Finite Group Databasebeta

← #34055847

Subgroups — #34055847

Order 19072.

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

normal subgroups only

This group has 326 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#23750180C211yesmaximal, characteristic
2#23750180C211yesmaximal, characteristic
3#23750030C211yesmaximal, characteristic, complemented
4#23750021C211yesmaximal, characteristic, complemented
5#23749975C211yesmaximal, characteristic, complemented
6#23749965C214yesmaximal
7#23749934C214yesmaximal, complemented
8#23749676C211yesmaximal, characteristic
9#487×C149C211yesmaximal, characteristic, complemented
10#18011911C2211yescharacteristic
11#18011911C2211yescharacteristic
12#18011902—21no—
13#18011902C2212yes—
14#18011891—21no—
15#18011891—21no—
16#18011887—21no—
17#18011887C2212yes—
18Q8×Dic149—21no—
19Q8×Dic149C2212yes—
20#18011876—21no—
21#18011876—21no—
22#18011869—21no—
23#18011869C2212yes—
24C42×D149C2211yescharacteristic
25#18011845C2214yes—
26#18011842C2214yes—
27#18011840C2212yes—
28#18011835C2214yes—
29#18011823C2212yes—
30C4×#11924746C2211yescharacteristic
31C4⋊Q8×C149C2211yescharacteristic
32C4⋊Q8×C149C2211yescharacteristic
33#103×C149C2214yes—
34C4×C8×C149C2211yescharacteristic
35C2×Dic298—21no—
36C2×Dic298—21no—
37C2×C4×D149D411yescharacteristic
38C2×C4×D149D411yescharacteristic
39C2×C4×D149C2311yescharacteristic
40C2×#11924746C2312yes—
41C4⋊Dic149—22no—
42C4⋊Dic149—22no—
43C4⋊Dic149—21no—
44C4⋊Dic149—21no—
45C4⋊Dic149C2312yes—
46C4⋊Dic149C2312yes—
47#14558715—22no—
48#14558715—22no—
49#14558713—22no—
50#14558713—22no—
51C4×Dic149—22no—
52C4×Dic149—22no—
53C4×Dic149D411yescharacteristic
54C4×Dic149D411yescharacteristic
55C4×Dic149C2311yescharacteristic
56C2×Q8×C149—21no—
57C2×Q8×C149C2312yes—
58C2×C8×C149C2312yes—
59C4⋊C4×C149—21no—
60C4⋊C4×C149—21no—
61C4⋊C4×C149—21no—
62C4⋊C4×C149C2312yes—
63C42×C149C2311yescharacteristic
64C22×D149C2×D411yescharacteristic
65Dic298—42no—
66Dic298—42no—
67C2×Dic149—22no—
68C2×Dic149—22no—
69C2×Dic149C2×D411yescharacteristic
70C2×Dic149C2×D411yescharacteristic
71C2×Dic149C2×D411yescharacteristic
72C4×D149—22no—
73C4×D149—22no—
74C4×D149—22no—
75#11924746—22no—
76Q8×C149—24no—
77Q8×C149—42no—
78C2×C4×C149—22no—
79C2×C4×C149—22no—
80C2×C4×C149C2×D411yescharacteristic
81C2×C4×C149C2×D411yescharacteristic
82C2×C4×C149C2411yescharacteristic
83C8×C149—22no—
84C22×C149C22×D411yescharacteristic
85C2×D149—21no—
86C2×D149—21no—
87C2×D149—21no—
88C4×C149—42no—
89C4×C149—42no—
90C4×C149#13512yes—
91C4×C149#13712yes—
92C4×C1492−1+412yes—
93Dic149—21no—
94Dic149—21no—
95Dic149—21no—
96Dic149—42no—
97Dic149—42no—
98C2×C149#53611yescharacteristic
99C2×C149C2×#13511yescharacteristic
100C2×C149C2×#13711yescharacteristic
101D149—41no—
102C149#323111yescharacteristic, complemented, Sylow
103#3231—1491nomaximal, Sylow
104#533—1491no—
105#533—1491no—
106#500—1491no—
107#498—1491no—
108#487—1491no—
109#487D14911yescharacteristic, complemented
110#485—1494no—
111#479—1494no—
112#431—1491no—
113C4⋊Q8—1491no—
114C4⋊Q8—1491no—
115C4⋊Q8C2×D14911yescharacteristic
116C4⋊Q8C2×D14911yescharacteristic
117#125—2981no—
118#125—1492no—
119C22⋊Q8—2981no—
120C22⋊Q8—2981no—
121C22⋊Q8—2981no—
122C22⋊Q8—1492no—
123C4×Q8—2981no—
124C4×Q8—2981no—
125C4×Q8—2981no—
126C4×Q8—1492no—
127#117—2981no—
128#117—1492no—
129C2×C42—1491no—
130#107—1494no—
131#106—1494no—
132C4⋊C8—1492no—
133#103—1494no—
134#103C2×D14914yes—
135C22⋊C8—1492no—
136C4×C8—1491no—
137C4×C8C2×D14911yescharacteristic
138C2×Q8—21no—
139C2×Q8—2981no—
140C2×Q8—2981no—
141C2×Q8C22×D14912yes—
142C22×C4—1491no—
143C22×C4—1491no—
144C22×C4—1491no—
145C2×C8—1492no—
146C2×C8C22×D14912yes—
147C4⋊C4—21no—
148C4⋊C4—21no—
149C4⋊C4—21no—
150C4⋊C4—2982no—
151C4⋊C4—2982no—
152C4⋊C4—2982no—
153C4⋊C4—2982no—
154C4⋊C4—2981no—
155C4⋊C4—2981no—
156C4⋊C4—1492no—
157C4⋊C4—1492no—
158C4⋊C4C22×D14912yes—
159C22⋊C4—2982no—
160C22⋊C4—2982no—
161C42—2982no—
162C42—2982no—
163C42—1491no—
164C42—1491no—
165C42—1491no—
166C42C22×D14911yescharacteristic
167C23—1491no—
168Q8—24no—
169Q8—42no—
170Q8—5962no—
171Q8—5962no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—2982no—
175C2×C4—2982no—
176C2×C4—2982no—
177C2×C4—2982no—
178C2×C4—2982no—
179C2×C4—1491no—
180C2×C4—1491no—
181C2×C4—1491no—
182C2×C4D4×D14911yescharacteristic
183C2×C4D4×D14911yescharacteristic
184C2×C4C23×D14911yescharacteristic
185C8—22no—
186C8—2982no—
187C22—2981no—
188C22—2981no—
189C22—2981no—
190C22C2×D4×D14911yescharacteristic
191C4—42no—
192C4—42no—
193C4—2981no—
194C4—2981no—
195C4—2981no—
196C4—5962no—
197C4—5962no—
198C4#1801193412yes—
199C4#1801194012yes—
200C4#1801195812yes—
201C2—5961no—
202C2#2375019511yescharacteristic
203C2C2×#1801193411yescharacteristic
204C2C2×#1801194011yescharacteristic
subgroupG/H
centerC22C2×D4×D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#487×C149C2
socleC22×C149C22×D4
prime radicalC22×D149C2×D4
hypercenter#487D149
hypocenterC149#3231
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
3. #23750030C2semidirect
4. #23750021C2semidirect
5. #23749975C2semidirect
7. #23749934C2semidirect
9. #487×C149C2semidirect
102. C149#3231semidirect
109. #487D149semidirect

Maximal subgroups

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

subgroupindex
#237501802
#237500302
#237500212
#237499752
#237499652
#237499342
#237496762
#487×C1492
#3231149