Finite Group Databasebeta

← #33003355

Subgroups — #33003355

Order 18688.

558 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14556530C211yesmaximal, characteristic, complemented
2C2×#17988824C211yesmaximal, characteristic
3C2×#17988389C211yesmaximal, characteristic, complemented
4C2×#17988387C211yesmaximal, characteristic, complemented
5#22609957C211yesmaximal, characteristic, complemented
6#22608022C211yesmaximal, characteristic, complemented
7C73×#1380C211yesmaximal, characteristic, complemented
8C22×D584C2211yescharacteristic
9C22×C8×D73C2211yescharacteristic
10C2×#14556566—21no—
11C2×#14556530—22no—
12C2×#14556530—22no—
13C2×#14556530—22no—
14C22×#11924003C2211yescharacteristic
15#17988824—22no—
16#17988824C414yescomplemented
17C2×#14556525C2211yescharacteristic
18C2×#14556511—21no—
19#17988444—21no—
20#17988389—22no—
21#17988387—22no—
22#17988349—21no—
23C22×C16×C73C2211yescharacteristic
24C2×#107×C73C2211yescharacteristic
25C2×#107×C73C2211yescharacteristic
26C22×D292—21no—
27C22×D292—41no—
28C22×C4×D73D411yescharacteristic
29C2×D584—21no—
30C2×D584—21no—
31C2×D584—21no—
32C2×D584—21no—
33C2×D584—21no—
34C2×D584C2×C412yes—
35C22×#10596831D411yescharacteristic
36C2×C8×D73—21no—
37C2×C8×D73—21no—
38C2×C8×D73—21no—
39C2×C8×D73—21no—
40C2×C8×D73C2×C412yes—
41C2×C8×D73D412yes—
42#14556566—24no—
43#14556566—42no—
44C2×C4⋊Dic73—21no—
45C2×#11924005—41no—
46C2×#11924005—41no—
47#14556530—28no—
48C2×#11924003—21no—
49C2×#11924003—21no—
50C2×#11924003D412yescomplemented
51#14556525—21no—
52#14556525—21no—
53#14556511—24no—
54#14556498—41no—
55C22×C8×C73C2311yescharacteristic
56C2×C4⋊C4×C73—21no—
57C2×C4⋊C4×C73—21no—
58C2×C16×C73—21no—
59C2×C16×C73—21no—
60C2×C16×C73D412yescomplemented
61#107×C73—21no—
62#107×C73—21no—
63#107×C73—21no—
64#107×C73C2×C412yes—
65C23×D73—41no—
66C23×D73D811yescharacteristic
67C2×D292—22no—
68C2×D292—41no—
69C2×D292—41no—
70C2×D292—41no—
71C2×D292—41no—
72C2×D292—44no—
73C2×D292—41no—
74C2×D292—82no—
75C2×D292—82no—
76C22×Dic73—41no—
77C22×Dic73D811yescharacteristic
78C2×C4×D73—21no—
79C2×C4×D73—21no—
80C2×C4×D73—21no—
81C2×C4×D73—21no—
82C2×C4×D73C22⋊C412yes—
83C2×C4×D73SD1612yes—
84D584—22no—
85D584—22no—
86D584—41no—
87D584—41no—
88C2×#10596831—21no—
89C2×#10596831—21no—
90C2×#10596831C22⋊C412yes—
91C8×D73—22no—
92C8×D73—22no—
93C8×D73—22no—
94C8×D73—22no—
95C4⋊Dic73—41no—
96C4⋊Dic73—41no—
97#11924005—44no—
98#11924005—82no—
99#11924003—22no—
100C22×C4×C73—41no—
101C22×C4×C73—41no—
102C22×C4×C73C2×D411yescharacteristic
103C2×C8×C73C22×C411yescharacteristic
104C2×C8×C73C2×D411yescharacteristic
105C2×C8×C73C2×D411yescharacteristic
106C2×C8×C73C2×D411yescharacteristic
107C2×C8×C73#4111yescharacteristic
108C2×C8×C73#4111yescharacteristic
109C4⋊C4×C73—22no—
110C4⋊C4×C73—41no—
111C4⋊C4×C73—41no—
112C4⋊C4×C73—41no—
113C16×C73—22no—
114C23×C73C2×D811yescharacteristic
115C22×D73—21no—
116C22×D73—21no—
117C22×D73—21no—
118C22×D73—21no—
119C22×D73—21no—
120C22×D73—21no—
121C22×D73—42no—
122C22×D73—81no—
123C22×D73—81no—
124C22×D73—81no—
125C22×D73—81no—
126C22×D73—81no—
127C22×D73—81no—
128C22×D73#10212yes—
129C2×C4×C73—42no—
130C2×C4×C73—81no—
131C2×C4×C73—81no—
132C2×C4×C73—81no—
133C2×C4×C73—81no—
134C2×C4×C73—81no—
135C2×C4×C73C2×C22⋊C411yescharacteristic
136C2×C4×C73#12111yescharacteristic
137C2×C4×C73#12111yescharacteristic
138C2×C4×C73C2×SD1611yescharacteristic
139C2×C4×C73#13511yescharacteristic
140C2×C4×C73#13511yescharacteristic
141C2×Dic73—21no—
142C2×Dic73—21no—
143C2×Dic73—81no—
144C2×Dic73—81no—
145C2×Dic73—81no—
146C2×Dic73#10212yes—
147D292—42no—
148D292—42no—
149D292—88no—
150D292—81no—
151D292—81no—
152C4×D73—22no—
153C4×D73—22no—
154C4×D73—22no—
155C4×D73—22no—
156C8×C73C4×D411yescharacteristic
157C8×C73C4×D411yescharacteristic
158C8×C73#12011yescharacteristic
159C8×C73#12311yescharacteristic
160#10596831—22no—
161C22×C73C2×#10211yescharacteristic
162C22×C73#46611yescharacteristic
163C22×C73#46611yescharacteristic
164C22×C73#50911yescharacteristic
165C22×C73#50911yescharacteristic
166C22×C73#50911yescharacteristic
167C22×C73#51011yescharacteristic
168C2×D73—22no—
169C2×D73—22no—
170C2×D73—22no—
171C2×D73—41no—
172C2×D73—41no—
173C2×D73—41no—
174C2×D73—41no—
175C2×D73—82no—
176C2×D73—82no—
177C2×D73—82no—
178C2×D73—161no—
179C2×D73—161no—
180C2×D73—161no—
181C2×D73—161no—
182C4×C73—82no—
183C4×C73—161no—
184C4×C73#38611yescharacteristic
185C4×C73#41611yescharacteristic
186C4×C73#46511yescharacteristic
187C4×C73#46511yescharacteristic
188Dic73—22no—
189Dic73—161no—
190#13061—731nomaximal, Sylow
191C2×C73#189311yescharacteristic
192C2×C73#213911yescharacteristic
193C2×C73#214011yescharacteristic
194C2×C73#221811yescharacteristic
195C2×C73#221811yescharacteristic
196C2×C73#222011yescharacteristic
197C2×C73#222011yescharacteristic
198D73—42no—
199D73—162no—
200C22×M32—731no—
201C2×#466—731no—
202C2×#357—731no—
203C2×#357—731no—
204#1808—731no—
205#1380—731no—
206#1380D7311yescharacteristic, complemented
207C73#1306111yescharacteristic, complemented, Sylow
208C22×D8—731no—
209C23×C8—731no—
210C2×#121—1461no—
211C2×M32—1462no—
212C2×M32—1462no—
213C2×M32—1462no—
214C22×C16—731no—
215C22×C16C2×D7311yescharacteristic
216#466—734no—
217#466—1462no—
218C2×#107—731no—
219C2×#107C2×D7311yescharacteristic
220C2×#107C2×D7311yescharacteristic
221C2×#102—1461no—
222#380—1461no—
223#357—1462no—
224#357—1462no—
225#340—1461no—
226C22×D4—2921no—
227C22×D4—1461no—
228C23×C4—731no—
229C2×D8—732no—
230C2×D8—1461no—
231C2×D8—1461no—
232C2×D8—1461no—
233C2×D8—1461no—
234C2×D8—1461no—
235C22×C8—731no—
236C22×C8—732no—
237C22×C8—732no—
238C22×C8—1461no—
239C22×C8—1461no—
240C22×C8—1461no—
241C22×C8—1461no—
242C22×C8C22×D7311yescharacteristic
243#121—2922no—
244#121—1464no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—1461no—
248C2×C22⋊C4—2921no—
249C2×C22⋊C4—2921no—
250M32—1468no—
251C2×C16—21no—
252C2×C16—21no—
253C2×C16—732no—
254C2×C16—1461no—
255C2×C16—1461no—
256C2×C16D29212yescomplemented
257#107—21no—
258#107—21no—
259#107—21no—
260#107—1461no—
261#107—1461no—
262#107C4×D7312yes—
263#102—1464no—
264#95—2921no—
265C24—2921no—
266C24—731no—
267C2×D4—2924no—
268C2×D4—2921no—
269C2×D4—2921no—
270C2×D4—2921no—
271C2×D4—2921no—
272C2×D4—2921no—
273C2×D4—5842no—
274C2×D4—5842no—
275C2×D4—1462no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—2921no—
279C22×C4—731no—
280C22×C4—732no—
281C22×C4—732no—
282C22×C4—1461no—
283C22×C4—1461no—
284C22×C4—1461no—
285C22×C4—1461no—
286C22×C4D4×D7311yescharacteristic
287D8—2921no—
288D8—2921no—
289D8—1462no—
290D8—1462no—
291C2×C8—732no—
292C2×C8—1461no—
293C2×C8—1461no—
294C2×C8—1462no—
295C2×C8—1462no—
296C2×C8—1462no—
297C2×C8—1462no—
298C2×C8C2×C4×D7311yescharacteristic
299C2×C8D4×D7311yescharacteristic
300C2×C8D4×D7311yescharacteristic
301C2×C8C2×D29211yescharacteristic
302C2×C8#1192402811yescharacteristic
303C2×C8#1192402911yescharacteristic
304C4⋊C4—22no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—2921no—
309C4⋊C4—2921no—
310C22⋊C4—2924no—
311C22⋊C4—5842no—
312C16—22no—
313C16—1462no—
314C23—2922no—
315C23—5841no—
316C23—5841no—
317C23—5841no—
318C23—5841no—
319C23—5841no—
320C23—5841no—
321C23—732no—
322C23—1461no—
323C23—1461no—
324C23—1461no—
325C23—1461no—
326C23—1461no—
327C23—1461no—
328C23D8×D7311yescharacteristic
329D4—2922no—
330D4—2922no—
331D4—5848no—
332D4—5841no—
333D4—5841no—
334C2×C4—42no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4—81no—
340C2×C4—5841no—
341C2×C4—5841no—
342C2×C4—5841no—
343C2×C4—732no—
344C2×C4—1461no—
345C2×C4—1461no—
346C2×C4—1462no—
347C2×C4—1462no—
348C2×C4—1462no—
349C2×C4—1462no—
350C2×C4C22⋊C4×D7311yescharacteristic
351C2×C4#1455656411yescharacteristic
352C2×C4#1455656611yescharacteristic
353C2×C4SD16×D7311yescharacteristic
354C2×C4#1455661211yescharacteristic
355C2×C4#1455661411yescharacteristic
356C8—1462no—
357C8#1455655311yescharacteristic
358C8C4×D29211yescharacteristic
359C8#1455656111yescharacteristic
360C8#1455657611yescharacteristic
361C22—2921no—
362C22—2921no—
363C22—2921no—
364C22—2921no—
365C22—5842no—
366C22—5842no—
367C22—5842no—
368C22—11681no—
369C22—11681no—
370C22—11681no—
371C22—11681no—
372C22—1462no—
373C22—1462no—
374C22—1462no—
375C22#102×D7311yescharacteristic
376C22#1798882211yescharacteristic
377C22#1798882411yescharacteristic
378C22#1798902111yescharacteristic
379C22#1798902411yescharacteristic
380C22#1798902411yescharacteristic
381C22#1798902911yescharacteristic
382C4—82no—
383C4—161no—
384C4—11681no—
385C4—1462no—
386C4#1798846511yescharacteristic
387C4#1798858211yescharacteristic
388C4#1798881711yescharacteristic
389C4#1798881911yescharacteristic
390C2—2922no—
391C2—11682no—
392C2#2261035811yescharacteristic
393C2#2261164211yescharacteristic
394C2#2261164511yescharacteristic
395C2#2261203311yescharacteristic
396C2#2261203511yescharacteristic
397C2#2261204511yescharacteristic
398C2#2261204711yescharacteristic
subgroupG/H
centerC23D8×D73
commutatorC2×C8×C73C22×C4
FrattiniC22×C8C22×D73
FittingC73×#1380C2
socleC23×C73C2×D8
prime radicalC2×#17988824C2
cosocleC22×C8×C73C23
hypercenter#1380D73
hypocenterC73#13061
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8 ◁ #1380

Complements

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

NH ≅ G/Ntype
1. C22×#14556530C2semidirect
3. C2×#17988389C2semidirect
4. C2×#17988387C2semidirect
5. #22609957C2semidirect
6. #22608022C2semidirect
7. C73×#1380C2semidirect
16. #17988824C4semidirect
50. C2×#11924003D4semidirect
60. C2×C16×C73D4semidirect
206. #1380D73semidirect
207. C73#13061semidirect
256. C2×C16D292semidirect

Maximal subgroups

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

subgroupindex
C22×#145565302
C2×#179888242
C2×#179883892
C2×#179883872
#226099572
#226080222
C73×#13802
#1306173