Finite Group Databasebeta

← #32996898

Subgroups — #32996898

Order 18688.

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

normal subgroups only

This group has 396 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#22614218C211yesmaximal, characteristic
2#22610011C211yesmaximal, characteristic
3#22607959C212yesmaximal, complemented
4#22607938C212yesmaximal, complemented
5C2×#324×C73C211yesmaximal, characteristic
6C2×#14556550C411yescharacteristic
7#17988551—21no—
8#17988526C411yescharacteristic
9#17988526C2211yescharacteristic
10#17988517C411yescharacteristic
11#17988517C2211yescharacteristic
12#17988468—21no—
13#17988468—21no—
14#17988460—21no—
15#17988450—21no—
16#17988450C411yescharacteristic
17#17988356—21no—
18#17988356C2212yes—
19C22×C4⋊C4×C73C2211yescharacteristic
20C2×C22⋊C8×C73—21no—
21#324×C73—21no—
22#324×C73C2212yes—
23C2×#11924012—22no—
24#14556550—41no—
25#14556550—41no—
26C2×C4⋊Dic73C2×C411yescharacteristic
27C2×C4⋊Dic73C2×C411yescharacteristic
28C2×#11924007—21no—
29C2×#11924007—21no—
30C2×C22⋊Dic73—21no—
31C2×C22⋊Dic73—21no—
32C2×C4×Dic73—21no—
33C22⋊#10596831—41no—
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#14556498C2×C411yescharacteristic
43#14556498C2×C411yescharacteristic
44C23×C4×C73C2×C411yescharacteristic
45C23×C4×C73D411yescharacteristic
46C23×C4×C73D411yescharacteristic
47C22×C8×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—
56C22×Dic73—21no—
57C22×Dic73—21no—
58C22×Dic73—21no—
59C22×Dic73—21no—
60#11924012—42no—
61#11924012—42no—
62C2×#10596831—42no—
63C4⋊Dic73—41no—
64C4⋊Dic73—41no—
65#11924007—42no—
66#11924007—42no—
67C22⋊Dic73—42no—
68C22⋊Dic73—41no—
69C22⋊Dic73—41no—
70C4×Dic73—42no—
71C24×C73C22⋊C411yescharacteristic
72C22×C4×C73—21no—
73C22×C4×C73—21no—
74C22×C4×C73—21no—
75C22×C4×C73—21no—
76C22×C4×C73—21no—
77C22×C4×C73—21no—
78C22×C4×C73—22no—
79C22×C4×C73—41no—
80C22×C4×C73—42no—
81C22×C4×C73—42no—
82C22×C4×C73C22⋊C411yescharacteristic
83C22×C4×C73C22⋊C411yescharacteristic
84C22×C4×C73C22⋊C411yescharacteristic
85C22×C4×C73C22×C411yescharacteristic
86C22×C4×C73C2×D411yescharacteristic
87C22×C4×C73C2×D411yescharacteristic
88C2×C8×C73—42no—
89C2×C8×C73—82no—
90C4⋊C4×C73—41no—
91C4⋊C4×C73—41no—
92C4⋊C4×C73—41no—
93C4⋊C4×C73—41no—
94C23×C73—22no—
95C23×C73—21no—
96C23×C73—21no—
97C23×C73—41no—
98C23×C73#9911yescharacteristic
99C23×C73#9911yescharacteristic
100C23×C73C2×C22⋊C411yescharacteristic
101C2×C4×C73—21no—
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—42no—
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#13512yes—
121C2×Dic73—41no—
122C2×Dic73—41no—
123C2×Dic73—41no—
124C2×Dic73—41no—
125C2×Dic73—42no—
126C2×Dic73—41no—
127C2×Dic73—41no—
128C2×Dic73—41no—
129C2×Dic73—41no—
130C2×Dic73—41no—
131C2×Dic73—41no—
132C8×C73—82no—
133#10596831—82no—
134C22×C73—21no—
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#35212yes—
144C22×C73C2×#9911yescharacteristic
145C22×C73#41611yescharacteristic
146C22×C73#42111yescharacteristic
147C4×C73—22no—
148C4×C73—41no—
149C4×C73—41no—
150C4×C73—41no—
151C4×C73—41no—
152C4×C73—41no—
153Dic73—81no—
154Dic73—81no—
155Dic73—81no—
156Dic73—81no—
157#11850—731nomaximal, Sylow
158C2×C73—21no—
159C2×C73—21no—
160C2×C73—22no—
161C2×C73—41no—
162C2×C73#151611yescharacteristic
163C2×C73C2×#35211yescharacteristic
164C2×C73#213311yescharacteristic
165#2484—731no—
166#1821—731no—
167C2×#324D7311yescharacteristic
168#1353—732no—
169#1346—732no—
170C73#1185011yescharacteristic, complemented, Sylow
171C2×#117—731no—
172C22×C4⋊C4C2×D7311yescharacteristic
173#407—1461no—
174C2×C22⋊C8—21no—
175#400—731no—
176#400—731no—
177#398—731no—
178#398—731no—
179#387—1461no—
180#387—1461no—
181#385—1461no—
182#381—731no—
183#381—1461no—
184#343—732no—
185#343—1461no—
186#324—21no—
187#324C2×D7312yes—
188C23×C4C4×D7311yescharacteristic
189C23×C4D29211yescharacteristic
190C23×C4C22⋊D7311yescharacteristic
191C2×M16—1462no—
192C22×C8—41no—
193#117—2921no—
194#117—2921no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4—731no—
198C2×C4⋊C4—731no—
199C2×C4⋊C4—1461no—
200C2×C4⋊C4—1461no—
201C2×C4⋊C4C4×D7311yescharacteristic
202C2×C4⋊C4D29211yescharacteristic
203C2×C4⋊C4C22⋊D7311yescharacteristic
204C2×C4⋊C4C22×D7311yescharacteristic
205C2×C22⋊C4—1461no—
206C2×C22⋊C4—1461no—
207C2×C42—1461no—
208C22⋊C8—22no—
209C22⋊C8—41no—
210C22⋊C8—2921no—
211C22⋊C8—2921no—
212#95—731no—
213#95—731no—
214#95—1461no—
215#95—1461no—
216#95—1461no—
217#95—1461no—
218#95—1461no—
219#95—1461no—
220#95—1461no—
221C24#1192400511yescharacteristic
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—22no—
229C22×C4—41no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—1461no—
233C22×C4—1461no—
234C22×C4—1461no—
235C22×C4—1461no—
236C22×C4#1192400511yescharacteristic
237C22×C4#1192400511yescharacteristic
238C22×C4#1192400511yescharacteristic
239C22×C4C2×C4×D7311yescharacteristic
240C22×C4C2×C22⋊D7311yescharacteristic
241C22×C4C2×D29211yescharacteristic
242M16—2922no—
243M16—2922no—
244C2×C8—42no—
245C2×C8—82no—
246C2×C8—2922no—
247C4⋊C4—41no—
248C4⋊C4—41no—
249C4⋊C4—41no—
250C4⋊C4—41no—
251C4⋊C4—2921no—
252C4⋊C4—2921no—
253C4⋊C4—2922no—
254C4⋊C4—2922no—
255C22⋊C4—2922no—
256C22⋊C4—2921no—
257C22⋊C4—2921no—
258C42—2922no—
259C23—22no—
260C23—21no—
261C23—21no—
262C23—41no—
263C23#1455650511yescharacteristic
264C23#1455650511yescharacteristic
265C23C2×#1192400511yescharacteristic
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—81no—
285C2×C4—2921no—
286C2×C4—2921no—
287C2×C4—2921no—
288C2×C4—2921no—
289C2×C4—2922no—
290C2×C4—2921no—
291C2×C4—2921no—
292C2×C4—2921no—
293C2×C4—2921no—
294C2×C4—2921no—
295C2×C4—2921no—
296C2×C4#1455661312yes—
297C8—82no—
298C8—5842no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—42no—
307C22—42no—
308C22#1798837612yes—
309C22C2×#1455650511yescharacteristic
310C22#1798858411yescharacteristic
311C22#1798860311yescharacteristic
312C4—22no—
313C4—41no—
314C4—41no—
315C4—41no—
316C4—41no—
317C4—41no—
318C4—5841no—
319C4—5841no—
320C4—5841no—
321C4—5841no—
322C2—21no—
323C2—21no—
324C2—22no—
325C2—41no—
326C2#2260848811yescharacteristic
327C2C2×#1798837611yescharacteristic
328C2#2261161811yescharacteristic
subgroupG/H
centerC22C2×#14556505
commutatorC22×C4×C73C22×C4
FrattiniC2×C4⋊C4C22×D73
FittingC2×#324×C73C2
socleC22×C73C2×#99
prime radicalC24×C73C22⋊C4
cosocleC2×C4⋊C4×C73C23
hypercenterC2×#324D73
hypocenterC73#11850
LCSG ▷ C22×C4×C73 ▷ C23×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4 ◁ C2×#324

Complements

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

NH ≅ G/Ntype
3. #22607959C2semidirect
4. #22607938C2semidirect
170. C73#11850semidirect

Maximal subgroups

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

subgroupindex
#226142182
#226100112
#226079592
#226079382
C2×#324×C732
#1185073