Finite Group Databasebeta

← #33112015

Subgroups — #33112015

Order 18688.

1080 conjugacy classes of subgroups; only the 255 normal subgroups are shown.

normal subgroups only

This group has 1080 conjugacy classes of subgroups — too many to draw. Only its 255 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
1C4×#14556560C211yesmaximal, characteristic
2#22610723C213yesmaximal, complemented
3#22610540C213yesmaximal, complemented
4#22610534C213yesmaximal, complemented
5#22610220C213yesmaximal, complemented
6#22609841C211yesmaximal, characteristic, complemented
7C73×#1779C211yesmaximal, characteristic, complemented
8C22×C8×D73C2211yescharacteristic, complemented
9C2×#14556560C411yescharacteristic
10C2×C4×C22⋊D73C2213yes—
11C2×#14556520C2213yescomplemented
12C2×C22⋊#10596831C2213yes—
13C2×C22⋊#10596831C2213yes—
14C2×#14556503C2213yescomplemented
15C2×C8×Dic73C2213yescomplemented
16#17988466C413yes—
17#17988460C413yes—
18#17988443C411yescharacteristic
19C4×C22⋊Dic73C2213yes—
20#17988341C2213yescomplemented
21#17988341C2213yescomplemented
22C24×C4×C73C2211yescharacteristic
23C2×C22⋊C8×C73C2213yes—
24C2×C22⋊C8×C73C2213yes—
25C22×C22⋊D73C2×C413yes—
26C22×C4×D73C2311yescharacteristic
27C22×#10596831C2313yes—
28C22×#10596831C2311yescharacteristic
29C2×C8×D73—43no—
30#14556560C818yescomplemented
31C4×C22⋊D73—43no—
32C4×C22⋊D73—43no—
33C2×#11924007C2×C413yes—
34C2×C22⋊Dic73C2×C413yes—
35C2×C22⋊Dic73C2×C413yes—
36C2×C22⋊Dic73C2×C413yes—
37C2×#11924005C2×C413yes—
38C2×C4×Dic73C2313yes—
39#14556520—43no—
40C22⋊#10596831—43no—
41C22⋊#10596831—43no—
42#14556503—43no—
43C8×Dic73—43no—
44#14556498C2×C416yes—
45C25×C73C2×C411yescharacteristic
46C23×C4×C73—43no—
47C23×C4×C73—43no—
48C23×C4×C73C2×C413yes—
49C23×C4×C73C2313yes—
50C22×C8×C73C2313yes—
51C22×C8×C73C2311yescharacteristic
52C22⋊C8×C73—43no—
53C22⋊C8×C73—43no—
54C23×D73C22×C411yescharacteristic
55C2×C22⋊D73—43no—
56C2×C22⋊D73—43no—
57C2×C22⋊D73C2×C8112yes—
58C22×Dic73C22×C413yes—
59C22×Dic73C22×C411yescharacteristic
60C22×Dic73C22×C413yes—
61C2×C4×D73—23no—
62C2×C4×D73—23no—
63C2×#10596831—23no—
64C2×#10596831—23no—
65C2×#10596831—23no—
66C2×#10596831—23no—
67C8×D73—44no—
68#11924007—43no—
69C22⋊Dic73—43no—
70C22⋊Dic73—43no—
71C22⋊Dic73C2×C8112yescomplemented
72#11924005—43no—
73C4×Dic73—23no—
74C4×Dic73—23no—
75C24×C73—43no—
76C24×C73—43no—
77C24×C73C2×C814yescomplemented
78C24×C73C22×C413yes—
79C22×C4×C73—23no—
80C22×C4×C73—23no—
81C22×C4×C73—23no—
82C22×C4×C73—23no—
83C22×C4×C73—23no—
84C22×C4×C73—23no—
85C22×C4×C73—43no—
86C22×C4×C73—43no—
87C22×C4×C73—43no—
88C22×C4×C73—412no—
89C22×C4×C73—44no—
90C22×C4×C73C22×C413yes—
91C22×C4×C73C2411yescharacteristic
92C2×C8×C73—23no—
93C2×C8×C73—23no—
94C2×C8×C73—23no—
95C2×C8×C73—23no—
96C23×C73—23no—
97C23×C73—23no—
98C23×C73—23no—
99C23×C73—23no—
100C23×C73—23no—
101C23×C73—23no—
102C23×C73—412no—
103C23×C73—44no—
104C23×C73—412no—
105C23×C73C22×C816yes—
106C23×C73C23×C411yescharacteristic
107C22×D73—23no—
108C22×D73—23no—
109C22×D73C22×C812yes—
110C2×C4×C73—23no—
111C2×C4×C73—23no—
112C2×C4×C73—23no—
113C2×C4×C73—412no—
114C2×C4×C73—43no—
115C2×C4×C73—43no—
116C2×C4×C73—43no—
117C2×C4×C73—43no—
118C2×C4×C73H(5,𝔽2)13yes—
119C2×C4×C73H(5,𝔽2)13yes—
120C22⋊D73—412no—
121C2×Dic73—23no—
122C2×Dic73—23no—
123C2×Dic73—23no—
124C2×Dic73—23no—
125C2×Dic73—26no—
126C2×Dic73C22×C816yes—
127C4×D73—43no—
128C4×D73—41no—
129C8×C73—43no—
130C8×C73—41no—
131#10596831—43no—
132#10596831—41no—
133C22×C73—26no—
134C22×C73—26no—
135C22×C73—26no—
136C22×C73—416no—
137C22×C73—43no—
138C22×C73—43no—
139C22×C73—43no—
140C22×C73—43no—
141C22×C73#51813yes—
142C22×C73C23×C811yescharacteristic
143C22×C73#56813yes—
144C2×D73—26no—
145C2×D73—41no—
146C2×D73—43no—
147C4×C73—43no—
148C4×C73#56013yes—
149C4×C73H(3,𝔽4)11yescharacteristic
150Dic73—26no—
151Dic73—41no—
152Dic73—43no—
153#23306—731nomaximal, Sylow
154C2×C73—46no—
155C2×C73#237511yescharacteristic
156C2×C73#297213yes—
157C2×C73#297813yes—
158D73—42no—
159C4×#120—731no—
160#1954—733no—
161#1923—733no—
162#1923—733no—
163#1869—733no—
164#1779—731no—
165#1779D7311yescharacteristic, complemented
166C73#2330611yescharacteristic, complemented, Sylow
167C24×C4C2×D7311yescharacteristic
168C23×C8—731no—
169C2×#120—731no—
170C2×C4×D4—733no—
171C2×C4⋊C8—733no—
172C2×C22⋊C8—733no—
173C2×C22⋊C8—733no—
174C2×C22⋊C8—733no—
175C2×C22⋊C8C2×D7313yes—
176C2×C22⋊C8C2×D7313yes—
177C2×C4×C8—733no—
178#386—733no—
179#385—733no—
180#379—731no—
181C4×C22⋊C4—733no—
182#336—733no—
183#336—733no—
184C25C4×D7311yescharacteristic
185C22×D4—733no—
186C23×C4—43no—
187C23×C4—43no—
188C23×C4—731no—
189C23×C4C4×D7313yes—
190C23×C4C22×D7313yes—
191C22×C8—2923no—
192C22×C8—731no—
193C22×C8—733no—
194C22×C8C22×D7311yescharacteristic
195C22×C8C22×D7313yes—
196#120—738no—
197C4×D4—2923no—
198C4×D4—2923no—
199C2×C4⋊C4—733no—
200C2×C22⋊C4—733no—
201C2×C22⋊C4—733no—
202C2×C22⋊C4—733no—
203C2×C22⋊C4—733no—
204C2×C42—733no—
205C4⋊C8—2923no—
206C22⋊C8—43no—
207C22⋊C8—43no—
208C22⋊C8—2923no—
209C22⋊C8—2923no—
210C22⋊C8—2923no—
211C4×C8—2923no—
212#95—736no—
213C24—43no—
214C24—43no—
215C24—731no—
216C24C8×D7314yescomplemented
217C24C2×C4×D7313yes—
218C2×D4—2923no—
219C2×D4—2923no—
220C2×D4—7312no—
221C22×C4—23no—
222C22×C4—23no—
223C22×C4—23no—
224C22×C4—23no—
225C22×C4—23no—
226C22×C4—23no—
227C22×C4—43no—
228C22×C4—43no—
229C22×C4—43no—
230C22×C4—412no—
231C22×C4—44no—
232C22×C4—733no—
233C22×C4—733no—
234C22×C4—731no—
235C22×C4—1463no—
236C22×C4—1463no—
237C22×C4C2×C4×D7313yes—
238C22×C4C23×D7311yescharacteristic
239C2×C8—23no—
240C2×C8—23no—
241C2×C8—23no—
242C2×C8—23no—
243C2×C8—2924no—
244C2×C8—1463no—
245C2×C8—1463no—
246C2×C8—1463no—
247C2×C8—1463no—
248C4⋊C4—2923no—
249C22⋊C4—2923no—
250C22⋊C4—2923no—
251C22⋊C4—2923no—
252C22⋊C4—7312no—
253C42—1463no—
254C42—1463no—
255C23—23no—
256C23—23no—
257C23—23no—
258C23—23no—
259C23—23no—
260C23—23no—
261C23—412no—
262C23—44no—
263C23—412no—
264C23—732no—
265C23—1463no—
266C23—1463no—
267C23C2×C8×D7316yes—
268C23C22×C4×D7311yescharacteristic
269D4—29212no—
270C2×C4—23no—
271C2×C4—23no—
272C2×C4—23no—
273C2×C4—412no—
274C2×C4—43no—
275C2×C4—43no—
276C2×C4—43no—
277C2×C4—43no—
278C2×C4—2923no—
279C2×C4—2921no—
280C2×C4—736no—
281C2×C4—1463no—
282C2×C4—1463no—
283C2×C4—1466no—
284C2×C4—1463no—
285C2×C4—1463no—
286C2×C4#1455663613yes—
287C2×C4#1455663613yes—
288C8—43no—
289C8—41no—
290C8—2923no—
291C8—2921no—
292C22—26no—
293C22—26no—
294C22—26no—
295C22—416no—
296C22—43no—
297C22—43no—
298C22—43no—
299C22—43no—
300C22—2921no—
301C22—2923no—
302C22—1466no—
303C22#1798906013yes—
304C22C22×C8×D7311yescharacteristic
305C22#1798935313yes—
306C4—43no—
307C4—2921no—
308C4—2923no—
309C4—1466no—
310C4#1798932613yes—
311C4#1798933011yescharacteristic
312C2—46no—
313C2—2922no—
314C2#2261307011yescharacteristic
315C2#2261895313yes—
316C2#2261899113yes—
subgroupG/H
center, FrattiniC22×C4C23×D73
commutatorC22×C73C23×C8
FittingC73×#1779C2
socleC23×C73C23×C4
prime radicalC2×#14556560C4
cosocleC22×C4×C73C24
hypercenter#1779D73
hypocenterC73#23306
LCSG ▷ C22×C73 ▷ C73
UCSC1 ◁ C22×C4 ◁ #1779

Complements

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

NH ≅ G/Ntype
2. #22610723C2semidirect
3. #22610540C2semidirect
4. #22610534C2semidirect
5. #22610220C2semidirect
6. #22609841C2semidirect
7. C73×#1779C2semidirect
8. C22×C8×D73C22semidirect
11. C2×#14556520C22semidirect
14. C2×#14556503C22semidirect
15. C2×C8×Dic73C22semidirect
20. #17988341C22semidirect
21. #17988341C22semidirect
30. #14556560C8semidirect
71. C22⋊Dic73C2×C8semidirect
77. C24×C73C2×C8semidirect
165. #1779D73semidirect
166. C73#23306semidirect
216. C24C8×D73semidirect

Maximal subgroups

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

subgroupindex
C4×#145565602
#226107232
#226105402
#226105342
#226102202
#226098412
C73×#17792
#2330673