Finite Group Databasebeta

← #32982713

Subgroups — #32982713

Order 18688.

632 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22612657C211yesmaximal, characteristic, complemented
2#22610764C211yesmaximal, characteristic, complemented
3#22608599C211yesmaximal, characteristic, complemented
4#22608491C211yesmaximal, characteristic, complemented
5#22607790C211yesmaximal, characteristic, complemented
6#22607759C211yesmaximal, characteristic, complemented
7C73×#3205C211yesmaximal, characteristic
8C2×D4×Dic73—21no—
9#17988640C2211yescharacteristic
10#17988611C2211yescharacteristic
11C2×#14556518—21no—
12C2×#14556518—21no—
13#17988490—21no—
14#17988467—22no—
15#17988467—22no—
16#17988453C2211yescharacteristic
17#17988433C2211yescharacteristic
18#17988331—22no—
19#17988321—22no—
20#532×C73C2211yescharacteristic
21C2×#127×C73C2211yescharacteristic
22#493×C73C412yescomplemented
23#486×C73C412yescomplemented
24#459×C73—22no—
25#447×C73—22no—
26#431×C73C2211yescharacteristic
27C23×Dic73—22no—
28C2×#11924012—21no—
29D4×Dic73—42no—
30D4×Dic73—42no—
31C2×C4⋊Dic73—21no—
32C2×C22⋊Dic73—22no—
33C2×C22⋊Dic73—22no—
34C2×C22⋊Dic73—22no—
35C2×C4×Dic73D411yescharacteristic
36C2×C4×Dic73D411yescharacteristic
37C4⋊#10596831—21no—
38C4⋊#10596831—21no—
39C4⋊#10596831—21no—
40#14556518—42no—
41#14556518—42no—
42C22⋊#10596831—21no—
43C4×#10596831—21no—
44#14556498—21no—
45#14556498—21no—
46#14556498—21no—
47#14556498D412yes—
48C2×#41×C73—21no—
49C22×D4×C73—21no—
50C22×D4×C73—21no—
51C22×D4×C73D412yes—
52C2×D8×C73—24no—
53C4⋊Q8×C73C2×C411yescharacteristic, complemented
54#127×C73—21no—
55#127×C73—21no—
56#127×C73C2×C412yes—
57#127×C73C2×C411yescharacteristic, complemented
58#127×C73D412yes—
59#124×C73—21no—
60#121×C73—21no—
61#121×C73—21no—
62C4×D4×C73—21no—
63C4×D4×C73—21no—
64C2×C42×C73C2311yescharacteristic
65C4⋊C8×C73—21no—
66#102×C73—24no—
67C22⋊C8×C73—21no—
68C4×C8×C73C2×C412yescomplemented
69C22×Dic73—21no—
70C22×Dic73—21no—
71C22×Dic73—22no—
72C22×Dic73—44no—
73C22×Dic73—82no—
74C22×Dic73—82no—
75#11924012—41no—
76#11924012—41no—
77C2×#10596831—41no—
78C2×#10596831—41no—
79C4⋊Dic73—41no—
80C4⋊Dic73—41no—
81C22⋊Dic73—44no—
82C22⋊Dic73—82no—
83C22⋊Dic73—82no—
84C4×Dic73—21no—
85C4×Dic73—21no—
86C4×Dic73—21no—
87C4×Dic73D812yescomplemented
88C24×C73—22no—
89#41×C73—41no—
90#41×C73—41no—
91#41×C73—41no—
92#41×C73—41no—
93C2×Q8×C73—21no—
94C2×D4×C73—21no—
95C2×D4×C73—22no—
96C2×D4×C73—21no—
97C2×D4×C73—22no—
98C2×D4×C73—21no—
99C2×D4×C73—22no—
100C2×D4×C73—22no—
101C2×D4×C73—22no—
102C2×D4×C73—41no—
103C2×D4×C73—41no—
104C2×D4×C73—41no—
105C2×D4×C73—41no—
106C2×D4×C73—41no—
107C2×D4×C73—82no—
108C2×D4×C73—82no—
109C2×D4×C73C22⋊C414yes—
110C22×C4×C73—41no—
111C22×C4×C73C2×D411yescharacteristic
112C22×C4×C73C2×D411yescharacteristic
113C22×C4×C73C2×D411yescharacteristic
114D8×C73—84no—
115C2×C8×C73—22no—
116C4⋊C4×C73—21no—
117C4⋊C4×C73—21no—
118C22⋊C4×C73—41no—
119C22⋊C4×C73—41no—
120C42×C73C22×C411yescharacteristic
121C42×C73C2×D411yescharacteristic
122C42×C73#4111yescharacteristic
123C42×C73#4111yescharacteristic
124C23×C73—24no—
125C23×C73—42no—
126C23×C73—42no—
127C23×C73—42no—
128C23×C73—42no—
129C23×C73—42no—
130C23×C73—42no—
131C23×C73—41no—
132C23×C73#12011yescharacteristic
133Q8×C73—41no—
134Q8×C73—41no—
135D4×C73—44no—
136D4×C73—42no—
137D4×C73—42no—
138D4×C73—41no—
139D4×C73—42no—
140D4×C73—41no—
141D4×C73—42no—
142D4×C73—42no—
143D4×C73—82no—
144D4×C73—82no—
145D4×C73—82no—
146D4×C73—81no—
147D4×C73—81no—
148D4×C73—81no—
149C2×C4×C73—41no—
150C2×C4×C73—41no—
151C2×C4×C73—41no—
152C2×C4×C73—81no—
153C2×C4×C73—81no—
154C2×C4×C73C2×C22⋊C411yescharacteristic
155C2×C4×C73C4×D411yescharacteristic
156C2×C4×C73C4×D411yescharacteristic
157C2×C4×C73#12111yescharacteristic
158C2×C4×C73#12111yescharacteristic
159C2×C4×C73#12311yescharacteristic
160C2×C4×C73C2×D811yescharacteristic
161C2×C4×C73#13511yescharacteristic
162C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
163C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
164C2×Dic73—22no—
165C2×Dic73—41no—
166C2×Dic73—41no—
167C2×Dic73—41no—
168C2×Dic73—41no—
169C2×Dic73—42no—
170C2×Dic73—42no—
171C2×Dic73—42no—
172C2×Dic73—41no—
173C2×Dic73—88no—
174C8×C73—42no—
175#10596831—81no—
176#10596831—81no—
177C22×C73—21no—
178C22×C73—21no—
179C22×C73—44no—
180C22×C73—44no—
181C22×C73—44no—
182C22×C73—82no—
183C22×C73—82no—
184C22×C73—82no—
185C22×C73—82no—
186C22×C73—81no—
187C22×C73—81no—
188C22×C73—81no—
189C22×C73#38611yescharacteristic
190C22×C73#44711yescharacteristic
191C22×C73#44911yescharacteristic
192C4×C73—21no—
193C4×C73—21no—
194C4×C73—21no—
195C4×C73—21no—
196C4×C73—81no—
197C4×C73C4×D811yescharacteristic
198C4×C73#44211yescharacteristic
199C4×C73#45911yescharacteristic
200C4×C73#46311yescharacteristic
201Dic73—42no—
202Dic73—81no—
203Dic73—82no—
204#9299—731nomaximal, Sylow
205C2×C73—21no—
206C2×C73—21no—
207C2×C73—84no—
208C2×C73—161no—
209C2×C73#187911yescharacteristic
210C2×C73#190411yescharacteristic
211C2×C73#191111yescharacteristic
212#3205D7311yescharacteristic
213#2328—731no—
214#1963—731no—
215#1545—731no—
216#1517—731no—
217#1286—731no—
218#1274—731no—
219C73#929911yescharacteristic, complemented, Sylow
220#532C2×D7311yescharacteristic
221C2×#127C2×D7311yescharacteristic
222C2×C4×D4—1461no—
223#493Dic7312yescomplemented
224#486Dic7312yescomplemented
225#459—22no—
226#447—22no—
227#431—731no—
228#431C2×D7311yescharacteristic
229#423—731no—
230C2×#104—1461no—
231C2×#104—1461no—
232#390—1461no—
233#386—1462no—
234#386—1462no—
235#382—731no—
236#376—731no—
237#331—1462no—
238#327—1462no—
239C2×#41—21no—
240C22×D4—21no—
241C22×D4—21no—
242C22×D4C22⋊D7312yes—
243C23×C4—1462no—
244C2×D8—24no—
245C2×M16—1461no—
246C4⋊Q8C2×Dic7311yescharacteristic, complemented
247#127—21no—
248#127—21no—
249#127C2×Dic7312yes—
250#127C2×Dic7311yescharacteristic, complemented
251#127C22⋊D7312yes—
252#124—21no—
253#121—21no—
254#121—21no—
255C4×D4—21no—
256C4×D4—21no—
257C4×D4—2922no—
258C4×D4—2922no—
259C2×C4⋊C4—1461no—
260C2×C22⋊C4—1462no—
261C2×C22⋊C4—1462no—
262C2×C22⋊C4—1462no—
263C2×C42—731no—
264C2×C42—731no—
265C2×C42C22×D7311yescharacteristic
266C4⋊C8—21no—
267C4⋊C8—1461no—
268C4⋊C8—1461no—
269C4⋊C8—1461no—
270#104—2922no—
271#104—2922no—
272#102—24no—
273C22⋊C8—21no—
274C22⋊C8—1461no—
275C4×C8—1461no—
276C4×C8C2×Dic7312yescomplemented
277#95—732no—
278#95—1461no—
279#95—1461no—
280#95—1461no—
281C24—22no—
282#41—41no—
283#41—41no—
284#41—41no—
285#41—41no—
286C2×Q8—21no—
287C2×D4—21no—
288C2×D4—22no—
289C2×D4—21no—
290C2×D4—22no—
291C2×D4—21no—
292C2×D4—22no—
293C2×D4—22no—
294C2×D4—22no—
295C2×D4—41no—
296C2×D4—41no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—41no—
300C2×D4—82no—
301C2×D4—82no—
302C2×D4C22⋊Dic7314yes—
303C22×C4—41no—
304C22×C4—2924no—
305C22×C4—5842no—
306C22×C4—5842no—
307C22×C4—1462no—
308C22×C4—1461no—
309C22×C4—1461no—
310C22×C4D4×D7311yescharacteristic
311C22×C4D4×D7311yescharacteristic
312C22×C4C2×C22⋊D7311yescharacteristic
313D8—84no—
314M16—2921no—
315M16—2921no—
316C2×C8—22no—
317C2×C8—2921no—
318C2×C8—2921no—
319C4⋊C4—21no—
320C4⋊C4—21no—
321C4⋊C4—2921no—
322C4⋊C4—2921no—
323C22⋊C4—41no—
324C22⋊C4—41no—
325C22⋊C4—2924no—
326C22⋊C4—5842no—
327C22⋊C4—5842no—
328C42—732no—
329C42—1461no—
330C42—1461no—
331C42—1461no—
332C42C22×Dic7311yescharacteristic
333C42C2×C22⋊D7311yescharacteristic
334C42#1192402711yescharacteristic
335C42#1192402711yescharacteristic
336C23—24no—
337C23—42no—
338C23—42no—
339C23—42no—
340C23—42no—
341C23—42no—
342C23—42no—
343C23—41no—
344C23#1455655911yescharacteristic
345Q8—41no—
346Q8—41no—
347D4—44no—
348D4—42no—
349D4—42no—
350D4—41no—
351D4—42no—
352D4—41no—
353D4—42no—
354D4—42no—
355D4—82no—
356D4—82no—
357D4—82no—
358D4—81no—
359D4—81no—
360D4—81no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—2921no—
367C2×C4—2921no—
368C2×C4—2921no—
369C2×C4—2921no—
370C2×C4—2922no—
371C2×C4—2922no—
372C2×C4—2922no—
373C2×C4—2921no—
374C2×C4—5848no—
375C2×C4—1462no—
376C2×C4C2×C22⋊Dic7311yescharacteristic
377C2×C4D4×Dic7311yescharacteristic
378C2×C4D4×Dic7311yescharacteristic
379C2×C4#1455656311yescharacteristic
380C2×C4#1455656311yescharacteristic
381C2×C4#1455657211yescharacteristic
382C2×C4D8×D7311yescharacteristic
383C2×C4#1455661011yescharacteristic
384C2×C4#1455661511yescharacteristic
385C2×C4#1455661511yescharacteristic
386C8—42no—
387C8—5841no—
388C8—5841no—
389C22—21no—
390C22—21no—
391C22—44no—
392C22—44no—
393C22—44no—
394C22—82no—
395C22—82no—
396C22—82no—
397C22—82no—
398C22—81no—
399C22—81no—
400C22—81no—
401C22#1798846711yescharacteristic
402C22#1798871911yescharacteristic
403C22#1798872911yescharacteristic
404C4—21no—
405C4—21no—
406C4—21no—
407C4—21no—
408C4—81no—
409C4—2922no—
410C4—5842no—
411C4—5841no—
412C4D8×Dic7311yescharacteristic
413C4#1798869911yescharacteristic
414C4#1798878011yescharacteristic
415C4#1798880011yescharacteristic
416C2—21no—
417C2—21no—
418C2—84no—
419C2—161no—
420C2#2261028111yescharacteristic
421C2#2261042011yescharacteristic
422C2#2261046411yescharacteristic
subgroupG/H
centerC22#17988467
commutatorC42×C73C22×C4
FrattiniC2×C42C22×D73
Fitting, prime radicalC73×#3205C2
socleC22×C73#386
cosocleC2×C42×C73C23
hypercenter#3205D73
hypocenterC73#9299
LCSG ▷ C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3205

Complements

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

NH ≅ G/Ntype
1. #22612657C2semidirect
2. #22610764C2semidirect
3. #22608599C2semidirect
4. #22608491C2semidirect
5. #22607790C2semidirect
6. #22607759C2semidirect
22. #493×C73C4semidirect
23. #486×C73C4semidirect
53. C4⋊Q8×C73C2×C4semidirect
57. #127×C73C2×C4semidirect
68. C4×C8×C73C2×C4semidirect
87. C4×Dic73D8semidirect
219. C73#9299semidirect
223. #493Dic73semidirect
224. #486Dic73semidirect
246. C4⋊Q8C2×Dic73semidirect
250. #127C2×Dic73semidirect
276. C4×C8C2×Dic73semidirect

Maximal subgroups

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

subgroupindex
#226126572
#226107642
#226085992
#226084912
#226077902
#226077592
C73×#32052
#929973