Finite Group Databasebeta

← #32983984

Subgroups — #32983984

Order 18688.

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

normal subgroups only

This group has 628 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#22608269C211yesmaximal, characteristic, complemented
3#22608258C212yesmaximal, complemented
4#22607837C212yesmaximal, complemented
5C73×#2534C211yesmaximal, characteristic
6#17988659—24no—
7#17988644—21no—
8#17988551C2212yes—
9#17988551C2212yes—
10C2×C22⋊#10596831—21no—
11#17988313—21no—
12#17988311—24no—
13#17988311—21no—
14C2×#123×C73C411yescharacteristic
15C2×#120×C73C411yescharacteristic
16C2×C4×D4×C73C2211yescharacteristic
17C22×C22⋊C4×C73C2211yescharacteristic
18#396×C73—21no—
19#394×C73—21no—
20#388×C73—21no—
21#386×C73—21no—
22#381×C73C411yescharacteristic
23#380×C73C2211yescharacteristic
24#379×C73C411yescharacteristic
25C2×#11924012—22no—
26C2×#11924012—22no—
27C22×#10596831—22no—
28C4⋊#10596831—22no—
29C22⋊#10596831—22no—
30C22⋊#10596831—22no—
31C22⋊#10596831—22no—
32C22⋊#10596831—22no—
33C22⋊#10596831D414yescomplemented
34#14556502—22no—
35C25×C73C2×C411yescharacteristic
36C22×D4×C73—21no—
37C22×D4×C73C2×C411yescharacteristic
38C23×C4×C73C2×C411yescharacteristic
39C23×C4×C73D411yescharacteristic
40C23×C4×C73C2311yescharacteristic
41#123×C73—42no—
42#120×C73—42no—
43C4×D4×C73—22no—
44C4×D4×C73—22no—
45C2×C4⋊C4×C73—21no—
46C2×C4⋊C4×C73—21no—
47C2×C4⋊C4×C73D411yescharacteristic
48C2×C22⋊C4×C73—21no—
49C2×C22⋊C4×C73—21no—
50C2×C22⋊C4×C73—21no—
51C2×C22⋊C4×C73—21no—
52C2×C22⋊C4×C73—21no—
53C2×C22⋊C4×C73—22no—
54C2×C22⋊C4×C73—22no—
55C2×C22⋊C4×C73—21no—
56C2×C22⋊C4×C73C2×C411yescharacteristic
57C2×C22⋊C4×C73C2×C411yescharacteristic
58C2×C22⋊C4×C73C2×C411yescharacteristic
59C2×C22⋊C4×C73D411yescharacteristic
60C2×C42×C73D411yescharacteristic
61#95×C73—21no—
62#95×C73—21no—
63#95×C73—21no—
64#11924012—82no—
65#11924012—82no—
66C2×#10596831—24no—
67C2×#10596831—42no—
68C2×#10596831—82no—
69C24×C73—21no—
70C24×C73—22no—
71C24×C73—42no—
72C24×C73—41no—
73C24×C73—41no—
74C24×C73—81no—
75C24×C73C22⋊C411yescharacteristic
76C24×C73C22×C411yescharacteristic
77C2×D4×C73—22no—
78C2×D4×C73—22no—
79C2×D4×C73—41no—
80C2×D4×C73—42no—
81C2×D4×C73—42no—
82C2×D4×C73—81no—
83C22×C4×C73—21no—
84C22×C4×C73—21no—
85C22×C4×C73—21no—
86C22×C4×C73—21no—
87C22×C4×C73—21no—
88C22×C4×C73—22no—
89C22×C4×C73—41no—
90C22×C4×C73—41no—
91C22×C4×C73—81no—
92C22×C4×C73—81no—
93C22×C4×C73C22⋊C411yescharacteristic
94C22×C4×C73C22⋊C411yescharacteristic
95C22×C4×C73C22⋊C411yescharacteristic
96C22×C4×C73C2×D412yes—
97C22×C4×C73C2×D411yescharacteristic
98C22×C4×C73C2×D411yescharacteristic
99C22×C4×C73#4112yes—
100C4⋊C4×C73—22no—
101C4⋊C4×C73—42no—
102C4⋊C4×C73—81no—
103C22⋊C4×C73—22no—
104C22⋊C4×C73—22no—
105C22⋊C4×C73—42no—
106C22⋊C4×C73—42no—
107C22⋊C4×C73—42no—
108C22⋊C4×C73—41no—
109C22⋊C4×C73—42no—
110C22⋊C4×C73—41no—
111C22⋊C4×C73—41no—
112C22⋊C4×C73—41no—
113C22⋊C4×C73—81no—
114C42×C73—21no—
115C42×C73—21no—
116C23×C73—21no—
117C23×C73—21no—
118C23×C73—21no—
119C23×C73—21no—
120C23×C73—22no—
121C23×C73—24no—
122C23×C73—41no—
123C23×C73—41no—
124C23×C73—41no—
125C23×C73—41no—
126C23×C73—41no—
127C23×C73—41no—
128C23×C73—41no—
129C23×C73—42no—
130C23×C73—41no—
131C23×C73—41no—
132C23×C73—41no—
133C23×C73—82no—
134C23×C73—82no—
135C23×C73—84no—
136C23×C73—84no—
137C23×C73#10012yes—
138C23×C73C2×C22⋊C411yescharacteristic
139C23×C73C4×D412yes—
140D4×C73—44no—
141D4×C73—84no—
142C2×C4×C73—21no—
143C2×C4×C73—21no—
144C2×C4×C73—21no—
145C2×C4×C73—21no—
146C2×C4×C73—21no—
147C2×C4×C73—41no—
148C2×C4×C73—41no—
149C2×C4×C73—42no—
150C2×C4×C73—42no—
151C2×C4×C73—42no—
152C2×C4×C73—41no—
153C2×C4×C73—41no—
154C2×C4×C73—41no—
155C2×C4×C73—41no—
156C2×C4×C73—41no—
157C2×C4×C73—41no—
158C2×C4×C73—41no—
159C2×C4×C73—41no—
160C2×C4×C73—41no—
161C2×C4×C73—41no—
162C2×C4×C73—41no—
163C2×C4×C73—81no—
164C2×C4×C73—82no—
165C2×C4×C73#12011yescharacteristic
166C2×C4×C73#12112yes—
167C2×C4×C73#12311yescharacteristic
168#10596831—44no—
169#10596831—82no—
170C22×C73—21no—
171C22×C73—21no—
172C22×C73—22no—
173C22×C73—22no—
174C22×C73—22no—
175C22×C73—22no—
176C22×C73—42no—
177C22×C73—42no—
178C22×C73—42no—
179C22×C73—42no—
180C22×C73—42no—
181C22×C73—41no—
182C22×C73—41no—
183C22×C73—42no—
184C22×C73—81no—
185C22×C73—81no—
186C22×C73—81no—
187C22×C73—84no—
188C22×C73—84no—
189C22×C73—81no—
190C22×C73—81no—
191C22×C73#38611yescharacteristic
192C22×C73C2×#10011yescharacteristic
193C22×C73#41311yescharacteristic
194C4×C73—21no—
195C4×C73—22no—
196C4×C73—21no—
197C4×C73—41no—
198C4×C73—41no—
199C4×C73—41no—
200C4×C73—81no—
201C4×C73—81no—
202C4×C73—81no—
203#9515—731nomaximal, Sylow
204C2×C73—21no—
205C2×C73—21no—
206C2×C73—42no—
207C2×C73—42no—
208C2×C73—82no—
209C2×C73#189111yescharacteristic
210C2×C73#190011yescharacteristic
211C2×C73#191311yescharacteristic
212#2973—731no—
213#2534D7311yescharacteristic
214#1465—731no—
215#1462—732no—
216#1305—732no—
217C73#951511yescharacteristic, complemented, Sylow
218C2×#123Dic7311yescharacteristic
219C2×#120Dic7311yescharacteristic
220C2×C4×D4C2×D7311yescharacteristic
221C22×C22⋊C4C2×D7311yescharacteristic
222#435—1464no—
223#432—1461no—
224#407—732no—
225#407—732no—
226C2×C22⋊C8—1461no—
227#396—21no—
228#394—21no—
229#388—21no—
230#386—21no—
231#381Dic7311yescharacteristic
232#380C2×D7311yescharacteristic
233#379Dic7311yescharacteristic
234#324—1461no—
235#323—1464no—
236#323—1461no—
237C25C2×Dic7311yescharacteristic
238C22×D4—21no—
239C22×D4C2×Dic7311yescharacteristic
240C23×C4C2×Dic7311yescharacteristic
241C23×C4C22⋊D7311yescharacteristic
242C23×C4C22×D7311yescharacteristic
243C2×M16—1462no—
244C2×M16—1462no—
245C22×C8—1462no—
246#123—42no—
247#120—42no—
248C4×D4—22no—
249C4×D4—22no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4C22⋊D7311yescharacteristic
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—21no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—21no—
261C2×C22⋊C4C2×Dic7311yescharacteristic
262C2×C22⋊C4C2×Dic7311yescharacteristic
263C2×C22⋊C4C2×Dic7311yescharacteristic
264C2×C22⋊C4C22⋊D7311yescharacteristic
265C2×C42C22⋊D7311yescharacteristic
266C4⋊C8—1462no—
267C22⋊C8—734no—
268C22⋊C8—1462no—
269C22⋊C8—1462no—
270C22⋊C8—1462no—
271C22⋊C8—1462no—
272#97—1462no—
273#95—21no—
274#95—21no—
275#95—21no—
276C24—21no—
277C24—22no—
278C24—42no—
279C24—41no—
280C24—41no—
281C24—81no—
282C24C22⋊Dic7311yescharacteristic
283C24C22×Dic7311yescharacteristic
284C2×D4—22no—
285C2×D4—22no—
286C2×D4—41no—
287C2×D4—42no—
288C2×D4—42no—
289C2×D4—81no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—22no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—81no—
299C22×C4—81no—
300C22×C4C22⋊Dic7311yescharacteristic
301C22×C4C22⋊Dic7311yescharacteristic
302C22×C4C22⋊Dic7311yescharacteristic
303C22×C4D4×D7312yes—
304C22×C4C2×C22⋊D7311yescharacteristic
305C22×C4C2×C22⋊D7311yescharacteristic
306C22×C4#1192402712yes—
307M16—5842no—
308M16—5842no—
309C2×C8—2922no—
310C2×C8—5842no—
311C2×C8—1464no—
312C4⋊C4—22no—
313C4⋊C4—42no—
314C4⋊C4—81no—
315C22⋊C4—22no—
316C22⋊C4—22no—
317C22⋊C4—42no—
318C22⋊C4—42no—
319C22⋊C4—42no—
320C22⋊C4—41no—
321C22⋊C4—42no—
322C22⋊C4—41no—
323C22⋊C4—41no—
324C22⋊C4—41no—
325C22⋊C4—81no—
326C42—21no—
327C42—21no—
328C23—21no—
329C23—21no—
330C23—21no—
331C23—21no—
332C23—22no—
333C23—24no—
334C23—41no—
335C23—41no—
336C23—41no—
337C23—41no—
338C23—41no—
339C23—41no—
340C23—41no—
341C23—42no—
342C23—41no—
343C23—41no—
344C23—41no—
345C23—82no—
346C23—82no—
347C23—84no—
348C23—84no—
349C23#1455650812yes—
350C23C2×C22⋊Dic7311yescharacteristic
351C23D4×Dic7312yes—
352D4—44no—
353D4—84no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—41no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—41no—
363C2×C4—42no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—81no—
376C2×C4—82no—
377C2×C4#1455655911yescharacteristic
378C2×C4#1455656312yes—
379C2×C4#1455657211yescharacteristic
380C8—2924no—
381C8—5842no—
382C22—21no—
383C22—21no—
384C22—22no—
385C22—22no—
386C22—22no—
387C22—22no—
388C22—42no—
389C22—42no—
390C22—42no—
391C22—42no—
392C22—42no—
393C22—41no—
394C22—41no—
395C22—42no—
396C22—81no—
397C22—81no—
398C22—81no—
399C22—84no—
400C22—84no—
401C22—81no—
402C22—81no—
403C22#1798846711yescharacteristic
404C22C2×#1455650811yescharacteristic
405C22#1798857111yescharacteristic
406C4—21no—
407C4—22no—
408C4—21no—
409C4—41no—
410C4—41no—
411C4—41no—
412C4—81no—
413C4—81no—
414C4—81no—
415C2—21no—
416C2—21no—
417C2—42no—
418C2—42no—
419C2—82no—
420C2#2261035011yescharacteristic
421C2#2261039511yescharacteristic
422C2#2261047411yescharacteristic
subgroupG/H
centerC22#17988467
commutatorC24×C73C22×C4
FrattiniC23×C4C22×D73
FittingC73×#2534C2
socleC22×C73#386
prime radicalC2×#120×C73C4
cosocleC23×C4×C73C23
hypercenter#2534D73
hypocenterC73#9515
LCSG ▷ C24×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2534

Complements

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

NH ≅ G/Ntype
1. #22618960C2semidirect
2. #22608269C2semidirect
3. #22608258C2semidirect
4. #22607837C2semidirect
33. C22⋊#10596831D4semidirect
217. C73#9515semidirect

Maximal subgroups

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

subgroupindex
#226189602
#226082692
#226082582
#226078372
C73×#25342
#951573