Finite Group Databasebeta

← #32997121

Subgroups — #32997121

Order 18688.

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

normal subgroups only

This group has 508 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#22614884C211yesmaximal, characteristic
2#22609919C211yesmaximal, characteristic
3#22607956C214yesmaximal, complemented
4C73×#1459C211yesmaximal, characteristic
5C2×#14556591C411yescharacteristic, complemented
6C2×#14556570—21no—
7C2×#14556570—21no—
8C2×#14556506—21no—
9#17988523C412yescomplemented
10#17988523C2212yes—
11#17988490—21no—
12#17988456—21no—
13#17988447—21no—
14C4×C4⋊Dic73C411yescharacteristic, complemented
15#17988354C2212yes—
16C2×C4×D4×C73C2211yescharacteristic
17C2×C22⋊C8×C73—21no—
18#324×C73—21no—
19#323×C73C2212yes—
20C22×Dic146—21no—
21#14556591—41no—
22#14556591—41no—
23#14556570—44no—
24#14556570—44no—
25C2×C4⋊Dic73—21no—
26C2×C4⋊Dic73—21no—
27C2×C4⋊Dic73—41no—
28C2×C4⋊Dic73C2×C412yes—
29C2×#11924007—22no—
30C2×#11924007—22no—
31C2×C22⋊Dic73—22no—
32C2×C22⋊Dic73—22no—
33C2×C22⋊Dic73—22no—
34C2×C4×Dic73—21no—
35#14556506—42no—
36#14556498—41no—
37#14556498C2×C412yes—
38C22×D4×C73D411yescharacteristic
39C23×C4×C73D411yescharacteristic
40C23×C4×C73D411yescharacteristic
41C22×C8×C73—41no—
42C4×D4×C73—41no—
43C4×D4×C73—41no—
44C2×C4⋊C4×C73D411yescharacteristic
45C2×C22⋊C4×C73C2×C411yescharacteristic
46C2×C22⋊C4×C73C2311yescharacteristic
47C2×C42×C73C2×C411yescharacteristic
48C22⋊C8×C73—22no—
49C22⋊C8×C73—41no—
50C2×Dic146—42no—
51C2×Dic146—42no—
52C2×Dic146—41no—
53C2×Dic146—41no—
54C22×Dic73—22no—
55C22×Dic73—22no—
56C22×Dic73—42no—
57C4⋊Dic73—24no—
58C4⋊Dic73—42no—
59C4⋊Dic73—42no—
60C4⋊Dic73—82no—
61#11924007—44no—
62#11924007—44no—
63C22⋊Dic73—44no—
64C22⋊Dic73—44no—
65C22⋊Dic73—42no—
66C22⋊Dic73—42no—
67C4×Dic73—42no—
68C24×C73C22⋊C411yescharacteristic
69C24×C73C2×D411yescharacteristic
70C2×D4×C73—41no—
71C2×D4×C73—41no—
72C2×D4×C73SD1612yes—
73C2×D4×C73Q1612yes—
74C22×C4×C73—21no—
75C22×C4×C73—21no—
76C22×C4×C73—21no—
77C22×C4×C73—21no—
78C22×C4×C73—42no—
79C22×C4×C73—42no—
80C22×C4×C73C22⋊C411yescharacteristic
81C22×C4×C73C22⋊C411yescharacteristic
82C22×C4×C73C22⋊C411yescharacteristic
83C22×C4×C73C22×C411yescharacteristic
84C22×C4×C73C2×D411yescharacteristic
85C2×C8×C73—42no—
86C2×C8×C73—82no—
87C4⋊C4×C73—41no—
88C22⋊C4×C73—41no—
89C22⋊C4×C73—41no—
90C42×C73—21no—
91C42×C73—21no—
92C23×C73—21no—
93C23×C73—21no—
94C23×C73—21no—
95C23×C73—21no—
96C23×C73—41no—
97C23×C73—41no—
98C23×C73—41no—
99C23×C73—41no—
100C23×C73#10312yes—
101C23×C73C2×C22⋊C411yescharacteristic
102C23×C73C2×SD1611yescharacteristic
103C23×C73C2×Q1611yescharacteristic
104D4×C73—42no—
105D4×C73—42no—
106C2×C4×C73—21no—
107C2×C4×C73—21no—
108C2×C4×C73—22no—
109C2×C4×C73—21no—
110C2×C4×C73—21no—
111C2×C4×C73—21no—
112C2×C4×C73—42no—
113C2×C4×C73—42no—
114C2×C4×C73—41no—
115C2×C4×C73—41no—
116C2×C4×C73—41no—
117C2×C4×C73—81no—
118C2×C4×C73—81no—
119C2×C4×C73#9911yescharacteristic
120C2×C4×C73#9911yescharacteristic
121C2×C4×C73#10312yes—
122C2×Dic73—24no—
123C2×Dic73—42no—
124C2×Dic73—42no—
125C2×Dic73—42no—
126C2×Dic73—42no—
127C2×Dic73—42no—
128C2×Dic73—82no—
129C2×Dic73—82no—
130C2×Dic73—82no—
131Dic146—82no—
132Dic146—82no—
133C8×C73—82no—
134C22×C73—21no—
135C22×C73—21no—
136C22×C73—21no—
137C22×C73—22no—
138C22×C73—21no—
139C22×C73—42no—
140C22×C73—41no—
141C22×C73—41no—
142C22×C73—41no—
143C22×C73—41no—
144C22×C73—41no—
145C22×C73—41no—
146C22×C73—81no—
147C22×C73—81no—
148C22×C73C2×#9911yescharacteristic
149C22×C73C2×#10311yescharacteristic
150C22×C73#42111yescharacteristic
151C4×C73—21no—
152C4×C73—22no—
153C4×C73—41no—
154C4×C73—41no—
155C4×C73—41no—
156C4×C73#35312yes—
157Dic73—44no—
158Dic73—82no—
159Dic73—162no—
160#11901—731nomaximal, Sylow
161C2×C73—21no—
162C2×C73—21no—
163C2×C73—41no—
164C2×C73—42no—
165C2×C73—41no—
166C2×C73#152011yescharacteristic
167C2×C73#212311yescharacteristic
168C2×C73C2×#35311yescharacteristic
169#2542—731no—
170#1799—731no—
171#1459D7311yescharacteristic
172#1352—734no—
173C73#1190111yescharacteristic, complemented, Sylow
174C2×C4⋊Q8—731no—
175C2×C22⋊Q8—1461no—
176C2×C22⋊Q8—1461no—
177C2×C4×D4C2×D7311yescharacteristic
178C2×#99—1461no—
179C2×C22⋊C8—21no—
180#399—732no—
181#399—732no—
182#390—1461no—
183#384—1461no—
184#380—1461no—
185C4×C4⋊C4—731no—
186#342—732no—
187#324—21no—
188#323C2×D7312yes—
189C22×Q8—1461no—
190C22×D4D29211yescharacteristic
191C23×C4D29211yescharacteristic
192C23×C4C22⋊D7311yescharacteristic
193C22×C8—41no—
194C4⋊Q8—2921no—
195C4⋊Q8—2921no—
196C22⋊Q8—2924no—
197C22⋊Q8—2924no—
198C4×D4—41no—
199C4×D4—41no—
200C2×C4⋊C4—2921no—
201C2×C4⋊C4—732no—
202C2×C4⋊C4—1462no—
203C2×C4⋊C4—1461no—
204C2×C4⋊C4—1462no—
205C2×C4⋊C4—1461no—
206C2×C4⋊C4C22⋊D7311yescharacteristic
207C2×C22⋊C4—1462no—
208C2×C22⋊C4—1462no—
209C2×C22⋊C4—1462no—
210C2×C22⋊C4C4×D7311yescharacteristic
211C2×C22⋊C4C22×D7311yescharacteristic
212C2×C42—1461no—
213C2×C42C4×D7311yescharacteristic
214#99—2922no—
215C22⋊C8—22no—
216C22⋊C8—41no—
217#95—2921no—
218#95—732no—
219C24#1192400511yescharacteristic
220C24C2×D29211yescharacteristic
221C2×Q8—2922no—
222C2×Q8—2922no—
223C2×Q8—2921no—
224C2×Q8—2921no—
225C2×D4—41no—
226C2×D4—41no—
227C2×D4#1192401712yes—
228C2×D4Dic29212yes—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—2922no—
236C22×C4—1462no—
237C22×C4—1462no—
238C22×C4#1192400511yescharacteristic
239C22×C4#1192400511yescharacteristic
240C22×C4#1192400511yescharacteristic
241C22×C4C2×C4×D7311yescharacteristic
242C22×C4C2×C22⋊D7311yescharacteristic
243C2×C8—42no—
244C2×C8—82no—
245C4⋊C4—41no—
246C4⋊C4—2922no—
247C4⋊C4—2922no—
248C4⋊C4—2924no—
249C4⋊C4—2924no—
250C4⋊C4—5842no—
251C4⋊C4—1464no—
252C22⋊C4—41no—
253C22⋊C4—41no—
254C22⋊C4—2924no—
255C22⋊C4—2924no—
256C22⋊C4—2922no—
257C22⋊C4—2922no—
258C42—21no—
259C42—21no—
260C42—2922no—
261C23—21no—
262C23—21no—
263C23—21no—
264C23—21no—
265C23—41no—
266C23—41no—
267C23—41no—
268C23—41no—
269C23#1455651612yes—
270C23C2×#1192400511yescharacteristic
271C23C2×#1192401711yescharacteristic
272C23C2×Dic29211yescharacteristic
273Q8—5842no—
274Q8—5842no—
275D4—42no—
276D4—42no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—22no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—81no—
289C2×C4—81no—
290C2×C4—2922no—
291C2×C4—2922no—
292C2×C4—2922no—
293C2×C4—2922no—
294C2×C4—2922no—
295C2×C4—5842no—
296C2×C4—5842no—
297C2×C4—5842no—
298C2×C4—1464no—
299C2×C4#1455650511yescharacteristic
300C2×C4#1455650511yescharacteristic
301C2×C4#1455651612yes—
302C8—82no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—22no—
307C22—21no—
308C22—42no—
309C22—41no—
310C22—41no—
311C22—41no—
312C22—41no—
313C22—41no—
314C22—41no—
315C22—81no—
316C22—81no—
317C22C2×#1455650511yescharacteristic
318C22C2×#1455651611yescharacteristic
319C22#1798860311yescharacteristic
320C4—22no—
321C4—21no—
322C4—41no—
323C4—41no—
324C4—41no—
325C4—2924no—
326C4—5842no—
327C4—11682no—
328C4#1798837912yes—
329C2—21no—
330C2—21no—
331C2—41no—
332C2—42no—
333C2—41no—
334C2#2260850411yescharacteristic
335C2#2261158611yescharacteristic
336C2C2×#1798837911yescharacteristic
subgroupG/H
centerC22C2×#14556505
commutatorC22×C4×C73C22×C4
FrattiniC2×C22⋊C4C22×D73
FittingC73×#1459C2
socleC22×C73C2×#99
prime radicalC22×D4×C73D4
cosocleC2×C22⋊C4×C73C23
hypercenter#1459D73
hypocenterC73#11901
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. #22607956C2semidirect
5. C2×#14556591C4semidirect
9. #17988523C4semidirect
14. C4×C4⋊Dic73C4semidirect
173. C73#11901semidirect

Maximal subgroups

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

subgroupindex
#226148842
#226099192
#226079562
C73×#14592
#1190173