Finite Group Databasebeta

← #32982676

Subgroups — #32982676

Order 18688.

490 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22620869C211yesmaximal, characteristic
2#22610755C211yesmaximal, characteristic
3#22608600C211yesmaximal, characteristic
4#22608495C211yesmaximal, characteristic, complemented
5#22607791C211yesmaximal, characteristic, complemented
6#22607762C211yesmaximal, characteristic, complemented
7C73×#2961C211yesmaximal, characteristic, complemented
8C2×#14556588C2211yescharacteristic
9#17988961C412yescomplemented
10#17988935C412yescomplemented
11#17988787—22no—
12#17988737—22no—
13#17988640C2211yescharacteristic
14#17988640C2211yescharacteristic
15C2×#14556513—21no—
16C2×#14556513—21no—
17#17988453C2211yescharacteristic
18#17988330—22no—
19#17988324—22no—
20#533×C73C2211yescharacteristic
21#446×C73—22no—
22#446×C73—22no—
23#424×C73—21no—
24#423×C73C2211yescharacteristic
25#405×C73—21no—
26C2×#97×C73C2211yescharacteristic
27C22×D292—21no—
28C22×D292—21no—
29C22×D292—21no—
30C2×#11924015—24no—
31#14556588—21no—
32#14556588—21no—
33#14556588—21no—
34#14556588C2×C412yes—
35C2×C4⋊Dic73—21no—
36C4⋊#10596831—21no—
37C4⋊#10596831—21no—
38#14556513—42no—
39#14556513—42no—
40#14556511—24no—
41C22⋊#10596831—21no—
42C22⋊#10596831—21no—
43C4×#10596831—21no—
44C4×#10596831C2×C412yes—
45#14556498—21no—
46#14556498—21no—
47C2×M16×C73—21no—
48C22×C8×C73D411yescharacteristic
49C22×C8×C73D411yescharacteristic
50C4⋊Q8×C73C2×C411yescharacteristic
51C4⋊Q8×C73C2×C411yescharacteristic
52#125×C73—21no—
53C22⋊Q8×C73—21no—
54C22⋊Q8×C73D412yes—
55C4×Q8×C73—21no—
56C4×Q8×C73D412yes—
57#117×C73—21no—
58C2×C42×C73C2311yescharacteristic
59C4⋊C8×C73—24no—
60C4⋊C8×C73—21no—
61C4⋊C8×C73D412yes—
62#97×C73—22no—
63#97×C73—21no—
64#97×C73—21no—
65C4×C8×C73—22no—
66C23×D73—41no—
67C2×D292—22no—
68C2×D292—22no—
69C2×D292—22no—
70C2×D292—41no—
71C2×D292—41no—
72C2×D292—41no—
73C2×D292—41no—
74C2×D292—41no—
75C2×D292—41no—
76C2×D292—41no—
77C2×D292—82no—
78C2×D292—82no—
79C22×Dic73—41no—
80#11924015—84no—
81C2×#10596831—22no—
82C2×#10596831—41no—
83C4⋊Dic73—41no—
84C4⋊Dic73—41no—
85C2×Q8×C73—21no—
86C2×Q8×C73C22⋊C412yes—
87C22×C4×C73C2×D411yescharacteristic
88C22×C4×C73C2×D411yescharacteristic
89C22×C4×C73#4111yescharacteristic
90M16×C73—41no—
91M16×C73—41no—
92C2×C8×C73—21no—
93C2×C8×C73—21no—
94C2×C8×C73—22no—
95C2×C8×C73—21no—
96C2×C8×C73—21no—
97C2×C8×C73—21no—
98C2×C8×C73SD1612yescomplemented
99C4⋊C4×C73—21no—
100C4⋊C4×C73—21no—
101C4⋊C4×C73—21no—
102C4⋊C4×C73—22no—
103C4⋊C4×C73—22no—
104C4⋊C4×C73C22⋊C412yes—
105C22⋊C4×C73—22no—
106C42×C73—22no—
107C42×C73C22×C411yescharacteristic
108C42×C73C2×D411yescharacteristic
109C42×C73C2×D411yescharacteristic
110C42×C73#4111yescharacteristic
111C23×C73#12111yescharacteristic
112C22×D73—42no—
113C22×D73—82no—
114C22×D73—82no—
115C22×D73—81no—
116C22×D73—81no—
117Q8×C73—42no—
118Q8×C73—81no—
119C2×C4×C73—22no—
120C2×C4×C73—22no—
121C2×C4×C73—41no—
122C2×C4×C73—41no—
123C2×C4×C73C2×C22⋊C411yescharacteristic
124C2×C4×C73C4×D411yescharacteristic
125C2×C4×C73C4×D411yescharacteristic
126C2×C4×C73#12011yescharacteristic
127C2×C4×C73#12111yescharacteristic
128C2×C4×C73#12311yescharacteristic
129C2×C4×C73C2×SD1611yescharacteristic
130C2×C4×C73#13511yescharacteristic
131C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
132C2×C4×C73#13711yescharacteristic
133C2×Dic73—81no—
134C2×Dic73—81no—
135C2×Dic73—81no—
136D292—44no—
137D292—42no—
138D292—42no—
139D292—82no—
140D292—82no—
141D292—82no—
142D292—81no—
143D292—81no—
144C8×C73—22no—
145C8×C73—41no—
146C8×C73—42no—
147#10596831—42no—
148#10596831—81no—
149C22×C73—21no—
150C22×C73—21no—
151C22×C73#38611yescharacteristic
152C22×C73#46011yescharacteristic
153C22×C73#46411yescharacteristic
154C2×D73—82no—
155C2×D73—82no—
156C2×D73—82no—
157C2×D73—162no—
158C2×D73—162no—
159C4×C73—21no—
160C4×C73—21no—
161C4×C73—21no—
162C4×C73—21no—
163C4×C73—42no—
164C4×C73—42no—
165C4×C73C4×SD1611yescharacteristic
166C4×C73#44011yescharacteristic
167C4×C73#44811yescharacteristic
168C4×C73#44911yescharacteristic
169Dic73—161no—
170#9294—731nomaximal, Sylow
171C2×C73—21no—
172C2×C73—21no—
173C2×C73#188011yescharacteristic
174C2×C73#190311yescharacteristic
175C2×C73#191011yescharacteristic
176D73—162no—
177#3208—731no—
178#2961D7311yescharacteristic, complemented
179#1961—731no—
180#1545—731no—
181#1518—731no—
182#1286—731no—
183#1275—731no—
184C73#929411yescharacteristic, complemented, Sylow
185#533C2×D7311yescharacteristic
186C2×#127—731no—
187#492—732no—
188#486—732no—
189#460—1462no—
190#450—1462no—
191#446—22no—
192#446—22no—
193#431—731no—
194#431—731no—
195#424—21no—
196#423C2×D7311yescharacteristic
197C2×#102—1461no—
198C2×#102—1461no—
199#405—21no—
200C2×#97C2×D7311yescharacteristic
201#382—731no—
202#331—1462no—
203#328—1462no—
204C22×D4—1461no—
205C22×D4—1461no—
206C22×D4—1461no—
207C2×SD16—1464no—
208C2×M16—21no—
209C22×C8C22⋊D7311yescharacteristic
210C22×C8C22⋊D7311yescharacteristic
211C4⋊Q8C4×D7311yescharacteristic
212C4⋊Q8C4×D7311yescharacteristic
213#127—732no—
214#127—1461no—
215#127—1461no—
216#127—1461no—
217#125—21no—
218C22⋊Q8—21no—
219C22⋊Q8D29212yes—
220C4×Q8—21no—
221C4×Q8C22⋊D7312yes—
222#117—21no—
223C2×C4⋊C4—1461no—
224C2×C42C22×D7311yescharacteristic
225C4⋊C8—24no—
226C4⋊C8—21no—
227C4⋊C8—1461no—
228C4⋊C8—1461no—
229C4⋊C8C22⋊D7312yes—
230#102—2922no—
231#102—2922no—
232#102—1464no—
233C22⋊C8—1461no—
234C22⋊C8—1461no—
235#97—22no—
236#97—21no—
237#97—21no—
238C4×C8—22no—
239C4×C8—732no—
240C4×C8—1461no—
241#95—1461no—
242#95—1461no—
243C24—2921no—
244C2×Q8—21no—
245C2×Q8#1192400512yes—
246C2×D4—2921no—
247C2×D4—2921no—
248C2×D4—2921no—
249C2×D4—2921no—
250C2×D4—2921no—
251C2×D4—2921no—
252C2×D4—2921no—
253C2×D4—5842no—
254C2×D4—5842no—
255C2×D4—1462no—
256C2×D4—1462no—
257C2×D4—1462no—
258C22×C4—2921no—
259C22×C4C2×C22⋊D7311yescharacteristic
260C22×C4C2×D29211yescharacteristic
261C22×C4#1192402811yescharacteristic
262SD16—5844no—
263M16—41no—
264M16—41no—
265C2×C8—21no—
266C2×C8—21no—
267C2×C8—22no—
268C2×C8—21no—
269C2×C8—21no—
270C2×C8—21no—
271C2×C8—2921no—
272C2×C8—1462no—
273C2×C8#1192401512yescomplemented
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4—22no—
278C4⋊C4—22no—
279C4⋊C4—2921no—
280C4⋊C4—2921no—
281C4⋊C4#1192400512yes—
282C22⋊C4—22no—
283C42—22no—
284C42C2×C4×D7311yescharacteristic
285C42C2×C22⋊D7311yescharacteristic
286C42C2×C22⋊D7311yescharacteristic
287C42#1192402811yescharacteristic
288C23—2922no—
289C23—5842no—
290C23—5842no—
291C23—5841no—
292C23—5841no—
293C23#1455656511yescharacteristic
294Q8—42no—
295Q8—81no—
296D4—2924no—
297D4—2922no—
298D4—2922no—
299D4—5842no—
300D4—5842no—
301D4—5842no—
302D4—5841no—
303D4—5841no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—5841no—
309C2×C4—5841no—
310C2×C4—5841no—
311C2×C4C2×#1192400511yescharacteristic
312C2×C4C4×C22⋊D7311yescharacteristic
313C2×C4C4×C22⋊D7311yescharacteristic
314C2×C4#1455656011yescharacteristic
315C2×C4#1455656511yescharacteristic
316C2×C4#1455657411yescharacteristic
317C2×C4C2×#1192401511yescharacteristic
318C2×C4#1455661111yescharacteristic
319C2×C4#1455661611yescharacteristic
320C2×C4#1455662211yescharacteristic
321C8—22no—
322C8—41no—
323C8—42no—
324C8—2922no—
325C8—5841no—
326C22—21no—
327C22—21no—
328C22—5842no—
329C22—5842no—
330C22—5842no—
331C22—11682no—
332C22—11682no—
333C22#1798846611yescharacteristic
334C22#1798878711yescharacteristic
335C22#1798880911yescharacteristic
336C4—21no—
337C4—21no—
338C4—21no—
339C4—21no—
340C4—42no—
341C4—42no—
342C4—11681no—
343C4C4×#1192401511yescharacteristic
344C4#1798868511yescharacteristic
345C4#1798872611yescharacteristic
346C4#1798873211yescharacteristic
347C2—21no—
348C2—21no—
349C2—11682no—
350C2#2261028611yescharacteristic
351C2#2261041111yescharacteristic
352C2#2261045811yescharacteristic
subgroupG/H
centerC22#17988466
commutatorC42×C73C22×C4
FrattiniC2×C42C22×D73
FittingC73×#2961C2
socleC22×C73#386
prime radicalC2×#14556588C22
cosocleC2×C42×C73C23
hypercenter#2961D73
hypocenterC73#9294
LCSG ▷ C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2961

Complements

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

NH ≅ G/Ntype
4. #22608495C2semidirect
5. #22607791C2semidirect
6. #22607762C2semidirect
7. C73×#2961C2semidirect
9. #17988961C4semidirect
10. #17988935C4semidirect
98. C2×C8×C73SD16semidirect
178. #2961D73semidirect
184. C73#9294semidirect
273. C2×C8#11924015semidirect

Maximal subgroups

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

subgroupindex
#226208692
#226107552
#226086002
#226084952
#226077912
#226077622
C73×#29612
#929473