Finite Group Databasebeta

← #32983398

Subgroups — #32983398

Order 18688.

570 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 570 conjugacy classes of subgroups — too many to draw. Only its 109 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#22618971C211yesmaximal, characteristic
2#22614880C211yesmaximal, characteristic
3#22609920C211yesmaximal, characteristic
4#22608617C211yesmaximal, characteristic, complemented
5#22608615C211yesmaximal, characteristic, complemented
6#22607834C211yesmaximal, characteristic, complemented
7C73×#1303C211yesmaximal, characteristic, complemented
8C2×#14556591C2211yescharacteristic
9C2×#14556571—22no—
10C2×C4×D292C2211yescharacteristic
11#17988666—22no—
12#17988664—22no—
13#17988640C2211yescharacteristic
14#17988611C2211yescharacteristic
15C2×#14556507—21no—
16#17988549—22no—
17#17988524C414yes—
18#17988489—21no—
19#17988457—21no—
20#17988355—21no—
21#17988334—22no—
22#17988332—22no—
23#431×C73C2211yescharacteristic
24#423×C73C2211yescharacteristic
25C4×C4⋊C4×C73C2211yescharacteristic
26C22×Dic146—21no—
27C22×D292D411yescharacteristic
28C22×C4×D73D412yes—
29C2×#11924012—21no—
30C2×#11924011—24no—
31#14556591—21no—
32#14556591—21no—
33#14556591—21no—
34#14556591—21no—
35#14556571—48no—
36C4×D292—21no—
37C4×D292—21no—
38C4×D292—21no—
39C4×D292—21no—
40C2×C4⋊Dic73—24no—
41C2×C4⋊Dic73—21no—
42C2×C4⋊Dic73C2×C412yes—
43C2×C4⋊Dic73D411yescharacteristic
44C2×#11924005—24no—
45C2×#11924005C2×C412yes—
46C4⋊#10596831—21no—
47C4⋊#10596831—21no—
48C4⋊#10596831D412yes—
49#14556507—42no—
50C22⋊#10596831—21no—
51#14556503—24no—
52C4×#10596831—21no—
53C2×M16×C73—21no—
54C2×C4⋊C4×C73—21no—
55C2×C42×C73—21no—
56C2×C42×C73C2311yescharacteristic
57C4⋊C8×C73—21no—
58C4⋊C8×C73—21no—
59C4⋊C8×C73D412yes—
60C22⋊C8×C73—21no—
61C4×C8×C73—21no—
62#95×C73C2×C412yes—
63C23×D73C22⋊C412yes—
64C2×Dic146—44no—
65C2×Dic146—41no—
66C2×Dic146—41no—
67C2×D292—21no—
68C2×D292—21no—
69C2×D292—21no—
70C2×D292—21no—
71C2×D292—21no—
72C2×D292—21no—
73C22×Dic73—22no—
74C22×Dic73C22⋊C412yes—
75C2×C4×D73—22no—
76C2×C4×D73—22no—
77C2×C4×D73—44no—
78#11924012—41no—
79#11924012—41no—
80#11924011—84no—
81C2×#10596831—22no—
82C2×#10596831—41no—
83C4⋊Dic73—21no—
84C4⋊Dic73—21no—
85C4⋊Dic73—22no—
86C4⋊Dic73—22no—
87C4⋊Dic73—48no—
88C4⋊Dic73—41no—
89C4⋊Dic73—41no—
90#11924005—22no—
91#11924005—22no—
92#11924005—48no—
93C22×C4×C73—22no—
94C22×C4×C73C22×C411yescharacteristic
95C22×C4×C73C2×D411yescharacteristic
96C22×C4×C73C2×D411yescharacteristic
97M16×C73—41no—
98M16×C73—41no—
99C2×C8×C73—22no—
100C2×C8×C73—41no—
101C4⋊C4×C73—24no—
102C42×C73—41no—
103C42×C73—41no—
104C42×C73C2×D411yescharacteristic
105C42×C73C2×D411yescharacteristic
106C42×C73#4111yescharacteristic
107C42×C73#4111yescharacteristic
108C23×C73C2×C22⋊C411yescharacteristic
109C22×D73—22no—
110C22×D73—22no—
111C22×D73—22no—
112C22×D73—42no—
113C22×D73—42no—
114C2×C4×C73—22no—
115C2×C4×C73—24no—
116C2×C4×C73—41no—
117C2×C4×C73—42no—
118C2×C4×C73—42no—
119C2×C4×C73C4×D411yescharacteristic
120C2×C4×C73C4×D411yescharacteristic
121C2×C4×C73#12011yescharacteristic
122C2×C4×C73#12111yescharacteristic
123C2×C4×C73#12111yescharacteristic
124C2×C4×C73#12311yescharacteristic
125C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
126C2×C4×C73#13711yescharacteristic
127C2×C4×C73#13711yescharacteristic
128C2×C4×C73#13711yescharacteristic
129C2×Dic73—22no—
130C2×Dic73—42no—
131C2×Dic73—44no—
132C2×Dic73—42no—
133D292—41no—
134D292—41no—
135D292—41no—
136D292—41no—
137C4×D73—44no—
138C4×D73—82no—
139Dic146—82no—
140Dic146—82no—
141C8×C73—42no—
142C8×C73—81no—
143#10596831—42no—
144#10596831—81no—
145C22×C73—21no—
146C22×C73—21no—
147C22×C73#38611yescharacteristic
148C22×C73#41711yescharacteristic
149C22×C73#41911yescharacteristic
150C2×D73—44no—
151C2×D73—42no—
152C2×D73—42no—
153C2×D73—42no—
154C2×D73—82no—
155C4×C73—21no—
156C4×C73—21no—
157C4×C73—21no—
158C4×C73—44no—
159C4×C73#45312yes—
160C4×C73#46811yescharacteristic
161C4×C73#46811yescharacteristic
162C4×C73#47011yescharacteristic
163C4×C73#47011yescharacteristic
164Dic73—42no—
165Dic73—82no—
166#9403—731nomaximal, Sylow
167C2×C73—21no—
168C2×C73—21no—
169C2×C73#188711yescharacteristic
170C2×C73#189411yescharacteristic
171C2×C73#191511yescharacteristic
172D73—82no—
173#2975—731no—
174#2542—731no—
175#1800—731no—
176#1549—731no—
177#1549—731no—
178#1303—731no—
179#1303D7311yescharacteristic, complemented
180C73#940311yescharacteristic, complemented, Sylow
181C2×C4⋊Q8—731no—
182C2×C22⋊Q8—1462no—
183C2×C4×D4—731no—
184#436—1462no—
185#436—1462no—
186#431—731no—
187#431C2×D7311yescharacteristic
188#423—731no—
189#423C2×D7311yescharacteristic
190C2×#100—1461no—
191#407—1462no—
192#399—734no—
193#390—1461no—
194#384—1461no—
195C4×C4⋊C4C2×D7311yescharacteristic
196#343—1461no—
197#332—1462no—
198#332—1462no—
199C22×Q8—1461no—
200C22×D4—731no—
201C23×C4—732no—
202C2×M16—21no—
203C2×M16—1464no—
204C2×M16—1461no—
205C4⋊Q8—1461no—
206C4⋊Q8—1461no—
207C4⋊Q8—1461no—
208C4⋊Q8—1461no—
209C22⋊Q8—2928no—
210C4×D4—1461no—
211C4×D4—1461no—
212C4×D4—1461no—
213C4×D4—1461no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—731no—
216C2×C4⋊C4—732no—
217C2×C4⋊C4—1464no—
218C2×C4⋊C4—1461no—
219C2×C22⋊C4—732no—
220C2×C22⋊C4—1464no—
221C2×C42—21no—
222C2×C42C22×D7311yescharacteristic
223C4⋊C8—21no—
224C4⋊C8—21no—
225C4⋊C8—732no—
226C4⋊C8—1461no—
227C4⋊C8—1461no—
228C4⋊C8D29212yes—
229#100—2922no—
230C22⋊C8—21no—
231C22⋊C8—1464no—
232C22⋊C8—1461no—
233C4×C8—21no—
234C4×C8—1461no—
235#95C4×D7312yes—
236C24—732no—
237C2×Q8—2924no—
238C2×Q8—2921no—
239C2×Q8—2921no—
240C2×D4—1461no—
241C2×D4—1461no—
242C2×D4—1461no—
243C2×D4—1461no—
244C2×D4—1461no—
245C2×D4—1461no—
246C22×C4—22no—
247C22×C4—2924no—
248C22×C4—732no—
249C22×C4—1462no—
250C22×C4—1462no—
251C22×C4—1462no—
252C22×C4C2×C4×D7311yescharacteristic
253C22×C4D4×D7311yescharacteristic
254C22×C4D4×D7311yescharacteristic
255M16—41no—
256M16—41no—
257M16—2921no—
258M16—2921no—
259M16—5844no—
260C2×C8—22no—
261C2×C8—41no—
262C2×C8—2921no—
263C2×C8—1462no—
264C4⋊C4—24no—
265C4⋊C4—2928no—
266C4⋊C4—2921no—
267C4⋊C4—2921no—
268C4⋊C4—1462no—
269C4⋊C4—1462no—
270C4⋊C4—1461no—
271C4⋊C4—1461no—
272C22⋊C4—2928no—
273C22⋊C4—1462no—
274C22⋊C4—1462no—
275C42—41no—
276C42—41no—
277C42D4×D7311yescharacteristic
278C42C2×D29211yescharacteristic
279C42#1192402811yescharacteristic
280C42#1192402911yescharacteristic
281C23—2922no—
282C23—2922no—
283C23—1462no—
284C23—1462no—
285C23—1462no—
286C23C22⋊C4×D7311yescharacteristic
287Q8—5842no—
288Q8—5842no—
289D4—2921no—
290D4—2921no—
291D4—2921no—
292D4—2921no—
293C2×C4—22no—
294C2×C4—24no—
295C2×C4—41no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—2924no—
299C2×C4—2922no—
300C2×C4—2924no—
301C2×C4—2922no—
302C2×C4—5842no—
303C2×C4—1462no—
304C2×C4#1455655311yescharacteristic
305C2×C4C4×D29211yescharacteristic
306C2×C4#1455656111yescharacteristic
307C2×C4#1455656411yescharacteristic
308C2×C4#1455656611yescharacteristic
309C2×C4#1455657611yescharacteristic
310C2×C4#1455661511yescharacteristic
311C2×C4#1455662011yescharacteristic
312C2×C4#1455662011yescharacteristic
313C2×C4#1455662311yescharacteristic
314C8—42no—
315C8—81no—
316C8—2922no—
317C8—5841no—
318C22—21no—
319C22—21no—
320C22—2924no—
321C22—2922no—
322C22—2922no—
323C22—2922no—
324C22—5842no—
325C22#1798846511yescharacteristic
326C22#1798858511yescharacteristic
327C22#1798859411yescharacteristic
328C4—21no—
329C4—21no—
330C4—21no—
331C4—44no—
332C4—2922no—
333C4—5842no—
334C4#1798875612yes—
335C4#1798883211yescharacteristic
336C4#1798883411yescharacteristic
337C4#1798884411yescharacteristic
338C4#1798884611yescharacteristic
339C2—21no—
340C2—21no—
341C2—5842no—
342C2#2261032811yescharacteristic
343C2#2261036311yescharacteristic
344C2#2261048211yescharacteristic
subgroupG/H
centerC22#17988465
commutatorC22×C4×C73C22×C4
FrattiniC2×C42C22×D73
FittingC73×#1303C2
socleC22×C73#386
prime radicalC22×D292D4
cosocleC2×C42×C73C23
hypercenter#1303D73
hypocenterC73#9403
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1303

Complements

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

NH ≅ G/Ntype
4. #22608617C2semidirect
5. #22608615C2semidirect
6. #22607834C2semidirect
7. C73×#1303C2semidirect
179. #1303D73semidirect
180. C73#9403semidirect

Maximal subgroups

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

subgroupindex
#226189712
#226148802
#226099202
#226086172
#226086152
#226078342
C73×#13032
#940373