Finite Group Databasebeta

← #33002981

Subgroups — #33002981

Order 18688.

528 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 127 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22619512C211yesmaximal, characteristic
2C2×#17988625C211yesmaximal, characteristic, complemented
3C2×#17988398C212yesmaximal, complemented
4#22608037C212yesmaximal, complemented
5C73×#2112C211yesmaximal, characteristic, complemented
6C2×#14556613C2211yescharacteristic
7#17989098—21no—
8C2×#14556550—21no—
9#17988828C412yescomplemented
10#17988823C412yescomplemented
11#17988816—22no—
12#17988625—24no—
13#17988625C2211yescharacteristic
14C2×#14556525—21no—
15C2×#14556523C2212yes—
16#17988584—21no—
17#17988398—24no—
18C23×C8×C73C2211yescharacteristic
19C2×M32×C73C2212yes—
20C22⋊C16×C73—22no—
21C2×#11924028—21no—
22#14556613—22no—
23#14556613—22no—
24C2×Dic292C2×C411yescharacteristic
25C2×#11924017—21no—
26C2×D584C2×C411yescharacteristic
27#14556574—41no—
28#14556570—22no—
29#14556565—22no—
30#14556560—41no—
31C4×C22⋊D73—42no—
32#14556550—24no—
33#14556550—21no—
34#14556550—42no—
35C2×C4⋊Dic73—22no—
36C2×C22⋊Dic73—41no—
37C2×C4×Dic73—41no—
38#14556525—24no—
39#14556525C2×C412yes—
40#14556523—22no—
41#14556523—21no—
42#14556523D414yescomplemented
43#14556516—22no—
44#14556513—22no—
45C23×C4×C73D411yescharacteristic
46C22×C8×C73—22no—
47C22×C8×C73—22no—
48C22×C8×C73C2×C412yes—
49C22×C8×C73D411yescharacteristic
50C22×C8×C73D412yes—
51C22×C8×C73C2311yescharacteristic
52M32×C73—22no—
53M32×C73—22no—
54C2×C16×C73—22no—
55#11924028—41no—
56#11924028—41no—
57#11924028—41no—
58#11924028—41no—
59C2×Dic146—21no—
60C2×D292—21no—
61C2×C22⋊D73—42no—
62C2×C22⋊D73—41no—
63C22×Dic73—42no—
64C2×C4×D73—41no—
65Dic292—22no—
66#11924017—42no—
67D584—22no—
68C4⋊Dic73—22no—
69C4⋊Dic73—24no—
70C4⋊Dic73—42no—
71#11924007—42no—
72C22⋊Dic73—44no—
73C22⋊Dic73—42no—
74C22⋊Dic73—41no—
75#11924005—42no—
76C4×Dic73—44no—
77C4×Dic73—41no—
78C24×C73D811yescharacteristic
79C22×C4×C73—21no—
80C22×C4×C73—21no—
81C22×C4×C73—21no—
82C22×C4×C73—21no—
83C22×C4×C73C22⋊C412yes—
84C22×C4×C73D811yescharacteristic
85C22×C4×C73SD1612yes—
86C22×C4×C73C2×D411yescharacteristic
87C2×C8×C73—21no—
88C2×C8×C73—21no—
89C2×C8×C73—24no—
90C2×C8×C73—24no—
91C2×C8×C73C22⋊C412yes—
92C2×C8×C73C22×C411yescharacteristic
93C2×C8×C73C2×D412yes—
94C2×C8×C73C2×D411yescharacteristic
95C2×C8×C73#4112yes—
96C16×C73—42no—
97C23×C73—21no—
98C23×C73—21no—
99C23×C73—42no—
100C23×C73#10212yes—
101C23×C73C2×D811yescharacteristic
102C22×D73—41no—
103C2×C4×C73—21no—
104C2×C4×C73—21no—
105C2×C4×C73—22no—
106C2×C4×C73—22no—
107C2×C4×C73—22no—
108C2×C4×C73—22no—
109C2×C4×C73#10212yes—
110C2×C4×C73C2×C22⋊C411yescharacteristic
111C2×C4×C73#12011yescharacteristic
112C2×C4×C73#12311yescharacteristic
113C2×C4×C73C2×SD1611yescharacteristic
114C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
115C2×C4×C73#13711yescharacteristic
116C22⋊D73—84no—
117C22⋊D73—81no—
118C22⋊D73—81no—
119C2×Dic73—42no—
120C2×Dic73—41no—
121C2×Dic73—44no—
122C2×Dic73—84no—
123D292—41no—
124D292—41no—
125C4×D73—81no—
126C4×D73—81no—
127Dic146—41no—
128Dic146—41no—
129C8×C73—22no—
130C8×C73C4×D412yes—
131C8×C73#12112yes—
132C22×C73—22no—
133C22×C73—22no—
134C22×C73—22no—
135C22×C73—22no—
136C22×C73—42no—
137C22×C73—42no—
138C22×C73C2×#10211yescharacteristic
139C22×C73#44711yescharacteristic
140C22×C73#48011yescharacteristic
141C2×D73—81no—
142C2×D73—82no—
143C4×C73—22no—
144C4×C73#38611yescharacteristic
145C4×C73#41711yescharacteristic
146C4×C73#44811yescharacteristic
147C4×C73#48111yescharacteristic
148Dic73—84no—
149Dic73—81no—
150Dic73—82no—
151#13002—731nomaximal, Sylow
152C2×C73—21no—
153C2×C73—21no—
154C2×C73—42no—
155C2×C73#187411yescharacteristic
156C2×C73#214512yes—
157D73—161no—
158#3050—731no—
159C2×#427—731no—
160C2×#360—732no—
161#2112D7311yescharacteristic, complemented
162#1386—732no—
163C73#1300211yescharacteristic, complemented, Sylow
164C2×#135—731no—
165C23×C8C2×D7311yescharacteristic
166#525—1461no—
167C2×#117—1461no—
168C2×M32C2×D7312yes—
169#467—732no—
170#466—732no—
171#465—1462no—
172#427—731no—
173#427—1464no—
174C2×#107—1461no—
175C2×#106—732no—
176#416—1461no—
177#360—1464no—
178C22⋊C16—22no—
179C2×#41—1461no—
180C23×C4D29211yescharacteristic
181#135—1462no—
182#135—1462no—
183C2×Q16—731no—
184C2×SD16—1461no—
185C2×D8—731no—
186C22×C8—22no—
187C22×C8—22no—
188C22×C8C4×D7312yes—
189C22×C8D29211yescharacteristic
190C22×C8C22⋊D7312yes—
191C22×C8C22×D7311yescharacteristic
192#123—2921no—
193C22⋊Q8—1462no—
194#121—1462no—
195#120—2921no—
196C4×D4—2922no—
197#117—2922no—
198#117—1464no—
199#117—1461no—
200C2×C4⋊C4—1462no—
201C2×C22⋊C4—2921no—
202C2×C42—2921no—
203M32—22no—
204M32—22no—
205C2×C16—22no—
206#107—732no—
207#107—1464no—
208#106—734no—
209#106—1462no—
210#106—1461no—
211#103—1462no—
212#102—1462no—
213C24D58411yescharacteristic
214#41—2921no—
215#41—2921no—
216#41—2921no—
217#41—2921no—
218C2×Q8—1461no—
219C2×D4—2922no—
220C2×D4—2921no—
221C2×D4—1461no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—2922no—
227C22×C4—2921no—
228C22×C4#1192400512yes—
229C22×C4D58411yescharacteristic
230C22×C4#1192401712yes—
231C22×C4C2×D29211yescharacteristic
232Q16—1462no—
233SD16—2922no—
234D8—1462no—
235C2×C8—21no—
236C2×C8—21no—
237C2×C8—24no—
238C2×C8—24no—
239C2×C8#1192400512yes—
240C2×C8C2×C4×D7311yescharacteristic
241C2×C8D4×D7312yes—
242C2×C8C2×C22⋊D7311yescharacteristic
243C2×C8#1192402712yes—
244C4⋊C4—2922no—
245C4⋊C4—2922no—
246C4⋊C4—1462no—
247C4⋊C4—1464no—
248C22⋊C4—2924no—
249C22⋊C4—2922no—
250C22⋊C4—2921no—
251C22⋊C4—2922no—
252C42—2924no—
253C42—2921no—
254C16—42no—
255C23—21no—
256C23—21no—
257C23—42no—
258C23—2921no—
259C23#1455651312yes—
260C23C2×D58411yescharacteristic
261Q8—2921no—
262Q8—2921no—
263D4—2921no—
264D4—2921no—
265D4—5844no—
266D4—5841no—
267D4—5841no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—2924no—
275C2×C4—2922no—
276C2×C4—2921no—
277C2×C4—5844no—
278C2×C4—5841no—
279C2×C4—5841no—
280C2×C4#1455651312yes—
281C2×C4C2×#1192400511yescharacteristic
282C2×C4#1455656111yescharacteristic
283C2×C4#1455657511yescharacteristic
284C2×C4C2×#1192401711yescharacteristic
285C2×C4#1455661911yescharacteristic
286C2×C4#1455662411yescharacteristic
287C8—22no—
288C8#1455655112yes—
289C8#1455656312yes—
290C22—22no—
291C22—22no—
292C22—22no—
293C22—22no—
294C22—42no—
295C22—42no—
296C22—5841no—
297C22—5842no—
298C22C2×#1455651311yescharacteristic
299C22#1798872311yescharacteristic
300C22#1798890111yescharacteristic
301C4—22no—
302C4—5844no—
303C4—5841no—
304C4—5842no—
305C4#1798846411yescharacteristic
306C4#1798858911yescharacteristic
307C4#1798872811yescharacteristic
308C4#1798890611yescharacteristic
309C2—21no—
310C2—21no—
311C2—42no—
312C2—11681no—
313C2#2261024611yescharacteristic
314C2#2261166812yes—
subgroupG/H
centerC22C2×#14556513
commutatorC2×C8×C73C22×C4
FrattiniC22×C8C22×D73
FittingC73×#2112C2
socleC22×C73C2×#102
prime radical#22619512C2
cosocleC22×C8×C73C23
hypercenter#2112D73
hypocenterC73#13002
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ #2112

Complements

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

NH ≅ G/Ntype
2. C2×#17988625C2semidirect
3. C2×#17988398C2semidirect
4. #22608037C2semidirect
5. C73×#2112C2semidirect
9. #17988828C4semidirect
10. #17988823C4semidirect
42. #14556523D4semidirect
161. #2112D73semidirect
163. C73#13002semidirect

Maximal subgroups

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

subgroupindex
#226195122
C2×#179886252
C2×#179883982
#226080372
C73×#21122
#1300273