Finite Group Databasebeta

← #34012478

Subgroups — #34012478

Order 18688.

548 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 102 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
1C2×#17989489C212yesmaximal
2C4×#14556651C211yesmaximal, characteristic
3C4×#14556649C211yesmaximal, characteristic
4#22625015C212yesmaximal
5#22612600C211yesmaximal, characteristic
6C2×#14556651C2212yes—
7C2×#14556649C2212yes—
8C4×#11924037—21no—
9C4×C22⋊#64408—21no—
10#17989489C418yescomplemented
11#17989467—21no—
12#17989465—21no—
13C2×#14556574C2211yescharacteristic
14C2×C4×C22⋊D73C2211yescharacteristic
15#17988482—21no—
16#17988477—21no—
17#17988468—21no—
18#17988466C2211yescharacteristic
19#17988460C411yescharacteristic, complemented
20#17988458—21no—
21#17988450C411yescharacteristic, complemented
22C4×C22⋊Dic73C411yescharacteristic, complemented
23C22×C4⋊C4×C73C411yescharacteristic, complemented
24C2×#11924037—22no—
25C2×C22⋊#64408—22no—
26C2×C4×#64408—41no—
27#14556651—22no—
28#14556651C2×C414yes—
29#14556649—22no—
30#14556649C2×C414yes—
31#14556646—41no—
32C4×#10596833—41no—
33(C2×#64408)⋊C4—41no—
34C22×C22⋊D73C2311yescharacteristic
35C22×C4×D73D411yescharacteristic
36#14556574—22no—
37#14556574C2×C414yes—
38C4×C22⋊D73—41no—
39C4×C22⋊D73—41no—
40C2×#11924007—21no—
41C2×#11924007—21no—
42C2×#11924007D411yescharacteristic
43C2×C22⋊Dic73C2×C411yescharacteristic
44C2×C22⋊Dic73C2×C411yescharacteristic
45C2×C22⋊Dic73C2×C411yescharacteristic
46C2×#11924005—21no—
47C2×#11924005—21no—
48C2×#11924005D411yescharacteristic
49C2×C4×Dic73—21no—
50C2×C4×Dic73D411yescharacteristic
51#14556498—21no—
52#14556498—21no—
53#14556498—21no—
54#14556498—21no—
55#14556498—21no—
56C23×C4×C73C2×C411yescharacteristic
57C23×C4×C73C2×C411yescharacteristic
58C23×C4×C73C2×C411yescharacteristic
59C2×C4⋊C4×C73—21no—
60C2×C4⋊C4×C73—21no—
61C2×C4⋊C4×C73—41no—
62C2×C4⋊C4×C73—41no—
63C22×#64408—42no—
64#11924037—24no—
65#11924037—42no—
66C2×#10596833—42no—
67C22⋊#64408—24no—
68C22⋊#64408—42no—
69C4×#64408—82no—
70C23×D73C2×D411yescharacteristic
71C2×C22⋊D73—41no—
72C2×C22⋊D73—41no—
73C2×C22⋊D73C22×C412yes—
74C2×C22⋊D73#4112yes—
75C22×Dic73—21no—
76C22×Dic73—21no—
77C22×Dic73C22⋊C411yescharacteristic
78C22×Dic73C22⋊C411yescharacteristic
79C22×Dic73C2×D411yescharacteristic
80C2×C4×D73—21no—
81C2×C4×D73—21no—
82C2×C4×D73—81no—
83#11924007—24no—
84#11924007—41no—
85#11924007—81no—
86C22⋊Dic73—21no—
87C22⋊Dic73—41no—
88C22⋊Dic73—41no—
89C22⋊Dic73C4212yescomplemented
90#11924005—24no—
91#11924005—41no—
92#11924005—81no—
93C4×Dic73—21no—
94C4×Dic73—21no—
95C4×Dic73—81no—
96C24×C73C22×C411yescharacteristic
97C22×C4×C73—21no—
98C22×C4×C73—21no—
99C22×C4×C73—21no—
100C22×C4×C73—21no—
101C22×C4×C73—21no—
102C22×C4×C73—41no—
103C22×C4×C73—41no—
104C22×C4×C73—41no—
105C22×C4×C73—81no—
106C22×C4×C73—81no—
107C22×C4×C73C4212yescomplemented
108C22×C4×C73C22⋊C411yescharacteristic
109C22×C4×C73C22⋊C411yescharacteristic
110C4⋊C4×C73—42no—
111C4⋊C4×C73—41no—
112C4⋊C4×C73—41no—
113C23×C73—21no—
114C23×C73—21no—
115C23×C73—41no—
116C23×C73—41no—
117C23×C73C2×C4211yescharacteristic
118C23×C73C2×C22⋊C411yescharacteristic
119C23×C73#11711yescharacteristic
120C22×D73—21no—
121C22×D73—21no—
122C22×D73—41no—
123C22×D73—41no—
124C22×D73C4×D412yes—
125C2×#64408—44no—
126C2×#64408—82no—
127C2×#64408—82no—
128C2×C4×C73—22no—
129C2×C4×C73—21no—
130C2×C4×C73—41no—
131C2×C4×C73—42no—
132C2×C4×C73—41no—
133C2×C4×C73—41no—
134C2×C4×C73—42no—
135C2×C4×C73—41no—
136C2×C4×C73—41no—
137C2×C4×C73—41no—
138C2×C4×C73—41no—
139C2×C4×C73—81no—
140C2×C4×C73—81no—
141C2×C4×C73—82no—
142C2×C4×C73#9911yescharacteristic
143C2×C4×C73#9911yescharacteristic
144C22⋊D73—42no—
145C22⋊D73—42no—
146C2×Dic73—21no—
147C2×Dic73—21no—
148C2×Dic73—22no—
149C2×Dic73—21no—
150C2×Dic73—21no—
151C2×Dic73—41no—
152C2×Dic73—41no—
153C2×Dic73—41no—
154C2×Dic73—41no—
155C2×Dic73—41no—
156C2×Dic73—41no—
157C2×Dic73—41no—
158C2×Dic73C4×D412yes—
159C4×D73—42no—
160C4×D73—81no—
161#10596833—44no—
162C22×C73—21no—
163C22×C73—21no—
164C22×C73—21no—
165C22×C73—21no—
166C22×C73—41no—
167C22×C73—41no—
168C22×C73—42no—
169C22×C73—41no—
170C22×C73—41no—
171C22×C73C4×C22⋊C411yescharacteristic
172C22×C73C2×#9911yescharacteristic
173C22×C73#41011yescharacteristic
174C2×D73—22no—
175C2×D73—41no—
176C2×D73—41no—
177C2×D73—42no—
178C2×D73—81no—
179#64408—84no—
180C4×C73—21no—
181C4×C73—21no—
182C4×C73—41no—
183C4×C73—42no—
184C4×C73—81no—
185Dic73—22no—
186Dic73—42no—
187Dic73—41no—
188Dic73—41no—
189Dic73—81no—
190#12621—731nomaximal, Sylow
191C2×C73—21no—
192C2×C73—21no—
193C2×C73—41no—
194C2×C73—41no—
195C2×C73C4×#9911yescharacteristic
196C2×C73#212111yescharacteristic
197C2×C73#212311yescharacteristic
198D73—42no—
199#2322—731no—
200C2×#352—732no—
201C4×#100—731no—
202C4×#99—731no—
203#1390—732no—
204C73#1262111yescharacteristic, complemented, Sylow
205C2×#123—731no—
206C2×C4×D4—731no—
207C22×C4⋊C4#6440811yescharacteristic, complemented
208C2×#100—732no—
209C2×#99—732no—
210C4×M16—1461no—
211#389—1461no—
212#388—1461no—
213#387—1461no—
214#386—731no—
215#385—731no—
216#385—1461no—
217#381—731no—
218C4×C22⋊C4—731no—
219C4×C22⋊C4—1461no—
220#352—738no—
221#343—1461no—
222#342—1461no—
223C22×D4—731no—
224C23×C4—731no—
225C23×C4C2×#6440811yescharacteristic
226C23×C4C2×#6440811yescharacteristic
227C23×C4C2×#6440811yescharacteristic
228C2×M16—1462no—
229#123—734no—
230#123—1462no—
231C4×D4—2921no—
232C4×D4—2921no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—41no—
236C2×C4⋊C4—41no—
237C2×C4⋊C4—731no—
238C2×C4⋊C4—1461no—
239C2×C4⋊C4—1461no—
240C2×C22⋊C4—731no—
241C2×C22⋊C4—731no—
242C2×C22⋊C4—731no—
243C2×C22⋊C4—731no—
244C2×C22⋊C4—1462no—
245C2×C22⋊C4—1461no—
246C2×C22⋊C4—1461no—
247C2×C42—2921no—
248C2×C42—731no—
249C2×C42—1461no—
250#100—734no—
251#100—1462no—
252#99—734no—
253#99—1462no—
254#97—2921no—
255C4×C8—2921no—
256#95—2921no—
257#95—1461no—
258#95—1461no—
259#95—1461no—
260#95—1461no—
261#95—1461no—
262C24—731no—
263C24C22×#6440811yescharacteristic
264C2×D4—2921no—
265C2×D4—2921no—
266C2×D4—732no—
267C2×D4—732no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—81no—
277C22×C4—81no—
278C22×C4—2922no—
279C22×C4—5841no—
280C22×C4—731no—
281C22×C4—731no—
282C22×C4—731no—
283C22×C4—1461no—
284C22×C4—1461no—
285C22×C4—1461no—
286C22×C4—1461no—
287C22×C4C4×#6440812yescomplemented
288C22×C4C22⋊#6440811yescharacteristic
289C22×C4C22⋊#6440811yescharacteristic
290M16—2922no—
291M16—1464no—
292C2×C8—2922no—
293C4⋊C4—42no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—2921no—
297C4⋊C4—5841no—
298C4⋊C4—1464no—
299C22⋊C4—2921no—
300C22⋊C4—2922no—
301C22⋊C4—2921no—
302C22⋊C4—2921no—
303C22⋊C4—5841no—
304C22⋊C4—732no—
305C22⋊C4—1464no—
306C22⋊C4—1464no—
307C22⋊C4—1461no—
308C42—5841no—
309C42—5842no—
310C42—1461no—
311C42—1461no—
312C23—21no—
313C23—21no—
314C23—41no—
315C23—41no—
316C23—2921no—
317C23—2921no—
318C23—732no—
319C23—1461no—
320C23—1461no—
321C23C2×C4×#6440811yescharacteristic
322C23C2×C22⋊#6440811yescharacteristic
323C23#1455667011yescharacteristic
324D4—2922no—
325D4—2922no—
326C2×C4—22no—
327C2×C4—21no—
328C2×C4—41no—
329C2×C4—42no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—42no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4—82no—
340C2×C4—2922no—
341C2×C4—2921no—
342C2×C4—2921no—
343C2×C4—2921no—
344C2×C4—2921no—
345C2×C4—2921no—
346C2×C4—2921no—
347C2×C4—2924no—
348C2×C4—2921no—
349C2×C4—5841no—
350C2×C4—5842no—
351C2×C4—5842no—
352C2×C4—732no—
353C2×C4—1462no—
354C2×C4—1461no—
355C2×C4—1461no—
356C2×C4—1461no—
357C2×C4—1461no—
358C2×C4#1455664911yescharacteristic
359C2×C4#1455664911yescharacteristic
360C8—2924no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—41no—
366C22—41no—
367C22—42no—
368C22—41no—
369C22—41no—
370C22—2922no—
371C22—2921no—
372C22—2921no—
373C22—5841no—
374C22—1462no—
375C22C4×C22⋊#6440811yescharacteristic
376C22C2×#1455664911yescharacteristic
377C22#1798956611yescharacteristic
378C4—21no—
379C4—21no—
380C4—41no—
381C4—42no—
382C4—81no—
383C4—2922no—
384C4—2921no—
385C4—2921no—
386C4—5841no—
387C4—5844no—
388C4—1462no—
389C2—21no—
390C2—21no—
391C2—41no—
392C2—41no—
393C2—2922no—
394C2C4×#1455664911yescharacteristic
395C2#2262618611yescharacteristic
396C2#2262619311yescharacteristic
subgroupG/H
centerC22C2×#14556649
commutatorC23×C73C2×C42
FrattiniC24C22×#64408
FittingC22×C4⋊C4×C73C4
socleC22×C73C2×#99
prime radical, cosocleC22×C22⋊D73C23
hypercenterC22×C4⋊C4#64408
hypocenterC73#12621
LCSG ▷ C23×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
10. #17989489C4semidirect
19. #17988460C4semidirect
21. #17988450C4semidirect
22. C4×C22⋊Dic73C4semidirect
23. C22×C4⋊C4×C73C4semidirect
89. C22⋊Dic73C42semidirect
107. C22×C4×C73C42semidirect
204. C73#12621semidirect
207. C22×C4⋊C4#64408semidirect
287. C22×C4C4×#64408semidirect

Maximal subgroups

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

subgroupindex
C2×#179894892
C4×#145566512
C4×#145566492
#226250152
#226126002
#1262173