no—
120C2×Dic146C2×C811yescharacteristic
121C2×C22⋊D73C2×C811yescharacteristic
122C2×C22⋊D73C2×C811yescharacteristic
123C22×Dic73C2×C811yescharacteristic
124C22×Dic73C2×C811yescharacteristic
125C22×Dic73C22×C411yescharacteristic
126C22×Dic73C22×C411yescharacteristic
127C2×C4×D73C2×C811yescharacteristic
128C2×C4×D73C2×C811yescharacteristic
129C2×C4×D73C22×C411yescharacteristic
130C4⋊Dic73C2×C811yescharacteristic
131#11924007C2×C811yescharacteristic
132#11924007C2×C811yescharacteristic
133#11924007C2×C811yescharacteristic
134#11924007C2×C811yescharacteristic
135C22⋊Dic73C2×C811yescharacteristic
136C22⋊Dic73C2×C811yescharacteristic
137#11924005C2×C811yescharacteristic
138#11924005C2×C811yescharacteristic
139C4×Dic73C2×C811yescharacteristic
140C4×Dic73C2×C811yescharacteristic
141C4×Dic73C2×C811yescharacteristic
142C4×Dic73C2×C811yescharacteristic
143C4×Dic73C2×C811yescharacteristic
144C4×Dic73C22×C411yescharacteristic
145C4×Dic73C22×C411yescharacteristic
146C4×Dic73C22×C411yescharacteristic
147C4×Dic73C22×C411yescharacteristic
148C2×D4×C73C2×C811yescharacteristic
149C22×C4×C73C2×C811yescharacteristic
150C22×C4×C73C2×C811yescharacteristic
151C4⋊C4×C73C2×C811yescharacteristic
152C22⋊C4×C73C2×C811yescharacteristic
153C22⋊C4×C73C2×C811yescharacteristic
154C42×C73C2×C811yescharacteristic
155C23×C73C22×C811yescharacteristic
156C23×C73C22×C811yescharacteristic
157C22×D73C22×C811yescharacteristic
158D4×C73—22no—
159C2×C4×C73—41no—
160C2×C4×C73—41no—
161C2×C4×C73C22×C811yescharacteristic
162C2×C4×C73C22×C811yescharacteristic
163C2×C4×C73C22×C811yescharacteristic
164C2×C4×C73C22×C811yescharacteristic
165C2×C4×C73C22×C811yescharacteristic
166C22⋊D73—22no—
167C22⋊D73—22no—
168C2×Dic73—21no—
169C2×Dic73—21no—
170C2×Dic73—41no—
171C2×Dic73—41no—
172C2×Dic73#11712yes—
173C2×Dic73C22×C811yescharacteristic
174C2×Dic73C22×C811yescharacteristic
175C2×Dic73C22×C811yescharacteristic
176C2×Dic73C22×C811yescharacteristic
177C2×Dic73C22×C811yescharacteristic
178C2×Dic73C22×C811yescharacteristic
179C2×Dic73C2×M1612yes—
180C2×Dic73C23×C411yescharacteristic
181C4×D73—21no—
182C4×D73—21no—
183C4×D73—41no—
184C4×D73—41no—
185Dic146—22no—
186#10596833—21no—
187#10596833—21no—
188#10596833—21no—
189#10596833—21no—
190#10596833#11712yes—
191#10596833C2×M1612yes—
192#10596833C2×M1612yes—
193#10596833C2×#4112yes—
194C22×C73—21no—
195C22×C73—21no—
196C22×C73—21no—
197C22×C73—21no—
198C22×C73—21no—
199C22×C73#43112yes—
200C22×C73C23×C811yescharacteristic
201C2×D73—21no—
202C2×D73—21no—
203C2×D73—21no—
204C4×C73—21no—
205C4×C73—21no—
206C4×C73—21no—
207C4×C73—21no—
208C4×C73—21no—
209Dic73—21no—
210Dic73—21no—
211Dic73—21no—
212Dic73—21no—
213Dic73—21no—
214Dic73#43112yes—
215Dic73C2×#11711yescharacteristic
216Dic73C22×M1611yescharacteristic
217#33421—731nomaximal, Sylow
218C2×C73—22no—
219C2×C73—41no—
220C2×C73C2×#43111yescharacteristic
221C2×C73#340711yescharacteristic
222C2×C73#340711yescharacteristic
223D73—41no—
224C4×#131—731no—
225#2164—732no—
226#2164—732no—
227#2163—731no—
228#2163—731no—
229#2163—731no—
230#2163—731no—
231#2151—732no—
232#2113—732no—
233C4×M32—731no—
234C2×#346—731no—
235C73#3342111yescharacteristic, complemented, Sylow
236C2×#131—731no—
237C2×#131—731no—
238C4×#41—731no—
239C2×M32—732no—
240C22×C16—732no—
241#405—731no—
242#405—731no—
243#405—731no—
244#405—731no—
245#405—731no—
246#405—731no—
247C4×M16—731no—
248C4×M16—731no—
249C4×M16—731no—
250C2×C4×C8—731no—
251C2×C4×C8—731no—
252C2×C4×C8—731no—
253C4⋊C16—732no—
254C4⋊C16—732no—
255C4⋊C16—732no—
256C4⋊C16—732no—
257C22⋊C16—732no—
258C22⋊C16—732no—
259C22⋊C16—732no—
260C22⋊C16—732no—
261#346—731no—
262#346—731no—
263#346—732no—
264#346—731no—
265#346—731no—
266C4×C16—732no—
267C2×#41—731no—
268#131—1464no—
269#131—1464no—
270C2×M16—731no—
271C2×M16—731no—
272C2×M16—731no—
273C2×M16—731no—
274C2×M16—731no—
275C2×M16—731no—
276C22×C8—731no—
277C22×C8—731no—
278C22×C8—731no—
279C22×C8—731no—
280C22×C8—731no—
281C22×C8—731no—
282C4×Q8—731no—
283C4×Q8—731no—
284C4×D4—731no—
285C4×D4—731no—
286C4×D4—731no—
287C4×D4—731no—
288C4×D4—731no—
289C4×D4#1059684211yescharacteristic
290#117—731no—
291#117—731no—
292#117—731no—
293C2×C42—731no—
294C2×C42—731no—
295C2×C42—731no—
296M32—1464no—
297C2×C16—732no—
298C2×C16—732no—
299C2×C16—732no—
300C2×C16—732no—
301C2×C16—1464no—
302#97—731no—
303#97—731no—
304#97—731no—
305#97—731no—
306#97—731no—
307#97—731no—
308C4×C8—731no—
309C4×C8—731no—
310C4×C8—731no—
311C4×C8—731no—
312C4×C8—731no—
313C4×C8—731no—
314C4×C8—731no—
315C4×C8—731no—
316C4×C8—731no—
317C4×C8—731no—
318#41—1462no—
319#41—1462no—
320C2×Q8—731no—
321C2×D4—731no—
322C2×D4—731no—
323C2×D4C2×#1059684211yescharacteristic
324C22×C4—731no—
325C22×C4—731no—
326C22×C4—731no—
327C22×C4—731no—
328C22×C4—731no—
329C22×C4—731no—
330C22×C4—731no—
331C22×C4C2×#1059684211yescharacteristic
332C22×C4C2×#1059684211yescharacteristic
333M16—1462no—
334M16—1462no—
335M16—1462no—
336M16—1462no—
337M16—1462no—
338M16—1462no—
339C2×C8—734no—
340C2×C8—731no—
341C2×C8—731no—
342C2×C8—731no—
343C2×C8—734no—
344C2×C8—731no—
345C2×C8—731no—
346C2×C8—731no—
347C2×C8—731no—
348C2×C8—731no—
349C2×C8—1462no—
350C2×C8—1462no—
351C2×C8—1462no—
352C2×C8—1462no—
353C4⋊C4—731no—
354C4⋊C4—731no—
355C4⋊C4—731no—
356C4⋊C4—731no—
357C4⋊C4—731no—
358C4⋊C4C2×#1059684211yescharacteristic
359C22⋊C4—731no—
360C22⋊C4—731no—
361C22⋊C4—731no—
362C22⋊C4—731no—
363C22⋊C4C2×#1059684211yescharacteristic
364C22⋊C4C2×#1059684211yescharacteristic
365C42—731no—
366C42—731no—
367C42—731no—
368C42—731no—
369C42—731no—
370C42—731no—
371C42—731no—
372C42—731no—
373C42—731no—
374C42C2×#1059684211yescharacteristic
375C16—1462no—
376C16—1462no—
377C16—1462no—
378C16—1462no—
379C23—731no—
380C23C22×#1059684211yescharacteristic
381C23C22×#1059684211yescharacteristic
382Q8—1462no—
383D4—22no—
384D4—1462no—
385D4—1462no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—2921no—
389C2×C4—2921no—
390C2×C4—2921no—
391C2×C4—2921no—
392C2×C4—732no—
393C2×C4—732no—
394C2×C4—731no—
395C2×C4—731no—
396C2×C4—731no—
397C2×C4—731no—
398C2×C4—731no—
399C2×C4—731no—
400C2×C4—731no—
401C2×C4—1461no—
402C2×C4—1461no—
403C2×C4—1461no—
404C2×C4—1461no—
405C2×C4C22×#1059684211yescharacteristic
406C2×C4C22×#1059684211yescharacteristic
407C2×C4C22×#1059684211yescharacteristic
408C2×C4C22×#1059684211yescharacteristic
409C2×C4C22×#1059684211yescharacteristic
410C8—732no—
411C8—732no—
412C8—732no—
413C8—732no—
414C8—1461no—
415C8—1461no—
416C8—1461no—
417C8—1461no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—1461no—
424C22—1461no—
425C22—1461no—
426C22#1798970012yes—
427C22C23×#1059684211yescharacteristic
428C4—21no—
429C4—21no—
430C4—21no—
431C4—21no—
432C4—21no—
433C4—731no—
434C4—731no—
435C4—732no—
436C4—1461no—
437C4—1461no—
438C4—1461no—
439C4—1461no—
440C4—1461no—
441C2—22no—
442C2—41no—
443C2—2921no—
444C2C2×#1798970011yescharacteristic
445C2#2262718411yescharacteristic
446C2#2262718511yescharacteristic
subgroupG/H
center, FrattiniC22C23×#10596842
commutator, socleC22×C73C23×C8
FittingC4×D4×C73C8
prime radicalC2×#11924027C2×C4
cosocleC2×#10596833C24
hypercenterC4×D4#10596842
hypocenterC73#33421
LCSG ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. #22627137C2semidirect
2. #22627136C2semidirect
3. #22627135C2semidirect
4. #22627134C2semidirect
5. #22627133C2semidirect
6. #22627132C2semidirect
7. #22627122C2semidirect
9. C4×#14556687C2semidirect
10. C2×#17989681C2semidirect
235. C73#33421semidirect

Maximal subgroups

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

subgroupindex
#226271372
#226271362
#226271352
#226271342
#226271332
#226271322
#226271222
#226271012
C4×#145566872
C2×#179896812
C4×#145566782
#3342173