Finite Group Databasebeta

← #32996500

Subgroups — #32996500

Order 18688.

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

normal subgroups only

This group has 470 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
1#22609670C211yesmaximal, characteristic, complemented
2#22608265C211yesmaximal, characteristic, complemented
3#22607961C212yesmaximal, complemented
4#22607934C212yesmaximal, complemented
5C73×#2610C211yesmaximal, characteristic
6C22×#11924012—21no—
7#17988644—21no—
8#17988356—21no—
9#17988356C2212yes—
10#17988313C2212yes—
11#17988311—21no—
12C2×#124×C73C411yescharacteristic
13C2×#117×C73C2211yescharacteristic
14#395×C73C2211yescharacteristic
15#394×C73C2211yescharacteristic
16#393×C73C411yescharacteristic
17#392×C73C411yescharacteristic
18#390×C73—21no—
19#388×C73—21no—
20#388×C73—21no—
21#386×C73—21no—
22#376×C73C411yescharacteristic
23C2×#11924012—24no—
24C2×#11924012—22no—
25C2×#11924012—41no—
26C2×#11924012—42no—
27C22×#10596831—41no—
28C4⋊#10596831—41no—
29C22⋊#10596831—22no—
30#14556502—41no—
31C22×Q8×C73C2×C411yescharacteristic
32C22×D4×C73—21no—
33C22×D4×C73C2×C411yescharacteristic
34C23×C4×C73D411yescharacteristic
35#124×C73—41no—
36#124×C73—41no—
37#117×C73—41no—
38#117×C73—41no—
39C2×C4⋊C4×C73—21no—
40C2×C4⋊C4×C73D411yescharacteristic
41C2×C4⋊C4×C73C2311yescharacteristic
42C2×C22⋊C4×C73—21no—
43C2×C22⋊C4×C73—21no—
44C2×C22⋊C4×C73—21no—
45C2×C22⋊C4×C73—21no—
46C2×C22⋊C4×C73—21no—
47C2×C22⋊C4×C73—21no—
48C2×C22⋊C4×C73C2×C411yescharacteristic
49C2×C22⋊C4×C73D411yescharacteristic
50C2×C42×C73—21no—
51C2×C42×C73C2×C411yescharacteristic
52C2×C42×C73D411yescharacteristic
53#95×C73—21no—
54#95×C73—21no—
55#95×C73—21no—
56#95×C73C2×C411yescharacteristic
57#95×C73C2×C411yescharacteristic
58#11924012—24no—
59#11924012—44no—
60#11924012—42no—
61C2×#10596831—44no—
62C2×#10596831—42no—
63C2×#10596831—42no—
64C24×C73—21no—
65C24×C73C22⋊C411yescharacteristic
66C2×Q8×C73—41no—
67C2×Q8×C73—81no—
68C2×D4×C73—41no—
69C2×D4×C73—42no—
70C2×D4×C73—42no—
71C2×D4×C73—81no—
72C2×D4×C73—81no—
73C22×C4×C73—22no—
74C22×C4×C73—21no—
75C22×C4×C73—21no—
76C22×C4×C73—21no—
77C22×C4×C73—21no—
78C22×C4×C73—21no—
79C22×C4×C73—21no—
80C22×C4×C73—21no—
81C22×C4×C73C22⋊C411yescharacteristic
82C22×C4×C73C22⋊C411yescharacteristic
83C22×C4×C73C22⋊C411yescharacteristic
84C22×C4×C73C22×C411yescharacteristic
85C22×C4×C73C2×D411yescharacteristic
86C22×C4×C73C2×D411yescharacteristic
87C4⋊C4×C73—41no—
88C4⋊C4×C73—41no—
89C4⋊C4×C73—81no—
90C22⋊C4×C73—41no—
91C22⋊C4×C73—41no—
92C22⋊C4×C73—42no—
93C22⋊C4×C73—42no—
94C22⋊C4×C73—41no—
95C22⋊C4×C73—41no—
96C22⋊C4×C73—41no—
97C22⋊C4×C73—41no—
98C22⋊C4×C73—81no—
99C22⋊C4×C73—81no—
100C42×C73—21no—
101C42×C73—21no—
102C42×C73—21no—
103C42×C73—21no—
104C42×C73—42no—
105C23×C73—21no—
106C23×C73—21no—
107C23×C73—41no—
108C23×C73—41no—
109C23×C73—41no—
110C23×C73—41no—
111C23×C73—41no—
112C23×C73—41no—
113C23×C73—41no—
114C23×C73—41no—
115C23×C73—41no—
116C23×C73#10412yes—
117C23×C73C2×C22⋊C411yescharacteristic
118Q8×C73—82no—
119D4×C73—84no—
120D4×C73—82no—
121C2×C4×C73—22no—
122C2×C4×C73—22no—
123C2×C4×C73—22no—
124C2×C4×C73—22no—
125C2×C4×C73—21no—
126C2×C4×C73—21no—
127C2×C4×C73—21no—
128C2×C4×C73—21no—
129C2×C4×C73—22no—
130C2×C4×C73—21no—
131C2×C4×C73—21no—
132C2×C4×C73—21no—
133C2×C4×C73—21no—
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—41no—
141C2×C4×C73—41no—
142C2×C4×C73—41no—
143C2×C4×C73—41no—
144C2×C4×C73—41no—
145C2×C4×C73—41no—
146C2×C4×C73#9911yescharacteristic
147C2×C4×C73#9911yescharacteristic
148C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
149C2×C4×C73#13711yescharacteristic
150#10596831—44no—
151#10596831—82no—
152C22×C73—21no—
153C22×C73—21no—
154C22×C73—22no—
155C22×C73—42no—
156C22×C73—41no—
157C22×C73—41no—
158C22×C73—81no—
159C22×C73—81no—
160C22×C73—81no—
161C22×C73—81no—
162C22×C73—81no—
163C22×C73—81no—
164C22×C73—81no—
165C22×C73—81no—
166C22×C73C2×#9911yescharacteristic
167C22×C73#41711yescharacteristic
168C22×C73C2×#10411yescharacteristic
169C4×C73—21no—
170C4×C73—21no—
171C4×C73—21no—
172C4×C73—21no—
173C4×C73—42no—
174C4×C73—41no—
175C4×C73—41no—
176C4×C73—41no—
177C4×C73—41no—
178C4×C73—81no—
179C4×C73—81no—
180#11765—731nomaximal, Sylow
181C2×C73—21no—
182C2×C73—21no—
183C2×C73—42no—
184C2×C73—161no—
185C2×C73#150411yescharacteristic
186C2×C73#212211yescharacteristic
187C2×C73#213211yescharacteristic
188#2610D7311yescharacteristic
189#1742—731no—
190#1464—731no—
191#1354—732no—
192#1345—732no—
193C73#1176511yescharacteristic, complemented, Sylow
194C22×M16—1461no—
195C2×#124Dic7311yescharacteristic
196C2×#117C2×D7311yescharacteristic
197#432—1461no—
198#395C2×D7311yescharacteristic
199#394C2×D7311yescharacteristic
200#393Dic7311yescharacteristic
201#392Dic7311yescharacteristic
202#390—21no—
203#388—21no—
204#388—21no—
205#386—21no—
206#376Dic7311yescharacteristic
207#343—732no—
208#343—1461no—
209#324—732no—
210#323—1461no—
211C22×Q8C2×Dic7311yescharacteristic
212C22×D4—21no—
213C22×D4C2×Dic7311yescharacteristic
214C23×C4C22⋊D7311yescharacteristic
215C2×M16—2921no—
216C2×M16—2922no—
217C2×M16—1464no—
218C2×M16—1462no—
219C22×C8—2921no—
220#124—41no—
221#124—41no—
222#117—41no—
223#117—41no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4C22⋊D7311yescharacteristic
226C2×C4⋊C4C22×D7311yescharacteristic
227C2×C22⋊C4—21no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—21no—
232C2×C22⋊C4—21no—
233C2×C22⋊C4C2×Dic7311yescharacteristic
234C2×C22⋊C4C22⋊D7311yescharacteristic
235C2×C42—21no—
236C2×C42C2×Dic7311yescharacteristic
237C2×C42C22⋊D7311yescharacteristic
238C4⋊C8—2921no—
239C22⋊C8—1462no—
240#97—2921no—
241#95—21no—
242#95—21no—
243#95—21no—
244#95C2×Dic7311yescharacteristic
245#95C2×Dic7311yescharacteristic
246C24—21no—
247C24C22⋊Dic7311yescharacteristic
248C2×Q8—41no—
249C2×Q8—81no—
250C2×D4—41no—
251C2×D4—42no—
252C2×D4—42no—
253C2×D4—81no—
254C2×D4—81no—
255C22×C4—22no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4C22⋊Dic7311yescharacteristic
264C22×C4C22⋊Dic7311yescharacteristic
265C22×C4C22⋊Dic7311yescharacteristic
266C22×C4C22×Dic7311yescharacteristic
267C22×C4C2×C22⋊D7311yescharacteristic
268C22×C4C2×C22⋊D7311yescharacteristic
269M16—2924no—
270M16—2922no—
271M16—1464no—
272C2×C8—2924no—
273C2×C8—2922no—
274C2×C8—2922no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—81no—
278C22⋊C4—41no—
279C22⋊C4—41no—
280C22⋊C4—42no—
281C22⋊C4—42no—
282C22⋊C4—41no—
283C22⋊C4—41no—
284C22⋊C4—41no—
285C22⋊C4—41no—
286C22⋊C4—81no—
287C22⋊C4—81no—
288C42—21no—
289C42—21no—
290C42—21no—
291C42—21no—
292C42—42no—
293C23—21no—
294C23—21no—
295C23—41no—
296C23—41no—
297C23—41no—
298C23—41no—
299C23—41no—
300C23—41no—
301C23—41no—
302C23—41no—
303C23—41no—
304C23#1455651812yes—
305C23C2×C22⋊Dic7311yescharacteristic
306Q8—82no—
307D4—84no—
308D4—82no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—22no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4#1455650611yescharacteristic
335C2×C4#1455650611yescharacteristic
336C2×C4#1455661611yescharacteristic
337C2×C4#1455662111yescharacteristic
338C8—2924no—
339C8—5842no—
340C22—21no—
341C22—21no—
342C22—22no—
343C22—42no—
344C22—41no—
345C22—41no—
346C22—81no—
347C22—81no—
348C22—81no—
349C22—81no—
350C22—81no—
351C22—81no—
352C22—81no—
353C22—81no—
354C22C2×#1455650611yescharacteristic
355C22#1798858811yescharacteristic
356C22C2×#1455651811yescharacteristic
357C4—21no—
358C4—21no—
359C4—21no—
360C4—21no—
361C4—42no—
362C4—41no—
363C4—41no—
364C4—41no—
365C4—41no—
366C4—81no—
367C4—81no—
368C2—21no—
369C2—21no—
370C2—42no—
371C2—161no—
372C2#2260843511yescharacteristic
373C2#2261158111yescharacteristic
374C2#2261161611yescharacteristic
subgroupG/H
centerC22C2×#14556506
commutatorC22×C4×C73C22×C4
FrattiniC2×C4⋊C4C22×D73
FittingC73×#2610C2
socleC22×C73C2×#99
prime radical#392×C73C4
cosocleC2×C4⋊C4×C73C23
hypercenter#2610D73
hypocenterC73#11765
LCSG ▷ C22×C4×C73 ▷ C23×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2610

Complements

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

NH ≅ G/Ntype
1. #22609670C2semidirect
2. #22608265C2semidirect
3. #22607961C2semidirect
4. #22607934C2semidirect
193. C73#11765semidirect

Maximal subgroups

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

subgroupindex
#226096702
#226082652
#226079612
#226079342
C73×#26102
#1176573