Finite Group Databasebeta

← #33003162

Subgroups — #33003162

Order 18688.

512 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988704C211yesmaximal, characteristic, complemented
2C2×#17988400C211yesmaximal, characteristic, complemented
3#22611643C211yesmaximal, characteristic, complemented
4#22610798C211yesmaximal, characteristic, complemented
5#22608040C211yesmaximal, characteristic, complemented
6#22608012C211yesmaximal, characteristic, complemented
7C2×#509×C73C211yesmaximal, characteristic
8C2×#14556598—21no—
9C2×#14556531C2211yescharacteristic
10#17988704—22no—
11#17988704—22no—
12C2×#14556527C2211yescharacteristic
13C2×#14556525C2211yescharacteristic
14C2×#14556518—21no—
15C2×#14556512—21no—
16C2×C8×Dic73C2211yescharacteristic
17#17988490—21no—
18#17988400—22no—
19#17988391—21no—
20#17988388—21no—
21C2×#135×C73C2211yescharacteristic
22C22×D8×C73C2211yescharacteristic
23#509×C73—22no—
24#509×C73C414yescomplemented
25C2×SD32×C73—21no—
26C2×D16×C73—21no—
27C2×M32×C73C2211yescharacteristic
28#14556598—24no—
29#14556598—42no—
30C2×#11924012—21no—
31C2×#11924012—41no—
32C22×#10596831—41no—
33C22×#10596831D411yescharacteristic
34C2×C4⋊Dic73—21no—
35C2×C22⋊Dic73—42no—
36C2×C4×Dic73D411yescharacteristic
37#14556531—21no—
38#14556531—21no—
39C2×#11924003—21no—
40#14556527—21no—
41#14556527D412yes—
42#14556525—21no—
43#14556525—21no—
44#14556518—24no—
45#14556512—41no—
46#14556512—41no—
47C8×Dic73—21no—
48C8×Dic73—21no—
49C2×#41×C73—21no—
50C22×D4×C73—21no—
51#135×C73—21no—
52#135×C73—21no—
53#135×C73C2×C412yes—
54#135×C73D412yes—
55C2×Q16×C73D411yescharacteristic
56C2×SD16×C73—21no—
57C2×D8×C73—21no—
58C2×D8×C73—21no—
59C2×D8×C73—21no—
60C2×D8×C73—21no—
61C2×D8×C73—21no—
62C2×D8×C73C2×C412yes—
63C2×D8×C73D411yescharacteristic
64C22×C8×C73C2311yescharacteristic
65SD32×C73—24no—
66D16×C73—24no—
67M32×C73—21no—
68M32×C73C2×C412yes—
69C2×C16×C73—21no—
70C22×Dic73—21no—
71C22×Dic73—41no—
72#11924012—22no—
73#11924012—44no—
74#11924012—41no—
75C2×#10596831—21no—
76C2×#10596831—21no—
77C2×#10596831—44no—
78C2×#10596831—41no—
79C2×#10596831D811yescharacteristic
80C2×#10596831D811yescharacteristic
81C4⋊Dic73—41no—
82C4⋊Dic73—41no—
83C22⋊Dic73—84no—
84C4×Dic73—21no—
85C4×Dic73—21no—
86#11924003—41no—
87C24×C73—41no—
88#41×C73—22no—
89#41×C73—22no—
90#41×C73—41no—
91#41×C73—41no—
92C2×Q8×C73—21no—
93C2×D4×C73—22no—
94C2×D4×C73—21no—
95C2×D4×C73—41no—
96C2×D4×C73—41no—
97C2×D4×C73—41no—
98C2×D4×C73—41no—
99C2×D4×C73—41no—
100C2×D4×C73—41no—
101C22×C4×C73—41no—
102C22×C4×C73C2×D411yescharacteristic
103Q16×C73—21no—
104Q16×C73C22⋊C412yescomplemented
105SD16×C73—22no—
106SD16×C73—22no—
107D8×C73—22no—
108D8×C73—22no—
109D8×C73—21no—
110D8×C73—41no—
111D8×C73—41no—
112D8×C73C22⋊C412yescomplemented
113C2×C8×C73C22×C411yescharacteristic
114C2×C8×C73C2×D411yescharacteristic
115C2×C8×C73C2×D411yescharacteristic
116C2×C8×C73C2×D411yescharacteristic
117C2×C8×C73#4111yescharacteristic
118C2×C8×C73#4111yescharacteristic
119C16×C73—22no—
120C23×C73—41no—
121C23×C73—42no—
122C23×C73—82no—
123C23×C73—82no—
124C23×C73—81no—
125C23×C73—81no—
126C23×C73AGL(1,ℤ/8)11yescharacteristic
127Q8×C73—22no—
128Q8×C73—41no—
129D4×C73—22no—
130D4×C73—42no—
131D4×C73—42no—
132D4×C73—42no—
133D4×C73—42no—
134D4×C73—41no—
135D4×C73—81no—
136D4×C73—81no—
137C2×C4×C73—41no—
138C2×C4×C73—42no—
139C2×C4×C73—42no—
140C2×C4×C73C4×D411yescharacteristic
141C2×C4×C73#12011yescharacteristic
142C2×C4×C73#12111yescharacteristic
143C2×C4×C73C2×D811yescharacteristic
144C2×C4×C73#13511yescharacteristic
145C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
146C2×Dic73—22no—
147C2×Dic73—41no—
148C2×Dic73—41no—
149C2×Dic73—82no—
150C2×Dic73—81no—
151C8×C73C2×C22⋊C411yescharacteristic
152C8×C73C4×D411yescharacteristic
153C8×C73#12111yescharacteristic
154C8×C73#12311yescharacteristic
155#10596831—21no—
156#10596831—21no—
157#10596831—42no—
158C22×C73—21no—
159C22×C73—21no—
160C22×C73—42no—
161C22×C73—84no—
162C22×C73—81no—
163C22×C73—81no—
164C22×C73—82no—
165C22×C73—162no—
166C22×C73—161no—
167C22×C73—161no—
168C22×C73#44211yescharacteristic
169C22×C73#44911yescharacteristic
170C22×C73#45911yescharacteristic
171C4×C73—42no—
172C4×C73#38611yescharacteristic
173C4×C73C4×D811yescharacteristic
174C4×C73#44711yescharacteristic
175C4×C73#46311yescharacteristic
176Dic73—42no—
177Dic73—161no—
178#13029—731nomaximal, Sylow
179C2×C73—21no—
180C2×C73—21no—
181C2×C73—82no—
182C2×C73—162no—
183C2×C73#187911yescharacteristic
184C2×C73#221211yescharacteristic
185C2×C73#221311yescharacteristic
186C2×#509D7311yescharacteristic
187C2×#443—731no—
188C2×#361—731no—
189#2139—731no—
190#1970—731no—
191#1387—731no—
192#1376—731no—
193C73#1302911yescharacteristic, complemented, Sylow
194C2×#135C2×D7311yescharacteristic
195C22×D8C2×D7311yescharacteristic
196C2×#131—1461no—
197#509—22no—
198#509Dic7314yescomplemented
199C2×SD32—21no—
200C2×D16—21no—
201C2×M32—731no—
202C2×M32C2×D7311yescharacteristic
203#443—1462no—
204#443—1462no—
205C2×#108—731no—
206C2×#107—731no—
207C2×#104—1461no—
208C2×#102—1461no—
209C2×C4×C8—731no—
210#390—1461no—
211#361—1462no—
212#358—1461no—
213#357—1461no—
214C2×#41—21no—
215C22×D4—21no—
216#135—21no—
217#135—21no—
218#135C2×Dic7312yes—
219#135C22⋊D7312yes—
220C2×Q16C22⋊D7311yescharacteristic
221C2×SD16—21no—
222C2×D8—21no—
223C2×D8—21no—
224C2×D8—21no—
225C2×D8—21no—
226C2×D8—21no—
227C2×D8C2×Dic7312yes—
228C2×D8C22⋊D7311yescharacteristic
229#131—2922no—
230#131—1464no—
231C2×M16—2921no—
232C2×M16—1461no—
233C22×C8—2921no—
234C22×C8—731no—
235C22×C8C22×D7311yescharacteristic
236C2×C4⋊C4—1461no—
237C2×C22⋊C4—2922no—
238C2×C42—731no—
239SD32—24no—
240D16—24no—
241M32—21no—
242M32—1461no—
243M32—1461no—
244M32C2×Dic7312yes—
245C2×C16—21no—
246C2×C16—1461no—
247#108—732no—
248#108—1461no—
249#107—1461no—
250#107—1461no—
251#104—1464no—
252#102—2921no—
253#102—2921no—
254C4×C8—1461no—
255C4×C8—1461no—
256C24—41no—
257#41—22no—
258#41—22no—
259#41—41no—
260#41—41no—
261C2×Q8—21no—
262C2×D4—22no—
263C2×D4—21no—
264C2×D4—41no—
265C2×D4—41no—
266C2×D4—41no—
267C2×D4—41no—
268C2×D4—41no—
269C2×D4—41no—
270C22×C4—41no—
271C22×C4—2921no—
272C22×C4—1461no—
273C22×C4D4×D7311yescharacteristic
274Q16—21no—
275Q16C22⋊Dic7312yescomplemented
276SD16—22no—
277SD16—22no—
278D8—22no—
279D8—22no—
280D8—21no—
281D8—41no—
282D8—41no—
283D8C22⋊Dic7312yescomplemented
284M16—2921no—
285M16—2924no—
286M16—1462no—
287C2×C8—2921no—
288C2×C8—2924no—
289C2×C8—731no—
290C2×C8—731no—
291C2×C8—1461no—
292C2×C8—1461no—
293C2×C8C22×Dic7311yescharacteristic
294C2×C8D4×D7311yescharacteristic
295C2×C8C2×C22⋊D7311yescharacteristic
296C2×C8C2×C22⋊D7311yescharacteristic
297C2×C8#1192402711yescharacteristic
298C2×C8#1192402711yescharacteristic
299C4⋊C4—2921no—
300C4⋊C4—2921no—
301C22⋊C4—5844no—
302C42—1461no—
303C42—1461no—
304C16—22no—
305C16—2921no—
306C23—41no—
307C23—42no—
308C23—82no—
309C23—82no—
310C23—81no—
311C23—81no—
312C23#1455661511yescharacteristic
313Q8—22no—
314Q8—41no—
315D4—22no—
316D4—42no—
317D4—42no—
318D4—42no—
319D4—42no—
320D4—41no—
321D4—81no—
322D4—81no—
323C2×C4—41no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—2921no—
327C2×C4—2921no—
328C2×C4—5842no—
329C2×C4—5841no—
330C2×C4—1462no—
331C2×C4D4×Dic7311yescharacteristic
332C2×C4#1455655911yescharacteristic
333C2×C4#1455656311yescharacteristic
334C2×C4D8×D7311yescharacteristic
335C2×C4#1455661011yescharacteristic
336C2×C4#1455661511yescharacteristic
337C8—2922no—
338C8—1461no—
339C8—1461no—
340C8C2×C22⋊Dic7311yescharacteristic
341C8D4×Dic7311yescharacteristic
342C8#1455656311yescharacteristic
343C8#1455657211yescharacteristic
344C22—21no—
345C22—21no—
346C22—42no—
347C22—84no—
348C22—81no—
349C22—81no—
350C22—82no—
351C22—162no—
352C22—161no—
353C22—161no—
354C22#1798869911yescharacteristic
355C22#1798872911yescharacteristic
356C22#1798878011yescharacteristic
357C4—42no—
358C4—2922no—
359C4—11681no—
360C4#1798846711yescharacteristic
361C4D8×Dic7311yescharacteristic
362C4#1798871911yescharacteristic
363C4#1798880011yescharacteristic
364C2—21no—
365C2—21no—
366C2—82no—
367C2—162no—
368C2#2261028111yescharacteristic
369C2#2261199811yescharacteristic
370C2#2261200511yescharacteristic
subgroupG/H
centerC22#17988780
commutatorC2×C8×C73C22×C4
FrattiniC22×C8C22×D73
Fitting, prime radicalC2×#509×C73C2
socleC22×C73#459
cosocleC22×C8×C73C23
hypercenterC2×#509D73
hypocenterC73#13029
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C2×#509

Complements

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

NH ≅ G/Ntype
1. C2×#17988704C2semidirect
2. C2×#17988400C2semidirect
3. #22611643C2semidirect
4. #22610798C2semidirect
5. #22608040C2semidirect
6. #22608012C2semidirect
24. #509×C73C4semidirect
104. Q16×C73C22⋊C4semidirect
112. D8×C73C22⋊C4semidirect
193. C73#13029semidirect
198. #509Dic73semidirect
275. Q16C22⋊Dic73semidirect
283. D8C22⋊Dic73semidirect

Maximal subgroups

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

subgroupindex
C2×#179887042
C2×#179884002
#226116432
#226107982
#226080402
#226080122
C2×#509×C732
#1302973