Finite Group Databasebeta

← #34060971

Subgroups — #34060971

Order 19072.

270 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 52 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#23750606C212yesmaximal, complemented
2#23750530C212yesmaximal, complemented
3#23750529C212yesmaximal, complemented
4#23750193C211yesmaximal, characteristic
5#18011990—22no—
6#18011970—22no—
7#18011970C2214yescomplemented
8C2×#14558737C2211yescharacteristic
9C2×D4×D149C411yescharacteristic, complemented
10#18011908C2211yescharacteristic
11#18011907C2211yescharacteristic
12#18011900C411yescharacteristic, complemented
13#18011884—22no—
14#18011881—22no—
15#18011868C411yescharacteristic, complemented
16#124×C149C411yescharacteristic, complemented
17C4⋊#10597010—42no—
18C22⋊#10597010—24no—
19C4×#10597010—42no—
20C23×D149—21no—
21#14558737—42no—
22C2×D596—22no—
23C2×D596C2×C411yescharacteristic
24C2×D596D412yes—
25C2×D596C2311yescharacteristic
26C2×C22⋊D149—22no—
27C2×C22⋊D149—21no—
28D4×D149—42no—
29C2×C4×D149C2×C411yescharacteristic
30C2×C4×D149D412yes—
31#14558715—21no—
32#14558713—21no—
33#14558713—22no—
34C4×Dic149C2×C411yescharacteristic
35C2×Q8×C149C2×C411yescharacteristic
36C2×D4×C149C2×C411yescharacteristic
37C22⋊C4×C149—22no—
38C42×C149C2×C411yescharacteristic
39C2×#10597010—44no—
40C22×D149—21no—
41C22×D149—42no—
42C22×D149—82no—
43C22×D149C22⋊C411yescharacteristic
44C22×D149C2×D412yes—
45D596—42no—
46D596—41no—
47D596—41no—
48D596—82no—
49C22⋊D149—42no—
50C22⋊D149—82no—
51C2×Dic149—21no—
52C2×Dic149C22⋊C411yescharacteristic
53C4×D149—22no—
54C4×D149—22no—
55C4×D149—22no—
56C23×C149—21no—
57Q8×C149—41no—
58D4×C149—41no—
59C2×C4×C149—21no—
60C2×C4×C149C22⋊C412yes—
61C2×C4×C149C22×C411yescharacteristic
62C22×C149—42no—
63C22×C149—41no—
64C22×C149C2×C22⋊C411yescharacteristic
65C2×D149—21no—
66C2×D149—22no—
67C2×D149—42no—
68C2×D149—41no—
69C2×D149—42no—
70C2×D149—41no—
71C2×D149—84no—
72#10597010—84no—
73C4×C149—22no—
74C4×C149—21no—
75C4×C149—41no—
76Dic149—41no—
77Dic149#9912yes—
78C2×C149—21no—
79C2×C149—81no—
80C2×C149C2×#9911yescharacteristic
81D149—42no—
82D149—41no—
83D149—81no—
84C149#212811yescharacteristic, complemented, Sylow
85#2128—1491nomaximal, Sylow
86#535—1491no—
87#410—1492no—
88#353—1492no—
89#353—1492no—
90C2×#41—1491no—
91C22×D4—1491no—
92#127—1491no—
93#127—1491no—
94#124—1491no—
95#124#1059701011yescharacteristic, complemented
96#121—2982no—
97#120—2982no—
98#117—2982no—
99#117—1491no—
100#99—2982no—
101#99—1494no—
102C24—2981no—
103#41—5962no—
104C2×Q8C2×#1059701011yescharacteristic
105C2×D4—2982no—
106C2×D4—2982no—
107C2×D4—2981no—
108C2×D4—5962no—
109C2×D4—1492no—
110C2×D4—1491no—
111C2×D4—1491no—
112C2×D4C2×#1059701011yescharacteristic
113C22×C4—1492no—
114C22×C4—1491no—
115C4⋊C4—2981no—
116C4⋊C4—5962no—
117C22⋊C4—22no—
118C22⋊C4—2981no—
119C22⋊C4—2982no—
120C22⋊C4—2984no—
121C42—5962no—
122C42—1491no—
123C42C2×#1059701011yescharacteristic
124C23—21no—
125C23—2981no—
126C23—5962no—
127C23—1491no—
128C23—1492no—
129C23—11922no—
130Q8—41no—
131D4—41no—
132D4—5962no—
133D4—5961no—
134D4—5961no—
135D4—5962no—
136D4—11922no—
137D4—11922no—
138C2×C4—21no—
139C2×C4—2982no—
140C2×C4—2982no—
141C2×C4—2982no—
142C2×C4—2981no—
143C2×C4—5964no—
144C2×C4—1491no—
145C2×C4C22⋊#1059701012yes—
146C2×C4C22×#1059701011yescharacteristic
147C22—42no—
148C22—41no—
149C22—2982no—
150C22—2981no—
151C22—5962no—
152C22—5961no—
153C22—5962no—
154C22—5961no—
155C22—11924no—
156C22C2×C22⋊#1059701011yescharacteristic
157C4—22no—
158C4—21no—
159C4—41no—
160C4—5961no—
161C4—1492no—
162C4—11924no—
163C2—21no—
164C2—81no—
165C2—5962no—
166C2—5961no—
167C2—11921no—
168C2D149⋊#9911yescharacteristic
subgroupG/H
centerC2D149⋊#99
commutatorC2×C4×C149C22×C4
FrattiniC2×C4C22×#10597010
Fitting#124×C149C4
socleC2×C149C2×#99
prime radical#23750193C2
cosocleC2×D596C23
hypercenter#124#10597010
hypocenterC149#2128
LCSG ▷ C2×C4×C149 ▷ C22×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #124

Complements

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

NH ≅ G/Ntype
1. #23750606C2semidirect
2. #23750530C2semidirect
3. #23750529C2semidirect
7. #18011970C22semidirect
9. C2×D4×D149C4semidirect
12. #18011900C4semidirect
15. #18011868C4semidirect
16. #124×C149C4semidirect
84. C149#2128semidirect
95. #124#10597010semidirect

Maximal subgroups

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

subgroupindex
#237506062
#237505302
#237505292
#237501932
#2128149