Finite Group Databasebeta

← #33003358

Subgroups — #33003358

Order 18688.

506 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14556530C211yesmaximal, characteristic
2C2×#17988825C211yesmaximal, characteristic
3C2×#17988391C211yesmaximal, characteristic, complemented
4#22609957C211yesmaximal, characteristic
5#22608023C211yesmaximal, characteristic, complemented
6#22608022C211yesmaximal, characteristic, complemented
7C2×#358×C73C211yesmaximal, characteristic, complemented
8C22×C8×D73C2211yescharacteristic
9C2×#14556567—21no—
10C2×#14556530—22no—
11C2×#14556530—22no—
12C2×#14556530—22no—
13C22×#11924003C2211yescharacteristic
14#17988825—22no—
15#17988825C414yescomplemented
16C2×#14556525C2211yescharacteristic
17C2×#14556525C2211yescharacteristic
18C2×#14556515—21no—
19#17988444—21no—
20#17988391—22no—
21#17988349—21no—
22C22×Q16×C73C2211yescharacteristic
23C22×C16×C73C2211yescharacteristic
24C2×#107×C73C2211yescharacteristic
25#358×C73—22no—
26C22×C4×D73D411yescharacteristic
27C22×#10596831D411yescharacteristic
28C2×C8×D73—21no—
29C2×C8×D73—21no—
30C2×C8×D73—21no—
31C2×C8×D73—21no—
32C2×C8×D73C2×C412yes—
33C2×C8×D73D412yes—
34#14556567—24no—
35#14556567—42no—
36C2×C4⋊Dic73—21no—
37C2×C4⋊Dic73—21no—
38C2×#11924007—41no—
39C2×#11924005—41no—
40C2×#11924005—41no—
41#14556530—28no—
42C2×#11924003—21no—
43C2×#11924003—21no—
44C2×#11924003D412yes—
45#14556525—21no—
46#14556525—21no—
47#14556525—21no—
48#14556525C2×C412yes—
49#14556515—24no—
50#14556498—41no—
51C22×Q8×C73—21no—
52C2×Q16×C73—21no—
53C2×Q16×C73—21no—
54C2×Q16×C73—21no—
55C2×Q16×C73—21no—
56C2×Q16×C73—21no—
57C2×Q16×C73C2×C412yes—
58C22×C8×C73C2311yescharacteristic
59C2×C4⋊C4×C73—21no—
60C2×C16×C73—21no—
61C2×C16×C73—21no—
62C2×C16×C73D412yes—
63#107×C73—21no—
64#107×C73—21no—
65C23×D73D811yescharacteristic
66C22×Dic73—41no—
67C22×Dic73—41no—
68C22×Dic73D811yescharacteristic
69C2×C4×D73—21no—
70C2×C4×D73—21no—
71C2×C4×D73—21no—
72C2×C4×D73—21no—
73C2×C4×D73C22⋊C412yes—
74C2×C4×D73SD1612yes—
75C2×#10596831—21no—
76C2×#10596831—21no—
77C2×#10596831C22⋊C412yes—
78C8×D73—22no—
79C8×D73—22no—
80C8×D73—22no—
81C8×D73—22no—
82C4⋊Dic73—22no—
83C4⋊Dic73—41no—
84C4⋊Dic73—41no—
85C4⋊Dic73—41no—
86#11924007—44no—
87#11924005—44no—
88#11924005—82no—
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—41no—
110C4⋊C4×C73—41no—
111C16×C73—22no—
112C23×C73C2×D811yescharacteristic
113C22×D73—21no—
114C22×D73—21no—
115C22×D73—21no—
116C22×D73—21no—
117C22×D73—21no—
118C22×D73—21no—
119C22×D73#10212yes—
120Q8×C73—42no—
121Q8×C73—42no—
122Q8×C73—81no—
123Q8×C73—81no—
124C2×C4×C73—42no—
125C2×C4×C73—81no—
126C2×C4×C73—81no—
127C2×C4×C73—81no—
128C2×C4×C73—81no—
129C2×C4×C73—81no—
130C2×C4×C73C2×C22⋊C411yescharacteristic
131C2×C4×C73#12111yescharacteristic
132C2×C4×C73#12111yescharacteristic
133C2×C4×C73C2×SD1611yescharacteristic
134C2×C4×C73#13511yescharacteristic
135C2×C4×C73#13511yescharacteristic
136C2×Dic73—21no—
137C2×Dic73—21no—
138C2×Dic73—42no—
139C2×Dic73—81no—
140C2×Dic73—81no—
141C2×Dic73—81no—
142C2×Dic73—81no—
143C2×Dic73—81no—
144C2×Dic73#10212yes—
145C4×D73—22no—
146C4×D73—22no—
147C4×D73—22no—
148C4×D73—22no—
149C8×C73C4×D411yescharacteristic
150C8×C73C4×D411yescharacteristic
151C8×C73#12011yescharacteristic
152C8×C73#12311yescharacteristic
153#10596831—22no—
154C22×C73C2×#10211yescharacteristic
155C22×C73#46611yescharacteristic
156C22×C73#46611yescharacteristic
157C22×C73#50911yescharacteristic
158C22×C73#51011yescharacteristic
159C22×C73#51011yescharacteristic
160C22×C73#51011yescharacteristic
161C2×D73—22no—
162C2×D73—22no—
163C2×D73—22no—
164C2×D73—41no—
165C2×D73—41no—
166C2×D73—41no—
167C2×D73—41no—
168C4×C73—82no—
169C4×C73—161no—
170C4×C73#38611yescharacteristic
171C4×C73#41611yescharacteristic
172C4×C73#46511yescharacteristic
173C4×C73#46511yescharacteristic
174Dic73—22no—
175Dic73—82no—
176Dic73—161no—
177#13062—731nomaximal, Sylow
178C2×C73#189311yescharacteristic
179C2×C73#213911yescharacteristic
180C2×C73#214111yescharacteristic
181C2×C73#221911yescharacteristic
182C2×C73#221911yescharacteristic
183C2×C73#222011yescharacteristic
184C2×C73#222011yescharacteristic
185D73—42no—
186C22×M32—731no—
187C2×#467—731no—
188C2×#358—731no—
189C2×#358D7311yescharacteristic, complemented
190#1808—731no—
191#1380—731no—
192#1380—731no—
193C73#1306211yescharacteristic, complemented, Sylow
194C22×Q16C2×D7311yescharacteristic
195C23×C8—731no—
196C2×C22⋊Q8—1461no—
197C2×M32—1462no—
198C2×M32—1462no—
199C2×M32—1462no—
200C22×C16—731no—
201C22×C16C2×D7311yescharacteristic
202#467—734no—
203#467—1462no—
204C2×#107—731no—
205C2×#107—731no—
206C2×#107C2×D7311yescharacteristic
207C2×#103—1461no—
208#380—1461no—
209#358—22no—
210#358—1462no—
211#340—1461no—
212C22×Q8—21no—
213C23×C4—731no—
214C2×Q16—21no—
215C2×Q16—21no—
216C2×Q16—21no—
217C2×Q16—21no—
218C2×Q16—21no—
219C2×Q16C4×D7312yes—
220C22×C8—731no—
221C22×C8—732no—
222C22×C8—732no—
223C22×C8—1461no—
224C22×C8—1461no—
225C22×C8—1461no—
226C22×C8—1461no—
227C22×C8C22×D7311yescharacteristic
228C22⋊Q8—2922no—
229C22⋊Q8—1464no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—2921no—
232C2×C4⋊C4—1461no—
233C2×C4⋊C4—1461no—
234C2×C22⋊C4—2921no—
235C2×C22⋊C4—2921no—
236M32—1468no—
237C2×C16—21no—
238C2×C16—21no—
239C2×C16—732no—
240C2×C16—1461no—
241C2×C16—1461no—
242C2×C16C22⋊D7312yes—
243#107—21no—
244#107—21no—
245#107—732no—
246#107—1461no—
247#107—1461no—
248#107—1461no—
249#103—1464no—
250#95—2921no—
251C24—731no—
252C2×Q8—22no—
253C2×Q8—41no—
254C2×Q8—41no—
255C2×Q8—41no—
256C2×Q8—41no—
257C2×Q8—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—2921no—
261C22×C4—2921no—
262C22×C4—731no—
263C22×C4—732no—
264C22×C4—732no—
265C22×C4—1461no—
266C22×C4—1461no—
267C22×C4—1461no—
268C22×C4—1461no—
269C22×C4D4×D7311yescharacteristic
270Q16—22no—
271Q16—22no—
272Q16—41no—
273Q16—41no—
274C2×C8—732no—
275C2×C8—1461no—
276C2×C8—1461no—
277C2×C8—1462no—
278C2×C8—1462no—
279C2×C8—1462no—
280C2×C8—1462no—
281C2×C8C2×C4×D7311yescharacteristic
282C2×C8D4×D7311yescharacteristic
283C2×C8D4×D7311yescharacteristic
284C2×C8C2×C22⋊D7311yescharacteristic
285C2×C8#1192402711yescharacteristic
286C2×C8#1192402811yescharacteristic
287C4⋊C4—41no—
288C4⋊C4—41no—
289C4⋊C4—2924no—
290C4⋊C4—2921no—
291C4⋊C4—2921no—
292C4⋊C4—2921no—
293C4⋊C4—1462no—
294C22⋊C4—2924no—
295C22⋊C4—5842no—
296C16—22no—
297C16—1462no—
298C23—732no—
299C23—1461no—
300C23—1461no—
301C23—1461no—
302C23—1461no—
303C23—1461no—
304C23—1461no—
305C23D8×D7311yescharacteristic
306Q8—42no—
307Q8—42no—
308Q8—81no—
309Q8—81no—
310C2×C4—42no—
311C2×C4—81no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4—2922no—
317C2×C4—5841no—
318C2×C4—5841no—
319C2×C4—5841no—
320C2×C4—5841no—
321C2×C4—5841no—
322C2×C4—732no—
323C2×C4—1461no—
324C2×C4—1461no—
325C2×C4—1462no—
326C2×C4—1462no—
327C2×C4—1462no—
328C2×C4—1462no—
329C2×C4C22⋊C4×D7311yescharacteristic
330C2×C4#1455656211yescharacteristic
331C2×C4#1455656411yescharacteristic
332C2×C4SD16×D7311yescharacteristic
333C2×C4#1455661011yescharacteristic
334C2×C4#1455661211yescharacteristic
335C8—1462no—
336C8#1455655111yescharacteristic
337C8C4×C22⋊D7311yescharacteristic
338C8#1455655911yescharacteristic
339C8#1455657311yescharacteristic
340C22—2921no—
341C22—2921no—
342C22—2921no—
343C22—2921no—
344C22—1462no—
345C22—1462no—
346C22—1462no—
347C22#102×D7311yescharacteristic
348C22#1798882011yescharacteristic
349C22#1798882211yescharacteristic
350C22#1798902411yescharacteristic
351C22#1798902611yescharacteristic
352C22#1798902911yescharacteristic
353C22#1798902911yescharacteristic
354C4—82no—
355C4—161no—
356C4—5842no—
357C4—11681no—
358C4—1462no—
359C4#1798846311yescharacteristic
360C4#1798858011yescharacteristic
361C4#1798881311yescharacteristic
362C4#1798881511yescharacteristic
363C2—2922no—
364C2#2261035611yescharacteristic
365C2#2261164211yescharacteristic
366C2#2261164911yescharacteristic
367C2#2261203611yescharacteristic
368C2#2261203811yescharacteristic
369C2#2261204111yescharacteristic
370C2#2261204711yescharacteristic
subgroupG/H
centerC23D8×D73
commutatorC2×C8×C73C22×C4
FrattiniC22×C8C22×D73
FittingC2×#358×C73C2
socleC23×C73C2×D8
prime radicalC23×D73D8
cosocleC22×C8×C73C23
hypercenterC2×#358D73
hypocenterC73#13062
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
3. C2×#17988391C2semidirect
5. #22608023C2semidirect
6. #22608022C2semidirect
7. C2×#358×C73C2semidirect
15. #17988825C4semidirect
189. C2×#358D73semidirect
193. C73#13062semidirect

Maximal subgroups

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

subgroupindex
C22×#145565302
C2×#179888252
C2×#179883912
#226099572
#226080232
#226080222
C2×#358×C732
#1306273