Finite Group Databasebeta

← #32996505

Subgroups — #32996505

Order 18688.

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

normal subgroups only

This group has 488 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
1C2×#17988356C211yesmaximal, characteristic, complemented
2#22608261C211yesmaximal, characteristic, complemented
3#22607958C212yesmaximal, complemented
4#22607934C212yesmaximal, complemented
5C73×#2591C211yesmaximal, characteristic
6C22×#11924012—21no—
7#17988551—21no—
8#17988356—21no—
9#17988356C2212yes—
10#17988313—21no—
11#17988313C2212yes—
12C2×#121×C73C411yescharacteristic
13C22×C4⋊C4×C73C2211yescharacteristic
14#400×C73C411yescharacteristic
15#400×C73C2211yescharacteristic
16#394×C73C411yescharacteristic
17#394×C73C2211yescharacteristic
18#388×C73—21no—
19#388×C73—21no—
20#386×C73—21no—
21#386×C73—21no—
22C4×C22⋊C4×C73C411yescharacteristic
23C2×#11924012—24no—
24C2×#11924012—22no—
25C2×#11924012—41no—
26C2×#11924012—42no—
27C2×#11924012—41no—
28C22×#10596831—41no—
29C22⋊#10596831—22no—
30C22⋊#10596831—41no—
31C22×D4×C73—21no—
32C22×D4×C73C2×C411yescharacteristic
33C23×C4×C73C2×C411yescharacteristic
34C23×C4×C73D411yescharacteristic
35C23×C4×C73D411yescharacteristic
36#121×C73—41no—
37#121×C73—41no—
38C2×C4⋊C4×C73—41no—
39C2×C4⋊C4×C73—41no—
40C2×C4⋊C4×C73C2×C411yescharacteristic
41C2×C4⋊C4×C73C2×C411yescharacteristic
42C2×C4⋊C4×C73D411yescharacteristic
43C2×C4⋊C4×C73D411yescharacteristic
44C2×C4⋊C4×C73C2311yescharacteristic
45C2×C22⋊C4×C73—21no—
46C2×C22⋊C4×C73—21no—
47C2×C22⋊C4×C73—21no—
48C2×C22⋊C4×C73—21no—
49C2×C22⋊C4×C73—21no—
50C2×C22⋊C4×C73—21no—
51C2×C42×C73—21no—
52#95×C73—21no—
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—
61#11924012—82no—
62C2×#10596831—44no—
63C2×#10596831—42no—
64C2×#10596831—42no—
65C24×C73—21no—
66C24×C73C22⋊C411yescharacteristic
67C2×D4×C73—41no—
68C2×D4×C73—42no—
69C2×D4×C73—42no—
70C2×D4×C73—41no—
71C2×D4×C73—41no—
72C2×D4×C73—81no—
73C22×C4×C73—21no—
74C22×C4×C73—21no—
75C22×C4×C73—22no—
76C22×C4×C73—21no—
77C22×C4×C73—21no—
78C22×C4×C73—21no—
79C22×C4×C73—21no—
80C22×C4×C73—21no—
81C22×C4×C73—21no—
82C22×C4×C73—21no—
83C22×C4×C73—21no—
84C22×C4×C73—21no—
85C22×C4×C73—81no—
86C22×C4×C73—81no—
87C22×C4×C73C22⋊C411yescharacteristic
88C22×C4×C73C22⋊C411yescharacteristic
89C22×C4×C73C22⋊C411yescharacteristic
90C22×C4×C73C22×C411yescharacteristic
91C22×C4×C73C2×D411yescharacteristic
92C22×C4×C73C2×D411yescharacteristic
93C4⋊C4×C73—41no—
94C4⋊C4×C73—41no—
95C4⋊C4×C73—41no—
96C4⋊C4×C73—41no—
97C4⋊C4×C73—41no—
98C22⋊C4×C73—42no—
99C22⋊C4×C73—42no—
100C22⋊C4×C73—42no—
101C22⋊C4×C73—41no—
102C22⋊C4×C73—41no—
103C22⋊C4×C73—81no—
104C22⋊C4×C73—81no—
105C42×C73—81no—
106C23×C73—22no—
107C23×C73—21no—
108C23×C73—21no—
109C23×C73—41no—
110C23×C73—41no—
111C23×C73—41no—
112C23×C73—41no—
113C23×C73—41no—
114C23×C73—41no—
115C23×C73—41no—
116C23×C73—41no—
117C23×C73#9911yescharacteristic
118C23×C73#9911yescharacteristic
119C23×C73C2×C22⋊C411yescharacteristic
120D4×C73—82no—
121D4×C73—82no—
122D4×C73—82no—
123C2×C4×C73—22no—
124C2×C4×C73—22no—
125C2×C4×C73—22no—
126C2×C4×C73—22no—
127C2×C4×C73—21no—
128C2×C4×C73—21no—
129C2×C4×C73—21no—
130C2×C4×C73—21no—
131C2×C4×C73—21no—
132C2×C4×C73—21no—
133C2×C4×C73—21no—
134C2×C4×C73—21no—
135C2×C4×C73—42no—
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—41no—
147C2×C4×C73—41no—
148C2×C4×C73—41no—
149C2×C4×C73—81no—
150C2×C4×C73—81no—
151C2×C4×C73—81no—
152C2×C4×C73#10412yes—
153C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
154C2×C4×C73#13711yescharacteristic
155#10596831—44no—
156#10596831—82no—
157C22×C73—21no—
158C22×C73—21no—
159C22×C73—21no—
160C22×C73—21no—
161C22×C73—21no—
162C22×C73—21no—
163C22×C73—21no—
164C22×C73—42no—
165C22×C73—42no—
166C22×C73—81no—
167C22×C73—81no—
168C22×C73—81no—
169C22×C73—81no—
170C22×C73—81no—
171C22×C73—81no—
172C22×C73—81no—
173C22×C73#35512yes—
174C22×C73C2×#9911yescharacteristic
175C22×C73#41711yescharacteristic
176C22×C73C2×#10411yescharacteristic
177C4×C73—21no—
178C4×C73—22no—
179C4×C73—21no—
180C4×C73—41no—
181C4×C73—41no—
182C4×C73—41no—
183C4×C73—41no—
184C4×C73—81no—
185C4×C73—81no—
186C4×C73—81no—
187#11766—731nomaximal, Sylow
188C2×C73—21no—
189C2×C73—21no—
190C2×C73—22no—
191C2×C73—41no—
192C2×C73—161no—
193C2×C73#150411yescharacteristic
194C2×C73#212111yescharacteristic
195C2×C73C2×#35511yescharacteristic
196#2591D7311yescharacteristic
197C2×#343—731no—
198#1463—731no—
199#1353—732no—
200#1345—732no—
201C73#1176611yescharacteristic, complemented, Sylow
202C22×M16—1461no—
203C2×#121Dic7311yescharacteristic
204C22×C4⋊C4C2×D7311yescharacteristic
205#407—1461no—
206#400Dic7311yescharacteristic
207#400C2×D7311yescharacteristic
208#394Dic7311yescharacteristic
209#394C2×D7311yescharacteristic
210#388—21no—
211#388—21no—
212#386—21no—
213#386—21no—
214C4×C22⋊C4Dic7311yescharacteristic
215#343—732no—
216#343—1461no—
217#324—732no—
218#324—1461no—
219C22×D4—21no—
220C22×D4C2×Dic7311yescharacteristic
221C23×C4C2×Dic7311yescharacteristic
222C23×C4C22⋊D7311yescharacteristic
223C23×C4C22⋊D7311yescharacteristic
224C2×M16—2921no—
225C2×M16—2922no—
226C2×M16—2921no—
227C2×M16—1464no—
228C2×M16—1462no—
229C22×C8—2921no—
230#121—41no—
231#121—41no—
232C2×C4⋊C4—41no—
233C2×C4⋊C4—41no—
234C2×C4⋊C4C2×Dic7311yescharacteristic
235C2×C4⋊C4C2×Dic7311yescharacteristic
236C2×C4⋊C4C22⋊D7311yescharacteristic
237C2×C4⋊C4C22⋊D7311yescharacteristic
238C2×C4⋊C4C22×D7311yescharacteristic
239C2×C22⋊C4—21no—
240C2×C22⋊C4—21no—
241C2×C22⋊C4—21no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—21no—
244C2×C22⋊C4—21no—
245C2×C42—21no—
246C22⋊C8—2921no—
247C22⋊C8—1462no—
248#95—21no—
249#95—21no—
250#95—21no—
251#95—21no—
252#95C2×Dic7311yescharacteristic
253#95C2×Dic7311yescharacteristic
254C24—21no—
255C24C22⋊Dic7311yescharacteristic
256C2×D4—41no—
257C2×D4—42no—
258C2×D4—42no—
259C2×D4—41no—
260C2×D4—41no—
261C2×D4—81no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—22no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—81no—
275C22×C4—81no—
276C22×C4C22⋊Dic7311yescharacteristic
277C22×C4C22⋊Dic7311yescharacteristic
278C22×C4C22⋊Dic7311yescharacteristic
279C22×C4C22×Dic7311yescharacteristic
280C22×C4C2×C22⋊D7311yescharacteristic
281C22×C4C2×C22⋊D7311yescharacteristic
282M16—2924no—
283M16—2922no—
284M16—5842no—
285M16—1464no—
286C2×C8—2924no—
287C2×C8—2922no—
288C2×C8—2922no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C22⋊C4—42no—
295C22⋊C4—42no—
296C22⋊C4—42no—
297C22⋊C4—41no—
298C22⋊C4—41no—
299C22⋊C4—81no—
300C22⋊C4—81no—
301C42—81no—
302C23—22no—
303C23—21no—
304C23—21no—
305C23—41no—
306C23—41no—
307C23—41no—
308C23—41no—
309C23—41no—
310C23—41no—
311C23—41no—
312C23—41no—
313C23#1455650611yescharacteristic
314C23#1455650611yescharacteristic
315C23C2×C22⋊Dic7311yescharacteristic
316D4—82no—
317D4—82no—
318D4—82no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—42no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—81no—
346C2×C4—81no—
347C2×C4—81no—
348C2×C4#1455651812yes—
349C2×C4#1455661611yescharacteristic
350C2×C4#1455662111yescharacteristic
351C8—2924no—
352C8—5842no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—42no—
361C22—42no—
362C22—81no—
363C22—81no—
364C22—81no—
365C22—81no—
366C22—81no—
367C22—81no—
368C22—81no—
369C22#1798838412yes—
370C22C2×#1455650611yescharacteristic
371C22#1798858811yescharacteristic
372C22C2×#1455651811yescharacteristic
373C4—21no—
374C4—22no—
375C4—21no—
376C4—41no—
377C4—41no—
378C4—41no—
379C4—41no—
380C4—81no—
381C4—81no—
382C4—81no—
383C2—21no—
384C2—21no—
385C2—22no—
386C2—41no—
387C2—161no—
388C2#2260843511yescharacteristic
389C2#2261157711yescharacteristic
390C2C2×#1798838411yescharacteristic
subgroupG/H
centerC22C2×#14556506
commutatorC22×C4×C73C22×C4
FrattiniC2×C4⋊C4C22×D73
FittingC73×#2591C2
socleC22×C73C2×#99
prime radicalC2×#121×C73C4
cosocleC2×C4⋊C4×C73C23
hypercenter#2591D73
hypocenterC73#11766
LCSG ▷ C22×C4×C73 ▷ C23×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ #2591

Complements

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

NH ≅ G/Ntype
1. C2×#17988356C2semidirect
2. #22608261C2semidirect
3. #22607958C2semidirect
4. #22607934C2semidirect
201. C73#11766semidirect

Maximal subgroups

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

subgroupindex
C2×#179883562
#226082612
#226079582
#226079342
C73×#25912
#1176673