Finite Group Databasebeta

← #32996702

Subgroups — #32996702

Order 18688.

402 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 79 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#22613199C211yesmaximal, characteristic
2#22610008C211yesmaximal, characteristic
3#22607956C212yesmaximal
4#22607953C212yesmaximal
5C73×#1464C211yesmaximal, characteristic
6C2×C4×Dic146C411yescharacteristic, complemented
7#17988526C411yescharacteristic, complemented
8#17988523C2211yescharacteristic
9#17988520—21no—
10#17988510—21no—
11#17988507C411yescharacteristic, complemented
12#17988499C2211yescharacteristic
13#17988450—21no—
14C4×#11924007—21no—
15C4×C4⋊Dic73C411yescharacteristic, complemented
16C4×C22⋊Dic73—21no—
17#17988354—21no—
18#17988354C2212yes—
19C2×#117×C73C2211yescharacteristic
20#432×C73—21no—
21#324×C73—21no—
22#323×C73C2212yes—
23C22×Dic146C2×C411yescharacteristic
24C4×Dic146—41no—
25C4×Dic146—41no—
26C2×C4⋊Dic73—21no—
27C2×C4⋊Dic73C2×C411yescharacteristic
28C2×#11924007—21no—
29C2×#11924007—21no—
30C2×#11924007—21no—
31C2×C22⋊Dic73—21no—
32C2×C22⋊Dic73—21no—
33C2×C22⋊Dic73—22no—
34C2×C4×Dic73—21no—
35C2×C4×Dic73—21no—
36#14556498—21no—
37#14556498—21no—
38#14556498—21no—
39#14556498—21no—
40#14556498—41no—
41#14556498—41no—
42#14556498C2×C411yescharacteristic
43#14556498C2×C411yescharacteristic
44C23×C4×C73D411yescharacteristic
45#117×C73—41no—
46#117×C73—41no—
47C2×C4⋊C4×C73C2×C411yescharacteristic
48C2×C4⋊C4×C73D411yescharacteristic
49C2×C22⋊C4×C73D411yescharacteristic
50C2×C22⋊C4×C73C2311yescharacteristic
51C2×C42×C73C2×C411yescharacteristic
52C2×C42×C73D411yescharacteristic
53C4⋊C8×C73—41no—
54C22⋊C8×C73—22no—
55#97×C73—41no—
56C2×Dic146—41no—
57C2×Dic146—42no—
58C22×Dic73—21no—
59C22×Dic73—21no—
60C22×Dic73—21no—
61C22×Dic73—21no—
62C22×Dic73—42no—
63C4⋊Dic73—42no—
64C4⋊Dic73—41no—
65#11924007—42no—
66#11924007—42no—
67#11924007—42no—
68C22⋊Dic73—42no—
69C22⋊Dic73—42no—
70C22⋊Dic73—42no—
71C22⋊Dic73—42no—
72C4×Dic73—42no—
73C4×Dic73—41no—
74C4×Dic73—41no—
75C24×C73C2×D411yescharacteristic
76C22×C4×C73—22no—
77C22×C4×C73—21no—
78C22×C4×C73—21no—
79C22×C4×C73—41no—
80C22×C4×C73C22⋊C411yescharacteristic
81C22×C4×C73C22⋊C411yescharacteristic
82C22×C4×C73C22⋊C411yescharacteristic
83C22×C4×C73C22⋊C411yescharacteristic
84C22×C4×C73C22×C411yescharacteristic
85C22×C4×C73C2×D411yescharacteristic
86C2×C8×C73—42no—
87C4⋊C4×C73—41no—
88C4⋊C4×C73—41no—
89C22⋊C4×C73—41no—
90C22⋊C4×C73—41no—
91C42×C73—21no—
92C42×C73—21no—
93C42×C73—21no—
94C42×C73—21no—
95C23×C73—21no—
96C23×C73—21no—
97C23×C73—41no—
98C23×C73—41no—
99C23×C73C2×C22⋊C411yescharacteristic
100C23×C73#13712yes—
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—41no—
113C2×C4×C73—41no—
114C2×C4×C73—41no—
115C2×C4×C73—41no—
116C2×C4×C73#9911yescharacteristic
117C2×C4×C73#9911yescharacteristic
118C2×Dic73—22no—
119C2×Dic73—22no—
120C2×Dic73—41no—
121C2×Dic73—41no—
122C2×Dic73—41no—
123C2×Dic73—41no—
124C2×Dic73—41no—
125C2×Dic73—41no—
126C2×Dic73—41no—
127C2×Dic73—41no—
128C2×Dic73—41no—
129C2×Dic73—41no—
130C2×Dic73—82no—
131C2×Dic73—82no—
132C2×Dic73—82no—
133Dic146—42no—
134Dic146—42no—
135C8×C73—82no—
136C22×C73—21no—
137C22×C73—21no—
138C22×C73—22no—
139C22×C73—42no—
140C22×C73—41no—
141C22×C73—41no—
142C22×C73—81no—
143C22×C73C2×#9911yescharacteristic
144C22×C73#41911yescharacteristic
145C22×C73#42111yescharacteristic
146C4×C73—21no—
147C4×C73—21no—
148C4×C73—41no—
149C4×C73—41no—
150C4×C73—41no—
151C4×C73—41no—
152C4×C73—41no—
153Dic73—42no—
154Dic73—42no—
155Dic73—81no—
156Dic73—81no—
157Dic73—162no—
158#11810—731nomaximal, Sylow
159C2×C73—21no—
160C2×C73—21no—
161C2×C73—42no—
162C2×C73#151111yescharacteristic
163C2×C73#212011yescharacteristic
164C2×C73#212611yescharacteristic
165#2396—731no—
166#1820—731no—
167#1464D7311yescharacteristic
168#1352—732no—
169#1351—732no—
170C73#1181011yescharacteristic, complemented, Sylow
171C2×C4×Q8—731no—
172C2×#117C2×D7311yescharacteristic
173#432—21no—
174#400—731no—
175#399—731no—
176#398—1461no—
177#396—1461no—
178#395—731no—
179#393—731no—
180#381—1461no—
181C4×C4⋊C4—731no—
182C4×C4⋊C4—1461no—
183C4×C22⋊C4—1461no—
184#342—732no—
185#342—1461no—
186#324—21no—
187#323C2×D7312yes—
188C22×Q8—731no—
189C23×C4D29211yescharacteristic
190C4×Q8—2921no—
191C4×Q8—2921no—
192#117—41no—
193#117—41no—
194C2×C4⋊C4—731no—
195C2×C4⋊C4—1461no—
196C2×C4⋊C4—1461no—
197C2×C4⋊C4—1461no—
198C2×C4⋊C4—1461no—
199C2×C4⋊C4C4×D7311yescharacteristic
200C2×C4⋊C4C22⋊D7311yescharacteristic
201C2×C22⋊C4—1462no—
202C2×C22⋊C4—1461no—
203C2×C22⋊C4—1461no—
204C2×C22⋊C4D29211yescharacteristic
205C2×C22⋊C4C22×D7311yescharacteristic
206C2×C42—1461no—
207C2×C42—1461no—
208C2×C42C4×D7311yescharacteristic
209C2×C42C22⋊D7311yescharacteristic
210C4⋊C8—41no—
211C22⋊C8—22no—
212#97—41no—
213#95—2921no—
214#95—2921no—
215#95—731no—
216#95—731no—
217#95—1461no—
218#95—1461no—
219#95—1461no—
220#95—1461no—
221C24C2×D29211yescharacteristic
222C2×Q8—2921no—
223C2×Q8—2922no—
224C22×C4—22no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—41no—
228C22×C4—2922no—
229C22×C4—1461no—
230C22×C4—1461no—
231C22×C4—1461no—
232C22×C4—1461no—
233C22×C4#1192400511yescharacteristic
234C22×C4#1192400511yescharacteristic
235C22×C4#1192400511yescharacteristic
236C22×C4#1192400511yescharacteristic
237C22×C4C2×C4×D7311yescharacteristic
238C22×C4C2×C22⋊D7311yescharacteristic
239C2×C8—42no—
240C4⋊C4—41no—
241C4⋊C4—41no—
242C4⋊C4—2921no—
243C4⋊C4—2922no—
244C4⋊C4—2922no—
245C4⋊C4—2922no—
246C4⋊C4—2922no—
247C22⋊C4—41no—
248C22⋊C4—41no—
249C22⋊C4—2922no—
250C22⋊C4—2922no—
251C22⋊C4—2922no—
252C22⋊C4—2922no—
253C42—21no—
254C42—21no—
255C42—21no—
256C42—21no—
257C42—2922no—
258C42—2921no—
259C42—2921no—
260C23—21no—
261C23—21no—
262C23—41no—
263C23—41no—
264C23C2×#1192400511yescharacteristic
265C23#1455662412yes—
266Q8—2922no—
267Q8—2922no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—2921no—
284C2×C4—2921no—
285C2×C4—2921no—
286C2×C4—2921no—
287C2×C4—2921no—
288C2×C4—2921no—
289C2×C4—2921no—
290C2×C4—2921no—
291C2×C4—2921no—
292C2×C4—2921no—
293C2×C4—5842no—
294C2×C4—5842no—
295C2×C4—5842no—
296C2×C4—1462no—
297C2×C4—1462no—
298C2×C4#1455650511yescharacteristic
299C2×C4#1455650511yescharacteristic
300C8—82no—
301C22—21no—
302C22—21no—
303C22—22no—
304C22—42no—
305C22—41no—
306C22—41no—
307C22—81no—
308C22C2×#1455650511yescharacteristic
309C22#1798859711yescharacteristic
310C22#1798860311yescharacteristic
311C4—21no—
312C4—21no—
313C4—41no—
314C4—41no—
315C4—41no—
316C4—41no—
317C4—41no—
318C4—2922no—
319C4—2922no—
320C4—5841no—
321C4—5841no—
322C4—11682no—
323C2—21no—
324C2—21no—
325C2—42no—
326C2#2260846511yescharacteristic
327C2#2261157311yescharacteristic
328C2#2261159411yescharacteristic
subgroupG/H
centerC22C2×#14556505
commutatorC22×C4×C73C22×C4
FrattiniC2×C22⋊C4C22×D73
FittingC73×#1464C2
socleC22×C73C2×#99
prime radicalC24×C73C2×D4
cosocleC2×C22⋊C4×C73C23
hypercenter#1464D73
hypocenterC73#11810
LCSG ▷ C22×C4×C73 ▷ C23×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1464

Complements

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

NH ≅ G/Ntype
6. C2×C4×Dic146C4semidirect
7. #17988526C4semidirect
11. #17988507C4semidirect
15. C4×C4⋊Dic73C4semidirect
170. C73#11810semidirect

Maximal subgroups

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

subgroupindex
#226131992
#226100082
#226079562
#226079532
C73×#14642
#1181073