Finite Group Databasebeta

← #34056916

Subgroups — #34056916

Order 19072.

448 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 123 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
1D4×D596C211yesmaximal, characteristic
2#23749973C211yesmaximal, characteristic, complemented
3#23749949C212yesmaximal, complemented
4#23749902C211yesmaximal, characteristic, complemented
5#23749873C212yesmaximal, complemented
6#23749825C211yesmaximal, characteristic, complemented
7#23749775C212yesmaximal, complemented
8C4×#14558723C211yesmaximal, characteristic, complemented
9#23749703C211yesmaximal, characteristic, complemented
10#23749629C212yesmaximal, complemented
11#556×C149C211yesmaximal, characteristic, complemented
12C22×D596C2212yes—
13C2×D4×D149—21no—
14C2×#14558723C2211yescharacteristic
15C2×#14558723C2212yescomplemented
16C2×#14558720C2212yescomplemented
17#18011908C2211yescharacteristic
18#18011886—21no—
19#18011885C2212yes—
20#18011881—22no—
21C4×D596C2211yescharacteristic
22#18011844C2212yescomplemented
23#18011842C2211yescharacteristic, complemented
24C4⋊#11924746C2211yescharacteristic, complemented
25#18011835C2211yescharacteristic, complemented
26#18011831C2212yes—
27#18011831C2211yescharacteristic
28#18011831C2212yescomplemented
29#18011831C2212yescomplemented
30#18011831C2212yescomplemented
31C22⋊#11924746C2212yescomplemented
32C4×#11924746C2211yescharacteristic, complemented
33C4⋊Q8×C149C2211yescharacteristic, complemented
34#126×C149—22no—
35#124×C149—21no—
36C22⋊Q8×C149C2212yescomplemented
37C4×Q8×C149—21no—
38C4×Q8×C149C2211yescharacteristic, complemented
39C4×D4×C149C2211yescharacteristic
40#117×C149C2212yescomplemented
41C23×D149—22no—
42C2×D596—24no—
43C2×D596—21no—
44C2×D596—22no—
45C2×D596—21no—
46C2×D596—42no—
47C2×D596C2312yes—
48C2×D596C2311yescharacteristic
49C2×C22⋊D149—22no—
50D4×D149—44no—
51C2×C4×D149—21no—
52#14558723—22no—
53#14558723—42no—
54#14558720—42no—
55C2×#11924746C2311yescharacteristic
56C2×#11924746C2312yes—
57C2×#11924746C2311yescharacteristic
58C4⋊Dic149C2311yescharacteristic
59#14558713—21no—
60#14558713—22no—
61C2×Q8×C149—21no—
62C2×Q8×C149C2311yescharacteristic
63C2×D4×C149D411yescharacteristic
64C22×C4×C149C2312yes—
65C4⋊C4×C149—22no—
66C4⋊C4×C149—21no—
67C4⋊C4×C149—21no—
68C4⋊C4×C149D411yescharacteristic
69C4⋊C4×C149C2311yescharacteristic
70C4⋊C4×C149C2312yes—
71C22⋊C4×C149—22no—
72C22⋊C4×C149—22no—
73C22⋊C4×C149D412yes—
74C42×C149—22no—
75C42×C149—21no—
76C42×C149C2311yescharacteristic
77C22×D149—21no—
78C22×D149—22no—
79C22×D149—44no—
80C22×D149—44no—
81C22×D149—82no—
82D596—22no—
83D596—21no—
84D596—22no—
85D596—44no—
86D596—42no—
87D596—81no—
88D596#4112yes—
89C22⋊D149—44no—
90C2×Dic149—21no—
91C4×D149—42no—
92#11924746—21no—
93#11924746—21no—
94#11924746—22no—
95C23×C149C2×D412yes—
96Q8×C149—21no—
97Q8×C149—21no—
98Q8×C149—42no—
99D4×C149—22no—
100C2×C4×C149—22no—
101C2×C4×C149—22no—
102C2×C4×C149—21no—
103C2×C4×C149—21no—
104C2×C4×C149—22no—
105C2×C4×C149C2×D412yes—
106C2×C4×C149C2×D411yescharacteristic
107C2×C4×C149C2×D411yescharacteristic
108C2×C4×C149C2411yescharacteristic
109C22×C149—22no—
110C22×C149—22no—
111C22×C149—22no—
112C22×C149C22×D411yescharacteristic
113C2×D149—22no—
114C2×D149—41no—
115C2×D149—41no—
116C2×D149—42no—
117C2×D149—42no—
118C2×D149—42no—
119C2×D149—88no—
120C4×C149—22no—
121C4×C149—21no—
122C4×C149—41no—
123C4×C149—42no—
124C4×C149—41no—
125C4×C149AGL(1,ℤ/8)12yes—
126C4×C149C2×#4111yescharacteristic
127C4×C149H(5,𝔽2)11yescharacteristic
128Dic149—41no—
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#556D14911yescharacteristic, complemented
138D42—1491no—
139#486—1491no—
140#482—1492no—
141#474—1491no—
142#468—1492no—
143#460—1491no—
144#450—1492no—
145C4×SD16—1491no—
146#436—1491no—
147#418—1492no—
148C22×D4—2981no—
149C22×D4—1492no—
150C2×SD16—1491no—
151C2×SD16—1492no—
152C2×M16—1492no—
153C4⋊Q8C2×D14911yescharacteristic, complemented
154#127—1491no—
155#126—22no—
156#124—21no—
157C22⋊Q8C2×D14912yescomplemented
158#121—2981no—
159#121—1492no—
160#120—2982no—
161C4×Q8—21no—
162C4×Q8C2×D14911yescharacteristic, complemented
163C4×D4—1491no—
164C4×D4C2×D14911yescharacteristic
165#117C2×D14912yescomplemented
166#107—1492no—
167#106—1491no—
168C4⋊C8—1491no—
169#103—1491no—
170#102—1491no—
171#102—1492no—
172#102—1492no—
173#102—1492no—
174#102—1492no—
175C22⋊C8—1492no—
176C4×C8—1491no—
177C24—2982no—
178C2×Q8—21no—
179C2×Q8C22×D14911yescharacteristic
180C2×D4—2984no—
181C2×D4—2982no—
182C2×D4—2982no—
183C2×D4—2981no—
184C2×D4—2981no—
185C2×D4—5962no—
186C2×D4—5964no—
187C2×D4—1492no—
188C2×D4—1491no—
189C2×D4C22⋊D14911yescharacteristic
190C22×C4—2981no—
191C22×C4C22×D14912yes—
192SD16—2982no—
193SD16—5962no—
194M16—5962no—
195C2×C8—1491no—
196C2×C8—1491no—
197C2×C8—1492no—
198C4⋊C4—22no—
199C4⋊C4—21no—
200C4⋊C4—21no—
201C4⋊C4—1491no—
202C4⋊C4C22⋊D14911yescharacteristic
203C4⋊C4C22×D14911yescharacteristic
204C4⋊C4C22×D14912yes—
205C22⋊C4—22no—
206C22⋊C4—22no—
207C22⋊C4—2981no—
208C22⋊C4—2982no—
209C22⋊C4C22⋊D14912yes—
210C42—22no—
211C42—21no—
212C42C22×D14911yescharacteristic
213C23—2981no—
214C23—2982no—
215C23—5964no—
216C23—5964no—
217C23—11922no—
218C23C2×C22⋊D14912yes—
219Q8—21no—
220Q8—21no—
221Q8—42no—
222D4—22no—
223D4—2982no—
224D4—2981no—
225D4—2982no—
226D4—5964no—
227D4—5964no—
228D4—5962no—
229D4—1492no—
230D4—11921no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—22no—
236C2×C4—2981no—
237C2×C4—5962no—
238C2×C4C2×C22⋊D14912yes—
239C2×C4C2×C22⋊D14911yescharacteristic
240C2×C4C2×C22⋊D14911yescharacteristic
241C2×C4C23×D14911yescharacteristic
242C8—2981no—
243C8—2981no—
244C8—2982no—
245C22—22no—
246C22—22no—
247C22—22no—
248C22—2982no—
249C22—5961no—
250C22—5961no—
251C22—5962no—
252C22—5962no—
253C22—5962no—
254C22—11928no—
255C22C22×C22⋊D14911yescharacteristic
256C4—21no—
257C4—22no—
258C4—42no—
259C4—41no—
260C4—41no—
261C4—5961no—
262C4#1801193612yes—
263C4#41×D14911yescharacteristic
264C4#1801195711yescharacteristic
265C2—42no—
266C2—5962no—
267C2—11922no—
268C2#2375026311yescharacteristic
269C2C2×#1801193611yescharacteristic
270C2#2375043311yescharacteristic
subgroupG/H
centerC22C22×C22⋊D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#556×C149C2
socleC22×C149C22×D4
prime radicalD4×D596C2
hypercenter#556D149
hypocenterC149#3324
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C4×D4 ◁ #556

Complements

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

NH ≅ G/Ntype
2. #23749973C2semidirect
3. #23749949C2semidirect
4. #23749902C2semidirect
5. #23749873C2semidirect
6. #23749825C2semidirect
7. #23749775C2semidirect
8. C4×#14558723C2semidirect
9. #23749703C2semidirect
10. #23749629C2semidirect
11. #556×C149C2semidirect
15. C2×#14558723C22semidirect
16. C2×#14558720C22semidirect
22. #18011844C22semidirect
23. #18011842C22semidirect
24. C4⋊#11924746C22semidirect
25. #18011835C22semidirect
28. #18011831C22semidirect
29. #18011831C22semidirect
30. #18011831C22semidirect
31. C22⋊#11924746C22semidirect
32. C4×#11924746C22semidirect
33. C4⋊Q8×C149C22semidirect
36. C22⋊Q8×C149C22semidirect
38. C4×Q8×C149C22semidirect
40. #117×C149C22semidirect
135. C149#3324semidirect
137. #556D149semidirect
153. C4⋊Q8C2×D149semidirect
157. C22⋊Q8C2×D149semidirect
162. C4×Q8C2×D149semidirect
165. #117C2×D149semidirect

Maximal subgroups

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

subgroupindex
D4×D5962
#237499732
#237499492
#237499022
#237498732
#237498252
#237497752
C4×#145587232
#237497032
#237496292
#556×C1492
#3324149