Finite Group Databasebeta

← #32989139

Subgroups — #32989139

Order 18688.

466 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 99 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#22614937C211yesmaximal, characteristic
2#22608503C211yesmaximal, characteristic
3C2×#17988324C211yesmaximal, characteristic
4#22607809C211yesmaximal, characteristic
5#22607798C211yesmaximal, characteristic
6#22607779C211yesmaximal, characteristic
7C73×#1996C211yesmaximal, characteristic
8C2×#14556568—21no—
9C2×C22⋊#10596831C2211yescharacteristic
10C2×C22⋊#10596831C2211yescharacteristic
11#17988519—21no—
12#17988508—21no—
13#17988493—21no—
14#17988485—21no—
15#17988462—21no—
16#17988447C2211yescharacteristic
17C2×#14556498C2211yescharacteristic
18#17988324—21no—
19#17988324—21no—
20#17988324—21no—
21#17988324—21no—
22C2×C22⋊Q8×C73C2211yescharacteristic
23#380×C73C2211yescharacteristic
24C4×C22⋊C4×C73C2211yescharacteristic
25#342×C73—21no—
26#342×C73—21no—
27C22×Dic146—21no—
28C23×Dic73—21no—
29C22×#10596831—21no—
30C22×#10596831—21no—
31#14556568—42no—
32#14556568—42no—
33C2×C4⋊Dic73—21no—
34C2×C4⋊Dic73—21no—
35C2×C4⋊Dic73—21no—
36C2×C4⋊Dic73—21no—
37C2×#11924007—21no—
38C2×#11924007—21no—
39C2×C22⋊Dic73—21no—
40C22⋊#10596831—21no—
41C22⋊#10596831—21no—
42C22⋊#10596831—21no—
43C22⋊#10596831—21no—
44#14556498—21no—
45#14556498—21no—
46#14556498—21no—
47#14556498D411yescharacteristic
48#14556498D411yescharacteristic
49#14556498D411yescharacteristic
50#14556498D411yescharacteristic
51C22×Q8×C73Q811yescharacteristic, complemented
52C23×C4×C73C2311yescharacteristic
53C22⋊Q8×C73—21no—
54C22⋊Q8×C73—21no—
55C22⋊Q8×C73—21no—
56C22⋊Q8×C73—21no—
57C2×C4⋊C4×C73—21no—
58C2×C4⋊C4×C73—21no—
59C2×C4⋊C4×C73Q811yescharacteristic, complemented
60C2×C22⋊C4×C73—21no—
61C2×C22⋊C4×C73—21no—
62C2×C22⋊C4×C73D411yescharacteristic
63C2×C22⋊C4×C73D411yescharacteristic
64C2×C42×C73—21no—
65#95×C73—21no—
66#95×C73—21no—
67C2×Dic146—42no—
68C2×Dic146—42no—
69C2×Dic146—81no—
70C22×Dic73—21no—
71C22×Dic73—21no—
72C22×Dic73—21no—
73C22×Dic73—21no—
74C22×Dic73—42no—
75C22×Dic73—41no—
76C22×Dic73—41no—
77C22×Dic73—81no—
78C2×#10596831—41no—
79C2×#10596831—41no—
80C2×#10596831—81no—
81C2×#10596831—81no—
82C4⋊Dic73—42no—
83C4⋊Dic73—42no—
84C4⋊Dic73—81no—
85C4⋊Dic73—81no—
86#11924007—42no—
87#11924007—42no—
88C22⋊Dic73—41no—
89C22⋊Dic73—41no—
90C24×C73C2×D411yescharacteristic
91C2×Q8×C73—21no—
92C2×Q8×C73—21no—
93C2×Q8×C73—81no—
94C22×C4×C73—21no—
95C22×C4×C73—21no—
96C22×C4×C73—21no—
97C22×C4×C73—21no—
98C22×C4×C73—41no—
99C22×C4×C73—81no—
100C22×C4×C73C2×D411yescharacteristic
101C22×C4×C73C2×D411yescharacteristic
102C22×C4×C73C2×Q811yescharacteristic
103C22×C4×C73#4111yescharacteristic
104C22×C4×C73#4111yescharacteristic
105C22×C4×C73#4111yescharacteristic
106C4⋊C4×C73—21no—
107C4⋊C4×C73—21no—
108C4⋊C4×C73—41no—
109C4⋊C4×C73—41no—
110C4⋊C4×C73—81no—
111C22⋊C4×C73—21no—
112C22⋊C4×C73—21no—
113C22⋊C4×C73—41no—
114C22⋊C4×C73—41no—
115C22⋊C4×C73—41no—
116C22⋊C4×C73—41no—
117C22⋊C4×C73SD1612yes—
118C22⋊C4×C73Q1612yes—
119C42×C73—81no—
120C23×C73—41no—
121C23×C73—41no—
122C23×C73#12011yescharacteristic
123C23×C73C22⋊Q811yescharacteristic
124C23×C73#12411yescharacteristic
125C23×C73C2×SD1611yescharacteristic
126C23×C73C2×Q1611yescharacteristic
127C23×C73AGL(1,ℤ/8)11yescharacteristic
128C23×C73#13711yescharacteristic
129Q8×C73—81no—
130Q8×C73—81no—
131C2×C4×C73—22no—
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×C73—41no—
139C2×C4×C73—41no—
140C2×C4×C73—41no—
141C2×C4×C73—41no—
142C2×C4×C73—41no—
143C2×C4×C73—41no—
144C2×C4×C73—41no—
145C2×C4×C73—81no—
146C2×C4×C73—81no—
147C2×C4×C73—81no—
148C2×C4×C73—81no—
149C2×C4×C73—81no—
150C2×C4×C73C22⋊Q811yescharacteristic
151C2×C4×C73C22⋊Q811yescharacteristic
152C2×C4×C73#12411yescharacteristic
153C2×C4×C73#12411yescharacteristic
154C2×Dic73—22no—
155C2×Dic73—41no—
156C2×Dic73—41no—
157C2×Dic73—41no—
158C2×Dic73—41no—
159C2×Dic73—41no—
160C2×Dic73—41no—
161C2×Dic73—41no—
162C2×Dic73—41no—
163C2×Dic73—41no—
164C2×Dic73—41no—
165C2×Dic73—41no—
166C2×Dic73—82no—
167C2×Dic73—82no—
168Dic146—84no—
169#10596831—81no—
170#10596831—81no—
171C22×C73—21no—
172C22×C73—21no—
173C22×C73—21no—
174C22×C73—21no—
175C22×C73—21no—
176C22×C73—21no—
177C22×C73—41no—
178C22×C73—41no—
179C22×C73—41no—
180C22×C73—41no—
181C22×C73#39311yescharacteristic
182C22×C73#45011yescharacteristic
183C22×C73#45111yescharacteristic
184C22×C73#47511yescharacteristic
185C22×C73#47711yescharacteristic
186C22×C73#48711yescharacteristic
187C22×C73#48911yescharacteristic
188C4×C73—21no—
189C4×C73—21no—
190C4×C73—21no—
191C4×C73—21no—
192C4×C73—42no—
193C4×C73—42no—
194C4×C73—81no—
195C4×C73—81no—
196C4×C73—161no—
197Dic73—42no—
198Dic73—81no—
199Dic73—81no—
200Dic73—81no—
201#10375—731nomaximal, Sylow
202C2×C73—21no—
203C2×C73—21no—
204C2×C73—21no—
205C2×C73—21no—
206C2×C73—21no—
207C2×C73—21no—
208C2×C73#202411yescharacteristic
209C2×C73#203111yescharacteristic
210C2×C73#203211yescharacteristic
211#2549—731no—
212#1996D7311yescharacteristic
213#1520—731no—
214C2×#328—731no—
215#1294—731no—
216#1289—731no—
217#1282—731no—
218C73#1037511yescharacteristic, complemented, Sylow
219C2×C22⋊Q8—1461no—
220C2×C22⋊Q8C2×D7311yescharacteristic
221C2×C22⋊C8—731no—
222C2×C22⋊C8—731no—
223#398—1461no—
224#395—1461no—
225#391—1461no—
226#389—1461no—
227#385—1461no—
228#380—731no—
229#380C2×D7311yescharacteristic
230C4×C22⋊C4C2×D7311yescharacteristic
231C2×#95—731no—
232#342—21no—
233#342—21no—
234#328—1461no—
235#328—1461no—
236#328—1461no—
237#328—1461no—
238C22×Q8—1461no—
239C22×Q8Dic14611yescharacteristic, complemented
240C23×C4—1461no—
241C23×C4C22×D7311yescharacteristic
242C22×C8—1461no—
243C22×C8—1461no—
244C22⋊Q8—21no—
245C22⋊Q8—21no—
246C22⋊Q8—21no—
247C22⋊Q8—21no—
248C22⋊Q8—2922no—
249C22⋊Q8—2922no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—1461no—
253C2×C4⋊C4—1461no—
254C2×C4⋊C4—1461no—
255C2×C4⋊C4—1461no—
256C2×C4⋊C4—1461no—
257C2×C4⋊C4—1461no—
258C2×C4⋊C4Dic14611yescharacteristic, complemented
259C2×C22⋊C4—21no—
260C2×C22⋊C4—21no—
261C2×C22⋊C4—1461no—
262C2×C22⋊C4C22⋊D7311yescharacteristic
263C2×C22⋊C4C22⋊D7311yescharacteristic
264C2×C42—21no—
265C22⋊C8—1461no—
266C22⋊C8—1461no—
267C22⋊C8—1461no—
268C22⋊C8—1461no—
269#95—21no—
270#95—21no—
271#95—731no—
272#95—731no—
273#95—731no—
274#95—731no—
275#95—1461no—
276#95—1461no—
277#95—1461no—
278C24C2×C22⋊D7311yescharacteristic
279C2×Q8—21no—
280C2×Q8—21no—
281C2×Q8—81no—
282C2×Q8—2922no—
283C2×Q8—2922no—
284C2×Q8—5841no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—41no—
290C22×C4—81no—
291C22×C4—2921no—
292C22×C4—2921no—
293C22×C4—2922no—
294C22×C4—5841no—
295C22×C4—1461no—
296C22×C4—1461no—
297C22×C4—1461no—
298C22×C4—1461no—
299C22×C4D4×D7311yescharacteristic
300C22×C4D4×D7311yescharacteristic
301C22×C4C2×Dic14611yescharacteristic
302C22×C4#1192402711yescharacteristic
303C22×C4#1192402711yescharacteristic
304C22×C4#1192402811yescharacteristic
305C2×C8—2921no—
306C2×C8—2921no—
307C2×C8—5841no—
308C2×C8—5841no—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—81no—
314C4⋊C4—2922no—
315C4⋊C4—2922no—
316C4⋊C4—2922no—
317C4⋊C4—2922no—
318C4⋊C4—5841no—
319C4⋊C4—5841no—
320C22⋊C4—21no—
321C22⋊C4—21no—
322C22⋊C4—41no—
323C22⋊C4—41no—
324C22⋊C4—41no—
325C22⋊C4—41no—
326C22⋊C4—2921no—
327C22⋊C4—2921no—
328C22⋊C4#1192401512yes—
329C22⋊C4#1192401812yes—
330C42—81no—
331C23—41no—
332C23—41no—
333C23#1455655911yescharacteristic
334C23#1455657011yescharacteristic
335C23#1455657811yescharacteristic
336C23C2×#1192401511yescharacteristic
337C23C2×#1192401811yescharacteristic
338C23#1455661611yescharacteristic
339C23#1455662111yescharacteristic
340Q8—81no—
341Q8—81no—
342Q8—5844no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—2921no—
363C2×C4—2921no—
364C2×C4—2921no—
365C2×C4—2921no—
366C2×C4—2921no—
367C2×C4—2921no—
368C2×C4—2921no—
369C2×C4—2921no—
370C2×C4—2921no—
371C2×C4—2921no—
372C2×C4—2921no—
373C2×C4—5842no—
374C2×C4—5842no—
375C2×C4—1462no—
376C2×C4#1455656811yescharacteristic
377C2×C4#1455656811yescharacteristic
378C2×C4#1455657711yescharacteristic
379C2×C4#1455657711yescharacteristic
380C8—5841no—
381C8—5841no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—41no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22#1798849911yescharacteristic
393C22#1798873711yescharacteristic
394C22#1798874211yescharacteristic
395C22#1798887211yescharacteristic
396C22#1798888211yescharacteristic
397C22#1798893711yescharacteristic
398C22#1798894611yescharacteristic
399C4—21no—
400C4—21no—
401C4—21no—
402C4—21no—
403C4—42no—
404C4—42no—
405C4—81no—
406C4—81no—
407C4—161no—
408C4—2922no—
409C4—5841no—
410C4—5841no—
411C4—5841no—
412C2—21no—
413C2—21no—
414C2—21no—
415C2—21no—
416C2—21no—
417C2—21no—
418C2#2261107211yescharacteristic
419C2#2261110511yescharacteristic
420C2#2261110811yescharacteristic
subgroupG/H
centerC22#17988499
commutator, cosocleC23×C4×C73C23
FrattiniC23×C4C22×D73
FittingC73×#1996C2
socleC22×C73#393
prime radicalC24×C73C2×D4
hypercenter#1996D73
hypocenterC73#10375
LCSG ▷ C23×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1996

Complements

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

NH ≅ G/Ntype
51. C22×Q8×C73Q8semidirect
59. C2×C4⋊C4×C73Q8semidirect
218. C73#10375semidirect
239. C22×Q8Dic146semidirect
258. C2×C4⋊C4Dic146semidirect

Maximal subgroups

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

subgroupindex
#226149372
#226085032
C2×#179883242
#226078092
#226077982
#226077792
C73×#19962
#1037573