Finite Group Databasebeta

← #32996858

Subgroups — #32996858

Order 18688.

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

normal subgroups only

This group has 390 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#22615508C211yesmaximal, characteristic
2#22610008C211yesmaximal, characteristic
3#22607953C212yesmaximal
4#22607932C212yesmaximal
5C73×#1458C211yesmaximal, characteristic
6C2×C4×Dic146C411yescharacteristic, complemented
7#17988517C411yescharacteristic, complemented
8#17988512C2211yescharacteristic
9#17988507C411yescharacteristic, complemented
10#17988499C2211yescharacteristic
11#17988475—21no—
12#17988471—21no—
13#17988468—21no—
14#17988460—21no—
15#17988455C411yescharacteristic, complemented
16#17988450—21no—
17#17988354—21no—
18#17988354C2212yes—
19C2×#117×C73C2211yescharacteristic
20#431×C73—21no—
21#324×C73—21no—
22#323×C73C2212yes—
23C22×Dic146C2×C411yescharacteristic
24C4×Dic146—41no—
25C4×Dic146—41no—
26C2×C4⋊Dic73C2×C411yescharacteristic
27C2×#11924007—21no—
28C2×#11924007—21no—
29C2×#11924007—21no—
30C2×C22⋊Dic73—21no—
31C2×C22⋊Dic73—21no—
32C2×C22⋊Dic73—22no—
33C2×C4×Dic73—21no—
34#14556498—21no—
35#14556498—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—
55C4×C8×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—41no—
64#11924007—42no—
65#11924007—42no—
66#11924007—41no—
67#11924007—41no—
68C22⋊Dic73—42no—
69C22⋊Dic73—42no—
70C22⋊Dic73—42no—
71C22⋊Dic73—42no—
72C4×Dic73—42no—
73C24×C73C2×D411yescharacteristic
74C22×C4×C73—22no—
75C22×C4×C73—21no—
76C22×C4×C73—21no—
77C22×C4×C73—41no—
78C22×C4×C73C22⋊C411yescharacteristic
79C22×C4×C73C22⋊C411yescharacteristic
80C22×C4×C73C22⋊C411yescharacteristic
81C22×C4×C73C22⋊C411yescharacteristic
82C22×C4×C73C22×C411yescharacteristic
83C22×C4×C73C2×D411yescharacteristic
84C2×C8×C73—42no—
85C4⋊C4×C73—41no—
86C4⋊C4×C73—41no—
87C22⋊C4×C73—41no—
88C22⋊C4×C73—41no—
89C42×C73—21no—
90C42×C73—21no—
91C42×C73—21no—
92C42×C73—21no—
93C23×C73—21no—
94C23×C73—21no—
95C23×C73—41no—
96C23×C73—41no—
97C23×C73C2×C22⋊C411yescharacteristic
98C23×C73#13512yes—
99C2×C4×C73—21no—
100C2×C4×C73—21no—
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—42no—
109C2×C4×C73—42no—
110C2×C4×C73—41no—
111C2×C4×C73—41no—
112C2×C4×C73—41no—
113C2×C4×C73—41no—
114C2×C4×C73#9911yescharacteristic
115C2×C4×C73#9911yescharacteristic
116C2×Dic73—22no—
117C2×Dic73—41no—
118C2×Dic73—41no—
119C2×Dic73—41no—
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—82no—
129C2×Dic73—82no—
130C2×Dic73—82no—
131Dic146—42no—
132Dic146—42no—
133C8×C73—82no—
134C22×C73—21no—
135C22×C73—21no—
136C22×C73—22no—
137C22×C73—42no—
138C22×C73—41no—
139C22×C73—41no—
140C22×C73—81no—
141C22×C73C2×#9911yescharacteristic
142C22×C73#41611yescharacteristic
143C22×C73#42111yescharacteristic
144C4×C73—21no—
145C4×C73—21no—
146C4×C73—41no—
147C4×C73—41no—
148C4×C73—41no—
149C4×C73—41no—
150C4×C73—41no—
151Dic73—42no—
152Dic73—81no—
153Dic73—81no—
154Dic73—81no—
155Dic73—162no—
156#11842—731nomaximal, Sylow
157C2×C73—21no—
158C2×C73—21no—
159C2×C73—42no—
160C2×C73#151611yescharacteristic
161C2×C73#212011yescharacteristic
162C2×C73#212911yescharacteristic
163#2599—731no—
164#1820—731no—
165#1458D7311yescharacteristic
166#1351—732no—
167#1344—732no—
168C73#1184211yescharacteristic, complemented, Sylow
169C2×C4×Q8—731no—
170C2×#117C2×D7311yescharacteristic
171#431—21no—
172#398—731no—
173#397—731no—
174#395—731no—
175#393—731no—
176#388—1461no—
177#387—1461no—
178#387—1461no—
179#385—1461no—
180#383—731no—
181#381—1461no—
182#342—732no—
183#342—1461no—
184#324—21no—
185#323C2×D7312yes—
186C22×Q8—731no—
187C23×C4D29211yescharacteristic
188C4×Q8—2921no—
189C4×Q8—2921no—
190#117—41no—
191#117—41no—
192C2×C4⋊C4—731no—
193C2×C4⋊C4—1461no—
194C2×C4⋊C4—1461no—
195C2×C4⋊C4—1461no—
196C2×C4⋊C4C4×D7311yescharacteristic
197C2×C4⋊C4C22⋊D7311yescharacteristic
198C2×C22⋊C4—1462no—
199C2×C22⋊C4—1461no—
200C2×C22⋊C4—1461no—
201C2×C22⋊C4D29211yescharacteristic
202C2×C22⋊C4C22×D7311yescharacteristic
203C2×C42—1461no—
204C2×C42C4×D7311yescharacteristic
205C2×C42C22⋊D7311yescharacteristic
206C4⋊C8—41no—
207C22⋊C8—22no—
208C4×C8—41no—
209#95—2921no—
210#95—2921no—
211#95—731no—
212#95—731no—
213#95—1461no—
214#95—1461no—
215#95—1461no—
216#95—1461no—
217#95—1461no—
218#95—1461no—
219C24C2×D29211yescharacteristic
220C2×Q8—2921no—
221C2×Q8—2922no—
222C22×C4—22no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—41no—
226C22×C4—2922no—
227C22×C4—1461no—
228C22×C4—1461no—
229C22×C4—1461no—
230C22×C4—1461no—
231C22×C4#1192400511yescharacteristic
232C22×C4#1192400511yescharacteristic
233C22×C4#1192400511yescharacteristic
234C22×C4#1192400511yescharacteristic
235C22×C4C2×C4×D7311yescharacteristic
236C22×C4C2×C22⋊D7311yescharacteristic
237C2×C8—42no—
238C4⋊C4—41no—
239C4⋊C4—41no—
240C4⋊C4—2921no—
241C4⋊C4—2922no—
242C4⋊C4—2922no—
243C4⋊C4—2921no—
244C4⋊C4—2921no—
245C22⋊C4—41no—
246C22⋊C4—41no—
247C22⋊C4—2922no—
248C22⋊C4—2922no—
249C22⋊C4—2922no—
250C22⋊C4—2922no—
251C42—21no—
252C42—21no—
253C42—21no—
254C42—21no—
255C42—2922no—
256C23—21no—
257C23—21no—
258C23—41no—
259C23—41no—
260C23C2×#1192400511yescharacteristic
261C23#1455661312yes—
262Q8—2922no—
263Q8—2922no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—2921no—
280C2×C4—2921no—
281C2×C4—2921no—
282C2×C4—2921no—
283C2×C4—2921no—
284C2×C4—2921no—
285C2×C4—2921no—
286C2×C4—2921no—
287C2×C4—2921no—
288C2×C4—2921no—
289C2×C4—2921no—
290C2×C4—5842no—
291C2×C4—5842no—
292C2×C4—5842no—
293C2×C4—1462no—
294C2×C4#1455650511yescharacteristic
295C2×C4#1455650511yescharacteristic
296C8—82no—
297C22—21no—
298C22—21no—
299C22—22no—
300C22—42no—
301C22—41no—
302C22—41no—
303C22—81no—
304C22C2×#1455650511yescharacteristic
305C22#1798858411yescharacteristic
306C22#1798860311yescharacteristic
307C4—21no—
308C4—21no—
309C4—41no—
310C4—41no—
311C4—41no—
312C4—41no—
313C4—41no—
314C4—2922no—
315C4—5841no—
316C4—5841no—
317C4—5841no—
318C4—11682no—
319C2—21no—
320C2—21no—
321C2—42no—
322C2#2260848811yescharacteristic
323C2#2261157311yescharacteristic
324C2#2261160511yescharacteristic
subgroupG/H
centerC22C2×#14556505
commutatorC22×C4×C73C22×C4
FrattiniC2×C22⋊C4C22×D73
FittingC73×#1458C2
socleC22×C73C2×#99
prime radicalC24×C73C2×D4
cosocleC2×C22⋊C4×C73C23
hypercenter#1458D73
hypocenterC73#11842
LCSG ▷ C22×C4×C73 ▷ C23×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1458

Complements

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

NH ≅ G/Ntype
6. C2×C4×Dic146C4semidirect
7. #17988517C4semidirect
9. #17988507C4semidirect
15. #17988455C4semidirect
168. C73#11842semidirect

Maximal subgroups

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

subgroupindex
#226155082
#226100082
#226079532
#226079322
C73×#14582
#1184273