Finite Group Databasebeta

← #33003239

Subgroups — #33003239

Order 18688.

508 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988709C211yesmaximal, characteristic, complemented
2C2×#17988400C211yesmaximal, characteristic, complemented
3C2×#17988397C211yesmaximal, characteristic, complemented
4#22610808C211yesmaximal, characteristic, complemented
5#22608030C211yesmaximal, characteristic, complemented
6#22608010C211yesmaximal, characteristic, complemented
7C2×#509×C73C211yesmaximal, characteristic
8C2×#14556598—21no—
9C2×#14556531C2211yescharacteristic
10#17988709—22no—
11#17988709—22no—
12C2×#14556527C2211yescharacteristic
13C2×#14556523C2211yescharacteristic
14C2×#14556518—21no—
15C2×#14556512—21no—
16C2×#14556501C2211yescharacteristic
17#17988490—21no—
18#17988400—22no—
19#17988397—22no—
20C2×#135×C73C2211yescharacteristic
21C22×D8×C73C2211yescharacteristic
22#509×C73—22no—
23#509×C73—22no—
24C2×SD32×C73C412yescomplemented
25C2×D16×C73C412yescomplemented
26C2×M32×C73C2211yescharacteristic
27#14556598—24no—
28#14556598—42no—
29C2×#11924012—21no—
30C2×#11924012—41no—
31C22×#10596831—41no—
32C22×#10596831D411yescharacteristic
33C2×C4⋊Dic73—21no—
34C2×C22⋊Dic73—42no—
35C2×C4×Dic73D411yescharacteristic
36#14556531—21no—
37#14556531—21no—
38C2×#11924003—21no—
39#14556527—21no—
40#14556527D412yes—
41#14556523—21no—
42#14556523—21no—
43#14556518—24no—
44#14556512—42no—
45#14556501—21no—
46#14556501—21no—
47C2×#41×C73—21no—
48C22×D4×C73—21no—
49#135×C73—21no—
50#135×C73—21no—
51#135×C73D412yes—
52#135×C73D412yes—
53C2×Q16×C73C2×C411yescharacteristic
54C2×SD16×C73—21no—
55C2×D8×C73—21no—
56C2×D8×C73—21no—
57C2×D8×C73—21no—
58C2×D8×C73—21no—
59C2×D8×C73—21no—
60C2×D8×C73C2×C412yes—
61C2×D8×C73C2×C411yescharacteristic
62C22×C8×C73C2311yescharacteristic
63SD32×C73—24no—
64D16×C73—24no—
65M32×C73—21no—
66M32×C73—21no—
67C2×C16×C73C2×C412yes—
68C22×Dic73—21no—
69C22×Dic73—41no—
70#11924012—22no—
71#11924012—44no—
72#11924012—41no—
73C2×#10596831—21no—
74C2×#10596831—21no—
75C2×#10596831—44no—
76C2×#10596831—41no—
77C2×#10596831D812yes—
78C4⋊Dic73—41no—
79C4⋊Dic73—41no—
80C22⋊Dic73—84no—
81C4×Dic73—21no—
82C4×Dic73—21no—
83#11924003—41no—
84C24×C73—41no—
85#41×C73—22no—
86#41×C73—22no—
87#41×C73—41no—
88#41×C73—41no—
89C2×Q8×C73—21no—
90C2×D4×C73—21no—
91C2×D4×C73—22no—
92C2×D4×C73—41no—
93C2×D4×C73—41no—
94C2×D4×C73—41no—
95C2×D4×C73—41no—
96C2×D4×C73—41no—
97C2×D4×C73—41no—
98C22×C4×C73—41no—
99C22×C4×C73C2×D411yescharacteristic
100Q16×C73—21no—
101Q16×C73C22⋊C412yescomplemented
102SD16×C73—22no—
103SD16×C73—22no—
104D8×C73—22no—
105D8×C73—21no—
106D8×C73—22no—
107D8×C73—41no—
108D8×C73—41no—
109D8×C73C22⋊C412yescomplemented
110C2×C8×C73C22×C411yescharacteristic
111C2×C8×C73C2×D411yescharacteristic
112C2×C8×C73C2×D411yescharacteristic
113C2×C8×C73C2×D411yescharacteristic
114C2×C8×C73#4111yescharacteristic
115C2×C8×C73#4111yescharacteristic
116C16×C73—22no—
117C23×C73—41no—
118C23×C73—42no—
119C23×C73—82no—
120C23×C73—82no—
121C23×C73—81no—
122C23×C73—81no—
123C23×C73AGL(1,ℤ/8)11yescharacteristic
124Q8×C73—22no—
125Q8×C73—41no—
126D4×C73—22no—
127D4×C73—42no—
128D4×C73—42no—
129D4×C73—42no—
130D4×C73—41no—
131D4×C73—42no—
132D4×C73—81no—
133D4×C73—81no—
134C2×C4×C73—41no—
135C2×C4×C73—42no—
136C2×C4×C73—42no—
137C2×C4×C73C4×D411yescharacteristic
138C2×C4×C73#12011yescharacteristic
139C2×C4×C73#12111yescharacteristic
140C2×C4×C73C2×D811yescharacteristic
141C2×C4×C73#13511yescharacteristic
142C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
143C2×Dic73—42no—
144C2×Dic73—41no—
145C2×Dic73—82no—
146C2×Dic73—81no—
147C8×C73C2×C22⋊C411yescharacteristic
148C8×C73C4×D411yescharacteristic
149C8×C73#12111yescharacteristic
150C8×C73#12311yescharacteristic
151#10596831—22no—
152#10596831—42no—
153C22×C73—22no—
154C22×C73—42no—
155C22×C73—82no—
156C22×C73—82no—
157C22×C73—84no—
158C22×C73—162no—
159C22×C73—161no—
160C22×C73—161no—
161C22×C73#44211yescharacteristic
162C22×C73#44911yescharacteristic
163C22×C73#45911yescharacteristic
164C4×C73—42no—
165C4×C73#38611yescharacteristic
166C4×C73C4×D811yescharacteristic
167C4×C73#44711yescharacteristic
168C4×C73#46311yescharacteristic
169Dic73—81no—
170Dic73—161no—
171#13040—731nomaximal, Sylow
172C2×C73—21no—
173C2×C73—21no—
174C2×C73—82no—
175C2×C73—162no—
176C2×C73#187911yescharacteristic
177C2×C73AGL(1,ℤ/16)12yes—
178C2×#509D7311yescharacteristic
179C2×#444—731no—
180C2×#361—731no—
181C2×#360—731no—
182#1972—731no—
183#1383—731no—
184#1375—731no—
185C73#1304011yescharacteristic, complemented, Sylow
186C2×#135C2×D7311yescharacteristic
187C22×D8C2×D7311yescharacteristic
188C2×#131—1461no—
189#509—22no—
190#509—22no—
191C2×SD32Dic7312yescomplemented
192C2×D16Dic7312yescomplemented
193C2×M32—731no—
194C2×M32C2×D7311yescharacteristic
195#444—1462no—
196#444—1462no—
197C2×#108—731no—
198C2×#106—731no—
199C2×#104—1461no—
200C2×#102—1461no—
201C2×#97—731no—
202#390—1461no—
203#361—1462no—
204#360—1462no—
205C2×#41—21no—
206C22×D4—21no—
207#135—21no—
208#135—21no—
209#135C22⋊D7312yes—
210#135C22⋊D7312yes—
211C2×Q16C2×Dic7311yescharacteristic
212C2×SD16—21no—
213C2×D8—21no—
214C2×D8—21no—
215C2×D8—21no—
216C2×D8—21no—
217C2×D8—21no—
218C2×D8C2×Dic7312yes—
219C2×D8C2×Dic7311yescharacteristic
220#131—2922no—
221#131—1464no—
222C2×M16—2921no—
223C2×M16—1461no—
224C22×C8—2921no—
225C22×C8—731no—
226C22×C8C22×D7311yescharacteristic
227C2×C4⋊C4—1461no—
228C2×C22⋊C4—2922no—
229C2×C42—731no—
230SD32—24no—
231D16—24no—
232M32—21no—
233M32—21no—
234M32—1461no—
235M32—1461no—
236C2×C16—1461no—
237C2×C16C2×Dic7312yes—
238#108—732no—
239#108—1461no—
240#106—1461no—
241#106—1461no—
242#104—1464no—
243#102—2922no—
244#97—1461no—
245#97—1461no—
246C24—41no—
247#41—22no—
248#41—22no—
249#41—41no—
250#41—41no—
251C2×Q8—21no—
252C2×D4—21no—
253C2×D4—22no—
254C2×D4—41no—
255C2×D4—41no—
256C2×D4—41no—
257C2×D4—41no—
258C2×D4—41no—
259C2×D4—41no—
260C22×C4—41no—
261C22×C4—2921no—
262C22×C4—1461no—
263C22×C4D4×D7311yescharacteristic
264Q16—21no—
265Q16C22⋊Dic7312yescomplemented
266SD16—22no—
267SD16—22no—
268D8—22no—
269D8—21no—
270D8—22no—
271D8—41no—
272D8—41no—
273D8C22⋊Dic7312yescomplemented
274M16—2921no—
275M16—2924no—
276M16—1462no—
277C2×C8—2921no—
278C2×C8—2924no—
279C2×C8—732no—
280C2×C8—1461no—
281C2×C8—1461no—
282C2×C8C22×Dic7311yescharacteristic
283C2×C8D4×D7311yescharacteristic
284C2×C8C2×C22⋊D7311yescharacteristic
285C2×C8C2×C22⋊D7311yescharacteristic
286C2×C8#1192402711yescharacteristic
287C2×C8#1192402711yescharacteristic
288C4⋊C4—2921no—
289C4⋊C4—2921no—
290C22⋊C4—5844no—
291C42—1461no—
292C42—1461no—
293C16—22no—
294C16—2921no—
295C23—41no—
296C23—42no—
297C23—82no—
298C23—82no—
299C23—81no—
300C23—81no—
301C23#1455661511yescharacteristic
302Q8—22no—
303Q8—41no—
304D4—22no—
305D4—42no—
306D4—42no—
307D4—41no—
308D4—42no—
309D4—42no—
310D4—81no—
311D4—81no—
312C2×C4—41no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—2922no—
316C2×C4—2921no—
317C2×C4—5842no—
318C2×C4—5841no—
319C2×C4D4×Dic7311yescharacteristic
320C2×C4#1455655911yescharacteristic
321C2×C4#1455656311yescharacteristic
322C2×C4D8×D7311yescharacteristic
323C2×C4#1455661011yescharacteristic
324C2×C4#1455661511yescharacteristic
325C8—2922no—
326C8—1462no—
327C8C2×C22⋊Dic7311yescharacteristic
328C8D4×Dic7311yescharacteristic
329C8#1455656311yescharacteristic
330C8#1455657211yescharacteristic
331C22—22no—
332C22—42no—
333C22—82no—
334C22—82no—
335C22—84no—
336C22—162no—
337C22—161no—
338C22—161no—
339C22#1798869911yescharacteristic
340C22#1798872911yescharacteristic
341C22#1798878011yescharacteristic
342C4—42no—
343C4—5841no—
344C4—11681no—
345C4#1798846711yescharacteristic
346C4D8×Dic7311yescharacteristic
347C4#1798871911yescharacteristic
348C4#1798880011yescharacteristic
349C2—21no—
350C2—21no—
351C2—82no—
352C2—162no—
353C2#2261028111yescharacteristic
354C2#2261183012yes—
subgroupG/H
centerC22#17988699
commutatorC2×C8×C73C22×C4
FrattiniC22×C8C22×D73
Fitting, prime radicalC2×#509×C73C2
socleC22×C73#442
cosocleC22×C8×C73C23
hypercenterC2×#509D73
hypocenterC73#13040
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×#17988709C2semidirect
2. C2×#17988400C2semidirect
3. C2×#17988397C2semidirect
4. #22610808C2semidirect
5. #22608030C2semidirect
6. #22608010C2semidirect
24. C2×SD32×C73C4semidirect
25. C2×D16×C73C4semidirect
101. Q16×C73C22⋊C4semidirect
109. D8×C73C22⋊C4semidirect
185. C73#13040semidirect
191. C2×SD32Dic73semidirect
192. C2×D16Dic73semidirect
265. Q16C22⋊Dic73semidirect
273. D8C22⋊Dic73semidirect

Maximal subgroups

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

subgroupindex
C2×#179887092
C2×#179884002
C2×#179883972
#226108082
#226080302
#226080102
C2×#509×C732
#1304073