Finite Group Databasebeta

← #32973014

Subgroups — #32973014

Order 18688.

422 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 75 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#22608261C211yesmaximal, characteristic
2#22608163C211yesmaximal, characteristic
3C73×#1821C211yesmaximal, characteristic
4C22×C4×Dic73C411yescharacteristic
5#17988551—21no—
6C2×#14556498—21no—
7C2×#14556498—21no—
8C2×#14556498C411yescharacteristic
9#17988313—21no—
10#17988313C412yescomplemented
11C22×C4⋊C4×C73C2211yescharacteristic
12#407×C73—21no—
13#343×C73—21no—
14#343×C73C412yescomplemented
15C23×Dic73—21no—
16C23×Dic73—21no—
17C2×#11924012—41no—
18C2×C4×Dic73—41no—
19C2×C4×Dic73—42no—
20C22⋊#10596831—22no—
21C22⋊#10596831—41no—
22#14556498—24no—
23#14556498—41no—
24#14556498—42no—
25C23×C4×C73C2×C411yescharacteristic
26C23×C4×C73D411yescharacteristic
27C23×C4×C73D411yescharacteristic
28C2×M16×C73—22no—
29C2×C4⋊C4×C73—41no—
30C2×C4⋊C4×C73—41no—
31C2×C4⋊C4×C73C2×C411yescharacteristic
32C2×C4⋊C4×C73C2×C411yescharacteristic
33C2×C4⋊C4×C73D411yescharacteristic
34C2×C4⋊C4×C73Q811yescharacteristic
35C22⋊C8×C73—41no—
36C22⋊C8×C73—41no—
37C22×Dic73—22no—
38C22×Dic73—22no—
39C22×Dic73—22no—
40C22×Dic73—22no—
41C22×Dic73—41no—
42C22×Dic73—41no—
43C22×Dic73—41no—
44C22×Dic73—41no—
45C22×Dic73—41no—
46C22×Dic73—41no—
47C22×Dic73—41no—
48C22×Dic73—41no—
49C22×Dic73—41no—
50C22×Dic73—41no—
51#11924012—81no—
52#11924012—81no—
53C2×#10596831—42no—
54C4×Dic73—42no—
55C4×Dic73—42no—
56C24×C73C22⋊C411yescharacteristic
57C22×C4×C73—21no—
58C22×C4×C73—21no—
59C22×C4×C73—21no—
60C22×C4×C73—21no—
61C22×C4×C73—21no—
62C22×C4×C73—21no—
63C22×C4×C73—22no—
64C22×C4×C73—41no—
65C22×C4×C73—41no—
66C22×C4×C73—41no—
67C22×C4×C73—41no—
68C22×C4×C73—41no—
69C22×C4×C73C4211yescharacteristic
70C22×C4×C73C22⋊C411yescharacteristic
71C22×C4×C73C22⋊C411yescharacteristic
72C22×C4×C73C4⋊C411yescharacteristic
73C22×C4×C73C4⋊C411yescharacteristic
74C22×C4×C73C4⋊C411yescharacteristic
75M16×C73—42no—
76M16×C73—42no—
77C2×C8×C73—42no—
78C4⋊C4×C73—41no—
79C4⋊C4×C73—41no—
80C4⋊C4×C73—41no—
81C4⋊C4×C73—41no—
82C23×C73—21no—
83C23×C73—21no—
84C23×C73—21no—
85C23×C73—21no—
86C23×C73—21no—
87C23×C73—21no—
88C23×C73#9511yescharacteristic
89C23×C73#9911yescharacteristic
90C23×C73#9911yescharacteristic
91C2×C4×C73—21no—
92C2×C4×C73—21no—
93C2×C4×C73—21no—
94C2×C4×C73—21no—
95C2×C4×C73—21no—
96C2×C4×C73—21no—
97C2×C4×C73—21no—
98C2×C4×C73—21no—
99C2×C4×C73—21no—
100C2×C4×C73—42no—
101C2×C4×C73—42no—
102C2×C4×C73—42no—
103C2×C4×C73—41no—
104C2×C4×C73—41no—
105C2×C4×C73—41no—
106C2×C4×C73—41no—
107C2×C4×C73—41no—
108C2×C4×C73—41no—
109C2×C4×C73—41no—
110C2×C4×C73—41no—
111C2×C4×C73—81no—
112C2×C4×C73#10412yes—
113C2×Dic73—24no—
114C2×Dic73—42no—
115C2×Dic73—42no—
116C2×Dic73—42no—
117C2×Dic73—42no—
118C2×Dic73—42no—
119C2×Dic73—42no—
120C2×Dic73—42no—
121C2×Dic73—42no—
122C2×Dic73—42no—
123C2×Dic73—81no—
124C2×Dic73—81no—
125C2×Dic73—81no—
126C2×Dic73—81no—
127C8×C73—82no—
128#10596831—82no—
129C22×C73—21no—
130C22×C73—21no—
131C22×C73—21no—
132C22×C73—21no—
133C22×C73—21no—
134C22×C73—21no—
135C22×C73—21no—
136C22×C73—21no—
137C22×C73—21no—
138C22×C73—21no—
139C22×C73—21no—
140C22×C73—21no—
141C22×C73—21no—
142C22×C73—21no—
143C22×C73H(3,ℤ/4)11yescharacteristic
144C22×C73#33911yescharacteristic
145C22×C73#34211yescharacteristic
146C22×C73AΣL(1,𝔽16)11yescharacteristic
147C22×C73#35211yescharacteristic
148C22×C73#35511yescharacteristic
149C22×C73#35611yescharacteristic
150C4×C73—22no—
151C4×C73—41no—
152C4×C73—41no—
153C4×C73—41no—
154C4×C73—41no—
155C4×C73—41no—
156Dic73—44no—
157Dic73—82no—
158#7215—731nomaximal, Sylow
159C2×C73—21no—
160C2×C73—21no—
161C2×C73—21no—
162C2×C73—21no—
163C2×C73—21no—
164C2×C73—21no—
165C2×C73#128411yescharacteristic
166C2×C73#139011yescharacteristic
167C2×C73#139411yescharacteristic
168#1821D7311yescharacteristic
169#1463—731no—
170#1436—731no—
171C73#721511yescharacteristic, complemented, Sylow
172C22×C4⋊C4C2×D7311yescharacteristic
173C22×C42—731no—
174#407—21no—
175#407—1461no—
176C2×#95—731no—
177C2×#95—1461no—
178C2×#95—1461no—
179#343—21no—
180#343Dic7312yescomplemented
181#324—732no—
182#324—1461no—
183C23×C4—1461no—
184C23×C4—1461no—
185C23×C4C4×D7311yescharacteristic
186C23×C4D29211yescharacteristic
187C23×C4C22⋊D7311yescharacteristic
188C2×M16—22no—
189C2×M16—2921no—
190C2×C4⋊C4—41no—
191C2×C4⋊C4—41no—
192C2×C4⋊C4Dic14611yescharacteristic
193C2×C4⋊C4C4×D7311yescharacteristic
194C2×C4⋊C4C2×Dic7311yescharacteristic
195C2×C4⋊C4C22⋊D7311yescharacteristic
196C2×C42—2921no—
197C2×C42—2922no—
198C22⋊C8—41no—
199C22⋊C8—41no—
200C22⋊C8—2921no—
201C22⋊C8—1462no—
202#95—2921no—
203#95—2922no—
204#95—1464no—
205C24#1192400511yescharacteristic
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—22no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—2921no—
219C22×C4—2921no—
220C22×C4—2921no—
221C22×C4—2921no—
222C22×C4—2921no—
223C22×C4—2921no—
224C22×C4—2921no—
225C22×C4—2921no—
226C22×C4—2921no—
227C22×C4—2921no—
228C22×C4—1462no—
229C22×C4—1462no—
230C22×C4—1462no—
231C22×C4—1462no—
232C22×C4C4×Dic7311yescharacteristic
233C22×C4#1192400511yescharacteristic
234C22×C4C22⋊Dic7311yescharacteristic
235C22×C4#1192400711yescharacteristic
236C22×C4#1192400711yescharacteristic
237C22×C4C4⋊Dic7311yescharacteristic
238M16—42no—
239M16—42no—
240M16—5841no—
241M16—5841no—
242C2×C8—42no—
243C2×C8—2922no—
244C4⋊C4—41no—
245C4⋊C4—41no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C42—2922no—
249C42—2922no—
250C23—21no—
251C23—21no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—21no—
256C23#1455649811yescharacteristic
257C23#1455650511yescharacteristic
258C23#1455650511yescharacteristic
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—81no—
280C2×C4—2922no—
281C2×C4—2922no—
282C2×C4—2922no—
283C2×C4—2922no—
284C2×C4—2922no—
285C2×C4—2922no—
286C2×C4—2922no—
287C2×C4—2922no—
288C2×C4—2922no—
289C2×C4—5841no—
290C2×C4—5841no—
291C2×C4—5841no—
292C2×C4—5841no—
293C2×C4—1464no—
294C2×C4#1455651812yes—
295C8—82no—
296C8—5842no—
297C22—21no—
298C22—21no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22#1798834311yescharacteristic
312C22#1798834611yescharacteristic
313C22#1798835311yescharacteristic
314C22#1798837511yescharacteristic
315C22#1798837811yescharacteristic
316C22#1798838311yescharacteristic
317C22#1798838511yescharacteristic
318C4—22no—
319C4—41no—
320C4—41no—
321C4—41no—
322C4—41no—
323C4—41no—
324C4—2924no—
325C4—5842no—
326C2—21no—
327C2—21no—
328C2—21no—
329C2—21no—
330C2—21no—
331C2—21no—
332C2#2260778311yescharacteristic
333C2#2260805011yescharacteristic
334C2#2260805711yescharacteristic
subgroupG/H
centerC22#17988353
commutatorC22×C4×C73C42
FrattiniC22×C4⋊C4C2×D73
FittingC73×#1821C2
socleC22×C73#342
prime radicalC24×C73C22⋊C4
cosocleC22×C4⋊C4×C73C22
hypercenter#1821D73
hypocenterC73#7215
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ #1821

Complements

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

NH ≅ G/Ntype
10. #17988313C4semidirect
14. #343×C73C4semidirect
171. C73#7215semidirect
180. #343Dic73semidirect

Maximal subgroups

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

subgroupindex
#226082612
#226081632
C73×#18212
#721573