Finite Group Databasebeta

← #32983704

Subgroups — #32983704

Order 18688.

904 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 904 conjugacy classes of subgroups — too many to draw. Only its 115 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#22612509C211yesmaximal, characteristic
2#22609852C211yesmaximal, characteristic
3#22608464C212yesmaximal, complemented
4#22607784C212yesmaximal, complemented
5C73×#2435C211yesmaximal, characteristic
6C22×C22⋊Dic73—21no—
7C2×#14556508—22no—
8#17988551C2212yes—
9#17988467—24no—
10#17988450C2212yes—
11#17988443—21no—
12#17988324—24no—
13C23×D4×C73C2211yescharacteristic
14C2×#124×C73C412yescomplemented
15C2×C22⋊Q8×C73C2211yescharacteristic
16#394×C73—22no—
17#390×C73C412yescomplemented
18#379×C73—21no—
19#379×C73—21no—
20C4×C22⋊C4×C73C2211yescharacteristic
21C23×Dic73—22no—
22C2×#11924012—22no—
23C2×C22⋊Dic73—22no—
24C2×C22⋊Dic73—22no—
25C2×C22⋊Dic73—22no—
26C2×C22⋊Dic73—22no—
27C2×C22⋊Dic73—44no—
28#14556508—44no—
29C22⋊#10596831—22no—
30C22⋊#10596831—22no—
31#14556498—22no—
32#14556498D414yes—
33C25×C73D411yescharacteristic
34C25×C73D411yescharacteristic
35C22×Q8×C73C2×C411yescharacteristic, complemented
36C22×D4×C73—22no—
37C22×D4×C73—22no—
38C22×D4×C73—44no—
39C22×D4×C73C2×C412yes—
40C22×D4×C73D412yes—
41C23×C4×C73C2311yescharacteristic
42#124×C73—44no—
43C22⋊Q8×C73—22no—
44C22⋊Q8×C73—22no—
45C2×C4⋊C4×C73—21no—
46C2×C4⋊C4×C73C2×C411yescharacteristic, complemented
47C2×C22⋊C4×C73—22no—
48C2×C22⋊C4×C73—22no—
49C2×C22⋊C4×C73—21no—
50C2×C22⋊C4×C73—21no—
51C2×C22⋊C4×C73—22no—
52C2×C22⋊C4×C73—22no—
53C2×C42×C73C2×C412yescomplemented
54#95×C73—21no—
55C22×Dic73—24no—
56C22×Dic73—44no—
57C22×Dic73—82no—
58C22×Dic73—82no—
59#11924012—44no—
60C2×#10596831—42no—
61C22⋊Dic73—44no—
62C22⋊Dic73—44no—
63C22⋊Dic73—82no—
64C22⋊Dic73—82no—
65C24×C73—21no—
66C24×C73—24no—
67C24×C73—21no—
68C24×C73—21no—
69C24×C73—21no—
70C24×C73—42no—
71C24×C73—42no—
72C24×C73—42no—
73C24×C73—81no—
74C24×C73—81no—
75C24×C73C22⋊C412yes—
76C24×C73C22⋊C412yes—
77C24×C73C2×D411yescharacteristic
78C2×Q8×C73—22no—
79C2×Q8×C73—42no—
80C2×D4×C73—24no—
81C2×D4×C73—24no—
82C2×D4×C73—24no—
83C2×D4×C73—24no—
84C2×D4×C73—42no—
85C2×D4×C73—42no—
86C2×D4×C73—44no—
87C2×D4×C73—44no—
88C2×D4×C73—48no—
89C22×C4×C73—22no—
90C22×C4×C73—21no—
91C22×C4×C73—21no—
92C22×C4×C73—42no—
93C22×C4×C73C22×C411yescharacteristic
94C22×C4×C73C2×D412yes—
95C22×C4×C73C2×D411yescharacteristic
96C22×C4×C73#4112yes—
97C4⋊C4×C73—22no—
98C4⋊C4×C73—42no—
99C22⋊C4×C73—42no—
100C22⋊C4×C73—44no—
101C22⋊C4×C73—44no—
102C22⋊C4×C73—41no—
103C22⋊C4×C73—41no—
104C22⋊C4×C73—82no—
105C22⋊C4×C73—82no—
106C42×C73—42no—
107C23×C73—22no—
108C23×C73—22no—
109C23×C73—24no—
110C23×C73—22no—
111C23×C73—22no—
112C23×C73—22no—
113C23×C73—22no—
114C23×C73—22no—
115C23×C73—22no—
116C23×C73—22no—
117C23×C73—41no—
118C23×C73—42no—
119C23×C73—42no—
120C23×C73—42no—
121C23×C73—42no—
122C23×C73—42no—
123C23×C73—42no—
124C23×C73—42no—
125C23×C73—42no—
126C23×C73—42no—
127C23×C73—42no—
128C23×C73—81no—
129C23×C73—81no—
130C23×C73—81no—
131C23×C73—81no—
132C23×C73—88no—
133C23×C73—88no—
134C23×C73#9914yes—
135C23×C73C2×C22⋊C411yescharacteristic
136C23×C73#12011yescharacteristic
137C23×C73#12311yescharacteristic
138Q8×C73—82no—
139D4×C73—48no—
140D4×C73—48no—
141C2×C4×C73—22no—
142C2×C4×C73—22no—
143C2×C4×C73—42no—
144C2×C4×C73—42no—
145C2×C4×C73—42no—
146C2×C4×C73—42no—
147C2×C4×C73—42no—
148C2×C4×C73—41no—
149C2×C4×C73—41no—
150C2×C4×C73—42no—
151C2×C4×C73—42no—
152C2×C4×C73C4×D412yes—
153C2×C4×C73#12112yes—
154C2×Dic73—44no—
155C2×Dic73—44no—
156C2×Dic73—44no—
157C2×Dic73—88no—
158#10596831—82no—
159C22×C73—21no—
160C22×C73—21no—
161C22×C73—21no—
162C22×C73—21no—
163C22×C73—21no—
164C22×C73—21no—
165C22×C73—24no—
166C22×C73—21no—
167C22×C73—44no—
168C22×C73—44no—
169C22×C73—44no—
170C22×C73—44no—
171C22×C73—44no—
172C22×C73—42no—
173C22×C73—42no—
174C22×C73—42no—
175C22×C73—42no—
176C22×C73—42no—
177C22×C73—82no—
178C22×C73—82no—
179C22×C73—84no—
180C22×C73—84no—
181C22×C73—84no—
182C22×C73#38611yescharacteristic
183C22×C73C2×#9912yes—
184C22×C73#45512yes—
185C4×C73—22no—
186C4×C73—22no—
187C4×C73—82no—
188C4×C73—81no—
189C4×C73—81no—
190Dic73—84no—
191#9459—731nomaximal, Sylow
192C2×C73—22no—
193C2×C73—22no—
194C2×C73—41no—
195C2×C73—44no—
196C2×C73—82no—
197C2×C73#188511yescharacteristic
198C2×C73#189811yescharacteristic
199C2×C73#191111yescharacteristic
200#2435D7311yescharacteristic
201#2308—731no—
202#1783—731no—
203#1511—732no—
204#1284—732no—
205C73#945911yescharacteristic, complemented, Sylow
206C23×D4C2×D7311yescharacteristic
207C2×#124Dic7312yescomplemented
208C2×C22⋊Q8C2×D7311yescharacteristic
209C22×C22⋊C4—1461no—
210C2×#100—1462no—
211#407—732no—
212#394—22no—
213#390Dic7312yescomplemented
214#386—1464no—
215#381—732no—
216#379—21no—
217#379—21no—
218#379—1461no—
219C4×C22⋊C4C2×D7311yescharacteristic
220#328—1464no—
221C25C22⋊D7311yescharacteristic
222C25C22⋊D7311yescharacteristic
223C22×Q8C2×Dic7311yescharacteristic, complemented
224C22×D4—22no—
225C22×D4—22no—
226C22×D4—44no—
227C22×D4C2×Dic7312yes—
228C22×D4C22⋊D7312yes—
229C23×C4—1462no—
230C23×C4C22×D7311yescharacteristic
231C2×M16—1462no—
232#124—44no—
233C22⋊Q8—22no—
234C22⋊Q8—22no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4C2×Dic7311yescharacteristic, complemented
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—21no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—2924no—
244C2×C22⋊C4—1462no—
245C2×C22⋊C4—1462no—
246C2×C22⋊C4—1462no—
247C2×C22⋊C4—1462no—
248C2×C42C2×Dic7312yescomplemented
249#100—2924no—
250C22⋊C8—1462no—
251C22⋊C8—1462no—
252#95—21no—
253#95—734no—
254#95—1462no—
255C24—21no—
256C24—24no—
257C24—21no—
258C24—21no—
259C24—21no—
260C24—42no—
261C24—42no—
262C24—42no—
263C24—81no—
264C24—81no—
265C24C22⋊Dic7312yes—
266C24C22⋊Dic7312yes—
267C24C2×C22⋊D7311yescharacteristic
268C2×Q8—22no—
269C2×Q8—42no—
270C2×D4—24no—
271C2×D4—24no—
272C2×D4—24no—
273C2×D4—24no—
274C2×D4—42no—
275C2×D4—42no—
276C2×D4—44no—
277C2×D4—44no—
278C2×D4—48no—
279C22×C4—22no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—42no—
283C22×C4—2924no—
284C22×C4—5842no—
285C22×C4—5842no—
286C22×C4—1464no—
287C22×C4C22×Dic7311yescharacteristic
288C22×C4D4×D7312yes—
289C22×C4C2×C22⋊D7311yescharacteristic
290C22×C4#1192402712yes—
291M16—2924no—
292C2×C8—2922no—
293C4⋊C4—22no—
294C4⋊C4—42no—
295C22⋊C4—42no—
296C22⋊C4—44no—
297C22⋊C4—44no—
298C22⋊C4—41no—
299C22⋊C4—41no—
300C22⋊C4—82no—
301C22⋊C4—82no—
302C22⋊C4—2924no—
303C22⋊C4—2924no—
304C22⋊C4—5842no—
305C22⋊C4—5842no—
306C42—42no—
307C23—22no—
308C23—22no—
309C23—24no—
310C23—22no—
311C23—22no—
312C23—22no—
313C23—22no—
314C23—22no—
315C23—22no—
316C23—22no—
317C23—41no—
318C23—42no—
319C23—42no—
320C23—42no—
321C23—42no—
322C23—42no—
323C23—42no—
324C23—42no—
325C23—42no—
326C23—42no—
327C23—42no—
328C23—81no—
329C23—81no—
330C23—81no—
331C23—81no—
332C23—88no—
333C23—88no—
334C23#1455650614yes—
335C23C2×C22⋊Dic7311yescharacteristic
336C23#1455655911yescharacteristic
337C23#1455657211yescharacteristic
338Q8—82no—
339D4—48no—
340D4—48no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—2924no—
353C2×C4—2924no—
354C2×C4—2924no—
355C2×C4—5848no—
356C2×C4D4×Dic7312yes—
357C2×C4#1455656312yes—
358C8—5842no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—24no—
367C22—44no—
368C22—44no—
369C22—44no—
370C22—44no—
371C22—44no—
372C22—42no—
373C22—42no—
374C22—42no—
375C22—42no—
376C22—42no—
377C22—82no—
378C22—82no—
379C22—84no—
380C22—84no—
381C22—84no—
382C22#1798846711yescharacteristic
383C22C2×#1455650612yes—
384C22#1798876412yes—
385C4—22no—
386C4—22no—
387C4—82no—
388C4—81no—
389C4—81no—
390C4—5844no—
391C2—22no—
392C2—22no—
393C2—41no—
394C2—44no—
395C2—82no—
396C2#2261032011yescharacteristic
397C2#2261038511yescharacteristic
398C2#2261046411yescharacteristic
subgroupG/H
centerC22#17988467
commutatorC22×C4×C73C22×C4
FrattiniC23×C4C22×D73
FittingC73×#2435C2
socleC22×C73#386
prime radicalC23×D4×C73C22
cosocleC23×C4×C73C23
hypercenter#2435D73
hypocenterC73#9459
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2435

Complements

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

NH ≅ G/Ntype
3. #22608464C2semidirect
4. #22607784C2semidirect
14. C2×#124×C73C4semidirect
17. #390×C73C4semidirect
35. C22×Q8×C73C2×C4semidirect
46. C2×C4⋊C4×C73C2×C4semidirect
53. C2×C42×C73C2×C4semidirect
205. C73#9459semidirect
207. C2×#124Dic73semidirect
213. #390Dic73semidirect
223. C22×Q8C2×Dic73semidirect
236. C2×C4⋊C4C2×Dic73semidirect
248. C2×C42C2×Dic73semidirect

Maximal subgroups

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

subgroupindex
#226125092
#226098522
#226084642
#226077842
C73×#24352
#945973