Finite Group Databasebeta

← #32996338

Subgroups — #32996338

Order 18688.

436 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 91 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
1C4×#14556506C211yesmaximal, characteristic
2#22608252C211yesmaximal, characteristic
3#22607952C212yesmaximal, complemented
4#22607931C212yesmaximal, complemented
5C73×#2638C211yesmaximal, characteristic
6#17988640—21no—
7C2×#14556506—21no—
8C4×C22⋊Dic73—21no—
9#17988354C2212yes—
10#17988311—21no—
11#17988311C2212yes—
12C2×C22⋊Q8×C73C411yescharacteristic, complemented
13C2×C4×D4×C73C2211yescharacteristic
14#398×C73C411yescharacteristic, complemented
15#397×C73—21no—
16#397×C73C2211yescharacteristic
17#395×C73C411yescharacteristic, complemented
18#394×C73—21no—
19#393×C73C2211yescharacteristic
20#387×C73—21no—
21#385×C73—21no—
22#381×C73C411yescharacteristic, complemented
23C2×C22⋊Dic73—22no—
24C2×C4×Dic73—41no—
25C4⋊#10596831—41no—
26#14556506—24no—
27C22⋊#10596831—22no—
28C4×#10596831—41no—
29#14556498—41no—
30C22×Q8×C73C2×C411yescharacteristic
31C22×D4×C73D411yescharacteristic
32C23×C4×C73C2×C411yescharacteristic
33C23×C4×C73D411yescharacteristic
34C22⋊Q8×C73—41no—
35C22⋊Q8×C73—41no—
36C4×D4×C73—41no—
37C4×D4×C73—41no—
38C2×C4⋊C4×C73—21no—
39C2×C4⋊C4×C73—21no—
40C2×C4⋊C4×C73—21no—
41C2×C4⋊C4×C73C2×C411yescharacteristic
42C2×C4⋊C4×C73C2×C411yescharacteristic
43C2×C22⋊C4×C73—21no—
44C2×C22⋊C4×C73—21no—
45C2×C22⋊C4×C73—21no—
46C2×C22⋊C4×C73—21no—
47C2×C22⋊C4×C73D411yescharacteristic
48C2×C22⋊C4×C73C2311yescharacteristic
49C2×C42×C73D411yescharacteristic
50#95×C73—21no—
51#95×C73—21no—
52#95×C73—21no—
53#95×C73—21no—
54#95×C73—21no—
55#95×C73C2×C411yescharacteristic
56#95×C73C2×C411yescharacteristic
57C22×Dic73—42no—
58C2×#10596831—42no—
59C22⋊Dic73—24no—
60C22⋊Dic73—42no—
61C4×Dic73—82no—
62C24×C73C22⋊C411yescharacteristic
63C24×C73C2×D411yescharacteristic
64C2×Q8×C73—41no—
65C2×Q8×C73—81no—
66C2×D4×C73—22no—
67C2×D4×C73—22no—
68C2×D4×C73—41no—
69C22×C4×C73—21no—
70C22×C4×C73—21no—
71C22×C4×C73—21no—
72C22×C4×C73—21no—
73C22×C4×C73—21no—
74C22×C4×C73—21no—
75C22×C4×C73—21no—
76C22×C4×C73—21no—
77C22×C4×C73—81no—
78C22×C4×C73—81no—
79C22×C4×C73C22⋊C411yescharacteristic
80C22×C4×C73C22⋊C411yescharacteristic
81C22×C4×C73C22⋊C411yescharacteristic
82C22×C4×C73C22×C411yescharacteristic
83C22×C4×C73C2×D411yescharacteristic
84C4⋊C4×C73—41no—
85C4⋊C4×C73—41no—
86C4⋊C4×C73—41no—
87C4⋊C4×C73—41no—
88C4⋊C4×C73—42no—
89C4⋊C4×C73—81no—
90C22⋊C4×C73—41no—
91C22⋊C4×C73—41no—
92C22⋊C4×C73—42no—
93C22⋊C4×C73—42no—
94C22⋊C4×C73—81no—
95C22⋊C4×C73—81no—
96C42×C73—21no—
97C42×C73—21no—
98C23×C73—21no—
99C23×C73—21no—
100C23×C73—21no—
101C23×C73—21no—
102C23×C73—21no—
103C23×C73—41no—
104C23×C73—41no—
105C23×C73—41no—
106C23×C73—41no—
107C23×C73C2×C22⋊C411yescharacteristic
108C23×C73#13512yes—
109Q8×C73—82no—
110D4×C73—44no—
111C2×C4×C73—21no—
112C2×C4×C73—21no—
113C2×C4×C73—21no—
114C2×C4×C73—21no—
115C2×C4×C73—21no—
116C2×C4×C73—41no—
117C2×C4×C73—42no—
118C2×C4×C73—41no—
119C2×C4×C73—41no—
120C2×C4×C73—41no—
121C2×C4×C73—41no—
122C2×C4×C73—41no—
123C2×C4×C73—41no—
124C2×C4×C73—41no—
125C2×C4×C73—41no—
126C2×C4×C73—41no—
127C2×C4×C73—41no—
128C2×C4×C73—41no—
129C2×C4×C73—41no—
130C2×C4×C73—41no—
131C2×C4×C73—41no—
132C2×C4×C73—41no—
133C2×C4×C73—81no—
134C2×C4×C73—81no—
135C2×C4×C73—81no—
136C2×C4×C73#9911yescharacteristic
137C2×C4×C73#9911yescharacteristic
138C2×C4×C73#10412yes—
139C2×Dic73—44no—
140C2×Dic73—82no—
141C2×Dic73—82no—
142#10596831—82no—
143C22×C73—21no—
144C22×C73—21no—
145C22×C73—22no—
146C22×C73—42no—
147C22×C73—41no—
148C22×C73—41no—
149C22×C73—41no—
150C22×C73—41no—
151C22×C73—41no—
152C22×C73—81no—
153C22×C73—81no—
154C22×C73—81no—
155C22×C73C2×#9911yescharacteristic
156C22×C73#41611yescharacteristic
157C22×C73C2×#10411yescharacteristic
158C4×C73—21no—
159C4×C73—21no—
160C4×C73—22no—
161C4×C73—41no—
162C4×C73—41no—
163C4×C73—41no—
164C4×C73—81no—
165C4×C73—81no—
166C4×C73—81no—
167C4×C73—81no—
168Dic73—84no—
169#11731—731nomaximal, Sylow
170C2×C73—21no—
171C2×C73—21no—
172C2×C73—42no—
173C2×C73—81no—
174C2×C73#150111yescharacteristic
175C2×C73#212111yescharacteristic
176C2×C73#212911yescharacteristic
177#2638D7311yescharacteristic
178C4×#99—731no—
179#1460—731no—
180#1351—732no—
181#1344—732no—
182C73#1173111yescharacteristic, complemented, Sylow
183C2×C22⋊Q8Dic7311yescharacteristic, complemented
184C2×C4×D4C2×D7311yescharacteristic
185#431—1461no—
186C2×#99—1461no—
187#398Dic7311yescharacteristic, complemented
188#397—21no—
189#397C2×D7311yescharacteristic
190#395Dic7311yescharacteristic, complemented
191#394—21no—
192#393C2×D7311yescharacteristic
193#387—21no—
194#385—21no—
195#381Dic7311yescharacteristic, complemented
196C4×C22⋊C4—1461no—
197#342—732no—
198#323—732no—
199#323—1461no—
200C22×Q8C2×Dic7311yescharacteristic
201C22×D4C22⋊D7311yescharacteristic
202C23×C4C2×Dic7311yescharacteristic
203C23×C4C22⋊D7311yescharacteristic
204C22⋊Q8—41no—
205C22⋊Q8—41no—
206C4×D4—41no—
207C4×D4—41no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4C2×Dic7311yescharacteristic
212C2×C4⋊C4C2×Dic7311yescharacteristic
213C2×C22⋊C4—21no—
214C2×C22⋊C4—21no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4—1462no—
218C2×C22⋊C4C22⋊D7311yescharacteristic
219C2×C22⋊C4C22×D7311yescharacteristic
220C2×C42—2921no—
221C2×C42C22⋊D7311yescharacteristic
222C4⋊C8—2921no—
223#99—1464no—
224C22⋊C8—1462no—
225C4×C8—2921no—
226#95—21no—
227#95—21no—
228#95—21no—
229#95—21no—
230#95—21no—
231#95—2921no—
232#95C2×Dic7311yescharacteristic
233#95C2×Dic7311yescharacteristic
234C24C22⋊Dic7311yescharacteristic
235C24C2×C22⋊D7311yescharacteristic
236C2×Q8—41no—
237C2×Q8—81no—
238C2×D4—22no—
239C2×D4—22no—
240C2×D4—41no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—81no—
250C22×C4—81no—
251C22×C4—2922no—
252C22×C4C22⋊Dic7311yescharacteristic
253C22×C4C22⋊Dic7311yescharacteristic
254C22×C4C22⋊Dic7311yescharacteristic
255C22×C4C22×Dic7311yescharacteristic
256C22×C4C2×C22⋊D7311yescharacteristic
257C2×C8—2922no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C4⋊C4—41no—
261C4⋊C4—41no—
262C4⋊C4—42no—
263C4⋊C4—81no—
264C22⋊C4—41no—
265C22⋊C4—41no—
266C22⋊C4—42no—
267C22⋊C4—42no—
268C22⋊C4—81no—
269C22⋊C4—81no—
270C22⋊C4—2922no—
271C22⋊C4—1464no—
272C42—21no—
273C42—21no—
274C42—5842no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23C2×C22⋊Dic7311yescharacteristic
285C23#1455661112yes—
286Q8—82no—
287D4—44no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—41no—
294C2×C4—42no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—81no—
311C2×C4—81no—
312C2×C4—81no—
313C2×C4—2924no—
314C2×C4—5842no—
315C2×C4—5842no—
316C2×C4#1455650611yescharacteristic
317C2×C4#1455650611yescharacteristic
318C2×C4#1455651812yes—
319C8—5842no—
320C22—21no—
321C22—21no—
322C22—22no—
323C22—42no—
324C22—41no—
325C22—41no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—81no—
330C22—81no—
331C22—81no—
332C22C2×#1455650611yescharacteristic
333C22#1798858311yescharacteristic
334C22C2×#1455651811yescharacteristic
335C4—22no—
336C4—21no—
337C4—21no—
338C4—41no—
339C4—41no—
340C4—41no—
341C4—81no—
342C4—81no—
343C4—81no—
344C4—81no—
345C4—5844no—
346C2—21no—
347C2—21no—
348C2—42no—
349C2—81no—
350C2#2260842211yescharacteristic
351C2#2261157711yescharacteristic
352C2#2261160611yescharacteristic
subgroupG/H
centerC22C2×#14556506
commutatorC22×C4×C73C22×C4
FrattiniC2×C22⋊C4C22×D73
FittingC73×#2638C2
socleC22×C73C2×#99
prime radicalC22×D4×C73D4
cosocleC2×C22⋊C4×C73C23
hypercenter#2638D73
hypocenterC73#11731
LCSG ▷ C22×C4×C73 ▷ C23×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2638

Complements

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

NH ≅ G/Ntype
3. #22607952C2semidirect
4. #22607931C2semidirect
12. C2×C22⋊Q8×C73C4semidirect
14. #398×C73C4semidirect
17. #395×C73C4semidirect
22. #381×C73C4semidirect
182. C73#11731semidirect
183. C2×C22⋊Q8Dic73semidirect
187. #398Dic73semidirect
190. #395Dic73semidirect
195. #381Dic73semidirect

Maximal subgroups

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

subgroupindex
C4×#145565062
#226082522
#226079522
#226079312
C73×#26382
#1173173