Finite Group Databasebeta

← #34057009

Subgroups — #34057009

Order 19072.

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

normal subgroups only

This group has 388 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#23750335C211yesmaximal, characteristic, complemented
2#23750279C211yesmaximal, characteristic
3#23749976C211yesmaximal, characteristic, complemented
4#23749957C212yesmaximal, complemented
5#23749915C211yesmaximal, characteristic, complemented
6#23749882C212yesmaximal, complemented
7#23749835C211yesmaximal, characteristic, complemented
8#23749784C212yesmaximal, complemented
9#23749704C211yesmaximal, characteristic, complemented
10#23749623C212yesmaximal, complemented
11C4×SD16×C149C211yesmaximal, characteristic, complemented
12C2×#14558735C2212yes—
13C2×#14558726C2212yes—
14C2×#14558719C2212yes—
15#18011911C2211yescharacteristic
16#18011911C2211yescharacteristic
17#18011903—21no—
18#18011895—22no—
19#18011891C2212yes—
20#18011889—22no—
21#18011887C2212yes—
22#18011885—21no—
23C4×D596—21no—
24C4×D596C2211yescharacteristic
25C2×C4⋊Dic149—21no—
26C4⋊C4×D149C2212yes—
27#18011842C2212yes—
28C4⋊#11924746C2211yescharacteristic
29#18011836C2212yes—
30#18011836C2212yes—
31#18011834C2212yes—
32#18011832C2212yes—
33#18011823C2212yes—
34C2×SD16×C149C2211yescharacteristic
35C4×Q8×C149C2211yescharacteristic
36C4×D4×C149C2211yescharacteristic
37#106×C149C2211yescharacteristic
38#103×C149C2211yescharacteristic
39#102×C149C2211yescharacteristic
40C4×C8×C149C2211yescharacteristic
41#14558735—24no—
42#14558735—42no—
43C2×Dic298—21no—
44C2×Dic298C2312yes—
45C2×D596D411yescharacteristic
46C2×C22⋊D149—22no—
47C22×Dic149—22no—
48C2×C4×D149—22no—
49C2×C4×D149C2312yes—
50#14558726—42no—
51#14558719—42no—
52C2×#11924746C2312yes—
53C4⋊Dic149—22no—
54C4⋊Dic149—21no—
55C4⋊Dic149—21no—
56C4⋊Dic149—21no—
57C4⋊Dic149D411yescharacteristic
58C4⋊Dic149C2312yes—
59#14558715—22no—
60#14558715—22no—
61#14558713—22no—
62#14558713—22no—
63#14558713D412yes—
64C2×Q8×C149C2311yescharacteristic
65C2×D4×C149C2311yescharacteristic
66C22×C4×C149—21no—
67SD16×C149—22no—
68C2×C8×C149C2311yescharacteristic
69C2×C8×C149C2311yescharacteristic
70C4⋊C4×C149—21no—
71C4⋊C4×C149C2311yescharacteristic
72C4⋊C4×C149C2311yescharacteristic
73C22⋊C4×C149—21no—
74C42×C149—21no—
75C42×C149C2311yescharacteristic
76C22×D149C2×D412yes—
77Dic298—22no—
78Dic298—22no—
79Dic298—81no—
80D596—22no—
81C22⋊D149—44no—
82C2×Dic149—22no—
83C2×Dic149—22no—
84C2×Dic149—44no—
85C2×Dic149C2×D412yes—
86C4×D149—22no—
87C4×D149—22no—
88C4×D149—82no—
89#11924746—22no—
90C23×C149—21no—
91Q8×C149—21no—
92Q8×C149—21no—
93D4×C149—21no—
94D4×C149#4112yes—
95C2×C4×C149—21no—
96C2×C4×C149—21no—
97C2×C4×C149—21no—
98C2×C4×C149—42no—
99C2×C4×C149C2×D411yescharacteristic
100C2×C4×C149C2×D411yescharacteristic
101C2×C4×C149C2411yescharacteristic
102C8×C149—21no—
103C8×C149—21no—
104C22×C149—22no—
105C22×C149—41no—
106C22×C149—41no—
107C22×C149C22×D411yescharacteristic
108C2×D149—22no—
109C2×D149—22no—
110C2×D149—22no—
111C4×C149—21no—
112C4×C149—41no—
113C4×C149—41no—
114C4×C149—41no—
115C4×C149#13712yes—
116C4×C149C2×#4111yescharacteristic
117C4×C149H(5,𝔽2)11yescharacteristic
118Dic149—22no—
119Dic149—42no—
120Dic149—42no—
121C2×C149—42no—
122C2×C149#54611yescharacteristic
123C2×C149C2×#13711yescharacteristic
124C2×C149#57711yescharacteristic
125D149—42no—
126C149#333111yescharacteristic, complemented, Sylow
127#3331—1491nomaximal, Sylow
128#554—1491no—
129#547—1491no—
130#487—1491no—
131#483—1492no—
132#476—1491no—
133#470—1492no—
134#462—1491no—
135#452—1492no—
136C4×SD16D14911yescharacteristic, complemented
137#436—1491no—
138#417—1492no—
139C2×#41—1492no—
140C2×Q16—1492no—
141C2×SD16C2×D14911yescharacteristic
142C2×M16—1492no—
143C4⋊Q8—1491no—
144C4⋊Q8—1491no—
145#125—2981no—
146#123—2982no—
147C22⋊Q8—2982no—
148C22⋊Q8—1492no—
149C22⋊Q8—1492no—
150#121—2981no—
151C4×Q8C2×D14911yescharacteristic
152C4×D4—2981no—
153C4×D4—1491no—
154C4×D4C2×D14911yescharacteristic
155C2×C4⋊C4—2981no—
156C2×C4⋊C4—1492no—
157#106—1492no—
158#106C2×D14911yescharacteristic
159C4⋊C8—1491no—
160#103—1492no—
161#103—1492no—
162#103—1492no—
163#103C2×D14911yescharacteristic
164#102—1492no—
165#102C2×D14911yescharacteristic
166C22⋊C8—1492no—
167C4×C8C2×D14911yescharacteristic
168#41—2984no—
169#41—5962no—
170C2×Q8—2981no—
171C2×Q8—1492no—
172C2×Q8C22×D14911yescharacteristic
173C2×D4—2982no—
174C2×D4—1491no—
175C2×D4C22×D14911yescharacteristic
176C22×C4—21no—
177C22×C4—2982no—
178C22×C4—2982no—
179C22×C4—1492no—
180Q16—5962no—
181SD16—22no—
182M16—5962no—
183C2×C8—1492no—
184C2×C8C22×D14911yescharacteristic
185C2×C8C22×D14911yescharacteristic
186C4⋊C4—21no—
187C4⋊C4—2982no—
188C4⋊C4—2982no—
189C4⋊C4—2982no—
190C4⋊C4—2981no—
191C4⋊C4—2981no—
192C4⋊C4—2981no—
193C4⋊C4—1492no—
194C4⋊C4—1491no—
195C4⋊C4C22×D14911yescharacteristic
196C4⋊C4C22×D14911yescharacteristic
197C22⋊C4—21no—
198C22⋊C4—2982no—
199C22⋊C4—2982no—
200C22⋊C4—1492no—
201C42—21no—
202C42C22×D14911yescharacteristic
203C23—21no—
204C23—1492no—
205Q8—21no—
206Q8—21no—
207Q8—2982no—
208Q8—2982no—
209Q8—11921no—
210D4—21no—
211D4—2982no—
212D4—5964no—
213D4#1455873612yes—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—42no—
218C2×C4—2982no—
219C2×C4—2982no—
220C2×C4—2982no—
221C2×C4—2982no—
222C2×C4—5964no—
223C2×C4—1492no—
224C2×C4—11922no—
225C2×C4D4×D14911yescharacteristic
226C2×C4D4×D14911yescharacteristic
227C2×C4C23×D14911yescharacteristic
228C8—21no—
229C8—21no—
230C8—2982no—
231C22—22no—
232C22—41no—
233C22—41no—
234C22—2982no—
235C22—2982no—
236C22—2982no—
237C22C2×D4×D14911yescharacteristic
238C4—21no—
239C4—41no—
240C4—41no—
241C4—41no—
242C4—2982no—
243C4—5962no—
244C4—5962no—
245C4#1801194012yes—
246C4C2×#1455873611yescharacteristic
247C4#1801195711yescharacteristic
248C2—42no—
249C2—5962no—
250C2#2375027111yescharacteristic
251C2C2×#1801194011yescharacteristic
252C2#2375043211yescharacteristic
subgroupG/H
centerC22C2×D4×D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
FittingC4×SD16×C149C2
socleC22×C149C22×D4
prime radical#23750279C2
hypercenterC4×SD16D149
hypocenterC149#3331
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. #23750335C2semidirect
3. #23749976C2semidirect
4. #23749957C2semidirect
5. #23749915C2semidirect
6. #23749882C2semidirect
7. #23749835C2semidirect
8. #23749784C2semidirect
9. #23749704C2semidirect
10. #23749623C2semidirect
11. C4×SD16×C149C2semidirect
126. C149#3331semidirect
136. C4×SD16D149semidirect

Maximal subgroups

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

subgroupindex
#237503352
#237502792
#237499762
#237499572
#237499152
#237498822
#237498352
#237497842
#237497042
#237496232
C4×SD16×C1492
#3331149