Finite Group Databasebeta

← #34053153

Subgroups — #34053153

Order 19072.

716 conjugacy classes of subgroups; only the 209 normal subgroups are shown.

normal subgroups only

This group has 716 conjugacy classes of subgroups — too many to draw. Only its 209 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
1C2×#18011958C211yesmaximal, characteristic
2#23750102C212yesmaximal, complemented
3#23749599C211yesmaximal, characteristic, complemented
4#23749598C212yesmaximal, complemented
5#23749597C218yesmaximal, complemented
6#410×C149C211yesmaximal, characteristic, complemented
7#18011958—24no—
8#18011958—24no—
9C2×#14558736C412yescomplemented
10C2×#14558736C2211yescharacteristic
11C2×#14558735C412yescomplemented
12C2×#14558735C2214yes—
13C22×Dic298C412yescomplemented
14C22×Dic298C2211yescharacteristic
15C2×Q8×D149C412yescomplemented
16C4×Dic298—24no—
17#18011876—24no—
18#18011870C2212yes—
19#18011867C2218yes—
20#18011826C2214yes—
21#18011825C2218yes—
22C2×#41×C149C2211yescharacteristic
23#117×C149C2212yes—
24#99×C149C2214yes—
25#14558736—24no—
26#14558736—24no—
27#14558736—22no—
28#14558736—22no—
29#14558735—28no—
30#14558735—216no—
31C2×Dic298—24no—
32C2×Dic298—24no—
33C2×Dic298C2×C414yes—
34C2×Dic298C2×C411yescharacteristic
35C2×Dic298C2×C412yes—
36C2×Dic298C2×C412yes—
37C2×Dic298D414yes—
38C2×Dic298D414yes—
39Q8×D149—28no—
40C2×D596C2×C411yescharacteristic
41C2×C22⋊D149C2×C414yes—
42C2×C22⋊D149C2312yes—
43C22×Dic149C2×C414yes—
44C22×Dic149C2312yes—
45C2×C4×D149C2×C412yes—
46C2×C4×D149C2×C414yes—
47C4⋊Dic149—22no—
48#14558715—28no—
49C22⋊Dic149C2314yes—
50C4×Dic149—28no—
51C4×Dic149—22no—
52#41×C149—22no—
53#41×C149—22no—
54C2×Q8×C149C2×C411yescharacteristic
55C2×D4×C149C2×C411yescharacteristic
56C2×D4×C149C2312yes—
57C22×C4×C149C2×C412yes—
58C22×C4×C149C2311yescharacteristic
59C4⋊C4×C149—22no—
60C22⋊C4×C149C2314yes—
61C42×C149—22no—
62C22×D149C22×C412yes—
63Dic298—21no—
64Dic298—21no—
65Dic298—28no—
66Dic298C22⋊C418yes—
67Dic298C22⋊C414yes—
68Dic298C22⋊C414yes—
69D596—21no—
70D596—21no—
71C22⋊D149—24no—
72C22⋊D149—28no—
73C2×Dic149—28no—
74C2×Dic149—24no—
75C2×Dic149C22×C412yes—
76C2×Dic149C22×C414yes—
77C2×Dic149C2×D418yes—
78C4×D149—24no—
79C4×D149—28no—
80C23×C149C22×C412yes—
81C23×C149C2411yescharacteristic
82Q8×C149—22no—
83D4×C149—24no—
84D4×C149—22no—
85C2×C4×C149—22no—
86C2×C4×C149—22no—
87C2×C4×C149—24no—
88C2×C4×C149C22×C412yes—
89C2×C4×C149C22×C411yescharacteristic
90C2×C4×C149C22×C411yescharacteristic
91C2×C4×C149C2×D412yes—
92C2×C4×C149C2×D412yes—
93C22×C149—22no—
94C22×C149—41no—
95C22×C149—42no—
96C22×C149C23×C411yescharacteristic
97C22×C149C22×D412yes—
98C2×D149—22no—
99C2×D149—42no—
100C4×C149—22no—
101C4×C149—44no—
102C4×C149C2×C22⋊C411yescharacteristic
103C4×C149C2×C22⋊C411yescharacteristic
104C4×C149C2×C22⋊C412yes—
105Dic149—22no—
106Dic149—44no—
107Dic149C2×C22⋊C418yes—
108C2×C149—22no—
109C2×C149—21no—
110C2×C149—42no—
111C2×C149C22×C22⋊C411yescharacteristic
112D149—42no—
113C149#288411yescharacteristic, complemented, Sylow
114#2884—1491nomaximal, Sylow
115C2×2−1+4—1491no—
116#519—1492no—
117#410—1491no—
118#410—1492no—
119#410—1498no—
120#410D14911yescharacteristic, complemented
1212−1+4—2984no—
1222−1+4—2984no—
123C2×#41—1492no—
124C2×#41—1494no—
125C2×#41—1492no—
126C2×#41—1491no—
127C2×#41C2×D14911yescharacteristic
128C22×Q8—1492no—
129C22×Q8—1492no—
130C22×Q8—1491no—
131C4×Q8—2984no—
132C4×Q8—2984no—
133#117—1492no—
134#117—1498no—
135#117C2×D14912yes—
136#99—1494no—
137#99—1498no—
138#99C2×D14914yes—
139#41—22no—
140#41—22no—
141#41—2984no—
142#41—2984no—
143#41—2988no—
144#41—29816no—
145#41—2982no—
146#41—2982no—
147C2×Q8—2984no—
148C2×Q8—2984no—
149C2×Q8—2988no—
150C2×Q8—1491no—
151C2×Q8—1494no—
152C2×Q8—1492no—
153C2×Q8—1492no—
154C2×Q8—1494no—
155C2×Q8—1494no—
156C2×Q8C4×D14911yescharacteristic
157C2×D4—1494no—
158C2×D4—1492no—
159C2×D4—1491no—
160C2×D4C4×D14911yescharacteristic
161C2×D4C22×D14912yes—
162C22×C4—1494no—
163C22×C4—1494no—
164C22×C4—1492no—
165C22×C4—1492no—
166C22×C4C4×D14912yes—
167C22×C4C22×D14911yescharacteristic
168C4⋊C4—22no—
169C4⋊C4—2988no—
170C4⋊C4—2982no—
171C22⋊C4—1494no—
172C22⋊C4C22×D14914yes—
173C42—22no—
174C42—2988no—
175C42—2982no—
176C23—1492no—
177C23C2×C4×D14912yes—
178C23C23×D14911yescharacteristic
179Q8—22no—
180Q8—2988no—
181Q8—2981no—
182Q8—2981no—
183Q8—1494no—
184Q8—1494no—
185Q8—1498no—
186D4—24no—
187D4—22no—
188D4—2984no—
189D4—2988no—
190D4—2981no—
191D4—2981no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—24no—
195C2×C4—2988no—
196C2×C4—2988no—
197C2×C4—2984no—
198C2×C4—2984no—
199C2×C4—1492no—
200C2×C4—1494no—
201C2×C4—1498no—
202C2×C4C2×C4×D14912yes—
203C2×C4C2×C4×D14911yescharacteristic
204C2×C4C2×C4×D14911yescharacteristic
205C2×C4D4×D14912yes—
206C2×C4D4×D14912yes—
207C22—22no—
208C22—41no—
209C22—42no—
210C22—2982no—
211C22—5962no—
212C22C22×C4×D14911yescharacteristic
213C22C2×D4×D14912yes—
214C4—22no—
215C4—44no—
216C4—2982no—
217C4—5964no—
218C4—1498no—
219C4C22⋊C4×D14911yescharacteristic
220C4C22⋊C4×D14911yescharacteristic
221C4C22⋊C4×D14912yes—
222C2—22no—
223C2—21no—
224C2—42no—
225C2—5962no—
226C2C2×C22⋊C4×D14911yescharacteristic
subgroupG/H
centerC2C2×C22⋊C4×D149
commutatorC22×C149C23×C4
FrattiniC23C23×D149
Fitting#410×C149C2
socleC2×C149C22×C22⋊C4
prime radicalC2×#18011958C2
cosocleC23×C149C24
hypercenter#410D149
hypocenterC149#2884
LCSG ▷ C22×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C2 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
2. #23750102C2semidirect
3. #23749599C2semidirect
4. #23749598C2semidirect
5. #23749597C2semidirect
6. #410×C149C2semidirect
9. C2×#14558736C4semidirect
11. C2×#14558735C4semidirect
13. C22×Dic298C4semidirect
15. C2×Q8×D149C4semidirect
113. C149#2884semidirect
120. #410D149semidirect

Maximal subgroups

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

subgroupindex
C2×#180119582
#237501022
#237495992
#237495982
#237495972
#410×C1492
#2884149