Finite Group Databasebeta

← #33003052

Subgroups — #33003052

Order 18688.

554 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14556534C211yesmaximal, characteristic
2C2×#17988669C211yesmaximal, characteristic
3C2×#17988389C211yesmaximal, characteristic
4#22610793C211yesmaximal, characteristic, complemented
5#22608022C211yesmaximal, characteristic, complemented
6#22608008C211yesmaximal, characteristic, complemented
7C2×#358×C73C211yesmaximal, characteristic, complemented
8C22×D584C2211yescharacteristic
9C2×#14556553—21no—
10C2×#14556534—22no—
11C2×#14556534—22no—
12C2×#14556534C414yescomplemented
13C22×#11924003C2211yescharacteristic
14#17988669—22no—
15#17988669—22no—
16C2×#14556525C2211yescharacteristic
17C2×#14556515—21no—
18C2×#14556511—21no—
19C2×C8×Dic73C2211yescharacteristic
20#17988494—21no—
21#17988389—22no—
22C22×Q16×C73C2211yescharacteristic
23C22×C16×C73C2211yescharacteristic
24C2×#107×C73C2211yescharacteristic
25#358×C73—22no—
26C22×D292—21no—
27C22×C4×D73—41no—
28C2×D584—21no—
29C2×D584—21no—
30C2×D584—21no—
31C2×D584C2×C412yes—
32C2×D584D412yes—
33C2×D584D412yes—
34C22×#10596831D411yescharacteristic
35#14556553—24no—
36#14556553—42no—
37C2×C4⋊Dic73—21no—
38C2×#11924005—41no—
39C2×C4×Dic73D411yescharacteristic
40#14556534—28no—
41C2×#11924003—21no—
42C2×#11924003—21no—
43C2×#11924003C2×C412yescomplemented
44#14556525—21no—
45#14556525—21no—
46#14556515—42no—
47#14556511—24no—
48C8×Dic73—21no—
49C8×Dic73—21no—
50#14556498—41no—
51#14556498—41no—
52C22×Q8×C73—21no—
53C2×Q16×C73—21no—
54C2×Q16×C73—21no—
55C2×Q16×C73—21no—
56C2×Q16×C73—21no—
57C2×Q16×C73—21no—
58C2×Q16×C73C2×C412yescomplemented
59C22×C8×C73C2311yescharacteristic
60C2×C4⋊C4×C73—21no—
61C2×C16×C73—21no—
62C2×C16×C73—21no—
63C2×C16×C73—21no—
64#107×C73—21no—
65#107×C73D412yes—
66C23×D73—41no—
67C2×D292—22no—
68C2×D292—22no—
69C2×D292—22no—
70C2×D292—41no—
71C2×D292—41no—
72C2×D292—41no—
73C22×Dic73—41no—
74C22×Dic73D811yescharacteristic
75C22×Dic73D811yescharacteristic
76C2×C4×D73—44no—
77C2×C4×D73—82no—
78C2×C4×D73—82no—
79D584—22no—
80D584—22no—
81D584—22no—
82D584C22⋊C414yes—
83C2×#10596831—21no—
84C2×#10596831—21no—
85C2×#10596831—21no—
86C4⋊Dic73—41no—
87C4⋊Dic73—41no—
88#11924005—44no—
89C4×Dic73—21no—
90C4×Dic73—21no—
91#11924003—22no—
92C2×Q8×C73—22no—
93C2×Q8×C73—41no—
94C2×Q8×C73—41no—
95C2×Q8×C73—41no—
96C2×Q8×C73—41no—
97C2×Q8×C73—41no—
98C22×C4×C73—41no—
99C22×C4×C73—41no—
100C22×C4×C73C2×D411yescharacteristic
101Q16×C73—22no—
102Q16×C73—22no—
103Q16×C73—41no—
104Q16×C73—41no—
105C2×C8×C73C22×C411yescharacteristic
106C2×C8×C73C2×D411yescharacteristic
107C2×C8×C73C2×D411yescharacteristic
108C2×C8×C73C2×D411yescharacteristic
109C2×C8×C73#4111yescharacteristic
110C2×C8×C73#4111yescharacteristic
111C4⋊C4×C73—22no—
112C4⋊C4×C73—41no—
113C16×C73—41no—
114C23×C73C2×D811yescharacteristic
115C22×D73—42no—
116C22×D73—42no—
117C22×D73—42no—
118C22×D73—81no—
119C22×D73—81no—
120C22×D73—81no—
121C22×D73—81no—
122Q8×C73—42no—
123Q8×C73—42no—
124Q8×C73—81no—
125Q8×C73—81no—
126C2×C4×C73—42no—
127C2×C4×C73—42no—
128C2×C4×C73—81no—
129C2×C4×C73—81no—
130C2×C4×C73—81no—
131C2×C4×C73—81no—
132C2×C4×C73C4×D411yescharacteristic
133C2×C4×C73#12011yescharacteristic
134C2×C4×C73#12111yescharacteristic
135C2×C4×C73#13511yescharacteristic
136C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
137C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
138C2×Dic73—21no—
139C2×Dic73—21no—
140C2×Dic73—21no—
141C2×Dic73—21no—
142C2×Dic73—21no—
143C2×Dic73—81no—
144C2×Dic73—81no—
145C2×Dic73—81no—
146C2×Dic73SD3212yescomplemented
147D292—24no—
148D292—42no—
149D292—42no—
150D292—42no—
151C4×D73—88no—
152C8×C73C2×C22⋊C411yescharacteristic
153C8×C73C4×D411yescharacteristic
154C8×C73#12111yescharacteristic
155C8×C73#12311yescharacteristic
156#10596831—41no—
157C22×C73C4×D811yescharacteristic
158C22×C73#44711yescharacteristic
159C22×C73#45911yescharacteristic
160C22×C73C2×SD3211yescharacteristic
161C22×C73#50811yescharacteristic
162C22×C73#50911yescharacteristic
163C22×C73#51011yescharacteristic
164C2×D73—44no—
165C2×D73—82no—
166C2×D73—82no—
167C2×D73—82no—
168C2×D73—82no—
169C2×D73—82no—
170C2×D73—82no—
171C4×C73—82no—
172C4×C73—82no—
173C4×C73#38611yescharacteristic
174C4×C73#44211yescharacteristic
175C4×C73#44911yescharacteristic
176C4×C73#46311yescharacteristic
177Dic73—22no—
178Dic73—41no—
179Dic73—161no—
180#13014—731nomaximal, Sylow
181C2×C73#187911yescharacteristic
182C2×C73C4×SD3211yescharacteristic
183C2×C73#217511yescharacteristic
184C2×C73#218711yescharacteristic
185C2×C73#218911yescharacteristic
186C2×C73#220811yescharacteristic
187C2×C73#221011yescharacteristic
188D73—84no—
189C22×SD32—731no—
190C2×C4×D8—731no—
191C2×#358D7311yescharacteristic, complemented
192C2×#357—731no—
193#1969—731no—
194#1380—731no—
195#1374—731no—
196C73#1301411yescharacteristic, complemented, Sylow
197C22×Q16C2×D7311yescharacteristic
198C22×D8—731no—
199C2×C4×D4—1461no—
200C2×SD32—734no—
201C2×SD32—1462no—
202C2×SD32—1462no—
203C22×C16—731no—
204C22×C16C2×D7311yescharacteristic
205C4×D8—1462no—
206C4×D8—1462no—
207C2×#107—731no—
208C2×#107C2×D7311yescharacteristic
209C2×#103—1461no—
210C2×#102—1461no—
211C2×C4×C8—731no—
212#391—1461no—
213#358—22no—
214#357—1462no—
215C22×Q8—21no—
216C22×D4—1461no—
217C23×C4—2921no—
218C2×Q16—21no—
219C2×Q16—21no—
220C2×Q16—21no—
221C2×Q16—21no—
222C2×Q16—21no—
223C2×Q16C4×D7312yescomplemented
224C2×D8—732no—
225C2×D8—732no—
226C2×D8—732no—
227C2×D8—1461no—
228C2×D8—1461no—
229C2×D8—1461no—
230C22×C8—731no—
231C22×C8C22×D7311yescharacteristic
232C4×D4—2922no—
233C4×D4—1464no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—1461no—
236C2×C22⋊C4—2921no—
237C2×C42—731no—
238SD32—1468no—
239C2×C16—21no—
240C2×C16—21no—
241C2×C16—21no—
242C2×C16—732no—
243C2×C16—1461no—
244C2×C16—1461no—
245#107—21no—
246#107—1461no—
247#107—1461no—
248#107C22⋊D7312yes—
249#103—2922no—
250#102—1464no—
251C4×C8—1461no—
252C4×C8—1461no—
253#95—2921no—
254#95—2921no—
255C24—2921no—
256C2×Q8—22no—
257C2×Q8—41no—
258C2×Q8—41no—
259C2×Q8—41no—
260C2×Q8—41no—
261C2×Q8—41no—
262C2×D4—2921no—
263C2×D4—2921no—
264C2×D4—2921no—
265C2×D4—1462no—
266C2×D4—1462no—
267C2×D4—1462no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—2924no—
271C22×C4—2921no—
272C22×C4—5842no—
273C22×C4—5842no—
274C22×C4—731no—
275C22×C4—731no—
276C22×C4D4×D7311yescharacteristic
277Q16—22no—
278Q16—22no—
279Q16—41no—
280Q16—41no—
281D8—734no—
282D8—1462no—
283D8—1462no—
284D8—1462no—
285C2×C8—1461no—
286C2×C8—1461no—
287C2×C8—1461no—
288C2×C8C2×C4×D7311yescharacteristic
289C2×C8D4×D7311yescharacteristic
290C2×C8D4×D7311yescharacteristic
291C2×C8C2×C22⋊D7311yescharacteristic
292C2×C8#1192402711yescharacteristic
293C2×C8#1192402811yescharacteristic
294C4⋊C4—22no—
295C4⋊C4—41no—
296C4⋊C4—2921no—
297C4⋊C4—2921no—
298C22⋊C4—2924no—
299C42—1461no—
300C42—1461no—
301C16—41no—
302C16—1462no—
303C23—2922no—
304C23—2922no—
305C23—2922no—
306C23—5841no—
307C23—5841no—
308C23—5841no—
309C23—5841no—
310C23D8×D7311yescharacteristic
311Q8—42no—
312Q8—42no—
313Q8—81no—
314Q8—81no—
315D4—2922no—
316D4—2922no—
317D4—2922no—
318D4—1464no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4—81no—
324C2×C4—81no—
325C2×C4—5848no—
326C2×C4—5841no—
327C2×C4—5841no—
328C2×C4—5841no—
329C2×C4—732no—
330C2×C4—1461no—
331C2×C4—1461no—
332C2×C4—1461no—
333C2×C4—1461no—
334C2×C4—1461no—
335C2×C4#1455655111yescharacteristic
336C2×C4#1455655911yescharacteristic
337C2×C4#1455656411yescharacteristic
338C2×C4#1455661211yescharacteristic
339C2×C4#1455661511yescharacteristic
340C2×C4#1455661811yescharacteristic
341C8—2921no—
342C8C22⋊C4×D7311yescharacteristic
343C8C4×C22⋊D7311yescharacteristic
344C8#1455656211yescharacteristic
345C8#1455657311yescharacteristic
346C22—2924no—
347C22—5842no—
348C22—5842no—
349C22—5842no—
350C22—5842no—
351C22—5842no—
352C22—5842no—
353C22#1798866711yescharacteristic
354C22#1798871911yescharacteristic
355C22#1798878111yescharacteristic
356C22SD32×D7311yescharacteristic
357C22#1798902011yescharacteristic
358C22#1798902411yescharacteristic
359C22#1798902911yescharacteristic
360C4—82no—
361C4—82no—
362C4—2921no—
363C4—11681no—
364C4—1462no—
365C4#1798846311yescharacteristic
366C4#1798869611yescharacteristic
367C4#1798873111yescharacteristic
368C4#1798880111yescharacteristic
369C2—5844no—
370C2#2261027611yescharacteristic
371C2#2261178011yescharacteristic
372C2#2261179811yescharacteristic
373C2#2261185811yescharacteristic
374C2#2261187011yescharacteristic
375C2#2261197511yescharacteristic
376C2#2261198911yescharacteristic
subgroupG/H
centerC23D8×D73
commutatorC2×C8×C73C22×C4
FrattiniC22×C8C22×D73
FittingC2×#358×C73C2
socleC23×C73C2×D8
prime radicalC22×D584C22
cosocleC22×C8×C73C23
hypercenterC2×#358D73
hypocenterC73#13014
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8 ◁ C2×#358

Complements

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

NH ≅ G/Ntype
4. #22610793C2semidirect
5. #22608022C2semidirect
6. #22608008C2semidirect
7. C2×#358×C73C2semidirect
12. C2×#14556534C4semidirect
43. C2×#11924003C2×C4semidirect
58. C2×Q16×C73C2×C4semidirect
146. C2×Dic73SD32semidirect
191. C2×#358D73semidirect
196. C73#13014semidirect
223. C2×Q16C4×D73semidirect

Maximal subgroups

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

subgroupindex
C22×#145565342
C2×#179886692
C2×#179883892
#226107932
#226080222
#226080082
C2×#358×C732
#1301473