Finite Group Databasebeta

← #32983758

Subgroups — #32983758

Order 18688.

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

normal subgroups only

This group has 598 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#22614239C211yesmaximal, characteristic
2#22610018C212yesmaximal, complemented
3#22609919C211yesmaximal, characteristic
4#22607799C212yesmaximal, complemented
5C73×#2972C211yesmaximal, characteristic
6C2×#14556572—21no—
7C22×C4⋊Dic73C2211yescharacteristic
8C2×#14556506—21no—
9#17988517C412yescomplemented
10#17988512C412yescomplemented
11#17988485—21no—
12#17988448—21no—
13#17988447—21no—
14#17988447C2212yes—
15#17988447C2211yescharacteristic
16#17988354—21no—
17#17988350—24no—
18C2×C4×D4×C73C2211yescharacteristic
19C8×D4×C73—24no—
20#431×C73—21no—
21C2×C22⋊C8×C73—21no—
22C2×C22⋊C8×C73C2212yes—
23C23×Dic73—21no—
24#14556572—42no—
25#14556572—42no—
26C2×C4⋊Dic73—22no—
27C2×C4⋊Dic73—21no—
28C2×C4⋊Dic73—21no—
29C2×C4⋊Dic73—22no—
30C2×C4⋊Dic73—22no—
31C2×C4⋊Dic73C2×C412yes—
32C2×#11924007—22no—
33C2×C22⋊Dic73—22no—
34C2×C22⋊Dic73—21no—
35C2×C22⋊Dic73—22no—
36#14556506—42no—
37#14556498—22no—
38#14556498—22no—
39#14556498—22no—
40#14556498C2×C412yes—
41C22×D4×C73D411yescharacteristic
42C23×C4×C73D411yescharacteristic
43C23×C4×C73C2311yescharacteristic
44C22×C8×C73—22no—
45C22×C8×C73D414yes—
46C4×D4×C73—22no—
47C4×D4×C73—22no—
48C2×C4⋊C4×C73C2×C411yescharacteristic
49C2×C22⋊C4×C73C2×C411yescharacteristic
50C2×C22⋊C4×C73D411yescharacteristic
51C2×C42×C73D411yescharacteristic
52C4⋊C8×C73—22no—
53C22⋊C8×C73—22no—
54C22⋊C8×C73—22no—
55C22⋊C8×C73—22no—
56C4×C8×C73—22no—
57C22×Dic73—22no—
58C22×Dic73—22no—
59C22×Dic73—42no—
60C22×Dic73—41no—
61C22×Dic73—41no—
62C22×Dic73—42no—
63C22×Dic73—42no—
64C4⋊Dic73—24no—
65C4⋊Dic73—42no—
66C4⋊Dic73—41no—
67C4⋊Dic73—41no—
68C4⋊Dic73—44no—
69#11924007—44no—
70C22⋊Dic73—44no—
71C22⋊Dic73—41no—
72C22⋊Dic73—41no—
73C22⋊Dic73—42no—
74C22⋊Dic73—42no—
75C24×C73C22⋊C411yescharacteristic
76C24×C73C2×D411yescharacteristic
77C2×D4×C73—22no—
78C2×D4×C73—22no—
79C2×D4×C73D812yes—
80C2×D4×C73SD1612yes—
81C22×C4×C73—21no—
82C22×C4×C73—21no—
83C22×C4×C73—42no—
84C22×C4×C73—42no—
85C22×C4×C73C22⋊C411yescharacteristic
86C22×C4×C73C22⋊C411yescharacteristic
87C22×C4×C73C22⋊C411yescharacteristic
88C22×C4×C73C22×C411yescharacteristic
89C22×C4×C73C2×D412yes—
90C22×C4×C73C2×D411yescharacteristic
91C22×C4×C73#4112yes—
92C2×C8×C73—24no—
93C2×C8×C73—44no—
94C2×C8×C73—82no—
95C4⋊C4×C73—22no—
96C22⋊C4×C73—22no—
97C22⋊C4×C73—22no—
98C42×C73—21no—
99C42×C73—21no—
100C23×C73—21no—
101C23×C73—21no—
102C23×C73—41no—
103C23×C73—41no—
104C23×C73—41no—
105C23×C73—41no—
106C23×C73#10212yes—
107C23×C73C2×C22⋊C411yescharacteristic
108C23×C73#12112yes—
109C23×C73C2×D811yescharacteristic
110C23×C73C2×SD1611yescharacteristic
111C23×C73#13512yes—
112D4×C73—24no—
113D4×C73—24no—
114C2×C4×C73—21no—
115C2×C4×C73—21no—
116C2×C4×C73—24no—
117C2×C4×C73—22no—
118C2×C4×C73—21no—
119C2×C4×C73—21no—
120C2×C4×C73—41no—
121C2×C4×C73—42no—
122C2×C4×C73—42no—
123C2×C4×C73—41no—
124C2×C4×C73—41no—
125C2×C4×C73—81no—
126C2×C4×C73#10212yes—
127C2×C4×C73C4×D412yes—
128C2×C4×C73#12011yescharacteristic
129C2×C4×C73#12311yescharacteristic
130C2×Dic73—42no—
131C2×Dic73—42no—
132C2×Dic73—42no—
133C2×Dic73—42no—
134C2×Dic73—42no—
135C2×Dic73—42no—
136C2×Dic73—82no—
137C2×Dic73—82no—
138C2×Dic73—82no—
139C2×Dic73—82no—
140C2×Dic73—82no—
141C8×C73—44no—
142C22×C73—21no—
143C22×C73—22no—
144C22×C73—21no—
145C22×C73—22no—
146C22×C73—22no—
147C22×C73—41no—
148C22×C73—41no—
149C22×C73—42no—
150C22×C73—42no—
151C22×C73—41no—
152C22×C73—41no—
153C22×C73—81no—
154C22×C73#38611yescharacteristic
155C22×C73C2×#10211yescharacteristic
156C22×C73#41611yescharacteristic
157C22×C73#46512yes—
158C22×C73#46612yes—
159C4×C73—21no—
160C4×C73—22no—
161C4×C73—22no—
162C4×C73—41no—
163C4×C73—41no—
164C4×C73#45812yes—
165Dic73—82no—
166Dic73—82no—
167Dic73—162no—
168#9470—731nomaximal, Sylow
169C2×C73—21no—
170C2×C73—22no—
171C2×C73—21no—
172C2×C73—22no—
173C2×C73—42no—
174C2×C73#189011yescharacteristic
175C2×C73#189311yescharacteristic
176C2×C73#190911yescharacteristic
177#2972D7311yescharacteristic
178#2486—731no—
179#1823—732no—
180#1799—731no—
181#1289—732no—
182C73#947011yescharacteristic, complemented, Sylow
183C2×#123—1461no—
184C2×C4×D4C2×D7311yescharacteristic
185C22×C4⋊C4—731no—
186C8×D4—24no—
187#431—21no—
188C2×#99—1461no—
189C2×C22⋊C8—21no—
190C2×C22⋊C8C2×D7312yes—
191#398—732no—
192#397—732no—
193#389—1461no—
194#381—1461no—
195#380—731no—
196#380—732no—
197#380—1461no—
198#342—1461no—
199#340—1464no—
200C22×D4D29211yescharacteristic
201C23×C4—1461no—
202C23×C4C22⋊D7311yescharacteristic
203C23×C4C22×D7311yescharacteristic
204C22×C8—22no—
205C22×C8C22⋊D7314yes—
206#123—2922no—
207#123—2922no—
208C4×D4—22no—
209C4×D4—22no—
210C2×C4⋊C4—732no—
211C2×C4⋊C4—1462no—
212C2×C4⋊C4—1462no—
213C2×C4⋊C4—1461no—
214C2×C4⋊C4—1461no—
215C2×C4⋊C4—1462no—
216C2×C4⋊C4—1462no—
217C2×C4⋊C4C4×D7311yescharacteristic
218C2×C22⋊C4—1462no—
219C2×C22⋊C4—1462no—
220C2×C22⋊C4—1461no—
221C2×C22⋊C4C4×D7311yescharacteristic
222C2×C22⋊C4D29211yescharacteristic
223C2×C42C22⋊D7311yescharacteristic
224C4⋊C8—22no—
225#99—2922no—
226C22⋊C8—22no—
227C22⋊C8—22no—
228C22⋊C8—22no—
229C4×C8—22no—
230#95—732no—
231#95—1462no—
232#95—1462no—
233#95—1462no—
234C24#1192400511yescharacteristic
235C24C2×D29211yescharacteristic
236C2×D4—22no—
237C2×D4—22no—
238C2×D4D58412yes—
239C2×D4#1192401712yes—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—2922no—
245C22×C4—2921no—
246C22×C4—2921no—
247C22×C4—2922no—
248C22×C4—2922no—
249C22×C4—1462no—
250C22×C4—1462no—
251C22×C4#1192400511yescharacteristic
252C22×C4#1192400511yescharacteristic
253C22×C4#1192400511yescharacteristic
254C22×C4C2×C4×D7311yescharacteristic
255C22×C4C2×C22⋊D7312yes—
256C22×C4C2×C22⋊D7311yescharacteristic
257C22×C4#1192402812yes—
258C2×C8—24no—
259C2×C8—44no—
260C2×C8—82no—
261C4⋊C4—22no—
262C4⋊C4—2922no—
263C4⋊C4—2924no—
264C4⋊C4—2924no—
265C4⋊C4—2921no—
266C4⋊C4—2921no—
267C4⋊C4—1464no—
268C22⋊C4—22no—
269C22⋊C4—22no—
270C22⋊C4—2924no—
271C22⋊C4—2921no—
272C22⋊C4—2921no—
273C22⋊C4—2922no—
274C22⋊C4—2922no—
275C42—21no—
276C42—21no—
277C23—21no—
278C23—21no—
279C23—41no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23#1455651312yes—
284C23C2×#1192400511yescharacteristic
285C23#1455656512yes—
286C23C2×D58411yescharacteristic
287C23C2×#1192401711yescharacteristic
288C23#1455661312yes—
289D4—24no—
290D4—24no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—24no—
294C2×C4—22no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—41no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—81no—
303C2×C4—2922no—
304C2×C4—2922no—
305C2×C4—2922no—
306C2×C4—2922no—
307C2×C4—2922no—
308C2×C4—2922no—
309C2×C4—5842no—
310C2×C4—5842no—
311C2×C4—5842no—
312C2×C4—5842no—
313C2×C4—5842no—
314C2×C4#1455651312yes—
315C2×C4C4×C22⋊D7312yes—
316C2×C4#1455656011yescharacteristic
317C2×C4#1455657411yescharacteristic
318C8—44no—
319C22—21no—
320C22—22no—
321C22—21no—
322C22—22no—
323C22—22no—
324C22—41no—
325C22—41no—
326C22—42no—
327C22—42no—
328C22—41no—
329C22—41no—
330C22—81no—
331C22#1798846611yescharacteristic
332C22C2×#1455651311yescharacteristic
333C22#1798858411yescharacteristic
334C22#1798881612yes—
335C22#1798882312yes—
336C4—21no—
337C4—22no—
338C4—22no—
339C4—41no—
340C4—41no—
341C4—5842no—
342C4—5842no—
343C4—11682no—
344C4#1798877712yes—
345C2—21no—
346C2—22no—
347C2—21no—
348C2—22no—
349C2—42no—
350C2#2261034411yescharacteristic
351C2#2261035911yescharacteristic
352C2#2261045311yescharacteristic
subgroupG/H
centerC22#17988466
commutatorC22×C4×C73C22×C4
FrattiniC23×C4C22×D73
FittingC73×#2972C2
socleC22×C73#386
prime radicalC22×D4×C73D4
cosocleC23×C4×C73C23
hypercenter#2972D73
hypocenterC73#9470
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2972

Complements

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

NH ≅ G/Ntype
2. #22610018C2semidirect
4. #22607799C2semidirect
9. #17988517C4semidirect
10. #17988512C4semidirect
182. C73#9470semidirect

Maximal subgroups

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

subgroupindex
#226142392
#226100182
#226099192
#226077992
C73×#29722
#947073