Finite Group Databasebeta

← #33005137

Subgroups — #33005137

Order 18688.

498 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988820C211yesmaximal, characteristic
2C2×#17988412C211yesmaximal, characteristic, complemented
3C2×#17988388C211yesmaximal, characteristic, complemented
4C2×#17988366C211yesmaximal, characteristic, complemented
5#22609957C211yesmaximal, characteristic, complemented
6#22608023C211yesmaximal, characteristic, complemented
7C2×#357×C73C211yesmaximal, characteristic, complemented
8C22×C8×D73C2211yescharacteristic
9C2×#14556562—21no—
10C22×#11924003C2211yescharacteristic
11#17988820—22no—
12#17988820—22no—
13C2×#14556525C2211yescharacteristic
14C2×#14556525C2211yescharacteristic
15C2×#14556512—21no—
16#17988444—21no—
17#17988412—22no—
18#17988388—22no—
19#17988366—22no—
20#17988349—21no—
21C22×D8×C73C2211yescharacteristic
22C22×C16×C73C2211yescharacteristic
23C2×#107×C73C2211yescharacteristic
24#357×C73—22no—
25C22×C22⋊D73—41no—
26C22×C4×D73D411yescharacteristic
27C22×#10596831D411yescharacteristic
28C2×C8×D73—21no—
29C2×C8×D73—21no—
30C2×C8×D73—21no—
31C2×C8×D73—21no—
32C2×C8×D73—21no—
33C2×C8×D73—21no—
34#14556562—24no—
35#14556562—42no—
36C2×C4⋊Dic73—21no—
37C2×C4⋊Dic73—21no—
38C2×C22⋊Dic73—41no—
39C2×#11924005—41no—
40C2×#11924003—21no—
41C2×#11924003—21no—
42C2×#11924003D412yescomplemented
43#14556525—21no—
44#14556525—21no—
45#14556525—21no—
46#14556525D412yes—
47#14556512—24no—
48#14556498—41no—
49C22×D4×C73—21no—
50C2×D8×C73—21no—
51C2×D8×C73—21no—
52C2×D8×C73—21no—
53C2×D8×C73—21no—
54C2×D8×C73—21no—
55C2×D8×C73D412yes—
56C22×C8×C73C2311yescharacteristic
57C2×C4⋊C4×C73—21no—
58C2×C16×C73—21no—
59C2×C16×C73—21no—
60C2×C16×C73—21no—
61#107×C73—21no—
62#107×C73—21no—
63C23×D73D811yescharacteristic
64C2×C22⋊D73—44no—
65C2×C22⋊D73—82no—
66C2×C22⋊D73—82no—
67C22×Dic73—41no—
68C22×Dic73—41no—
69C22×Dic73D811yescharacteristic
70C2×C4×D73—21no—
71C2×C4×D73—21no—
72C2×C4×D73—21no—
73C2×C4×D73—21no—
74C2×C4×D73—21no—
75C2×C4×D73—21no—
76C2×#10596831—21no—
77C2×#10596831—21no—
78C2×#10596831—21no—
79C8×D73—41no—
80C8×D73—41no—
81C8×D73—41no—
82C8×D73—41no—
83C4⋊Dic73—22no—
84C4⋊Dic73—41no—
85C4⋊Dic73—41no—
86C4⋊Dic73—41no—
87C22⋊Dic73—44no—
88#11924005—82no—
89#11924003—22no—
90C24×C73—41no—
91C2×D4×C73—22no—
92C2×D4×C73—41no—
93C2×D4×C73—41no—
94C2×D4×C73—41no—
95C2×D4×C73—41no—
96C2×D4×C73—41no—
97C22×C4×C73—41no—
98C22×C4×C73C2×D411yescharacteristic
99D8×C73—22no—
100D8×C73—22no—
101D8×C73—41no—
102D8×C73—41no—
103C2×C8×C73C2×D411yescharacteristic
104C2×C8×C73C2×D411yescharacteristic
105C2×C8×C73C2×D411yescharacteristic
106C2×C8×C73#4111yescharacteristic
107C2×C8×C73#4111yescharacteristic
108C2×C8×C73#4111yescharacteristic
109C4⋊C4×C73—41no—
110C4⋊C4×C73—41no—
111C16×C73—41no—
112C23×C73—42no—
113C23×C73—81no—
114C23×C73—81no—
115C23×C73—81no—
116C23×C73—81no—
117C23×C73—81no—
118C23×C73—81no—
119C23×C73C2×D811yescharacteristic
120C22×D73—21no—
121C22×D73—21no—
122C22×D73—21no—
123C22×D73—21no—
124C22×D73—21no—
125C22×D73—21no—
126C22×D73D1612yes—
127D4×C73—42no—
128D4×C73—42no—
129D4×C73—81no—
130D4×C73—81no—
131C2×C4×C73—81no—
132C2×C4×C73—81no—
133C2×C4×C73—81no—
134C2×C4×C73#12011yescharacteristic
135C2×C4×C73#12111yescharacteristic
136C2×C4×C73#12311yescharacteristic
137C2×C4×C73#13511yescharacteristic
138C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
139C2×C4×C73#13711yescharacteristic
140C22⋊D73—88no—
141C2×Dic73—21no—
142C2×Dic73—21no—
143C2×Dic73—21no—
144C2×Dic73—42no—
145C2×Dic73—81no—
146C2×Dic73—81no—
147C2×Dic73—81no—
148C2×Dic73—81no—
149C2×Dic73—81no—
150C4×D73—41no—
151C4×D73—41no—
152C4×D73—41no—
153C4×D73—41no—
154C8×C73#12111yescharacteristic
155C8×C73#12111yescharacteristic
156C8×C73#12311yescharacteristic
157C8×C73#12411yescharacteristic
158#10596831—41no—
159C22×C73—82no—
160C22×C73—82no—
161C22×C73—82no—
162C22×C73—161no—
163C22×C73—161no—
164C22×C73—161no—
165C22×C73—161no—
166C22×C73#44711yescharacteristic
167C22×C73#46611yescharacteristic
168C22×C73#48011yescharacteristic
169C22×C73C2×D1611yescharacteristic
170C22×C73#50811yescharacteristic
171C22×C73#50911yescharacteristic
172C22×C73#51011yescharacteristic
173C2×D73—22no—
174C2×D73—22no—
175C2×D73—22no—
176C2×D73—41no—
177C2×D73—41no—
178C2×D73—41no—
179C2×D73—41no—
180C4×C73—161no—
181C4×C73#39411yescharacteristic
182C4×C73#45211yescharacteristic
183C4×C73#46811yescharacteristic
184C4×C73#48211yescharacteristic
185Dic73—41no—
186Dic73—82no—
187Dic73—161no—
188#13380—731nomaximal, Sylow
189C2×C73—162no—
190C2×C73#203311yescharacteristic
191C2×C73#218411yescharacteristic
192C2×C73#218911yescharacteristic
193C2×C73#221511yescharacteristic
194C2×C73#222011yescharacteristic
195C2×C73#223211yescharacteristic
196C2×C73#223411yescharacteristic
197D73—42no—
198C2×#466—731no—
199C2×#366—731no—
200C2×#357—731no—
201C2×#357D7311yescharacteristic, complemented
202C2×C22⋊C16—731no—
203#1808—731no—
204#1380—731no—
205C73#1338011yescharacteristic, complemented, Sylow
206C22×D8C2×D7311yescharacteristic
207C23×C8—731no—
208C2×#121—1461no—
209C22×C16—731no—
210C22×C16C2×D7311yescharacteristic
211#466—1462no—
212#466—1462no—
213C2×#107—731no—
214C2×#107—731no—
215C2×#107C2×D7311yescharacteristic
216C2×#102—1461no—
217#380—1461no—
218#366—1462no—
219#357—22no—
220#357—1462no—
221C22⋊C16—1462no—
222#340—1461no—
223C22×D4—21no—
224C22×D4—2921no—
225C23×C4—731no—
226C2×D8—21no—
227C2×D8—21no—
228C2×D8—21no—
229C2×D8—21no—
230C2×D8—21no—
231C2×D8C22⋊D7312yes—
232C22×C8—731no—
233C22×C8—1461no—
234C22×C8—1461no—
235C22×C8—1461no—
236C22×C8—1461no—
237C22×C8—1461no—
238C22×C8—1461no—
239C22×C8C22×D7311yescharacteristic
240#121—2922no—
241#121—1464no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—1461no—
244C2×C4⋊C4—1461no—
245C2×C22⋊C4—2921no—
246C2×C22⋊C4—2921no—
247C2×C16—21no—
248C2×C16—21no—
249C2×C16—21no—
250C2×C16—732no—
251C2×C16—1461no—
252C2×C16—1461no—
253#107—21no—
254#107—21no—
255#107—732no—
256#107—1461no—
257#107—1461no—
258#107—1461no—
259#102—1464no—
260#95—2921no—
261C24—41no—
262C24—731no—
263C2×D4—22no—
264C2×D4—41no—
265C2×D4—41no—
266C2×D4—41no—
267C2×D4—41no—
268C2×D4—41no—
269C2×D4—2924no—
270C2×D4—5842no—
271C2×D4—5842no—
272C22×C4—41no—
273C22×C4—2921no—
274C22×C4—2921no—
275C22×C4—731no—
276C22×C4—1461no—
277C22×C4—1461no—
278C22×C4—1461no—
279C22×C4—1461no—
280C22×C4—1461no—
281C22×C4—1461no—
282C22×C4D4×D7311yescharacteristic
283D8—22no—
284D8—22no—
285D8—41no—
286D8—41no—
287C2×C8—2921no—
288C2×C8—2921no—
289C2×C8—2921no—
290C2×C8—2921no—
291C2×C8—1461no—
292C2×C8—1461no—
293C2×C8—1461no—
294C2×C8D4×D7311yescharacteristic
295C2×C8D4×D7311yescharacteristic
296C2×C8C2×C22⋊D7311yescharacteristic
297C2×C8#1192402711yescharacteristic
298C2×C8#1192402811yescharacteristic
299C2×C8#1192402811yescharacteristic
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—2921no—
303C4⋊C4—2921no—
304C4⋊C4—2921no—
305C4⋊C4—1462no—
306C22⋊C4—2924no—
307C22⋊C4—5842no—
308C16—41no—
309C16—1462no—
310C23—42no—
311C23—81no—
312C23—81no—
313C23—81no—
314C23—81no—
315C23—81no—
316C23—81no—
317C23—732no—
318C23—1461no—
319C23—1461no—
320C23—1461no—
321C23—1461no—
322C23—1461no—
323C23—1461no—
324C23D8×D7311yescharacteristic
325D4—42no—
326D4—42no—
327D4—81no—
328D4—81no—
329D4—5848no—
330C2×C4—81no—
331C2×C4—81no—
332C2×C4—81no—
333C2×C4—2921no—
334C2×C4—2921no—
335C2×C4—2921no—
336C2×C4—2921no—
337C2×C4—2922no—
338C2×C4—5841no—
339C2×C4—5841no—
340C2×C4—5841no—
341C2×C4—5841no—
342C2×C4—5841no—
343C2×C4—1461no—
344C2×C4—1461no—
345C2×C4—1461no—
346C2×C4#1455655911yescharacteristic
347C2×C4#1455656411yescharacteristic
348C2×C4#1455657311yescharacteristic
349C2×C4#1455661211yescharacteristic
350C2×C4#1455661511yescharacteristic
351C2×C4#1455662011yescharacteristic
352C8—2921no—
353C8#1455656311yescharacteristic
354C8#1455656411yescharacteristic
355C8#1455657411yescharacteristic
356C8#1455657711yescharacteristic
357C22—82no—
358C22—82no—
359C22—82no—
360C22—161no—
361C22—161no—
362C22—161no—
363C22—161no—
364C22—2921no—
365C22—2921no—
366C22—2921no—
367C22—2921no—
368C22—1462no—
369C22—1462no—
370C22—1462no—
371C22#1798871911yescharacteristic
372C22#1798882211yescharacteristic
373C22#1798889911yescharacteristic
374C22D16×D7311yescharacteristic
375C22#1798901811yescharacteristic
376C22#1798902111yescharacteristic
377C22#1798902611yescharacteristic
378C4—161no—
379C4—2921no—
380C4—5842no—
381C4—11681no—
382C4#1798850211yescharacteristic
383C4#1798874611yescharacteristic
384C4#1798883211yescharacteristic
385C4#1798891011yescharacteristic
386C2—162no—
387C2—2922no—
388C2#2261111111yescharacteristic
389C2#2261184111yescharacteristic
390C2#2261186811yescharacteristic
391C2#2261201611yescharacteristic
392C2#2261204311yescharacteristic
393C2#2261211011yescharacteristic
394C2#2261212111yescharacteristic
subgroupG/H
centerC23D8×D73
commutator, cosocleC22×C8×C73C23
FrattiniC22×C8C22×D73
FittingC2×#357×C73C2
socleC23×C73C2×D8
prime radicalC2×#17988820C2
hypercenterC2×#357D73
hypocenterC73#13380
LCSG ▷ C22×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8 ◁ C2×#357

Complements

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

NH ≅ G/Ntype
2. C2×#17988412C2semidirect
3. C2×#17988388C2semidirect
4. C2×#17988366C2semidirect
5. #22609957C2semidirect
6. #22608023C2semidirect
7. C2×#357×C73C2semidirect
42. C2×#11924003D4semidirect
201. C2×#357D73semidirect
205. C73#13380semidirect

Maximal subgroups

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

subgroupindex
C2×#179888202
C2×#179884122
C2×#179883882
C2×#179883662
#226099572
#226080232
C2×#357×C732
#1338073