Finite Group Databasebeta

← #33003426

Subgroups — #33003426

Order 18688.

478 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×#17988857C211yesmaximal, characteristic
2C2×#17988401C211yesmaximal, characteristic, complemented
3C2×#17988389C211yesmaximal, characteristic, complemented
4#22609995C211yesmaximal, characteristic, complemented
5#22608040C211yesmaximal, characteristic, complemented
6#22608025C211yesmaximal, characteristic, complemented
7C2×#504×C73C211yesmaximal, characteristic, complemented
8C2×#14556616—21no—
9C22×D584C2211yescharacteristic
10#17988857—22no—
11#17988857C414yescomplemented
12C2×#14556527C2211yescharacteristic
13C2×#14556527C2211yescharacteristic
14C2×#14556525C2211yescharacteristic
15#17988588—21no—
16C2×#14556507—21no—
17#17988401—22no—
18#17988389—21no—
19#17988389—21no—
20#17988355—21no—
21C2×#131×C73C2211yescharacteristic
22#504×C73—22no—
23#504×C73—22no—
24C2×M32×C73—21no—
25C2×M32×C73C2211yescharacteristic
26C22×C16×C73—21no—
27C22×C16×C73C2211yescharacteristic
28C22×D292—21no—
29#14556616—24no—
30#14556616—42no—
31C2×#11924015—41no—
32C2×D584—21no—
33C2×D584—21no—
34C2×D584—21no—
35C2×D584—21no—
36C2×D584—21no—
37C2×D584C2×C412yes—
38C2×#11924013—41no—
39C2×#11924012—21no—
40C2×#11924012—21no—
41C2×C4⋊Dic73—21no—
42#14556527—21no—
43#14556527—21no—
44#14556527—21no—
45#14556527C2×C412yes—
46#14556525—21no—
47#14556525—21no—
48#14556515—41no—
49#14556512—41no—
50#14556507—24no—
51C2×#41×C73D411yescharacteristic
52#131×C73—21no—
53#131×C73—21no—
54#131×C73C2×C412yes—
55#131×C73D412yes—
56C2×M16×C73—21no—
57C2×M16×C73D411yescharacteristic
58C22×C8×C73—21no—
59C22×C8×C73C2311yescharacteristic
60M32×C73—24no—
61M32×C73—21no—
62M32×C73D412yes—
63C2×C16×C73—24no—
64C2×C16×C73—21no—
65C2×C16×C73—21no—
66C2×C16×C73—21no—
67C2×C16×C73D411yescharacteristic
68C2×C16×C73D411yescharacteristic
69C23×D73—41no—
70C2×D292—22no—
71C2×D292—41no—
72C2×D292—41no—
73C2×D292—41no—
74C2×D292—41no—
75C2×D292—41no—
76C22×Dic73—41no—
77#11924015—44no—
78D584—22no—
79D584—22no—
80D584—41no—
81D584—41no—
82#11924013—44no—
83#11924012—22no—
84#11924012—41no—
85#11924012—41no—
86#11924012—41no—
87C2×#10596831—41no—
88C2×#10596831—41no—
89C4⋊Dic73—41no—
90C4⋊Dic73—41no—
91#41×C73—21no—
92#41×C73—21no—
93#41×C73C22⋊C412yes—
94#41×C73SD1612yes—
95C2×Q8×C73D811yescharacteristic
96C2×D4×C73—21no—
97C2×D4×C73D811yescharacteristic
98C22×C4×C73—21no—
99C22×C4×C73C2×D411yescharacteristic
100M16×C73—21no—
101M16×C73—22no—
102M16×C73—22no—
103M16×C73C22⋊C412yes—
104C2×C8×C73—21no—
105C2×C8×C73—22no—
106C2×C8×C73—22no—
107C2×C8×C73C22×C411yescharacteristic
108C2×C8×C73C2×D411yescharacteristic
109C2×C8×C73C2×D411yescharacteristic
110C2×C8×C73C2×D411yescharacteristic
111C2×C8×C73#4111yescharacteristic
112C2×C8×C73#4111yescharacteristic
113C16×C73—21no—
114C16×C73—22no—
115C16×C73—21no—
116C23×C73—21no—
117C23×C73#13511yescharacteristic
118C22×D73—42no—
119C22×D73—81no—
120C22×D73—81no—
121C22×D73—161no—
122C22×D73—161no—
123Q8×C73—21no—
124Q8×C73#10212yes—
125D4×C73—22no—
126D4×C73—22no—
127D4×C73—21no—
128D4×C73#10212yes—
129C2×C4×C73—21no—
130C2×C4×C73—22no—
131C2×C4×C73—22no—
132C2×C4×C73C2×C22⋊C411yescharacteristic
133C2×C4×C73#12111yescharacteristic
134C2×C4×C73#12111yescharacteristic
135C2×C4×C73C2×D811yescharacteristic
136C2×C4×C73C2×SD1611yescharacteristic
137C2×C4×C73#13511yescharacteristic
138C2×Dic73—81no—
139C2×Dic73—161no—
140D292—42no—
141D292—42no—
142D292—81no—
143D292—81no—
144C8×C73—22no—
145C8×C73C4×D411yescharacteristic
146C8×C73C4×D411yescharacteristic
147C8×C73#12011yescharacteristic
148C8×C73#12311yescharacteristic
149#10596831—42no—
150#10596831—81no—
151C22×C73—21no—
152C22×C73—21no—
153C22×C73—22no—
154C22×C73—41no—
155C22×C73—41no—
156C22×C73#41611yescharacteristic
157C22×C73#46511yescharacteristic
158C22×C73#46611yescharacteristic
159C2×D73—82no—
160C2×D73—162no—
161C2×D73—161no—
162C2×D73—161no—
163C2×D73—162no—
164C4×C73—22no—
165C4×C73#38611yescharacteristic
166C4×C73C2×#10211yescharacteristic
167C4×C73#46511yescharacteristic
168C4×C73#46611yescharacteristic
169Dic73—161no—
170#13073—731nomaximal, Sylow
171C2×C73—21no—
172C2×C73—21no—
173C2×C73—42no—
174C2×C73#189311yescharacteristic
175C2×C73#222111yescharacteristic
176C2×C73#222311yescharacteristic
177D73—162no—
178C2×#504D7311yescharacteristic, complemented
179C2×#472—731no—
180C2×#361—731no—
181C2×#357—731no—
182#1816—731no—
183#1387—731no—
184#1381—731no—
185C73#1307311yescharacteristic, complemented, Sylow
186C2×AGL(1,ℤ/8)—1461no—
187C22×D8—731no—
188C2×#131C2×D7311yescharacteristic
189#504—22no—
190#504—22no—
191C2×M32—21no—
192C2×M32C2×D7311yescharacteristic
193C22×C16—21no—
194C22×C16C2×D7311yescharacteristic
195#472—734no—
196#472—1462no—
197C2×#108—731no—
198C2×#108—731no—
199C2×#107—731no—
200#417—1461no—
201C2×#100—1461no—
202#361—1462no—
203#357—1461no—
204#357—1461no—
205#343—1461no—
206C2×#41D29211yescharacteristic
207C22×D4—1461no—
208AGL(1,ℤ/8)—2922no—
209AGL(1,ℤ/8)—1464no—
210C2×SD16—2921no—
211C2×D8—2921no—
212C2×D8—732no—
213C2×D8—1461no—
214C2×D8—1461no—
215C2×D8—1461no—
216C2×D8—1461no—
217C2×D8—1461no—
218#131—21no—
219#131—21no—
220#131C4×D7312yes—
221#131C22⋊D7312yes—
222C2×M16—21no—
223C2×M16—1461no—
224C2×M16—1461no—
225C2×M16D29211yescharacteristic
226C22×C8—21no—
227C22×C8C22×D7311yescharacteristic
228C2×C4⋊C4—1461no—
229M32—24no—
230M32—21no—
231M32C22⋊D7312yes—
232C2×C16—24no—
233C2×C16—21no—
234C2×C16—21no—
235C2×C16—21no—
236C2×C16C22⋊D7311yescharacteristic
237C2×C16C22⋊D7311yescharacteristic
238#108—732no—
239#108—1461no—
240#108—1461no—
241#108—1461no—
242#107—1461no—
243#107—1461no—
244#103—2921no—
245#102—2921no—
246#100—1464no—
247C24—2921no—
248#41—21no—
249#41—21no—
250#41#1192400512yes—
251#41#1192401712yes—
252C2×Q8D58411yescharacteristic
253C2×D4—21no—
254C2×D4—2921no—
255C2×D4—2921no—
256C2×D4—2921no—
257C2×D4—2921no—
258C2×D4—2921no—
259C2×D4—1462no—
260C2×D4D58411yescharacteristic
261C22×C4—21no—
262C22×C4—2921no—
263C22×C4C2×D29211yescharacteristic
264SD16—2924no—
265D8—2924no—
266D8—2921no—
267D8—2921no—
268D8—1462no—
269D8—1462no—
270M16—21no—
271M16—22no—
272M16—22no—
273M16—2921no—
274M16—2921no—
275M16—2921no—
276M16—1462no—
277M16#1192400512yes—
278C2×C8—21no—
279C2×C8—22no—
280C2×C8—22no—
281C2×C8—2921no—
282C2×C8—2921no—
283C2×C8C2×C4×D7311yescharacteristic
284C2×C8C2×C22⋊D7311yescharacteristic
285C2×C8C2×C22⋊D7311yescharacteristic
286C2×C8C2×C22⋊D7311yescharacteristic
287C2×C8#1192402811yescharacteristic
288C2×C8#1192402811yescharacteristic
289C4⋊C4—2921no—
290C4⋊C4—2921no—
291C16—21no—
292C16—22no—
293C16—21no—
294C23—21no—
295C23—2922no—
296C23—5841no—
297C23—5841no—
298C23—11681no—
299C23—11681no—
300C23#1455661311yescharacteristic
301Q8—21no—
302Q8#1455651312yes—
303D4—22no—
304D4—22no—
305D4—21no—
306D4—2922no—
307D4—2922no—
308D4—5841no—
309D4—5841no—
310D4#1455651312yes—
311C2×C4—21no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—5841no—
315C2×C4—11681no—
316C2×C4C2×#1192400511yescharacteristic
317C2×C4#1455656511yescharacteristic
318C2×C4#1455656511yescharacteristic
319C2×C4C2×D58411yescharacteristic
320C2×C4C2×#1192401711yescharacteristic
321C2×C4#1455661311yescharacteristic
322C8—22no—
323C8—2922no—
324C8—5841no—
325C8C4×C22⋊D7311yescharacteristic
326C8C4×C22⋊D7311yescharacteristic
327C8#1455656011yescharacteristic
328C8#1455657411yescharacteristic
329C22—21no—
330C22—21no—
331C22—22no—
332C22—41no—
333C22—41no—
334C22—5842no—
335C22—11682no—
336C22—11682no—
337C22—11681no—
338C22—11681no—
339C22#1798858411yescharacteristic
340C22#1798881611yescharacteristic
341C22#1798882311yescharacteristic
342C4—22no—
343C4—11681no—
344C4#1798846611yescharacteristic
345C4C2×#1455651311yescharacteristic
346C4#1798881611yescharacteristic
347C4#1798882311yescharacteristic
348C2—21no—
349C2—21no—
350C2—42no—
351C2—11682no—
352C2#2261035911yescharacteristic
353C2#2261205111yescharacteristic
354C2#2261206111yescharacteristic
subgroupG/H
centerC22#17988823
commutatorC2×C8×C73C22×C4
FrattiniC22×C8C22×D73
FittingC2×#504×C73C2
socleC22×C73#466
prime radicalC2×#17988857C2
cosocleC22×C8×C73C23
hypercenterC2×#504D73
hypocenterC73#13073
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#131 ◁ C2×#504

Complements

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

NH ≅ G/Ntype
2. C2×#17988401C2semidirect
3. C2×#17988389C2semidirect
4. #22609995C2semidirect
5. #22608040C2semidirect
6. #22608025C2semidirect
7. C2×#504×C73C2semidirect
11. #17988857C4semidirect
178. C2×#504D73semidirect
185. C73#13073semidirect

Maximal subgroups

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

subgroupindex
C2×#179888572
C2×#179884012
C2×#179883892
#226099952
#226080402
#226080252
C2×#504×C732
#1307373