Finite Group Databasebeta

← #32996790

Subgroups — #32996790

Order 18688.

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

normal subgroups only

This group has 416 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#22609927C211yesmaximal, characteristic, complemented
2#22608275C211yesmaximal, characteristic, complemented
3#22607958C212yesmaximal, complemented
4#22607934C212yesmaximal
5C73×#2398C211yesmaximal, characteristic
6#17988644—21no—
7#17988611—21no—
8C2×#14556510—21no—
9#17988356C2212yes—
10#17988313—21no—
11#17988313C2212yes—
12C2×C4×Q8×C73C2211yescharacteristic
13C2×C4×D4×C73C411yescharacteristic
14#400×C73C411yescharacteristic
15#400×C73C2211yescharacteristic
16#397×C73—21no—
17#394×C73C411yescharacteristic
18#394×C73C2211yescharacteristic
19#393×C73—21no—
20C4×C22⋊C4×C73—21no—
21C4×C22⋊C4×C73—21no—
22#376×C73C411yescharacteristic
23C2×#11924012—22no—
24C4⋊#10596831—41no—
25C4⋊#10596831—41no—
26C4⋊#10596831—41no—
27#14556510—42no—
28C22⋊#10596831—22no—
29#14556502—41no—
30C22×Q8×C73D411yescharacteristic
31C22×D4×C73C2×C411yescharacteristic
32C23×C4×C73—21no—
33C4×Q8×C73—41no—
34C4×Q8×C73—41no—
35C4×D4×C73—41no—
36C4×D4×C73—41no—
37C2×C4⋊C4×C73C2×C411yescharacteristic
38C2×C4⋊C4×C73C2×C411yescharacteristic
39C2×C4⋊C4×C73D411yescharacteristic
40C2×C4⋊C4×C73C2311yescharacteristic
41C2×C22⋊C4×C73—21no—
42C2×C22⋊C4×C73—21no—
43C2×C22⋊C4×C73—21no—
44C2×C22⋊C4×C73—21no—
45C2×C22⋊C4×C73—21no—
46C2×C42×C73—21no—
47C2×C42×C73C2×C411yescharacteristic
48C2×C42×C73D411yescharacteristic
49C2×C42×C73D411yescharacteristic
50#95×C73—21no—
51#95×C73—21no—
52#95×C73—21no—
53#95×C73—21no—
54#95×C73—21no—
55#95×C73C2×C411yescharacteristic
56#95×C73C2×C411yescharacteristic
57#11924012—42no—
58#11924012—42no—
59C2×#10596831—42no—
60C2×#10596831—42no—
61C24×C73—21no—
62C2×Q8×C73—21no—
63C2×Q8×C73—21no—
64C2×Q8×C73—41no—
65C2×Q8×C73—41no—
66C2×D4×C73—41no—
67C2×D4×C73—42no—
68C22×C4×C73—21no—
69C22×C4×C73—21no—
70C22×C4×C73—21no—
71C22×C4×C73—42no—
72C22×C4×C73—42no—
73C22×C4×C73—41no—
74C22×C4×C73—41no—
75C22×C4×C73C22⋊C411yescharacteristic
76C22×C4×C73C22⋊C411yescharacteristic
77C22×C4×C73C22⋊C411yescharacteristic
78C22×C4×C73C22⋊C411yescharacteristic
79C22×C4×C73C22×C411yescharacteristic
80C22×C4×C73C2×D411yescharacteristic
81C22×C4×C73C2×D411yescharacteristic
82C4⋊C4×C73—41no—
83C4⋊C4×C73—41no—
84C4⋊C4×C73—41no—
85C4⋊C4×C73—41no—
86C22⋊C4×C73—42no—
87C22⋊C4×C73—42no—
88C22⋊C4×C73—42no—
89C22⋊C4×C73—42no—
90C22⋊C4×C73—41no—
91C22⋊C4×C73—41no—
92C42×C73—21no—
93C42×C73—21no—
94C42×C73—21no—
95C42×C73—21no—
96C42×C73—21no—
97C42×C73—21no—
98C42×C73—42no—
99C23×C73—22no—
100C23×C73—41no—
101C23×C73—41no—
102C23×C73—41no—
103C23×C73—41no—
104C23×C73—41no—
105C23×C73—41no—
106C23×C73C2×C22⋊C411yescharacteristic
107Q8×C73—42no—
108Q8×C73—42no—
109D4×C73—42no—
110D4×C73—42no—
111C2×C4×C73—21no—
112C2×C4×C73—21no—
113C2×C4×C73—22no—
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—82no—
136C2×C4×C73—82no—
137C2×C4×C73#9911yescharacteristic
138C2×C4×C73#9911yescharacteristic
139C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
140C2×C4×C73#13711yescharacteristic
141#10596831—82no—
142#10596831—82no—
143C22×C73—21no—
144C22×C73—21no—
145C22×C73—42no—
146C22×C73—42no—
147C22×C73—42no—
148C22×C73—81no—
149C22×C73—81no—
150C22×C73—81no—
151C22×C73—81no—
152C22×C73C2×#9911yescharacteristic
153C22×C73#41711yescharacteristic
154C22×C73#42111yescharacteristic
155C4×C73—21no—
156C4×C73—21no—
157C4×C73—21no—
158C4×C73—41no—
159C4×C73—41no—
160C4×C73—41no—
161C4×C73—41no—
162C4×C73—41no—
163C4×C73—81no—
164C4×C73—81no—
165C4×C73—81no—
166C4×C73#35512yes—
167#11829—731nomaximal, Sylow
168C2×C73—21no—
169C2×C73—21no—
170C2×C73—82no—
171C2×C73#151311yescharacteristic
172C2×C73#212011yescharacteristic
173C2×C73C2×#35511yescharacteristic
174#2398D7311yescharacteristic
175#1801—731no—
176#1467—731no—
177#1353—732no—
178#1345—732no—
179C73#1182911yescharacteristic, complemented, Sylow
180C2×C4×Q8C2×D7311yescharacteristic
181C2×C4×D4Dic7311yescharacteristic
182#432—1461no—
183#423—1461no—
184C2×#101—1461no—
185#400Dic7311yescharacteristic
186#400C2×D7311yescharacteristic
187#397—21no—
188#394Dic7311yescharacteristic
189#394C2×D7311yescharacteristic
190#393—21no—
191C4×C22⋊C4—21no—
192C4×C22⋊C4—21no—
193#376Dic7311yescharacteristic
194#343—732no—
195#324—732no—
196#324—1461no—
197C22×Q8C22⋊D7311yescharacteristic
198C22×D4C2×Dic7311yescharacteristic
199C23×C4—21no—
200C2×M16—1462no—
201C4×Q8—41no—
202C4×Q8—41no—
203C4×D4—41no—
204C4×D4—41no—
205C2×C4⋊C4C2×Dic7311yescharacteristic
206C2×C4⋊C4C2×Dic7311yescharacteristic
207C2×C4⋊C4C22⋊D7311yescharacteristic
208C2×C4⋊C4C22×D7311yescharacteristic
209C2×C22⋊C4—21no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—21no—
212C2×C22⋊C4—21no—
213C2×C22⋊C4—21no—
214C2×C42—21no—
215C2×C42C2×Dic7311yescharacteristic
216C2×C42C22⋊D7311yescharacteristic
217C2×C42C22⋊D7311yescharacteristic
218C4⋊C8—2921no—
219C4⋊C8—2921no—
220C4⋊C8—2921no—
221#101—2922no—
222C22⋊C8—1462no—
223#97—2921no—
224#95—21no—
225#95—21no—
226#95—21no—
227#95—21no—
228#95—21no—
229#95C2×Dic7311yescharacteristic
230#95C2×Dic7311yescharacteristic
231C24—21no—
232C2×Q8—21no—
233C2×Q8—21no—
234C2×Q8—41no—
235C2×Q8—41no—
236C2×D4—41no—
237C2×D4—42no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4C22⋊Dic7311yescharacteristic
246C22×C4C22⋊Dic7311yescharacteristic
247C22×C4C22⋊Dic7311yescharacteristic
248C22×C4C22⋊Dic7311yescharacteristic
249C22×C4C22×Dic7311yescharacteristic
250C22×C4C2×C22⋊D7311yescharacteristic
251C22×C4C2×C22⋊D7311yescharacteristic
252M16—2922no—
253M16—2922no—
254C2×C8—2922no—
255C2×C8—2922no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C22⋊C4—42no—
261C22⋊C4—42no—
262C22⋊C4—42no—
263C22⋊C4—42no—
264C22⋊C4—41no—
265C22⋊C4—41no—
266C42—21no—
267C42—21no—
268C42—21no—
269C42—21no—
270C42—21no—
271C42—21no—
272C42—42no—
273C23—22no—
274C23—41no—
275C23—41no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23C2×C22⋊Dic7311yescharacteristic
281Q8—42no—
282Q8—42no—
283D4—42no—
284D4—42no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—22no—
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—21no—
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—82no—
310C2×C4—82no—
311C2×C4#1455650611yescharacteristic
312C2×C4#1455650611yescharacteristic
313C2×C4#1455661611yescharacteristic
314C2×C4#1455662111yescharacteristic
315C8—5842no—
316C8—5842no—
317C22—21no—
318C22—21no—
319C22—42no—
320C22—42no—
321C22—42no—
322C22—81no—
323C22—81no—
324C22—81no—
325C22—81no—
326C22C2×#1455650611yescharacteristic
327C22#1798858811yescharacteristic
328C22#1798860411yescharacteristic
329C4—21no—
330C4—21no—
331C4—21no—
332C4—41no—
333C4—41no—
334C4—41no—
335C4—41no—
336C4—41no—
337C4—81no—
338C4—81no—
339C4—81no—
340C4#1798838412yes—
341C2—21no—
342C2—21no—
343C2—82no—
344C2#2260847411yescharacteristic
345C2#2261157211yescharacteristic
346C2C2×#1798838411yescharacteristic
subgroupG/H
centerC22C2×#14556506
commutatorC22×C4×C73C22×C4
FrattiniC2×C4⋊C4C22×D73
FittingC73×#2398C2
socleC22×C73C2×#99
prime radicalC22×D4×C73C2×C4
cosocleC2×C4⋊C4×C73C23
hypercenter#2398D73
hypocenterC73#11829
LCSG ▷ C22×C4×C73 ▷ C23×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #2398

Complements

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

NH ≅ G/Ntype
1. #22609927C2semidirect
2. #22608275C2semidirect
3. #22607958C2semidirect
179. C73#11829semidirect

Maximal subgroups

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

subgroupindex
#226099272
#226082752
#226079582
#226079342
C73×#23982
#1182973