Finite Group Databasebeta

← #32996600

Subgroups — #32996600

Order 18688.

460 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 87 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#22615104C211yesmaximal, characteristic
2#22608269C211yesmaximal, characteristic
3#22607951C214yesmaximal, complemented
4C4×#99×C73C211yesmaximal, characteristic
5C2×#14556591C411yescharacteristic, complemented
6#17988551—21no—
7#17988499C412yescomplemented
8#17988499C2212yes—
9#17988490—21no—
10#17988452—21no—
11#17988447—21no—
12#17988447—21no—
13C4×C4⋊Dic73C411yescharacteristic, complemented
14#17988313—21no—
15#17988311C2212yes—
16C2×C4×D4×C73C2211yescharacteristic
17C2×#99×C73—21no—
18C4×C22⋊C4×C73—21no—
19#342×C73C2212yes—
20C22×Dic146C2×C412yes—
21C2×#11924012—41no—
22#14556591—41no—
23#14556591—41no—
24C2×C4⋊Dic73—21no—
25C2×C4⋊Dic73—21no—
26C2×C4⋊Dic73—21no—
27C2×C22⋊Dic73—22no—
28C2×C22⋊Dic73—22no—
29C2×C4×Dic73—21no—
30C22⋊#10596831—22no—
31C22⋊#10596831—41no—
32#14556498—22no—
33#14556498—22no—
34#14556498C2×C412yes—
35C22×D4×C73D411yescharacteristic
36C23×C4×C73D411yescharacteristic
37C23×C4×C73D411yescharacteristic
38C4×D4×C73—41no—
39C4×D4×C73—41no—
40C2×C4⋊C4×C73D411yescharacteristic
41C2×C22⋊C4×C73—22no—
42C2×C22⋊C4×C73C2×C411yescharacteristic
43C2×C22⋊C4×C73C2311yescharacteristic
44C2×C42×C73—41no—
45C2×C42×C73C2×C411yescharacteristic
46#99×C73—24no—
47#95×C73—41no—
48C2×Dic146—42no—
49C2×Dic146—82no—
50C22×Dic73—22no—
51C22×Dic73—22no—
52#11924012—82no—
53C2×#10596831—42no—
54C4⋊Dic73—42no—
55C4⋊Dic73—42no—
56C4⋊Dic73—41no—
57C4⋊Dic73—41no—
58C22⋊Dic73—82no—
59C22⋊Dic73—82no—
60C4×Dic73—42no—
61C24×C73C22⋊C411yescharacteristic
62C24×C73C2×D411yescharacteristic
63C2×D4×C73—22no—
64C2×D4×C73—22no—
65C2×D4×C73—41no—
66C22×C4×C73—21no—
67C22×C4×C73—21no—
68C22×C4×C73—21no—
69C22×C4×C73—21no—
70C22×C4×C73—42no—
71C22×C4×C73—42no—
72C22×C4×C73—42no—
73C22×C4×C73C22⋊C411yescharacteristic
74C22×C4×C73C22⋊C411yescharacteristic
75C22×C4×C73C22⋊C411yescharacteristic
76C22×C4×C73C22×C411yescharacteristic
77C22×C4×C73C2×D411yescharacteristic
78C4⋊C4×C73—41no—
79C22⋊C4×C73—24no—
80C22⋊C4×C73—41no—
81C22⋊C4×C73—41no—
82C22⋊C4×C73—42no—
83C42×C73—21no—
84C42×C73—21no—
85C42×C73—82no—
86C23×C73—21no—
87C23×C73—21no—
88C23×C73—21no—
89C23×C73—21no—
90C23×C73—21no—
91C23×C73—41no—
92C23×C73—41no—
93C23×C73—41no—
94C23×C73—41no—
95C23×C73C2×C22⋊C411yescharacteristic
96C23×C73#13712yes—
97D4×C73—44no—
98C2×C4×C73—21no—
99C2×C4×C73—21no—
100C2×C4×C73—22no—
101C2×C4×C73—21no—
102C2×C4×C73—21no—
103C2×C4×C73—21no—
104C2×C4×C73—42no—
105C2×C4×C73—42no—
106C2×C4×C73—41no—
107C2×C4×C73—41no—
108C2×C4×C73—41no—
109C2×C4×C73—44no—
110C2×C4×C73—81no—
111C2×C4×C73—81no—
112C2×C4×C73—82no—
113C2×C4×C73—82no—
114C2×C4×C73#9911yescharacteristic
115C2×C4×C73#9911yescharacteristic
116C2×C4×C73#10412yes—
117C2×Dic73—42no—
118C2×Dic73—42no—
119C2×Dic73—42no—
120C2×Dic73—42no—
121C2×Dic73—42no—
122C2×Dic73—42no—
123Dic146—84no—
124#10596831—82no—
125C22×C73—21no—
126C22×C73—21no—
127C22×C73—22no—
128C22×C73—42no—
129C22×C73—42no—
130C22×C73—41no—
131C22×C73—41no—
132C22×C73—41no—
133C22×C73—41no—
134C22×C73—41no—
135C22×C73—81no—
136C22×C73—81no—
137C22×C73C2×#9911yescharacteristic
138C22×C73#41911yescharacteristic
139C22×C73C2×#10411yescharacteristic
140C4×C73—21no—
141C4×C73—22no—
142C4×C73—41no—
143C4×C73—41no—
144C4×C73—41no—
145C4×C73—84no—
146C4×C73#35412yes—
147Dic73—82no—
148Dic73—82no—
149#11787—731nomaximal, Sylow
150C2×C73—21no—
151C2×C73—21no—
152C2×C73—42no—
153C2×C73—81no—
154C2×C73#150811yescharacteristic
155C2×C73#212211yescharacteristic
156C2×C73C2×#35411yescharacteristic
157#2563—731no—
158C4×#99D7311yescharacteristic
159#1465—731no—
160#1351—734no—
161C73#1178711yescharacteristic, complemented, Sylow
162C2×C4⋊Q8—731no—
163C2×C4×D4C2×D7311yescharacteristic
164C2×#99—21no—
165#407—1461no—
166#393—732no—
167#393—732no—
168#390—1461no—
169#382—1461no—
170#380—1461no—
171#380—1461no—
172C4×C4⋊C4—731no—
173C4×C22⋊C4—21no—
174#342C2×D7312yes—
175#324—1461no—
176#323—732no—
177C22×Q8—732no—
178C22×D4C22⋊D7311yescharacteristic
179C23×C4D29211yescharacteristic
180C23×C4C22⋊D7311yescharacteristic
181C2×M16—2921no—
182C4⋊Q8—2921no—
183C4⋊Q8—2921no—
184C4×D4—41no—
185C4×D4—41no—
186C2×C4⋊C4—1461no—
187C2×C4⋊C4—1461no—
188C2×C4⋊C4—1461no—
189C2×C4⋊C4D29211yescharacteristic
190C2×C22⋊C4—22no—
191C2×C22⋊C4—1462no—
192C2×C22⋊C4—1462no—
193C2×C22⋊C4C4×D7311yescharacteristic
194C2×C22⋊C4C22×D7311yescharacteristic
195C2×C42—41no—
196C2×C42—1461no—
197C2×C42C4×D7311yescharacteristic
198#99—24no—
199C22⋊C8—2921no—
200C22⋊C8—1462no—
201#95—41no—
202#95—732no—
203#95—1462no—
204#95—1462no—
205C24#1192400511yescharacteristic
206C24C2×C22⋊D7311yescharacteristic
207C2×Q8—2922no—
208C2×Q8—5842no—
209C2×D4—22no—
210C2×D4—22no—
211C2×D4—41no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—1462no—
220C22×C4—1462no—
221C22×C4#1192400511yescharacteristic
222C22×C4#1192400511yescharacteristic
223C22×C4#1192400511yescharacteristic
224C22×C4C2×C4×D7311yescharacteristic
225C22×C4C2×D29211yescharacteristic
226M16—5842no—
227C2×C8—2922no—
228C4⋊C4—41no—
229C4⋊C4—2922no—
230C4⋊C4—2922no—
231C4⋊C4—2921no—
232C4⋊C4—2921no—
233C22⋊C4—24no—
234C22⋊C4—41no—
235C22⋊C4—41no—
236C22⋊C4—42no—
237C22⋊C4—5842no—
238C22⋊C4—5842no—
239C42—21no—
240C42—21no—
241C42—82no—
242C42—2922no—
243C23—21no—
244C23—21no—
245C23—21no—
246C23—21no—
247C23—21no—
248C23—41no—
249C23—41no—
250C23—41no—
251C23—41no—
252C23C2×#1192400511yescharacteristic
253C23#1455662112yes—
254Q8—5844no—
255D4—44no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—44no—
268C2×C4—81no—
269C2×C4—81no—
270C2×C4—82no—
271C2×C4—82no—
272C2×C4—2922no—
273C2×C4—2922no—
274C2×C4—2922no—
275C2×C4—2922no—
276C2×C4—2922no—
277C2×C4—2922no—
278C2×C4#1455650511yescharacteristic
279C2×C4#1455650511yescharacteristic
280C2×C4#1455651912yes—
281C8—5842no—
282C22—21no—
283C22—21no—
284C22—22no—
285C22—42no—
286C22—42no—
287C22—41no—
288C22—41no—
289C22—41no—
290C22—41no—
291C22—41no—
292C22—81no—
293C22—81no—
294C22C2×#1455650511yescharacteristic
295C22#1798859511yescharacteristic
296C22C2×#1455651911yescharacteristic
297C4—22no—
298C4—21no—
299C4—41no—
300C4—41no—
301C4—41no—
302C4—84no—
303C4—5842no—
304C4—5842no—
305C4#1798838112yes—
306C2—21no—
307C2—21no—
308C2—42no—
309C2—81no—
310C2#2260845011yescharacteristic
311C2#2261158011yescharacteristic
312C2C2×#1798838111yescharacteristic
subgroupG/H
centerC22C2×#14556505
commutatorC22×C4×C73C22×C4
FrattiniC2×C22⋊C4C22×D73
FittingC4×#99×C73C2
socleC22×C73C2×#99
prime radicalC22×D4×C73D4
cosocleC2×C22⋊C4×C73C23
hypercenterC4×#99D73
hypocenterC73#11787
LCSG ▷ C22×C4×C73 ▷ C23×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ C4×#99

Complements

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

NH ≅ G/Ntype
3. #22607951C2semidirect
5. C2×#14556591C4semidirect
7. #17988499C4semidirect
13. C4×C4⋊Dic73C4semidirect
161. C73#11787semidirect

Maximal subgroups

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

subgroupindex
#226151042
#226082692
#226079512
C4×#99×C732
#1178773