Finite Group Databasebeta

← #34047274

Subgroups — #34047274

Order 19072.

784 conjugacy classes of subgroups; only the 229 normal subgroups are shown.

normal subgroups only

This group has 784 conjugacy classes of subgroups — too many to draw. Only its 229 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
1C22×Q8×D149C211yesmaximal, characteristic
2C2×#18011869C211yesmaximal, characteristic
3C2×#18011868C211yesmaximal, characteristic
4#23749532C211yesmaximal, characteristic, complemented
5#23749531C211yesmaximal, characteristic, complemented
6#23749530C211yesmaximal, characteristic, complemented
7#23749529C214yesmaximal, complemented
8#23749487C214yesmaximal
9#391×C149C211yesmaximal, characteristic, complemented
10C22×Dic298C2212yes—
11C22×Dic298C2211yescharacteristic
12C2×Q8×D149—24no—
13C2×Q8×D149—24no—
14C2×Q8×D149C418yescomplemented
15C22×C4×D149C2212yes—
16C22×C4×D149C2211yescharacteristic
17#18011869—24no—
18#18011868—24no—
19C2×C4⋊Dic149C2211yescharacteristic
20C2×#14558715C2212yes—
21C2×#14558713C2212yes—
22C2×#14558713C2212yes—
23C2×C4×Dic149C2212yes—
24C2×C4×Dic149C2211yescharacteristic
25#18011818C2214yes—
26#18011818C2214yes—
27#18011818C2214yes—
28C22×Q8×C149C2211yescharacteristic
29C2×C4⋊C4×C149C2211yescharacteristic
30C2×C42×C149C2211yescharacteristic
31#95×C149C2214yes—
32C23×D149C2311yescharacteristic
33C2×Dic298—24no—
34C2×Dic298—24no—
35C2×Dic298—22no—
36C2×Dic298—22no—
37C2×Dic298C2×C418yes—
38C2×Dic298C2×C414yes—
39Q8×D149—232no—
40C22×Dic149C2312yes—
41C22×Dic149C2312yes—
42C22×Dic149C2312yes—
43C22×Dic149C2311yescharacteristic
44C2×C4×D149—24no—
45C2×C4×D149—24no—
46C2×C4×D149C2×C418yes—
47C2×C4×D149C2×C414yes—
48C2×C4×D149D414yes—
49C2×C4×D149D414yes—
50C4⋊Dic149—22no—
51#14558715—24no—
52#14558713—24no—
53#14558713—24no—
54C4×Dic149—24no—
55C4×Dic149—22no—
56C2×Q8×C149—22no—
57C2×Q8×C149—22no—
58C2×Q8×C149C2×C414yes—
59C22×C4×C149C2312yes—
60C22×C4×C149C2312yes—
61C22×C4×C149C2312yes—
62C22×C4×C149C2311yescharacteristic
63C4⋊C4×C149—22no—
64C42×C149—22no—
65C22×D149—22no—
66C22×D149—22no—
67C22×D149C22×C412yes—
68C22×D149C2×D412yes—
69C22×D149C2×D412yes—
70Dic298—28no—
71Dic298—216no—
72C2×Dic149—22no—
73C2×Dic149—22no—
74C2×Dic149—22no—
75C2×Dic149—22no—
76C2×Dic149—22no—
77C2×Dic149—22no—
78C2×Dic149—24no—
79C2×Dic149C22×C414yes—
80C2×Dic149C22×C412yes—
81C2×Dic149C2×D412yes—
82C2×Dic149C2×D412yes—
83C4×D149—216no—
84C4×D149C22⋊C4116yes—
85C23×C149C2411yescharacteristic
86Q8×C149—28no—
87C2×C4×C149—22no—
88C2×C4×C149—22no—
89C2×C4×C149—22no—
90C2×C4×C149—22no—
91C2×C4×C149—22no—
92C2×C4×C149—24no—
93C2×C4×C149—22no—
94C2×C4×C149C22×C414yes—
95C2×C4×C149C22×C412yes—
96C2×C4×C149C2×D412yes—
97C2×C4×C149C2×D412yes—
98C22×C149C23×C411yescharacteristic
99C22×C149C22×D411yescharacteristic
100C22×C149C22×D411yescharacteristic
101C22×C1492−1+412yes—
102C22×C1492−1+412yes—
103C2×D149—24no—
104C2×D149—22no—
105C2×D149—24no—
106C2×D149—22no—
107C2×D149C2×C22⋊C414yes—
108C4×C149—24no—
109C4×C149—42no—
110C4×C149—42no—
111C4×C149C2×C22⋊C414yes—
112Dic149—24no—
113Dic149—42no—
114Dic149—42no—
115Dic149C2×C22⋊C414yes—
116C2×C149C22×C22⋊C411yescharacteristic
117C2×C149#51912yes—
118C2×C149#53611yescharacteristic
119C2×C149#53612yes—
120C2×C149#53611yescharacteristic
121D149—24no—
122C149#231011yescharacteristic, complemented, Sylow
123#2310—1491nomaximal, Sylow
124C23×Q8—1491no—
125C2×#117—1491no—
126C2×#117—1491no—
127#391—1491no—
128#391—1491no—
129#391—1491no—
130#391—1494no—
131#391D14911yescharacteristic, complemented
132#381—1494no—
133C22×Q8—2984no—
134C22×Q8—2984no—
135C22×Q8—1498no—
136C22×Q8—1491no—
137C22×Q8—1492no—
138C22×Q8C2×D14911yescharacteristic
139C23×C4—1492no—
140C23×C4—1491no—
141#117—2984no—
142#117—2984no—
143C2×C4⋊C4—1492no—
144C2×C4⋊C4—1491no—
145C2×C4⋊C4C2×D14911yescharacteristic
146C2×C22⋊C4—1492no—
147C2×C22⋊C4—1492no—
148C2×C42—1492no—
149C2×C42—1491no—
150C2×C42C2×D14911yescharacteristic
151#95—1494no—
152#95—1494no—
153#95—1494no—
154#95C2×D14914yes—
155C24—1491no—
156C2×Q8—22no—
157C2×Q8—22no—
158C2×Q8—2984no—
159C2×Q8—2984no—
160C2×Q8—29832no—
161C2×Q8—2982no—
162C2×Q8—2982no—
163C2×Q8—1494no—
164C2×Q8—1498no—
165C2×Q8C4×D14914yes—
166C22×C4—2984no—
167C22×C4—2984no—
168C22×C4—1492no—
169C22×C4—1492no—
170C22×C4—1492no—
171C22×C4—1498no—
172C22×C4—1491no—
173C22×C4—1494no—
174C22×C4—1494no—
175C22×C4—1494no—
176C22×C4C22×D14912yes—
177C22×C4C22×D14912yes—
178C22×C4C22×D14912yes—
179C22×C4C22×D14911yescharacteristic
180C4⋊C4—22no—
181C4⋊C4—2984no—
182C4⋊C4—2982no—
183C22⋊C4—2984no—
184C22⋊C4—2984no—
185C42—22no—
186C42—2984no—
187C42—2982no—
188C23—2982no—
189C23—2982no—
190C23—1492no—
191C23—1492no—
192C23—1492no—
193C23C23×D14911yescharacteristic
194Q8—28no—
195Q8—29816no—
196Q8—2988no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—24no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—29816no—
205C2×C4—2982no—
206C2×C4—2982no—
207C2×C4—2982no—
208C2×C4—2982no—
209C2×C4—2982no—
210C2×C4—2982no—
211C2×C4—2984no—
212C2×C4—1494no—
213C2×C4—1492no—
214C2×C4—1492no—
215C2×C4—1492no—
216C2×C4—14916no—
217C2×C4C2×C4×D14914yes—
218C2×C4C2×C4×D14912yes—
219C2×C4D4×D14912yes—
220C2×C4D4×D14912yes—
221C22—2982no—
222C22—2984no—
223C22—2982no—
224C22—2984no—
225C22—1494no—
226C22C22×C4×D14911yescharacteristic
227C22C2×D4×D14911yescharacteristic
228C22C2×D4×D14911yescharacteristic
229C22#1801195812yes—
230C22#1801195912yes—
231C4—24no—
232C4—42no—
233C4—42no—
234C4—2984no—
235C4—5962no—
236C4—5962no—
237C4—1494no—
238C4C22⋊C4×D14914yes—
239C2—2984no—
240C2C2×C22⋊C4×D14911yescharacteristic
241C2#2375010112yes—
242C2#2375019511yescharacteristic
243C2#2375019812yes—
244C2#2375020011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D149
commutatorC22×C149C23×C4
Fitting#391×C149C2
socle, cosocleC23×C149C24
prime radicalC23×D149C23
hypercenter#391D149
hypocenterC149#2310
LCSG ▷ C22×C149 ▷ C149
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
4. #23749532C2semidirect
5. #23749531C2semidirect
6. #23749530C2semidirect
7. #23749529C2semidirect
9. #391×C149C2semidirect
14. C2×Q8×D149C4semidirect
122. C149#2310semidirect
131. #391D149semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×D1492
C2×#180118692
C2×#180118682
#237495322
#237495312
#237495302
#237495292
#237494872
#391×C1492
#2310149