Finite Group Databasebeta

← #34012409

Subgroups — #34012409

Order 18688.

444 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22626501C211yesmaximal, characteristic
2#22626174C211yesmaximal, characteristic
3#22626146C211yesmaximal, characteristic, complemented
4C2×#17989477C211yesmaximal, characteristic, complemented
5#22624983C211yesmaximal, characteristic, complemented
6#22624980C211yesmaximal, characteristic, complemented
7#22611557C211yesmaximal, characteristic
8D73⋊#131C411yescharacteristic
9#41×#64408C411yescharacteristic
10C2×#14556666C2211yescharacteristic
11#17989643C2211yescharacteristic
12#17989553—21no—
13#17989553C411yescharacteristic
14#17989550—21no—
15#17989550C2211yescharacteristic
16M16×#64408—21no—
17M16×#64408C411yescharacteristic
18C2×#14556645—21no—
19C2×#14556645C2211yescharacteristic
20#17989486C412yescomplemented
21#17989485C412yescomplemented
22#17989478—21no—
23#17989477—21no—
24#17989477—21no—
25#17989477—21no—
26#131×D73C2211yescharacteristic
27C2×#14556530C2211yescharacteristic
28M32×D73C2211yescharacteristic
29#17988372C411yescharacteristic, complemented
30#17988371C411yescharacteristic, complemented
31#17988370C411yescharacteristic, complemented
32#350×C73C411yescharacteristic, complemented
33#14556679—21no—
34#14556678—22no—
35#14556678—21no—
36D73⋊M16—21no—
37D73⋊M16C2×C411yescharacteristic
38C2×#11924036—21no—
39C2×#11924036C2×C411yescharacteristic
40Q8×#64408—21no—
41D4×#64408—22no—
42D4×#64408—21no—
43#14556670—21no—
44#14556670C2×C411yescharacteristic
45C2×C4×#64408—21no—
46C2×C4×#64408C2×C411yescharacteristic
47#14556666—21no—
48#14556666—21no—
49#14556666—21no—
50#14556665C2×C412yescomplemented
51#14556664—21no—
52C2×#11924031C2×C412yescomplemented
53#14556645—21no—
54#14556645—21no—
55#14556645—21no—
56#14556645D411yescharacteristic
57#14556645D411yescharacteristic
58#14556645D411yescharacteristic
59#14556645D411yescharacteristic
60C8×#64408—21no—
61C8×#64408—21no—
62C8×#64408—21no—
63#41×D73C2×C411yescharacteristic
64#14556599—21no—
65#14556599C2×C411yescharacteristic
66#14556598C2×C411yescharacteristic
67#14556597—21no—
68#14556597C2×C411yescharacteristic
69M16×D73—21no—
70M16×D73C2×C411yescharacteristic
71C2×C8×D73—21no—
72C2×C8×D73C2311yescharacteristic
73#14556531C2×C411yescharacteristic, complemented
74#14556530—21no—
75#14556530—21no—
76#14556530—21no—
77C2×#11924003C2×C411yescharacteristic, complemented
78C16×D73—21no—
79#131×C73C2×C411yescharacteristic
80M32×C73C2×C411yescharacteristic, complemented
81C2×C16×C73C2×C411yescharacteristic, complemented
82C22×#64408—22no—
83C22×#64408—21no—
84#11924038—22no—
85#11924038—21no—
86#11924037—22no—
87#11924037M1612yes—
88#11924036—22no—
89#11924036—21no—
90#11924036—21no—
91#11924036C22⋊C411yescharacteristic
92#11924036C22⋊C411yescharacteristic
93C2×#10596833—22no—
94C2×#10596833—21no—
95C4⋊#64408—22no—
96C4⋊#64408—21no—
97C22⋊#64408—22no—
98C22⋊#64408M1612yes—
99C4×#64408—22no—
100C4×#64408—21no—
101C4×#64408—21no—
102C4×#64408C22⋊C411yescharacteristic
103C4×#64408C22⋊C411yescharacteristic
104#11924031—22no—
105#11924031—22no—
106#11924029M1611yescharacteristic
107#11924028—21no—
108#11924028C4211yescharacteristic
109#11924027—21no—
110#11924027M1611yescharacteristic
111Q8×D73M1611yescharacteristic
112D4×D73—21no—
113D4×D73M1611yescharacteristic
114C2×C4×D73—21no—
115C2×C4×D73C22×C411yescharacteristic
116#11924012—21no—
117#11924012C4211yescharacteristic
118#11924011—21no—
119#11924011—21no—
120#11924011—21no—
121C2×#10596831—21no—
122C2×#10596831C22×C411yescharacteristic
123C8×D73—21no—
124C8×D73—21no—
125C8×D73—21no—
126C8×D73C2×D411yescharacteristic
127C8×D73C2×D411yescharacteristic
128C8×D73#4111yescharacteristic
129C8×D73#4111yescharacteristic
130#11924003—21no—
131#11924003—21no—
132#41×C73C4211yescharacteristic
133M16×C73—21no—
134M16×C73C4211yescharacteristic
135C2×C8×C73—21no—
136C2×C8×C73C22×C411yescharacteristic
137C16×C73—21no—
138C16×C73—21no—
139C22×D73—21no—
140C22×D73C2×M1611yescharacteristic
141C2×#64408—21no—
142C2×#64408—22no—
143C2×#64408—21no—
144C2×#64408—81no—
145C2×#64408—82no—
146Q8×C73#9711yescharacteristic
147D4×C73—21no—
148D4×C73#9711yescharacteristic
149C2×C4×C73—21no—
150C2×C4×C73C2×C4211yescharacteristic
151C22⋊D73—21no—
152C22⋊D73—21no—
153C22⋊D73—21no—
154C2×Dic73—21no—
155C2×Dic73C2×M1611yescharacteristic
156D292—21no—
157D292#9711yescharacteristic
158C4×D73—21no—
159C4×D73—21no—
160C4×D73—21no—
161C4×D73C2×C22⋊C411yescharacteristic
162C4×D73#11711yescharacteristic
163C4×D73C2×M1611yescharacteristic
164C4×D73C2×M1611yescharacteristic
165Dic146—21no—
166Dic146#9711yescharacteristic
167#10596833—21no—
168#10596833—22no—
169#10596833—21no—
170C8×C73—21no—
171C8×C73C4×D411yescharacteristic
172C8×C73C4×D411yescharacteristic
173#10596831—21no—
174#10596831C4×D411yescharacteristic
175#10596831C4×D411yescharacteristic
176C22×C73—21no—
177C22×C73C4×M1611yescharacteristic
178C2×D73—21no—
179C2×D73—41no—
180C2×D73—42no—
181C2×D73#40711yescharacteristic
182C2×D73#43211yescharacteristic
183#64408—42no—
184#64408—41no—
185#64408—41no—
186C4×C73—21no—
187C4×C73C4×C22⋊C411yescharacteristic
188C4×C73C2×#9711yescharacteristic
189Dic73—21no—
190Dic73#40711yescharacteristic
191Dic73#43211yescharacteristic
192#12609—731nomaximal, Sylow
193C2×C73—21no—
194C2×C73—41no—
195C2×C73#174911yescharacteristic
196D73—21no—
197D73—21no—
198D73—41no—
199#2875—731no—
200#2117—731no—
201#2117—731no—
202C2×#347—731no—
203C2×#347—731no—
204#1378—731no—
205#1378—731no—
206C73#1260911yescharacteristic, complemented, Sylow
207C2×#131—731no—
208C2×#131—731no—
209C4×#41—731no—
210C2×M32—731no—
211C2×M32—731no—
212C2×M32—731no—
213C2×M32—731no—
214#405—731no—
215#405—731no—
216#405—1461no—
217#405—1461no—
218C4×M16—731no—
219C4×M16—1461no—
220C2×#97—731no—
221C2×#97—1461no—
222#350—732no—
223#350—732no—
224#350—731no—
225#350—731no—
226#350—731no—
227#350#6440811yescharacteristic, complemented
228#347—1461no—
229#347—1461no—
230#347—1461no—
231#347—1461no—
232C2×#41—731no—
233#131—731no—
234#131—731no—
235#131—731no—
236#131—1461no—
237#131—1461no—
238#131—1462no—
239#131—1461no—
240#131—1461no—
241#131C2×#6440811yescharacteristic
242C2×M16—731no—
243C2×M16—731no—
244C2×M16—1461no—
245C2×M16—1461no—
246C22×C8—731no—
247C22×C8—731no—
248C22×C8—1461no—
249C22×C8—1461no—
250C4×Q8—1461no—
251C4×D4—1462no—
252C4×D4—1461no—
253#117—731no—
254#117—1461no—
255C2×C42—731no—
256C2×C42—1461no—
257M32—732no—
258M32—731no—
259M32—1461no—
260M32—1461no—
261M32—1461no—
262M32—1461no—
263M32—1461no—
264M32—1461no—
265M32C2×#6440811yescharacteristic, complemented
266C2×C16—732no—
267C2×C16—731no—
268C2×C16—1461no—
269C2×C16—1461no—
270C2×C16C2×#6440811yescharacteristic, complemented
271#97—731no—
272#97—731no—
273#97—731no—
274#97—731no—
275#97—1461no—
276#97—1461no—
277#97—1461no—
278C4×C8—1461no—
279C4×C8—1461no—
280C4×C8—1461no—
281#41—731no—
282#41—731no—
283#41—731no—
284#41—1461no—
285#41—1461no—
286#41C4×#6440811yescharacteristic
287C2×Q8—731no—
288C2×D4—731no—
289C2×D4—1461no—
290C22×C4—731no—
291C22×C4—1461no—
292C22×C4—1461no—
293C22×C4—1462no—
294M16—21no—
295M16—732no—
296M16—731no—
297M16—1461no—
298M16—1461no—
299M16—1461no—
300M16—1461no—
301M16—1462no—
302M16—1462no—
303M16—1461no—
304M16C4×#6440811yescharacteristic
305C2×C8—21no—
306C2×C8—731no—
307C2×C8—731no—
308C2×C8—731no—
309C2×C8—731no—
310C2×C8—731no—
311C2×C8—731no—
312C2×C8—731no—
313C2×C8—1461no—
314C2×C8—1461no—
315C2×C8—1461no—
316C2×C8—1461no—
317C2×C8—1461no—
318C2×C8—1461no—
319C2×C8—1462no—
320C2×C8—1462no—
321C2×C8—1461no—
322C2×C8C22×#6440811yescharacteristic
323C4⋊C4—1461no—
324C4⋊C4—1462no—
325C22⋊C4—732no—
326C22⋊C4—1462no—
327C42—731no—
328C42—731no—
329C42—1461no—
330C42—1461no—
331C42—1462no—
332C16—21no—
333C16—21no—
334C16—1462no—
335C16—1462no—
336C16—1461no—
337C16—1461no—
338C23—731no—
339C23—1461no—
340Q8—731no—
341Q8—1461no—
342Q8#1455664511yescharacteristic
343D4—21no—
344D4—731no—
345D4—1461no—
346D4—1461no—
347D4—1461no—
348D4—1461no—
349D4#1455664511yescharacteristic
350C2×C4—21no—
351C2×C4—5841no—
352C2×C4—5842no—
353C2×C4—731no—
354C2×C4—731no—
355C2×C4—731no—
356C2×C4—731no—
357C2×C4—731no—
358C2×C4—1461no—
359C2×C4—1461no—
360C2×C4—1461no—
361C2×C4—1461no—
362C2×C4—1462no—
363C2×C4—1461no—
364C2×C4—1461no—
365C2×C4C2×C4×#6440811yescharacteristic
366C8—21no—
367C8—731no—
368C8—731no—
369C8—1461no—
370C8—1461no—
371C8—1462no—
372C8—1461no—
373C8D4×#6440811yescharacteristic
374C8D4×#6440811yescharacteristic
375C22—21no—
376C22—2921no—
377C22—2922no—
378C22—731no—
379C22—731no—
380C22—1461no—
381C22M16×#6440811yescharacteristic
382C4—21no—
383C4—2921no—
384C4—2921no—
385C4—2922no—
386C4—731no—
387C4—731no—
388C4—1461no—
389C4C22⋊C4×#6440811yescharacteristic
390C4C2×#1455664511yescharacteristic
391C2—21no—
392C2—41no—
393C2—2921no—
394C2—1461no—
395C2—1461no—
396C2#2262565911yescharacteristic
subgroupG/H
centerC4C22⋊C4×#64408
commutatorC2×C4×C73C2×C42
FrattiniC2×C8C22×#64408
Fitting#350×C73C4
socleC2×C73#1749
prime radical#41×D73C2×C4
cosocleC2×C8×D73C23
hypercenter#350#64408
hypocenterC73#12609
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C4 ◁ C2×C8 ◁ #350

Complements

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

NH ≅ G/Ntype
3. #22626146C2semidirect
4. C2×#17989477C2semidirect
5. #22624983C2semidirect
6. #22624980C2semidirect
20. #17989486C4semidirect
21. #17989485C4semidirect
29. #17988372C4semidirect
30. #17988371C4semidirect
31. #17988370C4semidirect
32. #350×C73C4semidirect
50. #14556665C2×C4semidirect
52. C2×#11924031C2×C4semidirect
73. #14556531C2×C4semidirect
77. C2×#11924003C2×C4semidirect
80. M32×C73C2×C4semidirect
81. C2×C16×C73C2×C4semidirect
206. C73#12609semidirect
227. #350#64408semidirect
265. M32C2×#64408semidirect
270. C2×C16C2×#64408semidirect

Maximal subgroups

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

subgroupindex
#226265012
#226261742
#226261462
C2×#179894772
#226249832
#226249802
#226115572
#1260973