Finite Group Databasebeta

← #34012461

Subgroups — #34012461

Order 18688.

382 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22626174C211yesmaximal, characteristic
2#22626150C211yesmaximal, characteristic, complemented
3C2×#17989477C211yesmaximal, characteristic, complemented
4#22625709C211yesmaximal, characteristic
5#22624983C211yesmaximal, characteristic, complemented
6#22624977C211yesmaximal, characteristic, complemented
7#22623032C211yesmaximal, characteristic
8C2×#14556666C2211yescharacteristic
9#17989643C2211yescharacteristic
10D73⋊#104C411yescharacteristic
11#17989577C411yescharacteristic
12#17989550C2211yescharacteristic
13C2×#14556645C2211yescharacteristic
14#17989486C412yescomplemented
15#17989485C412yescomplemented
16#17989478—21no—
17#17989477—21no—
18#17989477—21no—
19#17989477—21no—
20#17989462C411yescharacteristic
21#17989459C411yescharacteristic
22#131×D73C2211yescharacteristic
23#17989005—21no—
24#17989005C411yescharacteristic, complemented
25#17989004C411yescharacteristic, complemented
26#17989003—21no—
27#17989003C411yescharacteristic, complemented
28C2×#14556530—21no—
29C2×#14556530C2211yescharacteristic
30M32×D73—21no—
31M32×D73C2211yescharacteristic
32#504×C73C411yescharacteristic, complemented
33D73⋊M16C2×C411yescharacteristic
34C2×#11924036C2×C411yescharacteristic
35#14556670C2×C411yescharacteristic
36C2×C4×#64408C2×C411yescharacteristic
37#14556666—21no—
38#14556666—21no—
39#14556666—21no—
40#14556665C2×C412yes—
41#14556664—21no—
42C2×#11924031C2×C412yes—
43#14556656—21no—
44#14556655—21no—
45#14556654—21no—
46#14556653—21no—
47#14556645—21no—
48#14556645—21no—
49#14556645—21no—
50C8×#64408—21no—
51#41×D73C2×C411yescharacteristic
52#14556599—21no—
53#14556599C2×C411yescharacteristic
54#14556598C2×C411yescharacteristic
55#14556597—21no—
56#14556597C2×C411yescharacteristic
57M16×D73—21no—
58M16×D73C2×C411yescharacteristic
59C2×C8×D73—21no—
60C2×C8×D73C2311yescharacteristic
61#14556531—21no—
62#14556531C2×C411yescharacteristic
63#14556530—21no—
64#14556530—21no—
65#14556530—21no—
66#14556530D411yescharacteristic
67#14556530D411yescharacteristic
68#14556530D411yescharacteristic
69#14556530D411yescharacteristic
70C2×#11924003—21no—
71C2×#11924003C2×C411yescharacteristic
72C16×D73—21no—
73C16×D73—21no—
74C16×D73—21no—
75#131×C73C2×C411yescharacteristic
76M32×C73—21no—
77M32×C73C2×C411yescharacteristic
78C2×C16×C73—21no—
79C2×C16×C73C2×C411yescharacteristic
80C22×#64408—21no—
81#11924038—21no—
82#11924037—21no—
83#11924036—21no—
84#11924036—21no—
85#11924036—21no—
86C2×#10596833—21no—
87C4⋊#64408—21no—
88C22⋊#64408—21no—
89C4×#64408—21no—
90C4×#64408—21no—
91C4×#64408—21no—
92#11924031—22no—
93#11924031—22no—
94#11924029M1611yescharacteristic
95#11924028—21no—
96#11924028C4211yescharacteristic
97#11924027—21no—
98#11924027M1611yescharacteristic
99Q8×D73M1611yescharacteristic
100D4×D73—21no—
101D4×D73M1611yescharacteristic
102C2×C4×D73—21no—
103C2×C4×D73C22×C411yescharacteristic
104#11924012—21no—
105#11924012C4211yescharacteristic
106#11924011—21no—
107#11924011—21no—
108#11924011—21no—
109C2×#10596831—21no—
110C2×#10596831C22×C411yescharacteristic
111C8×D73—21no—
112C8×D73—21no—
113C8×D73—21no—
114C8×D73C2×D411yescharacteristic
115C8×D73C2×D411yescharacteristic
116C8×D73#4111yescharacteristic
117C8×D73#4111yescharacteristic
118#11924003—21no—
119#11924003C22⋊C411yescharacteristic
120#11924003C22⋊C411yescharacteristic
121#41×C73C4211yescharacteristic
122M16×C73—21no—
123M16×C73C4211yescharacteristic
124C2×C8×C73—21no—
125C2×C8×C73C22×C411yescharacteristic
126C16×C73—21no—
127C16×C73C22⋊C411yescharacteristic
128C16×C73C22⋊C411yescharacteristic
129C22×D73—21no—
130C22×D73#13111yescharacteristic
131C2×#64408—41no—
132C2×#64408—41no—
133C2×#64408—81no—
134Q8×C73#9711yescharacteristic
135D4×C73—21no—
136D4×C73#9711yescharacteristic
137C2×C4×C73—21no—
138C2×C4×C73C2×C4211yescharacteristic
139C22⋊D73—21no—
140C22⋊D73—21no—
141C22⋊D73—21no—
142C2×Dic73—21no—
143C2×Dic73#13111yescharacteristic
144D292—21no—
145D292#9711yescharacteristic
146C4×D73—21no—
147C4×D73—21no—
148C4×D73—21no—
149C4×D73C4×D411yescharacteristic
150C4×D73C4×D411yescharacteristic
151C4×D73C2×M1611yescharacteristic
152C4×D73C2×M1611yescharacteristic
153Dic146—21no—
154Dic146#9711yescharacteristic
155#10596833—41no—
156#10596833—41no—
157C8×C73—21no—
158C8×C73C2×C22⋊C411yescharacteristic
159C8×C73#11711yescharacteristic
160#10596831—21no—
161#10596831C4×D411yescharacteristic
162#10596831C4×D411yescharacteristic
163C22×C73—21no—
164C22×C73#40511yescharacteristic
165C2×D73—21no—
166C2×D73—41no—
167C2×D73—81no—
168C2×D73#43511yescharacteristic
169C2×D73#43511yescharacteristic
170#64408—81no—
171#64408—81no—
172C4×C73—21no—
173C4×C73C4×C22⋊C411yescharacteristic
174C4×C73C2×#9711yescharacteristic
175Dic73—21no—
176Dic73#43511yescharacteristic
177Dic73#43511yescharacteristic
178#12618—731nomaximal, Sylow
179C2×C73—21no—
180C2×C73—41no—
181C2×C73#175211yescharacteristic
182D73—21no—
183D73—21no—
184D73—41no—
185#3407—731no—
186#2117—731no—
187#2110—731no—
188C2×#347—731no—
189#1765—731no—
190#1378—731no—
191#1377—731no—
192C73#1261811yescharacteristic, complemented, Sylow
193C2×#131—731no—
194#504—731no—
195#504—731no—
196#504—731no—
197#504—1461no—
198#504—1461no—
199#504#6440811yescharacteristic, complemented
200C2×M32—731no—
201C2×M32—731no—
202C2×M32—731no—
203C2×M32—731no—
204C2×M32—1461no—
205C2×M32—1461no—
206C2×#104—731no—
207#416—731no—
208#405—731no—
209C2×#97—731no—
210#350—732no—
211#350—732no—
212#347—1461no—
213#347—1461no—
214#347—1461no—
215#347—1461no—
216#341—731no—
217#339—731no—
218C2×#41—731no—
219#131—731no—
220#131—731no—
221#131—731no—
222#131—1461no—
223#131—1461no—
224#131C2×#6440811yescharacteristic
225C2×M16—731no—
226C2×M16—731no—
227C2×M16—1461no—
228C22×C8—731no—
229C22×C8—731no—
230C22×C8—1461no—
231#117—731no—
232C2×C42—731no—
233M32—21no—
234M32—731no—
235M32—731no—
236M32—731no—
237M32—731no—
238M32—732no—
239M32—731no—
240M32—1461no—
241M32—1461no—
242M32—1461no—
243M32—1461no—
244M32—1461no—
245M32—1461no—
246M32—1461no—
247M32C2×#6440811yescharacteristic
248C2×C16—21no—
249C2×C16—731no—
250C2×C16—732no—
251C2×C16—1461no—
252C2×C16—1461no—
253C2×C16—1461no—
254C2×C16—1461no—
255C2×C16—1461no—
256C2×C16C2×#6440811yescharacteristic
257#104—1461no—
258#104—1461no—
259#103—1461no—
260#102—1461no—
261#97—1461no—
262#97—1461no—
263#97—1461no—
264C4×C8—1461no—
265#41—731no—
266#41—731no—
267#41—731no—
268#41—1461no—
269#41—1461no—
270#41C4×#6440811yescharacteristic
271C2×Q8—731no—
272C2×D4—731no—
273C2×D4—1461no—
274C22×C4—731no—
275C22×C4—1461no—
276C22×C4—1461no—
277M16—21no—
278M16—731no—
279M16—1461no—
280M16—1461no—
281M16—1461no—
282M16—1461no—
283M16—1461no—
284M16—1461no—
285M16C4×#6440811yescharacteristic
286C2×C8—21no—
287C2×C8—731no—
288C2×C8—731no—
289C2×C8—731no—
290C2×C8—731no—
291C2×C8—731no—
292C2×C8—1461no—
293C2×C8—1461no—
294C2×C8—1461no—
295C2×C8—1461no—
296C2×C8—1461no—
297C2×C8—1461no—
298C2×C8—1461no—
299C2×C8—1461no—
300C2×C8C22×#6440811yescharacteristic
301C4⋊C4—1461no—
302C22⋊C4—1461no—
303C42—1461no—
304C42—1461no—
305C42—1461no—
306C16—21no—
307C16—731no—
308C16—731no—
309C16—1462no—
310C16—1462no—
311C16—1461no—
312C16C22⋊#6440811yescharacteristic
313C16C22⋊#6440811yescharacteristic
314C23—731no—
315C23—1461no—
316Q8—731no—
317Q8—1461no—
318Q8#1455664511yescharacteristic
319D4—21no—
320D4—731no—
321D4—1461no—
322D4—1461no—
323D4—1461no—
324D4—1461no—
325D4#1455664511yescharacteristic
326C2×C4—21no—
327C2×C4—2921no—
328C2×C4—2921no—
329C2×C4—5841no—
330C2×C4—731no—
331C2×C4—731no—
332C2×C4—731no—
333C2×C4—731no—
334C2×C4—731no—
335C2×C4—1461no—
336C2×C4—1461no—
337C2×C4—1461no—
338C2×C4—1461no—
339C2×C4C2×C4×#6440811yescharacteristic
340C8—21no—
341C8—2921no—
342C8—2921no—
343C8—731no—
344C8—731no—
345C8—1461no—
346C8C2×C22⋊#6440811yescharacteristic
347C8#1455667011yescharacteristic
348C22—21no—
349C22—2921no—
350C22—5841no—
351C22—731no—
352C22—731no—
353C22—1461no—
354C22#1798955011yescharacteristic
355C4—21no—
356C4—5841no—
357C4—5841no—
358C4—731no—
359C4—731no—
360C4—1461no—
361C4C4×C22⋊#6440811yescharacteristic
362C4C2×#1455664511yescharacteristic
363C2—21no—
364C2—41no—
365C2—2921no—
366C2—1461no—
367C2—1461no—
368C2#2262566711yescharacteristic
subgroupG/H
centerC4C4×C22⋊#64408
commutatorC2×C4×C73C2×C42
FrattiniC2×C8C22×#64408
Fitting#504×C73C4
socleC2×C73#1752
prime radical#41×D73C2×C4
cosocleC2×C8×D73C23
hypercenter#504#64408
hypocenterC73#12618
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C4 ◁ C2×C16 ◁ #504

Complements

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

NH ≅ G/Ntype
2. #22626150C2semidirect
3. C2×#17989477C2semidirect
5. #22624983C2semidirect
6. #22624977C2semidirect
14. #17989486C4semidirect
15. #17989485C4semidirect
24. #17989005C4semidirect
25. #17989004C4semidirect
27. #17989003C4semidirect
32. #504×C73C4semidirect
192. C73#12618semidirect
199. #504#64408semidirect

Maximal subgroups

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

subgroupindex
#226261742
#226261502
C2×#179894772
#226257092
#226249832
#226249772
#226230322
#1261873