Finite Group Databasebeta

← #34055558

Subgroups — #34055558

Order 19072.

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

normal subgroups only

This group has 410 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#23750189C211yesmaximal, characteristic
2#23749963C212yesmaximal, complemented
3#23749949C212yesmaximal, complemented
4#23749873C212yesmaximal, complemented
5#23749856C212yesmaximal, complemented
6#23749775C212yesmaximal, complemented
7#23749764C212yesmaximal, complemented
8#23749592C211yesmaximal, characteristic, complemented
9#536×C149C211yesmaximal, characteristic, complemented
10C22×D596C2211yescharacteristic
11C2×#14558723C2212yes—
12C2×#14558723C2212yes—
13C2×#14558720C2211yescharacteristic, complemented
14C22×#11924746C2211yescharacteristic, complemented
15#18011907—21no—
16#18011900—21no—
17#18011885C2212yes—
18#18011885C2212yes—
19#18011879—21no—
20#18011879—21no—
21#18011870C2211yescharacteristic
22#18011844C2212yescomplemented
23#18011842C2212yescomplemented
24#18011835C2212yescomplemented
25#18011835C2212yescomplemented
26#18011831C2212yes—
27#18011831C2212yes—
28C22⋊#11924746C2212yes—
29C22⋊#11924746C2212yescomplemented
30C2×#41×C149C2211yescharacteristic
31C22×Q8×C149C2211yescharacteristic, complemented
32C4⋊Q8×C149—21no—
33#124×C149—21no—
34#123×C149—21no—
35#123×C149—21no—
36C22⋊Q8×C149C2212yes—
37C22⋊Q8×C149C2212yescomplemented
38#117×C149C2211yescharacteristic, complemented
39C23×D149—21no—
40C2×D596—21no—
41C2×D596—41no—
42C2×D596—41no—
43C2×D596C2312yes—
44C2×C22⋊D149—22no—
45C2×C22⋊D149—22no—
46#14558723—42no—
47#14558723—42no—
48#14558720—41no—
49C2×#11924746—41no—
50C2×#11924746C2312yes—
51C2×#11924746C2312yes—
52C4⋊Dic149C2312yes—
53C22⋊Dic149—21no—
54#14558713—22no—
55#14558713—22no—
56C4×Dic149—21no—
57#41×C149—41no—
58#41×C149—41no—
59C2×Q8×C149—21no—
60C2×Q8×C149—41no—
61C2×Q8×C149—41no—
62C2×Q8×C149D411yescharacteristic
63C2×Q8×C149C2312yes—
64C2×D4×C149D411yescharacteristic
65C2×D4×C149D411yescharacteristic
66C2×D4×C149D411yescharacteristic
67C22×C4×C149—21no—
68C22×C4×C149C2311yescharacteristic
69C22×C4×C149C2311yescharacteristic
70C22×C4×C149C2311yescharacteristic
71C4⋊C4×C149—22no—
72C4⋊C4×C149—22no—
73C4⋊C4×C149C2312yes—
74C22⋊C4×C149—21no—
75C22⋊C4×C149—22no—
76C22⋊C4×C149—22no—
77C42×C149—21no—
78C22×D149—22no—
79C22×D149—42no—
80C22×D149—82no—
81D596—22no—
82D596—22no—
83D596—42no—
84C22⋊D149—82no—
85C22⋊D149—82no—
86C2×Dic149—22no—
87#11924746—22no—
88#11924746—22no—
89C23×C149C2×D411yescharacteristic
90C23×C149C2×D411yescharacteristic
91C23×C149C2×D411yescharacteristic
92Q8×C149—22no—
93Q8×C149—22no—
94Q8×C149—41no—
95Q8×C149—42no—
96D4×C149—41no—
97D4×C149—41no—
98D4×C149—41no—
99C2×C4×C149—21no—
100C2×C4×C149—21no—
101C2×C4×C149—21no—
102C2×C4×C149—21no—
103C2×C4×C149—21no—
104C2×C4×C149—21no—
105C2×C4×C149—22no—
106C2×C4×C149—22no—
107C2×C4×C149—42no—
108C2×C4×C149C2×D411yescharacteristic
109C2×C4×C149C2×D411yescharacteristic
110C2×C4×C149C2×D411yescharacteristic
111C2×C4×C149C2411yescharacteristic
112C22×C149—21no—
113C22×C149—21no—
114C22×C149—21no—
115C22×C149—22no—
116C22×C149—22no—
117C22×C149—22no—
118C22×C149C22×D411yescharacteristic
119C2×D149—42no—
120C2×D149—42no—
121C2×D149—42no—
122C2×D149—84no—
123C4×C149—21no—
124C4×C149—21no—
125C4×C149—21no—
126C4×C149—42no—
127C4×C149—42no—
128C4×C149H(5,𝔽2)11yescharacteristic
129C4×C149H(5,𝔽2)11yescharacteristic
130Dic149—42no—
131C2×C149—41no—
132C2×C149—41no—
133C2×C149—41no—
134C2×C149#53411yescharacteristic
135C2×C149#57712yes—
136D149—82no—
137C149#320011yescharacteristic, complemented, Sylow
138#3200—1491nomaximal, Sylow
139#536D14911yescharacteristic, complemented
140#535—1491no—
141#485—1492no—
142#482—1492no—
143#468—1492no—
144#465—1492no—
145#450—1492no—
146#448—1492no—
147#408—1491no—
148C2×#41C2×D14911yescharacteristic
149C22×Q8C2×D14911yescharacteristic, complemented
150C22×D4—1491no—
151C2×SD16—1492no—
152C2×SD16—1492no—
153C2×M16—1491no—
154C22×C8—1491no—
155C4⋊Q8—21no—
156#127—2981no—
157#124—21no—
158#124—2981no—
159#123—21no—
160#123—21no—
161C22⋊Q8C2×D14912yes—
162C22⋊Q8C2×D14912yescomplemented
163#121—1492no—
164#121—1492no—
165#120—2981no—
166#120—2981no—
167#117—1491no—
168#117C2×D14911yescharacteristic, complemented
169#107—1492no—
170#106—1492no—
171#103—1492no—
172#103—1492no—
173#102—1492no—
174#102—1492no—
175C22⋊C8—1492no—
176C22⋊C8—1492no—
177C24—2981no—
178#41—41no—
179#41—41no—
180C2×Q8—21no—
181C2×Q8—41no—
182C2×Q8—41no—
183C2×Q8C22⋊D14911yescharacteristic
184C2×Q8C22×D14912yes—
185C2×D4—2982no—
186C2×D4—2982no—
187C2×D4—2981no—
188C2×D4—5961no—
189C2×D4—5961no—
190C2×D4—1492no—
191C2×D4C22⋊D14911yescharacteristic
192C2×D4C22⋊D14911yescharacteristic
193C2×D4C22⋊D14911yescharacteristic
194C22×C4—21no—
195C22×C4C22×D14911yescharacteristic
196C22×C4C22×D14911yescharacteristic
197C22×C4C22×D14911yescharacteristic
198SD16—5962no—
199SD16—5962no—
200M16—5961no—
201C2×C8—5961no—
202C2×C8—1492no—
203C2×C8—1492no—
204C4⋊C4—22no—
205C4⋊C4—22no—
206C4⋊C4—1492no—
207C4⋊C4C22×D14912yes—
208C22⋊C4—21no—
209C22⋊C4—22no—
210C22⋊C4—22no—
211C22⋊C4—2981no—
212C22⋊C4—2982no—
213C22⋊C4—2982no—
214C42—21no—
215C42—2981no—
216C23—2982no—
217C23—5962no—
218C23—11922no—
219C23C2×C22⋊D14911yescharacteristic
220C23C2×C22⋊D14911yescharacteristic
221C23C2×C22⋊D14911yescharacteristic
222Q8—22no—
223Q8—22no—
224Q8—41no—
225Q8—42no—
226D4—41no—
227D4—41no—
228D4—41no—
229D4—2982no—
230D4—2982no—
231D4—5962no—
232D4—11922no—
233D4—11922no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—42no—
243C2×C4—2982no—
244C2×C4C2×C22⋊D14911yescharacteristic
245C2×C4C2×C22⋊D14911yescharacteristic
246C2×C4C2×C22⋊D14911yescharacteristic
247C2×C4C23×D14911yescharacteristic
248C8—2982no—
249C8—2982no—
250C22—22no—
251C22—22no—
252C22—22no—
253C22—21no—
254C22—21no—
255C22—21no—
256C22—5962no—
257C22—5962no—
258C22—5962no—
259C22—11924no—
260C22C22×C22⋊D14911yescharacteristic
261C4—21no—
262C4—21no—
263C4—21no—
264C4—42no—
265C4—42no—
266C4—5962no—
267C4#1801195611yescharacteristic
268C4#1801195611yescharacteristic
269C2—41no—
270C2—41no—
271C2—41no—
272C2—11922no—
273C2#2375018411yescharacteristic
274C2#2375043312yes—
subgroupG/H
centerC22C22×C22⋊D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#536×C149C2
socleC22×C149C22×D4
prime radical#23750189C2
hypercenter#536D149
hypocenterC149#3200
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C2×#41 ◁ #536

Complements

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

NH ≅ G/Ntype
2. #23749963C2semidirect
3. #23749949C2semidirect
4. #23749873C2semidirect
5. #23749856C2semidirect
6. #23749775C2semidirect
7. #23749764C2semidirect
8. #23749592C2semidirect
9. #536×C149C2semidirect
13. C2×#14558720C22semidirect
14. C22×#11924746C22semidirect
22. #18011844C22semidirect
23. #18011842C22semidirect
24. #18011835C22semidirect
25. #18011835C22semidirect
29. C22⋊#11924746C22semidirect
31. C22×Q8×C149C22semidirect
37. C22⋊Q8×C149C22semidirect
38. #117×C149C22semidirect
137. C149#3200semidirect
139. #536D149semidirect
149. C22×Q8C2×D149semidirect
162. C22⋊Q8C2×D149semidirect
168. #117C2×D149semidirect

Maximal subgroups

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

subgroupindex
#237501892
#237499632
#237499492
#237498732
#237498562
#237497752
#237497642
#237495922
#536×C1492
#3200149