Finite Group Databasebeta

← #32983779

Subgroups — #32983779

Order 18688.

576 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 576 conjugacy classes of subgroups — too many to draw. Only its 123 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#22618960C211yesmaximal, characteristic
2#22610017C212yesmaximal, complemented
3#22609919C211yesmaximal, characteristic
4#22607798C212yesmaximal, complemented
5C73×#2486C211yesmaximal, characteristic
6#17988659—24no—
7#17988644—21no—
8C2×#14556506—21no—
9#17988551—21no—
10C2×C22⋊#10596831C2212yes—
11#17988447C2212yes—
12#17988354—21no—
13#17988349—24no—
14C2×#123×C73—21no—
15C2×C4×D4×C73C2211yescharacteristic
16C22×C4⋊C4×C73C2211yescharacteristic
17#398×C73C412yescomplemented
18#397×C73C412yescomplemented
19#389×C73—21no—
20#381×C73—21no—
21#380×C73—21no—
22#380×C73C2211yescharacteristic
23C2×#11924012—22no—
24C22×#10596831D414yes—
25C2×C4⋊Dic73—22no—
26C2×C22⋊Dic73—22no—
27C4⋊#10596831—22no—
28#14556506—42no—
29C22⋊#10596831—22no—
30C22⋊#10596831—22no—
31C22⋊#10596831—22no—
32#14556502—22no—
33#14556498—22no—
34C22×D4×C73D411yescharacteristic
35C23×C4×C73—21no—
36C23×C4×C73D411yescharacteristic
37C23×C4×C73C2311yescharacteristic
38#123×C73—44no—
39C4×D4×C73—22no—
40C4×D4×C73—22no—
41C2×C4⋊C4×C73—22no—
42C2×C4⋊C4×C73—22no—
43C2×C4⋊C4×C73—22no—
44C2×C4⋊C4×C73—21no—
45C2×C4⋊C4×C73—21no—
46C2×C4⋊C4×C73C2×C412yes—
47C2×C4⋊C4×C73C2×C411yescharacteristic
48C2×C22⋊C4×C73—22no—
49C2×C22⋊C4×C73—21no—
50C2×C22⋊C4×C73C2×C411yescharacteristic
51C2×C22⋊C4×C73D411yescharacteristic
52C2×C42×C73D411yescharacteristic
53#95×C73—22no—
54#95×C73—22no—
55#95×C73C2×C412yes—
56C22×Dic73—42no—
57#11924012—82no—
58C2×#10596831—24no—
59C2×#10596831—44no—
60C4⋊Dic73—44no—
61C22⋊Dic73—42no—
62C22⋊Dic73—42no—
63C24×C73C22⋊C411yescharacteristic
64C24×C73C2×D411yescharacteristic
65C2×D4×C73—22no—
66C2×D4×C73—22no—
67C2×D4×C73—21no—
68C2×D4×C73—21no—
69C22×C4×C73—21no—
70C22×C4×C73—21no—
71C22×C4×C73—22no—
72C22×C4×C73—22no—
73C22×C4×C73—42no—
74C22×C4×C73—42no—
75C22×C4×C73—42no—
76C22×C4×C73—42no—
77C22×C4×C73—41no—
78C22×C4×C73—41no—
79C22×C4×C73C22⋊C411yescharacteristic
80C22×C4×C73C22⋊C411yescharacteristic
81C22×C4×C73C22⋊C411yescharacteristic
82C22×C4×C73C22×C411yescharacteristic
83C22×C4×C73C2×D412yes—
84C22×C4×C73C2×D411yescharacteristic
85C22×C4×C73#4112yes—
86C4⋊C4×C73—22no—
87C4⋊C4×C73—24no—
88C4⋊C4×C73—42no—
89C4⋊C4×C73—44no—
90C4⋊C4×C73—41no—
91C4⋊C4×C73—41no—
92C22⋊C4×C73—22no—
93C22⋊C4×C73—22no—
94C22⋊C4×C73—44no—
95C22⋊C4×C73—42no—
96C42×C73—21no—
97C42×C73—21no—
98C23×C73—21no—
99C23×C73—21no—
100C23×C73—21no—
101C23×C73—41no—
102C23×C73—41no—
103C23×C73—41no—
104C23×C73—41no—
105C23×C73C2×C22⋊C411yescharacteristic
106C23×C73#12112yes—
107C23×C73AGL(1,ℤ/8)11yescharacteristic
108C23×C73AGL(1,ℤ/8)12yes—
109C23×C73#13711yescharacteristic
110D4×C73—42no—
111D4×C73—42no—
112C2×C4×C73—22no—
113C2×C4×C73—24no—
114C2×C4×C73—21no—
115C2×C4×C73—21no—
116C2×C4×C73—21no—
117C2×C4×C73—21no—
118C2×C4×C73—21no—
119C2×C4×C73—41no—
120C2×C4×C73—42no—
121C2×C4×C73—42no—
122C2×C4×C73—41no—
123C2×C4×C73—41no—
124C2×C4×C73—42no—
125C2×C4×C73—42no—
126C2×C4×C73—42no—
127C2×C4×C73—42no—
128C2×C4×C73—42no—
129C2×C4×C73—42no—
130C2×C4×C73—81no—
131C2×C4×C73—82no—
132C2×C4×C73—82no—
133C2×C4×C73C4×D412yes—
134C2×C4×C73#12011yescharacteristic
135C2×C4×C73#12311yescharacteristic
136C2×Dic73—82no—
137C2×Dic73—82no—
138C2×Dic73—82no—
139#10596831—44no—
140C22×C73—22no—
141C22×C73—21no—
142C22×C73—22no—
143C22×C73—21no—
144C22×C73—41no—
145C22×C73—42no—
146C22×C73—41no—
147C22×C73—42no—
148C22×C73—41no—
149C22×C73—41no—
150C22×C73—41no—
151C22×C73—81no—
152C22×C73#38611yescharacteristic
153C22×C73#41711yescharacteristic
154C22×C73#41811yescharacteristic
155C22×C73#46812yes—
156C22×C73#46912yes—
157C4×C73—21no—
158C4×C73—22no—
159C4×C73—41no—
160C4×C73—41no—
161C4×C73—41no—
162C4×C73—82no—
163C4×C73—82no—
164C4×C73#45812yes—
165Dic73—162no—
166#9474—731nomaximal, Sylow
167C2×C73—22no—
168C2×C73—21no—
169C2×C73—21no—
170C2×C73—22no—
171C2×C73—81no—
172C2×C73#189011yescharacteristic
173C2×C73#189511yescharacteristic
174C2×C73#190911yescharacteristic
175#2973—731no—
176#2486D7311yescharacteristic
177#1823—732no—
178#1799—731no—
179#1289—732no—
180C73#947411yescharacteristic, complemented, Sylow
181C2×#123—21no—
182C2×C4×D4C2×D7311yescharacteristic
183C22×C4⋊C4C2×D7311yescharacteristic
184#435—1464no—
185#432—1461no—
186C2×#99—1461no—
187#407—1461no—
188C2×C22⋊C8—732no—
189#398Dic7312yescomplemented
190#397Dic7312yescomplemented
191#389—21no—
192#381—21no—
193#380—21no—
194#380—732no—
195#380C2×D7311yescharacteristic
196#342—1461no—
197#340—1464no—
198C22×D4C22⋊D7311yescharacteristic
199C23×C4—21no—
200C23×C4C22⋊D7311yescharacteristic
201C23×C4C22×D7311yescharacteristic
202C2×M16—1462no—
203C22×C8—734no—
204#123—44no—
205C4×D4—22no—
206C4×D4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—1462no—
213C2×C4⋊C4C2×Dic7312yes—
214C2×C4⋊C4C2×Dic7311yescharacteristic
215C2×C22⋊C4—22no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4—1462no—
218C2×C22⋊C4C2×Dic7311yescharacteristic
219C2×C22⋊C4C22⋊D7311yescharacteristic
220C2×C42C22⋊D7311yescharacteristic
221C4⋊C8—1462no—
222#99—2922no—
223C22⋊C8—1462no—
224C22⋊C8—1462no—
225C22⋊C8—1462no—
226#97—1462no—
227#95—22no—
228#95—22no—
229#95—1462no—
230#95C2×Dic7312yes—
231C24C22⋊Dic7311yescharacteristic
232C24C2×C22⋊D7311yescharacteristic
233C2×D4—22no—
234C2×D4—21no—
235C2×D4—22no—
236C2×D4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—41no—
245C22×C4—42no—
246C22×C4—41no—
247C22×C4—2922no—
248C22×C4C22⋊Dic7311yescharacteristic
249C22×C4C22⋊Dic7311yescharacteristic
250C22×C4C22⋊Dic7311yescharacteristic
251C22×C4C22×Dic7311yescharacteristic
252C22×C4D4×D7312yes—
253C22×C4C2×C22⋊D7311yescharacteristic
254C22×C4#1192402712yes—
255M16—5842no—
256C2×C8—2924no—
257C2×C8—1464no—
258C4⋊C4—22no—
259C4⋊C4—24no—
260C4⋊C4—42no—
261C4⋊C4—44no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—2924no—
265C22⋊C4—22no—
266C22⋊C4—22no—
267C22⋊C4—44no—
268C22⋊C4—42no—
269C22⋊C4—2922no—
270C22⋊C4—2922no—
271C42—21no—
272C42—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23C2×C22⋊Dic7311yescharacteristic
281C23#1455656312yes—
282C23#1455661611yescharacteristic
283C23#1455661712yes—
284C23#1455662111yescharacteristic
285D4—42no—
286D4—42no—
287C2×C4—22no—
288C2×C4—24no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—41no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—81no—
306C2×C4—82no—
307C2×C4—82no—
308C2×C4—5842no—
309C2×C4—5842no—
310C2×C4—5842no—
311C2×C4D4×Dic7312yes—
312C2×C4#1455655911yescharacteristic
313C2×C4#1455657211yescharacteristic
314C8—2924no—
315C22—21no—
316C22—22no—
317C22—21no—
318C22—22no—
319C22—41no—
320C22—42no—
321C22—41no—
322C22—42no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—81no—
327C22#1798846711yescharacteristic
328C22#1798858811yescharacteristic
329C22#1798859211yescharacteristic
330C22#1798883112yes—
331C22#1798883812yes—
332C4—21no—
333C4—22no—
334C4—41no—
335C4—41no—
336C4—41no—
337C4—82no—
338C4—82no—
339C4—11682no—
340C4#1798877612yes—
341C2—21no—
342C2—22no—
343C2—21no—
344C2—22no—
345C2—81no—
346C2#2261034511yescharacteristic
347C2#2261037011yescharacteristic
348C2#2261045411yescharacteristic
subgroupG/H
centerC22#17988467
commutatorC22×C4×C73C22×C4
FrattiniC23×C4C22×D73
FittingC73×#2486C2
socleC22×C73#386
prime radicalC22×D4×C73D4
cosocleC23×C4×C73C23
hypercenter#2486D73
hypocenterC73#9474
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2486

Complements

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

NH ≅ G/Ntype
2. #22610017C2semidirect
4. #22607798C2semidirect
17. #398×C73C4semidirect
18. #397×C73C4semidirect
180. C73#9474semidirect
189. #398Dic73semidirect
190. #397Dic73semidirect

Maximal subgroups

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

subgroupindex
#226189602
#226100172
#226099192
#226077982
C73×#24862
#947473