Finite Group Databasebeta

← #34078827

Subgroups — #34078827

Order 19328.

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#23753010C211yesmaximal, characteristic, complemented
2C2×#18012759C211yesmaximal, characteristic, complemented
3C2×#18012746C211yesmaximal, characteristic, complemented
4#23752485C212yesmaximal, complemented
5#23752480C212yesmaximal, complemented
6#23752477C212yesmaximal, complemented
7#23752473C212yesmaximal, complemented
8C2×#18012674C211yesmaximal, characteristic, complemented
9#23752196C211yesmaximal, characteristic, complemented
10#23752194C211yesmaximal, characteristic, complemented
11#575×C151C211yesmaximal, characteristic, complemented
12C2×#14559004C2211yescharacteristic
13#18012770—21no—
14#18012769C2212yes—
15#18012765—21no—
16#18012764C2212yes—
17#18012759—21no—
18#18012759—21no—
19#18012759—21no—
20#18012759—21no—
21#18012759—22no—
22C2×#14558994C2211yescharacteristic, complemented
23C2×#14558992C2211yescharacteristic, complemented
24C2×#14558991C2211yescharacteristic, complemented
25C2×#14558989C2211yescharacteristic, complemented
26#18012746—21no—
27#18012746—21no—
28#18012746C2212yes—
29#18012746C2212yes—
30C2×#14558988—21no—
31C2×#14558988C2211yescharacteristic
32C2×#14558988C2211yescharacteristic
33C22×C151⋊C8—21no—
34C22×C151⋊C8C2211yescharacteristic
35#18012674C2212yescomplemented
36#18012674C2212yescomplemented
37#18012658C2212yescomplemented
38#18012657C2212yescomplemented
39#18012656C2212yescomplemented
40#18012655C2212yescomplemented
41C2×#41×C151C2211yescharacteristic
42C2×#41×C151C2211yescharacteristic
43#137×C151—21no—
44#137×C151C2212yes—
45AGL(1,ℤ/8)×C151—21no—
46AGL(1,ℤ/8)×C151C2212yes—
47#135×C151—22no—
48C2×M16×C151C2211yescharacteristic
49#14559004—21no—
50#14559004—21no—
51#14559004—21no—
52#14559004—21no—
53C2×Dic302C2311yescharacteristic
54C2×D604C2311yescharacteristic
55C2×C22⋊D151—21no—
56C2×C4×D151—21no—
57#14558994—21no—
58#14558994—21no—
59#14558994—22no—
60#14558992—21no—
61#14558992—21no—
62#14558992—22no—
63#14558991—21no—
64#14558991—21no—
65#14558991—22no—
66#14558989—21no—
67#14558989—21no—
68#14558989—22no—
69#14558988—22no—
70#14558988—22no—
71#14558988—21no—
72#14558988—21no—
73#14558988C2312yes—
74#14558988C2312yes—
75C2×C151⋊C8—22no—
76C2×C151⋊C8—22no—
77C2×C151⋊C8—21no—
78C2×C151⋊C8—21no—
79C2×C151⋊C8D411yescharacteristic
80C2×C151⋊C8D411yescharacteristic
81C2×C151⋊C8D411yescharacteristic
82C2×C151⋊C8D411yescharacteristic
83C2×C151⋊C8C2311yescharacteristic
84C2×C151⋊C8C2311yescharacteristic
85#41×C151—21no—
86#41×C151—21no—
87#41×C151—21no—
88#41×C151—21no—
89#41×C151—21no—
90#41×C151—21no—
91#41×C151D412yes—
92#41×C151C2312yes—
93C2×Q8×C151D411yescharacteristic
94C2×Q8×C151C2311yescharacteristic
95C2×D4×C151—21no—
96C2×D4×C151—21no—
97C2×D4×C151D411yescharacteristic
98C2×D4×C151C2311yescharacteristic
99C22×C4×C151—21no—
100C22×C4×C151—21no—
101C22×C4×C151C2311yescharacteristic
102Q16×C151—22no—
103SD16×C151—22no—
104SD16×C151—22no—
105D8×C151—22no—
106M16×C151—21no—
107M16×C151C2312yes—
108C2×C8×C151—21no—
109C22×D151—21no—
110Dic302—21no—
111Dic302—21no—
112D604—21no—
113D604—21no—
114C22⋊D151—41no—
115C22⋊D151—41no—
116C2×Dic151—21no—
117C4×D151—41no—
118C4×D151—41no—
119C151⋊C8—22no—
120C151⋊C8—22no—
121C151⋊C8C2×D411yescharacteristic
122C151⋊C8C2×D411yescharacteristic
123C151⋊C8C2×D411yescharacteristic
124C151⋊C8C2×D411yescharacteristic
125C23×C151—21no—
126C23×C151—21no—
127C23×C151#4111yescharacteristic
128Q8×C151—21no—
129Q8×C151—21no—
130Q8×C151—21no—
131Q8×C151C2×D412yes—
132D4×C151—22no—
133D4×C151—22no—
134D4×C151—21no—
135D4×C151—21no—
136D4×C151—21no—
137D4×C151—41no—
138D4×C151—41no—
139D4×C151C2×D412yes—
140C2×C4×C151—21no—
141C2×C4×C151—22no—
142C2×C4×C151—22no—
143C2×C4×C151—21no—
144C2×C4×C151—41no—
145C2×C4×C151—41no—
146C2×C4×C151C2×D411yescharacteristic
147C2×C4×C151C2×D411yescharacteristic
148C2×C4×C151C2×D411yescharacteristic
149C2×C4×C151C2×D411yescharacteristic
150C2×C4×C151#4111yescharacteristic
151C2×C4×C151C2411yescharacteristic
152C8×C151—22no—
153C22×C151—22no—
154C22×C151—41no—
155C22×C151—42no—
156C22×C151—41no—
157C22×C151—81no—
158C22×C151#12111yescharacteristic
159C22×C151#12111yescharacteristic
160C22×C151C2×#4111yescharacteristic
161C2×D151—41no—
162C2×D151—81no—
163C4×C151—22no—
164C4×C151—41no—
165C4×C151#12111yescharacteristic
166C4×C151#12111yescharacteristic
167C4×C151C22×D411yescharacteristic
168C4×C151C22×D411yescharacteristic
169Dic151—41no—
170C2×C151—21no—
171C2×C151—21no—
172C2×C151—21no—
173C2×C151—42no—
174C2×C151—81no—
175C2×C151C2×#12111yescharacteristic
176D151—81no—
177C151#306711yescharacteristic, complemented, Sylow
178#3067—1511nomaximal, Sylow
179#575—1511no—
180#575D15111yescharacteristic, complemented
181C2×#135—1511no—
182C2×#131—1511no—
183#473—1512no—
184#472—1512no—
185#471—1512no—
186#471—1512no—
187C2×#108—1511no—
188#413—1511no—
189#413—1511no—
190C2×#41—1511no—
191C2×#41C2×D15111yescharacteristic
192C2×#41C2×D15111yescharacteristic
193#137—21no—
194#137—3021no—
195#137—1512no—
196#137C2×D15112yes—
197AGL(1,ℤ/8)—21no—
198AGL(1,ℤ/8)—3021no—
199AGL(1,ℤ/8)—1512no—
200AGL(1,ℤ/8)C2×D15112yes—
201#135—22no—
202#135—3022no—
203#135—3021no—
204#135—3021no—
205#135—3021no—
206#135—3021no—
207C2×Q16—1511no—
208C2×SD16—1511no—
209C2×SD16—1511no—
210C2×D8—1511no—
211#131—3021no—
212#131—3021no—
213#131—1512no—
214#131—1512no—
215C2×M16—3021no—
216C2×M16—1511no—
217C2×M16—1511no—
218C2×M16C2×D15111yescharacteristic
219C22×C8—3021no—
220C22×C8—1511no—
221#108—1512no—
222#108—1512no—
223#101—1512no—
224#101—1512no—
225#100—1512no—
226#100—1512no—
227#41—21no—
228#41—21no—
229#41—21no—
230#41—21no—
231#41—21no—
232#41—21no—
233#41—3021no—
234#41—3021no—
235#41—3021no—
236#41—3021no—
237#41C22⋊D15112yes—
238#41C22×D15112yes—
239C2×Q8—1511no—
240C2×Q8C22⋊D15111yescharacteristic
241C2×Q8C22×D15111yescharacteristic
242C2×D4—21no—
243C2×D4—21no—
244C2×D4—3021no—
245C2×D4—1511no—
246C2×D4C22⋊D15111yescharacteristic
247C2×D4C22×D15111yescharacteristic
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—3021no—
251C22×C4C22×D15111yescharacteristic
252Q16—22no—
253Q16—3022no—
254Q16—3021no—
255Q16—3021no—
256SD16—22no—
257SD16—22no—
258SD16—3022no—
259SD16—3022no—
260SD16—3021no—
261SD16—3021no—
262SD16—3021no—
263SD16—3021no—
264D8—22no—
265D8—3022no—
266D8—3021no—
267D8—3021no—
268M16—21no—
269M16—3021no—
270M16—3021no—
271M16—3022no—
272M16—3022no—
273M16—1512no—
274M16—1512no—
275M16C22×D15112yes—
276C2×C8—21no—
277C2×C8—3021no—
278C2×C8—3021no—
279C2×C8—3022no—
280C2×C8—3022no—
281C2×C8—1511no—
282C2×C8—1511no—
283C2×C8—1511no—
284C2×C8—1511no—
285C2×C8—1511no—
286C2×C8—1511no—
287C23—21no—
288C23—21no—
289C23—3021no—
290C23#1455900311yescharacteristic
291Q8—21no—
292Q8—21no—
293Q8—21no—
294Q8—3021no—
295Q8—3021no—
296Q8C2×C22⋊D15112yes—
297D4—22no—
298D4—22no—
299D4—21no—
300D4—21no—
301D4—21no—
302D4—41no—
303D4—41no—
304D4—3021no—
305D4—3021no—
306D4—6041no—
307D4—6041no—
308D4C2×C22⋊D15112yes—
309C2×C4—21no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—21no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—3021no—
316C2×C4—6041no—
317C2×C4—6041no—
318C2×C4D4×D15111yescharacteristic
319C2×C4D4×D15111yescharacteristic
320C2×C4C2×C22⋊D15111yescharacteristic
321C2×C4C2×C22⋊D15111yescharacteristic
322C2×C4#1455900311yescharacteristic
323C2×C4C23×D15111yescharacteristic
324C8—22no—
325C8—3022no—
326C8—3022no—
327C8—1511no—
328C8—1511no—
329C8—1511no—
330C8—1511no—
331C22—22no—
332C22—41no—
333C22—42no—
334C22—41no—
335C22—81no—
336C22—6041no—
337C22—12081no—
338C22#1801271111yescharacteristic
339C22#1801271111yescharacteristic
340C22C2×#1455900311yescharacteristic
341C4—22no—
342C4—41no—
343C4—6041no—
344C4#1801271111yescharacteristic
345C4#1801271111yescharacteristic
346C4C2×D4×D15111yescharacteristic
347C4C22×C22⋊D15111yescharacteristic
348C2—21no—
349C2—21no—
350C2—21no—
351C2—42no—
352C2—81no—
353C2—12081no—
354C2C2×#1801271111yescharacteristic
subgroupG/H
centerC4#18012711
commutator, cosocleC2×C4×C151C24
FrattiniC2×C4C23×D151
Fitting#575×C151C2
socleC2×C151C2×#121
hypercenter#575D151
hypocenterC151#3067
LCSG ▷ C2×C4×C151 ▷ C2×C151 ▷ C151
UCSC1 ◁ C4 ◁ C22×C4 ◁ #575

Complements

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

NH ≅ G/Ntype
1. #23753010C2semidirect
2. C2×#18012759C2semidirect
3. C2×#18012746C2semidirect
4. #23752485C2semidirect
5. #23752480C2semidirect
6. #23752477C2semidirect
7. #23752473C2semidirect
8. C2×#18012674C2semidirect
9. #23752196C2semidirect
10. #23752194C2semidirect
11. #575×C151C2semidirect
22. C2×#14558994C22semidirect
23. C2×#14558992C22semidirect
24. C2×#14558991C22semidirect
25. C2×#14558989C22semidirect
35. #18012674C22semidirect
36. #18012674C22semidirect
37. #18012658C22semidirect
38. #18012657C22semidirect
39. #18012656C22semidirect
40. #18012655C22semidirect
177. C151#3067semidirect
180. #575D151semidirect

Maximal subgroups

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

subgroupindex
#237530102
C2×#180127592
C2×#180127462
#237524852
#237524802
#237524772
#237524732
C2×#180126742
#237521962
#237521942
#575×C1512
#3067151