Finite Group Databasebeta

← #32973071

Subgroups — #32973071

Order 18688.

372 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 87 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22610224C211yesmaximal, characteristic
2#22609923C211yesmaximal, characteristic
3C73×#1799C211yesmaximal, characteristic
4C23×#10596831C411yescharacteristic, complemented
5#17988611—21no—
6C2×#14556508—21no—
7C2×C22⋊#10596831—21no—
8C2×C22⋊#10596831C411yescharacteristic, complemented
9#17988356C412yescomplemented
10#17988342—21no—
11C2×C4×D4×C73C2211yescharacteristic
12C2×#99×C73—21no—
13#380×C73—21no—
14#342×C73C412yes—
15C2×#11924012—22no—
16C22×#10596831—21no—
17C22×#10596831—21no—
18C22×#10596831—41no—
19C22×#10596831—42no—
20C4⋊#10596831—41no—
21C4⋊#10596831—41no—
22#14556508—41no—
23#14556508—41no—
24C22⋊#10596831—41no—
25C22⋊#10596831—42no—
26C22×D4×C73D411yescharacteristic
27C23×C4×C73C2×C411yescharacteristic
28C23×C4×C73D411yescharacteristic
29C4×D4×C73—41no—
30C4×D4×C73—41no—
31C2×C4⋊C4×C73—41no—
32C2×C4⋊C4×C73C2×C411yescharacteristic
33C2×C22⋊C4×C73—22no—
34C2×C22⋊C4×C73C2×C411yescharacteristic
35C2×C22⋊C4×C73Q811yescharacteristic
36C2×C42×C73D411yescharacteristic
37#99×C73—41no—
38#99×C73—41no—
39#95×C73—41no—
40#11924012—42no—
41#11924012—42no—
42C2×#10596831—22no—
43C2×#10596831—41no—
44C2×#10596831—42no—
45C2×#10596831—41no—
46C2×#10596831—42no—
47C2×#10596831—41no—
48C2×#10596831—42no—
49C2×#10596831—41no—
50C2×#10596831—41no—
51C24×C73C22⋊C411yescharacteristic
52C24×C73C4⋊C411yescharacteristic
53C2×D4×C73—41no—
54C2×D4×C73—41no—
55C2×D4×C73D811yescharacteristic
56C2×D4×C73SD1611yescharacteristic
57C2×D4×C73SD1611yescharacteristic
58C2×D4×C73Q1611yescharacteristic
59C22×C4×C73—21no—
60C22×C4×C73—21no—
61C22×C4×C73—21no—
62C22×C4×C73—21no—
63C22×C4×C73—41no—
64C22×C4×C73—41no—
65C22×C4×C73—41no—
66C22×C4×C73—41no—
67C22×C4×C73—42no—
68C22×C4×C73C4211yescharacteristic
69C22×C4×C73C22⋊C411yescharacteristic
70C22×C4×C73C22⋊C411yescharacteristic
71C22×C4×C73C4⋊C411yescharacteristic
72C22×C4×C73C4⋊C411yescharacteristic
73C4⋊C4×C73—41no—
74C4⋊C4×C73—81no—
75C4⋊C4×C73—81no—
76C22⋊C4×C73—41no—
77C22⋊C4×C73—41no—
78C22⋊C4×C73—42no—
79C22⋊C4×C73—42no—
80C42×C73—21no—
81C42×C73—21no—
82C23×C73—21no—
83C23×C73—21no—
84C23×C73—21no—
85C23×C73—21no—
86C23×C73—41no—
87C23×C73—41no—
88C23×C73—41no—
89C23×C73—41no—
90C23×C73#9511yescharacteristic
91C23×C73#10211yescharacteristic
92C23×C73#10311yescharacteristic
93C23×C73#10611yescharacteristic
94C23×C73#10711yescharacteristic
95D4×C73—41no—
96D4×C73—41no—
97D4×C73—41no—
98D4×C73—41no—
99C2×C4×C73—21no—
100C2×C4×C73—21no—
101C2×C4×C73—21no—
102C2×C4×C73—21no—
103C2×C4×C73—21no—
104C2×C4×C73—21no—
105C2×C4×C73—21no—
106C2×C4×C73—42no—
107C2×C4×C73—41no—
108C2×C4×C73—41no—
109C2×C4×C73—41no—
110C2×C4×C73—41no—
111C2×C4×C73—41no—
112C2×C4×C73—41no—
113C2×C4×C73—41no—
114C2×C4×C73—81no—
115C2×C4×C73—82no—
116C2×C4×C73—82no—
117C2×C4×C73—82no—
118C2×C4×C73#9911yescharacteristic
119C2×C4×C73#9911yescharacteristic
120C2×C4×C73#10211yescharacteristic
121C2×C4×C73#10311yescharacteristic
122#10596831—42no—
123#10596831—81no—
124#10596831—82no—
125C22×C73—21no—
126C22×C73—21no—
127C22×C73—21no—
128C22×C73—21no—
129C22×C73—21no—
130C22×C73—21no—
131C22×C73—41no—
132C22×C73—41no—
133C22×C73—41no—
134C22×C73—41no—
135C22×C73—42no—
136C22×C73—41no—
137C22×C73—41no—
138C22×C73—41no—
139C22×C73—41no—
140C22×C73—81no—
141C22×C73H(3,ℤ/4)11yescharacteristic
142C22×C73#34011yescharacteristic
143C22×C73#34211yescharacteristic
144C4×C73—21no—
145C4×C73—21no—
146C4×C73—21no—
147C4×C73—41no—
148C4×C73—41no—
149C4×C73—41no—
150C4×C73—162no—
151C4×C73AΣL(1,𝔽16)11yescharacteristic
152C4×C73#35211yescharacteristic
153#7236—731nomaximal, Sylow
154C2×C73—21no—
155C2×C73—21no—
156C2×C73—41no—
157C2×C73—41no—
158C2×C73—41no—
159C2×C73—41no—
160C2×C73#128911yescharacteristic
161C2×C73#139011yescharacteristic
162C2×C73#139511yescharacteristic
163#1869—731no—
164#1800—731no—
165#1799D7311yescharacteristic
166C73#723611yescharacteristic, complemented, Sylow
167C23×C8—731no—
168C2×C4×D4C2×D7311yescharacteristic
169#423—1461no—
170C2×#100—1461no—
171C2×#99—21no—
172C2×C22⋊C8—731no—
173C2×C22⋊C8—1461no—
174#380—21no—
175#343—732no—
176#342Dic7312yes—
177#336—1461no—
178C22×D4C22⋊D7311yescharacteristic
179C23×C4C4×D7311yescharacteristic
180C23×C4C22⋊D7311yescharacteristic
181C2×M16—1462no—
182C22×C8—2921no—
183C22×C8—2922no—
184C22×C8—1461no—
185C22×C8—1461no—
186C4×D4—41no—
187C4×D4—41no—
188C2×C4⋊C4—41no—
189C2×C4⋊C4C4×D7311yescharacteristic
190C2×C22⋊C4—22no—
191C2×C22⋊C4Dic14611yescharacteristic
192C2×C22⋊C4C2×Dic7311yescharacteristic
193C2×C42D29211yescharacteristic
194C4⋊C8—2921no—
195C4⋊C8—2921no—
196#100—2921no—
197#100—2921no—
198#99—41no—
199#99—41no—
200C22⋊C8—2922no—
201C22⋊C8—2921no—
202#95—41no—
203C24C22⋊Dic7311yescharacteristic
204C24#1192400711yescharacteristic
205C2×D4—41no—
206C2×D4—41no—
207C2×D4#1192401311yescharacteristic
208C2×D4#1192401511yescharacteristic
209C2×D4#1192401611yescharacteristic
210C2×D4#1192401811yescharacteristic
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—42no—
220C22×C4C4×Dic7311yescharacteristic
221C22×C4#1192400511yescharacteristic
222C22×C4#1192400511yescharacteristic
223C22×C4#1192400711yescharacteristic
224C22×C4C4⋊Dic7311yescharacteristic
225M16—2922no—
226M16—2922no—
227C2×C8—2921no—
228C2×C8—2922no—
229C2×C8—2921no—
230C2×C8—2922no—
231C2×C8—2921no—
232C2×C8—2922no—
233C2×C8—2921no—
234C2×C8—2921no—
235C2×C8—1462no—
236C4⋊C4—41no—
237C4⋊C4—81no—
238C4⋊C4—81no—
239C22⋊C4—41no—
240C22⋊C4—41no—
241C22⋊C4—42no—
242C22⋊C4—42no—
243C42—21no—
244C42—21no—
245C23—21no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—41no—
250C23—41no—
251C23—41no—
252C23—41no—
253C23#1455649811yescharacteristic
254C23#1455651211yescharacteristic
255C23#1455651511yescharacteristic
256C23#1455652211yescharacteristic
257C23#1455652411yescharacteristic
258D4—41no—
259D4—41no—
260D4—41no—
261D4—41no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—42no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—81no—
278C2×C4—82no—
279C2×C4—82no—
280C2×C4—82no—
281C2×C4#1455650511yescharacteristic
282C2×C4#1455650511yescharacteristic
283C2×C4#1455651111yescharacteristic
284C2×C4#1455651411yescharacteristic
285C8—2922no—
286C8—5841no—
287C8—5842no—
288C22—21no—
289C22—21no—
290C22—21no—
291C22—21no—
292C22—21no—
293C22—21no—
294C22—41no—
295C22—41no—
296C22—41no—
297C22—41no—
298C22—42no—
299C22—41no—
300C22—41no—
301C22—41no—
302C22—41no—
303C22—81no—
304C22#1798834311yescharacteristic
305C22#1798834911yescharacteristic
306C22#1798835311yescharacteristic
307C4—21no—
308C4—21no—
309C4—21no—
310C4—41no—
311C4—41no—
312C4—41no—
313C4—162no—
314C4#1798837311yescharacteristic
315C4#1798837611yescharacteristic
316C2—21no—
317C2—21no—
318C2—41no—
319C2—41no—
320C2—41no—
321C2—41no—
322C2#2260779811yescharacteristic
323C2#2260804811yescharacteristic
324C2#2260805911yescharacteristic
subgroupG/H
centerC22#17988353
commutatorC22×C4×C73C42
FrattiniC2×C4×D4C2×D73
FittingC73×#1799C2
socleC22×C73#342
prime radicalC22×D4×C73D4
cosocleC2×C4×D4×C73C22
hypercenter#1799D73
hypocenterC73#7236
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1799

Complements

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

NH ≅ G/Ntype
4. C23×#10596831C4semidirect
8. C2×C22⋊#10596831C4semidirect
9. #17988356C4semidirect
166. C73#7236semidirect

Maximal subgroups

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

subgroupindex
#226102242
#226099232
C73×#17992
#723673