Finite Group Databasebeta

← #32983626

Subgroups — #32983626

Order 18688.

474 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 95 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#22618569C211yesmaximal, characteristic
2#22614380C211yesmaximal, characteristic
3#22608463C212yesmaximal
4#22607783C212yesmaximal
5C73×#1820C211yesmaximal, characteristic
6C2×#14556570C2211yescharacteristic
7#17988648—24no—
8#17988611—21no—
9#17988551C2212yes—
10C4×#11924012—21no—
11#17988515—21no—
12#17988515—21no—
13#17988492C411yescharacteristic, complemented
14#17988492C411yescharacteristic, complemented
15#17988471—21no—
16#17988471—21no—
17#17988456C411yescharacteristic, complemented
18#17988452C411yescharacteristic, complemented
19C4×C22⋊Dic73C2211yescharacteristic
20#17988323—24no—
21C2×#117×C73C2211yescharacteristic
22#381×C73C2212yes—
23#342×C73—21no—
24#342×C73—21no—
25C22×Dic146C2×C411yescharacteristic
26C2×#11924012—22no—
27#14556570—22no—
28#14556570—22no—
29C2×C4⋊Dic73C2×C411yescharacteristic
30C2×#11924007—21no—
31C2×#11924007—21no—
32C2×#11924007—21no—
33C2×C22⋊Dic73—21no—
34C2×C22⋊Dic73—21no—
35C2×C4×Dic73C2×C411yescharacteristic
36C2×C4×Dic73C2×C411yescharacteristic
37C4⋊#10596831—24no—
38C22⋊#10596831—22no—
39C22⋊#10596831D414yes—
40#14556502—22no—
41C4×#10596831—22no—
42#14556498—21no—
43#14556498—21no—
44#14556498—21no—
45#14556498—21no—
46#14556498—21no—
47#14556498—21no—
48#14556498—21no—
49C23×C4×C73C2311yescharacteristic
50#117×C73—22no—
51#117×C73—22no—
52C2×C4⋊C4×C73C2×C411yescharacteristic
53C2×C4⋊C4×C73C2×C411yescharacteristic
54C2×C22⋊C4×C73—22no—
55C2×C22⋊C4×C73D411yescharacteristic
56C2×C22⋊C4×C73D411yescharacteristic
57C2×C42×C73D411yescharacteristic
58C2×C42×C73D411yescharacteristic
59#95×C73—22no—
60#95×C73—22no—
61C2×Dic146—22no—
62C2×Dic146—41no—
63C2×Dic146—41no—
64C22×Dic73—21no—
65C22×Dic73—21no—
66C22×Dic73—21no—
67C22×Dic73—21no—
68#11924012—42no—
69#11924012—42no—
70C2×#10596831—24no—
71C4⋊Dic73—22no—
72#11924007—42no—
73#11924007—42no—
74#11924007—41no—
75#11924007—41no—
76C22⋊Dic73—42no—
77C22⋊Dic73—42no—
78C4×Dic73—41no—
79C4×Dic73—41no—
80C24×C73C2×D411yescharacteristic
81C22×C4×C73—41no—
82C22×C4×C73—41no—
83C22×C4×C73—42no—
84C22×C4×C73C22⋊C411yescharacteristic
85C22×C4×C73C22⋊C411yescharacteristic
86C22×C4×C73C22⋊C411yescharacteristic
87C22×C4×C73C22⋊C411yescharacteristic
88C22×C4×C73C22×C411yescharacteristic
89C22×C4×C73C2×D412yes—
90C22×C4×C73C2×D411yescharacteristic
91C22×C4×C73#4112yes—
92C4⋊C4×C73—22no—
93C4⋊C4×C73—22no—
94C22⋊C4×C73—22no—
95C22⋊C4×C73—22no—
96C22⋊C4×C73—42no—
97C22⋊C4×C73—42no—
98C42×C73—21no—
99C42×C73—21no—
100C42×C73—21no—
101C42×C73—21no—
102C23×C73—21no—
103C23×C73—21no—
104C23×C73—41no—
105C23×C73—41no—
106C23×C73C2×C22⋊C411yescharacteristic
107C23×C73#12112yes—
108C2×C4×C73—21no—
109C2×C4×C73—21no—
110C2×C4×C73—21no—
111C2×C4×C73—21no—
112C2×C4×C73—21no—
113C2×C4×C73—21no—
114C2×C4×C73—21no—
115C2×C4×C73—21no—
116C2×C4×C73—42no—
117C2×C4×C73—42no—
118C2×C4×C73—41no—
119C2×C4×C73—41no—
120C2×C4×C73—41no—
121C2×C4×C73—41no—
122C2×C4×C73—82no—
123C2×C4×C73—82no—
124C2×C4×C73—82no—
125C2×C4×C73C4×D412yes—
126C2×C4×C73#12011yescharacteristic
127C2×C4×C73#12311yescharacteristic
128C2×Dic73—22no—
129C2×Dic73—41no—
130C2×Dic73—41no—
131C2×Dic73—41no—
132C2×Dic73—41no—
133C2×Dic73—41no—
134C2×Dic73—41no—
135C2×Dic73—41no—
136C2×Dic73—41no—
137C2×Dic73—41no—
138C2×Dic73—41no—
139C2×Dic73—41no—
140Dic146—82no—
141#10596831—44no—
142C22×C73—21no—
143C22×C73—21no—
144C22×C73—22no—
145C22×C73—22no—
146C22×C73—22no—
147C22×C73—81no—
148C22×C73—81no—
149C22×C73#38611yescharacteristic
150C22×C73#41011yescharacteristic
151C22×C73#41011yescharacteristic
152C4×C73—21no—
153C4×C73—22no—
154C4×C73—21no—
155C4×C73—41no—
156C4×C73—41no—
157C4×C73—41no—
158C4×C73—41no—
159C4×C73—162no—
160Dic73—42no—
161Dic73—81no—
162Dic73—81no—
163Dic73—81no—
164#9444—731nomaximal, Sylow
165C2×C73—21no—
166C2×C73—21no—
167C2×C73—42no—
168C2×C73#188411yescharacteristic
169C2×C73#189911yescharacteristic
170C2×C73#190911yescharacteristic
171#2921—731no—
172#2499—731no—
173#1820D7311yescharacteristic
174#1511—732no—
175#1284—732no—
176C73#944411yescharacteristic, complemented, Sylow
177C2×C22⋊Q8—731no—
178C2×#117C2×D7311yescharacteristic
179#433—1464no—
180#423—1461no—
181#407—732no—
182C4×M16—1461no—
183#397—1461no—
184#397—1461no—
185#391—731no—
186#391—731no—
187#387—1461no—
188#387—1461no—
189#384—731no—
190#382—731no—
191#381C2×D7312yes—
192C4×C22⋊C4—731no—
193#342—21no—
194#342—21no—
195#328—1464no—
196C22×Q8—731no—
197C23×C4C22×D7311yescharacteristic
198C2×M16—1462no—
199C22⋊Q8—1462no—
200C22⋊Q8—1462no—
201#117—22no—
202#117—22no—
203C2×C4⋊C4—731no—
204C2×C4⋊C4—1461no—
205C2×C4⋊C4—1461no—
206C2×C4⋊C4—1461no—
207C2×C4⋊C4C4×D7311yescharacteristic
208C2×C4⋊C4C4×D7311yescharacteristic
209C2×C22⋊C4—22no—
210C2×C22⋊C4—1461no—
211C2×C22⋊C4—1461no—
212C2×C22⋊C4C22⋊D7311yescharacteristic
213C2×C22⋊C4C22⋊D7311yescharacteristic
214C2×C42—731no—
215C2×C42—731no—
216C2×C42D29211yescharacteristic
217C2×C42D29211yescharacteristic
218C4⋊C8—1464no—
219C22⋊C8—734no—
220C22⋊C8—1462no—
221#97—1462no—
222C4×C8—1462no—
223#95—22no—
224#95—22no—
225#95—1461no—
226#95—1461no—
227#95—1461no—
228#95—1461no—
229#95—1461no—
230#95—1461no—
231#95—1461no—
232C24C2×C22⋊D7311yescharacteristic
233C2×Q8—2921no—
234C2×Q8—2921no—
235C2×Q8—1462no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—42no—
239C22×C4—1461no—
240C22×C4—1461no—
241C22×C4—1461no—
242C22×C4—1461no—
243C22×C4#1192400511yescharacteristic
244C22×C4#1192400511yescharacteristic
245C22×C4#1192400511yescharacteristic
246C22×C4#1192400511yescharacteristic
247C22×C4C2×C4×D7311yescharacteristic
248C22×C4D4×D7312yes—
249C22×C4C2×D29211yescharacteristic
250C22×C4#1192402712yes—
251M16—2922no—
252M16—2922no—
253C2×C8—1464no—
254C4⋊C4—22no—
255C4⋊C4—22no—
256C4⋊C4—2922no—
257C4⋊C4—2922no—
258C4⋊C4—2921no—
259C4⋊C4—2921no—
260C4⋊C4—1462no—
261C22⋊C4—22no—
262C22⋊C4—22no—
263C22⋊C4—42no—
264C22⋊C4—42no—
265C22⋊C4—2922no—
266C22⋊C4—2922no—
267C42—21no—
268C42—21no—
269C42—21no—
270C42—21no—
271C42—2921no—
272C42—2921no—
273C23—21no—
274C23—21no—
275C23—41no—
276C23—41no—
277C23C2×#1192400511yescharacteristic
278C23#1455656312yes—
279Q8—5842no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—82no—
295C2×C4—82no—
296C2×C4—82no—
297C2×C4—2921no—
298C2×C4—2921no—
299C2×C4—2921no—
300C2×C4—2921no—
301C2×C4—2921no—
302C2×C4—2921no—
303C2×C4—2921no—
304C2×C4—2921no—
305C2×C4—2921no—
306C2×C4—2921no—
307C2×C4—2921no—
308C2×C4—1462no—
309C2×C4#1455655112yes—
310C2×C4#1455656111yescharacteristic
311C2×C4#1455657511yescharacteristic
312C8—2924no—
313C22—21no—
314C22—21no—
315C22—22no—
316C22—22no—
317C22—22no—
318C22—81no—
319C22—81no—
320C22#1798846411yescharacteristic
321C22#1798856011yescharacteristic
322C22#1798856011yescharacteristic
323C4—21no—
324C4—22no—
325C4—21no—
326C4—41no—
327C4—41no—
328C4—41no—
329C4—41no—
330C4—162no—
331C4—2922no—
332C4—5841no—
333C4—5841no—
334C4—5841no—
335C2—21no—
336C2—21no—
337C2—42no—
338C2#2261031211yescharacteristic
339C2#2261038711yescharacteristic
340C2#2261045111yescharacteristic
subgroupG/H
centerC22#17988464
commutatorC22×C4×C73C22×C4
FrattiniC23×C4C22×D73
FittingC73×#1820C2
socleC22×C73#386
prime radicalC24×C73C2×D4
cosocleC23×C4×C73C23
hypercenter#1820D73
hypocenterC73#9444
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1820

Complements

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

NH ≅ G/Ntype
13. #17988492C4semidirect
14. #17988492C4semidirect
17. #17988456C4semidirect
18. #17988452C4semidirect
176. C73#9444semidirect

Maximal subgroups

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

subgroupindex
#226185692
#226143802
#226084632
#226077832
C73×#18202
#944473