Finite Group Databasebeta

← #34045856

Subgroups — #34045856

Order 19072.

466 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 67 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#23750251C211yesmaximal, characteristic, complemented
2#23749480C212yesmaximal
3#23749480C211yesmaximal, characteristic
4#23749391C212yesmaximal, complemented
5#342×C149C211yesmaximal, characteristic, complemented
6C24×D149C2211yescharacteristic
7#18011890—22no—
8#18011888—24no—
9C2×C22⋊Dic149C2211yescharacteristic
10C2×C22⋊Dic149C2212yes—
11C2×#14558713—22no—
12C2×#14558713—24no—
13C2×C22⋊C4×C149C2212yes—
14C2×C22⋊C4×C149C2211yescharacteristic
15C23×D149—22no—
16C23×D149—21no—
17C23×D149—21no—
18C23×D149—22no—
19C23×D149—44no—
20C23×D149Q812yescomplemented
21C2×Dic298—22no—
22C22×Dic149—21no—
23C22×Dic149D414yes—
24C4⋊Dic149—22no—
25#14558715—24no—
26C22⋊Dic149—22no—
27C22⋊Dic149—22no—
28C22⋊Dic149—22no—
29#14558713—44no—
30#14558713—48no—
31C24×C149C2311yescharacteristic
32C22×C4×C149—22no—
33C22×C4×C149D412yes—
34C22⋊C4×C149—22no—
35C22⋊C4×C149—22no—
36C22⋊C4×C149—22no—
37C22×D149—28no—
38C22×D149—24no—
39C22×D149—24no—
40C22×D149—22no—
41C22×D149—44no—
42C22×D149—44no—
43C22×D149—48no—
44C22×D149—44no—
45C22×D149—42no—
46C22×D149—41no—
47C22×D149—42no—
48C22×D149—41no—
49Dic298—82no—
50C2×Dic149—24no—
51C2×Dic149—44no—
52C2×Dic149—42no—
53C2×Dic149—41no—
54C23×C149—42no—
55C23×C149C2×D412yes—
56C23×C149C2×D411yescharacteristic
57C23×C149C2×Q811yescharacteristic
58C23×C149#4111yescharacteristic
59C23×C149#4112yes—
60C2×C4×C149—22no—
61C2×C4×C149—42no—
62C2×C4×C149—44no—
63C2×C4×C149—42no—
64C22×C149—24no—
65C22×C149—22no—
66C22×C149—41no—
67C22×C149—41no—
68C22×C149—42no—
69C22×C149#12011yescharacteristic
70C22×C149C22⋊Q812yes—
71C22×C149C22⋊Q811yescharacteristic
72C22×C149#12412yes—
73C22×C149#12411yescharacteristic
74C2×D149—28no—
75C2×D149—24no—
76C2×D149—44no—
77C2×D149—48no—
78C2×D149—44no—
79C2×D149—42no—
80C2×D149—42no—
81C2×D149—44no—
82C4×C149—42no—
83C4×C149—82no—
84Dic149—44no—
85Dic149—81no—
86C2×C149—21no—
87C2×C149—22no—
88C2×C149—22no—
89C2×C149—21no—
90C2×C149#39311yescharacteristic
91C2×C149B(4,𝔽2)12yes—
92D149—44no—
93C149#202911yescharacteristic, complemented, Sylow
94#2029—1491nomaximal, Sylow
95#543—1491no—
96#379—1492no—
97#379—1491no—
98#342—1492no—
99#342D14911yescharacteristic, complemented
100C25—1491no—
101C22⋊Q8—2984no—
102C22⋊Q8—2982no—
103C2×C22⋊C4—2984no—
104C2×C22⋊C4—2982no—
105C2×C22⋊C4—1491no—
106C2×C22⋊C4—1492no—
107C2×C22⋊C4C2×D14912yes—
108C2×C22⋊C4C2×D14911yescharacteristic
109C24—2982no—
110C24—2981no—
111C24—2981no—
112C24—2982no—
113C24—5964no—
114C24—1492no—
115C24C22×D14911yescharacteristic
116C2×Q8—2982no—
117C22×C4—22no—
118C22×C4—2981no—
119C22×C4—1494no—
120C22×C4D59612yes—
121C4⋊C4—2984no—
122C4⋊C4—2982no—
123C22⋊C4—22no—
124C22⋊C4—22no—
125C22⋊C4—22no—
126C22⋊C4—2982no—
127C22⋊C4—2982no—
128C22⋊C4—2982no—
129C22⋊C4—5964no—
130C22⋊C4—5968no—
131C23—42no—
132C23—2988no—
133C23—2984no—
134C23—2984no—
135C23—2982no—
136C23—5964no—
137C23—5962no—
138C23—5964no—
139C23—5964no—
140C23—5968no—
141C23—5961no—
142C23—5962no—
143C23—5961no—
144C23D4×D14912yes—
145C23C2×D59611yescharacteristic
146C23Q8×D14911yescharacteristic
147C23#1455873511yescharacteristic
148C23#1455873612yes—
149Q8—11922no—
150C2×C4—22no—
151C2×C4—42no—
152C2×C4—44no—
153C2×C4—42no—
154C2×C4—2984no—
155C2×C4—5964no—
156C2×C4—5962no—
157C2×C4—5961no—
158C22—24no—
159C22—22no—
160C22—41no—
161C22—41no—
162C22—42no—
163C22—2988no—
164C22—2984no—
165C22—5968no—
166C22—5964no—
167C22—5964no—
168C22—5962no—
169C22—5962no—
170C22—5964no—
171C22#1801188111yescharacteristic
172C22#1801188912yes—
173C22#1801189111yescharacteristic
174C22#1801189712yes—
175C22#1801189911yescharacteristic
176C4—42no—
177C4—82no—
178C4—5964no—
179C4—11921no—
180C2—22no—
181C2—21no—
182C2—22no—
183C2—21no—
184C2—5964no—
185C2#2374953811yescharacteristic
186C2#2374981312yes—
subgroupG/H
centerC22#18011881
commutator, cosocleC24×C149C23
FrattiniC24C22×D149
Fitting#342×C149C2
socleC22×C149#120
prime radicalC24×D149C22
hypercenter#342D149
hypocenterC149#2029
LCSG ▷ C24×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C24 ◁ #342

Complements

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

NH ≅ G/Ntype
1. #23750251C2semidirect
4. #23749391C2semidirect
5. #342×C149C2semidirect
20. C23×D149Q8semidirect
93. C149#2029semidirect
99. #342D149semidirect

Maximal subgroups

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

subgroupindex
#237502512
#237494802
#237493912
#342×C1492
#2029149