Finite Group Databasebeta

← #34060734

Subgroups — #34060734

Order 19072.

344 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 66 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#23750703C211yesmaximal, characteristic, complemented
2#23750617C212yesmaximal, complemented
3#23750605C211yesmaximal, characteristic, complemented
4#23750492C212yesmaximal, complemented
5#23750188C211yesmaximal, characteristic
6#18011998—24no—
7D149⋊M16C2212yes—
8C2×#14558745—22no—
9#18011990C2212yes—
10#18011973—24no—
11#18011969—22no—
12C2×#14558735C2211yescharacteristic
13C2×D4×D149C2211yescharacteristic
14#18011907C411yescharacteristic, complemented
15#18011899C411yescharacteristic, complemented
16#18011898C411yescharacteristic, complemented
17#18011883—22no—
18#18011879—22no—
19#18011868C2211yescharacteristic
20#127×C149C411yescharacteristic, complemented
21#14558746—22no—
22#14558745—24no—
23#14558745—22no—
24D149⋊C8D414yescomplemented
25C2×#11924753—24no—
26C4⋊#10597010—22no—
27C22⋊#10597010—22no—
28C4×#10597010—22no—
29C23×D149—21no—
30#14558735—22no—
31#14558735—22no—
32C2×Dic298C2×C411yescharacteristic
33C2×D596C2×C411yescharacteristic
34C2×C22⋊D149—21no—
35C2×C22⋊D149—22no—
36C2×C22⋊D149D412yes—
37D4×D149—22no—
38D4×D149—22no—
39C22×Dic149D412yes—
40C2×C4×D149C2311yescharacteristic
41#14558715—21no—
42C22⋊Dic149—22no—
43#14558713—21no—
44#14558713—22no—
45C4×Dic149C2×C411yescharacteristic
46C2×D4×C149—22no—
47C2×D4×C149C2×C411yescharacteristic
48C2×D4×C149C2×C411yescharacteristic
49C42×C149C2×C411yescharacteristic
50C2×#10597010—42no—
51#11924753—24no—
52C22×D149—21no—
53C22×D149—42no—
54C22×D149—82no—
55C22×D149C2×D411yescharacteristic
56Dic298—22no—
57D596—22no—
58C22⋊D149—24no—
59C22⋊D149—41no—
60C22⋊D149—41no—
61C22⋊D149—82no—
62C2×Dic149—22no—
63C2×Dic149—22no—
64C2×Dic149—21no—
65C2×Dic149C22⋊C412yes—
66C2×Dic149C2×D411yescharacteristic
67C4×D149C2×D412yes—
68C4×D149#4112yes—
69C23×C149—21no—
70C23×C149C22⋊C412yes—
71D4×C149—22no—
72D4×C149—22no—
73D4×C149—82no—
74C2×C4×C149—21no—
75C2×C4×C149C22×C411yescharacteristic
76C22×C149—22no—
77C22×C149—42no—
78C22×C149—42no—
79C22×C149—41no—
80C22×C149C2×C22⋊C411yescharacteristic
81C2×D149—41no—
82C2×D149—42no—
83C2×D149—41no—
84C2×D149—84no—
85C2×D149#12112yes—
86#10597010—82no—
87C4×C149—41no—
88C4×C149C4×D412yes—
89Dic149—22no—
90Dic149—41no—
91Dic149#12011yescharacteristic
92Dic149#12311yescharacteristic
93C2×C149—21no—
94C2×C149—42no—
95C2×C149—81no—
96C2×C149#38611yescharacteristic
97D149—22no—
98D149—81no—
99C149#191011yescharacteristic, complemented, Sylow
100#1910—1491nomaximal, Sylow
101#568—1491no—
102#535—1491no—
103#418—1492no—
104#410—1491no—
105H(3,ℤ/4)—1492no—
106C2×#41—1491no—
107C22×D4—1491no—
108#131—2984no—
109C2×M16—2982no—
110C2×M16—1492no—
111#127—1491no—
112#127#1059701011yescharacteristic, complemented
113#124—1491no—
114#124—1491no—
115#121—2982no—
116#120—2982no—
117#117—1492no—
118#117—1491no—
119#102—2984no—
120#99—2982no—
121C24—2981no—
122#41—2982no—
123#41—2982no—
124C2×Q8—1491no—
125C2×D4—22no—
126C2×D4—2982no—
127C2×D4—2982no—
128C2×D4—2982no—
129C2×D4—2981no—
130C2×D4—1492no—
131C2×D4—1491no—
132C2×D4C2×#1059701011yescharacteristic
133C2×D4C2×#1059701011yescharacteristic
134C22×C4—1492no—
135C22×C4—1491no—
136M16—2984no—
137M16—2982no—
138M16—2982no—
139C2×C8—2984no—
140C2×C8—1494no—
141C4⋊C4—2981no—
142C4⋊C4—2982no—
143C22⋊C4—2981no—
144C22⋊C4—2982no—
145C22⋊C4—2982no—
146C22⋊C4—2982no—
147C42—2982no—
148C42—1491no—
149C42C2×#1059701011yescharacteristic
150C23—21no—
151C23—2981no—
152C23—5962no—
153C23—1491no—
154C23—11922no—
155C23C22⋊#1059701012yes—
156Q8—2982no—
157D4—22no—
158D4—22no—
159D4—82no—
160D4—2984no—
161D4—2982no—
162D4—5961no—
163D4—5961no—
164D4—11922no—
165C2×C4—21no—
166C2×C4—2982no—
167C2×C4—2982no—
168C2×C4—2981no—
169C2×C4—5962no—
170C2×C4—1492no—
171C2×C4—1491no—
172C2×C4—1492no—
173C2×C4—1492no—
174C2×C4C22×#1059701011yescharacteristic
175C8—2984no—
176C22—22no—
177C22—42no—
178C22—42no—
179C22—41no—
180C22—5961no—
181C22—5962no—
182C22—5961no—
183C22—1492no—
184C22—11924no—
185C22C2×C22⋊#1059701011yescharacteristic
186C4—41no—
187C4—2982no—
188C4—5961no—
189C4—1491no—
190C4—1491no—
191C4—11922no—
192C4D4×#1059701012yes—
193C2—21no—
194C2—42no—
195C2—81no—
196C2—2982no—
197C2—11921no—
198C2#2375057311yescharacteristic
subgroupG/H
centerC2#23750573
commutatorC2×C4×C149C22×C4
FrattiniC2×C4C22×#10597010
Fitting#127×C149C4
socleC2×C149#386
prime radical#23750188C2
cosocleC2×C4×D149C23
hypercenter#127#10597010
hypocenterC149#1910
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C2 ◁ C2×C4 ◁ #127

Complements

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

NH ≅ G/Ntype
1. #23750703C2semidirect
2. #23750617C2semidirect
3. #23750605C2semidirect
4. #23750492C2semidirect
14. #18011907C4semidirect
15. #18011899C4semidirect
16. #18011898C4semidirect
20. #127×C149C4semidirect
24. D149⋊C8D4semidirect
99. C149#1910semidirect
112. #127#10597010semidirect

Maximal subgroups

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

subgroupindex
#237507032
#237506172
#237506052
#237504922
#237501882
#1910149