Finite Group Databasebeta

← #32996265

Subgroups — #32996265

Order 18688.

468 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 107 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#22613199C211yesmaximal, characteristic
2#22609664C211yesmaximal, characteristic
3#22607956C212yesmaximal
4#22607953C212yesmaximal
5C73×#1458C211yesmaximal, characteristic
6C2×C4×Dic146C411yescharacteristic, complemented
7C2×#14556550—21no—
8#17988526C411yescharacteristic, complemented
9#17988523C2211yescharacteristic
10#17988520—21no—
11#17988510—21no—
12#17988507C411yescharacteristic, complemented
13#17988499C2211yescharacteristic
14C4×#11924007—21no—
15C4×C4⋊Dic73C411yescharacteristic, complemented
16C4×C22⋊Dic73—21no—
17#17988354—21no—
18#17988354C2212yes—
19C2×#117×C73C2211yescharacteristic
20#431×C73—21no—
21#324×C73—21no—
22#323×C73C2212yes—
23C22×Dic146C2×C411yescharacteristic
24C4×Dic146—41no—
25C4×Dic146—41no—
26#14556550—24no—
27#14556550—42no—
28C2×C4⋊Dic73—21no—
29C2×C4⋊Dic73—41no—
30C2×C4⋊Dic73C2×C411yescharacteristic
31C2×#11924007—21no—
32C2×#11924007—21no—
33C2×#11924007—21no—
34C2×C22⋊Dic73—22no—
35C2×C22⋊Dic73—21no—
36C2×C22⋊Dic73—21no—
37C2×C4×Dic73—21no—
38C2×C4×Dic73—21no—
39C2×C4×Dic73—41no—
40#14556498—21no—
41#14556498—21no—
42#14556498—21no—
43#14556498—21no—
44#14556498C2×C411yescharacteristic
45#14556498C2×C411yescharacteristic
46C23×C4×C73D411yescharacteristic
47#117×C73—41no—
48#117×C73—41no—
49C2×C4⋊C4×C73C2×C411yescharacteristic
50C2×C4⋊C4×C73D411yescharacteristic
51C2×C22⋊C4×C73D411yescharacteristic
52C2×C22⋊C4×C73C2311yescharacteristic
53C2×C42×C73C2×C411yescharacteristic
54C2×C42×C73D411yescharacteristic
55C4⋊C8×C73—41no—
56C22⋊C8×C73—22no—
57C4×C8×C73—41no—
58C2×Dic146—41no—
59C2×Dic146—42no—
60C22×Dic73—21no—
61C22×Dic73—21no—
62C22×Dic73—21no—
63C22×Dic73—21no—
64C22×Dic73—42no—
65C4⋊Dic73—42no—
66C4⋊Dic73—44no—
67C4⋊Dic73—41no—
68#11924007—42no—
69#11924007—42no—
70#11924007—42no—
71C22⋊Dic73—24no—
72C22⋊Dic73—42no—
73C22⋊Dic73—42no—
74C22⋊Dic73—42no—
75C4×Dic73—44no—
76C4×Dic73—41no—
77C4×Dic73—41no—
78C4×Dic73—42no—
79C24×C73C2×D411yescharacteristic
80C22×C4×C73—21no—
81C22×C4×C73—21no—
82C22×C4×C73—22no—
83C22×C4×C73C22⋊C411yescharacteristic
84C22×C4×C73C22⋊C411yescharacteristic
85C22×C4×C73C22⋊C411yescharacteristic
86C22×C4×C73C22⋊C411yescharacteristic
87C22×C4×C73SD1612yes—
88C22×C4×C73Q1612yes—
89C22×C4×C73C22×C411yescharacteristic
90C22×C4×C73C2×D411yescharacteristic
91C2×C8×C73—42no—
92C4⋊C4×C73—41no—
93C4⋊C4×C73—41no—
94C22⋊C4×C73—41no—
95C22⋊C4×C73—41no—
96C42×C73—21no—
97C42×C73—21no—
98C42×C73—21no—
99C42×C73—21no—
100C23×C73—21no—
101C23×C73—21no—
102C23×C73—41no—
103C23×C73—41no—
104C23×C73C2×C22⋊C411yescharacteristic
105C23×C73C2×SD1611yescharacteristic
106C23×C73C2×Q1611yescharacteristic
107C2×C4×C73—21no—
108C2×C4×C73—21no—
109C2×C4×C73—22no—
110C2×C4×C73—22no—
111C2×C4×C73—22no—
112C2×C4×C73—22no—
113C2×C4×C73—21no—
114C2×C4×C73—21no—
115C2×C4×C73—21no—
116C2×C4×C73—21no—
117C2×C4×C73—41no—
118C2×C4×C73—41no—
119C2×C4×C73—41no—
120C2×C4×C73—41no—
121C2×C4×C73#9911yescharacteristic
122C2×C4×C73#9911yescharacteristic
123C2×C4×C73#10312yes—
124C2×C4×C73#10312yes—
125C2×C4×C73#10412yes—
126C2×Dic73—22no—
127C2×Dic73—22no—
128C2×Dic73—41no—
129C2×Dic73—41no—
130C2×Dic73—41no—
131C2×Dic73—41no—
132C2×Dic73—44no—
133C2×Dic73—41no—
134C2×Dic73—41no—
135C2×Dic73—41no—
136C2×Dic73—41no—
137C2×Dic73—41no—
138C2×Dic73—41no—
139C2×Dic73—82no—
140C2×Dic73—82no—
141Dic146—42no—
142Dic146—42no—
143C8×C73—82no—
144C22×C73—21no—
145C22×C73—21no—
146C22×C73—21no—
147C22×C73—21no—
148C22×C73—41no—
149C22×C73—41no—
150C22×C73—41no—
151C22×C73—41no—
152C22×C73—81no—
153C22×C73C2×#9911yescharacteristic
154C22×C73C2×#10311yescharacteristic
155C22×C73C2×#10411yescharacteristic
156C4×C73—22no—
157C4×C73—41no—
158C4×C73—41no—
159C4×C73—41no—
160C4×C73—41no—
161C4×C73#32812yes—
162C4×C73#32812yes—
163Dic73—42no—
164Dic73—42no—
165Dic73—81no—
166Dic73—84no—
167Dic73—81no—
168#11715—731nomaximal, Sylow
169C2×C73—21no—
170C2×C73—21no—
171C2×C73—41no—
172C2×C73—41no—
173C2×C73C2×#32811yescharacteristic
174C2×C73#212011yescharacteristic
175C2×C73#212611yescharacteristic
176#2396—731no—
177#1740—731no—
178#1458D7311yescharacteristic
179#1352—732no—
180#1351—732no—
181C73#1171511yescharacteristic, complemented, Sylow
182C2×C4×Q8—731no—
183C2×#117—1461no—
184C2×#117C2×D7311yescharacteristic
185#431—21no—
186#400—731no—
187#399—731no—
188#398—1461no—
189#396—1461no—
190#395—731no—
191#393—731no—
192C4×C4⋊C4—731no—
193C4×C4⋊C4—1461no—
194C4×C22⋊C4—1461no—
195#342—732no—
196#342—1461no—
197#324—21no—
198#323C2×D7312yes—
199C22×Q8—731no—
200C23×C4D29211yescharacteristic
201C4×Q8—2921no—
202C4×Q8—2921no—
203#117—41no—
204#117—41no—
205#117—2922no—
206#117—1464no—
207C2×C4⋊C4—2921no—
208C2×C4⋊C4—731no—
209C2×C4⋊C4—1461no—
210C2×C4⋊C4—1461no—
211C2×C4⋊C4—1461no—
212C2×C4⋊C4—1461no—
213C2×C4⋊C4C4×D7311yescharacteristic
214C2×C4⋊C4C22⋊D7311yescharacteristic
215C2×C22⋊C4—1461no—
216C2×C22⋊C4—1461no—
217C2×C22⋊C4—1462no—
218C2×C22⋊C4D29211yescharacteristic
219C2×C22⋊C4C22×D7311yescharacteristic
220C2×C42—2921no—
221C2×C42—1461no—
222C2×C42—1461no—
223C2×C42C4×D7311yescharacteristic
224C2×C42C22⋊D7311yescharacteristic
225C4⋊C8—41no—
226C22⋊C8—22no—
227C4×C8—41no—
228#95—731no—
229#95—731no—
230#95—1461no—
231#95—1461no—
232#95—1461no—
233#95—1461no—
234C24C2×D29211yescharacteristic
235C2×Q8—2921no—
236C2×Q8—2922no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—22no—
240C22×C4—2922no—
241C22×C4—1461no—
242C22×C4—1461no—
243C22×C4—1461no—
244C22×C4—1461no—
245C22×C4#1192400511yescharacteristic
246C22×C4#1192400511yescharacteristic
247C22×C4#1192400511yescharacteristic
248C22×C4#1192400511yescharacteristic
249C22×C4#1192401712yes—
250C22×C4Dic29212yes—
251C22×C4C2×C4×D7311yescharacteristic
252C22×C4C2×C22⋊D7311yescharacteristic
253C2×C8—42no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—2921no—
257C4⋊C4—2922no—
258C4⋊C4—2924no—
259C4⋊C4—2922no—
260C4⋊C4—2922no—
261C4⋊C4—2922no—
262C22⋊C4—41no—
263C22⋊C4—41no—
264C22⋊C4—2922no—
265C22⋊C4—2922no—
266C22⋊C4—2922no—
267C22⋊C4—1464no—
268C42—21no—
269C42—21no—
270C42—21no—
271C42—21no—
272C42—2924no—
273C42—2921no—
274C42—2921no—
275C42—2922no—
276C23—21no—
277C23—21no—
278C23—41no—
279C23—41no—
280C23C2×#1192400511yescharacteristic
281C23C2×#1192401711yescharacteristic
282C23C2×Dic29211yescharacteristic
283Q8—2922no—
284Q8—2922no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—2921no—
300C2×C4—2921no—
301C2×C4—2921no—
302C2×C4—2921no—
303C2×C4—2924no—
304C2×C4—2921no—
305C2×C4—2921no—
306C2×C4—2921no—
307C2×C4—2921no—
308C2×C4—2921no—
309C2×C4—2921no—
310C2×C4—5842no—
311C2×C4—5842no—
312C2×C4—1462no—
313C2×C4—1462no—
314C2×C4#1455650511yescharacteristic
315C2×C4#1455650511yescharacteristic
316C2×C4#1455651612yes—
317C2×C4#1455651612yes—
318C2×C4#1455651712yes—
319C8—82no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—41no—
325C22—41no—
326C22—41no—
327C22—41no—
328C22—81no—
329C22C2×#1455650511yescharacteristic
330C22C2×#1455651611yescharacteristic
331C22C2×#1455651711yescharacteristic
332C4—22no—
333C4—41no—
334C4—41no—
335C4—41no—
336C4—41no—
337C4—2922no—
338C4—2922no—
339C4—5841no—
340C4—5844no—
341C4—5841no—
342C4#1798832512yes—
343C4#1798832512yes—
344C2—21no—
345C2—21no—
346C2—41no—
347C2—41no—
348C2C2×#1798832511yescharacteristic
349C2#2261157311yescharacteristic
350C2#2261159411yescharacteristic
subgroupG/H
centerC2×C4#14556505
commutatorC22×C4×C73C22×C4
FrattiniC2×C22⋊C4C22×D73
FittingC73×#1458C2
socleC22×C73C2×#99
prime radicalC24×C73C2×D4
cosocleC2×C22⋊C4×C73C23
hypercenter#1458D73
hypocenterC73#11715
LCSG ▷ C22×C4×C73 ▷ C23×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#117 ◁ #1458

Complements

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

NH ≅ G/Ntype
6. C2×C4×Dic146C4semidirect
8. #17988526C4semidirect
12. #17988507C4semidirect
15. C4×C4⋊Dic73C4semidirect
181. C73#11715semidirect

Maximal subgroups

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

subgroupindex
#226131992
#226096642
#226079562
#226079532
C73×#14582
#1171573