Finite Group Databasebeta

← #33090136

Subgroups — #33090136

Order 18688.

838 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988645C212yesmaximal, complemented
2#22612396C211yesmaximal, characteristic
3#22610730C211yesmaximal, characteristic
4#22610654C211yesmaximal, characteristic, complemented
5#22610652C211yesmaximal, characteristic, complemented
6#22610540C212yesmaximal, complemented
7#22610534C212yesmaximal, complemented
8C4⋊C8×Dic73C211yesmaximal, characteristic, complemented
9#22609699C212yesmaximal, complemented
10#97×Dic73C211yesmaximal, characteristic, complemented
11C73×#1955C211yesmaximal, characteristic
12C2×#14556550C2212yes—
13#17988645—28no—
14C2×C4⋊#10596831C2211yescharacteristic
15C2×#14556520C2212yes—
16C2×#14556520C2212yes—
17C2×C22⋊#10596831C2212yes—
18C2×C22⋊#10596831C2212yes—
19C2×#14556502C2211yescharacteristic
20C2×#14556501C2211yescharacteristic
21C2×#14556501C2211yescharacteristic
22C2×C8×Dic73C2212yes—
23C2×C8×Dic73C2211yescharacteristic
24C2×C8×Dic73C2211yescharacteristic
25#17988490C412yes—
26#17988486C412yes—
27#17988480C414yes—
28C4×C4⋊Dic73C2211yescharacteristic
29C4×C22⋊Dic73C2212yes—
30C42×Dic73C2211yescharacteristic
31#17988341C2212yes—
32#17988341C2212yes—
33#17988341C2212yes—
34C2×C4×D4×C73C2211yescharacteristic
35C2×C4⋊C8×C73C2211yescharacteristic
36C2×C22⋊C8×C73C2212yes—
37C2×C22⋊C8×C73C2212yes—
38C2×#97×C73C2211yescharacteristic
39C22×#10596831C2312yes—
40C22×#10596831C2311yescharacteristic
41C22×#10596831C2311yescharacteristic
42#14556550—24no—
43#14556550—24no—
44C2×C4⋊Dic73C2×C414yes—
45C2×C4⋊Dic73C2×C412yes—
46C2×C22⋊Dic73C2×C414yes—
47C2×C22⋊Dic73C2×C414yes—
48C2×C4×Dic73C2×C412yes—
49C2×C4×Dic73C2×C412yes—
50C2×C4×Dic73C2311yescharacteristic
51C2×C4×Dic73C2312yes—
52C2×C4×Dic73C2311yescharacteristic
53C4⋊#10596831—22no—
54#14556520—24no—
55#14556520—24no—
56C22⋊#10596831—24no—
57C22⋊#10596831—24no—
58#14556502—22no—
59#14556501—22no—
60#14556501D414yescomplemented
61C8×Dic73—22no—
62C8×Dic73—24no—
63C8×Dic73—22no—
64#14556498C2×C414yes—
65#14556498C2×C412yes—
66C22×D4×C73C2×C411yescharacteristic
67C23×C4×C73C2312yes—
68C22×C8×C73C2312yes—
69C22×C8×C73C2311yescharacteristic
70C22×C8×C73C2311yescharacteristic
71C4×D4×C73—22no—
72C4×D4×C73—22no—
73C2×C4⋊C4×C73C2×C411yescharacteristic
74C2×C22⋊C4×C73C2×C412yes—
75C2×C42×C73C2311yescharacteristic
76C4⋊C8×C73—22no—
77C22⋊C8×C73—24no—
78C22⋊C8×C73—24no—
79#97×C73—22no—
80C22×Dic73C22×C414yes—
81C22×Dic73C22×C412yes—
82C22×Dic73C22×C412yes—
83C2×#10596831—21no—
84C2×#10596831—21no—
85C2×#10596831—21no—
86C2×#10596831—21no—
87C2×#10596831—22no—
88C2×#10596831—22no—
89C2×#10596831—22no—
90C2×#10596831C2×D412yes—
91C2×#10596831#4112yes—
92C4⋊Dic73—28no—
93C4⋊Dic73—42no—
94C22⋊Dic73—28no—
95C22⋊Dic73—28no—
96C4×Dic73—22no—
97C4×Dic73—21no—
98C4×Dic73—21no—
99C4×Dic73—22no—
100C4×Dic73—24no—
101C4×Dic73—42no—
102C4×Dic73C2×D412yes—
103C4×Dic73#4112yes—
104C24×C73C22×C412yes—
105C2×D4×C73—22no—
106C2×D4×C73—42no—
107C22×C4×C73—22no—
108C22×C4×C73—22no—
109C22×C4×C73—22no—
110C22×C4×C73—22no—
111C22×C4×C73—22no—
112C22×C4×C73—22no—
113C22×C4×C73C22×C412yes—
114C22×C4×C73C22×C411yescharacteristic
115C22×C4×C73C22×C411yescharacteristic
116C22×C4×C73C2411yescharacteristic
117C2×C8×C73—21no—
118C2×C8×C73—21no—
119C2×C8×C73—21no—
120C2×C8×C73—21no—
121C2×C8×C73—22no—
122C2×C8×C73—22no—
123C2×C8×C73—22no—
124C2×C8×C73C2×D412yes—
125C2×C8×C73#4112yes—
126C4⋊C4×C73—22no—
127C22⋊C4×C73—24no—
128C42×C73#4112yes—
129C42×C73#4112yes—
130C23×C73—22no—
131C23×C73—22no—
132C23×C73—22no—
133C23×C73—24no—
134C23×C73—24no—
135C23×C73C23×C411yescharacteristic
136D4×C73—44no—
137C2×C4×C73—22no—
138C2×C4×C73—22no—
139C2×C4×C73—22no—
140C2×C4×C73—22no—
141C2×C4×C73—22no—
142C2×C4×C73—42no—
143C2×C4×C73—42no—
144C2×C4×C73—42no—
145C2×C4×C73—42no—
146C2×C4×C73#11712yes—
147C2×C4×C73#11712yes—
148C2×C4×C73C22×D411yescharacteristic
149C2×C4×C73C2×#4111yescharacteristic
150C2×C4×C73C2×#4111yescharacteristic
151C2×C4×C73C2×#4111yescharacteristic
152C2×C4×C73C2×#4111yescharacteristic
153C2×C4×C73C2×#4111yescharacteristic
154C2×Dic73—22no—
155C2×Dic73—22no—
156C2×Dic73—22no—
157C2×Dic73—22no—
158C2×Dic73—24no—
159C2×Dic73—24no—
160C2×Dic73—24no—
161C2×Dic73C4×D414yes—
162C2×Dic73#13114yes—
163C8×C73—22no—
164C8×C73—22no—
165C8×C73—42no—
166#10596831—22no—
167#10596831—22no—
168#10596831—42no—
169C22×C73—42no—
170C22×C73—42no—
171C22×C73—44no—
172C22×C73—44no—
173C22×C73—42no—
174C22×C73C2×#11711yescharacteristic
175C22×C73C2×C4×D411yescharacteristic
176C22×C73C4×#4111yescharacteristic
177C22×C73C2×#13112yes—
178C22×C73#56812yes—
179C4×C73—22no—
180C4×C73—22no—
181C4×C73—42no—
182C4×C73#52911yescharacteristic
183C4×C73#52911yescharacteristic
184C4×C73#53211yescharacteristic
185C4×C73#53211yescharacteristic
186Dic73—24no—
187Dic73—24no—
188Dic73—44no—
189#21269—731nomaximal, Sylow
190C2×C73—82no—
191C2×C73#229311yescharacteristic
192C2×C73#292111yescharacteristic
193C2×C73#292211yescharacteristic
194C2×C73#293512yes—
195C2×C73#296812yes—
196C2×#433—732no—
197#2293—731no—
198#1955—731no—
199#1955D7311yescharacteristic
200#1940—731no—
201#1940—731no—
202#1923—732no—
203#1923—732no—
204C4×C4⋊C8—731no—
205#1750—732no—
206C4×#97—731no—
207C73#2126911yescharacteristic, complemented, Sylow
208C2×C4×D4C2×D7311yescharacteristic
209C2×#117—732no—
210#433—1468no—
211C2×C4⋊C8—732no—
212C2×C4⋊C8—732no—
213C2×C4⋊C8—731no—
214C2×C4⋊C8C2×D7311yescharacteristic
215C2×C22⋊C8—732no—
216C2×C22⋊C8—732no—
217C2×C22⋊C8C2×D7312yes—
218C2×C22⋊C8C2×D7312yes—
219C2×#97—731no—
220C2×#97—731no—
221C2×#97—731no—
222C2×#97C2×D7311yescharacteristic
223C2×C4×C8—732no—
224C2×C4×C8—731no—
225C2×C4×C8—731no—
226#390—732no—
227#389—732no—
228#388—734no—
229C4×C4⋊C4—731no—
230C4×C22⋊C4—732no—
231C43—731no—
232#336—732no—
233#336—732no—
234#336—732no—
235C22×D4C4×D7311yescharacteristic
236C23×C4C22×D7312yes—
237C22×C8—731no—
238C22×C8—731no—
239C22×C8—732no—
240C22×C8C22×D7312yes—
241C22×C8C22×D7311yescharacteristic
242C22×C8C22×D7311yescharacteristic
243C4×D4—22no—
244C4×D4—22no—
245#117—1464no—
246#117—1464no—
247C2×C4⋊C4—732no—
248C2×C4⋊C4—734no—
249C2×C4⋊C4C4×D7311yescharacteristic
250C2×C22⋊C4—734no—
251C2×C22⋊C4—734no—
252C2×C22⋊C4C4×D7312yes—
253C2×C42—732no—
254C2×C42—731no—
255C2×C42—732no—
256C2×C42—732no—
257C2×C42—731no—
258C2×C42C22×D7311yescharacteristic
259C4⋊C8—22no—
260C4⋊C8—1464no—
261C4⋊C8—1464no—
262C4⋊C8—1462no—
263C22⋊C8—24no—
264C22⋊C8—24no—
265C22⋊C8—1464no—
266C22⋊C8—1464no—
267#97—22no—
268#97—734no—
269#97—1462no—
270#97—1462no—
271C4×C8—1464no—
272C4×C8—1462no—
273C4×C8—1462no—
274#95—734no—
275#95—732no—
276C24C2×C4×D7312yes—
277C2×D4—22no—
278C2×D4—42no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—732no—
286C22×C4—732no—
287C22×C4—734no—
288C22×C4C2×C4×D7312yes—
289C22×C4C2×C4×D7311yescharacteristic
290C22×C4C2×C4×D7311yescharacteristic
291C22×C4C23×D7311yescharacteristic
292C2×C8—21no—
293C2×C8—21no—
294C2×C8—21no—
295C2×C8—21no—
296C2×C8—22no—
297C2×C8—22no—
298C2×C8—22no—
299C2×C8—732no—
300C2×C8—732no—
301C2×C8—1462no—
302C2×C8—1462no—
303C2×C8—1462no—
304C2×C8—1461no—
305C2×C8—1461no—
306C2×C8—1461no—
307C2×C8—1461no—
308C2×C8D4×D7312yes—
309C2×C8#1192402712yes—
310C4⋊C4—22no—
311C4⋊C4—2922no—
312C4⋊C4—1468no—
313C22⋊C4—24no—
314C22⋊C4—1468no—
315C22⋊C4—1468no—
316C42—2922no—
317C42—732no—
318C42—732no—
319C42—1464no—
320C42—1462no—
321C42—1461no—
322C42—1462no—
323C42—1461no—
324C42#1192402712yes—
325C42#1192402712yes—
326C23—22no—
327C23—22no—
328C23—22no—
329C23—24no—
330C23—24no—
331C23C22×C4×D7311yescharacteristic
332D4—44no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—734no—
343C2×C4—734no—
344C2×C4—1464no—
345C2×C4—1464no—
346C2×C4—1464no—
347C2×C4—1462no—
348C2×C4—1462no—
349C2×C4—1462no—
350C2×C4—1462no—
351C2×C4#1455654712yes—
352C2×C4#1455654712yes—
353C2×C4C2×D4×D7311yescharacteristic
354C2×C4#41×D7311yescharacteristic
355C2×C4#41×D7311yescharacteristic
356C2×C4C2×#1192402711yescharacteristic
357C2×C4C2×#1192402711yescharacteristic
358C2×C4C2×#1192402711yescharacteristic
359C8—22no—
360C8—22no—
361C8—42no—
362C8—2922no—
363C8—1462no—
364C8—1462no—
365C22—44no—
366C22—44no—
367C22—42no—
368C22—42no—
369C22—42no—
370C22C2×#1455654711yescharacteristic
371C22C4×D4×D7311yescharacteristic
372C22C4×#1192402711yescharacteristic
373C22#131×D7312yes—
374C22#1798935312yes—
375C4—22no—
376C4—22no—
377C4—42no—
378C4—2924no—
379C4—1464no—
380C4—1464no—
381C4#1798911611yescharacteristic
382C4#1798911811yescharacteristic
383C4#1798913211yescharacteristic
384C4#1798913511yescharacteristic
385C2—82no—
386C2#2261239311yescharacteristic
387C2#2261856611yescharacteristic
388C2#2261857111yescharacteristic
389C2#2261867212yes—
390C2#2261892212yes—
subgroupG/H
center, FrattiniC22×C4C23×D73
commutator, socleC23×C73C23×C4
FittingC73×#1955C2
prime radicalC22×D4×C73C2×C4
cosocleC22×C4×C73C24
hypercenter#1955D73
hypocenterC73#21269
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C22×C4 ◁ #1955

Complements

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

NH ≅ G/Ntype
1. C2×#17988645C2semidirect
4. #22610654C2semidirect
5. #22610652C2semidirect
6. #22610540C2semidirect
7. #22610534C2semidirect
8. C4⋊C8×Dic73C2semidirect
9. #22609699C2semidirect
10. #97×Dic73C2semidirect
60. #14556501D4semidirect
207. C73#21269semidirect

Maximal subgroups

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

subgroupindex
C2×#179886452
#226123962
#226107302
#226106542
#226106522
#226105402
#226105342
C4⋊C8×Dic732
#226096992
#97×Dic732
C73×#19552
#2126973