Finite Group Databasebeta

← #34056924

Subgroups — #34056924

Order 19072.

448 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 117 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23750348C211yesmaximal, characteristic, complemented
2D4×D596C211yesmaximal, characteristic
3#23749972C211yesmaximal, characteristic, complemented
4#23749951C212yesmaximal, complemented
5#23749905C211yesmaximal, characteristic, complemented
6#23749874C212yesmaximal, complemented
7#23749825C211yesmaximal, characteristic, complemented
8#23749774C212yesmaximal, complemented
9#23749704C211yesmaximal, characteristic, complemented
10#23749628C212yesmaximal, complemented
11C4×SD16×C149C211yesmaximal, characteristic, complemented
12C22×D596—21no—
13C2×D4×D149C2212yes—
14C2×#14558723C2212yescomplemented
15C2×#14558719C2212yescomplemented
16#18011911C2211yescharacteristic, complemented
17#18011908C2211yescharacteristic
18#18011906—22no—
19#18011899—21no—
20#18011887C2212yescomplemented
21#18011886C2212yes—
22#18011885—21no—
23#18011881—22no—
24C4×Dic298—21no—
25C4×D596C2211yescharacteristic
26#18011869C2212yescomplemented
27#18011844C2212yescomplemented
28C4⋊#11924746C2211yescharacteristic, complemented
29#18011833C2212yescomplemented
30#18011833C2212yescomplemented
31#18011832C2212yes—
32#18011831C2212yescomplemented
33#18011823C2212yescomplemented
34C2×SD16×C149C2211yescharacteristic
35C4×Q8×C149C2211yescharacteristic, complemented
36C4×D4×C149C2211yescharacteristic
37#106×C149C2211yescharacteristic, complemented
38#103×C149C2211yescharacteristic, complemented
39#102×C149C2211yescharacteristic
40C4×C8×C149C2211yescharacteristic, complemented
41C23×D149—22no—
42C2×Dic298—21no—
43C2×D596—22no—
44C2×D596—21no—
45C2×D596—21no—
46C2×D596—44no—
47C2×D596D411yescharacteristic
48C2×D596C2312yes—
49C2×C22⋊D149—22no—
50D4×D149—24no—
51D4×D149—42no—
52C2×C4×D149C2312yes—
53#14558723—42no—
54#14558719—42no—
55C2×#11924746C2312yes—
56C4⋊Dic149—21no—
57C4⋊Dic149D411yescharacteristic
58C4⋊Dic149C2312yes—
59#14558715—22no—
60#14558713—22no—
61#14558713—22no—
62#14558713—22no—
63#14558713D412yes—
64C4×Dic149—22no—
65C2×Q8×C149C2311yescharacteristic
66C2×D4×C149C2311yescharacteristic
67C22×C4×C149—21no—
68SD16×C149—22no—
69C2×C8×C149C2311yescharacteristic
70C2×C8×C149C2311yescharacteristic
71C4⋊C4×C149—21no—
72C4⋊C4×C149C2311yescharacteristic
73C4⋊C4×C149C2311yescharacteristic
74C22⋊C4×C149—21no—
75C42×C149—21no—
76C42×C149C2311yescharacteristic
77C22×D149—22no—
78C22×D149—44no—
79C22×D149—44no—
80C22×D149—82no—
81C22×D149C2×D412yes—
82Dic298—42no—
83D596—22no—
84D596—22no—
85D596—22no—
86D596—44no—
87D596—42no—
88D596—81no—
89C22⋊D149—44no—
90C2×Dic149—22no—
91C2×Dic149C2×D412yes—
92C4×D149—22no—
93C4×D149—22no—
94#11924746—22no—
95C23×C149—21no—
96Q8×C149—21no—
97Q8×C149—21no—
98D4×C149—21no—
99D4×C149#4112yes—
100C2×C4×C149—21no—
101C2×C4×C149—21no—
102C2×C4×C149—21no—
103C2×C4×C149—42no—
104C2×C4×C149C2×D411yescharacteristic
105C2×C4×C149C2×D411yescharacteristic
106C2×C4×C149C2411yescharacteristic
107C8×C149—21no—
108C8×C149—21no—
109C22×C149—22no—
110C22×C149—41no—
111C22×C149—41no—
112C22×C149C22×D411yescharacteristic
113C2×D149—22no—
114C2×D149—22no—
115C2×D149—22no—
116C2×D149—42no—
117C2×D149—42no—
118C2×D149—42no—
119C2×D149—88no—
120C4×C149—21no—
121C4×C149—41no—
122C4×C149—41no—
123C4×C149—41no—
124C4×C149AGL(1,ℤ/8)12yes—
125C4×C149C2×#4111yescharacteristic
126C4×C149H(5,𝔽2)11yescharacteristic
127Dic149—22no—
128Dic149—42no—
129C2×C149—42no—
130C2×C149#54611yescharacteristic
131C2×C149C2×AGL(1,ℤ/8)11yescharacteristic
132C2×C149#57711yescharacteristic
133D149—42no—
134D149—82no—
135C149#332411yescharacteristic, complemented, Sylow
136#3324—1491nomaximal, Sylow
137#556—1491no—
138D42—1491no—
139#486—1491no—
140#482—1492no—
141#474—1491no—
142#468—1492no—
143#460—1491no—
144#450—1492no—
145C4×SD16D14911yescharacteristic, complemented
146#436—1491no—
147#418—1492no—
148C22×D4—2981no—
149C22×D4—1492no—
150C2×SD16—1492no—
151C2×SD16C2×D14911yescharacteristic
152C2×M16—1492no—
153C4⋊Q8—1491no—
154#127—1491no—
155#126—2982no—
156#124—2981no—
157C22⋊Q8—1492no—
158#121—2981no—
159#121—1492no—
160#120—2982no—
161C4×Q8—2981no—
162C4×Q8C2×D14911yescharacteristic, complemented
163C4×D4—1491no—
164C4×D4C2×D14911yescharacteristic
165#117—1492no—
166#107—1492no—
167#106C2×D14911yescharacteristic, complemented
168C4⋊C8—1491no—
169#103C2×D14911yescharacteristic, complemented
170#102—1492no—
171#102—1492no—
172#102—1492no—
173#102—1492no—
174#102C2×D14911yescharacteristic
175C22⋊C8—1492no—
176C4×C8C2×D14911yescharacteristic, complemented
177C24—2982no—
178C2×Q8—2981no—
179C2×Q8C22×D14911yescharacteristic
180C2×D4—2982no—
181C2×D4—2984no—
182C2×D4—2982no—
183C2×D4—2981no—
184C2×D4—2981no—
185C2×D4—5964no—
186C2×D4—5962no—
187C2×D4—1491no—
188C2×D4—1492no—
189C2×D4C22×D14911yescharacteristic
190C22×C4—21no—
191C22×C4—1492no—
192SD16—22no—
193SD16—5962no—
194M16—5962no—
195C2×C8—1492no—
196C2×C8C22×D14911yescharacteristic
197C2×C8C22×D14911yescharacteristic
198C4⋊C4—21no—
199C4⋊C4—2982no—
200C4⋊C4—2981no—
201C4⋊C4—1492no—
202C4⋊C4—1491no—
203C4⋊C4C22×D14911yescharacteristic
204C4⋊C4C22×D14911yescharacteristic
205C22⋊C4—21no—
206C22⋊C4—2982no—
207C22⋊C4—2982no—
208C22⋊C4—2982no—
209C22⋊C4—1492no—
210C42—21no—
211C42—2982no—
212C42C22×D14911yescharacteristic
213C23—21no—
214C23—2982no—
215C23—5964no—
216C23—5964no—
217C23—1492no—
218C23—11922no—
219Q8—21no—
220Q8—21no—
221Q8—5962no—
222D4—21no—
223D4—2982no—
224D4—2982no—
225D4—2982no—
226D4—5964no—
227D4—5964no—
228D4—5962no—
229D4—11921no—
230D4#1455873612yes—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—42no—
235C2×C4—2982no—
236C2×C4—2982no—
237C2×C4—2982no—
238C2×C4—1492no—
239C2×C4D4×D14911yescharacteristic
240C2×C4D4×D14911yescharacteristic
241C2×C4C23×D14911yescharacteristic
242C8—21no—
243C8—21no—
244C8—2982no—
245C22—22no—
246C22—41no—
247C22—41no—
248C22—2982no—
249C22—2982no—
250C22—2982no—
251C22—5962no—
252C22—5962no—
253C22—5962no—
254C22—11928no—
255C22C2×D4×D14911yescharacteristic
256C4—21no—
257C4—41no—
258C4—41no—
259C4—41no—
260C4—2982no—
261C4—5962no—
262C4#1801193812yes—
263C4C2×#1455873611yescharacteristic
264C4#1801195711yescharacteristic
265C2—42no—
266C2—5962no—
267C2—11922no—
268C2#2375027111yescharacteristic
269C2C2×#1801193811yescharacteristic
270C2#2375043211yescharacteristic
subgroupG/H
centerC22C2×D4×D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
FittingC4×SD16×C149C2
socleC22×C149C22×D4
prime radicalD4×D596C2
hypercenterC4×SD16D149
hypocenterC149#3324
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C42 ◁ C4×SD16

Complements

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

NH ≅ G/Ntype
1. #23750348C2semidirect
3. #23749972C2semidirect
4. #23749951C2semidirect
5. #23749905C2semidirect
6. #23749874C2semidirect
7. #23749825C2semidirect
8. #23749774C2semidirect
9. #23749704C2semidirect
10. #23749628C2semidirect
11. C4×SD16×C149C2semidirect
14. C2×#14558723C22semidirect
15. C2×#14558719C22semidirect
16. #18011911C22semidirect
20. #18011887C22semidirect
26. #18011869C22semidirect
27. #18011844C22semidirect
28. C4⋊#11924746C22semidirect
29. #18011833C22semidirect
30. #18011833C22semidirect
32. #18011831C22semidirect
33. #18011823C22semidirect
35. C4×Q8×C149C22semidirect
37. #106×C149C22semidirect
38. #103×C149C22semidirect
40. C4×C8×C149C22semidirect
135. C149#3324semidirect
145. C4×SD16D149semidirect
162. C4×Q8C2×D149semidirect
167. #106C2×D149semidirect
169. #103C2×D149semidirect
176. C4×C8C2×D149semidirect

Maximal subgroups

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

subgroupindex
#237503482
D4×D5962
#237499722
#237499512
#237499052
#237498742
#237498252
#237497742
#237497042
#237496282
C4×SD16×C1492
#3324149