Finite Group Databasebeta

← #32981973

Subgroups — #32981973

Order 18688.

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

normal subgroups only

This group has 478 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
2C4×#14556510C211yesmaximal, characteristic, complemented
3#22608529C212yesmaximal
4#22607795C212yesmaximal, complemented
5C73×#2632C211yesmaximal, characteristic
6#17988718—24no—
7#17988644—21no—
8#17988644C2212yes—
9#17988640—21no—
10C2×#14556510—21no—
11C4×#11924012C2212yes—
12#17988356—21no—
13#17988327—24no—
14C2×#124×C73C2211yescharacteristic
15C2×C4×Q8×C73C2211yescharacteristic
16#399×C73—21no—
17#395×C73C411yescharacteristic
18#395×C73C411yescharacteristic
19#394×C73C411yescharacteristic
20#394×C73C411yescharacteristic
21#393×C73—21no—
22#388×C73—21no—
23#388×C73—21no—
24#383×C73C2211yescharacteristic
25C2×#11924012—22no—
26C4⋊#10596831—22no—
27C4⋊#10596831—22no—
28C4⋊#10596831—22no—
29#14556510—24no—
30C22⋊#10596831—22no—
31#14556502—22no—
32#14556502—22no—
33#14556502D414yes—
34C4×#10596831—22no—
35C4×#10596831—22no—
36C22×Q8×C73C2×C411yescharacteristic
37C22×Q8×C73D411yescharacteristic
38C22×D4×C73C2×C411yescharacteristic
39#124×C73—22no—
40#124×C73—22no—
41C4×Q8×C73—22no—
42C4×Q8×C73—22no—
43C2×C4⋊C4×C73—21no—
44C2×C4⋊C4×C73—21no—
45C2×C4⋊C4×C73C2×C411yescharacteristic
46C2×C4⋊C4×C73C2×C411yescharacteristic
47C2×C4⋊C4×C73D411yescharacteristic
48C2×C22⋊C4×C73—21no—
49C2×C22⋊C4×C73—21no—
50C2×C22⋊C4×C73—21no—
51C2×C22⋊C4×C73—21no—
52C2×C22⋊C4×C73—21no—
53C2×C22⋊C4×C73—21no—
54C2×C42×C73D411yescharacteristic
55C2×C42×C73D411yescharacteristic
56C2×C42×C73C2311yescharacteristic
57#95×C73—21no—
58#95×C73—21no—
59#95×C73—21no—
60#95×C73—21no—
61#95×C73C2×C411yescharacteristic
62#95×C73C2×C411yescharacteristic
63#11924012—24no—
64#11924012—42no—
65C2×#10596831—24no—
66C2×#10596831—42no—
67C24×C73—21no—
68C2×Q8×C73—22no—
69C2×Q8×C73—22no—
70C2×Q8×C73—22no—
71C2×Q8×C73—41no—
72C2×Q8×C73—81no—
73C2×D4×C73—22no—
74C2×D4×C73—81no—
75C22×C4×C73—21no—
76C22×C4×C73—21no—
77C22×C4×C73—21no—
78C22×C4×C73C22⋊C411yescharacteristic
79C22×C4×C73C22⋊C411yescharacteristic
80C22×C4×C73C22⋊C411yescharacteristic
81C22×C4×C73C22⋊C411yescharacteristic
82C22×C4×C73C22×C411yescharacteristic
83C22×C4×C73C2×D411yescharacteristic
84C22×C4×C73C2×D411yescharacteristic
85C4⋊C4×C73—22no—
86C4⋊C4×C73—22no—
87C4⋊C4×C73—22no—
88C4⋊C4×C73—81no—
89C4⋊C4×C73—81no—
90C22⋊C4×C73—42no—
91C22⋊C4×C73—42no—
92C22⋊C4×C73—41no—
93C22⋊C4×C73—42no—
94C22⋊C4×C73—41no—
95C22⋊C4×C73—81no—
96C22⋊C4×C73—81no—
97C42×C73—21no—
98C42×C73—21no—
99C42×C73—21no—
100C42×C73—21no—
101C42×C73C2×D412yes—
102C42×C73#4112yes—
103C23×C73—41no—
104C23×C73—41no—
105C23×C73—41no—
106C23×C73—41no—
107C23×C73—41no—
108C23×C73—41no—
109C23×C73—41no—
110C23×C73C2×C22⋊C411yescharacteristic
111Q8×C73—44no—
112Q8×C73—82no—
113D4×C73—82no—
114C2×C4×C73—21no—
115C2×C4×C73—21no—
116C2×C4×C73—21no—
117C2×C4×C73—21no—
118C2×C4×C73—21no—
119C2×C4×C73—21no—
120C2×C4×C73—21no—
121C2×C4×C73—21no—
122C2×C4×C73—21no—
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—41no—
134C2×C4×C73—41no—
135C2×C4×C73—41no—
136C2×C4×C73#10412yes—
137C2×C4×C73C4×D412yes—
138C2×C4×C73#12011yescharacteristic
139C2×C4×C73#12112yes—
140C2×C4×C73#12311yescharacteristic
141#10596831—44no—
142#10596831—44no—
143C22×C73—21no—
144C22×C73—21no—
145C22×C73—81no—
146C22×C73—81no—
147C22×C73—81no—
148C22×C73—81no—
149C22×C73—81no—
150C22×C73—81no—
151C22×C73—81no—
152C22×C73#38611yescharacteristic
153C22×C73C2×#10411yescharacteristic
154C22×C73#42111yescharacteristic
155C4×C73—22no—
156C4×C73—22no—
157C4×C73—21no—
158C4×C73—21no—
159C4×C73—22no—
160C4×C73—41no—
161C4×C73—41no—
162C4×C73—41no—
163C4×C73—81no—
164C4×C73—81no—
165C4×C73—81no—
166#9188—731nomaximal, Sylow
167C2×C73—21no—
168C2×C73—21no—
169C2×C73—161no—
170C2×C73#187111yescharacteristic
171C2×C73#191211yescharacteristic
172C2×C73#191411yescharacteristic
173#2983—731no—
174#2632D7311yescharacteristic
175C4×#101—731no—
176#1527—732no—
177#1288—732no—
178C73#918811yescharacteristic, complemented, Sylow
179C2×#124C2×D7311yescharacteristic
180C2×C4×Q8C2×D7311yescharacteristic
181#446—1464no—
182#432—732no—
183#432—1461no—
184#431—1461no—
185C2×#101—1461no—
186C4×M16—732no—
187#399—21no—
188#395Dic7311yescharacteristic
189#395Dic7311yescharacteristic
190#394Dic7311yescharacteristic
191#394Dic7311yescharacteristic
192#393—21no—
193#388—21no—
194#388—21no—
195#383C2×D7311yescharacteristic
196#343—1461no—
197#329—1464no—
198C22×Q8C2×Dic7311yescharacteristic
199C22×Q8C22⋊D7311yescharacteristic
200C22×D4C2×Dic7311yescharacteristic
201C2×M16—1462no—
202#124—22no—
203#124—22no—
204C4×Q8—22no—
205C4×Q8—22no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4C2×Dic7311yescharacteristic
209C2×C4⋊C4C2×Dic7311yescharacteristic
210C2×C4⋊C4C22⋊D7311yescharacteristic
211C2×C22⋊C4—21no—
212C2×C22⋊C4—21no—
213C2×C22⋊C4—21no—
214C2×C22⋊C4—21no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—21no—
217C2×C42C22⋊D7311yescharacteristic
218C2×C42C22⋊D7311yescharacteristic
219C2×C42C22×D7311yescharacteristic
220C4⋊C8—1462no—
221C4⋊C8—1462no—
222C4⋊C8—1462no—
223#101—1464no—
224C22⋊C8—1462no—
225#97—734no—
226#97—1462no—
227#97—1462no—
228C4×C8—1462no—
229C4×C8—1462no—
230#95—21no—
231#95—21no—
232#95—21no—
233#95—21no—
234#95C2×Dic7311yescharacteristic
235#95C2×Dic7311yescharacteristic
236C24—21no—
237C2×Q8—22no—
238C2×Q8—22no—
239C2×Q8—22no—
240C2×Q8—41no—
241C2×Q8—81no—
242C2×D4—22no—
243C2×D4—81no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4C22⋊Dic7311yescharacteristic
248C22×C4C22⋊Dic7311yescharacteristic
249C22×C4C22⋊Dic7311yescharacteristic
250C22×C4C22⋊Dic7311yescharacteristic
251C22×C4C22×Dic7311yescharacteristic
252C22×C4C2×C22⋊D7311yescharacteristic
253C22×C4C2×C22⋊D7311yescharacteristic
254M16—2922no—
255M16—1464no—
256C2×C8—2922no—
257C2×C8—1464no—
258C4⋊C4—22no—
259C4⋊C4—22no—
260C4⋊C4—22no—
261C4⋊C4—81no—
262C4⋊C4—81no—
263C22⋊C4—42no—
264C22⋊C4—42no—
265C22⋊C4—41no—
266C22⋊C4—42no—
267C22⋊C4—41no—
268C22⋊C4—81no—
269C22⋊C4—81no—
270C42—21no—
271C42—21no—
272C42—21no—
273C42—21no—
274C42D4×D7312yes—
275C42#1192402712yes—
276C23—41no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23C2×C22⋊Dic7311yescharacteristic
284Q8—44no—
285Q8—82no—
286D4—82no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
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#1455651812yes—
310C2×C4D4×Dic7312yes—
311C2×C4#1455655911yescharacteristic
312C2×C4#1455656312yes—
313C2×C4#1455657211yescharacteristic
314C8—2924no—
315C8—2924no—
316C22—21no—
317C22—21no—
318C22—81no—
319C22—81no—
320C22—81no—
321C22—81no—
322C22—81no—
323C22—81no—
324C22—81no—
325C22#1798846711yescharacteristic
326C22C2×#1455651811yescharacteristic
327C22#1798860411yescharacteristic
328C4—21no—
329C4—21no—
330C4—22no—
331C4—22no—
332C4—22no—
333C4—41no—
334C4—41no—
335C4—41no—
336C4—81no—
337C4—81no—
338C4—81no—
339C2—21no—
340C2—21no—
341C2—161no—
342C2#2261023411yescharacteristic
343C2#2261046911yescharacteristic
344C2#2261047911yescharacteristic
subgroupG/H
centerC22#17988467
commutatorC22×C4×C73C22×C4
FrattiniC2×C42C22×D73
FittingC73×#2632C2
socleC22×C73#386
prime radicalC22×D4×C73C2×C4
cosocleC2×C42×C73C23
hypercenter#2632D73
hypocenterC73#9188
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2632

Complements

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

NH ≅ G/Ntype
1. #22619016C2semidirect
2. C4×#14556510C2semidirect
4. #22607795C2semidirect
178. C73#9188semidirect

Maximal subgroups

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

subgroupindex
#226190162
C4×#145565102
#226085292
#226077952
C73×#26322
#918873