Finite Group Databasebeta

← #32974192

Subgroups — #32974192

Order 18688.

506 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 506 conjugacy classes of subgroups — too many to draw. Only its 101 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#22612469C211yesmaximal, characteristic
2#22609998C211yesmaximal, characteristic
3#22609692C211yesmaximal, characteristic
4#22607757C211yesmaximal, characteristic, complemented
5#22607755C211yesmaximal, characteristic, complemented
6#22607753C211yesmaximal, characteristic, complemented
7C73×#1272C211yesmaximal, characteristic, complemented
8C2×#14556567C411yescharacteristic
9C2×Q8×Dic73C411yescharacteristic
10C2×C42×D73C2211yescharacteristic
11C2×#14556503—21no—
12C2×#14556503—21no—
13C2×#14556501—21no—
14#17988485C2211yescharacteristic
15#17988474—21no—
16#17988461—21no—
17#17988444C411yescharacteristic
18C4⋊C4×Dic73C411yescharacteristic
19#17988438—21no—
20#17988434—21no—
21#17988342C2211yescharacteristic
22#17988342C2211yescharacteristic
23#17988341—21no—
24#391×C73C2211yescharacteristic
25#336×C73C2211yescharacteristic
26#336×C73C2211yescharacteristic
27C22×C4×D73C2×C411yescharacteristic
28C22×C4×D73D411yescharacteristic
29C22×C4×D73D411yescharacteristic
30C22×#10596831—21no—
31C22×#10596831—21no—
32#14556567—22no—
33#14556567—22no—
34Q8×Dic73—42no—
35C2×C4⋊Dic73—21no—
36C2×C4⋊Dic73C2×C411yescharacteristic
37C2×C4⋊Dic73C2×C411yescharacteristic
38C2×#11924007—21no—
39C2×#11924007—21no—
40C2×#11924005—21no—
41C2×#11924005—21no—
42C2×C4×Dic73—21no—
43C2×C4×Dic73—21no—
44C2×C4×Dic73C2×C411yescharacteristic
45C2×C4×Dic73D411yescharacteristic
46C2×C4×Dic73D411yescharacteristic
47C42×D73—42no—
48#14556503—24no—
49#14556503—42no—
50#14556501—42no—
51#14556498—21no—
52#14556498—21no—
53#14556498—21no—
54C22×Q8×C73C2×C411yescharacteristic
55C22×C8×C73—21no—
56C22×C8×C73—21no—
57C2×C4⋊C4×C73C2×C411yescharacteristic
58C2×C42×C73C2311yescharacteristic
59#95×C73—21no—
60#95×C73—21no—
61C23×D73C22⋊C411yescharacteristic
62C22×Dic73—21no—
63C22×Dic73—21no—
64C22×Dic73C22⋊C411yescharacteristic
65C22×Dic73C22⋊C411yescharacteristic
66C22×Dic73C22⋊C411yescharacteristic
67C2×C4×D73—22no—
68C2×C4×D73—21no—
69C2×C4×D73—21no—
70C2×C4×D73—21no—
71C2×C4×D73—21no—
72C2×C4×D73—21no—
73C2×C4×D73—21no—
74C2×C4×D73—22no—
75C2×C4×D73—41no—
76C2×C4×D73—41no—
77C2×C4×D73—41no—
78C2×C4×D73—41no—
79C2×#10596831—22no—
80C2×#10596831—41no—
81C2×#10596831—41no—
82C2×#10596831—41no—
83C2×#10596831—41no—
84C2×#10596831—41no—
85C4⋊Dic73—21no—
86C4⋊Dic73—42no—
87C4⋊Dic73—41no—
88C4⋊Dic73M1612yes—
89#11924007—24no—
90#11924007—41no—
91#11924007—41no—
92#11924005—24no—
93#11924005—42no—
94C4×Dic73—22no—
95C4×Dic73—22no—
96C4×Dic73—41no—
97C4×Dic73—42no—
98C4×Dic73—42no—
99C2×Q8×C73—21no—
100C2×Q8×C73—41no—
101C2×Q8×C73—41no—
102C2×Q8×C73M1612yes—
103C22×C4×C73—21no—
104C22×C4×C73—21no—
105C22×C4×C73C22×C411yescharacteristic
106C22×C4×C73C2×D411yescharacteristic
107C22×C4×C73C2×D411yescharacteristic
108C2×C8×C73—22no—
109C2×C8×C73—41no—
110C2×C8×C73—41no—
111C2×C8×C73—41no—
112C2×C8×C73—41no—
113C2×C8×C73—41no—
114C4⋊C4×C73—41no—
115C42×C73—41no—
116C23×C73C2×C22⋊C411yescharacteristic
117C22×D73—21no—
118C22×D73—21no—
119C22×D73—21no—
120C22×D73—21no—
121C22×D73—21no—
122C22×D73—21no—
123C22×D73#10412yes—
124Q8×C73—42no—
125Q8×C73—42no—
126C2×C4×C73—21no—
127C2×C4×C73—21no—
128C2×C4×C73—21no—
129C2×C4×C73—21no—
130C2×C4×C73—21no—
131C2×C4×C73—21no—
132C2×C4×C73—22no—
133C2×C4×C73—41no—
134C2×C4×C73—41no—
135C2×C4×C73—41no—
136C2×C4×C73—41no—
137C2×C4×C73—41no—
138C2×C4×C73C2×M1611yescharacteristic
139C2×C4×C73C2×M1611yescharacteristic
140C2×C4×C73#13511yescharacteristic
141C2×C4×C73#13511yescharacteristic
142C2×C4×C73#13711yescharacteristic
143C2×C4×C73#13711yescharacteristic
144C2×Dic73—21no—
145C2×Dic73—21no—
146C2×Dic73—22no—
147C2×Dic73—21no—
148C2×Dic73—21no—
149C2×Dic73—21no—
150C2×Dic73—21no—
151C2×Dic73—21no—
152C2×Dic73—21no—
153C2×Dic73—41no—
154C2×Dic73—41no—
155C2×Dic73—41no—
156C2×Dic73—41no—
157C2×Dic73—41no—
158C2×Dic73#9912yes—
159C4×D73—42no—
160C4×D73—42no—
161C4×D73—42no—
162C4×D73—42no—
163C4×D73—42no—
164C4×D73—42no—
165C8×C73—42no—
166C8×C73—81no—
167#10596831—42no—
168#10596831—81no—
169C22×C73#40711yescharacteristic
170C22×C73C2×#9911yescharacteristic
171C22×C73#41311yescharacteristic
172C22×C73#41611yescharacteristic
173C22×C73#41911yescharacteristic
174C22×C73C2×#10411yescharacteristic
175C22×C73#42111yescharacteristic
176C2×D73—22no—
177C2×D73—22no—
178C2×D73—22no—
179C2×D73—41no—
180C2×D73—41no—
181C2×D73—41no—
182C2×D73—41no—
183C4×C73—21no—
184C4×C73—21no—
185C4×C73—21no—
186C4×C73—21no—
187C4×C73—21no—
188C4×C73—21no—
189C4×C73—42no—
190C4×C73—81no—
191Dic73—22no—
192Dic73—41no—
193Dic73—42no—
194Dic73—41no—
195Dic73—81no—
196#7623—731nomaximal, Sylow
197C2×C73#146511yescharacteristic
198C2×C73#148811yescharacteristic
199C2×C73#149111yescharacteristic
200C2×C73#150111yescharacteristic
201C2×C73#151111yescharacteristic
202C2×C73#153211yescharacteristic
203C2×C73#155411yescharacteristic
204D73—42no—
205C4×C22⋊Q8—731no—
206#1817—731no—
207#1749—731no—
208#1272—731no—
209#1272D7311yescharacteristic, complemented
210#1271—731no—
211#1271—731no—
212C73#762311yescharacteristic, complemented, Sylow
213C2×C22⋊Q8—731no—
214C2×C4×Q8—731no—
215C22×C42—731no—
216C2×C22⋊C8—1461no—
217C2×C22⋊C8—1461no—
218C2×#97—1461no—
219#391C2×D7311yescharacteristic
220#389—731no—
221#387—1461no—
222#385—1461no—
223#380—731no—
224C4×C4⋊C4—731no—
225C4×C4⋊C4—1461no—
226C4×C22⋊C4—1461no—
227#336—731no—
228#336—731no—
229#336—1461no—
230#336C2×D7311yescharacteristic
231#336C2×D7311yescharacteristic
232C22×Q8C4×D7311yescharacteristic
233C23×C4—731no—
234C23×C4—731no—
235C23×C4—731no—
236C22×C8—21no—
237C22×C8—21no—
238C22×C8—1461no—
239C22×C8—1461no—
240C22⋊Q8—1462no—
241C22⋊Q8—1462no—
242C4×Q8—2922no—
243C2×C4⋊C4—731no—
244C2×C4⋊C4—731no—
245C2×C4⋊C4—1461no—
246C2×C4⋊C4—1461no—
247C2×C4⋊C4—1461no—
248C2×C4⋊C4C4×D7311yescharacteristic
249C2×C22⋊C4—1461no—
250C2×C22⋊C4—1461no—
251C2×C42—2922no—
252C2×C42—731no—
253C2×C42—731no—
254C2×C42—731no—
255C2×C42—1461no—
256C2×C42—1461no—
257C2×C42C22×D7311yescharacteristic
258C22⋊C8—2922no—
259C22⋊C8—1464no—
260#97—2922no—
261#95—21no—
262#95—21no—
263#95—1461no—
264#95—1461no—
265#95—1461no—
266C24—731no—
267C2×Q8—21no—
268C2×Q8—41no—
269C2×Q8—41no—
270C2×Q8#1192401112yes—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—2921no—
274C22×C4—2921no—
275C22×C4—2921no—
276C22×C4—2921no—
277C22×C4—731no—
278C22×C4—731no—
279C22×C4—731no—
280C22×C4—1462no—
281C22×C4—1462no—
282C22×C4—1461no—
283C22×C4—1461no—
284C22×C4—1461no—
285C22×C4—1461no—
286C22×C4—1461no—
287C22×C4—1461no—
288C22×C4—1461no—
289C22×C4—1461no—
290C22×C4C2×C4×D7311yescharacteristic
291C22×C4D4×D7311yescharacteristic
292C22×C4D4×D7311yescharacteristic
293C2×C8—22no—
294C2×C8—41no—
295C2×C8—41no—
296C2×C8—41no—
297C2×C8—41no—
298C2×C8—41no—
299C2×C8—2921no—
300C2×C8—2921no—
301C2×C8—2921no—
302C2×C8—2921no—
303C2×C8—2921no—
304C2×C8—1462no—
305C4⋊C4—41no—
306C4⋊C4—2921no—
307C4⋊C4—2922no—
308C4⋊C4—2921no—
309C4⋊C4—2921no—
310C4⋊C4—732no—
311C4⋊C4—1464no—
312C4⋊C4—1461no—
313C22⋊C4—2922no—
314C22⋊C4—1464no—
315C42—41no—
316C42—2921no—
317C42—2922no—
318C42—2922no—
319C42—1462no—
320C42—1462no—
321C23—732no—
322C23—1461no—
323C23—1461no—
324C23—1461no—
325C23—1461no—
326C23—1461no—
327C23—1461no—
328C23C22⋊C4×D7311yescharacteristic
329Q8—42no—
330Q8—42no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—22no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—2922no—
344C2×C4—2922no—
345C2×C4—2922no—
346C2×C4—2922no—
347C2×C4—2922no—
348C2×C4—2922no—
349C2×C4—2921no—
350C2×C4—2921no—
351C2×C4—2921no—
352C2×C4—2921no—
353C2×C4—2921no—
354C2×C4—732no—
355C2×C4—1462no—
356C2×C4—1461no—
357C2×C4—1461no—
358C2×C4—1461no—
359C2×C4—1461no—
360C2×C4—1461no—
361C2×C4—1461no—
362C2×C4—1461no—
363C2×C4—1461no—
364C2×C4M16×D7311yescharacteristic
365C2×C4C2×#1192401111yescharacteristic
366C2×C4#1455661211yescharacteristic
367C2×C4#1455661411yescharacteristic
368C2×C4#1455662011yescharacteristic
369C2×C4#1455662311yescharacteristic
370C8—42no—
371C8—81no—
372C8—2922no—
373C8—5841no—
374C22—2921no—
375C22—2921no—
376C22—2921no—
377C22—2921no—
378C22—1462no—
379C22—1462no—
380C22—1462no—
381C22#1798854911yescharacteristic
382C22#99×D7311yescharacteristic
383C22#1798856811yescharacteristic
384C22#1798858211yescharacteristic
385C22#1798859411yescharacteristic
386C22#104×D7311yescharacteristic
387C22#1798860211yescharacteristic
388C4—21no—
389C4—21no—
390C4—21no—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—42no—
395C4—81no—
396C4—2921no—
397C4—2921no—
398C4—2922no—
399C4—5841no—
400C4—1462no—
401C2—2922no—
402C2#2260826611yescharacteristic
403C2#2260836411yescharacteristic
404C2#2260837611yescharacteristic
405C2#2260842111yescharacteristic
406C2#2260846211yescharacteristic
407C2#2260854811yescharacteristic
408C2#2260864011yescharacteristic
subgroupG/H
centerC23C22⋊C4×D73
commutatorC22×C4×C73C22×C4
FrattiniC2×C42C22×D73
FittingC73×#1272C2
socleC23×C73C2×C22⋊C4
prime radicalC23×D73C22⋊C4
cosocleC2×C42×C73C23
hypercenter#1272D73
hypocenterC73#7623
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1272

Complements

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

NH ≅ G/Ntype
4. #22607757C2semidirect
5. #22607755C2semidirect
6. #22607753C2semidirect
7. C73×#1272C2semidirect
209. #1272D73semidirect
212. C73#7623semidirect

Maximal subgroups

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

subgroupindex
#226124692
#226099982
#226096922
#226077572
#226077552
#226077532
C73×#12722
#762373