Finite Group Databasebeta

← #32983834

Subgroups — #32983834

Order 18688.

626 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 626 conjugacy classes of subgroups — too many to draw. Only its 135 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#22618955C211yesmaximal, characteristic, complemented
2#22609923C211yesmaximal, characteristic, complemented
3#22608356C212yesmaximal, complemented
4#22607789C212yesmaximal, complemented
5C73×#2535C211yesmaximal, characteristic
6D4×#10596831—24no—
7#17988640C2212yes—
8#17988611C2212yes—
9C2×#14556508—21no—
10C2×C22⋊#10596831—22no—
11#17988356—21no—
12D4⋊#10596831—24no—
13C2×#121×C73—21no—
14C2×C4×D4×C73C2211yescharacteristic
15C2×C4×D4×C73C2211yescharacteristic
16#397×C73C412yes—
17#392×C73C412yes—
18#386×C73—22no—
19#382×C73C2211yescharacteristic
20#380×C73—21no—
21C2×#11924012—22no—
22C22×#10596831—24no—
23C4⋊#10596831—22no—
24C4⋊#10596831—22no—
25C4⋊#10596831—22no—
26#14556508—42no—
27C22⋊#10596831—24no—
28C22⋊#10596831—22no—
29C4×#10596831D414yescomplemented
30C22×D4×C73—22no—
31C22×D4×C73C2×C411yescharacteristic
32C22×D4×C73D411yescharacteristic
33C23×C4×C73—21no—
34C23×C4×C73D412yes—
35#121×C73—42no—
36#121×C73—42no—
37C4×D4×C73—22no—
38C4×D4×C73—22no—
39C4×D4×C73—22no—
40C4×D4×C73—22no—
41C2×C4⋊C4×C73—21no—
42C2×C4⋊C4×C73C2×C411yescharacteristic
43C2×C4⋊C4×C73D411yescharacteristic
44C2×C22⋊C4×C73—22no—
45C2×C22⋊C4×C73—21no—
46C2×C22⋊C4×C73—22no—
47C2×C22⋊C4×C73C2×C412yes—
48C2×C42×C73C2311yescharacteristic
49#95×C73—22no—
50#95×C73—21no—
51#95×C73C2×C412yes—
52#11924012—42no—
53#11924012—42no—
54C2×#10596831—24no—
55C2×#10596831—42no—
56C2×#10596831—84no—
57C24×C73—21no—
58C24×C73C22⋊C412yes—
59C2×D4×C73—22no—
60C2×D4×C73—22no—
61C2×D4×C73—22no—
62C2×D4×C73—42no—
63C2×D4×C73—44no—
64C2×D4×C73—44no—
65C2×D4×C73—82no—
66C2×D4×C73D812yes—
67C2×D4×C73SD1612yes—
68C22×C4×C73—22no—
69C22×C4×C73—22no—
70C22×C4×C73—22no—
71C22×C4×C73—21no—
72C22×C4×C73—42no—
73C22×C4×C73—41no—
74C22×C4×C73—41no—
75C22×C4×C73—42no—
76C22×C4×C73—82no—
77C22×C4×C73C22⋊C412yes—
78C22×C4×C73C22×C411yescharacteristic
79C22×C4×C73C2×D411yescharacteristic
80C22×C4×C73C2×D411yescharacteristic
81C4⋊C4×C73—22no—
82C4⋊C4×C73—22no—
83C4⋊C4×C73—41no—
84C4⋊C4×C73—41no—
85C22⋊C4×C73—24no—
86C22⋊C4×C73—44no—
87C22⋊C4×C73—41no—
88C22⋊C4×C73—41no—
89C22⋊C4×C73—82no—
90C42×C73C2×D412yes—
91C42×C73#4112yes—
92C23×C73—22no—
93C23×C73—22no—
94C23×C73—22no—
95C23×C73—42no—
96C23×C73—42no—
97C23×C73—41no—
98C23×C73—41no—
99C23×C73—41no—
100C23×C73—41no—
101C23×C73—41no—
102C23×C73—41no—
103C23×C73#10214yes—
104C23×C73C2×C22⋊C411yescharacteristic
105D4×C73—24no—
106D4×C73—24no—
107D4×C73—24no—
108D4×C73—42no—
109D4×C73—88no—
110C2×C4×C73—22no—
111C2×C4×C73—41no—
112C2×C4×C73—41no—
113C2×C4×C73—44no—
114C2×C4×C73—42no—
115C2×C4×C73—42no—
116C2×C4×C73—42no—
117C2×C4×C73—41no—
118C2×C4×C73—41no—
119C2×C4×C73—42no—
120C2×C4×C73—41no—
121C2×C4×C73—82no—
122C2×C4×C73—82no—
123C2×C4×C73—82no—
124C2×C4×C73C4×D412yes—
125C2×C4×C73#12011yescharacteristic
126C2×C4×C73#12112yes—
127C2×C4×C73#12311yescharacteristic
128C2×C4×C73C2×D811yescharacteristic
129C2×C4×C73C2×SD1611yescharacteristic
130C2×C4×C73#13512yes—
131#10596831—44no—
132#10596831—82no—
133C22×C73—21no—
134C22×C73—21no—
135C22×C73—24no—
136C22×C73—44no—
137C22×C73—42no—
138C22×C73—42no—
139C22×C73—42no—
140C22×C73—42no—
141C22×C73—42no—
142C22×C73—82no—
143C22×C73—81no—
144C22×C73—81no—
145C22×C73—81no—
146C22×C73—81no—
147C22×C73#38611yescharacteristic
148C22×C73C2×#10211yescharacteristic
149C22×C73#41611yescharacteristic
150C4×C73—22no—
151C4×C73—21no—
152C4×C73—21no—
153C4×C73—42no—
154C4×C73—82no—
155C4×C73—81no—
156C4×C73#46512yes—
157C4×C73#46612yes—
158#9485—731nomaximal, Sylow
159C2×C73—21no—
160C2×C73—21no—
161C2×C73—44no—
162C2×C73—82no—
163C2×C73#189111yescharacteristic
164C2×C73#189311yescharacteristic
165C2×C73#191111yescharacteristic
166#2972—731no—
167#2535D7311yescharacteristic
168#1800—731no—
169#1486—732no—
170#1286—732no—
171C73#948511yescharacteristic, complemented, Sylow
172C2×#121—21no—
173C2×C4×D4C2×D7311yescharacteristic
174C2×C4×D4C2×D7311yescharacteristic
175C8×D4—1464no—
176#431—732no—
177#423—732no—
178C2×#100—1461no—
179C2×C22⋊C8—1462no—
180#397Dic7312yes—
181#392Dic7312yes—
182#386—22no—
183#382C2×D7311yescharacteristic
184#380—21no—
185#343—1461no—
186D4⋊C8—1464no—
187C22×D4—22no—
188C22×D4C2×Dic7311yescharacteristic
189C22×D4C22⋊D7311yescharacteristic
190C23×C4—21no—
191C23×C4C22⋊D7312yes—
192C2×M16—1462no—
193C22×C8—1464no—
194#121—42no—
195#121—42no—
196C4×D4—22no—
197C4×D4—22no—
198C4×D4—22no—
199C4×D4—22no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4C2×Dic7311yescharacteristic
202C2×C4⋊C4C22⋊D7311yescharacteristic
203C2×C22⋊C4—22no—
204C2×C22⋊C4—21no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4C2×Dic7312yes—
207C2×C42C22×D7311yescharacteristic
208C4⋊C8—1462no—
209C4⋊C8—1462no—
210C4⋊C8—1462no—
211#100—2922no—
212C22⋊C8—1464no—
213C22⋊C8—1462no—
214C4×C8—734no—
215#95—22no—
216#95—21no—
217#95C2×Dic7312yes—
218C24—21no—
219C24C22⋊Dic7312yes—
220C2×D4—22no—
221C2×D4—22no—
222C2×D4—22no—
223C2×D4—42no—
224C2×D4—44no—
225C2×D4—44no—
226C2×D4—82no—
227C2×D4#1192401312yes—
228C2×D4#1192401612yes—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—21no—
233C22×C4—42no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—42no—
237C22×C4—82no—
238C22×C4C22⋊Dic7312yes—
239C22×C4C22×Dic7311yescharacteristic
240C22×C4C2×C22⋊D7311yescharacteristic
241C22×C4C2×C22⋊D7311yescharacteristic
242M16—2922no—
243M16—2922no—
244C2×C8—2922no—
245C2×C8—5844no—
246C2×C8—1464no—
247C4⋊C4—22no—
248C4⋊C4—22no—
249C4⋊C4—41no—
250C4⋊C4—41no—
251C22⋊C4—24no—
252C22⋊C4—44no—
253C22⋊C4—41no—
254C22⋊C4—41no—
255C22⋊C4—82no—
256C42D4×D7312yes—
257C42#1192402712yes—
258C23—22no—
259C23—22no—
260C23—22no—
261C23—42no—
262C23—42no—
263C23—41no—
264C23—41no—
265C23—41no—
266C23—41no—
267C23—41no—
268C23—41no—
269C23#1455651214yes—
270C23C2×C22⋊Dic7311yescharacteristic
271D4—24no—
272D4—24no—
273D4—24no—
274D4—42no—
275D4—88no—
276C2×C4—22no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—44no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—42no—
286C2×C4—41no—
287C2×C4—82no—
288C2×C4—82no—
289C2×C4—82no—
290C2×C4D4×Dic7312yes—
291C2×C4#1455655911yescharacteristic
292C2×C4#1455656312yes—
293C2×C4#1455657211yescharacteristic
294C2×C4C2×#1192401311yescharacteristic
295C2×C4C2×#1192401611yescharacteristic
296C2×C4#1455661112yes—
297C8—2924no—
298C8—5842no—
299C22—21no—
300C22—21no—
301C22—24no—
302C22—44no—
303C22—42no—
304C22—42no—
305C22—42no—
306C22—42no—
307C22—42no—
308C22—82no—
309C22—81no—
310C22—81no—
311C22—81no—
312C22—81no—
313C22#1798846711yescharacteristic
314C22C2×#1455651211yescharacteristic
315C22#1798858311yescharacteristic
316C4—21no—
317C4—21no—
318C4—22no—
319C4—42no—
320C4—82no—
321C4—81no—
322C4#1798881412yes—
323C4#1798882112yes—
324C2—21no—
325C2—21no—
326C2—44no—
327C2—82no—
328C2#2261035011yescharacteristic
329C2#2261036011yescharacteristic
330C2#2261046411yescharacteristic
subgroupG/H
centerC22#17988467
commutatorC22×C4×C73C22×C4
FrattiniC2×C42C22×D73
Fitting, prime radicalC73×#2535C2
socleC22×C73#386
cosocleC2×C42×C73C23
hypercenter#2535D73
hypocenterC73#9485
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2535

Complements

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

NH ≅ G/Ntype
1. #22618955C2semidirect
2. #22609923C2semidirect
3. #22608356C2semidirect
4. #22607789C2semidirect
29. C4×#10596831D4semidirect
171. C73#9485semidirect

Maximal subgroups

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

subgroupindex
#226189552
#226099232
#226083562
#226077892
C73×#25352
#948573