Finite Group Databasebeta

← #32983644

Subgroups — #32983644

Order 18688.

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

normal subgroups only

This group has 568 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#22618960C211yesmaximal, characteristic, complemented
2#22608555C212yesmaximal, complemented
3#22608272C211yesmaximal, characteristic, complemented
4#22607787C212yesmaximal, complemented
5C73×#2544C211yesmaximal, characteristic
6#17988659—24no—
7#17988644C2212yes—
8#17988644C2212yes—
9#17988551—21no—
10C2×C22⋊#10596831—21no—
11#17988327—24no—
12#17988311—21no—
13#17988311—21no—
14C2×#127×C73C411yescharacteristic
15C2×#124×C73C411yescharacteristic
16C2×#124×C73C2211yescharacteristic
17C2×C4×D4×C73C2211yescharacteristic
18#394×C73—21no—
19#394×C73—21no—
20#391×C73C411yescharacteristic
21#390×C73C411yescharacteristic
22#386×C73—21no—
23#386×C73—21no—
24C4×C4⋊C4×C73C2211yescharacteristic
25C2×#11924012—22no—
26C22×#10596831—22no—
27C4⋊#10596831—22no—
28C4⋊#10596831—22no—
29C22⋊#10596831—22no—
30C22⋊#10596831—22no—
31C22⋊#10596831—22no—
32C22⋊#10596831—22no—
33#14556502—22no—
34#14556502D414yescomplemented
35C22×Q8×C73C2×C411yescharacteristic
36C22×D4×C73—21no—
37C22×D4×C73—21no—
38C22×D4×C73C2×C411yescharacteristic
39C22×D4×C73C2×C411yescharacteristic
40C23×C4×C73D411yescharacteristic
41C23×C4×C73D411yescharacteristic
42#127×C73—42no—
43#124×C73—22no—
44#124×C73—22no—
45#124×C73—42no—
46C4×D4×C73—22no—
47C4×D4×C73—22no—
48C2×C4⋊C4×C73—21no—
49C2×C4⋊C4×C73C2×C411yescharacteristic
50C2×C22⋊C4×C73—21no—
51C2×C22⋊C4×C73—21no—
52C2×C22⋊C4×C73—21no—
53C2×C22⋊C4×C73—21no—
54C2×C22⋊C4×C73—21no—
55C2×C22⋊C4×C73—21no—
56C2×C22⋊C4×C73D411yescharacteristic
57C2×C22⋊C4×C73D411yescharacteristic
58C2×C42×C73C2×C411yescharacteristic
59C2×C42×C73C2×C411yescharacteristic
60C2×C42×C73C2311yescharacteristic
61#95×C73—21no—
62#95×C73—21no—
63#95×C73—21no—
64#11924012—82no—
65C2×#10596831—24no—
66C2×#10596831—42no—
67C2×#10596831—82no—
68C24×C73—21no—
69C24×C73C22⋊C411yescharacteristic
70C24×C73C22⋊C411yescharacteristic
71C2×Q8×C73—22no—
72C2×Q8×C73—41no—
73C2×Q8×C73—41no—
74C2×D4×C73—22no—
75C2×D4×C73—22no—
76C2×D4×C73—22no—
77C2×D4×C73—41no—
78C2×D4×C73—41no—
79C2×D4×C73—41no—
80C2×D4×C73—42no—
81C2×D4×C73—42no—
82C2×D4×C73—42no—
83C2×D4×C73—42no—
84C2×D4×C73—81no—
85C2×D4×C73—81no—
86C22×C4×C73—21no—
87C22×C4×C73—21no—
88C22×C4×C73—21no—
89C22×C4×C73—21no—
90C22×C4×C73—21no—
91C22×C4×C73—21no—
92C22×C4×C73—21no—
93C22×C4×C73—81no—
94C22×C4×C73—81no—
95C22×C4×C73C22⋊C411yescharacteristic
96C22×C4×C73C22⋊C411yescharacteristic
97C22×C4×C73C22×C411yescharacteristic
98C22×C4×C73C2×D411yescharacteristic
99C22×C4×C73C2×D411yescharacteristic
100C4⋊C4×C73—22no—
101C4⋊C4×C73—42no—
102C22⋊C4×C73—22no—
103C22⋊C4×C73—22no—
104C22⋊C4×C73—42no—
105C22⋊C4×C73—42no—
106C22⋊C4×C73—41no—
107C22⋊C4×C73—42no—
108C22⋊C4×C73—41no—
109C22⋊C4×C73—81no—
110C22⋊C4×C73—81no—
111C42×C73—41no—
112C42×C73—41no—
113C42×C73C2×D412yes—
114C42×C73#4112yes—
115C23×C73—21no—
116C23×C73—21no—
117C23×C73—21no—
118C23×C73—21no—
119C23×C73—21no—
120C23×C73—41no—
121C23×C73—41no—
122C23×C73—41no—
123C23×C73—41no—
124C23×C73—41no—
125C23×C73—41no—
126C23×C73—41no—
127C23×C73—41no—
128C23×C73—41no—
129C23×C73—41no—
130C23×C73—41no—
131C23×C73#10012yes—
132C23×C73C2×C22⋊C411yescharacteristic
133Q8×C73—82no—
134D4×C73—44no—
135D4×C73—82no—
136D4×C73—84no—
137D4×C73—84no—
138C2×C4×C73—21no—
139C2×C4×C73—21no—
140C2×C4×C73—22no—
141C2×C4×C73—21no—
142C2×C4×C73—21no—
143C2×C4×C73—41no—
144C2×C4×C73—41no—
145C2×C4×C73—42no—
146C2×C4×C73—42no—
147C2×C4×C73—41no—
148C2×C4×C73—41no—
149C2×C4×C73—41no—
150C2×C4×C73—41no—
151C2×C4×C73—41no—
152C2×C4×C73—41no—
153C2×C4×C73—41no—
154C2×C4×C73—41no—
155C2×C4×C73—41no—
156C2×C4×C73—41no—
157C2×C4×C73—81no—
158C2×C4×C73—81no—
159C2×C4×C73C4×D412yes—
160C2×C4×C73#12011yescharacteristic
161C2×C4×C73#12112yes—
162C2×C4×C73#12311yescharacteristic
163#10596831—44no—
164#10596831—82no—
165C22×C73—21no—
166C22×C73—21no—
167C22×C73—22no—
168C22×C73—42no—
169C22×C73—42no—
170C22×C73—41no—
171C22×C73—41no—
172C22×C73—41no—
173C22×C73—41no—
174C22×C73—41no—
175C22×C73—81no—
176C22×C73—81no—
177C22×C73—81no—
178C22×C73—81no—
179C22×C73—81no—
180C22×C73—81no—
181C22×C73—81no—
182C22×C73—81no—
183C22×C73—81no—
184C22×C73#38611yescharacteristic
185C22×C73C2×#10011yescharacteristic
186C22×C73#41311yescharacteristic
187C4×C73—22no—
188C4×C73—22no—
189C4×C73—21no—
190C4×C73—21no—
191C4×C73—42no—
192C4×C73—41no—
193C4×C73—41no—
194C4×C73—81no—
195C4×C73—81no—
196#9447—731nomaximal, Sylow
197C2×C73—21no—
198C2×C73—21no—
199C2×C73—42no—
200C2×C73—81no—
201C2×C73—161no—
202C2×C73#188411yescharacteristic
203C2×C73#190011yescharacteristic
204C2×C73#191111yescharacteristic
205#2973—731no—
206#2544D7311yescharacteristic
207#1533—732no—
208#1466—731no—
209#1285—732no—
210C73#944711yescharacteristic, complemented, Sylow
211C2×#127Dic7311yescharacteristic
212C2×#124Dic7311yescharacteristic
213C2×#124C2×D7311yescharacteristic
214C2×C4×D4C2×D7311yescharacteristic
215#435—1464no—
216#432—732no—
217#432—732no—
218#407—1461no—
219C2×C22⋊C8—1461no—
220#394—21no—
221#394—21no—
222#391Dic7311yescharacteristic
223#390Dic7311yescharacteristic
224#386—21no—
225#386—21no—
226C4×C4⋊C4C2×D7311yescharacteristic
227#329—1464no—
228#323—1461no—
229#323—1461no—
230C22×Q8C2×Dic7311yescharacteristic
231C22×D4—21no—
232C22×D4—21no—
233C22×D4C2×Dic7311yescharacteristic
234C22×D4C2×Dic7311yescharacteristic
235C23×C4C22⋊D7311yescharacteristic
236C23×C4C22⋊D7311yescharacteristic
237C2×M16—1462no—
238C22×C8—1462no—
239#127—42no—
240#124—22no—
241#124—22no—
242#124—42no—
243C4×D4—22no—
244C4×D4—22no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4C2×Dic7311yescharacteristic
247C2×C22⋊C4—21no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4C22⋊D7311yescharacteristic
254C2×C22⋊C4C22⋊D7311yescharacteristic
255C2×C42C2×Dic7311yescharacteristic
256C2×C42C2×Dic7311yescharacteristic
257C2×C42C22×D7311yescharacteristic
258C4⋊C8—1462no—
259C4⋊C8—1462no—
260C22⋊C8—1462no—
261C22⋊C8—1462no—
262C22⋊C8—1462no—
263C22⋊C8—1462no—
264#97—734no—
265#97—1462no—
266#95—21no—
267#95—21no—
268#95—21no—
269C24—21no—
270C24C22⋊Dic7311yescharacteristic
271C24C22⋊Dic7311yescharacteristic
272C2×Q8—22no—
273C2×Q8—41no—
274C2×Q8—41no—
275C2×D4—22no—
276C2×D4—22no—
277C2×D4—22no—
278C2×D4—41no—
279C2×D4—41no—
280C2×D4—41no—
281C2×D4—42no—
282C2×D4—42no—
283C2×D4—42no—
284C2×D4—42no—
285C2×D4—81no—
286C2×D4—81no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—81no—
295C22×C4—81no—
296C22×C4C22⋊Dic7311yescharacteristic
297C22×C4C22⋊Dic7311yescharacteristic
298C22×C4C22×Dic7311yescharacteristic
299C22×C4C2×C22⋊D7311yescharacteristic
300C22×C4C2×C22⋊D7311yescharacteristic
301M16—5842no—
302C2×C8—2922no—
303C2×C8—5842no—
304C2×C8—1464no—
305C4⋊C4—22no—
306C4⋊C4—42no—
307C22⋊C4—22no—
308C22⋊C4—22no—
309C22⋊C4—42no—
310C22⋊C4—42no—
311C22⋊C4—41no—
312C22⋊C4—42no—
313C22⋊C4—41no—
314C22⋊C4—81no—
315C22⋊C4—81no—
316C42—41no—
317C42—41no—
318C42D4×D7312yes—
319C42#1192402712yes—
320C23—21no—
321C23—21no—
322C23—21no—
323C23—21no—
324C23—21no—
325C23—41no—
326C23—41no—
327C23—41no—
328C23—41no—
329C23—41no—
330C23—41no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23—41no—
335C23—41no—
336C23#1455650812yes—
337C23C2×C22⋊Dic7311yescharacteristic
338Q8—82no—
339D4—44no—
340D4—82no—
341D4—84no—
342D4—84no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—22no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4D4×Dic7312yes—
365C2×C4#1455655911yescharacteristic
366C2×C4#1455656312yes—
367C2×C4#1455657211yescharacteristic
368C8—2924no—
369C8—5842no—
370C22—21no—
371C22—21no—
372C22—22no—
373C22—42no—
374C22—42no—
375C22—41no—
376C22—41no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22—81no—
381C22—81no—
382C22—81no—
383C22—81no—
384C22—81no—
385C22—81no—
386C22—81no—
387C22—81no—
388C22—81no—
389C22#1798846711yescharacteristic
390C22C2×#1455650811yescharacteristic
391C22#1798857111yescharacteristic
392C4—21no—
393C4—21no—
394C4—22no—
395C4—22no—
396C4—42no—
397C4—41no—
398C4—41no—
399C4—81no—
400C4—81no—
401C2—21no—
402C2—21no—
403C2—42no—
404C2—81no—
405C2—161no—
406C2#2261031511yescharacteristic
407C2#2261039511yescharacteristic
408C2#2261046411yescharacteristic
subgroupG/H
centerC22#17988467
commutatorC22×C4×C73C22×C4
FrattiniC2×C42C22×D73
FittingC73×#2544C2
socleC22×C73#386
prime radicalC2×#127×C73C4
cosocleC2×C42×C73C23
hypercenter#2544D73
hypocenterC73#9447
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2544

Complements

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

NH ≅ G/Ntype
1. #22618960C2semidirect
2. #22608555C2semidirect
3. #22608272C2semidirect
4. #22607787C2semidirect
34. #14556502D4semidirect
210. C73#9447semidirect

Maximal subgroups

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

subgroupindex
#226189602
#226085552
#226082722
#226077872
C73×#25442
#944773