Finite Group Databasebeta

← #32997134

Subgroups — #32997134

Order 18688.

568 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 99 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#22613296C211yesmaximal, characteristic
2#22609923C211yesmaximal, characteristic
3#22607962C214yesmaximal, complemented
4C73×#1459C211yesmaximal, characteristic
5C23×Dic146C411yescharacteristic
6C2×#14556570—21no—
7#17988611—21no—
8C2×#14556508—21no—
9#17988507C2212yes—
10#17988499C412yes—
11#17988493—21no—
12#17988467—21no—
13#17988459—21no—
14#17988447C411yescharacteristic
15#17988356C2212yes—
16C2×C4×D4×C73C2211yescharacteristic
17C2×C22⋊C8×C73—21no—
18#324×C73C2212yes—
19#323×C73—21no—
20C22×Dic146—21no—
21C22×Dic146—41no—
22C22×Dic146—41no—
23C22×Dic146—44no—
24C22×Dic146C2×C412yes—
25C23×Dic73—21no—
26C2×#11924012—22no—
27#14556570—44no—
28C2×C4⋊Dic73—21no—
29C2×#11924007—22no—
30C2×C22⋊Dic73—21no—
31C2×C22⋊Dic73—22no—
32C4⋊#10596831—41no—
33C4⋊#10596831—41no—
34#14556508—42no—
35#14556498—22no—
36#14556498—22no—
37#14556498C2×C412yes—
38C22×D4×C73D411yescharacteristic
39C23×C4×C73C2×C411yescharacteristic
40C23×C4×C73D411yescharacteristic
41C22×C8×C73—41no—
42C4×D4×C73—41no—
43C4×D4×C73—41no—
44C2×C4⋊C4×C73C2311yescharacteristic
45C2×C22⋊C4×C73C2×C411yescharacteristic
46C2×C22⋊C4×C73D411yescharacteristic
47C2×C42×C73D411yescharacteristic
48C22⋊C8×C73—22no—
49C22⋊C8×C73—41no—
50C2×Dic146—24no—
51C2×Dic146—42no—
52C2×Dic146—48no—
53C2×Dic146—44no—
54C2×Dic146—41no—
55C2×Dic146—41no—
56C2×Dic146—42no—
57C2×Dic146—44no—
58C2×Dic146—44no—
59C22×Dic73—22no—
60C22×Dic73—22no—
61C22×Dic73—44no—
62C22×Dic73—42no—
63#11924012—42no—
64#11924012—42no—
65C2×#10596831—42no—
66C4⋊Dic73—42no—
67#11924007—44no—
68C22⋊Dic73—44no—
69C22⋊Dic73—42no—
70C24×C73C22⋊C411yescharacteristic
71C24×C73C22⋊C411yescharacteristic
72C2×D4×C73—41no—
73C2×D4×C73—41no—
74C2×D4×C73SD1612yes—
75C2×D4×C73Q1612yes—
76C22×C4×C73—21no—
77C22×C4×C73—21no—
78C22×C4×C73—21no—
79C22×C4×C73—21no—
80C22×C4×C73—42no—
81C22×C4×C73—42no—
82C22×C4×C73C22⋊C411yescharacteristic
83C22×C4×C73C22⋊C411yescharacteristic
84C22×C4×C73C22×C411yescharacteristic
85C22×C4×C73C2×D411yescharacteristic
86C22×C4×C73C2×D411yescharacteristic
87C2×C8×C73—42no—
88C2×C8×C73—82no—
89C4⋊C4×C73—41no—
90C22⋊C4×C73—41no—
91C22⋊C4×C73—41no—
92C42×C73—21no—
93C42×C73—21no—
94C23×C73—21no—
95C23×C73—21no—
96C23×C73—21no—
97C23×C73—21no—
98C23×C73—41no—
99C23×C73—41no—
100C23×C73—41no—
101C23×C73—41no—
102C23×C73#10312yes—
103C23×C73#10312yes—
104C23×C73C2×C22⋊C411yescharacteristic
105D4×C73—42no—
106D4×C73—42no—
107C2×C4×C73—21no—
108C2×C4×C73—21no—
109C2×C4×C73—22no—
110C2×C4×C73—21no—
111C2×C4×C73—21no—
112C2×C4×C73—21no—
113C2×C4×C73—42no—
114C2×C4×C73—42no—
115C2×C4×C73—42no—
116C2×C4×C73—41no—
117C2×C4×C73—41no—
118C2×C4×C73—41no—
119C2×C4×C73—81no—
120C2×C4×C73#9911yescharacteristic
121C2×C4×C73#9911yescharacteristic
122C2×C4×C73C2×SD1611yescharacteristic
123C2×C4×C73C2×Q1611yescharacteristic
124C2×Dic73—24no—
125C2×Dic73—42no—
126C2×Dic73—48no—
127C2×Dic73—42no—
128C2×Dic73—42no—
129C2×Dic73—42no—
130C2×Dic73—42no—
131Dic146—44no—
132Dic146—44no—
133Dic146—44no—
134Dic146—44no—
135C8×C73—82no—
136#10596831—82no—
137C22×C73—21no—
138C22×C73—21no—
139C22×C73—22no—
140C22×C73—22no—
141C22×C73—42no—
142C22×C73—42no—
143C22×C73—42no—
144C22×C73—41no—
145C22×C73—41no—
146C22×C73—41no—
147C22×C73—41no—
148C22×C73—81no—
149C22×C73C2×#9911yescharacteristic
150C22×C73C2×#10311yescharacteristic
151C22×C73#42111yescharacteristic
152C4×C73—21no—
153C4×C73—21no—
154C4×C73—21no—
155C4×C73—41no—
156C4×C73—41no—
157C4×C73—41no—
158C4×C73AΣL(1,𝔽16)12yes—
159Dic73—44no—
160Dic73—82no—
161#11904—731nomaximal, Sylow
162C2×C73—21no—
163C2×C73—21no—
164C2×C73—42no—
165C2×C73—42no—
166C2×C73#152011yescharacteristic
167C2×C73C2×AΣL(1,𝔽16)11yescharacteristic
168C2×C73#213511yescharacteristic
169#2409—731no—
170#1800—731no—
171#1459D7311yescharacteristic
172#1354—734no—
173C73#1190411yescharacteristic, complemented, Sylow
174C23×Q8—731no—
175C2×C22⋊Q8—1461no—
176C2×C4×D4C2×D7311yescharacteristic
177#423—1461no—
178C2×#100—1461no—
179C2×C22⋊C8—21no—
180#395—732no—
181#393—732no—
182#391—1461no—
183#386—1461no—
184#385—1461no—
185#380—731no—
186#343—732no—
187#324C2×D7312yes—
188#323—21no—
189C22×Q8—2921no—
190C22×Q8—2921no—
191C22×Q8—2924no—
192C22×Q8—732no—
193C22×Q8—1461no—
194C22×D4D29211yescharacteristic
195C23×C4—1461no—
196C23×C4C4×D7311yescharacteristic
197C23×C4C22⋊D7311yescharacteristic
198C2×M16—1462no—
199C22×C8—41no—
200C22⋊Q8—2924no—
201C4×D4—41no—
202C4×D4—41no—
203C2×C4⋊C4—1461no—
204C2×C4⋊C4—1462no—
205C2×C4⋊C4C22×D7311yescharacteristic
206C2×C22⋊C4—1462no—
207C2×C22⋊C4—1461no—
208C2×C22⋊C4C4×D7311yescharacteristic
209C2×C22⋊C4C22⋊D7311yescharacteristic
210C2×C42D29211yescharacteristic
211C4⋊C8—2921no—
212C4⋊C8—2921no—
213#100—2922no—
214C22⋊C8—22no—
215C22⋊C8—41no—
216#95—732no—
217#95—1462no—
218#95—1462no—
219C24#1192400511yescharacteristic
220C24#1192400511yescharacteristic
221C2×Q8—2922no—
222C2×Q8—2922no—
223C2×Q8—2924no—
224C2×Q8—2928no—
225C2×Q8—2924no—
226C2×Q8—2924no—
227C2×Q8—2921no—
228C2×Q8—2921no—
229C2×Q8—1464no—
230C2×D4—41no—
231C2×D4—41no—
232C2×D4#1192401712yes—
233C2×D4Dic29212yes—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—2922no—
241C22×C4—2924no—
242C22×C4—1462no—
243C22×C4—1462no—
244C22×C4#1192400511yescharacteristic
245C22×C4#1192400511yescharacteristic
246C22×C4C2×C4×D7311yescharacteristic
247C22×C4C2×C22⋊D7311yescharacteristic
248C22×C4C2×D29211yescharacteristic
249M16—2922no—
250M16—2922no—
251C2×C8—42no—
252C2×C8—82no—
253C2×C8—2922no—
254C4⋊C4—41no—
255C4⋊C4—2922no—
256C4⋊C4—2924no—
257C22⋊C4—41no—
258C22⋊C4—41no—
259C22⋊C4—2922no—
260C22⋊C4—2924no—
261C42—21no—
262C42—21no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—41no—
268C23—41no—
269C23—41no—
270C23—41no—
271C23#1455651612yes—
272C23#1455651612yes—
273C23C2×#1192400511yescharacteristic
274Q8—2924no—
275Q8—2924no—
276Q8—2924no—
277Q8—2924no—
278D4—42no—
279D4—42no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—22no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—81no—
293C2×C4—2922no—
294C2×C4—2928no—
295C2×C4—2922no—
296C2×C4—2922no—
297C2×C4—2922no—
298C2×C4—2922no—
299C2×C4—1464no—
300C2×C4#1455650511yescharacteristic
301C2×C4#1455650511yescharacteristic
302C2×C4C2×#1192401711yescharacteristic
303C2×C4C2×Dic29211yescharacteristic
304C8—82no—
305C8—5842no—
306C22—21no—
307C22—21no—
308C22—22no—
309C22—22no—
310C22—42no—
311C22—42no—
312C22—42no—
313C22—41no—
314C22—41no—
315C22—41no—
316C22—41no—
317C22—81no—
318C22C2×#1455650511yescharacteristic
319C22C2×#1455651611yescharacteristic
320C22#1798860311yescharacteristic
321C4—21no—
322C4—21no—
323C4—21no—
324C4—41no—
325C4—41no—
326C4—41no—
327C4—2924no—
328C4—5842no—
329C4#1798837312yes—
330C2—21no—
331C2—21no—
332C2—42no—
333C2—42no—
334C2#2260850411yescharacteristic
335C2C2×#1798837311yescharacteristic
336C2#2261162511yescharacteristic
subgroupG/H
centerC22C2×#14556505
commutatorC22×C4×C73C22×C4
FrattiniC2×C4⋊C4C22×D73
FittingC73×#1459C2
socleC22×C73C2×#99
prime radicalC22×D4×C73D4
cosocleC2×C4⋊C4×C73C23
hypercenter#1459D73
hypocenterC73#11904
LCSG ▷ C22×C4×C73 ▷ C23×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1459

Complements

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

NH ≅ G/Ntype
3. #22607962C2semidirect
173. C73#11904semidirect

Maximal subgroups

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

subgroupindex
#226132962
#226099232
#226079622
C73×#14592
#1190473