Finite Group Databasebeta

← #32985099

Subgroups — #32985099

Order 18688.

590 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22618289C211yesmaximal, characteristic
2#22610047C211yesmaximal, characteristic, complemented
3#22609786C211yesmaximal, characteristic, complemented
4C2×#17988333C211yesmaximal, characteristic, complemented
5C2×#17988330C211yesmaximal, characteristic
6C2×#17988330C211yesmaximal, characteristic
7C2×#332×C73C211yesmaximal, characteristic, complemented
8C2×#14556588C2211yescharacteristic
9C2×#14556580—22no—
10C2×C4⋊#10596831C2211yescharacteristic
11C2×C4⋊#10596831C2211yescharacteristic
12C2×#14556501—21no—
13#17988488—21no—
14#17988488—21no—
15#17988488C414yes—
16C42×Dic73C2211yescharacteristic
17#17988341—21no—
18#17988341—22no—
19#17988333—22no—
20#17988330—22no—
21#17988330—22no—
22C2×C4⋊Q8×C73C2211yescharacteristic
23C2×C4⋊C8×C73C2211yescharacteristic
24C2×C4⋊C8×C73C2211yescharacteristic
25#332×C73—22no—
26C22×D292—21no—
27C22×D292—21no—
28C22×D292C2×C412yes—
29C22×#10596831—21no—
30C22×#10596831—21no—
31#14556588—21no—
32#14556588—21no—
33#14556588—21no—
34#14556588—21no—
35#14556580—44no—
36#14556580—44no—
37C2×#11924005—24no—
38C2×#11924005—24no—
39C2×C4×Dic73C2×C412yes—
40C2×C4×Dic73D411yescharacteristic
41C2×C4×Dic73D411yescharacteristic
42C2×C4×Dic73D412yes—
43C4⋊#10596831—21no—
44C4⋊#10596831—21no—
45C4⋊#10596831—21no—
46C4⋊#10596831D412yes—
47#14556501—42no—
48C22×Q8×C73—21no—
49C22×C8×C73—21no—
50C22×C8×C73—21no—
51C4⋊Q8×C73—21no—
52C4⋊Q8×C73—21no—
53C4⋊Q8×C73—21no—
54C4⋊Q8×C73—21no—
55C2×C4⋊C4×C73—21no—
56C2×C4⋊C4×C73C2×C412yes—
57C2×C42×C73C2311yescharacteristic
58C4⋊C8×C73—21no—
59C4⋊C8×C73—21no—
60C4⋊C8×C73—21no—
61C4⋊C8×C73D412yes—
62C23×D73—22no—
63C2×D292—22no—
64C2×D292—22no—
65C2×D292—44no—
66C2×D292—41no—
67C2×D292—41no—
68C2×D292—41no—
69C2×D292—41no—
70C2×D292—41no—
71C2×D292—41no—
72C2×D292—42no—
73C2×D292—42no—
74C2×D292—41no—
75C2×D292—41no—
76C22×Dic73C22⋊C412yes—
77C22×Dic73C22⋊C412yes—
78C2×#10596831—22no—
79C2×#10596831—41no—
80C2×#10596831—41no—
81C2×#10596831—41no—
82C2×#10596831—42no—
83#11924005—48no—
84#11924005—48no—
85C4×Dic73—21no—
86C4×Dic73—21no—
87C4×Dic73—21no—
88C4×Dic73—22no—
89C4×Dic73—22no—
90C4×Dic73—21no—
91C4×Dic73—42no—
92C2×Q8×C73—41no—
93C2×Q8×C73—41no—
94C2×Q8×C73—42no—
95C2×Q8×C73—42no—
96C22×C4×C73—22no—
97C22×C4×C73C22×C411yescharacteristic
98C22×C4×C73C2×D411yescharacteristic
99C22×C4×C73C2×D411yescharacteristic
100C2×C8×C73—22no—
101C2×C8×C73—41no—
102C2×C8×C73—41no—
103C2×C8×C73—41no—
104C2×C8×C73—42no—
105C4⋊C4×C73—22no—
106C4⋊C4×C73—22no—
107C4⋊C4×C73—41no—
108C4⋊C4×C73—41no—
109C42×C73C2×D411yescharacteristic
110C42×C73C2×D411yescharacteristic
111C42×C73#4111yescharacteristic
112C42×C73#4111yescharacteristic
113C23×C73C2×C22⋊C411yescharacteristic
114C22×D73—42no—
115C22×D73—44no—
116C22×D73—42no—
117C22×D73—42no—
118C22×D73—44no—
119Q8×C73—82no—
120Q8×C73—81no—
121Q8×C73—81no—
122C2×C4×C73—22no—
123C2×C4×C73—42no—
124C2×C4×C73—44no—
125C2×C4×C73C4×D411yescharacteristic
126C2×C4×C73C4×D411yescharacteristic
127C2×C4×C73#12111yescharacteristic
128C2×C4×C73#12111yescharacteristic
129C2×C4×C73#12411yescharacteristic
130C2×C4×C73#12711yescharacteristic
131C2×C4×C73#13511yescharacteristic
132C2×C4×C73#13511yescharacteristic
133C2×C4×C73AGL(1,ℤ/8)12yes—
134C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
135C2×C4×C73AGL(1,ℤ/8)12yes—
136C2×C4×C73#13711yescharacteristic
137C2×Dic73—22no—
138C2×Dic73—22no—
139C2×Dic73—22no—
140C2×Dic73—22no—
141C2×Dic73—22no—
142C2×Dic73—22no—
143D292—82no—
144D292—82no—
145D292—81no—
146D292—81no—
147D292—81no—
148D292—81no—
149D292—82no—
150D292—81no—
151D292—81no—
152C8×C73—42no—
153C8×C73—81no—
154#10596831—42no—
155#10596831—81no—
156C22×C73#39011yescharacteristic
157C22×C73#41312yes—
158C22×C73#41611yescharacteristic
159C22×C73#41711yescharacteristic
160C22×C73#41812yes—
161C2×D73—84no—
162C2×D73—84no—
163C2×D73—82no—
164C2×D73—82no—
165C2×D73—82no—
166C4×C73—21no—
167C4×C73—21no—
168C4×C73—82no—
169C4×C73#46311yescharacteristic
170C4×C73#46311yescharacteristic
171C4×C73#46411yescharacteristic
172C4×C73#46411yescharacteristic
173C4×C73#49012yes—
174C4×C73#49611yescharacteristic
175C4×C73#49611yescharacteristic
176Dic73—42no—
177Dic73—42no—
178#9710—731nomaximal, Sylow
179C2×C73#155312yes—
180C2×C73#155712yes—
181C2×C73#196111yescharacteristic
182C2×C73#196711yescharacteristic
183C2×C73#198311yescharacteristic
184D73—162no—
185#2858—731no—
186#1831—731no—
187#1767—731no—
188C2×#332—731no—
189C2×#332D7311yescharacteristic, complemented
190C2×#331—731no—
191C2×#331—731no—
192C73#971011yescharacteristic, complemented, Sylow
193C2×C4⋊Q8C2×D7311yescharacteristic
194C2×#127—731no—
195C2×#124—1462no—
196C2×C4⋊C8—731no—
197C2×C4⋊C8—731no—
198C2×C4⋊C8C2×D7311yescharacteristic
199C2×C4⋊C8C2×D7311yescharacteristic
200C2×#97—1461no—
201#390—734no—
202#390—1461no—
203#390—1461no—
204C43—731no—
205#336—1462no—
206#336—1461no—
207#332—22no—
208#332—1462no—
209#331—1462no—
210#331—1462no—
211C22×Q8—21no—
212C22×D4—732no—
213C22×D4—1461no—
214C22×D4—1461no—
215C22×C8—21no—
216C22×C8—21no—
217C22×C8—1461no—
218C22×C8—1461no—
219C4⋊Q8—21no—
220C4⋊Q8—21no—
221C4⋊Q8—21no—
222C4⋊Q8—21no—
223#127—1461no—
224#127—1461no—
225#127—1461no—
226#127—1461no—
227#124—2924no—
228#124—2924no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4C4×D7312yes—
231C2×C22⋊C4—1464no—
232C2×C22⋊C4—1464no—
233C2×C42—732no—
234C2×C42—732no—
235C2×C42—731no—
236C2×C42—731no—
237C2×C42C22×D7311yescharacteristic
238C4⋊C8—21no—
239C4⋊C8—21no—
240C4⋊C8—21no—
241C4⋊C8—732no—
242C4⋊C8—1461no—
243C4⋊C8—1461no—
244C4⋊C8—1461no—
245C4⋊C8C22⋊D7312yes—
246#97—2922no—
247C24—1462no—
248C2×Q8—41no—
249C2×Q8—41no—
250C2×Q8—42no—
251C2×Q8—42no—
252C2×D4—2924no—
253C2×D4—2921no—
254C2×D4—2921no—
255C2×D4—2921no—
256C2×D4—2921no—
257C2×D4—2921no—
258C2×D4—2921no—
259C2×D4—2922no—
260C2×D4—2922no—
261C2×D4—2921no—
262C2×D4—2921no—
263C2×D4—1462no—
264C2×D4—1462no—
265C22×C4—22no—
266C22×C4—732no—
267C22×C4—732no—
268C22×C4C2×C4×D7311yescharacteristic
269C22×C4D4×D7311yescharacteristic
270C22×C4D4×D7311yescharacteristic
271C2×C8—22no—
272C2×C8—41no—
273C2×C8—41no—
274C2×C8—41no—
275C2×C8—42no—
276C2×C8—2921no—
277C2×C8—2921no—
278C2×C8—2921no—
279C2×C8—2922no—
280C2×C8—1462no—
281C4⋊C4—22no—
282C4⋊C4—22no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C22⋊C4—2928no—
286C22⋊C4—2928no—
287C42—2922no—
288C42—1461no—
289C42—1461no—
290C42—1462no—
291C42—1461no—
292C42—1462no—
293C42—1461no—
294C42D4×D7311yescharacteristic
295C42C2×C22⋊D7311yescharacteristic
296C42#1192402711yescharacteristic
297C42#1192402811yescharacteristic
298C23—2922no—
299C23—2924no—
300C23—2922no—
301C23—2922no—
302C23—2924no—
303C23C22⋊C4×D7311yescharacteristic
304Q8—82no—
305Q8—81no—
306Q8—81no—
307D4—5842no—
308D4—5842no—
309D4—5842no—
310D4—5841no—
311D4—5841no—
312D4—5841no—
313D4—5841no—
314D4—5841no—
315D4—5841no—
316C2×C4—22no—
317C2×C4—42no—
318C2×C4—44no—
319C2×C4—1462no—
320C2×C4—1462no—
321C2×C4—1462no—
322C2×C4—1462no—
323C2×C4—1462no—
324C2×C4—1462no—
325C2×C4#1455655111yescharacteristic
326C2×C4C4×C22⋊D7311yescharacteristic
327C2×C4#1455656311yescharacteristic
328C2×C4#1455656411yescharacteristic
329C2×C4#1455657711yescharacteristic
330C2×C4#1455658711yescharacteristic
331C2×C4#1455661211yescharacteristic
332C2×C4#1455661411yescharacteristic
333C2×C4#1455661512yes—
334C2×C4#1455661812yes—
335C2×C4#1455661811yescharacteristic
336C2×C4#1455662311yescharacteristic
337C8—42no—
338C8—81no—
339C8—2922no—
340C8—5841no—
341C22—5844no—
342C22—5842no—
343C22—5842no—
344C22—5842no—
345C22—5844no—
346C22#1798848711yescharacteristic
347C22#1798857012yes—
348C22#1798858211yescharacteristic
349C22#1798858711yescharacteristic
350C22#1798859012yes—
351C4—21no—
352C4—21no—
353C4—82no—
354C4—2922no—
355C4—2922no—
356C4#1798880411yescharacteristic
357C4#1798880511yescharacteristic
358C4#1798881111yescharacteristic
359C4#1798881211yescharacteristic
360C4#1798895012yes—
361C4#1798897311yescharacteristic
362C4#1798897511yescharacteristic
363C2—11682no—
364C2#2260863512yes—
365C2#2260865212yes—
366C2#2261075311yescharacteristic
367C2#2261077911yescharacteristic
368C2#2261086011yescharacteristic
subgroupG/H
centerC23C22⋊C4×D73
commutatorC22×C4×C73C22×C4
FrattiniC2×C42C22×D73
FittingC2×#332×C73C2
socleC23×C73C2×C22⋊C4
prime radicalC2×#14556588C22
cosocleC2×C42×C73C23
hypercenterC2×#332D73
hypocenterC73#9710
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#332

Complements

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

NH ≅ G/Ntype
2. #22610047C2semidirect
3. #22609786C2semidirect
4. C2×#17988333C2semidirect
7. C2×#332×C73C2semidirect
189. C2×#332D73semidirect
192. C73#9710semidirect

Maximal subgroups

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

subgroupindex
#226182892
#226100472
#226097862
C2×#179883332
C2×#179883302
C2×#332×C732
#971073