Finite Group Databasebeta

← #33007601

Subgroups — #33007601

Order 18688.

2878 conjugacy classes of subgroups; only the 427 normal subgroups are shown.

normal subgroups only

This group has 2878 conjugacy classes of subgroups — too many to draw. Only its 427 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
1C2×#17988442C213yesmaximal
2#22608169C212yesmaximal, complemented
3#22608168C213yesmaximal, complemented
4#22608167C216yesmaximal, complemented
5C73×#1437C211yesmaximal, characteristic, complemented
6C25×D73C2211yescharacteristic
7C22×C22⋊Dic73C2213yes—
8C22×#11924005C2216yes—
9C22×#11924005C2216yes—
10C22×C4×Dic73C2213yescomplemented
11#17988442C4124yescomplemented
12C2×#14556498C2213yescomplemented
13C2×#14556498C2216yescomplemented
14C22×C22⋊C4×C73C2213yes—
15C22×C42×C73C2211yescharacteristic, complemented
16C2×#95×C73C2213yescomplemented
17C24×D73—43no—
18C24×D73—43no—
19C24×D73—46no—
20C24×D73C2×C4112yes—
21C24×D73C2312yes—
22C23×Dic73C2316yes—
23C2×C22⋊Dic73—46no—
24C2×C22⋊Dic73C2×C4112yes—
25C2×#11924005—46no—
26C2×#11924005—46no—
27C2×#11924005—46no—
28C2×#11924005—46no—
29C2×#11924005C2×C4124yes—
30C2×#11924005C2×C4124yes—
31C2×C4×Dic73—43no—
32C2×C4×Dic73—46no—
33#14556498—43no—
34#14556498—46no—
35C25×C73C2311yescharacteristic
36C23×C4×C73C2313yes—
37C23×C4×C73C2313yes—
38C2×C22⋊C4×C73—43no—
39C2×C22⋊C4×C73—43no—
40C2×C22⋊C4×C73C2×C4112yes—
41C2×C42×C73—43no—
42#95×C73—43no—
43C23×D73—26no—
44C23×D73—26no—
45C23×D73—26no—
46C23×D73—26no—
47C23×D73—424no—
48C23×D73—424no—
49C23×D73—412no—
50C23×D73—412no—
51C23×D73—44no—
52C23×D73—424no—
53C23×D73—412no—
54C23×D73—48no—
55C23×D73—412no—
56C23×D73—44no—
57C23×D73C42116yescomplemented
58C23×D73C22×C4112yes—
59C22×Dic73—26no—
60C22×Dic73—26no—
61C22×Dic73—26no—
62C22×Dic73—26no—
63C22×Dic73—26no—
64C22×Dic73—26no—
65C22×Dic73C22×C4112yes—
66C22⋊Dic73—412no—
67#11924005—424no—
68#11924005—424no—
69C4×Dic73—412no—
70C24×C73—23no—
71C24×C73—23no—
72C24×C73—26no—
73C24×C73C22×C416yes—
74C24×C73C2411yescharacteristic
75C22×C4×C73—23no—
76C22×C4×C73—26no—
77C22×C4×C73—23no—
78C22×C4×C73—26no—
79C22×C4×C73—23no—
80C22×C4×C73—23no—
81C22×C4×C73—26no—
82C22×C4×C73—26no—
83C22×C4×C73C22×C416yes—
84C22×C4×C73C22×C416yes—
85C22⋊C4×C73—412no—
86C42×C73—44no—
87C23×C73—26no—
88C23×C73—26no—
89C23×C73—212no—
90C23×C73—212no—
91C23×C73—41no—
92C23×C73—46no—
93C23×C73—43no—
94C23×C73—42no—
95C23×C73—43no—
96C23×C73—41no—
97C23×C73C2×C4214yes—
98C23×C73C23×C413yes—
99C23×C73H(5,𝔽2)13yes—
100C23×C73H(5,𝔽2)13yes—
101C23×C73H(5,𝔽2)16yes—
102C22×D73—212no—
103C22×D73—212no—
104C22×D73—212no—
105C22×D73—212no—
106C22×D73—212no—
107C22×D73—212no—
108C22×D73—448no—
109C22×D73—448no—
110C22×D73—448no—
111C22×D73—496no—
112C22×D73—42no—
113C22×D73—46no—
114C22×D73—46no—
115C22×D73—46no—
116C22×D73—42no—
117C22×D73—42no—
118C22×D73—46no—
119C22×D73—42no—
120C22×D73C2×C4218yes—
121C2×C4×C73—26no—
122C2×C4×C73—26no—
123C2×C4×C73—212no—
124C2×C4×C73—212no—
125C2×C4×C73—43no—
126C2×C4×C73—46no—
127C2×C4×C73—46no—
128C2×C4×C73—43no—
129C2×C4×C73—43no—
130C2×C4×C73—43no—
131C2×Dic73—212no—
132C2×Dic73—212no—
133C2×Dic73—212no—
134C2×Dic73—46no—
135C2×Dic73—46no—
136C2×Dic73—46no—
137C2×Dic73—46no—
138C22×C73—212no—
139C22×C73—46no—
140C22×C73—46no—
141C22×C73—412no—
142C22×C73C22×C4211yescharacteristic
143C22×C73#51816yes—
144C22×C73#51816yes—
145C22×C73#51813yes—
146C22×C73#51813yes—
147C22×C73#56011yescharacteristic
148C22×C73#56016yes—
149C22×C73#56013yes—
150C22×C73#56012yes—
151C22×C73H(3,𝔽4)13yes—
152C22×C73H(3,𝔽4)11yescharacteristic
153C2×D73—224no—
154C2×D73—464no—
155C2×D73—412no—
156C2×D73—412no—
157C2×D73—412no—
158C2×D73—412no—
159C4×C73—46no—
160C4×C73—46no—
161Dic73—412no—
162#13827—731nomaximal, Sylow
163C2×C73—44no—
164C2×C73#227313yes—
165C2×C73#237513yes—
166C2×C73#237513yes—
167C2×C73#237516yes—
168D73—48no—
169C2×#379—733no—
170#1437—732no—
171#1437—736no—
172#1437—733no—
173#1437D7311yescharacteristic, complemented
174C73#1382711yescharacteristic, complemented, Sylow
175C26—731no—
176C22×C22⋊C4—733no—
177C22×C22⋊C4—736no—
178C22×C22⋊C4—736no—
179C22×C22⋊C4C2×D7313yes—
180C22×C42—733no—
181C22×C42C2×D7311yescharacteristic, complemented
182#379—7324no—
183C2×#95—736no—
184C2×#95—733no—
185C2×#95C2×D7313yescomplemented
186C25—2923no—
187C25—2923no—
188C25—2926no—
189C25—7312no—
190C25—732no—
191C25C22×D7311yescharacteristic
192C23×C4—736no—
193C23×C4C22×D7313yes—
194C23×C4C22×D7313yes—
195C2×C22⋊C4—43no—
196C2×C22⋊C4—43no—
197C2×C22⋊C4—2926no—
198C2×C22⋊C4—2926no—
199C2×C22⋊C4—2926no—
200C2×C22⋊C4—2926no—
201C2×C22⋊C4—2926no—
202C2×C22⋊C4—7312no—
203C2×C22⋊C4—7324no—
204C2×C22⋊C4—7324no—
205C2×C22⋊C4C4×D73112yes—
206C2×C42—43no—
207C2×C42—2923no—
208C2×C42—2926no—
209#95—43no—
210#95—2923no—
211#95—2926no—
212C24—23no—
213C24—23no—
214C24—26no—
215C24—29224no—
216C24—29224no—
217C24—29212no—
218C24—29212no—
219C24—2924no—
220C24—29224no—
221C24—29212no—
222C24—2928no—
223C24—29212no—
224C24—2924no—
225C24—7316no—
226C24—7312no—
227C24—1466no—
228C24—1466no—
229C24—1466no—
230C24—1466no—
231C24C2×C4×D7316yes—
232C24C23×D7311yescharacteristic
233C22×C4—23no—
234C22×C4—26no—
235C22×C4—23no—
236C22×C4—23no—
237C22×C4—26no—
238C22×C4—23no—
239C22×C4—26no—
240C22×C4—26no—
241C22×C4—7312no—
242C22×C4—1466no—
243C22×C4—1466no—
244C22×C4—1466no—
245C22×C4—1466no—
246C22×C4—1466no—
247C22×C4—1466no—
248C22×C4C2×C4×D7316yes—
249C22×C4C2×C4×D7316yes—
250C22⋊C4—412no—
251C22⋊C4—29224no—
252C22⋊C4—29224no—
253C22⋊C4—29212no—
254C42—44no—
255C42—29212no—
256C23—212no—
257C23—212no—
258C23—26no—
259C23—26no—
260C23—41no—
261C23—46no—
262C23—43no—
263C23—42no—
264C23—43no—
265C23—41no—
266C23—29248no—
267C23—29248no—
268C23—29248no—
269C23—29296no—
270C23—2922no—
271C23—2926no—
272C23—2926no—
273C23—2926no—
274C23—2922no—
275C23—2922no—
276C23—2926no—
277C23—2922no—
278C23—738no—
279C23—14612no—
280C23—14612no—
281C23—14612no—
282C23—14612no—
283C23—14612no—
284C23—14612no—
285C23C42×D7314yes—
286C23C22×C4×D7313yes—
287C23#1455663613yes—
288C23#1455663613yes—
289C23#1455663716yes—
290C2×C4—212no—
291C2×C4—212no—
292C2×C4—26no—
293C2×C4—26no—
294C2×C4—43no—
295C2×C4—46no—
296C2×C4—46no—
297C2×C4—43no—
298C2×C4—43no—
299C2×C4—43no—
300C2×C4—2926no—
301C2×C4—2926no—
302C2×C4—2926no—
303C2×C4—2926no—
304C2×C4—14612no—
305C2×C4—14612no—
306C2×C4—14612no—
307C22—212no—
308C22—46no—
309C22—46no—
310C22—412no—
311C22—29264no—
312C22—29212no—
313C22—29212no—
314C22—29212no—
315C22—29212no—
316C22—14624no—
317C22C2×C42×D7311yescharacteristic
318C22#1798905916yes—
319C22#1798905916yes—
320C22#1798906013yes—
321C22#1798906113yes—
322C22#1798932411yescharacteristic
323C22#1798932516yes—
324C22#1798932713yes—
325C22#1798932812yes—
326C22#1798932913yes—
327C22#1798933011yescharacteristic
328C4—46no—
329C4—46no—
330C4—29212no—
331C2—44no—
332C2—2928no—
333C2#2261228313yes—
334C2#2261306813yes—
335C2#2261306913yes—
336C2#2261307116yes—
subgroupG/H
center, FrattiniC24C23×D73
commutatorC22×C73C22×C42
FittingC73×#1437C2
socle, cosocleC24×C73C24
prime radicalC25×D73C22
hypercenter#1437D73
hypocenterC73#13827
LCSG ▷ C22×C73 ▷ C73
UCSC1 ◁ C24 ◁ #1437

Complements

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

NH ≅ G/Ntype
2. #22608169C2semidirect
3. #22608168C2semidirect
4. #22608167C2semidirect
5. C73×#1437C2semidirect
10. C22×C4×Dic73C22semidirect
11. #17988442C4semidirect
12. C2×#14556498C22semidirect
13. C2×#14556498C22semidirect
15. C22×C42×C73C22semidirect
16. C2×#95×C73C22semidirect
57. C23×D73C42semidirect
173. #1437D73semidirect
174. C73#13827semidirect
181. C22×C42C2×D73semidirect
185. C2×#95C2×D73semidirect

Maximal subgroups

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

subgroupindex
C2×#179884422
#226081692
#226081682
#226081672
C73×#14372
#1382773