Finite Group Databasebeta

← #33069062

Subgroups — #33069062

Order 18688.

714 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22621049C212yesmaximal
2C2×#17988967C211yesmaximal, characteristic, complemented
3#22609420C214yesmaximal, complemented
4#22609296C214yesmaximal, complemented
5#22608679C211yesmaximal, characteristic, complemented
6#22608604C212yesmaximal, complemented
7C73×#2931C211yesmaximal, characteristic
8C22×Dic292—21no—
9C22×Dic292—21no—
10C2×#14556591C2212yes—
11#17988990—22no—
12#17988967—21no—
13#17988967—21no—
14#17988967C2212yes—
15#17988967C2212yes—
16#17988933—22no—
17#17988898—24no—
18#17988867C2218yes—
19#17988640C2212yes—
20#17988340C2212yes—
21#17988338C2212yes—
22#17988334C2218yes—
23C2×#131×C73—21no—
24#532×C73C2211yescharacteristic
25#436×C73—21no—
26#436×C73C2212yes—
27C8×D4×C73—21no—
28C8×D4×C73C2212yes—
29#423×C73C2211yescharacteristic
30#408×C73—21no—
31C2×C4×C8×C73C2211yescharacteristic
32C22×Dic146—22no—
33C2×Dic292—22no—
34C2×Dic292—22no—
35C2×Dic292—22no—
36C2×Dic292—22no—
37C2×Dic292—44no—
38C2×Dic292—44no—
39#14556591—22no—
40#14556591—22no—
41#14556591—22no—
42#14556591C2314yes—
43C2×C4⋊Dic73—22no—
44C2×C4⋊Dic73—22no—
45#14556524—28no—
46C4⋊#10596831C2314yes—
47#14556512—28no—
48C22⋊#10596831—22no—
49C4×#10596831—22no—
50C2×#41×C73—21no—
51#131×C73—44no—
52C2×M16×C73—21no—
53C2×M16×C73—22no—
54C22×C8×C73—22no—
55C22×C8×C73D411yescharacteristic
56C22×C8×C73D411yescharacteristic
57C4⋊Q8×C73D411yescharacteristic
58#127×C73D411yescharacteristic
59#124×C73—21no—
60#121×C73—21no—
61#121×C73D412yes—
62C4×D4×C73—21no—
63C4×D4×C73C2312yes—
64C2×C42×C73C2311yescharacteristic
65C4⋊C8×C73—21no—
66C4⋊C8×C73C2312yes—
67C22⋊C8×C73—22no—
68C22⋊C8×C73—22no—
69C4×C8×C73D411yescharacteristic
70C4×C8×C73D411yescharacteristic
71C4×C8×C73C2311yescharacteristic
72C4×C8×C73C2311yescharacteristic
73C2×Dic146—24no—
74C2×Dic146—44no—
75C2×Dic146—44no—
76C2×Dic146—42no—
77C22×Dic73—42no—
78Dic292—44no—
79Dic292—44no—
80Dic292—44no—
81Dic292—44no—
82C2×#10596831—24no—
83C4⋊Dic73—24no—
84C4⋊Dic73—24no—
85C4⋊Dic73—42no—
86C4⋊Dic73—42no—
87#41×C73—42no—
88#41×C73—42no—
89C2×Q8×C73—21no—
90C2×D4×C73—22no—
91C2×D4×C73—21no—
92C2×D4×C73—21no—
93C2×D4×C73C2×D412yes—
94C22×C4×C73—22no—
95C22×C4×C73C2×D411yescharacteristic
96C22×C4×C73C2×D411yescharacteristic
97C22×C4×C73#4111yescharacteristic
98M16×C73—42no—
99M16×C73—44no—
100C2×C8×C73—22no—
101C2×C8×C73—22no—
102C2×C8×C73—44no—
103C2×C8×C73D814yes—
104C2×C8×C73C2×D411yescharacteristic
105C2×C8×C73C2×D411yescharacteristic
106C2×C8×C73C2×D411yescharacteristic
107C2×C8×C73C2×D411yescharacteristic
108C4⋊C4×C73—21no—
109C4⋊C4×C73C2×D412yes—
110C22⋊C4×C73—22no—
111C22⋊C4×C73—22no—
112C42×C73C2×D411yescharacteristic
113C42×C73C2×D411yescharacteristic
114C42×C73#4111yescharacteristic
115C42×C73C2411yescharacteristic
116C23×C73—22no—
117C23×C73#12111yescharacteristic
118Q8×C73—42no—
119D4×C73—24no—
120D4×C73—44no—
121D4×C73—42no—
122D4×C73—42no—
123C2×C4×C73—22no—
124C2×C4×C73—22no—
125C2×C4×C73—22no—
126C2×C4×C73—42no—
127C2×C4×C73—42no—
128C2×C4×C73#12111yescharacteristic
129C2×C4×C73#12111yescharacteristic
130C2×C4×C73#12111yescharacteristic
131C2×C4×C73C2×D812yes—
132C2×C4×C73AGL(1,ℤ/8)12yes—
133C2×C4×C73C22×D411yescharacteristic
134C2×C4×C73C22×D411yescharacteristic
135C2×C4×C73C2×#4111yescharacteristic
136C2×Dic73—44no—
137C2×Dic73—84no—
138Dic146—44no—
139Dic146—44no—
140Dic146—84no—
141C8×C73—21no—
142C8×C73—21no—
143C8×C73—42no—
144C8×C73C2×D812yes—
145C8×C73C2×D812yes—
146#10596831—44no—
147C22×C73—21no—
148C22×C73—21no—
149C22×C73—42no—
150C22×C73—42no—
151C22×C73—42no—
152C22×C73#45912yes—
153C22×C73C2×#12111yescharacteristic
154C4×C73—21no—
155C4×C73—21no—
156C4×C73—42no—
157C4×C73#45912yes—
158C4×C73#47312yes—
159C4×C73C22×D811yescharacteristic
160C4×C73C2×AGL(1,ℤ/8)11yescharacteristic
161C4×C73#57812yes—
162Dic73—84no—
163#19188—731nomaximal, Sylow
164C2×C73—22no—
165C2×C73—82no—
166C2×C73C2×#45911yescharacteristic
167C2×C73#305911yescharacteristic
168C2×C73C2×#47311yescharacteristic
169#3226—732no—
170C2×#494—731no—
171#2931D7311yescharacteristic
172#1689—734no—
173#1669—734no—
174#1564—731no—
175#1546—732no—
176C73#1918811yescharacteristic, complemented, Sylow
177C22×Q16—1461no—
178C22×Q16—1461no—
179C2×#131—21no—
180#532C2×D7311yescharacteristic
181C2×C4⋊Q8—732no—
182#500—1462no—
183#494—732no—
184#494—732no—
185#494—1461no—
186#494—1461no—
187#486—1462no—
188#480—1464no—
189#474—738no—
190#436—21no—
191#436C2×D7312yes—
192C8×D4—21no—
193C8×D4C2×D7312yes—
194#431—732no—
195#423C2×D7311yescharacteristic
196#408—21no—
197C2×C4×C8C2×D7311yescharacteristic
198#335—732no—
199#334—732no—
200#332—738no—
201C2×#41—21no—
202C22×Q8—1462no—
203C2×Q16—2924no—
204C2×Q16—2924no—
205C2×Q16—1462no—
206C2×Q16—1462no—
207C2×Q16—1462no—
208C2×Q16—1462no—
209#131—44no—
210C2×M16—21no—
211C2×M16—22no—
212C22×C8—22no—
213C22×C8C22⋊D7311yescharacteristic
214C22×C8C22⋊D7311yescharacteristic
215C4⋊Q8—734no—
216C4⋊Q8—1462no—
217C4⋊Q8—1462no—
218C4⋊Q8—1462no—
219C4⋊Q8D29211yescharacteristic
220#127D29211yescharacteristic
221#124—21no—
222#121—21no—
223#121D29212yes—
224C4×D4—21no—
225C4×D4C22×D7312yes—
226C2×C4⋊C4—1462no—
227C2×C4⋊C4—1462no—
228C2×C42C22×D7311yescharacteristic
229#107—1468no—
230C4⋊C8—21no—
231C4⋊C8—734no—
232C4⋊C8C22×D7312yes—
233#102—1468no—
234C22⋊C8—22no—
235C22⋊C8—22no—
236C22⋊C8—1462no—
237C4×C8—1462no—
238C4×C8C22⋊D7311yescharacteristic
239C4×C8C22⋊D7311yescharacteristic
240C4×C8C22×D7311yescharacteristic
241C4×C8C22×D7311yescharacteristic
242#41—42no—
243#41—42no—
244C2×Q8—21no—
245C2×Q8—2922no—
246C2×Q8—2924no—
247C2×Q8—2924no—
248C2×Q8—1464no—
249C2×D4—22no—
250C2×D4—21no—
251C2×D4—21no—
252C2×D4C2×D29212yes—
253C22×C4—22no—
254C22×C4—2922no—
255C22×C4C2×C22⋊D7311yescharacteristic
256C22×C4C2×D29211yescharacteristic
257C22×C4#1192402811yescharacteristic
258Q16—2924no—
259Q16—2924no—
260Q16—2924no—
261Q16—2924no—
262M16—42no—
263M16—44no—
264C2×C8—22no—
265C2×C8—22no—
266C2×C8—44no—
267C2×C8—1464no—
268C2×C8#1192401314yes—
269C2×C8C2×C22⋊D7311yescharacteristic
270C2×C8C2×C22⋊D7311yescharacteristic
271C2×C8C2×C22⋊D7311yescharacteristic
272C2×C8C2×C22⋊D7311yescharacteristic
273C4⋊C4—21no—
274C4⋊C4—2922no—
275C4⋊C4—2922no—
276C4⋊C4—1464no—
277C4⋊C4—1464no—
278C4⋊C4C2×D29212yes—
279C22⋊C4—22no—
280C22⋊C4—22no—
281C42C2×C22⋊D7311yescharacteristic
282C42C2×D29211yescharacteristic
283C42#1192402811yescharacteristic
284C42C23×D7311yescharacteristic
285C23—22no—
286C23#1455656511yescharacteristic
287Q8—42no—
288Q8—2924no—
289Q8—2924no—
290Q8—5844no—
291D4—24no—
292D4—44no—
293D4—42no—
294D4—42no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—2924no—
301C2×C4—5844no—
302C2×C4#1455656511yescharacteristic
303C2×C4#1455656511yescharacteristic
304C2×C4#1455656511yescharacteristic
305C2×C4C2×#1192401312yes—
306C2×C4#1455661612yes—
307C2×C4C22×C22⋊D7311yescharacteristic
308C2×C4C22×D29211yescharacteristic
309C2×C4C2×#1192402811yescharacteristic
310C8—21no—
311C8—21no—
312C8—42no—
313C8—2924no—
314C8C2×#1192401312yes—
315C8C2×#1192401312yes—
316C22—21no—
317C22—21no—
318C22—42no—
319C22—42no—
320C22—42no—
321C22#1798878212yes—
322C22C2×#1455656511yescharacteristic
323C4—21no—
324C4—21no—
325C4—42no—
326C4—5844no—
327C4#1798878212yes—
328C4#1798886212yes—
329C4C22×#1192401311yescharacteristic
330C4C2×#1455661611yescharacteristic
331C4#1798940412yes—
332C2—22no—
333C2—82no—
334C2C2×#1798878211yescharacteristic
335C2#2261958611yescharacteristic
336C2C2×#1798886211yescharacteristic
subgroupG/H
centerC22C2×#14556565
commutator, cosocleC42×C73C24
FrattiniC42C23×D73
FittingC73×#2931C2
socleC22×C73C2×#121
prime radical#532×C73C22
hypercenter#2931D73
hypocenterC73#19188
LCSG ▷ C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2931

Complements

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

NH ≅ G/Ntype
2. C2×#17988967C2semidirect
3. #22609420C2semidirect
4. #22609296C2semidirect
5. #22608679C2semidirect
6. #22608604C2semidirect
176. C73#19188semidirect

Maximal subgroups

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

subgroupindex
#226210492
C2×#179889672
#226094202
#226092962
#226086792
#226086042
C73×#29312
#1918873