Finite Group Databasebeta

← #32989400

Subgroups — #32989400

Order 18688.

520 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22614926C211yesmaximal, characteristic
2#22610891C211yesmaximal, characteristic, complemented
3#22609928C211yesmaximal, characteristic
4C2×#17988332C211yesmaximal, characteristic, complemented
5C2×#17988314C211yesmaximal, characteristic
6#22607772C211yesmaximal, characteristic, complemented
7C73×#1947C211yesmaximal, characteristic, complemented
8C2×#14556591C2211yescharacteristic
9C2×#14556579—21no—
10C2×C4×D292C2211yescharacteristic
11C2×C4⋊#10596831C2211yescharacteristic
12C2×#14556511—21no—
13C2×C4×#10596831C2211yescharacteristic
14#17988514—22no—
15#17988493—21no—
16#17988444—21no—
17#17988444—21no—
18#17988349—21no—
19#17988349—22no—
20#17988332—22no—
21#17988314—22no—
22C2×#107×C73—21no—
23C2×C4⋊C8×C73C2211yescharacteristic
24#384×C73C2211yescharacteristic
25C4×C4⋊C4×C73C2211yescharacteristic
26#340×C73—21no—
27C22×Dic146—21no—
28C22×D292D411yescharacteristic
29C22×C4×D73—21no—
30C22×#10596831—21no—
31C22×#10596831—21no—
32#14556591—21no—
33#14556591—21no—
34#14556591—21no—
35#14556591—21no—
36#14556579—42no—
37#14556579—42no—
38C4×D292—21no—
39C4×D292—21no—
40C4×D292—21no—
41C4×D292—21no—
42C2×C4⋊Dic73—21no—
43C2×C4⋊Dic73D412yes—
44C2×C4⋊Dic73D411yescharacteristic
45C2×#11924005—22no—
46C2×#11924005—22no—
47C2×#11924005—21no—
48C4⋊#10596831—21no—
49C4⋊#10596831—21no—
50#14556511—42no—
51C4×#10596831—21no—
52C4×#10596831—21no—
53#14556498—22no—
54#14556498—22no—
55C22×C8×C73—21no—
56C2×C4⋊C4×C73—21no—
57C2×C4⋊C4×C73—21no—
58C2×C4⋊C4×C73—21no—
59C2×C4⋊C4×C73D411yescharacteristic
60C2×C4⋊C4×C73D411yescharacteristic
61C2×C42×C73—21no—
62C2×C42×C73C2311yescharacteristic
63#107×C73—42no—
64C4⋊C8×C73—21no—
65C4⋊C8×C73D412yes—
66#95×C73—21no—
67C23×D73—21no—
68C2×Dic146—41no—
69C2×Dic146—41no—
70C2×Dic146—41no—
71C2×Dic146—41no—
72C2×Dic146—41no—
73C2×Dic146—41no—
74C2×D292—21no—
75C2×D292—21no—
76C2×D292—21no—
77C2×D292—21no—
78C2×D292—21no—
79C2×D292D812yes—
80C22×Dic73—21no—
81C22×Dic73—22no—
82C2×C4×D73—41no—
83C2×C4×D73—41no—
84C2×C4×D73—41no—
85C2×C4×D73—41no—
86C2×C4×D73—41no—
87C2×C4×D73—41no—
88C2×#10596831—22no—
89C2×#10596831—41no—
90C2×#10596831—41no—
91C2×#10596831—41no—
92C2×#10596831—41no—
93C2×#10596831—41no—
94C4⋊Dic73—21no—
95C4⋊Dic73—22no—
96C4⋊Dic73—21no—
97C4⋊Dic73—22no—
98C4⋊Dic73—41no—
99C4⋊Dic73—41no—
100#11924005—44no—
101#11924005—44no—
102#11924005—41no—
103#11924005—41no—
104C22×C4×C73—21no—
105C22×C4×C73—21no—
106C22×C4×C73—41no—
107C22×C4×C73C2×D411yescharacteristic
108C22×C4×C73C2×D411yescharacteristic
109C22×C4×C73C2×D411yescharacteristic
110C2×C8×C73—22no—
111C2×C8×C73—41no—
112C2×C8×C73—41no—
113C4⋊C4×C73—21no—
114C4⋊C4×C73—21no—
115C4⋊C4×C73—21no—
116C4⋊C4×C73—21no—
117C4⋊C4×C73—42no—
118C4⋊C4×C73—41no—
119C4⋊C4×C73—41no—
120C4⋊C4×C73—41no—
121C4⋊C4×C73—41no—
122C42×C73—41no—
123C42×C73—41no—
124C42×C73C2×D411yescharacteristic
125C42×C73#4111yescharacteristic
126C42×C73#4111yescharacteristic
127C42×C73#4111yescharacteristic
128C23×C73#12011yescharacteristic
129C22×D73—22no—
130C22×D73—41no—
131C22×D73—41no—
132C22×D73—41no—
133C22×D73—41no—
134C22×D73—41no—
135C22×D73—41no—
136C2×C4×C73—21no—
137C2×C4×C73—21no—
138C2×C4×C73—41no—
139C2×C4×C73—41no—
140C2×C4×C73—41no—
141C2×C4×C73—41no—
142C2×C4×C73—41no—
143C2×C4×C73—41no—
144C2×C4×C73—81no—
145C2×C4×C73—81no—
146C2×C4×C73—81no—
147C2×C4×C73#12111yescharacteristic
148C2×C4×C73#12111yescharacteristic
149C2×C4×C73#12111yescharacteristic
150C2×C4×C73#12311yescharacteristic
151C2×C4×C73#12311yescharacteristic
152C2×C4×C73#12411yescharacteristic
153C2×C4×C73C2×D811yescharacteristic
154C2×C4×C73#13511yescharacteristic
155C2×C4×C73#13511yescharacteristic
156C2×C4×C73#13511yescharacteristic
157C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
158C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
159C2×C4×C73#13711yescharacteristic
160C2×C4×C73#13711yescharacteristic
161C2×Dic73—41no—
162C2×Dic73—42no—
163C2×Dic73—42no—
164C2×Dic73—41no—
165C2×Dic73—41no—
166C2×Dic73—42no—
167D292—22no—
168D292—22no—
169D292—41no—
170D292—41no—
171C4×D73—81no—
172C4×D73—81no—
173C4×D73—81no—
174C4×D73—81no—
175Dic146—81no—
176Dic146—81no—
177Dic146—81no—
178Dic146—81no—
179C8×C73—42no—
180#10596831—42no—
181#10596831—81no—
182C22×C73#39411yescharacteristic
183C22×C73#44711yescharacteristic
184C22×C73#44911yescharacteristic
185C22×C73#45211yescharacteristic
186C22×C73#45211yescharacteristic
187C22×C73#45411yescharacteristic
188C22×C73#45611yescharacteristic
189C2×D73—42no—
190C2×D73—42no—
191C2×D73—42no—
192C2×D73—81no—
193C2×D73—81no—
194C2×D73—81no—
195C2×D73—81no—
196C4×C73—21no—
197C4×C73—21no—
198C4×C73—81no—
199C4×C73—81no—
200C4×C73—161no—
201C4×C73#46311yescharacteristic
202C4×C73#46411yescharacteristic
203C4×C73#46611yescharacteristic
204C4×C73#46811yescharacteristic
205C4×C73#48011yescharacteristic
206C4×C73#48211yescharacteristic
207C4×C73#48811yescharacteristic
208C4×C73#48911yescharacteristic
209Dic73—82no—
210Dic73—81no—
211#10416—731nomaximal, Sylow
212C2×C73#162511yescharacteristic
213C2×C73#163811yescharacteristic
214C2×C73#164711yescharacteristic
215C2×C73#165211yescharacteristic
216C2×C73#203311yescharacteristic
217C2×C73#203811yescharacteristic
218C2×C73#204211yescharacteristic
219D73—82no—
220#2547—731no—
221#1990—731no—
222#1947D7311yescharacteristic, complemented
223#1802—731no—
224C2×#332—731no—
225C2×D4⋊C8—731no—
226#1279—731no—
227C73#1041611yescharacteristic, complemented, Sylow
228C2×C4⋊Q8—731no—
229C2×#124—1461no—
230C2×C4×D4—731no—
231C2×#107—21no—
232C2×C4⋊C8—731no—
233C2×C4⋊C8C2×D7311yescharacteristic
234C2×#102—1461no—
235C2×C4×C8—731no—
236#397—1462no—
237#391—1461no—
238#384C2×D7311yescharacteristic
239#380—1461no—
240#380—1461no—
241C4×C4⋊C4C2×D7311yescharacteristic
242#340—21no—
243#340—1461no—
244#340—1462no—
245#332—1462no—
246D4⋊C8—1462no—
247C22×Q8—1461no—
248C22×D4—731no—
249C23×C4—1461no—
250C22×C8—21no—
251C22×C8—1461no—
252C22×C8—1461no—
253C4⋊Q8—1461no—
254C4⋊Q8—1461no—
255C4⋊Q8—1461no—
256C4⋊Q8—1461no—
257#124—2922no—
258#124—2922no—
259C4×D4—1461no—
260C4×D4—1461no—
261C4×D4—1461no—
262C4×D4—1461no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—731no—
267C2×C4⋊C4—732no—
268C2×C4⋊C4—1461no—
269C2×C4⋊C4C22⋊D7311yescharacteristic
270C2×C4⋊C4C22⋊D7311yescharacteristic
271C2×C22⋊C4—1462no—
272C2×C22⋊C4—1462no—
273C2×C22⋊C4—1461no—
274C2×C42—21no—
275C2×C42C22×D7311yescharacteristic
276#107—42no—
277C4⋊C8—21no—
278C4⋊C8—1461no—
279C4⋊C8—1461no—
280C4⋊C8D29212yes—
281#102—2922no—
282C4×C8—1461no—
283C4×C8—1461no—
284#95—21no—
285#95—1462no—
286#95—1462no—
287C24—1461no—
288C2×Q8—2921no—
289C2×Q8—2921no—
290C2×Q8—2921no—
291C2×Q8—2921no—
292C2×Q8—2921no—
293C2×Q8—2921no—
294C2×D4—732no—
295C2×D4—1461no—
296C2×D4—1461no—
297C2×D4—1461no—
298C2×D4—1461no—
299C2×D4—1461no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—41no—
303C22×C4—2921no—
304C22×C4—2921no—
305C22×C4—2921no—
306C22×C4—2921no—
307C22×C4—2921no—
308C22×C4—2921no—
309C22×C4—1461no—
310C22×C4—1462no—
311C22×C4D4×D7311yescharacteristic
312C22×C4D4×D7311yescharacteristic
313C22×C4C2×C22⋊D7311yescharacteristic
314C2×C8—22no—
315C2×C8—41no—
316C2×C8—41no—
317C2×C8—2921no—
318C2×C8—2921no—
319C2×C8—2921no—
320C2×C8—2921no—
321C2×C8—2921no—
322C2×C8—1462no—
323C4⋊C4—21no—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—42no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—2921no—
333C4⋊C4—2921no—
334C4⋊C4—1461no—
335C4⋊C4—1462no—
336C4⋊C4—1461no—
337C4⋊C4—1462no—
338C22⋊C4—2924no—
339C22⋊C4—2924no—
340C22⋊C4—2921no—
341C22⋊C4—2921no—
342C42—41no—
343C42—41no—
344C42C2×D29211yescharacteristic
345C42#1192402811yescharacteristic
346C42#1192402911yescharacteristic
347C42#1192402911yescharacteristic
348C23—2921no—
349C23—2921no—
350C23—2921no—
351C23—2921no—
352C23—2921no—
353C23—2921no—
354C23—1462no—
355C23#1455655911yescharacteristic
356Q8—5841no—
357Q8—5841no—
358Q8—5841no—
359Q8—5841no—
360D4—2921no—
361D4—2921no—
362D4—1462no—
363D4—1462no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—2922no—
376C2×C4—2921no—
377C2×C4—2921no—
378C2×C4—2922no—
379C2×C4—2921no—
380C2×C4—2922no—
381C2×C4—5841no—
382C2×C4—5841no—
383C2×C4—5841no—
384C2×C4—5841no—
385C2×C4#1455656511yescharacteristic
386C2×C4#1455656611yescharacteristic
387C2×C4#1455656611yescharacteristic
388C2×C4#1455657611yescharacteristic
389C2×C4#1455657611yescharacteristic
390C2×C4#1455658011yescharacteristic
391C2×C4D8×D7311yescharacteristic
392C2×C4#1455661111yescharacteristic
393C2×C4#1455661111yescharacteristic
394C2×C4#1455661411yescharacteristic
395C2×C4#1455661511yescharacteristic
396C2×C4#1455661711yescharacteristic
397C2×C4#1455662211yescharacteristic
398C2×C4#1455662311yescharacteristic
399C8—42no—
400C8—2922no—
401C8—5841no—
402C22—2922no—
403C22—2922no—
404C22—2922no—
405C22—5841no—
406C22—5841no—
407C22—5841no—
408C22—5841no—
409C22#1798850411yescharacteristic
410C22#1798871911yescharacteristic
411C22#1798873011yescharacteristic
412C22#1798874711yescharacteristic
413C22#1798874811yescharacteristic
414C22#1798875811yescharacteristic
415C22#1798876811yescharacteristic
416C4—21no—
417C4—21no—
418C4—81no—
419C4—81no—
420C4—161no—
421C4—5842no—
422C4—5841no—
423C4#1798880211yescharacteristic
424C4#1798880911yescharacteristic
425C4#1798882411yescharacteristic
426C4#1798883411yescharacteristic
427C4#1798890211yescharacteristic
428C4#1798891411yescharacteristic
429C4#1798894411yescharacteristic
430C4#1798894911yescharacteristic
431C2—5842no—
432C2#2260901911yescharacteristic
433C2#2260910211yescharacteristic
434C2#2260916511yescharacteristic
435C2#2260919911yescharacteristic
436C2#2261111511yescharacteristic
437C2#2261115011yescharacteristic
438C2#2261117211yescharacteristic
subgroupG/H
centerC23#14556559
commutator, cosocleC2×C42×C73C23
FrattiniC2×C42C22×D73
FittingC73×#1947C2
socleC23×C73#120
prime radicalC22×D292D4
hypercenter#1947D73
hypocenterC73#10416
LCSG ▷ C2×C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1947

Complements

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

NH ≅ G/Ntype
2. #22610891C2semidirect
4. C2×#17988332C2semidirect
6. #22607772C2semidirect
7. C73×#1947C2semidirect
222. #1947D73semidirect
227. C73#10416semidirect

Maximal subgroups

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

subgroupindex
#226149262
#226108912
#226099282
C2×#179883322
C2×#179883142
#226077722
C73×#19472
#1041673