Finite Group Databasebeta

← #32980250

Subgroups — #32980250

Order 18688.

530 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#22612882C211yesmaximal, characteristic
2#22610690C211yesmaximal, characteristic, complemented
3#22610058C211yesmaximal, characteristic, complemented
4#22609932C211yesmaximal, characteristic
5C2×#17988314C211yesmaximal, characteristic
6#22607770C211yesmaximal, characteristic, complemented
7C73×#1296C211yesmaximal, characteristic, complemented
8C2×C4×D292C2211yescharacteristic
9C2×#14556549—21no—
10C2×#14556525—21no—
11C2×#14556513—21no—
12C2×C4×#10596831C2211yescharacteristic
13#17988488—21no—
14#17988477—22no—
15#17988457C2211yescharacteristic
16#17988444—21no—
17C4⋊C4×Dic73—21no—
18C4×C4⋊Dic73C2211yescharacteristic
19#17988350—21no—
20#17988350—21no—
21#17988349—22no—
22#17988314C414yescomplemented
23C2×C4⋊C8×C73C2211yescharacteristic
24C2×C4⋊C8×C73C2211yescharacteristic
25#384×C73C2211yescharacteristic
26C22×D292D411yescharacteristic
27C22×C4×D73—21no—
28C22×#10596831—21no—
29C4×D292—21no—
30C4×D292—21no—
31C4×D292C2×C412yes—
32C4×D292Q812yes—
33#14556549—42no—
34#14556549—42no—
35C2×C4⋊Dic73—21no—
36C2×C4⋊Dic73—21no—
37C2×C4⋊Dic73—21no—
38C2×C4⋊Dic73D411yescharacteristic
39C2×C4⋊Dic73D411yescharacteristic
40C2×C4⋊Dic73D411yescharacteristic
41C2×#11924005—22no—
42C2×#11924005—22no—
43C2×#11924005—21no—
44C2×C4×Dic73—22no—
45C2×C4×Dic73—21no—
46#14556525—42no—
47#14556513—42no—
48C4×#10596831—21no—
49C4×#10596831C2×C412yes—
50#14556498—22no—
51#14556498—21no—
52C22×C8×C73—21no—
53C22×C8×C73—21no—
54C2×C4⋊C4×C73—21no—
55C2×C4⋊C4×C73—21no—
56C2×C4⋊C4×C73D412yes—
57C2×C42×C73C2311yescharacteristic
58C4⋊C8×C73—21no—
59C4⋊C8×C73—21no—
60C4⋊C8×C73—21no—
61C4⋊C8×C73C2×C412yes—
62C23×D73—21no—
63C2×D292—21no—
64C2×D292—21no—
65C2×D292—21no—
66C2×D292—21no—
67C2×D292—21no—
68C2×D292C4⋊C412yes—
69C22×Dic73—21no—
70C22×Dic73—21no—
71C22×Dic73—21no—
72C22×Dic73—41no—
73C2×C4×D73—22no—
74C2×C4×D73—22no—
75C2×C4×D73—41no—
76C2×C4×D73—41no—
77C2×C4×D73—41no—
78C2×C4×D73—41no—
79C2×#10596831—22no—
80C2×#10596831—41no—
81C2×#10596831—41no—
82C4⋊Dic73—21no—
83C4⋊Dic73—21no—
84C4⋊Dic73—21no—
85C4⋊Dic73—21no—
86C4⋊Dic73—42no—
87C4⋊Dic73—41no—
88C4⋊Dic73—41no—
89C4⋊Dic73—41no—
90C4⋊Dic73—41no—
91C4⋊Dic73C4⋊C412yes—
92C4⋊Dic73D812yes—
93#11924005—22no—
94#11924005—22no—
95#11924005—44no—
96#11924005—82no—
97C4×Dic73—44no—
98C4×Dic73—41no—
99C4×Dic73—41no—
100C22×C4×C73—22no—
101C22×C4×C73C2×D411yescharacteristic
102C22×C4×C73C2×D411yescharacteristic
103C22×C4×C73C2×D411yescharacteristic
104C2×C8×C73—22no—
105C2×C8×C73—41no—
106C2×C8×C73—41no—
107C2×C8×C73—41no—
108C2×C8×C73—41no—
109C2×C8×C73—41no—
110C4⋊C4×C73—42no—
111C4⋊C4×C73—41no—
112C4⋊C4×C73—41no—
113C4⋊C4×C73—41no—
114C4⋊C4×C73—41no—
115C42×C73C22×C411yescharacteristic
116C42×C73C2×Q811yescharacteristic
117C42×C73#4111yescharacteristic
118C42×C73#4111yescharacteristic
119C23×C73#12011yescharacteristic
120C22×D73—22no—
121C22×D73—41no—
122C22×D73—41no—
123C22×D73—41no—
124C22×D73—41no—
125C22×D73—41no—
126C22×D73—41no—
127C2×C4×C73—21no—
128C2×C4×C73—21no—
129C2×C4×C73—42no—
130C2×C4×C73—42no—
131C2×C4×C73—42no—
132C2×C4×C73C2×C4⋊C411yescharacteristic
133C2×C4×C73C4×D411yescharacteristic
134C2×C4×C73C4×D411yescharacteristic
135C2×C4×C73C22⋊Q811yescharacteristic
136C2×C4×C73C22⋊Q811yescharacteristic
137C2×C4×C73#12311yescharacteristic
138C2×C4×C73C2×D811yescharacteristic
139C2×C4×C73#13511yescharacteristic
140C2×C4×C73#13511yescharacteristic
141C2×C4×C73#13511yescharacteristic
142C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
143C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
144C2×C4×C73#13711yescharacteristic
145C2×C4×C73#13711yescharacteristic
146C2×Dic73—22no—
147C2×Dic73—22no—
148C2×Dic73—41no—
149C2×Dic73—41no—
150C2×Dic73—41no—
151C2×Dic73—41no—
152C2×Dic73—41no—
153C2×Dic73—41no—
154C2×Dic73—41no—
155C2×Dic73—81no—
156C2×Dic73—81no—
157C2×Dic73—81no—
158D292—22no—
159D292—22no—
160D292—41no—
161D292—41no—
162C4×D73—42no—
163C4×D73—42no—
164C4×D73—42no—
165C4×D73—42no—
166C8×C73—42no—
167C8×C73—81no—
168#10596831—42no—
169C22×C73#38511yescharacteristic
170C22×C73#44711yescharacteristic
171C22×C73#44911yescharacteristic
172C22×C73#45211yescharacteristic
173C22×C73#45211yescharacteristic
174C22×C73#45411yescharacteristic
175C22×C73#45511yescharacteristic
176C2×D73—42no—
177C2×D73—42no—
178C2×D73—42no—
179C2×D73—81no—
180C2×D73—81no—
181C2×D73—81no—
182C2×D73—81no—
183C4×C73—21no—
184C4×C73—21no—
185C4×C73—82no—
186C4×C73#42711yescharacteristic
187C4×C73#42811yescharacteristic
188C4×C73C4×D811yescharacteristic
189C4×C73#44011yescharacteristic
190C4×C73#47411yescharacteristic
191C4×C73#47811yescharacteristic
192C4×C73#48211yescharacteristic
193C4×C73#48511yescharacteristic
194Dic73—42no—
195Dic73—42no—
196Dic73—81no—
197Dic73—161no—
198#8897—731nomaximal, Sylow
199C2×C73#162511yescharacteristic
200C2×C73#163811yescharacteristic
201C2×C73#164111yescharacteristic
202C2×C73#164611yescharacteristic
203C2×C73#185411yescharacteristic
204C2×C73#186011yescharacteristic
205C2×C73#186211yescharacteristic
206D73—82no—
207#2354—731no—
208#1947—731no—
209#1834—731no—
210#1802—731no—
211C2×D4⋊C8—731no—
212#1296D7311yescharacteristic, complemented
213#1279—731no—
214C73#889711yescharacteristic, complemented, Sylow
215C2×C4×D4—731no—
216C2×#117—1461no—
217C2×#107—1461no—
218C2×C4⋊C8C2×D7311yescharacteristic
219C2×C4⋊C8C2×D7311yescharacteristic
220C2×#102—1461no—
221C2×C4×C8—731no—
222#390—1461no—
223#388—1462no—
224#384—731no—
225#384C2×D7311yescharacteristic
226#380—1461no—
227C4×C4⋊C4—731no—
228C4×C4⋊C4—1461no—
229#340—1461no—
230#340—1461no—
231#340—1462no—
232D4⋊C8—734no—
233C22×D4—731no—
234C23×C4—1461no—
235C22×C8—21no—
236C22×C8—21no—
237C22×C8—1461no—
238C4×D4—732no—
239C4×D4—732no—
240C4×D4—1461no—
241C4×D4—1461no—
242#117—2922no—
243#117—2922no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—731no—
247C2×C4⋊C4—731no—
248C2×C4⋊C4—731no—
249C2×C4⋊C4—1461no—
250C2×C4⋊C4—1461no—
251C2×C4⋊C4—1461no—
252C2×C4⋊C4D29212yes—
253C2×C22⋊C4—1462no—
254C2×C22⋊C4—1462no—
255C2×C22⋊C4—1461no—
256C2×C42—1462no—
257C2×C42—1461no—
258C2×C42C22×D7311yescharacteristic
259#107—2922no—
260C4⋊C8—21no—
261C4⋊C8—21no—
262C4⋊C8—21no—
263C4⋊C8C4×D7312yes—
264#102—2922no—
265C4×C8—732no—
266C4×C8—1461no—
267#95—1462no—
268#95—1461no—
269C24—1461no—
270C2×D4—732no—
271C2×D4—1461no—
272C2×D4—1461no—
273C2×D4—1461no—
274C2×D4—1461no—
275C2×D4—1461no—
276C22×C4—22no—
277C22×C4—2921no—
278C22×C4—2921no—
279C22×C4—2921no—
280C22×C4—2921no—
281C22×C4—2921no—
282C22×C4—1461no—
283C22×C4—1461no—
284C22×C4—1461no—
285C22×C4—1462no—
286C22×C4—1462no—
287C22×C4D4×D7311yescharacteristic
288C22×C4D4×D7311yescharacteristic
289C22×C4C2×D29211yescharacteristic
290C2×C8—22no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8—41no—
294C2×C8—41no—
295C2×C8—41no—
296C2×C8—2921no—
297C2×C8—2921no—
298C2×C8—1462no—
299C4⋊C4—42no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—2922no—
305C4⋊C4—2921no—
306C4⋊C4—2921no—
307C4⋊C4—2921no—
308C4⋊C4—2921no—
309C4⋊C4—732no—
310C4⋊C4—732no—
311C4⋊C4—1461no—
312C4⋊C4—1461no—
313C4⋊C4—1461no—
314C4⋊C4—1461no—
315C22⋊C4—2924no—
316C22⋊C4—5842no—
317C22⋊C4—1462no—
318C22⋊C4—1462no—
319C42—2924no—
320C42—2921no—
321C42—2921no—
322C42C2×C4×D7311yescharacteristic
323C42Q8×D7311yescharacteristic
324C42#1192402711yescharacteristic
325C42#1192402811yescharacteristic
326C23—2921no—
327C23—2921no—
328C23—2921no—
329C23—2921no—
330C23—2921no—
331C23—2921no—
332C23—1462no—
333C23#1455656111yescharacteristic
334D4—2921no—
335D4—2921no—
336D4—1462no—
337D4—1462no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—2921no—
344C2×C4—2921no—
345C2×C4—2921no—
346C2×C4—2921no—
347C2×C4—2922no—
348C2×C4—2922no—
349C2×C4—2922no—
350C2×C4—2922no—
351C2×C4—2921no—
352C2×C4—2921no—
353C2×C4—2921no—
354C2×C4—5841no—
355C2×C4—5841no—
356C2×C4—5841no—
357C2×C4—1462no—
358C2×C4—1462no—
359C2×C4C4⋊C4×D7311yescharacteristic
360C2×C4#1455655111yescharacteristic
361C2×C4C4×D29211yescharacteristic
362C2×C4#1455656911yescharacteristic
363C2×C4#1455657111yescharacteristic
364C2×C4#1455657311yescharacteristic
365C2×C4D8×D7311yescharacteristic
366C2×C4#1455661011yescharacteristic
367C2×C4#1455661211yescharacteristic
368C2×C4#1455661411yescharacteristic
369C2×C4#1455661811yescharacteristic
370C2×C4#1455661811yescharacteristic
371C2×C4#1455662011yescharacteristic
372C2×C4#1455662311yescharacteristic
373C8—42no—
374C8—81no—
375C8—2922no—
376C22—2922no—
377C22—2922no—
378C22—2922no—
379C22—5841no—
380C22—5841no—
381C22—5841no—
382C22—5841no—
383C22#1798846111yescharacteristic
384C22#1798872211yescharacteristic
385C22#1798873511yescharacteristic
386C22#1798875011yescharacteristic
387C22#1798875211yescharacteristic
388C22#1798876211yescharacteristic
389C22#1798876711yescharacteristic
390C4—21no—
391C4—21no—
392C4—82no—
393C4—2922no—
394C4—2922no—
395C4—5841no—
396C4—11681no—
397C4#1798862411yescharacteristic
398C4#1798862611yescharacteristic
399C4#1798866711yescharacteristic
400C4#1798869011yescharacteristic
401C4#1798886611yescharacteristic
402C4#1798888611yescharacteristic
403C4#1798890911yescharacteristic
404C4#1798892711yescharacteristic
405C2—5842no—
406C2#2260902111yescharacteristic
407C2#2260910411yescharacteristic
408C2#2260912511yescharacteristic
409C2#2260916011yescharacteristic
410C2#2261013311yescharacteristic
411C2#2261016711yescharacteristic
412C2#2261017911yescharacteristic
subgroupG/H
centerC23#14556561
commutatorC42×C73C22×C4
FrattiniC2×C42C22×D73
FittingC73×#1296C2
socleC23×C73#120
prime radicalC22×D292D4
cosocleC2×C42×C73C23
hypercenter#1296D73
hypocenterC73#8897
LCSG ▷ C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1296

Complements

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

NH ≅ G/Ntype
2. #22610690C2semidirect
3. #22610058C2semidirect
6. #22607770C2semidirect
7. C73×#1296C2semidirect
22. #17988314C4semidirect
212. #1296D73semidirect
214. C73#8897semidirect

Maximal subgroups

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

subgroupindex
#226128822
#226106902
#226100582
#226099322
C2×#179883142
#226077702
C73×#12962
#889773