Finite Group Databasebeta

← #33003059

Subgroups — #33003059

Order 18688.

490 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14556537C211yesmaximal, characteristic
2C2×#17988681C211yesmaximal, characteristic
3C2×#17988392C211yesmaximal, characteristic
4#22610793C211yesmaximal, characteristic
5#22608022C211yesmaximal, characteristic
6#22608008C211yesmaximal, characteristic
7C2×#358×C73C211yesmaximal, characteristic
8C22×Dic292C2211yescharacteristic
9C2×#14556556—21no—
10C2×#14556537—22no—
11C2×#14556537—22no—
12C2×#14556537C414yescomplemented
13C22×#11924003C2211yescharacteristic
14#17988681—22no—
15#17988681—22no—
16C2×#14556525C2211yescharacteristic
17C2×#14556515—21no—
18C2×#14556514—21no—
19C2×C8×Dic73C2211yescharacteristic
20#17988494—21no—
21#17988392—22no—
22C22×Q16×C73C2211yescharacteristic
23C22×C16×C73C2211yescharacteristic
24C2×#107×C73C2211yescharacteristic
25#358×C73—22no—
26C22×Dic146—21no—
27C2×Dic292—21no—
28C2×Dic292—21no—
29C2×Dic292—21no—
30C2×Dic292C2×C412yes—
31C2×Dic292D412yes—
32C2×Dic292D412yes—
33C22×#10596831D411yescharacteristic
34#14556556—24no—
35#14556556—42no—
36C2×C4⋊Dic73—21no—
37C2×#11924007—41no—
38C2×C4×Dic73—41no—
39C2×C4×Dic73D411yescharacteristic
40#14556537—28no—
41C2×#11924003—21no—
42C2×#11924003—21no—
43C2×#11924003C2×C412yes—
44#14556525—21no—
45#14556525—21no—
46#14556515—42no—
47#14556514—24no—
48C8×Dic73—21no—
49C8×Dic73—21no—
50#14556498—41no—
51#14556498—41no—
52C22×Q8×C73—21no—
53C2×Q16×C73—21no—
54C2×Q16×C73—21no—
55C2×Q16×C73—21no—
56C2×Q16×C73—21no—
57C2×Q16×C73—21no—
58C2×Q16×C73C2×C412yes—
59C22×C8×C73C2311yescharacteristic
60C2×C4⋊C4×C73—21no—
61C2×C16×C73—21no—
62C2×C16×C73—21no—
63C2×C16×C73—21no—
64#107×C73—21no—
65#107×C73D412yes—
66C2×Dic146—22no—
67C2×Dic146—22no—
68C2×Dic146—22no—
69C2×Dic146—41no—
70C2×Dic146—41no—
71C2×Dic146—41no—
72C22×Dic73—41no—
73C22×Dic73—41no—
74C22×Dic73D811yescharacteristic
75C22×Dic73D811yescharacteristic
76Dic292—22no—
77Dic292—22no—
78Dic292—22no—
79Dic292C22⋊C414yes—
80C2×#10596831—21no—
81C2×#10596831—21no—
82C2×#10596831—21no—
83C4⋊Dic73—41no—
84C4⋊Dic73—41no—
85#11924007—44no—
86C4×Dic73—21no—
87C4×Dic73—21no—
88C4×Dic73—44no—
89#11924003—22no—
90C2×Q8×C73—22no—
91C2×Q8×C73—41no—
92C2×Q8×C73—41no—
93C2×Q8×C73—41no—
94C2×Q8×C73—41no—
95C2×Q8×C73—41no—
96C22×C4×C73—41no—
97C22×C4×C73—41no—
98C22×C4×C73C2×D411yescharacteristic
99Q16×C73—22no—
100Q16×C73—22no—
101Q16×C73—41no—
102Q16×C73—41no—
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—
111C16×C73—41no—
112C23×C73C2×D811yescharacteristic
113Q8×C73—42no—
114Q8×C73—42no—
115Q8×C73—81no—
116Q8×C73—81no—
117C2×C4×C73—42no—
118C2×C4×C73—42no—
119C2×C4×C73—81no—
120C2×C4×C73—81no—
121C2×C4×C73—81no—
122C2×C4×C73—81no—
123C2×C4×C73C4×D411yescharacteristic
124C2×C4×C73#12011yescharacteristic
125C2×C4×C73#12111yescharacteristic
126C2×C4×C73#13511yescharacteristic
127C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
128C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
129C2×Dic73—21no—
130C2×Dic73—21no—
131C2×Dic73—21no—
132C2×Dic73—21no—
133C2×Dic73—21no—
134C2×Dic73—42no—
135C2×Dic73—42no—
136C2×Dic73—42no—
137C2×Dic73—81no—
138C2×Dic73—81no—
139C2×Dic73—81no—
140C2×Dic73Q3212yescomplemented
141Dic146—24no—
142Dic146—42no—
143Dic146—42no—
144Dic146—42no—
145C8×C73C2×C22⋊C411yescharacteristic
146C8×C73C4×D411yescharacteristic
147C8×C73#12111yescharacteristic
148C8×C73#12311yescharacteristic
149#10596831—41no—
150C22×C73C4×D811yescharacteristic
151C22×C73#44711yescharacteristic
152C22×C73#45911yescharacteristic
153C22×C73C2×Q3211yescharacteristic
154C22×C73#50811yescharacteristic
155C22×C73#51011yescharacteristic
156C22×C73#51011yescharacteristic
157C4×C73—82no—
158C4×C73—82no—
159C4×C73#38611yescharacteristic
160C4×C73#44211yescharacteristic
161C4×C73#44911yescharacteristic
162C4×C73#46311yescharacteristic
163Dic73—22no—
164Dic73—44no—
165Dic73—41no—
166Dic73—161no—
167#13015—731nomaximal, Sylow
168C2×C73#187911yescharacteristic
169C2×C73C4×Q3211yescharacteristic
170C2×C73#217611yescharacteristic
171C2×C73#218811yescharacteristic
172C2×C73#218911yescharacteristic
173C2×C73#220911yescharacteristic
174C2×C73#221111yescharacteristic
175C22×Q32—731no—
176C2×C4×Q16—731no—
177C2×#358—731no—
178C2×#358D7311yescharacteristic
179#1969—731no—
180#1380—731no—
181#1374—731no—
182C73#1301511yescharacteristic, complemented, Sylow
183C22×Q16—731no—
184C22×Q16C2×D7311yescharacteristic
185C2×C4×Q8—1461no—
186C2×Q32—734no—
187C2×Q32—1462no—
188C2×Q32—1462no—
189C22×C16—731no—
190C22×C16C2×D7311yescharacteristic
191C4×Q16—1462no—
192C4×Q16—1462no—
193C2×#107—731no—
194C2×#107C2×D7311yescharacteristic
195C2×#103—1461no—
196C2×#103—1461no—
197C2×C4×C8—731no—
198#391—1461no—
199#358—22no—
200#358—1462no—
201C22×Q8—21no—
202C22×Q8—1461no—
203C2×Q16—21no—
204C2×Q16—21no—
205C2×Q16—21no—
206C2×Q16—21no—
207C2×Q16—21no—
208C2×Q16—732no—
209C2×Q16—732no—
210C2×Q16—732no—
211C2×Q16—1461no—
212C2×Q16—1461no—
213C2×Q16—1461no—
214C2×Q16C4×D7312yes—
215C22×C8—731no—
216C22×C8C22×D7311yescharacteristic
217C4×Q8—2922no—
218C4×Q8—1464no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—2921no—
221C2×C4⋊C4—1461no—
222C2×C42—2921no—
223C2×C42—731no—
224Q32—1468no—
225C2×C16—21no—
226C2×C16—21no—
227C2×C16—21no—
228C2×C16—732no—
229C2×C16—1461no—
230C2×C16—1461no—
231#107—21no—
232#107—1461no—
233#107—1461no—
234#107C22⋊D7312yes—
235#103—2922no—
236#103—1464no—
237C4×C8—1461no—
238C4×C8—1461no—
239#95—2921no—
240#95—2921no—
241C2×Q8—22no—
242C2×Q8—41no—
243C2×Q8—41no—
244C2×Q8—41no—
245C2×Q8—41no—
246C2×Q8—41no—
247C2×Q8—2921no—
248C2×Q8—2921no—
249C2×Q8—2921no—
250C2×Q8—1462no—
251C2×Q8—1462no—
252C2×Q8—1462no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—2921no—
256C22×C4—2921no—
257C22×C4—731no—
258C22×C4—731no—
259C22×C4D4×D7311yescharacteristic
260Q16—22no—
261Q16—22no—
262Q16—41no—
263Q16—41no—
264Q16—734no—
265Q16—1462no—
266Q16—1462no—
267Q16—1462no—
268C2×C8—1461no—
269C2×C8—1461no—
270C2×C8—1461no—
271C2×C8C2×C4×D7311yescharacteristic
272C2×C8D4×D7311yescharacteristic
273C2×C8D4×D7311yescharacteristic
274C2×C8C2×C22⋊D7311yescharacteristic
275C2×C8#1192402711yescharacteristic
276C2×C8#1192402811yescharacteristic
277C4⋊C4—22no—
278C4⋊C4—41no—
279C4⋊C4—2924no—
280C4⋊C4—2921no—
281C4⋊C4—2921no—
282C42—2924no—
283C42—1461no—
284C42—1461no—
285C16—41no—
286C16—1462no—
287C23D8×D7311yescharacteristic
288Q8—42no—
289Q8—42no—
290Q8—81no—
291Q8—81no—
292Q8—2922no—
293Q8—2922no—
294Q8—2922no—
295Q8—1464no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—81no—
299C2×C4—81no—
300C2×C4—81no—
301C2×C4—81no—
302C2×C4—2922no—
303C2×C4—2922no—
304C2×C4—2922no—
305C2×C4—5841no—
306C2×C4—5841no—
307C2×C4—5841no—
308C2×C4—732no—
309C2×C4—1461no—
310C2×C4—1461no—
311C2×C4—1461no—
312C2×C4—1461no—
313C2×C4—1461no—
314C2×C4#1455655111yescharacteristic
315C2×C4#1455655911yescharacteristic
316C2×C4#1455656411yescharacteristic
317C2×C4#1455661211yescharacteristic
318C2×C4#1455661511yescharacteristic
319C2×C4#1455661811yescharacteristic
320C8—2921no—
321C8C22⋊C4×D7311yescharacteristic
322C8C4×C22⋊D7311yescharacteristic
323C8#1455656211yescharacteristic
324C8#1455657311yescharacteristic
325C22#1798866711yescharacteristic
326C22#1798871911yescharacteristic
327C22#1798878111yescharacteristic
328C22Q32×D7311yescharacteristic
329C22#1798901811yescharacteristic
330C22#1798902611yescharacteristic
331C22#1798902911yescharacteristic
332C4—82no—
333C4—82no—
334C4—2924no—
335C4—2921no—
336C4—11681no—
337C4—1462no—
338C4#1798846311yescharacteristic
339C4#1798869611yescharacteristic
340C4#1798873111yescharacteristic
341C4#1798880111yescharacteristic
342C2#2261027611yescharacteristic
343C2#2261178711yescharacteristic
344C2#2261179911yescharacteristic
345C2#2261186311yescharacteristic
346C2#2261186811yescharacteristic
347C2#2261198011yescharacteristic
348C2#2261199611yescharacteristic
subgroupG/H
centerC23D8×D73
commutatorC2×C8×C73C22×C4
FrattiniC22×C8C22×D73
FittingC2×#358×C73C2
socle, prime radicalC23×C73C2×D8
cosocleC22×C8×C73C23
hypercenterC2×#358D73
hypocenterC73#13015
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8 ◁ C2×#358

Complements

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

NH ≅ G/Ntype
12. C2×#14556537C4semidirect
140. C2×Dic73Q32semidirect
182. C73#13015semidirect

Maximal subgroups

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

subgroupindex
C22×#145565372
C2×#179886812
C2×#179883922
#226107932
#226080222
#226080082
C2×#358×C732
#1301573