Finite Group Databasebeta

← #33012752

Subgroups — #33012752

Order 18688.

1830 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988498C211yesmaximal, characteristic
2C2×#17988478C213yesmaximal
3#22608202C211yesmaximal, characteristic, complemented
4#22608194C213yesmaximal, complemented
5#22608173C213yesmaximal
6#22608158C213yesmaximal, complemented
7C22×C4×Q8×C73C211yesmaximal, characteristic, complemented
8C22×#11924007C2213yes—
9C22×#11924005C2211yescharacteristic
10C22×#11924005C2213yes—
11C22×#11924005C2213yes—
12#17988498C418yescomplemented
13#17988478—212no—
14C2×#14556498C2213yes—
15C2×#14556498C2213yes—
16C2×#14556498C2213yes—
17C2×#14556498C2216yes—
18C2×#14556498C2213yes—
19C23×Q8×C73C2211yescharacteristic
20C2×C4×Q8×C73—212no—
21C22×C4⋊C4×C73C2213yes—
22C22×C42×C73C2213yes—
23C24×D73C2311yescharacteristic
24C23×Dic73C2313yes—
25C23×Dic73C2311yescharacteristic
26C23×Dic73C2313yes—
27C2×#11924007—26no—
28C2×#11924007—46no—
29C2×#11924005—26no—
30C2×#11924005—26no—
31C2×#11924005—26no—
32C2×#11924005—26no—
33C2×#11924005—46no—
34C2×#11924005C2×C4112yes—
35#14556498—26no—
36#14556498—26no—
37#14556498—26no—
38#14556498—46no—
39#14556498C2×C4112yes—
40C22×Q8×C73—26no—
41C22×Q8×C73—26no—
42C22×Q8×C73C2×C414yes—
43C23×C4×C73C2313yes—
44C23×C4×C73C2313yes—
45C23×C4×C73C2311yescharacteristic
46C4×Q8×C73—232no—
47C2×C4⋊C4×C73—212no—
48C2×C4⋊C4×C73—26no—
49C2×C42×C73—212no—
50C2×C42×C73D4112yes—
51C23×D73—23no—
52C23×D73—23no—
53C23×D73—23no—
54C23×D73—21no—
55C23×D73—21no—
56C23×D73—23no—
57C23×D73C22×C412yes—
58C22×Dic73—23no—
59C22×Dic73—23no—
60C22×Dic73—26no—
61C22×Dic73—26no—
62C22×Dic73—23no—
63C22×Dic73—23no—
64C22×Dic73—26no—
65C22×Dic73—26no—
66C22×Dic73—23no—
67C22×Dic73—21no—
68C22×Dic73—23no—
69C22×Dic73—23no—
70C22×Dic73C22×C416yes—
71#11924007—412no—
72#11924005—224no—
73#11924005—412no—
74#11924005—44no—
75C24×C73C2411yescharacteristic
76C2×Q8×C73—28no—
77C2×Q8×C73—224no—
78C2×Q8×C73—224no—
79C22×C4×C73—26no—
80C22×C4×C73—26no—
81C22×C4×C73—21no—
82C22×C4×C73—23no—
83C22×C4×C73—26no—
84C22×C4×C73—23no—
85C22×C4×C73—26no—
86C22×C4×C73—23no—
87C22×C4×C73C22×C416yes—
88C22×C4×C73C2×D416yes—
89C22×C4×C73C2×D416yes—
90C22×C4×C73C2×D416yes—
91C22×C4×C73#4116yes—
92C4⋊C4×C73—224no—
93C42×C73—224no—
94C23×C73C23×C411yescharacteristic
95C23×C73C22×D413yes—
96C23×C73C2×#4113yes—
97C23×C73H(5,𝔽2)13yes—
98C23×C732−1+411yescharacteristic
99C23×C732−1+411yescharacteristic
100C23×C732−1+413yes—
101C22×D73—26no—
102C22×D73—22no—
103C22×D73—26no—
104C22×D73—41no—
105C22×D73—43no—
106C22×D73—43no—
107C22×D73—43no—
108C22×D73—43no—
109C22×D73—43no—
110C22×D73—43no—
111C22×D73—43no—
112C22×D73—46no—
113Q8×C73—232no—
114C2×C4×C73—212no—
115C2×C4×C73—26no—
116C2×C4×C73—26no—
117C2×C4×C73—212no—
118C2×C4×C73—26no—
119C2×C4×C73—26no—
120C2×C4×C73—26no—
121C2×C4×C73—26no—
122C2×C4×C73—26no—
123C2×C4×C73—46no—
124C2×C4×C73—46no—
125C2×C4×C73C4×D4112yes—
126C2×C4×C73#12014yes—
127C2×Dic73—26no—
128C2×Dic73—212no—
129C2×Dic73—43no—
130C2×Dic73—43no—
131C2×Dic73—46no—
132C2×Dic73—41no—
133C2×Dic73—43no—
134C2×Dic73—43no—
135C2×Dic73—43no—
136C2×Dic73—43no—
137C2×Dic73—43no—
138C2×Dic73—46no—
139C2×Dic73—46no—
140C22×C73C2×C4×D413yes—
141C22×C73#51911yescharacteristic
142C22×C73#52013yes—
143C22×C73C2×#12011yescharacteristic
144C22×C73#52513yes—
145C22×C73#53513yes—
146C22×C73#53613yes—
147C22×C73#53613yes—
148C22×C73#53713yes—
149C22×C73#54113yes—
150C22×C73#54213yes—
151C22×C73#54216yes—
152C2×D73—46no—
153C2×D73—46no—
154C2×D73—42no—
155C2×D73—81no—
156C2×D73—81no—
157C2×D73—83no—
158C2×D73—83no—
159C4×C73—24no—
160C4×C73—212no—
161C4×C73—46no—
162Dic73—46no—
163Dic73—81no—
164Dic73—83no—
165#14301—731nomaximal, Sylow
166C2×C73C4×#12011yescharacteristic
167C2×C73#232313yes—
168C2×C73#232613yes—
169C2×C73#261411yescharacteristic
170C2×C73#261511yescharacteristic
171C2×C73#263213yes—
172C2×C73#263713yes—
173D73—82no—
174C22×C4×Q8D7311yescharacteristic, complemented
175C2×#393—731no—
176C2×#388—733no—
177#1446—731no—
178#1444—733no—
179#1439—733no—
180#1434—733no—
181C73#1430111yescharacteristic, complemented, Sylow
182C23×Q8C2×D7311yescharacteristic
183C2×C4×Q8—212no—
184C22×C4⋊C4—733no—
185C22×C4⋊C4C2×D7313yes—
186C22×C22⋊C4—733no—
187C22×C22⋊C4—733no—
188C22×C22⋊C4—731no—
189C22×C42C2×D7313yes—
190#393—738no—
191#388—14612no—
192C2×#95—736no—
193C2×#95—733no—
194C2×#95—733no—
195C2×#95—733no—
196C2×#95—733no—
197C25—731no—
198C22×Q8—26no—
199C22×Q8—26no—
200C22×Q8C4×D7314yes—
201C23×C4—733no—
202C23×C4—731no—
203C23×C4—733no—
204C23×C4C22×D7313yes—
205C23×C4C22×D7313yes—
206C23×C4C22×D7311yescharacteristic
207C4×Q8—232no—
208C2×C4⋊C4—212no—
209C2×C4⋊C4—26no—
210C2×C4⋊C4—2926no—
211C2×C4⋊C4—1466no—
212C2×C22⋊C4—2926no—
213C2×C22⋊C4—7312no—
214C2×C22⋊C4—1466no—
215C2×C22⋊C4—1466no—
216C2×C22⋊C4—1466no—
217C2×C22⋊C4—1466no—
218C2×C42—212no—
219C2×C42C22⋊D73112yes—
220#95—2926no—
221#95—7312no—
222#95—1466no—
223#95—1466no—
224#95—1466no—
225C24—732no—
226C24—1463no—
227C24—1463no—
228C24—1463no—
229C24—1461no—
230C24—1463no—
231C24—1461no—
232C24C23×D7311yescharacteristic
233C2×Q8—28no—
234C2×Q8—224no—
235C2×Q8—224no—
236C22×C4—26no—
237C22×C4—26no—
238C22×C4—23no—
239C22×C4—21no—
240C22×C4—26no—
241C22×C4—23no—
242C22×C4—26no—
243C22×C4—23no—
244C22×C4—736no—
245C22×C4—1463no—
246C22×C4—1463no—
247C22×C4—1466no—
248C22×C4—1466no—
249C22×C4—1463no—
250C22×C4—1461no—
251C22×C4—1463no—
252C22×C4—1463no—
253C22×C4—1463no—
254C22×C4—1463no—
255C22×C4—1466no—
256C22×C4—1466no—
257C22×C4C2×C4×D7316yes—
258C22×C4C2×C22⋊D7316yes—
259C22×C4C2×C22⋊D7316yes—
260C22×C4C2×C22⋊D7316yes—
261C22×C4#1192402816yes—
262C4⋊C4—224no—
263C4⋊C4—29212no—
264C22⋊C4—29212no—
265C22⋊C4—2924no—
266C22⋊C4—14624no—
267C42—224no—
268C23—2923no—
269C23—2923no—
270C23—2923no—
271C23—2923no—
272C23—2923no—
273C23—2923no—
274C23—2926no—
275C23—2921no—
276C23—2923no—
277C23—1466no—
278C23—1462no—
279C23—1466no—
280C23C22×C4×D7311yescharacteristic
281C23C22×C22⋊D7313yes—
282C23C2×#1192402813yes—
283C23#1455663713yes—
284C23#1455663811yescharacteristic
285C23#1455663913yes—
286C23#1455663911yescharacteristic
287Q8—232no—
288C2×C4—26no—
289C2×C4—26no—
290C2×C4—212no—
291C2×C4—26no—
292C2×C4—26no—
293C2×C4—212no—
294C2×C4—26no—
295C2×C4—26no—
296C2×C4—26no—
297C2×C4—46no—
298C2×C4—46no—
299C2×C4—2923no—
300C2×C4—2923no—
301C2×C4—2926no—
302C2×C4—2923no—
303C2×C4—2923no—
304C2×C4—2923no—
305C2×C4—2923no—
306C2×C4—2926no—
307C2×C4—2926no—
308C2×C4—2921no—
309C2×C4—2923no—
310C2×C4—1466no—
311C2×C4—14612no—
312C2×C4C4×C22⋊D73112yes—
313C2×C4#1455656014yes—
314C22—2926no—
315C22—2926no—
316C22—2922no—
317C22—5843no—
318C22—5841no—
319C22—5841no—
320C22—5843no—
321C22C2×C4×C22⋊D7313yes—
322C22#1798906311yescharacteristic
323C22#1798906913yes—
324C22C2×#1455656011yescharacteristic
325C22#1798909813yes—
326C22#1798915113yes—
327C22#1798916113yes—
328C22#1798916113yes—
329C22#1798916513yes—
330C22#1798919513yes—
331C22#1798920916yes—
332C22#1798920913yes—
333C4—24no—
334C4—212no—
335C4—46no—
336C4—2926no—
337C4—5841no—
338C4—5843no—
339C2—5842no—
340C2C4×#1455656011yescharacteristic
341C2#2261260713yes—
342C2#2261263813yes—
343C2#2261565011yescharacteristic
344C2#2261565511yescharacteristic
345C2#2261579113yes—
346C2#2261583713yes—
subgroupG/H
center, FrattiniC24C23×D73
commutatorC23×C73C23×C4
FittingC22×C4×Q8×C73C2
socle, cosocleC24×C73C24
prime radicalC24×D73C23
hypercenterC22×C4×Q8D73
hypocenterC73#14301
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C24 ◁ C22×C4×Q8

Complements

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

NH ≅ G/Ntype
3. #22608202C2semidirect
4. #22608194C2semidirect
6. #22608158C2semidirect
7. C22×C4×Q8×C73C2semidirect
12. #17988498C4semidirect
174. C22×C4×Q8D73semidirect
181. C73#14301semidirect

Maximal subgroups

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

subgroupindex
C2×#179884982
C2×#179884782
#226082022
#226081942
#226081732
#226081582
C22×C4×Q8×C732
#1430173