Finite Group Databasebeta

← #32983649

Subgroups — #32983649

Order 18688.

486 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 486 conjugacy classes of subgroups — too many to draw. Only its 107 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#22619016C211yesmaximal, characteristic, complemented
2#22608555C212yesmaximal
3#22608275C211yesmaximal, characteristic, complemented
4#22607787C212yesmaximal, complemented
5C73×#2545C211yesmaximal, characteristic
6#17988718—24no—
7#17988644—21no—
8#17988644C2212yes—
9#17988644C2212yes—
10#17988640—21no—
11#17988327—24no—
12#17988313—21no—
13#17988313—21no—
14C2×C4⋊Q8×C73C411yescharacteristic
15C2×#124×C73C411yescharacteristic
16C2×#124×C73C2211yescharacteristic
17C2×C4×Q8×C73C2211yescharacteristic
18#396×C73—21no—
19#396×C73—21no—
20#391×C73C411yescharacteristic
21#390×C73C411yescharacteristic
22#388×C73—21no—
23#388×C73—21no—
24C4×C4⋊C4×C73C2211yescharacteristic
25C4⋊#10596831—22no—
26C4⋊#10596831—22no—
27C4⋊#10596831—22no—
28C4⋊#10596831—22no—
29C22⋊#10596831—22no—
30C22⋊#10596831—22no—
31#14556502—22no—
32#14556502—22no—
33#14556502D414yes—
34C4×#10596831—22no—
35C22×Q8×C73C2×C411yescharacteristic
36C22×Q8×C73C2×C411yescharacteristic
37C22×D4×C73C2×C411yescharacteristic
38C4⋊Q8×C73—42no—
39#124×C73—22no—
40#124×C73—22no—
41#124×C73—42no—
42C4×Q8×C73—22no—
43C4×Q8×C73—22no—
44C2×C4⋊C4×C73—21no—
45C2×C4⋊C4×C73—21no—
46C2×C4⋊C4×C73—21no—
47C2×C4⋊C4×C73C2×C411yescharacteristic
48C2×C4⋊C4×C73D411yescharacteristic
49C2×C4⋊C4×C73D411yescharacteristic
50C2×C22⋊C4×C73—21no—
51C2×C22⋊C4×C73—21no—
52C2×C22⋊C4×C73—21no—
53C2×C22⋊C4×C73—21no—
54C2×C22⋊C4×C73—21no—
55C2×C22⋊C4×C73—21no—
56C2×C42×C73C2×C411yescharacteristic
57C2×C42×C73C2×C411yescharacteristic
58C2×C42×C73D411yescharacteristic
59C2×C42×C73D411yescharacteristic
60C2×C42×C73C2311yescharacteristic
61#95×C73—21no—
62#95×C73—21no—
63#95×C73—21no—
64C2×#10596831—24no—
65C2×#10596831—42no—
66C24×C73—21no—
67C2×Q8×C73—22no—
68C2×Q8×C73—22no—
69C2×Q8×C73—22no—
70C2×Q8×C73—41no—
71C2×Q8×C73—41no—
72C2×Q8×C73—41no—
73C2×D4×C73—22no—
74C2×D4×C73—41no—
75C2×D4×C73—41no—
76C22×C4×C73—21no—
77C22×C4×C73—21no—
78C22×C4×C73—21no—
79C22×C4×C73C22⋊C411yescharacteristic
80C22×C4×C73C22⋊C411yescharacteristic
81C22×C4×C73C22⋊C411yescharacteristic
82C22×C4×C73C22⋊C411yescharacteristic
83C22×C4×C73C22×C411yescharacteristic
84C22×C4×C73C2×D411yescharacteristic
85C22×C4×C73C2×D411yescharacteristic
86C4⋊C4×C73—22no—
87C4⋊C4×C73—22no—
88C4⋊C4×C73—22no—
89C4⋊C4×C73—42no—
90C4⋊C4×C73—42no—
91C4⋊C4×C73—42no—
92C22⋊C4×C73—42no—
93C22⋊C4×C73—42no—
94C22⋊C4×C73—42no—
95C22⋊C4×C73—42no—
96C22⋊C4×C73—41no—
97C22⋊C4×C73—42no—
98C22⋊C4×C73—41no—
99C42×C73—21no—
100C42×C73—21no—
101C42×C73—21no—
102C42×C73—21no—
103C42×C73—41no—
104C42×C73—41no—
105C42×C73C2×D412yes—
106C42×C73#4112yes—
107C23×C73—41no—
108C23×C73—41no—
109C23×C73—41no—
110C23×C73—41no—
111C23×C73—41no—
112C23×C73—41no—
113C23×C73—41no—
114C23×C73C2×C22⋊C411yescharacteristic
115Q8×C73—44no—
116Q8×C73—82no—
117D4×C73—82no—
118C2×C4×C73—21no—
119C2×C4×C73—21no—
120C2×C4×C73—21no—
121C2×C4×C73—21no—
122C2×C4×C73—22no—
123C2×C4×C73—21no—
124C2×C4×C73—21no—
125C2×C4×C73—21no—
126C2×C4×C73—21no—
127C2×C4×C73—21no—
128C2×C4×C73—41no—
129C2×C4×C73—41no—
130C2×C4×C73—41no—
131C2×C4×C73—41no—
132C2×C4×C73—41no—
133C2×C4×C73—41no—
134C2×C4×C73—41no—
135C2×C4×C73—41no—
136C2×C4×C73—41no—
137C2×C4×C73—41no—
138C2×C4×C73—41no—
139C2×C4×C73—41no—
140C2×C4×C73#10012yes—
141C2×C4×C73C4×D412yes—
142C2×C4×C73#12011yescharacteristic
143C2×C4×C73#12112yes—
144C2×C4×C73#12311yescharacteristic
145#10596831—44no—
146#10596831—82no—
147C22×C73—21no—
148C22×C73—21no—
149C22×C73—81no—
150C22×C73—81no—
151C22×C73—81no—
152C22×C73—81no—
153C22×C73—81no—
154C22×C73—81no—
155C22×C73—81no—
156C22×C73#38611yescharacteristic
157C22×C73C2×#10011yescharacteristic
158C22×C73#41311yescharacteristic
159C4×C73—22no—
160C4×C73—22no—
161C4×C73—22no—
162C4×C73—21no—
163C4×C73—21no—
164C4×C73—42no—
165C4×C73—41no—
166C4×C73—41no—
167C4×C73—41no—
168C4×C73—81no—
169C4×C73—81no—
170#9448—731nomaximal, Sylow
171C2×C73—21no—
172C2×C73—21no—
173C2×C73—161no—
174C2×C73#188411yescharacteristic
175C2×C73#190011yescharacteristic
176C2×C73#191211yescharacteristic
177#2983—731no—
178#2545D7311yescharacteristic
179#1533—732no—
180#1467—731no—
181#1285—732no—
182C73#944811yescharacteristic, complemented, Sylow
183C2×C4⋊Q8Dic7311yescharacteristic
184C2×#124Dic7311yescharacteristic
185C2×#124C2×D7311yescharacteristic
186C2×C4×Q8C2×D7311yescharacteristic
187#446—1464no—
188#432—732no—
189#432—732no—
190#432—1461no—
191#431—1461no—
192#396—21no—
193#396—21no—
194#391Dic7311yescharacteristic
195#390Dic7311yescharacteristic
196#388—21no—
197#388—21no—
198C4×C4⋊C4C2×D7311yescharacteristic
199#329—1464no—
200#324—1461no—
201#324—1461no—
202C22×Q8C2×Dic7311yescharacteristic
203C22×Q8C2×Dic7311yescharacteristic
204C22×D4C2×Dic7311yescharacteristic
205C4⋊Q8—42no—
206#124—22no—
207#124—22no—
208#124—42no—
209C4×Q8—22no—
210C4×Q8—22no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4C2×Dic7311yescharacteristic
215C2×C4⋊C4C22⋊D7311yescharacteristic
216C2×C4⋊C4C22⋊D7311yescharacteristic
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C42C2×Dic7311yescharacteristic
224C2×C42C2×Dic7311yescharacteristic
225C2×C42C22⋊D7311yescharacteristic
226C2×C42C22⋊D7311yescharacteristic
227C2×C42C22×D7311yescharacteristic
228C4⋊C8—1462no—
229C4⋊C8—1462no—
230C4⋊C8—1462no—
231C4⋊C8—1462no—
232C22⋊C8—1462no—
233C22⋊C8—1462no—
234#97—734no—
235#97—1462no—
236#97—1462no—
237C4×C8—1462no—
238#95—21no—
239#95—21no—
240#95—21no—
241C24—21no—
242C2×Q8—22no—
243C2×Q8—22no—
244C2×Q8—22no—
245C2×Q8—41no—
246C2×Q8—41no—
247C2×Q8—41no—
248C2×D4—22no—
249C2×D4—41no—
250C2×D4—41no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4C22⋊Dic7311yescharacteristic
255C22×C4C22⋊Dic7311yescharacteristic
256C22×C4C22⋊Dic7311yescharacteristic
257C22×C4C22⋊Dic7311yescharacteristic
258C22×C4C22×Dic7311yescharacteristic
259C22×C4C2×C22⋊D7311yescharacteristic
260C22×C4C2×C22⋊D7311yescharacteristic
261C2×C8—2922no—
262C2×C8—1464no—
263C4⋊C4—22no—
264C4⋊C4—22no—
265C4⋊C4—22no—
266C4⋊C4—42no—
267C4⋊C4—42no—
268C4⋊C4—42no—
269C22⋊C4—42no—
270C22⋊C4—42no—
271C22⋊C4—42no—
272C22⋊C4—42no—
273C22⋊C4—41no—
274C22⋊C4—42no—
275C22⋊C4—41no—
276C42—21no—
277C42—21no—
278C42—21no—
279C42—21no—
280C42—41no—
281C42—41no—
282C42D4×D7312yes—
283C42#1192402712yes—
284C23—41no—
285C23—41no—
286C23—41no—
287C23—41no—
288C23—41no—
289C23—41no—
290C23—41no—
291C23C2×C22⋊Dic7311yescharacteristic
292Q8—44no—
293Q8—82no—
294D4—82no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—22no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4#1455650812yes—
318C2×C4D4×Dic7312yes—
319C2×C4#1455655911yescharacteristic
320C2×C4#1455656312yes—
321C2×C4#1455657211yescharacteristic
322C8—2924no—
323C8—5842no—
324C22—21no—
325C22—21no—
326C22—81no—
327C22—81no—
328C22—81no—
329C22—81no—
330C22—81no—
331C22—81no—
332C22—81no—
333C22#1798846711yescharacteristic
334C22C2×#1455650811yescharacteristic
335C22#1798857111yescharacteristic
336C4—21no—
337C4—21no—
338C4—22no—
339C4—22no—
340C4—22no—
341C4—42no—
342C4—41no—
343C4—41no—
344C4—41no—
345C4—81no—
346C4—81no—
347C2—21no—
348C2—21no—
349C2—161no—
350C2#2261031511yescharacteristic
351C2#2261039511yescharacteristic
352C2#2261046911yescharacteristic
subgroupG/H
centerC22#17988467
commutatorC22×C4×C73C22×C4
FrattiniC2×C42C22×D73
FittingC73×#2545C2
socleC22×C73#386
prime radicalC22×D4×C73C2×C4
cosocleC2×C42×C73C23
hypercenter#2545D73
hypocenterC73#9448
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2545

Complements

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

NH ≅ G/Ntype
1. #22619016C2semidirect
3. #22608275C2semidirect
4. #22607787C2semidirect
182. C73#9448semidirect

Maximal subgroups

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

subgroupindex
#226190162
#226085552
#226082752
#226077872
C73×#25452
#944873