Finite Group Databasebeta

← #37582237

Subgroups — #37582237

Order 24448.

452 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27133461C211yesmaximal, characteristic, complemented
2C2×#20259342C211yesmaximal, characteristic, complemented
3C2×#20259329C211yesmaximal, characteristic, complemented
4#27132936C212yesmaximal, complemented
5#27132931C212yesmaximal, complemented
6#27132928C212yesmaximal, complemented
7#27132924C212yesmaximal, complemented
8C2×#20259257C211yesmaximal, characteristic, complemented
9#27132647C211yesmaximal, characteristic, complemented
10#27132645C211yesmaximal, characteristic, complemented
11#575×C191C211yesmaximal, characteristic, complemented
12C2×#15690287C2211yescharacteristic
13#20259353—21no—
14#20259352C2212yes—
15#20259348—21no—
16#20259347C2212yes—
17#20259342—21no—
18#20259342—21no—
19#20259342—21no—
20#20259342—21no—
21#20259342—22no—
22C2×#15690277C2211yescharacteristic, complemented
23C2×#15690275C2211yescharacteristic, complemented
24C2×#15690274C2211yescharacteristic, complemented
25C2×#15690272C2211yescharacteristic, complemented
26#20259329—21no—
27#20259329—21no—
28#20259329C2212yes—
29#20259329C2212yes—
30C2×#15690271—21no—
31C2×#15690271C2211yescharacteristic
32C2×#15690271C2211yescharacteristic
33C22×C191⋊C8—21no—
34C22×C191⋊C8C2211yescharacteristic
35#20259257C2212yescomplemented
36#20259257C2212yescomplemented
37#20259241C2212yescomplemented
38#20259240C2212yescomplemented
39#20259239C2212yescomplemented
40#20259238C2212yescomplemented
41C2×#41×C191C2211yescharacteristic
42C2×#41×C191C2211yescharacteristic
43#137×C191—21no—
44#137×C191C2212yes—
45AGL(1,ℤ/8)×C191—21no—
46AGL(1,ℤ/8)×C191C2212yes—
47#135×C191—22no—
48C2×M16×C191C2211yescharacteristic
49#15690287—21no—
50#15690287—21no—
51#15690287—21no—
52#15690287—21no—
53C2×Dic382C2311yescharacteristic
54C2×D764C2311yescharacteristic
55C2×C22⋊D191—21no—
56C2×C4×D191—21no—
57#15690277—21no—
58#15690277—21no—
59#15690277—22no—
60#15690275—21no—
61#15690275—21no—
62#15690275—22no—
63#15690274—21no—
64#15690274—21no—
65#15690274—22no—
66#15690272—21no—
67#15690272—21no—
68#15690272—22no—
69#15690271—22no—
70#15690271—22no—
71#15690271—21no—
72#15690271—21no—
73#15690271C2312yes—
74#15690271C2312yes—
75C2×C191⋊C8—22no—
76C2×C191⋊C8—22no—
77C2×C191⋊C8—21no—
78C2×C191⋊C8—21no—
79C2×C191⋊C8D411yescharacteristic
80C2×C191⋊C8D411yescharacteristic
81C2×C191⋊C8D411yescharacteristic
82C2×C191⋊C8D411yescharacteristic
83C2×C191⋊C8C2311yescharacteristic
84C2×C191⋊C8C2311yescharacteristic
85#41×C191—21no—
86#41×C191—21no—
87#41×C191—21no—
88#41×C191—21no—
89#41×C191—21no—
90#41×C191—21no—
91#41×C191D412yes—
92#41×C191C2312yes—
93C2×Q8×C191D411yescharacteristic
94C2×Q8×C191C2311yescharacteristic
95C2×D4×C191—21no—
96C2×D4×C191—21no—
97C2×D4×C191D411yescharacteristic
98C2×D4×C191C2311yescharacteristic
99C22×C4×C191—21no—
100C22×C4×C191—21no—
101C22×C4×C191C2311yescharacteristic
102Q16×C191—22no—
103SD16×C191—22no—
104SD16×C191—22no—
105D8×C191—22no—
106M16×C191—21no—
107M16×C191C2312yes—
108C2×C8×C191—21no—
109C22×D191—21no—
110Dic382—21no—
111Dic382—21no—
112D764—21no—
113D764—21no—
114C22⋊D191—41no—
115C22⋊D191—41no—
116C2×Dic191—21no—
117C4×D191—41no—
118C4×D191—41no—
119C191⋊C8—22no—
120C191⋊C8—22no—
121C191⋊C8C2×D411yescharacteristic
122C191⋊C8C2×D411yescharacteristic
123C191⋊C8C2×D411yescharacteristic
124C191⋊C8C2×D411yescharacteristic
125C23×C191—21no—
126C23×C191—21no—
127C23×C191#4111yescharacteristic
128Q8×C191—21no—
129Q8×C191—21no—
130Q8×C191—21no—
131Q8×C191C2×D412yes—
132D4×C191—22no—
133D4×C191—22no—
134D4×C191—21no—
135D4×C191—21no—
136D4×C191—21no—
137D4×C191—41no—
138D4×C191—41no—
139D4×C191C2×D412yes—
140C2×C4×C191—21no—
141C2×C4×C191—22no—
142C2×C4×C191—22no—
143C2×C4×C191—21no—
144C2×C4×C191—41no—
145C2×C4×C191—41no—
146C2×C4×C191C2×D411yescharacteristic
147C2×C4×C191C2×D411yescharacteristic
148C2×C4×C191C2×D411yescharacteristic
149C2×C4×C191C2×D411yescharacteristic
150C2×C4×C191#4111yescharacteristic
151C2×C4×C191C2411yescharacteristic
152C8×C191—22no—
153C22×C191—22no—
154C22×C191—41no—
155C22×C191—42no—
156C22×C191—41no—
157C22×C191—81no—
158C22×C191#12111yescharacteristic
159C22×C191#12111yescharacteristic
160C22×C191C2×#4111yescharacteristic
161C2×D191—41no—
162C2×D191—81no—
163C4×C191—22no—
164C4×C191—41no—
165C4×C191#12111yescharacteristic
166C4×C191#12111yescharacteristic
167C4×C191C22×D411yescharacteristic
168C4×C191C22×D411yescharacteristic
169Dic191—41no—
170C2×C191—21no—
171C2×C191—21no—
172C2×C191—21no—
173C2×C191—42no—
174C2×C191—81no—
175C2×C191C2×#12111yescharacteristic
176D191—81no—
177C191#306711yescharacteristic, complemented, Sylow
178#3067—1911nomaximal, Sylow
179#575—1911no—
180#575D19111yescharacteristic, complemented
181C2×#135—1911no—
182C2×#131—1911no—
183#473—1912no—
184#472—1912no—
185#471—1912no—
186#471—1912no—
187C2×#108—1911no—
188#413—1911no—
189#413—1911no—
190C2×#41—1911no—
191C2×#41C2×D19111yescharacteristic
192C2×#41C2×D19111yescharacteristic
193#137—21no—
194#137—3821no—
195#137—1912no—
196#137C2×D19112yes—
197AGL(1,ℤ/8)—21no—
198AGL(1,ℤ/8)—3821no—
199AGL(1,ℤ/8)—1912no—
200AGL(1,ℤ/8)C2×D19112yes—
201#135—22no—
202#135—3822no—
203#135—3821no—
204#135—3821no—
205#135—3821no—
206#135—3821no—
207C2×Q16—1911no—
208C2×SD16—1911no—
209C2×SD16—1911no—
210C2×D8—1911no—
211#131—3821no—
212#131—3821no—
213#131—1912no—
214#131—1912no—
215C2×M16—3821no—
216C2×M16—1911no—
217C2×M16—1911no—
218C2×M16C2×D19111yescharacteristic
219C22×C8—3821no—
220C22×C8—1911no—
221#108—1912no—
222#108—1912no—
223#101—1912no—
224#101—1912no—
225#100—1912no—
226#100—1912no—
227#41—21no—
228#41—21no—
229#41—21no—
230#41—21no—
231#41—21no—
232#41—21no—
233#41—3821no—
234#41—3821no—
235#41—3821no—
236#41—3821no—
237#41C22⋊D19112yes—
238#41C22×D19112yes—
239C2×Q8—1911no—
240C2×Q8C22⋊D19111yescharacteristic
241C2×Q8C22×D19111yescharacteristic
242C2×D4—21no—
243C2×D4—21no—
244C2×D4—3821no—
245C2×D4—1911no—
246C2×D4C22⋊D19111yescharacteristic
247C2×D4C22×D19111yescharacteristic
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—3821no—
251C22×C4C22×D19111yescharacteristic
252Q16—22no—
253Q16—3822no—
254Q16—3821no—
255Q16—3821no—
256SD16—22no—
257SD16—22no—
258SD16—3822no—
259SD16—3822no—
260SD16—3821no—
261SD16—3821no—
262SD16—3821no—
263SD16—3821no—
264D8—22no—
265D8—3822no—
266D8—3821no—
267D8—3821no—
268M16—21no—
269M16—3821no—
270M16—3821no—
271M16—3822no—
272M16—3822no—
273M16—1912no—
274M16—1912no—
275M16C22×D19112yes—
276C2×C8—21no—
277C2×C8—3821no—
278C2×C8—3821no—
279C2×C8—3822no—
280C2×C8—3822no—
281C2×C8—1911no—
282C2×C8—1911no—
283C2×C8—1911no—
284C2×C8—1911no—
285C2×C8—1911no—
286C2×C8—1911no—
287C23—21no—
288C23—21no—
289C23—3821no—
290C23#1569028611yescharacteristic
291Q8—21no—
292Q8—21no—
293Q8—21no—
294Q8—3821no—
295Q8—3821no—
296Q8C2×C22⋊D19112yes—
297D4—22no—
298D4—22no—
299D4—21no—
300D4—21no—
301D4—21no—
302D4—41no—
303D4—41no—
304D4—3821no—
305D4—3821no—
306D4—7641no—
307D4—7641no—
308D4C2×C22⋊D19112yes—
309C2×C4—21no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—21no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—3821no—
316C2×C4—7641no—
317C2×C4—7641no—
318C2×C4D4×D19111yescharacteristic
319C2×C4D4×D19111yescharacteristic
320C2×C4C2×C22⋊D19111yescharacteristic
321C2×C4C2×C22⋊D19111yescharacteristic
322C2×C4#1569028611yescharacteristic
323C2×C4C23×D19111yescharacteristic
324C8—22no—
325C8—3822no—
326C8—3822no—
327C8—1911no—
328C8—1911no—
329C8—1911no—
330C8—1911no—
331C22—22no—
332C22—41no—
333C22—42no—
334C22—41no—
335C22—81no—
336C22—15281no—
337C22—7641no—
338C22#2025929411yescharacteristic
339C22#2025929411yescharacteristic
340C22C2×#1569028611yescharacteristic
341C4—22no—
342C4—41no—
343C4—7641no—
344C4#2025929411yescharacteristic
345C4#2025929411yescharacteristic
346C4C2×D4×D19111yescharacteristic
347C4C22×C22⋊D19111yescharacteristic
348C2—21no—
349C2—21no—
350C2—21no—
351C2—42no—
352C2—81no—
353C2—15281no—
354C2C2×#2025929411yescharacteristic
subgroupG/H
centerC4#20259294
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
Fitting#575×C191C2
socleC2×C191C2×#121
hypercenter#575D191
hypocenterC191#3067
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C4 ◁ C22×C4 ◁ #575

Complements

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

NH ≅ G/Ntype
1. #27133461C2semidirect
2. C2×#20259342C2semidirect
3. C2×#20259329C2semidirect
4. #27132936C2semidirect
5. #27132931C2semidirect
6. #27132928C2semidirect
7. #27132924C2semidirect
8. C2×#20259257C2semidirect
9. #27132647C2semidirect
10. #27132645C2semidirect
11. #575×C191C2semidirect
22. C2×#15690277C22semidirect
23. C2×#15690275C22semidirect
24. C2×#15690274C22semidirect
25. C2×#15690272C22semidirect
35. #20259257C22semidirect
36. #20259257C22semidirect
37. #20259241C22semidirect
38. #20259240C22semidirect
39. #20259239C22semidirect
40. #20259238C22semidirect
177. C191#3067semidirect
180. #575D191semidirect

Maximal subgroups

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

subgroupindex
#271334612
C2×#202593422
C2×#202593292
#271329362
#271329312
#271329282
#271329242
C2×#202592572
#271326472
#271326452
#575×C1912
#3067191