Finite Group Databasebeta

← #33003237

Subgroups — #33003237

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×#17989022C211yesmaximal, characteristic
2C2×#17988399C211yesmaximal, characteristic, complemented
3C2×#17988397C211yesmaximal, characteristic, complemented
4#22610807C211yesmaximal, characteristic, complemented
5#22608031C211yesmaximal, characteristic, complemented
6#22608011C211yesmaximal, characteristic, complemented
7C2×#444×C73C211yesmaximal, characteristic, complemented
8C22×D584C2211yescharacteristic
9#17989022—22no—
10#17989022—22no—
11C2×#14556534C412yescomplemented
12C2×#14556532C412yescomplemented
13C2×#14556531C2211yescharacteristic
14C2×#14556531C2211yescharacteristic
15C2×#14556523C2211yescharacteristic
16C2×#14556513—21no—
17#17988489—21no—
18#17988399—22no—
19#17988397—22no—
20C2×#135×C73C2211yescharacteristic
21C2×#131×C73—21no—
22#444×C73—22no—
23#444×C73—22no—
24C2×#108×C73C2211yescharacteristic
25C2×#104×C73—21no—
26C2×#97×C73C2211yescharacteristic
27C22×D292—21no—
28C2×D584—21no—
29C2×D584—21no—
30C2×D584—21no—
31C2×D584—21no—
32C2×D584—21no—
33C2×D584C2×C412yes—
34C2×C4⋊Dic73—21no—
35C2×#11924005—42no—
36#14556534—24no—
37#14556532—24no—
38#14556531—21no—
39#14556531—21no—
40#14556531—21no—
41#14556531—21no—
42C2×#11924003—21no—
43C2×#11924003C2×C412yes—
44#14556523—21no—
45#14556523—21no—
46#14556513—42no—
47C2×#41×C73—21no—
48#135×C73—21no—
49#135×C73—21no—
50#135×C73D412yes—
51#135×C73D412yes—
52C2×Q16×C73C2×C411yescharacteristic
53C2×SD16×C73—21no—
54C2×D8×C73C2×C411yescharacteristic
55#131×C73—24no—
56#131×C73—42no—
57C2×M16×C73—21no—
58C2×M16×C73—41no—
59C22×C8×C73—41no—
60C22×C8×C73D411yescharacteristic
61C22×C8×C73C2311yescharacteristic
62C2×C42×C73D411yescharacteristic
63#108×C73—21no—
64#108×C73D412yes—
65#104×C73—24no—
66#97×C73—21no—
67#97×C73—21no—
68C23×D73—41no—
69C2×D292—22no—
70C2×D292—41no—
71C2×D292—41no—
72C2×D292—41no—
73C2×D292—41no—
74C2×D292—41no—
75C22×Dic73—41no—
76D584—22no—
77D584—22no—
78D584—41no—
79D584—41no—
80C4⋊Dic73—41no—
81C4⋊Dic73—41no—
82#11924005—84no—
83#11924003—22no—
84#11924003—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—41no—
92C22×C4×C73—21no—
93C22×C4×C73—41no—
94C22×C4×C73C2×D411yescharacteristic
95Q16×C73—21no—
96Q16×C73C22⋊C412yescomplemented
97SD16×C73—22no—
98SD16×C73—22no—
99D8×C73—21no—
100D8×C73C22⋊C412yescomplemented
101M16×C73—22no—
102M16×C73—44no—
103M16×C73—41no—
104C2×C8×C73—21no—
105C2×C8×C73—21no—
106C2×C8×C73—44no—
107C2×C8×C73—41no—
108C2×C8×C73D812yes—
109C2×C8×C73C22×C411yescharacteristic
110C2×C8×C73C2×D411yescharacteristic
111C2×C8×C73C2×D411yescharacteristic
112C2×C8×C73C2×D411yescharacteristic
113C2×C8×C73#4111yescharacteristic
114C2×C8×C73#4111yescharacteristic
115C42×C73—21no—
116C42×C73—21no—
117C23×C73—41no—
118C23×C73AGL(1,ℤ/8)11yescharacteristic
119C22×D73—42no—
120C22×D73—82no—
121C22×D73—82no—
122C22×D73—81no—
123C22×D73—81no—
124Q8×C73—22no—
125Q8×C73—41no—
126D4×C73—22no—
127D4×C73—42no—
128D4×C73—42no—
129D4×C73—41no—
130C2×C4×C73—41no—
131C2×C4×C73—42no—
132C2×C4×C73—42no—
133C2×C4×C73—42no—
134C2×C4×C73—41no—
135C2×C4×C73C4×D411yescharacteristic
136C2×C4×C73#12011yescharacteristic
137C2×C4×C73#12111yescharacteristic
138C2×C4×C73C2×D811yescharacteristic
139C2×C4×C73#13511yescharacteristic
140C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
141C2×Dic73—82no—
142C2×Dic73—81no—
143D292—42no—
144D292—42no—
145D292—81no—
146D292—81no—
147C8×C73—22no—
148C8×C73—42no—
149C8×C73C2×C22⋊C411yescharacteristic
150C8×C73C4×D411yescharacteristic
151C8×C73#12111yescharacteristic
152C8×C73#12311yescharacteristic
153C22×C73—22no—
154C22×C73—42no—
155C22×C73—82no—
156C22×C73#44211yescharacteristic
157C22×C73#44911yescharacteristic
158C22×C73#45911yescharacteristic
159C2×D73—82no—
160C2×D73—84no—
161C2×D73—161no—
162C2×D73—161no—
163C2×D73—162no—
164C4×C73—42no—
165C4×C73—81no—
166C4×C73#38611yescharacteristic
167C4×C73C4×D811yescharacteristic
168C4×C73#44711yescharacteristic
169C4×C73#46311yescharacteristic
170Dic73—161no—
171#13040—731nomaximal, Sylow
172C2×C73—21no—
173C2×C73—21no—
174C2×C73—82no—
175C2×C73#187911yescharacteristic
176C2×C73AGL(1,ℤ/16)12yes—
177D73—162no—
178C2×#509—731no—
179C2×#444D7311yescharacteristic, complemented
180C2×#361—731no—
181C2×#360—731no—
182#1972—731no—
183#1383—731no—
184#1375—731no—
185C73#1304011yescharacteristic, complemented, Sylow
186C2×#135C2×D7311yescharacteristic
187C22×D8—731no—
188C2×#131—21no—
189#509—1462no—
190#509—1462no—
191C2×SD32—732no—
192C2×D16—732no—
193C2×M32—731no—
194C2×M32—731no—
195#444—22no—
196#444—22no—
197C2×#108C2×D7311yescharacteristic
198C2×#106—731no—
199C2×#104—21no—
200C2×#102—1461no—
201C2×#97C2×D7311yescharacteristic
202#390—1461no—
203#361—1462no—
204#360—1462no—
205C2×#41—21no—
206C22×D4—1461no—
207#135—21no—
208#135—21no—
209#135D29212yes—
210#135C22⋊D7312yes—
211C2×Q16C4×D7311yescharacteristic
212C2×SD16—21no—
213C2×D8—732no—
214C2×D8—1461no—
215C2×D8—1461no—
216C2×D8—1461no—
217C2×D8—1461no—
218C2×D8—1461no—
219C2×D8C4×D7311yescharacteristic
220#131—24no—
221#131—42no—
222C2×M16—21no—
223C2×M16—41no—
224C22×C8—41no—
225C22×C8C22⋊D7311yescharacteristic
226C22×C8C22×D7311yescharacteristic
227C2×C4⋊C4—1461no—
228C2×C22⋊C4—2922no—
229C2×C42C22⋊D7311yescharacteristic
230SD32—1464no—
231D16—1464no—
232M32—1461no—
233M32—1461no—
234M32—1461no—
235M32—1461no—
236C2×C16—732no—
237C2×C16—1461no—
238#108—21no—
239#108C22⋊D7312yes—
240#106—1461no—
241#106—1461no—
242#104—24no—
243#102—2922no—
244#97—21no—
245#97—21no—
246C24—2921no—
247#41—22no—
248#41—22no—
249#41—41no—
250#41—41no—
251C2×Q8—21no—
252C2×D4—21no—
253C2×D4—41no—
254C2×D4—2921no—
255C2×D4—2921no—
256C2×D4—2921no—
257C2×D4—2921no—
258C2×D4—2921no—
259C2×D4—1462no—
260C22×C4—21no—
261C22×C4—41no—
262C22×C4—2921no—
263C22×C4C2×C22⋊D7311yescharacteristic
264Q16—21no—
265Q16#1192400512yescomplemented
266SD16—22no—
267SD16—22no—
268D8—21no—
269D8—2921no—
270D8—2921no—
271D8—1462no—
272D8—1462no—
273D8#1192400512yescomplemented
274M16—22no—
275M16—44no—
276M16—41no—
277C2×C8—21no—
278C2×C8—21no—
279C2×C8—44no—
280C2×C8—41no—
281C2×C8#1192401312yes—
282C2×C8C2×C4×D7311yescharacteristic
283C2×C8C2×C22⋊D7311yescharacteristic
284C2×C8C2×C22⋊D7311yescharacteristic
285C2×C8C2×D29211yescharacteristic
286C2×C8#1192402811yescharacteristic
287C2×C8#1192402811yescharacteristic
288C4⋊C4—2921no—
289C4⋊C4—2921no—
290C22⋊C4—5844no—
291C42—21no—
292C42—21no—
293C16—2921no—
294C16—1462no—
295C23—41no—
296C23—2922no—
297C23—5842no—
298C23—5842no—
299C23—5841no—
300C23—5841no—
301C23#1455661611yescharacteristic
302Q8—22no—
303Q8—41no—
304D4—22no—
305D4—42no—
306D4—42no—
307D4—41no—
308D4—2922no—
309D4—2922no—
310D4—5841no—
311D4—5841no—
312C2×C4—41no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—41no—
317C2×C4—5842no—
318C2×C4—5841no—
319C2×C4C4×C22⋊D7311yescharacteristic
320C2×C4#1455656011yescharacteristic
321C2×C4#1455656511yescharacteristic
322C2×C4C2×#1192401311yescharacteristic
323C2×C4#1455661111yescharacteristic
324C2×C4#1455661711yescharacteristic
325C8—22no—
326C8—42no—
327C8C2×#1192400511yescharacteristic
328C8C4×C22⋊D7311yescharacteristic
329C8#1455656511yescharacteristic
330C8#1455657411yescharacteristic
331C22—22no—
332C22—42no—
333C22—82no—
334C22—5842no—
335C22—5844no—
336C22—11682no—
337C22—11681no—
338C22—11681no—
339C22#1798869711yescharacteristic
340C22#1798873211yescharacteristic
341C22#1798878211yescharacteristic
342C4—42no—
343C4—81no—
344C4—11681no—
345C4#1798846611yescharacteristic
346C4C4×#1192401311yescharacteristic
347C4#1798872111yescharacteristic
348C4#1798880211yescharacteristic
349C2—21no—
350C2—21no—
351C2—82no—
352C2—11682no—
353C2#2261027911yescharacteristic
354C2#2261182912yes—
subgroupG/H
centerC22#17988697
commutatorC2×C8×C73C22×C4
FrattiniC22×C8C22×D73
FittingC2×#444×C73C2
socleC22×C73#442
prime radicalC2×#17989022C2
cosocleC22×C8×C73C23
hypercenterC2×#444D73
hypocenterC73#13040
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#97 ◁ C2×#444

Complements

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

NH ≅ G/Ntype
2. C2×#17988399C2semidirect
3. C2×#17988397C2semidirect
4. #22610807C2semidirect
5. #22608031C2semidirect
6. #22608011C2semidirect
7. C2×#444×C73C2semidirect
11. C2×#14556534C4semidirect
12. C2×#14556532C4semidirect
96. Q16×C73C22⋊C4semidirect
100. D8×C73C22⋊C4semidirect
179. C2×#444D73semidirect
185. C73#13040semidirect
265. Q16#11924005semidirect
273. D8#11924005semidirect

Maximal subgroups

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

subgroupindex
C2×#179890222
C2×#179883992
C2×#179883972
#226108072
#226080312
#226080112
C2×#444×C732
#1304073