Finite Group Databasebeta

← #33002863

Subgroups — #33002863

Order 18688.

470 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989004C211yesmaximal, characteristic, complemented
2C2×#17988371C211yesmaximal, characteristic, complemented
3C2×#17988371C211yesmaximal, characteristic, complemented
4#22611545C211yesmaximal, characteristic, complemented
5#22608013C211yesmaximal, characteristic, complemented
6#22608009C211yesmaximal, characteristic, complemented
7C2×#443×C73C211yesmaximal, characteristic
8#17989004—22no—
9#17989004—22no—
10C2×#14556531—21no—
11C2×#14556531C2211yescharacteristic
12C2×#14556531C2211yescharacteristic
13C22×#11924003—21no—
14C22×#11924003C2211yescharacteristic
15C22×#11924003C2211yescharacteristic
16#17988371—22no—
17#17988371—22no—
18C22⋊#11924003—21no—
19C22⋊#11924003—21no—
20C2×#135×C73C411yescharacteristic
21C2×#131×C73C2211yescharacteristic
22C2×#131×C73C2211yescharacteristic
23#443×C73—22no—
24#443×C73—22no—
25C2×#108×C73C411yescharacteristic
26C2×#104×C73C411yescharacteristic
27C2×#104×C73C411yescharacteristic
28C2×C4×C8×C73C2211yescharacteristic
29#14556531—24no—
30#14556531—21no—
31#14556531—21no—
32#14556531—21no—
33#14556531D412yes—
34C2×#11924003—24no—
35C2×#11924003—21no—
36C2×#11924003—21no—
37C2×#11924003—21no—
38C2×#11924003—21no—
39C2×#11924003—21no—
40C2×#11924003—21no—
41C2×#11924003—21no—
42C2×#11924003D411yescharacteristic
43C2×#11924003D411yescharacteristic
44C2×#41×C73C2×C411yescharacteristic
45C2×#41×C73C2×C411yescharacteristic
46#135×C73—22no—
47#135×C73—22no—
48C2×Q16×C73C811yescharacteristic
49C2×SD16×C73C811yescharacteristic
50C2×SD16×C73C811yescharacteristic
51C2×D8×C73C811yescharacteristic
52#131×C73—21no—
53#131×C73—21no—
54#131×C73—21no—
55#131×C73—21no—
56#131×C73—21no—
57#131×C73—21no—
58#131×C73D412yes—
59#131×C73D412yes—
60C2×M16×C73—21no—
61C2×M16×C73—21no—
62C2×M16×C73C2×C411yescharacteristic
63C2×M16×C73C2×C411yescharacteristic
64C22×C8×C73—21no—
65C22×C8×C73—21no—
66C22×C8×C73C2×C411yescharacteristic
67C22×C8×C73C2311yescharacteristic
68C2×C42×C73C2×C411yescharacteristic
69#108×C73—22no—
70#104×C73—22no—
71#104×C73—22no—
72C4×C8×C73—21no—
73C4×C8×C73—21no—
74#11924003—21no—
75#11924003—22no—
76#11924003—21no—
77#11924003—41no—
78#11924003—41no—
79#41×C73—21no—
80#41×C73—21no—
81#41×C73—21no—
82#41×C73—21no—
83#41×C73—21no—
84#41×C73—21no—
85#41×C73C22⋊C412yes—
86#41×C73M1612yes—
87C2×Q8×C73C2×C811yescharacteristic
88C2×Q8×C73C2×C811yescharacteristic
89C2×D4×C73—21no—
90C2×D4×C73—21no—
91C2×D4×C73C2×C811yescharacteristic
92C2×D4×C73C2×C811yescharacteristic
93C22×C4×C73—21no—
94C22×C4×C73—21no—
95C22×C4×C73—21no—
96C22×C4×C73C22×C411yescharacteristic
97Q16×C73—22no—
98SD16×C73—22no—
99SD16×C73—22no—
100D8×C73—22no—
101M16×C73—21no—
102M16×C73—22no—
103M16×C73—22no—
104M16×C73—21no—
105M16×C73—21no—
106M16×C73—41no—
107M16×C73—41no—
108M16×C73C22⋊C412yes—
109C2×C8×C73—21no—
110C2×C8×C73—21no—
111C2×C8×C73—21no—
112C2×C8×C73—22no—
113C2×C8×C73—22no—
114C2×C8×C73—21no—
115C2×C8×C73—41no—
116C2×C8×C73—41no—
117C2×C8×C73C2×C811yescharacteristic
118C2×C8×C73C2×C811yescharacteristic
119C2×C8×C73C2×D411yescharacteristic
120C2×C8×C73C2×D411yescharacteristic
121C2×C8×C73C2×D411yescharacteristic
122C2×C8×C73C2×D411yescharacteristic
123C2×C8×C73#4111yescharacteristic
124C2×C8×C73#4111yescharacteristic
125C42×C73—21no—
126C42×C73—21no—
127C23×C73—21no—
128C23×C73—21no—
129C23×C73#13111yescharacteristic
130Q8×C73—21no—
131Q8×C73—21no—
132Q8×C73—21no—
133Q8×C73C22⋊C812yes—
134D4×C73—22no—
135D4×C73—22no—
136D4×C73—21no—
137D4×C73—21no—
138D4×C73—21no—
139D4×C73—41no—
140D4×C73—41no—
141D4×C73C22⋊C812yes—
142C2×C4×C73—21no—
143C2×C4×C73—22no—
144C2×C4×C73—22no—
145C2×C4×C73—22no—
146C2×C4×C73—21no—
147C2×C4×C73—41no—
148C2×C4×C73—41no—
149C2×C4×C73—41no—
150C2×C4×C73—41no—
151C2×C4×C73C2×C22⋊C411yescharacteristic
152C2×C4×C73C4×D411yescharacteristic
153C2×C4×C73C4×D411yescharacteristic
154C2×C4×C73C22×C811yescharacteristic
155C2×C4×C73C2×M1611yescharacteristic
156C2×C4×C73#13111yescharacteristic
157C8×C73—22no—
158C8×C73—21no—
159C8×C73—21no—
160C8×C73—41no—
161C8×C73#12011yescharacteristic
162C8×C73#12111yescharacteristic
163C8×C73#12111yescharacteristic
164C8×C73#12311yescharacteristic
165C22×C73—21no—
166C22×C73—21no—
167C22×C73—22no—
168C22×C73—41no—
169C22×C73—41no—
170C22×C73—41no—
171C22×C73—41no—
172C22×C73—41no—
173C22×C73#40811yescharacteristic
174C22×C73C8×D411yescharacteristic
175C22×C73#43511yescharacteristic
176C4×C73—22no—
177C4×C73—42no—
178C4×C73—41no—
179C4×C73#38611yescharacteristic
180C4×C73C2×C22⋊C811yescharacteristic
181C4×C73C8×D411yescharacteristic
182C4×C73#43511yescharacteristic
183#12980—731nomaximal, Sylow
184C2×C73—21no—
185C2×C73—21no—
186C2×C73—42no—
187C2×C73—81no—
188C2×C73#186911yescharacteristic
189C2×C73#217011yescharacteristic
190C2×C73#217111yescharacteristic
191C2×#504—731no—
192C2×#443D7311yescharacteristic
193C2×#350—731no—
194C2×#350—731no—
195#2113—731no—
196#1376—731no—
197#1374—731no—
198C73#1298011yescharacteristic, complemented, Sylow
199C2×#135Dic7311yescharacteristic
200C2×#131C2×D7311yescharacteristic
201C2×#131C2×D7311yescharacteristic
202#504—1462no—
203#504—1462no—
204C2×M32—731no—
205C2×M32—731no—
206C2×M32—1461no—
207C22×C16—731no—
208C22×C16—731no—
209C22×C16—1461no—
210#443—22no—
211#443—22no—
212C2×#108Dic7311yescharacteristic
213C2×#104Dic7311yescharacteristic
214C2×#104Dic7311yescharacteristic
215C2×C4×C8C2×D7311yescharacteristic
216#350—1462no—
217#350—1462no—
218C22⋊C16—1461no—
219C22⋊C16—1461no—
220C2×#41C2×Dic7311yescharacteristic
221C2×#41C2×Dic7311yescharacteristic
222#135—22no—
223#135—22no—
224C2×Q16#1059683111yescharacteristic
225C2×SD16#1059683111yescharacteristic
226C2×SD16#1059683111yescharacteristic
227C2×D8#1059683111yescharacteristic
228#131—21no—
229#131—21no—
230#131—21no—
231#131—21no—
232#131—21no—
233#131—21no—
234#131C22⋊D7312yes—
235#131C22⋊D7312yes—
236C2×M16—21no—
237C2×M16—21no—
238C2×M16C2×Dic7311yescharacteristic
239C2×M16C2×Dic7311yescharacteristic
240C22×C8—21no—
241C22×C8—21no—
242C22×C8C2×Dic7311yescharacteristic
243C22×C8C22×D7311yescharacteristic
244C2×C42C2×Dic7311yescharacteristic
245M32—732no—
246M32—1461no—
247M32—1461no—
248M32—1464no—
249M32—1461no—
250C2×C16—731no—
251C2×C16—731no—
252C2×C16—1461no—
253C2×C16—1461no—
254C2×C16—1461no—
255C2×C16—1461no—
256C2×C16—1464no—
257C2×C16—1461no—
258C2×C16—1461no—
259C2×C16—1461no—
260#108—22no—
261#104—22no—
262#104—22no—
263C4×C8—21no—
264C4×C8—21no—
265#41—21no—
266#41—21no—
267#41—21no—
268#41—21no—
269#41—21no—
270#41—21no—
271#41C22⋊Dic7312yes—
272#41#1192401212yes—
273C2×Q8C2×#1059683111yescharacteristic
274C2×Q8C2×#1059683111yescharacteristic
275C2×D4—21no—
276C2×D4—21no—
277C2×D4C2×#1059683111yescharacteristic
278C2×D4C2×#1059683111yescharacteristic
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4C22×Dic7311yescharacteristic
283Q16—22no—
284SD16—22no—
285SD16—22no—
286D8—22no—
287M16—21no—
288M16—22no—
289M16—22no—
290M16—21no—
291M16—21no—
292M16—41no—
293M16—41no—
294M16C22⋊Dic7312yes—
295C2×C8—21no—
296C2×C8—21no—
297C2×C8—21no—
298C2×C8—22no—
299C2×C8—22no—
300C2×C8—21no—
301C2×C8—41no—
302C2×C8—41no—
303C2×C8C2×#1059683111yescharacteristic
304C2×C8C2×#1059683111yescharacteristic
305C2×C8D4×D7311yescharacteristic
306C2×C8D4×D7311yescharacteristic
307C2×C8C2×C22⋊D7311yescharacteristic
308C2×C8C2×C22⋊D7311yescharacteristic
309C2×C8#1192402711yescharacteristic
310C2×C8#1192402711yescharacteristic
311C42—21no—
312C42—21no—
313C16—2921no—
314C16—2921no—
315C16—1461no—
316C16—1461no—
317C16—1462no—
318C23—21no—
319C23—21no—
320C23#1455659811yescharacteristic
321Q8—21no—
322Q8—21no—
323Q8—21no—
324Q8C22⋊#1059683112yes—
325D4—22no—
326D4—22no—
327D4—21no—
328D4—21no—
329D4—21no—
330D4—41no—
331D4—41no—
332D4C22⋊#1059683112yes—
333C2×C4—21no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—21no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4C2×C22⋊Dic7311yescharacteristic
343C2×C4D4×Dic7311yescharacteristic
344C2×C4D4×Dic7311yescharacteristic
345C2×C4C22×#1059683111yescharacteristic
346C2×C4C2×#1192401211yescharacteristic
347C2×C4#1455659811yescharacteristic
348C8—22no—
349C8—21no—
350C8—21no—
351C8—41no—
352C8#1455655911yescharacteristic
353C8#1455656311yescharacteristic
354C8#1455656311yescharacteristic
355C8#1455657211yescharacteristic
356C22—21no—
357C22—21no—
358C22—22no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22#1798855411yescharacteristic
365C22D4×#1059683111yescharacteristic
366C22#1798865911yescharacteristic
367C4—22no—
368C4—42no—
369C4—41no—
370C4#1798846711yescharacteristic
371C4C2×C22⋊#1059683111yescharacteristic
372C4D4×#1059683111yescharacteristic
373C4#1798865911yescharacteristic
374C2—21no—
375C2—21no—
376C2—42no—
377C2—81no—
378C2#2261022411yescharacteristic
379C2#2261176811yescharacteristic
380C2#2261177311yescharacteristic
subgroupG/H
centerC2×C8D4×D73
commutatorC2×C4×C73C22×C8
FrattiniC22×C8C22×D73
FittingC2×#443×C73C2
socleC22×C73C8×D4
prime radicalC2×#135×C73C4
cosocleC22×C8×C73C23
hypercenterC2×#443D73
hypocenterC73#12980
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2×C8 ◁ C22×C8 ◁ C2×#443

Complements

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

NH ≅ G/Ntype
1. C2×#17989004C2semidirect
2. C2×#17988371C2semidirect
3. C2×#17988371C2semidirect
4. #22611545C2semidirect
5. #22608013C2semidirect
6. #22608009C2semidirect
198. C73#12980semidirect

Maximal subgroups

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

subgroupindex
C2×#179890042
C2×#179883712
#226115452
#226080132
#226080092
C2×#443×C732
#1298073