Finite Group Databasebeta

← #32982671

Subgroups — #32982671

Order 18688.

558 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
1#22618649C211yesmaximal, characteristic, complemented
2#22610756C211yesmaximal, characteristic, complemented
3#22608599C211yesmaximal, characteristic, complemented
4#22608491C211yesmaximal, characteristic, complemented
5#22607790C211yesmaximal, characteristic, complemented
6#22607761C211yesmaximal, characteristic, complemented
7C73×#3205C211yesmaximal, characteristic
8C2×#14556598—21no—
9#17988659—22no—
10#17988659—22no—
11#17988640C2211yescharacteristic
12#17988611C2211yescharacteristic
13C2×#14556512—21no—
14C2×#14556512—21no—
15#17988554—21no—
16C2×#14556502C2211yescharacteristic
17#17988453C2211yescharacteristic
18#17988331—22no—
19#17988321—22no—
20#532×C73C2211yescharacteristic
21C2×#127×C73C2211yescharacteristic
22#493×C73C412yescomplemented
23#486×C73C412yescomplemented
24#459×C73—22no—
25#447×C73—22no—
26#431×C73C2211yescharacteristic
27#14556598—42no—
28#14556598—42no—
29C2×#11924012—22no—
30C2×#11924012—21no—
31C22×#10596831—22no—
32C22×#10596831D411yescharacteristic
33C22×#10596831D411yescharacteristic
34C2×C4⋊Dic73—21no—
35C4⋊#10596831—21no—
36C4⋊#10596831—21no—
37C4⋊#10596831D412yes—
38#14556512—42no—
39#14556512—42no—
40C22⋊#10596831—24no—
41C22⋊#10596831—21no—
42#14556502—21no—
43#14556502—21no—
44C4×#10596831—21no—
45#14556498—21no—
46#14556498—21no—
47C2×#41×C73—21no—
48C22×D4×C73—21no—
49C22×D4×C73—21no—
50C22×D4×C73—21no—
51C2×D8×C73—24no—
52C4⋊Q8×C73C2×C411yescharacteristic
53#127×C73—21no—
54#127×C73—21no—
55#127×C73—21no—
56#127×C73C2×C412yes—
57#127×C73C2×C411yescharacteristic
58#124×C73—21no—
59#121×C73—21no—
60#121×C73D412yes—
61C4×D4×C73—21no—
62C4×D4×C73D412yes—
63C2×C42×C73C2311yescharacteristic
64C4⋊C8×C73—21no—
65#102×C73—24no—
66C22⋊C8×C73—21no—
67C4×C8×C73C2×C412yes—
68C22×Dic73—41no—
69#11924012—44no—
70#11924012—41no—
71#11924012—41no—
72C2×#10596831—21no—
73C2×#10596831—21no—
74C2×#10596831—21no—
75C2×#10596831—22no—
76C2×#10596831—21no—
77C2×#10596831—21no—
78C2×#10596831—44no—
79C2×#10596831—41no—
80C2×#10596831D812yescomplemented
81C4⋊Dic73—41no—
82C4⋊Dic73—41no—
83C24×C73—41no—
84#41×C73—41no—
85#41×C73—41no—
86#41×C73—41no—
87#41×C73—41no—
88C2×Q8×C73—21no—
89C2×D4×C73—22no—
90C2×D4×C73—22no—
91C2×D4×C73—21no—
92C2×D4×C73—22no—
93C2×D4×C73—21no—
94C2×D4×C73—22no—
95C2×D4×C73—41no—
96C2×D4×C73—41no—
97C2×D4×C73—41no—
98C2×D4×C73—41no—
99C2×D4×C73—41no—
100C2×D4×C73—41no—
101C2×D4×C73—41no—
102C2×D4×C73—82no—
103C2×D4×C73—82no—
104C2×D4×C73C22⋊C412yes—
105C22×C4×C73—22no—
106C22×C4×C73C2×D411yescharacteristic
107C22×C4×C73C2×D411yescharacteristic
108C22×C4×C73#4111yescharacteristic
109D8×C73—84no—
110C2×C8×C73—22no—
111C4⋊C4×C73—21no—
112C4⋊C4×C73C22⋊C412yes—
113C22⋊C4×C73—22no—
114C22⋊C4×C73—22no—
115C42×C73C22×C411yescharacteristic
116C42×C73C2×D411yescharacteristic
117C42×C73C2×D411yescharacteristic
118C42×C73#4111yescharacteristic
119C23×C73—22no—
120C23×C73—42no—
121C23×C73—82no—
122C23×C73—82no—
123C23×C73—81no—
124C23×C73—81no—
125C23×C73#12111yescharacteristic
126Q8×C73—41no—
127Q8×C73—41no—
128D4×C73—42no—
129D4×C73—44no—
130D4×C73—42no—
131D4×C73—42no—
132D4×C73—41no—
133D4×C73—42no—
134D4×C73—41no—
135D4×C73—42no—
136D4×C73—81no—
137D4×C73—82no—
138D4×C73—82no—
139D4×C73—82no—
140D4×C73—81no—
141D4×C73—81no—
142C2×C4×C73—22no—
143C2×C4×C73—41no—
144C2×C4×C73—41no—
145C2×C4×C73—42no—
146C2×C4×C73—42no—
147C2×C4×C73C2×C22⋊C411yescharacteristic
148C2×C4×C73C4×D411yescharacteristic
149C2×C4×C73C4×D411yescharacteristic
150C2×C4×C73#12011yescharacteristic
151C2×C4×C73#12111yescharacteristic
152C2×C4×C73#12311yescharacteristic
153C2×C4×C73C2×D811yescharacteristic
154C2×C4×C73#13511yescharacteristic
155C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
156C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
157C2×Dic73—81no—
158C2×Dic73—81no—
159C2×Dic73—81no—
160C8×C73—42no—
161#10596831—22no—
162#10596831—42no—
163#10596831—41no—
164#10596831—81no—
165C22×C73—21no—
166C22×C73—21no—
167C22×C73—42no—
168C22×C73—42no—
169C22×C73—42no—
170C22×C73—82no—
171C22×C73—82no—
172C22×C73—82no—
173C22×C73—162no—
174C22×C73—162no—
175C22×C73#38611yescharacteristic
176C22×C73#45911yescharacteristic
177C22×C73#46311yescharacteristic
178C4×C73—21no—
179C4×C73—21no—
180C4×C73—21no—
181C4×C73—21no—
182C4×C73—42no—
183C4×C73C4×D811yescharacteristic
184C4×C73#44211yescharacteristic
185C4×C73#44711yescharacteristic
186C4×C73#44911yescharacteristic
187Dic73—161no—
188#9293—731nomaximal, Sylow
189C2×C73—21no—
190C2×C73—21no—
191C2×C73—82no—
192C2×C73—162no—
193C2×C73#187911yescharacteristic
194C2×C73#190311yescharacteristic
195C2×C73#191011yescharacteristic
196#3205D7311yescharacteristic
197#2932—731no—
198#1961—731no—
199#1545—731no—
200#1517—731no—
201#1286—731no—
202#1275—731no—
203C73#929311yescharacteristic, complemented, Sylow
204C2×#131—1461no—
205#532C2×D7311yescharacteristic
206C2×#127C2×D7311yescharacteristic
207#493Dic7312yescomplemented
208#486Dic7312yescomplemented
209#459—22no—
210#447—22no—
211#435—1462no—
212#435—1462no—
213#431—731no—
214#431C2×D7311yescharacteristic
215#423—731no—
216C2×#102—1461no—
217C2×#102—1461no—
218#408—1461no—
219C2×#97—731no—
220#382—731no—
221#331—1462no—
222#327—1462no—
223C2×#41—21no—
224C22×D4—21no—
225C22×D4—21no—
226C22×D4—21no—
227C2×D8—24no—
228#131—2922no—
229#131—2922no—
230C2×M16—1462no—
231C2×M16—1461no—
232C22×C8—731no—
233C22×C8—731no—
234C22×C8—1462no—
235C4⋊Q8C2×Dic7311yescharacteristic
236#127—21no—
237#127—21no—
238#127—21no—
239#127C2×Dic7312yes—
240#127C2×Dic7311yescharacteristic
241#124—21no—
242#121—21no—
243#121C22⋊D7312yes—
244C4×D4—21no—
245C4×D4C22⋊D7312yes—
246C2×C4⋊C4—1461no—
247C2×C42C22×D7311yescharacteristic
248C4⋊C8—21no—
249C4⋊C8—732no—
250C4⋊C8—1461no—
251C4⋊C8—1461no—
252#102—24no—
253#102—2922no—
254#102—2922no—
255C22⋊C8—21no—
256C22⋊C8—1461no—
257C22⋊C8—1464no—
258#97—1461no—
259#97—1461no—
260C4×C8—1461no—
261C4×C8C2×Dic7312yes—
262#95—1461no—
263#95—1461no—
264C24—41no—
265#41—41no—
266#41—41no—
267#41—41no—
268#41—41no—
269C2×Q8—21no—
270C2×D4—22no—
271C2×D4—22no—
272C2×D4—21no—
273C2×D4—22no—
274C2×D4—21no—
275C2×D4—22no—
276C2×D4—41no—
277C2×D4—41no—
278C2×D4—41no—
279C2×D4—41no—
280C2×D4—41no—
281C2×D4—41no—
282C2×D4—41no—
283C2×D4—82no—
284C2×D4—82no—
285C2×D4C22⋊Dic7312yes—
286C22×C4—22no—
287C22×C4—2921no—
288C22×C4D4×D7311yescharacteristic
289C22×C4C2×C22⋊D7311yescharacteristic
290C22×C4#1192402711yescharacteristic
291D8—84no—
292M16—2921no—
293M16—2921no—
294M16—2924no—
295C2×C8—22no—
296C2×C8—2924no—
297C2×C8—2921no—
298C2×C8—732no—
299C2×C8—1461no—
300C2×C8—1462no—
301C2×C8—1461no—
302C2×C8—1461no—
303C2×C8—1461no—
304C2×C8—1461no—
305C4⋊C4—21no—
306C4⋊C4—2921no—
307C4⋊C4—2921no—
308C4⋊C4C22⋊Dic7312yes—
309C22⋊C4—22no—
310C22⋊C4—22no—
311C42C22×Dic7311yescharacteristic
312C42D4×D7311yescharacteristic
313C42C2×C22⋊D7311yescharacteristic
314C42#1192402711yescharacteristic
315C23—22no—
316C23—42no—
317C23—82no—
318C23—82no—
319C23—81no—
320C23—81no—
321C23#1455656311yescharacteristic
322Q8—41no—
323Q8—41no—
324D4—42no—
325D4—44no—
326D4—42no—
327D4—42no—
328D4—41no—
329D4—42no—
330D4—41no—
331D4—42no—
332D4—81no—
333D4—82no—
334D4—82no—
335D4—82no—
336D4—81no—
337D4—81no—
338C2×C4—22no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—5841no—
344C2×C4—5841no—
345C2×C4—5841no—
346C2×C4C2×C22⋊Dic7311yescharacteristic
347C2×C4D4×Dic7311yescharacteristic
348C2×C4D4×Dic7311yescharacteristic
349C2×C4#1455655911yescharacteristic
350C2×C4#1455656311yescharacteristic
351C2×C4#1455657211yescharacteristic
352C2×C4D8×D7311yescharacteristic
353C2×C4#1455661011yescharacteristic
354C2×C4#1455661511yescharacteristic
355C2×C4#1455661511yescharacteristic
356C8—42no—
357C8—2922no—
358C8—2921no—
359C8—5841no—
360C8—1462no—
361C22—21no—
362C22—21no—
363C22—42no—
364C22—42no—
365C22—42no—
366C22—82no—
367C22—82no—
368C22—82no—
369C22—162no—
370C22—162no—
371C22#1798846711yescharacteristic
372C22#1798878011yescharacteristic
373C22#1798880011yescharacteristic
374C4—21no—
375C4—21no—
376C4—21no—
377C4—21no—
378C4—42no—
379C4—11681no—
380C4D8×Dic7311yescharacteristic
381C4#1798869911yescharacteristic
382C4#1798871911yescharacteristic
383C4#1798872911yescharacteristic
384C2—21no—
385C2—21no—
386C2—82no—
387C2—162no—
388C2#2261028111yescharacteristic
389C2#2261041311yescharacteristic
390C2#2261045911yescharacteristic
subgroupG/H
centerC22#17988467
commutatorC42×C73C22×C4
FrattiniC2×C42C22×D73
Fitting, prime radicalC73×#3205C2
socleC22×C73#386
cosocleC2×C42×C73C23
hypercenter#3205D73
hypocenterC73#9293
LCSG ▷ C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3205

Complements

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

NH ≅ G/Ntype
1. #22618649C2semidirect
2. #22610756C2semidirect
3. #22608599C2semidirect
4. #22608491C2semidirect
5. #22607790C2semidirect
6. #22607761C2semidirect
22. #493×C73C4semidirect
23. #486×C73C4semidirect
80. C2×#10596831D8semidirect
203. C73#9293semidirect
207. #493Dic73semidirect
208. #486Dic73semidirect

Maximal subgroups

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

subgroupindex
#226186492
#226107562
#226085992
#226084912
#226077902
#226077612
C73×#32052
#929373