Finite Group Databasebeta

← #32975979

Subgroups — #32975979

Order 18688.

850 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988640C211yesmaximal, characteristic
2C2×C4×C4⋊Dic73C211yesmaximal, characteristic
3#22607809C214yesmaximal
4C73×#2285C211yesmaximal, characteristic
5C22×C4⋊Dic73—21no—
6C22×C4×Dic73—21no—
7#17988640—22no—
8#17988640C414yescomplemented
9C2×C4⋊#10596831—21no—
10C2×C22⋊#10596831C2212yes—
11C2×C4×#10596831—21no—
12C4×C4⋊Dic73—22no—
13C4×C4⋊Dic73C414yescomplemented
14C2×#14556498C2212yes—
15C2×#117×C73—21no—
16C22×C42×C73C2211yescharacteristic
17#384×C73C412yescomplemented
18#382×C73—21no—
19#380×C73—21no—
20#380×C73C2212yes—
21C4×C4⋊C4×C73—21no—
22C4×C4⋊C4×C73C412yescomplemented
23C23×Dic73—22no—
24C22×#10596831—22no—
25C2×C4⋊Dic73—28no—
26C2×C4⋊Dic73—24no—
27C2×C4×Dic73—24no—
28C2×C4×Dic73—42no—
29C2×C4×Dic73—42no—
30C4⋊#10596831—24no—
31C22⋊#10596831—22no—
32C22⋊#10596831C2×C414yes—
33C4×#10596831—24no—
34#14556498—22no—
35#14556498C2×C414yes—
36C23×C4×C73D411yescharacteristic
37C23×C4×C73D411yescharacteristic
38C23×C4×C73C2311yescharacteristic
39#117×C73—42no—
40#117×C73—42no—
41C2×C4⋊C4×C73—22no—
42C2×C4⋊C4×C73—22no—
43C2×C4⋊C4×C73C2×C414yes—
44C2×C22⋊C4×C73—22no—
45C2×C42×C73—22no—
46C2×C42×C73C2×C412yes—
47C2×C42×C73C2×C412yes—
48C2×C42×C73C2×C411yescharacteristic
49C2×C42×C73C2×C411yescharacteristic
50C2×C42×C73D412yes—
51C2×C42×C73D411yescharacteristic
52C2×C42×C73D411yescharacteristic
53C2×C42×C73Q812yes—
54#95×C73—22no—
55#95×C73—22no—
56C22×Dic73—28no—
57C22×Dic73—24no—
58C22×Dic73—44no—
59C22×Dic73—42no—
60C22×Dic73—42no—
61C2×#10596831—24no—
62C2×#10596831—44no—
63C4⋊Dic73—216no—
64C4×Dic73—48no—
65C24×C73C2×D411yescharacteristic
66C22×C4×C73—24no—
67C22×C4×C73—24no—
68C22×C4×C73—42no—
69C22×C4×C73C22⋊C412yes—
70C22×C4×C73C22⋊C412yes—
71C22×C4×C73C22⋊C411yescharacteristic
72C22×C4×C73C22⋊C411yescharacteristic
73C22×C4×C73C22⋊C411yescharacteristic
74C22×C4×C73C22⋊C411yescharacteristic
75C22×C4×C73C4⋊C412yes—
76C22×C4×C73C4⋊C412yes—
77C22×C4×C73D812yes—
78C22×C4×C73SD1612yes—
79C22×C4×C73SD1612yes—
80C22×C4×C73Q1612yes—
81C22×C4×C73C22×C411yescharacteristic
82C22×C4×C73C22×C411yescharacteristic
83C22×C4×C73C22×C411yescharacteristic
84C22×C4×C73C2×D411yescharacteristic
85C22×C4×C73C2×D411yescharacteristic
86C22×C4×C73C2×Q811yescharacteristic
87C4⋊C4×C73—44no—
88C4⋊C4×C73—44no—
89C4⋊C4×C73—44no—
90C22⋊C4×C73—42no—
91C22⋊C4×C73—42no—
92C42×C73—44no—
93C42×C73C4212yes—
94C42×C73C4212yes—
95C42×C73C22⋊C412yes—
96C42×C73C22⋊C412yes—
97C42×C73C4⋊C412yes—
98C42×C73C4⋊C412yes—
99C42×C73C4⋊C412yes—
100C42×C73C4⋊C412yes—
101C23×C73—22no—
102C23×C73—22no—
103C23×C73C2×C22⋊C411yescharacteristic
104C23×C73C2×C22⋊C411yescharacteristic
105C23×C73C2×C4⋊C411yescharacteristic
106C23×C73C2×D811yescharacteristic
107C23×C73C2×SD1611yescharacteristic
108C23×C73C2×SD1611yescharacteristic
109C23×C73C2×Q1611yescharacteristic
110C2×C4×C73—22no—
111C2×C4×C73—22no—
112C2×C4×C73—22no—
113C2×C4×C73—24no—
114C2×C4×C73—24no—
115C2×C4×C73—41no—
116C2×C4×C73—41no—
117C2×C4×C73—41no—
118C2×C4×C73—41no—
119C2×C4×C73—44no—
120C2×C4×C73—44no—
121C2×C4×C73—44no—
122C2×C4×C73#9512yes—
123C2×C4×C73#9512yes—
124C2×C4×C73#9512yes—
125C2×C4×C73#9512yes—
126C2×C4×C73#10214yes—
127C2×C4×C73#10212yes—
128C2×C4×C73#10212yes—
129C2×C4×C73#10314yes—
130C2×C4×C73#10312yes—
131C2×C4×C73#10312yes—
132C2×C4×C73#10414yes—
133C2×C4×C73#10614yes—
134C2×C4×C73#10714yes—
135C2×C4×C73C2×C4211yescharacteristic
136C2×C4×C73C2×C22⋊C411yescharacteristic
137C2×C4×C73C2×C4⋊C411yescharacteristic
138C2×C4×C73C2×C4⋊C411yescharacteristic
139C2×Dic73—28no—
140C2×Dic73—44no—
141C2×Dic73—48no—
142C2×Dic73—48no—
143C2×Dic73—44no—
144#10596831—44no—
145C22×C73—21no—
146C22×C73—21no—
147C22×C73—21no—
148C22×C73—21no—
149C22×C73—22no—
150C22×C73—22no—
151C22×C73—22no—
152C22×C73—22no—
153C22×C73C2×#9511yescharacteristic
154C22×C73#41012yes—
155C22×C73C2×#10211yescharacteristic
156C22×C73C2×#10211yescharacteristic
157C22×C73C2×#10311yescharacteristic
158C22×C73C2×#10311yescharacteristic
159C22×C73C2×#10412yes—
160C22×C73C2×#10611yescharacteristic
161C22×C73C2×#10711yescharacteristic
162C4×C73—22no—
163C4×C73—21no—
164C4×C73—21no—
165C4×C73—21no—
166C4×C73—21no—
167C4×C73—84no—
168C4×C73#33914yes—
169C4×C73#34014yes—
170C4×C73#34014yes—
171Dic73—48no—
172#8014—731nomaximal, Sylow
173C2×C73—21no—
174C2×C73—21no—
175C2×C73—21no—
176C2×C73—21no—
177C2×C73#150212yes—
178C2×C73#150312yes—
179C2×C73C2×#33911yescharacteristic
180C2×C73C2×#34011yescharacteristic
181C2×C73#174011yescharacteristic
182C2×#431—731no—
183#2285D7311yescharacteristic
184C2×C4×C4⋊C4—731no—
185#1294—734no—
186C73#801411yescharacteristic, complemented, Sylow
187C2×#117—21no—
188C22×C4⋊C4—1461no—
189C22×C42—1461no—
190C22×C42C2×D7311yescharacteristic
191#431—734no—
192#431—1462no—
193C2×C4⋊C8—1461no—
194C2×C22⋊C8—732no—
195C2×C4×C8—1461no—
196#384Dic7312yescomplemented
197#382—21no—
198#380—21no—
199#380C2×D7312yes—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—734no—
202C4×C4⋊C4—1462no—
203C4×C4⋊C4Dic7312yescomplemented
204C2×#95—732no—
205C23×C4—1462no—
206C23×C4C22⋊D7311yescharacteristic
207C23×C4C22⋊D7311yescharacteristic
208C23×C4C22×D7311yescharacteristic
209C22×C8—1462no—
210#117—42no—
211#117—42no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—1468no—
215C2×C4⋊C4—1464no—
216C2×C4⋊C4C2×Dic7314yes—
217C2×C22⋊C4—22no—
218C2×C42—22no—
219C2×C42—2922no—
220C2×C42—2922no—
221C2×C42—1464no—
222C2×C42Dic14612yes—
223C2×C42C4×D7312yes—
224C2×C42C4×D7312yes—
225C2×C42D29212yes—
226C2×C42C2×Dic7311yescharacteristic
227C2×C42C2×Dic7311yescharacteristic
228C2×C42C22⋊D7311yescharacteristic
229C2×C42C22⋊D7311yescharacteristic
230C4⋊C8—1464no—
231C22⋊C8—734no—
232C22⋊C8—1462no—
233C4×C8—1464no—
234#95—22no—
235#95—22no—
236#95—734no—
237#95—1462no—
238C24C2×C22⋊D7311yescharacteristic
239C22×C4—24no—
240C22×C4—24no—
241C22×C4—42no—
242C22×C4—2924no—
243C22×C4—2922no—
244C22×C4—2922no—
245C22×C4—1468no—
246C22×C4—1464no—
247C22×C4#1192400512yes—
248C22×C4#1192400512yes—
249C22×C4C22⋊Dic7311yescharacteristic
250C22×C4C22⋊Dic7311yescharacteristic
251C22×C4C22⋊Dic7311yescharacteristic
252C22×C4C22⋊Dic7311yescharacteristic
253C22×C4#1192400712yes—
254C22×C4#1192400712yes—
255C22×C4#1192401312yes—
256C22×C4#1192401512yes—
257C22×C4#1192401612yes—
258C22×C4#1192401812yes—
259C22×C4C2×C4×D7311yescharacteristic
260C22×C4C2×C4×D7311yescharacteristic
261C22×C4C22×Dic7311yescharacteristic
262C22×C4C2×C22⋊D7311yescharacteristic
263C22×C4C2×D29211yescharacteristic
264C22×C4C2×Dic14611yescharacteristic
265C2×C8—2924no—
266C2×C8—1464no—
267C4⋊C4—44no—
268C4⋊C4—44no—
269C4⋊C4—44no—
270C4⋊C4—14616no—
271C22⋊C4—42no—
272C22⋊C4—42no—
273C42—44no—
274C42—2928no—
275C42C4×Dic7312yes—
276C42C4×Dic7312yes—
277C42#1192400512yes—
278C42#1192400512yes—
279C42#1192400712yes—
280C42#1192400712yes—
281C42C4⋊Dic7312yes—
282C42C4⋊Dic7312yes—
283C23—22no—
284C23—22no—
285C23C2×#1192400511yescharacteristic
286C23C2×C22⋊Dic7311yescharacteristic
287C23C2×#1192400711yescharacteristic
288C23C2×#1192401311yescharacteristic
289C23C2×#1192401511yescharacteristic
290C23C2×#1192401611yescharacteristic
291C23C2×#1192401811yescharacteristic
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—24no—
296C2×C4—24no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—44no—
302C2×C4—44no—
303C2×C4—44no—
304C2×C4—2924no—
305C2×C4—2928no—
306C2×C4—2928no—
307C2×C4—2924no—
308C2×C4—1468no—
309C2×C4#1455649812yes—
310C2×C4#1455649812yes—
311C2×C4#1455649812yes—
312C2×C4#1455649812yes—
313C2×C4#1455651114yes—
314C2×C4#1455651212yes—
315C2×C4#1455651212yes—
316C2×C4#1455651414yes—
317C2×C4#1455651512yes—
318C2×C4#1455651512yes—
319C2×C4#1455651814yes—
320C2×C4#1455652214yes—
321C2×C4#1455652414yes—
322C2×C4C2×C4×Dic7311yescharacteristic
323C2×C4C2×#1192400511yescharacteristic
324C2×C4C2×#1192400711yescharacteristic
325C2×C4C2×C4⋊Dic7311yescharacteristic
326C8—2924no—
327C22—21no—
328C22—21no—
329C22—22no—
330C22—22no—
331C22—22no—
332C22—22no—
333C22—21no—
334C22—21no—
335C22C2×#1455649811yescharacteristic
336C22#1798856112yes—
337C22C2×#1455651111yescharacteristic
338C22C2×#1455651211yescharacteristic
339C22C2×#1455651411yescharacteristic
340C22C2×#1455651511yescharacteristic
341C22C2×#1455651812yes—
342C22C2×#1455652211yescharacteristic
343C22C2×#1455652411yescharacteristic
344C4—21no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—22no—
349C4—84no—
350C4—2928no—
351C4#1798834614yes—
352C4#1798834914yes—
353C4#1798834914yes—
354C2—21no—
355C2—21no—
356C2—21no—
357C2—21no—
358C2#2260842612yes—
359C2#2260843012yes—
360C2C2×#1798834611yescharacteristic
361C2C2×#1798834911yescharacteristic
362C2#2260966411yescharacteristic
subgroupG/H
centerC22×C4C22⋊Dic73
commutatorC2×C4×C73C2×C42
FrattiniC23×C4C22×D73
FittingC73×#2285C2
socleC23×C73C2×C22⋊C4
prime radicalC24×C73C2×D4
cosocleC23×C4×C73C23
hypercenter#2285D73
hypocenterC73#8014
LCSG ▷ C2×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ #2285

Complements

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

NH ≅ G/Ntype
8. #17988640C4semidirect
13. C4×C4⋊Dic73C4semidirect
17. #384×C73C4semidirect
22. C4×C4⋊C4×C73C4semidirect
186. C73#8014semidirect
196. #384Dic73semidirect
203. C4×C4⋊C4Dic73semidirect

Maximal subgroups

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

subgroupindex
C2×#179886402
C2×C4×C4⋊Dic732
#226078092
C73×#22852
#801473