Finite Group Databasebeta

← #32995551

Subgroups — #32995551

Order 18688.

398 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 83 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#22614412C211yesmaximal, characteristic
2C2×#17988313C211yesmaximal, characteristic
3#22607874C212yesmaximal, complemented
4#22607864C212yesmaximal, complemented
5C73×#1463C211yesmaximal, characteristic
6C2×#14556570C411yescharacteristic
7C2×C22⋊#10596831—21no—
8#17988525—21no—
9#17988525—21no—
10#17988492C411yescharacteristic
11#17988492C2211yescharacteristic
12#17988452C411yescharacteristic
13#17988452C2211yescharacteristic
14#17988450—21no—
15#17988450—21no—
16C4×C22⋊Dic73C411yescharacteristic
17#17988313—21no—
18#17988313C2212yes—
19C22×C4⋊C4×C73C2211yescharacteristic
20#407×C73—21no—
21#324×C73—21no—
22#324×C73C2212yes—
23C22×Dic146C2×C411yescharacteristic
24C22×#10596831—41no—
25#14556570—41no—
26#14556570—41no—
27C2×C4⋊Dic73C2×C411yescharacteristic
28C2×#11924007—21no—
29C2×C22⋊Dic73—21no—
30C2×C22⋊Dic73—21no—
31C2×C4×Dic73C2×C411yescharacteristic
32C2×C4×Dic73C2×C411yescharacteristic
33C22⋊#10596831—22no—
34C22⋊#10596831—41no—
35#14556498—21no—
36#14556498—21no—
37#14556498—21no—
38#14556498—21no—
39#14556498—21no—
40#14556498—21no—
41#14556498—21no—
42#14556498—21no—
43#14556498—21no—
44C23×C4×C73C2×C411yescharacteristic
45C23×C4×C73D411yescharacteristic
46C23×C4×C73D411yescharacteristic
47C2×M16×C73—41no—
48C2×C4⋊C4×C73—41no—
49C2×C4⋊C4×C73—41no—
50C2×C4⋊C4×C73C2×C411yescharacteristic
51C2×C4⋊C4×C73D411yescharacteristic
52C2×C4⋊C4×C73D411yescharacteristic
53C2×C4⋊C4×C73C2311yescharacteristic
54C22⋊C8×C73—22no—
55C22⋊C8×C73—41no—
56C2×Dic146—22no—
57C2×Dic146—41no—
58C2×Dic146—41no—
59C22×Dic73—21no—
60C22×Dic73—21no—
61C22×Dic73—21no—
62C22×Dic73—21no—
63C2×#10596831—42no—
64C2×#10596831—82no—
65C4⋊Dic73—41no—
66#11924007—42no—
67C22⋊Dic73—42no—
68C22⋊Dic73—41no—
69C22⋊Dic73—41no—
70C4×Dic73—22no—
71C4×Dic73—41no—
72C24×C73C22⋊C411yescharacteristic
73C22×C4×C73—21no—
74C22×C4×C73—21no—
75C22×C4×C73—21no—
76C22×C4×C73—21no—
77C22×C4×C73—21no—
78C22×C4×C73—21no—
79C22×C4×C73—22no—
80C22×C4×C73—41no—
81C22×C4×C73—42no—
82C22×C4×C73—42no—
83C22×C4×C73C22⋊C411yescharacteristic
84C22×C4×C73C22⋊C411yescharacteristic
85C22×C4×C73C22⋊C411yescharacteristic
86C22×C4×C73C22×C411yescharacteristic
87C22×C4×C73C2×D411yescharacteristic
88C22×C4×C73C2×D411yescharacteristic
89M16×C73—82no—
90C2×C8×C73—42no—
91C4⋊C4×C73—41no—
92C4⋊C4×C73—41no—
93C4⋊C4×C73—41no—
94C4⋊C4×C73—41no—
95C23×C73—22no—
96C23×C73—21no—
97C23×C73—21no—
98C23×C73—41no—
99C23×C73#9911yescharacteristic
100C23×C73#9911yescharacteristic
101C23×C73C2×C22⋊C411yescharacteristic
102C2×C4×C73—21no—
103C2×C4×C73—21no—
104C2×C4×C73—21no—
105C2×C4×C73—21no—
106C2×C4×C73—21no—
107C2×C4×C73—21no—
108C2×C4×C73—21no—
109C2×C4×C73—21no—
110C2×C4×C73—21no—
111C2×C4×C73—42no—
112C2×C4×C73—42no—
113C2×C4×C73—42no—
114C2×C4×C73—42no—
115C2×C4×C73—41no—
116C2×C4×C73—41no—
117C2×C4×C73—41no—
118C2×C4×C73—41no—
119C2×C4×C73—81no—
120C2×C4×C73—81no—
121C2×C4×C73#13112yes—
122C2×Dic73—41no—
123C2×Dic73—41no—
124C2×Dic73—41no—
125C2×Dic73—41no—
126C2×Dic73—42no—
127C2×Dic73—41no—
128C2×Dic73—41no—
129C2×Dic73—41no—
130C2×Dic73—41no—
131C2×Dic73—41no—
132C2×Dic73—41no—
133Dic146—82no—
134C8×C73—82no—
135#10596831—82no—
136C22×C73—21no—
137C22×C73—21no—
138C22×C73—21no—
139C22×C73—21no—
140C22×C73—21no—
141C22×C73—21no—
142C22×C73—21no—
143C22×C73—42no—
144C22×C73—42no—
145C22×C73#35412yes—
146C22×C73#40811yescharacteristic
147C22×C73C2×#9911yescharacteristic
148C22×C73#41311yescharacteristic
149C4×C73—22no—
150C4×C73—41no—
151C4×C73—41no—
152C4×C73—41no—
153C4×C73—41no—
154C4×C73—41no—
155Dic73—81no—
156Dic73—81no—
157Dic73—81no—
158Dic73—81no—
159#11551—731nomaximal, Sylow
160C2×C73—21no—
161C2×C73—21no—
162C2×C73—22no—
163C2×C73—41no—
164C2×C73#146811yescharacteristic
165C2×C73C2×#35411yescharacteristic
166C2×C73#213311yescharacteristic
167#2503—731no—
168#1463D7311yescharacteristic
169C2×#324—731no—
170#1324—732no—
171#1319—732no—
172C73#1155111yescharacteristic, complemented, Sylow
173C2×C22⋊Q8—731no—
174C22×C4⋊C4C2×D7311yescharacteristic
175#407—21no—
176C2×C22⋊C8—1461no—
177#400—1461no—
178#400—1461no—
179#391—731no—
180#391—731no—
181#382—731no—
182#382—731no—
183#381—1461no—
184#381—1461no—
185C4×C22⋊C4—731no—
186#324—21no—
187#324—732no—
188#324—1461no—
189#324C2×D7312yes—
190C22×Q8—731no—
191C23×C4C4×D7311yescharacteristic
192C23×C4D29211yescharacteristic
193C23×C4C22⋊D7311yescharacteristic
194C2×M16—41no—
195C22×C8—2921no—
196C22⋊Q8—2921no—
197C22⋊Q8—2921no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4—731no—
201C2×C4⋊C4—1461no—
202C2×C4⋊C4C4×D7311yescharacteristic
203C2×C4⋊C4D29211yescharacteristic
204C2×C4⋊C4C22⋊D7311yescharacteristic
205C2×C4⋊C4C22×D7311yescharacteristic
206C2×C22⋊C4—1461no—
207C2×C22⋊C4—1461no—
208C2×C42—731no—
209C2×C42—731no—
210C22⋊C8—22no—
211C22⋊C8—41no—
212C22⋊C8—2921no—
213C22⋊C8—1462no—
214#95—1461no—
215#95—1461no—
216#95—1461no—
217#95—1461no—
218#95—1461no—
219#95—1461no—
220#95—1461no—
221#95—1461no—
222#95—1461no—
223C24#1192400511yescharacteristic
224C2×Q8—2921no—
225C2×Q8—2921no—
226C2×Q8—1462no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—22no—
234C22×C4—41no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—1461no—
238C22×C4—1461no—
239C22×C4—1461no—
240C22×C4—1461no—
241C22×C4#1192400511yescharacteristic
242C22×C4#1192400511yescharacteristic
243C22×C4#1192400511yescharacteristic
244C22×C4C2×C4×D7311yescharacteristic
245C22×C4C2×C22⋊D7311yescharacteristic
246C22×C4C2×D29211yescharacteristic
247M16—82no—
248C2×C8—42no—
249C2×C8—2922no—
250C2×C8—5842no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—2921no—
256C4⋊C4—2922no—
257C22⋊C4—2922no—
258C22⋊C4—2921no—
259C22⋊C4—2921no—
260C42—2921no—
261C42—1462no—
262C23—22no—
263C23—21no—
264C23—21no—
265C23—41no—
266C23#1455650511yescharacteristic
267C23#1455650511yescharacteristic
268C23C2×#1192400511yescharacteristic
269Q8—5842no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—81no—
288C2×C4—81no—
289C2×C4—2921no—
290C2×C4—2921no—
291C2×C4—2921no—
292C2×C4—2921no—
293C2×C4—2922no—
294C2×C4—2921no—
295C2×C4—2921no—
296C2×C4—2921no—
297C2×C4—2921no—
298C2×C4—2921no—
299C2×C4—2921no—
300C2×C4#1455659912yes—
301C8—82no—
302C8—5842no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—42no—
311C22—42no—
312C22#1798838112yes—
313C22#1798855511yescharacteristic
314C22C2×#1455650511yescharacteristic
315C22#1798856911yescharacteristic
316C4—22no—
317C4—41no—
318C4—41no—
319C4—41no—
320C4—41no—
321C4—41no—
322C4—5841no—
323C4—5841no—
324C4—5841no—
325C4—5841no—
326C2—21no—
327C2—21no—
328C2—22no—
329C2—41no—
330C2#2260828011yescharacteristic
331C2C2×#1798838111yescharacteristic
332C2#2261161811yescharacteristic
subgroupG/H
centerC22C2×#14556505
commutatorC22×C4×C73C22×C4
FrattiniC2×C4⋊C4C22×D73
FittingC73×#1463C2
socleC22×C73C2×#99
prime radicalC24×C73C22⋊C4
cosocleC2×C4⋊C4×C73C23
hypercenter#1463D73
hypocenterC73#11551
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ #1463

Complements

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

NH ≅ G/Ntype
3. #22607874C2semidirect
4. #22607864C2semidirect
172. C73#11551semidirect

Maximal subgroups

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

subgroupindex
#226144122
C2×#179883132
#226078742
#226078642
C73×#14632
#1155173