Finite Group Databasebeta

← #32983909

Subgroups — #32983909

Order 18688.

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

normal subgroups only

This group has 660 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#22612529C211yesmaximal, characteristic
2#22610005C212yesmaximal
3#22610005C211yesmaximal, characteristic
4#22607835C212yesmaximal
5C73×#2483C211yesmaximal, characteristic
6#17988480—24no—
7#17988447—21no—
8#17988443C2212yes—
9#17988443C2212yes—
10C4×C22⋊Dic73—21no—
11#17988354—24no—
12#17988354—22no—
13C22×C22⋊C4×C73C2211yescharacteristic
14C22×C22⋊C4×C73C2211yescharacteristic
15#397×C73—22no—
16#385×C73—21no—
17#385×C73C412yescomplemented
18#381×C73—21no—
19#381×C73C412yescomplemented
20#379×C73C2211yescharacteristic
21C2×C4⋊Dic73—22no—
22C2×C22⋊Dic73—22no—
23C2×C22⋊Dic73—22no—
24C2×C22⋊Dic73—22no—
25C2×C22⋊Dic73—22no—
26C2×C22⋊Dic73—22no—
27C2×C22⋊Dic73D414yes—
28C2×C4×Dic73—22no—
29#14556498—22no—
30#14556498—22no—
31C25×C73C2311yescharacteristic
32C23×C4×C73C2×C411yescharacteristic
33C23×C4×C73C2×C411yescharacteristic
34C23×C4×C73D411yescharacteristic
35C23×C4×C73D411yescharacteristic
36C2×C4⋊C4×C73—22no—
37C2×C22⋊C4×C73—22no—
38C2×C22⋊C4×C73—22no—
39C2×C22⋊C4×C73—21no—
40C2×C22⋊C4×C73—22no—
41C2×C22⋊C4×C73—21no—
42C2×C22⋊C4×C73—21no—
43C2×C22⋊C4×C73—22no—
44C2×C22⋊C4×C73—21no—
45C2×C22⋊C4×C73C2×C412yes—
46C2×C22⋊C4×C73C2×C412yes—
47C2×C22⋊C4×C73D412yes—
48#95×C73—22no—
49#95×C73—22no—
50#95×C73—22no—
51C22×Dic73—24no—
52C22×Dic73—42no—
53C4⋊Dic73—82no—
54C22⋊Dic73—44no—
55C22⋊Dic73—44no—
56C22⋊Dic73—44no—
57C22⋊Dic73—44no—
58C22⋊Dic73—82no—
59C22⋊Dic73—82no—
60C4×Dic73—82no—
61C24×C73—42no—
62C24×C73—42no—
63C24×C73—81no—
64C24×C73C22×C411yescharacteristic
65C24×C73C2×D412yes—
66C24×C73C2×D411yescharacteristic
67C24×C73C2×D411yescharacteristic
68C24×C73#4112yes—
69C22×C4×C73—21no—
70C22×C4×C73—21no—
71C22×C4×C73—21no—
72C22×C4×C73—21no—
73C22×C4×C73—21no—
74C22×C4×C73—22no—
75C22×C4×C73—21no—
76C22×C4×C73—22no—
77C22×C4×C73—22no—
78C22×C4×C73—42no—
79C22×C4×C73—42no—
80C22×C4×C73—81no—
81C22×C4×C73—81no—
82C22×C4×C73C22⋊C412yes—
83C22×C4×C73C22⋊C412yes—
84C4⋊C4×C73—82no—
85C22⋊C4×C73—24no—
86C22⋊C4×C73—24no—
87C22⋊C4×C73—42no—
88C22⋊C4×C73—41no—
89C22⋊C4×C73—42no—
90C22⋊C4×C73—41no—
91C22⋊C4×C73—42no—
92C22⋊C4×C73—81no—
93C23×C73—22no—
94C23×C73—22no—
95C23×C73—22no—
96C23×C73—22no—
97C23×C73—24no—
98C23×C73—41no—
99C23×C73—41no—
100C23×C73—41no—
101C23×C73—41no—
102C23×C73—42no—
103C23×C73—41no—
104C23×C73—42no—
105C23×C73—42no—
106C23×C73—42no—
107C23×C73—42no—
108C23×C73—88no—
109C23×C73C2×C22⋊C411yescharacteristic
110C23×C73C4×D412yes—
111C23×C73#12011yescharacteristic
112C23×C73#12112yes—
113C23×C73#12311yescharacteristic
114C2×C4×C73—22no—
115C2×C4×C73—22no—
116C2×C4×C73—42no—
117C2×C4×C73—42no—
118C2×C4×C73—42no—
119C2×C4×C73—42no—
120C2×C4×C73—42no—
121C2×C4×C73—42no—
122C2×C4×C73—42no—
123C2×C4×C73—41no—
124C2×C4×C73—41no—
125C2×C4×C73—42no—
126C2×C4×C73—42no—
127C2×C4×C73—81no—
128C2×C4×C73—81no—
129C2×C4×C73—82no—
130C2×C4×C73—82no—
131C2×Dic73—44no—
132C2×Dic73—44no—
133C2×Dic73—44no—
134C2×Dic73—82no—
135C2×Dic73—82no—
136C2×Dic73—82no—
137C22×C73—21no—
138C22×C73—21no—
139C22×C73—21no—
140C22×C73—21no—
141C22×C73—22no—
142C22×C73—21no—
143C22×C73—22no—
144C22×C73—22no—
145C22×C73—22no—
146C22×C73—21no—
147C22×C73—22no—
148C22×C73—22no—
149C22×C73—21no—
150C22×C73—44no—
151C22×C73—44no—
152C22×C73—84no—
153C22×C73—82no—
154C22×C73—84no—
155C22×C73#38611yescharacteristic
156C22×C73#41012yes—
157C22×C73#45812yes—
158C4×C73—42no—
159C4×C73—42no—
160C4×C73—82no—
161C4×C73—81no—
162C4×C73—81no—
163Dic73—84no—
164Dic73—162no—
165#9500—731nomaximal, Sylow
166C2×C73—22no—
167C2×C73—22no—
168C2×C73—42no—
169C2×C73—41no—
170C2×C73—42no—
171C2×C73#189011yescharacteristic
172C2×C73#189911yescharacteristic
173C2×C73#191311yescharacteristic
174#2483D7311yescharacteristic
175#2312—731no—
176#1819—732no—
177#1819—731no—
178#1304—732no—
179C73#950011yescharacteristic, complemented, Sylow
180C22×C22⋊C4C2×D7311yescharacteristic
181C22×C22⋊C4C2×D7311yescharacteristic
182#397—22no—
183#388—1464no—
184#385—21no—
185#385Dic7312yescomplemented
186#381—21no—
187#381Dic7312yescomplemented
188#380—1461no—
189#379—732no—
190#379—732no—
191#379C2×D7311yescharacteristic
192C4×C22⋊C4—1461no—
193#342—1464no—
194#342—1462no—
195C25C22×D7311yescharacteristic
196C23×C4C2×Dic7311yescharacteristic
197C23×C4C2×Dic7311yescharacteristic
198C23×C4C22⋊D7311yescharacteristic
199C23×C4C22⋊D7311yescharacteristic
200C2×C4⋊C4—22no—
201C2×C4⋊C4—1462no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—21no—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—21no—
210C2×C22⋊C4—734no—
211C2×C22⋊C4—1462no—
212C2×C22⋊C4—1462no—
213C2×C22⋊C4—1462no—
214C2×C22⋊C4—1462no—
215C2×C22⋊C4—1462no—
216C2×C22⋊C4C2×Dic7312yes—
217C2×C22⋊C4C2×Dic7312yes—
218C2×C22⋊C4C22⋊D7312yes—
219C2×C42—1462no—
220#95—22no—
221#95—22no—
222#95—22no—
223#95—1462no—
224#95—1462no—
225C24—42no—
226C24—42no—
227C24—81no—
228C24C22×Dic7311yescharacteristic
229C24D4×D7312yes—
230C24C2×C22⋊D7311yescharacteristic
231C24C2×C22⋊D7311yescharacteristic
232C24#1192402712yes—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—22no—
239C22×C4—21no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—81no—
245C22×C4—81no—
246C22×C4—2922no—
247C22×C4—1464no—
248C22×C4C22⋊Dic7312yes—
249C22×C4C22⋊Dic7312yes—
250C4⋊C4—82no—
251C4⋊C4—5842no—
252C22⋊C4—24no—
253C22⋊C4—24no—
254C22⋊C4—42no—
255C22⋊C4—41no—
256C22⋊C4—42no—
257C22⋊C4—41no—
258C22⋊C4—42no—
259C22⋊C4—81no—
260C22⋊C4—2924no—
261C22⋊C4—2924no—
262C22⋊C4—2924no—
263C22⋊C4—2924no—
264C22⋊C4—5842no—
265C22⋊C4—5842no—
266C42—5842no—
267C23—22no—
268C23—22no—
269C23—22no—
270C23—24no—
271C23—22no—
272C23—41no—
273C23—41no—
274C23—41no—
275C23—41no—
276C23—42no—
277C23—42no—
278C23—42no—
279C23—41no—
280C23—42no—
281C23—42no—
282C23—88no—
283C23C2×C22⋊Dic7311yescharacteristic
284C23D4×Dic7312yes—
285C23#1455655911yescharacteristic
286C23#1455656312yes—
287C23#1455657211yescharacteristic
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—81no—
302C2×C4—81no—
303C2×C4—82no—
304C2×C4—82no—
305C2×C4—2924no—
306C2×C4—2924no—
307C2×C4—2924no—
308C2×C4—5842no—
309C2×C4—5842no—
310C2×C4—5842no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—22no—
316C22—21no—
317C22—22no—
318C22—21no—
319C22—22no—
320C22—22no—
321C22—22no—
322C22—22no—
323C22—21no—
324C22—44no—
325C22—44no—
326C22—82no—
327C22—84no—
328C22—84no—
329C22#1798846711yescharacteristic
330C22#1798856112yes—
331C22#1798877612yes—
332C4—42no—
333C4—42no—
334C4—82no—
335C4—81no—
336C4—81no—
337C4—5844no—
338C4—11682no—
339C2—22no—
340C2—22no—
341C2—42no—
342C2—42no—
343C2—41no—
344C2#2261034511yescharacteristic
345C2#2261039011yescharacteristic
346C2#2261047411yescharacteristic
subgroupG/H
centerC22#17988467
commutatorC24×C73C22×C4
FrattiniC25C22×D73
FittingC73×#2483C2
socleC22×C73#386
prime radical, cosocleC25×C73C23
hypercenter#2483D73
hypocenterC73#9500
LCSG ▷ C24×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C25 ◁ #2483

Complements

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

NH ≅ G/Ntype
17. #385×C73C4semidirect
19. #381×C73C4semidirect
179. C73#9500semidirect
185. #385Dic73semidirect
187. #381Dic73semidirect

Maximal subgroups

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

subgroupindex
#226125292
#226100052
#226078352
C73×#24832
#950073