Finite Group Databasebeta

← #32982037

Subgroups — #32982037

Order 18688.

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

normal subgroups only

This group has 612 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#22615736C211yesmaximal, characteristic
2#22608502C212yesmaximal, complemented
3C2×#17988311C211yesmaximal, characteristic
4#22607798C212yesmaximal, complemented
5C73×#2312C211yesmaximal, characteristic
6C2×#14556570C2211yescharacteristic
7C2×C22⋊#10596831C2212yes—
8#17988523C412yescomplemented
9#17988510—22no—
10#17988499C412yescomplemented
11#17988460—21no—
12#17988447C2211yescharacteristic
13#17988443—21no—
14#17988323—24no—
15#17988311—22no—
16C22×C22⋊C4×C73C2211yescharacteristic
17#388×C73—24no—
18#380×C73C2212yes—
19#379×C73—21no—
20C4×C22⋊C4×C73—21no—
21C22×Dic146C2×C411yescharacteristic
22C22×#10596831—22no—
23#14556570—22no—
24#14556570—22no—
25C2×C4⋊Dic73—21no—
26C2×C4⋊Dic73C2×C411yescharacteristic
27C2×#11924007—22no—
28C2×#11924007—21no—
29C2×C22⋊Dic73—22no—
30C2×C22⋊Dic73—22no—
31C2×C22⋊Dic73—21no—
32C2×C22⋊Dic73—21no—
33C22⋊#10596831—22no—
34C22⋊#10596831—22no—
35#14556498—22no—
36#14556498C2×C412yes—
37C25×C73D411yescharacteristic
38C23×C4×C73D411yescharacteristic
39C23×C4×C73C2311yescharacteristic
40C2×C4⋊C4×C73—22no—
41C2×C22⋊C4×C73—22no—
42C2×C22⋊C4×C73—22no—
43C2×C22⋊C4×C73—22no—
44C2×C22⋊C4×C73—22no—
45C2×C22⋊C4×C73—22no—
46C2×C22⋊C4×C73C2×C412yes—
47C2×C22⋊C4×C73D412yes—
48C2×C42×C73—22no—
49#95×C73—22no—
50#95×C73D414yes—
51C2×Dic146—22no—
52C2×Dic146—81no—
53C22×Dic73—21no—
54C22×Dic73—22no—
55C22×Dic73—21no—
56C2×#10596831—42no—
57C2×#10596831—82no—
58C4⋊Dic73—22no—
59C4⋊Dic73—81no—
60#11924007—42no—
61#11924007—82no—
62C22⋊Dic73—41no—
63C22⋊Dic73—41no—
64C22⋊Dic73—42no—
65C22⋊Dic73—82no—
66C22⋊Dic73—82no—
67C24×C73—21no—
68C24×C73—21no—
69C24×C73—22no—
70C24×C73—22no—
71C24×C73—81no—
72C24×C73—81no—
73C24×C73C22⋊C412yes—
74C24×C73C2×D411yescharacteristic
75C22×C4×C73—21no—
76C22×C4×C73—21no—
77C22×C4×C73—24no—
78C22×C4×C73—81no—
79C22×C4×C73—81no—
80C22×C4×C73C22⋊C412yes—
81C22×C4×C73C22×C411yescharacteristic
82C22×C4×C73C2×D412yes—
83C22×C4×C73C2×D411yescharacteristic
84C22×C4×C73#4112yes—
85C4⋊C4×C73—82no—
86C22⋊C4×C73—24no—
87C22⋊C4×C73—24no—
88C22⋊C4×C73—42no—
89C22⋊C4×C73—42no—
90C22⋊C4×C73—82no—
91C22⋊C4×C73—82no—
92C42×C73—82no—
93C23×C73—22no—
94C23×C73—22no—
95C23×C73—22no—
96C23×C73—41no—
97C23×C73—41no—
98C23×C73—42no—
99C23×C73—42no—
100C23×C73—42no—
101C23×C73—42no—
102C23×C73—42no—
103C23×C73—42no—
104C23×C73—81no—
105C23×C73—81no—
106C23×C73—84no—
107C23×C73—84no—
108C23×C73C2×C22⋊C411yescharacteristic
109C23×C73#12011yescharacteristic
110C23×C73#12311yescharacteristic
111C23×C73#13512yes—
112C23×C73#13712yes—
113C2×C4×C73—22no—
114C2×C4×C73—42no—
115C2×C4×C73—41no—
116C2×C4×C73—41no—
117C2×C4×C73—42no—
118C2×C4×C73—42no—
119C2×C4×C73—42no—
120C2×C4×C73—44no—
121C2×C4×C73—44no—
122C2×C4×C73—44no—
123C2×C4×C73—81no—
124C2×C4×C73C4×D412yes—
125C2×C4×C73#12112yes—
126C2×Dic73—42no—
127C2×Dic73—42no—
128C2×Dic73—42no—
129C2×Dic73—42no—
130C2×Dic73—41no—
131C2×Dic73—41no—
132C2×Dic73—42no—
133Dic146—82no—
134#10596831—82no—
135C22×C73—21no—
136C22×C73—21no—
137C22×C73—21no—
138C22×C73—21no—
139C22×C73—21no—
140C22×C73—21no—
141C22×C73—42no—
142C22×C73—42no—
143C22×C73—44no—
144C22×C73—42no—
145C22×C73—42no—
146C22×C73—42no—
147C22×C73—82no—
148C22×C73—84no—
149C22×C73—84no—
150C22×C73#38611yescharacteristic
151C22×C73#41611yescharacteristic
152C22×C73#41911yescharacteristic
153C22×C73#45212yes—
154C22×C73#48512yes—
155C4×C73—22no—
156C4×C73—22no—
157C4×C73—42no—
158C4×C73—84no—
159Dic73—82no—
160Dic73—81no—
161Dic73—81no—
162#9201—731nomaximal, Sylow
163C2×C73—21no—
164C2×C73—22no—
165C2×C73—21no—
166C2×C73—22no—
167C2×C73—82no—
168C2×C73#187311yescharacteristic
169C2×C73#191111yescharacteristic
170C2×C73#191311yescharacteristic
171#2626—731no—
172#2312D7311yescharacteristic
173#1520—732no—
174C2×#323—731no—
175#1289—732no—
176C73#920111yescharacteristic, complemented, Sylow
177C2×C22⋊Q8—731no—
178C22×C22⋊C4C2×D7311yescharacteristic
179C2×C22⋊C8—732no—
180#399—732no—
181#396—1462no—
182#393—732no—
183#388—24no—
184#385—1461no—
185#380—731no—
186#380C2×D7312yes—
187#379—21no—
188#379—1461no—
189C4×C22⋊C4—21no—
190#328—1464no—
191#323—1462no—
192C25C22⋊D7311yescharacteristic
193C22×Q8—731no—
194C23×C4C22⋊D7311yescharacteristic
195C23×C4C22×D7311yescharacteristic
196C22×C8—1462no—
197C22⋊Q8—1462no—
198C22⋊Q8—1462no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—731no—
201C2×C4⋊C4—1461no—
202C2×C4⋊C4—1462no—
203C2×C4⋊C4—1461no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—1461no—
210C2×C22⋊C4—1461no—
211C2×C22⋊C4—1462no—
212C2×C22⋊C4—1462no—
213C2×C22⋊C4C4×D7312yes—
214C2×C22⋊C4D29212yes—
215C2×C42—22no—
216C22⋊C8—1462no—
217C22⋊C8—1462no—
218#95—22no—
219#95—732no—
220#95—1462no—
221#95C22⋊D7314yes—
222C24—21no—
223C24—21no—
224C24—22no—
225C24—22no—
226C24—81no—
227C24—81no—
228C24#1192400512yes—
229C24C2×C22⋊D7311yescharacteristic
230C2×Q8—5841no—
231C2×Q8—1462no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—24no—
235C22×C4—81no—
236C22×C4—81no—
237C22×C4—1462no—
238C22×C4—1461no—
239C22×C4—1461no—
240C22×C4#1192400512yes—
241C22×C4C2×C4×D7311yescharacteristic
242C22×C4C2×C22⋊D7312yes—
243C22×C4C2×D29211yescharacteristic
244C22×C4#1192402812yes—
245C2×C8—2922no—
246C2×C8—5842no—
247C4⋊C4—82no—
248C4⋊C4—2922no—
249C4⋊C4—5842no—
250C4⋊C4—5841no—
251C4⋊C4—1462no—
252C22⋊C4—24no—
253C22⋊C4—24no—
254C22⋊C4—42no—
255C22⋊C4—42no—
256C22⋊C4—82no—
257C22⋊C4—82no—
258C22⋊C4—2921no—
259C22⋊C4—2921no—
260C22⋊C4—2922no—
261C22⋊C4—5842no—
262C22⋊C4—5842no—
263C42—82no—
264C23—22no—
265C23—22no—
266C23—22no—
267C23—41no—
268C23—41no—
269C23—42no—
270C23—42no—
271C23—42no—
272C23—42no—
273C23—42no—
274C23—42no—
275C23—81no—
276C23—81no—
277C23—84no—
278C23—84no—
279C23C2×#1192400511yescharacteristic
280C23#1455656011yescharacteristic
281C23#1455657411yescharacteristic
282C23#1455661112yes—
283C23#1455662112yes—
284Q8—5842no—
285C2×C4—22no—
286C2×C4—42no—
287C2×C4—41no—
288C2×C4—42no—
289C2×C4—41no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—44no—
293C2×C4—44no—
294C2×C4—44no—
295C2×C4—81no—
296C2×C4—2922no—
297C2×C4—2922no—
298C2×C4—2922no—
299C2×C4—2922no—
300C2×C4—2921no—
301C2×C4—2921no—
302C2×C4—2922no—
303C2×C4C4×C22⋊D7312yes—
304C2×C4#1455656512yes—
305C8—5842no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—42no—
313C22—42no—
314C22—44no—
315C22—42no—
316C22—42no—
317C22—42no—
318C22—82no—
319C22—84no—
320C22—84no—
321C22#1798846611yescharacteristic
322C22#1798858111yescharacteristic
323C22#1798859511yescharacteristic
324C22#1798874912yes—
325C22#1798892812yes—
326C4—22no—
327C4—22no—
328C4—42no—
329C4—84no—
330C4—5842no—
331C4—5841no—
332C4—5841no—
333C2—22no—
334C2—21no—
335C2—21no—
336C2—22no—
337C2—82no—
338C2#2261024311yescharacteristic
339C2#2261046311yescharacteristic
340C2#2261047311yescharacteristic
subgroupG/H
centerC22#17988466
commutatorC22×C4×C73C22×C4
FrattiniC23×C4C22×D73
FittingC73×#2312C2
socleC22×C73#386
prime radicalC25×C73D4
cosocleC23×C4×C73C23
hypercenter#2312D73
hypocenterC73#9201
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2312

Complements

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

NH ≅ G/Ntype
2. #22608502C2semidirect
4. #22607798C2semidirect
8. #17988523C4semidirect
10. #17988499C4semidirect
176. C73#9201semidirect

Maximal subgroups

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

subgroupindex
#226157362
#226085022
C2×#179883112
#226077982
C73×#23122
#920173