Finite Group Databasebeta

← #32980964

Subgroups — #32980964

Order 18688.

560 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 111 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#22618960C211yesmaximal, characteristic
2#22610030C212yesmaximal, complemented
3C4×#14556506C211yesmaximal, characteristic
4#22607817C212yesmaximal, complemented
5C73×#2395C211yesmaximal, characteristic
6#17988659—24no—
7#17988644—21no—
8C2×#14556506—21no—
9#17988551C2212yes—
10C2×C22⋊#10596831—21no—
11C4×C22⋊Dic73C2212yes—
12#17988354—21no—
13#17988346—24no—
14C2×C4×D4×C73C2211yescharacteristic
15C2×#117×C73C2211yescharacteristic
16#400×C73C411yescharacteristic, complemented
17#399×C73C411yescharacteristic, complemented
18#398×C73C411yescharacteristic, complemented
19#397×C73—21no—
20#397×C73C411yescharacteristic, complemented
21#394×C73—21no—
22C4×C4⋊C4×C73—21no—
23C4×C22⋊C4×C73—21no—
24C4×C22⋊C4×C73C2211yescharacteristic
25C2×#11924012D414yes—
26C22×#10596831—22no—
27C2×C22⋊Dic73—22no—
28C2×C4×Dic73—22no—
29C4⋊#10596831—22no—
30#14556506—24no—
31C22⋊#10596831—22no—
32C22⋊#10596831—22no—
33C22⋊#10596831—22no—
34#14556502—22no—
35#14556498—22no—
36C22×D4×C73D411yescharacteristic
37C23×C4×C73D411yescharacteristic
38C23×C4×C73C2311yescharacteristic
39C4×D4×C73—22no—
40C4×D4×C73—22no—
41#117×C73—22no—
42#117×C73—22no—
43C2×C4⋊C4×C73—21no—
44C2×C4⋊C4×C73—21no—
45C2×C4⋊C4×C73C2×C411yescharacteristic
46C2×C4⋊C4×C73C2×C411yescharacteristic
47C2×C4⋊C4×C73C2×C411yescharacteristic
48C2×C22⋊C4×C73—21no—
49C2×C22⋊C4×C73—21no—
50C2×C22⋊C4×C73—21no—
51C2×C22⋊C4×C73—21no—
52C2×C22⋊C4×C73C2×C411yescharacteristic
53C2×C22⋊C4×C73D411yescharacteristic
54C2×C42×C73—21no—
55C2×C42×C73—21no—
56C2×C42×C73D411yescharacteristic
57#95×C73—21no—
58#95×C73—21no—
59#95×C73—21no—
60#95×C73—21no—
61#95×C73C2×C411yescharacteristic
62#95×C73C2×C411yescharacteristic
63C22×Dic73—42no—
64#11924012—44no—
65C2×#10596831—24no—
66C2×#10596831—82no—
67C22⋊Dic73—24no—
68C22⋊Dic73—42no—
69C4×Dic73—44no—
70C24×C73C22⋊C411yescharacteristic
71C24×C73C2×D411yescharacteristic
72C2×D4×C73—22no—
73C2×D4×C73—22no—
74C2×D4×C73—41no—
75C22×C4×C73—21no—
76C22×C4×C73—21no—
77C22×C4×C73—21no—
78C22×C4×C73—21no—
79C22×C4×C73—21no—
80C22×C4×C73—21no—
81C22×C4×C73—42no—
82C22×C4×C73—42no—
83C22×C4×C73C22⋊C411yescharacteristic
84C22×C4×C73C22⋊C411yescharacteristic
85C22×C4×C73C22⋊C411yescharacteristic
86C22×C4×C73C22×C411yescharacteristic
87C22×C4×C73C2×D412yes—
88C22×C4×C73C2×D411yescharacteristic
89C22×C4×C73#4112yes—
90C4⋊C4×C73—22no—
91C4⋊C4×C73—22no—
92C4⋊C4×C73—22no—
93C4⋊C4×C73—42no—
94C4⋊C4×C73—42no—
95C22⋊C4×C73—22no—
96C22⋊C4×C73—22no—
97C22⋊C4×C73—42no—
98C22⋊C4×C73—42no—
99C22⋊C4×C73—42no—
100C22⋊C4×C73—42no—
101C42×C73—21no—
102C42×C73—21no—
103C42×C73—42no—
104C42×C73—41no—
105C42×C73—41no—
106C23×C73—21no—
107C23×C73—21no—
108C23×C73—21no—
109C23×C73—21no—
110C23×C73—41no—
111C23×C73—41no—
112C23×C73—41no—
113C23×C73—41no—
114C23×C73#10412yes—
115C23×C73C2×C22⋊C411yescharacteristic
116C23×C73#12112yes—
117D4×C73—44no—
118C2×C4×C73—22no—
119C2×C4×C73—24no—
120C2×C4×C73—22no—
121C2×C4×C73—21no—
122C2×C4×C73—21no—
123C2×C4×C73—21no—
124C2×C4×C73—21no—
125C2×C4×C73—21no—
126C2×C4×C73—41no—
127C2×C4×C73—42no—
128C2×C4×C73—41no—
129C2×C4×C73—41no—
130C2×C4×C73—42no—
131C2×C4×C73—41no—
132C2×C4×C73—41no—
133C2×C4×C73—41no—
134C2×C4×C73—41no—
135C2×C4×C73—41no—
136C2×C4×C73—41no—
137C2×C4×C73—41no—
138C2×C4×C73—41no—
139C2×C4×C73—41no—
140C2×C4×C73—41no—
141C2×C4×C73—41no—
142C2×C4×C73—81no—
143C2×C4×C73C4×D412yes—
144C2×C4×C73#12011yescharacteristic
145C2×C4×C73#12311yescharacteristic
146C2×Dic73—44no—
147C2×Dic73—82no—
148C2×Dic73—82no—
149#10596831—44no—
150C22×C73—21no—
151C22×C73—21no—
152C22×C73—22no—
153C22×C73—22no—
154C22×C73—22no—
155C22×C73—22no—
156C22×C73—42no—
157C22×C73—41no—
158C22×C73—41no—
159C22×C73—81no—
160C22×C73—81no—
161C22×C73—81no—
162C22×C73#38611yescharacteristic
163C22×C73C2×#10411yescharacteristic
164C22×C73#42111yescharacteristic
165C4×C73—21no—
166C4×C73—22no—
167C4×C73—42no—
168C4×C73—41no—
169C4×C73—41no—
170C4×C73—41no—
171C4×C73—81no—
172C4×C73—81no—
173C4×C73—81no—
174C4×C73#45812yes—
175Dic73—84no—
176#9011—731nomaximal, Sylow
177C2×C73—21no—
178C2×C73—21no—
179C2×C73—42no—
180C2×C73—42no—
181C2×C73#187111yescharacteristic
182C2×C73#188411yescharacteristic
183C2×C73#189011yescharacteristic
184#2973—731no—
185#2395D7311yescharacteristic
186#1826—732no—
187C4×#99—731no—
188#1297—732no—
189C73#901111yescharacteristic, complemented, Sylow
190C2×C4×D4C2×D7311yescharacteristic
191C2×#117C2×D7311yescharacteristic
192#435—1464no—
193#432—1461no—
194C2×#99—1461no—
195#407—732no—
196C2×C22⋊C8—1461no—
197#400Dic7311yescharacteristic, complemented
198#399Dic7311yescharacteristic, complemented
199#398Dic7311yescharacteristic, complemented
200#397—21no—
201#397Dic7311yescharacteristic, complemented
202#394—21no—
203C4×C4⋊C4—21no—
204C4×C22⋊C4—21no—
205C4×C22⋊C4—732no—
206C4×C22⋊C4C2×D7311yescharacteristic
207#342—1461no—
208#339—1464no—
209C22×D4C22⋊D7311yescharacteristic
210C23×C4C22⋊D7311yescharacteristic
211C23×C4C22×D7311yescharacteristic
212C2×M16—734no—
213C22×C8—1462no—
214C4×D4—22no—
215C4×D4—22no—
216#117—22no—
217#117—22no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4C2×Dic7311yescharacteristic
221C2×C4⋊C4C2×Dic7311yescharacteristic
222C2×C4⋊C4C2×Dic7311yescharacteristic
223C2×C22⋊C4—21no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—1462no—
228C2×C22⋊C4C2×Dic7311yescharacteristic
229C2×C22⋊C4C22⋊D7311yescharacteristic
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—1462no—
233C2×C42C22⋊D7311yescharacteristic
234C4⋊C8—1462no—
235#99—1464no—
236C22⋊C8—1462no—
237C22⋊C8—1462no—
238C22⋊C8—1462no—
239#97—1462no—
240#95—21no—
241#95—21no—
242#95—21no—
243#95—21no—
244#95—1462no—
245#95C2×Dic7311yescharacteristic
246#95C2×Dic7311yescharacteristic
247C24C22⋊Dic7311yescharacteristic
248C24C2×C22⋊D7311yescharacteristic
249C2×D4—22no—
250C2×D4—22no—
251C2×D4—41no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—2922no—
261C22×C4C22⋊Dic7311yescharacteristic
262C22×C4C22⋊Dic7311yescharacteristic
263C22×C4C22⋊Dic7311yescharacteristic
264C22×C4C22×Dic7311yescharacteristic
265C22×C4D4×D7312yes—
266C22×C4C2×C22⋊D7311yescharacteristic
267C22×C4#1192402712yes—
268M16—2924no—
269C2×C8—5842no—
270C2×C8—1464no—
271C4⋊C4—22no—
272C4⋊C4—22no—
273C4⋊C4—22no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C22⋊C4—22no—
277C22⋊C4—22no—
278C22⋊C4—42no—
279C22⋊C4—42no—
280C22⋊C4—42no—
281C22⋊C4—42no—
282C22⋊C4—2922no—
283C22⋊C4—1464no—
284C42—21no—
285C42—21no—
286C42—42no—
287C42—41no—
288C42—41no—
289C42—2924no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—41no—
295C23—41no—
296C23—41no—
297C23—41no—
298C23#1455651812yes—
299C23C2×C22⋊Dic7311yescharacteristic
300C23#1455656312yes—
301D4—44no—
302C2×C4—22no—
303C2×C4—24no—
304C2×C4—22no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—41no—
311C2×C4—42no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—42no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—81no—
327C2×C4—2924no—
328C2×C4—5842no—
329C2×C4—5842no—
330C2×C4D4×Dic7312yes—
331C2×C4#1455655911yescharacteristic
332C2×C4#1455657211yescharacteristic
333C8—2924no—
334C22—21no—
335C22—21no—
336C22—22no—
337C22—22no—
338C22—22no—
339C22—22no—
340C22—42no—
341C22—41no—
342C22—41no—
343C22—81no—
344C22—81no—
345C22—81no—
346C22#1798846711yescharacteristic
347C22C2×#1455651811yescharacteristic
348C22#1798860411yescharacteristic
349C4—21no—
350C4—22no—
351C4—42no—
352C4—41no—
353C4—41no—
354C4—41no—
355C4—81no—
356C4—81no—
357C4—81no—
358C4—5844no—
359C4#1798877612yes—
360C2—21no—
361C2—21no—
362C2—42no—
363C2—42no—
364C2#2261023411yescharacteristic
365C2#2261031511yescharacteristic
366C2#2261034511yescharacteristic
subgroupG/H
centerC2×C4#14556559
commutatorC22×C4×C73C22×C4
FrattiniC23×C4C22×D73
FittingC73×#2395C2
socleC22×C73#386
prime radicalC22×D4×C73D4
cosocleC23×C4×C73C23
hypercenter#2395D73
hypocenterC73#9011
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #2395

Complements

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

NH ≅ G/Ntype
2. #22610030C2semidirect
4. #22607817C2semidirect
16. #400×C73C4semidirect
17. #399×C73C4semidirect
18. #398×C73C4semidirect
20. #397×C73C4semidirect
189. C73#9011semidirect
197. #400Dic73semidirect
198. #399Dic73semidirect
199. #398Dic73semidirect
201. #397Dic73semidirect

Maximal subgroups

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

subgroupindex
#226189602
#226100302
C4×#145565062
#226078172
C73×#23952
#901173