Finite Group Databasebeta

← #32985301

Subgroups — #32985301

Order 18688.

526 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988935C211yesmaximal, characteristic
2#22610857C211yesmaximal, characteristic
3#22610855C211yesmaximal, characteristic, complemented
4#22609801C211yesmaximal, characteristic, complemented
5C2×#17988337C211yesmaximal, characteristic, complemented
6C2×#17988330C211yesmaximal, characteristic
7C2×#332×C73C211yesmaximal, characteristic, complemented
8C2×#14556588C2211yescharacteristic
9#17988935—22no—
10#17988935C414yescomplemented
11C2×C4⋊#10596831C2211yescharacteristic
12C2×#14556515—22no—
13C2×#14556513—22no—
14C2×#14556511—21no—
15C2×#14556511—21no—
16C2×#14556501—21no—
17C2×C4×#10596831C2211yescharacteristic
18C4×C4⋊Dic73C2211yescharacteristic
19#17988341—21no—
20#17988337—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×D292—21no—
29C22×#10596831—21no—
30C22×#10596831D411yescharacteristic
31C22×#10596831D411yescharacteristic
32#14556588—21no—
33#14556588—21no—
34#14556588—21no—
35#14556588C2×C412yes—
36C2×C4⋊Dic73D412yes—
37C2×C4×Dic73—21no—
38C4⋊#10596831—21no—
39C4⋊#10596831—21no—
40#14556515—44no—
41#14556513—44no—
42#14556511—24no—
43#14556511—24no—
44#14556501—42no—
45C4×#10596831C2×C412yes—
46C4×#10596831D412yes—
47#14556498—21no—
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×C73C2×C412yes—
55C2×C4⋊C4×C73—21no—
56C2×C4⋊C4×C73—21no—
57C2×C42×C73C2311yescharacteristic
58C4⋊C8×C73—21no—
59C4⋊C8×C73—21no—
60C4⋊C8×C73—21no—
61C4⋊C8×C73D412yes—
62C23×D73—41no—
63C2×D292—22no—
64C2×D292—22no—
65C2×D292—22no—
66C2×D292—41no—
67C2×D292—41no—
68C2×D292—42no—
69C2×D292—41no—
70C2×D292—42no—
71C2×D292—41no—
72C2×D292—41no—
73C2×D292—41no—
74C2×D292—41no—
75C2×D292—82no—
76C22×Dic73—22no—
77C2×#10596831—21no—
78C2×#10596831—21no—
79C2×#10596831—21no—
80C2×#10596831—21no—
81C2×#10596831—41no—
82C2×#10596831—42no—
83C2×#10596831C22⋊C412yes—
84C2×#10596831C22⋊C412yes—
85C4⋊Dic73—22no—
86C4⋊Dic73—22no—
87C4×Dic73—41no—
88C4×Dic73—41no—
89C2×Q8×C73—22no—
90C2×Q8×C73—41no—
91C2×Q8×C73—42no—
92C2×Q8×C73—42no—
93C22×C4×C73—41no—
94C22×C4×C73C2×D411yescharacteristic
95C22×C4×C73C2×D411yescharacteristic
96C22×C4×C73#4111yescharacteristic
97C2×C8×C73—22no—
98C2×C8×C73—41no—
99C2×C8×C73—41no—
100C2×C8×C73—41no—
101C2×C8×C73—42no—
102C4⋊C4×C73—22no—
103C4⋊C4×C73—22no—
104C4⋊C4×C73—41no—
105C4⋊C4×C73—41no—
106C42×C73C22×C411yescharacteristic
107C42×C73C2×D411yescharacteristic
108C42×C73C2×D411yescharacteristic
109C42×C73#4111yescharacteristic
110C23×C73#12111yescharacteristic
111C22×D73—42no—
112C22×D73—81no—
113C22×D73—81no—
114C22×D73—82no—
115C22×D73—82no—
116Q8×C73—44no—
117Q8×C73—81no—
118Q8×C73—81no—
119C2×C4×C73—22no—
120C2×C4×C73—42no—
121C2×C4×C73—82no—
122C2×C4×C73C2×C22⋊C411yescharacteristic
123C2×C4×C73C4×D411yescharacteristic
124C2×C4×C73C4×D411yescharacteristic
125C2×C4×C73#12111yescharacteristic
126C2×C4×C73#12411yescharacteristic
127C2×C4×C73#12711yescharacteristic
128C2×C4×C73#13511yescharacteristic
129C2×C4×C73#13511yescharacteristic
130C2×C4×C73AGL(1,ℤ/8)12yes—
131C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
132C2×C4×C73AGL(1,ℤ/8)12yes—
133C2×C4×C73#13711yescharacteristic
134C2×Dic73—42no—
135C2×Dic73—42no—
136C2×Dic73—42no—
137D292—42no—
138D292—42no—
139D292—44no—
140D292—82no—
141D292—82no—
142D292—81no—
143D292—81no—
144D292—81no—
145D292—81no—
146C8×C73—42no—
147C8×C73—81no—
148#10596831—22no—
149#10596831—22no—
150#10596831—81no—
151C22×C73#39011yescharacteristic
152C22×C73#46311yescharacteristic
153C22×C73#46411yescharacteristic
154C22×C73#46912yes—
155C22×C73#47112yes—
156C2×D73—82no—
157C2×D73—82no—
158C2×D73—82no—
159C2×D73—162no—
160C2×D73—162no—
161C4×C73—21no—
162C4×C73—21no—
163C4×C73—82no—
164C4×C73#41611yescharacteristic
165C4×C73#41711yescharacteristic
166C4×C73#44212yes—
167C4×C73#46311yescharacteristic
168C4×C73#46411yescharacteristic
169C4×C73#49611yescharacteristic
170C4×C73#49611yescharacteristic
171Dic73—82no—
172#9746—731nomaximal, Sylow
173C2×C73#170112yes—
174C2×C73#170212yes—
175C2×C73#197211yescharacteristic
176C2×C73#197711yescharacteristic
177C2×C73#198311yescharacteristic
178D73—162no—
179C2×#486—731no—
180#1982—731no—
181#1981—731no—
182#1770—731no—
183C2×#334—731no—
184C2×#332D7311yescharacteristic, complemented
185C2×#331—731no—
186C73#974611yescharacteristic, complemented, Sylow
187C2×C4⋊Q8C2×D7311yescharacteristic
188C2×#127—731no—
189#486—734no—
190#486—1462no—
191C2×C4⋊C8—731no—
192C2×C4⋊C8C2×D7311yescharacteristic
193C2×C4⋊C8C2×D7311yescharacteristic
194C2×#103—1462no—
195C2×#102—1462no—
196C2×#102—1461no—
197C2×#102—1461no—
198C2×#97—1461no—
199C2×C4×C8—731no—
200C4×C4⋊C4—731no—
201#336—1461no—
202#334—1462no—
203#332—22no—
204#331—1462no—
205C22×Q8—21no—
206C22×D4—1461no—
207C22×D4—1461no—
208C22×D4—1461no—
209C22×C8—21no—
210C22×C8—21no—
211C22×C8—731no—
212C22×C8—731no—
213C22×C8—1461no—
214C4⋊Q8—21no—
215C4⋊Q8—21no—
216C4⋊Q8—21no—
217C4⋊Q8C4×D7312yes—
218#127—732no—
219#127—1461no—
220#127—1461no—
221#127—1461no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—732no—
225C2×C42—1461no—
226C2×C42C22×D7311yescharacteristic
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—21no—
230C4⋊C8—1461no—
231C4⋊C8—1461no—
232C4⋊C8C22⋊D7312yes—
233#103—2924no—
234#102—2924no—
235#102—1464no—
236#102—1464no—
237#97—2922no—
238C4×C8—732no—
239C4×C8—732no—
240#95—1461no—
241C24—2921no—
242C2×Q8—22no—
243C2×Q8—41no—
244C2×Q8—42no—
245C2×Q8—42no—
246C2×D4—2921no—
247C2×D4—2921no—
248C2×D4—2921no—
249C2×D4—2921no—
250C2×D4—2921no—
251C2×D4—2922no—
252C2×D4—2922no—
253C2×D4—2921no—
254C2×D4—2921no—
255C2×D4—5842no—
256C2×D4—1462no—
257C2×D4—1462no—
258C2×D4—1462no—
259C22×C4—41no—
260C22×C4—1462no—
261C22×C4D4×D7311yescharacteristic
262C22×C4D4×D7311yescharacteristic
263C22×C4#1192402811yescharacteristic
264C2×C8—22no—
265C2×C8—41no—
266C2×C8—41no—
267C2×C8—41no—
268C2×C8—42no—
269C2×C8—2921no—
270C2×C8—2922no—
271C2×C8—732no—
272C2×C8—732no—
273C2×C8—1461no—
274C2×C8—1461no—
275C2×C8—1461no—
276C2×C8—1461no—
277C4⋊C4—22no—
278C4⋊C4—22no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4—1462no—
282C4⋊C4—1462no—
283C42—2921no—
284C42—2921no—
285C42C2×C4×D7311yescharacteristic
286C42D4×D7311yescharacteristic
287C42C2×C22⋊D7311yescharacteristic
288C42#1192402711yescharacteristic
289C23—2922no—
290C23—5841no—
291C23—5841no—
292C23—5842no—
293C23—5842no—
294C23#1455656411yescharacteristic
295Q8—44no—
296Q8—81no—
297Q8—81no—
298D4—2924no—
299D4—2922no—
300D4—2922no—
301D4—5842no—
302D4—5842no—
303D4—5841no—
304D4—5841no—
305D4—5841no—
306D4—5841no—
307C2×C4—22no—
308C2×C4—42no—
309C2×C4—82no—
310C2×C4—2922no—
311C2×C4—2922no—
312C2×C4—2922no—
313C2×C4C22⋊C4×D7311yescharacteristic
314C2×C4#1455655111yescharacteristic
315C2×C4C4×C22⋊D7311yescharacteristic
316C2×C4#1455656311yescharacteristic
317C2×C4#1455657711yescharacteristic
318C2×C4#1455658711yescharacteristic
319C2×C4#1455661211yescharacteristic
320C2×C4#1455661411yescharacteristic
321C2×C4#1455661512yes—
322C2×C4#1455661812yes—
323C2×C4#1455661811yescharacteristic
324C2×C4#1455662311yescharacteristic
325C8—42no—
326C8—81no—
327C8—5841no—
328C8—1462no—
329C8—1462no—
330C22—5842no—
331C22—5842no—
332C22—5842no—
333C22—11682no—
334C22—11682no—
335C22#1798848711yescharacteristic
336C22#1798880511yescharacteristic
337C22#1798881211yescharacteristic
338C22#1798883912yes—
339C22#1798885312yes—
340C4—21no—
341C4—21no—
342C4—82no—
343C4—5842no—
344C4#1798858211yescharacteristic
345C4#1798858711yescharacteristic
346C4#1798869612yes—
347C4#1798880411yescharacteristic
348C4#1798881111yescharacteristic
349C4#1798897311yescharacteristic
350C4#1798897511yescharacteristic
351C2—11682no—
352C2#2260949512yes—
353C2#2260950212yes—
354C2#2261080411yescharacteristic
355C2#2261083511yescharacteristic
356C2#2261086011yescharacteristic
subgroupG/H
centerC23#14556564
commutatorC42×C73C22×C4
FrattiniC2×C42C22×D73
FittingC2×#332×C73C2
socleC23×C73#121
prime radicalC2×#14556588C22
cosocleC2×C42×C73C23
hypercenterC2×#332D73
hypocenterC73#9746
LCSG ▷ C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#332

Complements

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

NH ≅ G/Ntype
3. #22610855C2semidirect
4. #22609801C2semidirect
5. C2×#17988337C2semidirect
7. C2×#332×C73C2semidirect
10. #17988935C4semidirect
184. C2×#332D73semidirect
186. C73#9746semidirect

Maximal subgroups

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

subgroupindex
C2×#179889352
#226108572
#226108552
#226098012
C2×#179883372
C2×#179883302
C2×#332×C732
#974673