Finite Group Databasebeta

← #32983771

Subgroups — #32983771

Order 18688.

556 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 115 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#22614248C211yesmaximal, characteristic
3#22610019C212yesmaximal, complemented
4#22607798C212yesmaximal, complemented
5C73×#1799C211yesmaximal, characteristic
6C22×C4⋊Dic73C2211yescharacteristic
7#17988659—24no—
8#17988644—21no—
9#17988551—21no—
10C2×C22⋊#10596831C2212yes—
11#17988526C412yescomplemented
12#17988523C412yescomplemented
13#17988485—21no—
14#17988467—21no—
15#17988459—21no—
16#17988447—21no—
17#17988447C2211yescharacteristic
18#17988349—24no—
19C2×C4×D4×C73C2211yescharacteristic
20C2×#99×C73—21no—
21#380×C73C2212yes—
22#342×C73—21no—
23C23×Dic73—21no—
24C2×#11924012—22no—
25C22×#10596831D414yes—
26C2×C4⋊Dic73—22no—
27C2×C4⋊Dic73—21no—
28C2×C4⋊Dic73—21no—
29C2×C4⋊Dic73—22no—
30C2×C4⋊Dic73C2×C412yes—
31C2×#11924007—22no—
32C2×C22⋊Dic73—22no—
33C2×C22⋊Dic73—21no—
34C4⋊#10596831—22no—
35C22⋊#10596831—22no—
36C22⋊#10596831—22no—
37C22⋊#10596831—22no—
38#14556502—22no—
39#14556498—22no—
40#14556498—22no—
41#14556498C2×C412yes—
42C22×D4×C73D411yescharacteristic
43C23×C4×C73D411yescharacteristic
44C23×C4×C73C2311yescharacteristic
45C4×D4×C73—22no—
46C4×D4×C73—22no—
47C2×C4⋊C4×C73—22no—
48C2×C4⋊C4×C73C2×C411yescharacteristic
49C2×C22⋊C4×C73—22no—
50C2×C22⋊C4×C73C2×C411yescharacteristic
51C2×C22⋊C4×C73D411yescharacteristic
52C2×C42×C73D411yescharacteristic
53#99×C73—42no—
54#95×C73—22no—
55C22×Dic73—22no—
56C22×Dic73—22no—
57C22×Dic73—42no—
58C22×Dic73—82no—
59#11924012—82no—
60C2×#10596831—24no—
61C2×#10596831—44no—
62C4⋊Dic73—24no—
63C4⋊Dic73—42no—
64C4⋊Dic73—41no—
65C4⋊Dic73—41no—
66#11924007—82no—
67C22⋊Dic73—41no—
68C22⋊Dic73—41no—
69C22⋊Dic73—82no—
70C24×C73C22⋊C411yescharacteristic
71C24×C73C2×D411yescharacteristic
72C2×D4×C73—22no—
73C2×D4×C73—21no—
74C2×D4×C73—22no—
75C2×D4×C73—21no—
76C22×C4×C73—21no—
77C22×C4×C73—21no—
78C22×C4×C73—42no—
79C22×C4×C73—42no—
80C22×C4×C73—42no—
81C22×C4×C73C22⋊C411yescharacteristic
82C22×C4×C73C22⋊C411yescharacteristic
83C22×C4×C73C22⋊C411yescharacteristic
84C22×C4×C73C22×C411yescharacteristic
85C22×C4×C73C2×D412yes—
86C22×C4×C73C2×D411yescharacteristic
87C22×C4×C73#4112yes—
88C4⋊C4×C73—22no—
89C4⋊C4×C73—44no—
90C22⋊C4×C73—22no—
91C22⋊C4×C73—22no—
92C22⋊C4×C73—42no—
93C22⋊C4×C73—42no—
94C42×C73—21no—
95C42×C73—21no—
96C23×C73—21no—
97C23×C73—21no—
98C23×C73—21no—
99C23×C73—41no—
100C23×C73—41no—
101C23×C73—41no—
102C23×C73—41no—
103C23×C73C2×C22⋊C411yescharacteristic
104C23×C73#12112yes—
105C23×C73AGL(1,ℤ/8)11yescharacteristic
106C23×C73#13712yes—
107C23×C73#13711yescharacteristic
108D4×C73—42no—
109D4×C73—42no—
110C2×C4×C73—24no—
111C2×C4×C73—22no—
112C2×C4×C73—21no—
113C2×C4×C73—21no—
114C2×C4×C73—21no—
115C2×C4×C73—21no—
116C2×C4×C73—21no—
117C2×C4×C73—41no—
118C2×C4×C73—42no—
119C2×C4×C73—42no—
120C2×C4×C73—41no—
121C2×C4×C73—41no—
122C2×C4×C73—81no—
123C2×C4×C73—82no—
124C2×C4×C73—82no—
125C2×C4×C73—82no—
126C2×C4×C73C4×D412yes—
127C2×C4×C73#12011yescharacteristic
128C2×C4×C73#12311yescharacteristic
129C2×Dic73—42no—
130C2×Dic73—42no—
131C2×Dic73—42no—
132C2×Dic73—42no—
133C2×Dic73—42no—
134C2×Dic73—42no—
135C2×Dic73—84no—
136#10596831—44no—
137C22×C73—21no—
138C22×C73—22no—
139C22×C73—21no—
140C22×C73—22no—
141C22×C73—41no—
142C22×C73—42no—
143C22×C73—41no—
144C22×C73—42no—
145C22×C73—41no—
146C22×C73—41no—
147C22×C73—41no—
148C22×C73—81no—
149C22×C73#38611yescharacteristic
150C22×C73#41711yescharacteristic
151C22×C73#41911yescharacteristic
152C22×C73#46812yes—
153C22×C73#47012yes—
154C4×C73—21no—
155C4×C73—22no—
156C4×C73—41no—
157C4×C73—41no—
158C4×C73—41no—
159C4×C73—162no—
160C4×C73#45812yes—
161Dic73—82no—
162Dic73—82no—
163#9473—731nomaximal, Sylow
164C2×C73—21no—
165C2×C73—22no—
166C2×C73—21no—
167C2×C73—22no—
168C2×C73—81no—
169C2×C73#189011yescharacteristic
170C2×C73#189411yescharacteristic
171C2×C73#191011yescharacteristic
172#2973—731no—
173#2487—731no—
174#1824—732no—
175#1799D7311yescharacteristic
176#1289—732no—
177C73#947311yescharacteristic, complemented, Sylow
178C2×C4×D4C2×D7311yescharacteristic
179C22×C4⋊C4—731no—
180#435—1464no—
181#432—1461no—
182C2×#99—21no—
183#407—1461no—
184C2×C22⋊C8—732no—
185#400—732no—
186#399—732no—
187#389—1461no—
188#386—1461no—
189#385—1461no—
190#380—731no—
191#380—1461no—
192#380C2×D7312yes—
193#342—21no—
194#340—1464no—
195C22×D4C22⋊D7311yescharacteristic
196C23×C4—1461no—
197C23×C4D29211yescharacteristic
198C23×C4C22×D7311yescharacteristic
199C2×M16—1462no—
200C22×C8—734no—
201C4×D4—22no—
202C4×D4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—732no—
205C2×C4⋊C4—1461no—
206C2×C4⋊C4—1461no—
207C2×C4⋊C4—1462no—
208C2×C4⋊C4—1462no—
209C2×C4⋊C4—1462no—
210C2×C4⋊C4C4×D7311yescharacteristic
211C2×C22⋊C4—22no—
212C2×C22⋊C4—1461no—
213C2×C22⋊C4—1462no—
214C2×C22⋊C4C4×D7311yescharacteristic
215C2×C22⋊C4C22⋊D7311yescharacteristic
216C2×C42D29211yescharacteristic
217C4⋊C8—1462no—
218#99—42no—
219C22⋊C8—1462no—
220C22⋊C8—1462no—
221C22⋊C8—1462no—
222#97—1462no—
223#95—22no—
224#95—732no—
225#95—1462no—
226#95—1462no—
227C24#1192400511yescharacteristic
228C24C2×C22⋊D7311yescharacteristic
229C2×D4—22no—
230C2×D4—21no—
231C2×D4—22no—
232C2×D4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—2922no—
239C22×C4—5842no—
240C22×C4—1462no—
241C22×C4—1462no—
242C22×C4#1192400511yescharacteristic
243C22×C4#1192400511yescharacteristic
244C22×C4#1192400511yescharacteristic
245C22×C4C2×C4×D7311yescharacteristic
246C22×C4D4×D7312yes—
247C22×C4C2×D29211yescharacteristic
248C22×C4#1192402712yes—
249M16—5842no—
250C2×C8—2924no—
251C2×C8—1464no—
252C4⋊C4—22no—
253C4⋊C4—44no—
254C4⋊C4—2922no—
255C4⋊C4—2921no—
256C4⋊C4—2921no—
257C4⋊C4—5842no—
258C4⋊C4—1464no—
259C22⋊C4—22no—
260C22⋊C4—22no—
261C22⋊C4—42no—
262C22⋊C4—42no—
263C22⋊C4—2921no—
264C22⋊C4—2921no—
265C22⋊C4—5842no—
266C42—21no—
267C42—21no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—41no—
272C23—41no—
273C23—41no—
274C23—41no—
275C23C2×#1192400511yescharacteristic
276C23#1455656312yes—
277C23#1455661711yescharacteristic
278C23#1455662112yes—
279C23#1455662211yescharacteristic
280D4—42no—
281D4—42no—
282C2×C4—24no—
283C2×C4—22no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—41no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—81no—
295C2×C4—82no—
296C2×C4—82no—
297C2×C4—82no—
298C2×C4—2922no—
299C2×C4—2922no—
300C2×C4—2922no—
301C2×C4—2922no—
302C2×C4—2922no—
303C2×C4—2922no—
304C2×C4—5844no—
305C2×C4#1455655112yes—
306C2×C4#1455656111yescharacteristic
307C2×C4#1455657511yescharacteristic
308C8—2924no—
309C22—21no—
310C22—22no—
311C22—21no—
312C22—22no—
313C22—41no—
314C22—42no—
315C22—41no—
316C22—42no—
317C22—41no—
318C22—41no—
319C22—41no—
320C22—81no—
321C22#1798846411yescharacteristic
322C22#1798858611yescharacteristic
323C22#1798859511yescharacteristic
324C22#1798883112yes—
325C22#1798884312yes—
326C4—21no—
327C4—22no—
328C4—41no—
329C4—41no—
330C4—41no—
331C4—162no—
332C4—5842no—
333C4—5842no—
334C4#1798877812yes—
335C2—22no—
336C2—21no—
337C2—21no—
338C2—22no—
339C2—81no—
340C2#2261034211yescharacteristic
341C2#2261036211yescharacteristic
342C2#2261045611yescharacteristic
subgroupG/H
centerC22#17988464
commutatorC22×C4×C73C22×C4
FrattiniC23×C4C22×D73
FittingC73×#1799C2
socleC22×C73#386
prime radicalC22×D4×C73D4
cosocleC23×C4×C73C23
hypercenter#1799D73
hypocenterC73#9473
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1799

Complements

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

NH ≅ G/Ntype
3. #22610019C2semidirect
4. #22607798C2semidirect
11. #17988526C4semidirect
12. #17988523C4semidirect
177. C73#9473semidirect

Maximal subgroups

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

subgroupindex
#226189602
#226142482
#226100192
#226077982
C73×#17992
#947373