Finite Group Databasebeta

← #34045109

Subgroups — #34045109

Order 19072.

342 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 89 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#23750132C211yesmaximal, characteristic
2#23749609C211yesmaximal, characteristic, complemented
3#23749588C212yesmaximal, complemented
4#23749396C212yesmaximal, complemented
5#568×C149C211yesmaximal, characteristic, complemented
6C22×C4×D149C2211yescharacteristic
7C2×#14558720C2212yes—
8#18011892C412yes—
9#18011887—21no—
10#18011882—21no—
11#18011879C412yes—
12C4×C22⋊D149—22no—
13#18011870C2211yescharacteristic
14#18011830—21no—
15#18011828—21no—
16#18011823—24no—
17C2×#41×C149C2211yescharacteristic
18#131×C149—24no—
19C2×M16×C149—22no—
20C2×M16×C149C2212yes—
21C23×D149C2×C411yescharacteristic
22C2×C22⋊D149—22no—
23C22×Dic149C2×C411yescharacteristic
24C2×C4×D149—22no—
25C2×C4×D149—21no—
26C2×C4×D149—21no—
27C2×C4×D149—22no—
28#14558720—22no—
29#14558720—22no—
30C2×#11924746—22no—
31C4⋊Dic149—21no—
32#14558715—22no—
33C22⋊Dic149—22no—
34C22⋊Dic149C2×C412yes—
35#14558713—22no—
36C4×Dic149—21no—
37#41×C149—22no—
38#41×C149—22no—
39C2×Q8×C149D411yescharacteristic
40C2×D4×C149C2×C412yes—
41C2×D4×C149D411yescharacteristic
42C22×C4×C149D412yes—
43C22×C4×C149C2311yescharacteristic
44M16×C149—24no—
45M16×C149—22no—
46M16×C149—22no—
47C2×C8×C149—24no—
48C2×C8×C149D414yescomplemented
49C22×D149—21no—
50C22×D149—22no—
51C22×D149—42no—
52C22⋊D149—44no—
53C2×Dic149—21no—
54C2×Dic149—22no—
55C2×Dic149—22no—
56C4×D149—41no—
57C4×D149—42no—
58C4×D149—41no—
59#11924746—42no—
60C23×C149C22⋊C412yes—
61C23×C149C22×C411yescharacteristic
62Q8×C149—22no—
63D4×C149—24no—
64D4×C149—22no—
65C2×C4×C149—22no—
66C2×C4×C149—22no—
67C2×C4×C149C22⋊C412yes—
68C2×C4×C149C2×D412yes—
69C2×C4×C149C2×D411yescharacteristic
70C2×C4×C149C2×D411yescharacteristic
71C2×C4×C149#4112yes—
72C8×C149—24no—
73C22×C149—22no—
74C22×C149—41no—
75C22×C149—42no—
76C22×C149C2×C22⋊C411yescharacteristic
77C22×C149C4×D412yes—
78C2×D149—22no—
79C2×D149—24no—
80C2×D149—41no—
81C2×D149—42no—
82C2×D149—41no—
83C4×C149—22no—
84C4×C149#12011yescharacteristic
85C4×C149#12112yes—
86C4×C149#12311yescharacteristic
87Dic149—42no—
88Dic149—41no—
89C2×C149—21no—
90C2×C149—22no—
91C2×C149—42no—
92C2×C149#38611yescharacteristic
93D149—42no—
94C149#189111yescharacteristic, complemented, Sylow
95#1891—1491nomaximal, Sylow
96#568D14911yescharacteristic, complemented
97#525—1491no—
98#413—1491no—
99#407—1492no—
100#344—1492no—
101C2×#41C2×D14911yescharacteristic
102C23×C4—1491no—
103#131—24no—
104C2×M16—22no—
105C2×M16—1492no—
106C2×M16C2×D14912yes—
107#123—1492no—
108C22⋊Q8—2981no—
109#121—2981no—
110#120—1492no—
111C4×D4—2982no—
112#117—1491no—
113#101—2981no—
114#100—2981no—
115C22⋊C8—2984no—
116C24—1491no—
117#41—22no—
118#41—22no—
119C2×Q8D59611yescharacteristic
120C2×D4—2982no—
121C2×D4C4×D14912yes—
122C2×D4D59611yescharacteristic
123C22×C4—2982no—
124C22×C4—2981no—
125C22×C4—2981no—
126C22×C4—2982no—
127C22×C4—1491no—
128C22×C4C22⋊D14912yes—
129C22×C4C22×D14911yescharacteristic
130M16—24no—
131M16—22no—
132M16—22no—
133M16—2982no—
134M16—2982no—
135C2×C8—24no—
136C2×C8—2982no—
137C2×C8C22⋊D14914yescomplemented
138C4⋊C4—2981no—
139C4⋊C4—2982no—
140C22⋊C4—2982no—
141C22⋊C4—2982no—
142C22⋊C4—1492no—
143C42—2981no—
144C23—2981no—
145C23—2982no—
146C23—5962no—
147C23#1455871312yes—
148C23C2×C4×D14911yescharacteristic
149Q8—22no—
150D4—24no—
151D4—22no—
152D4—5964no—
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—2982no—
156C2×C4—2981no—
157C2×C4—2982no—
158C2×C4—5961no—
159C2×C4—5962no—
160C2×C4—5961no—
161C2×C4#1455871312yes—
162C2×C4C2×C22⋊D14912yes—
163C2×C4C2×C22⋊D14911yescharacteristic
164C2×C4C2×D59611yescharacteristic
165C2×C4#1455873612yes—
166C8—24no—
167C8—5962no—
168C22—22no—
169C22—41no—
170C22—42no—
171C22—2982no—
172C22—2984no—
173C22—5961no—
174C22—5962no—
175C22—5961no—
176C22C2×#1455871311yescharacteristic
177C22C4×C22⋊D14912yes—
178C4—22no—
179C4—5962no—
180C4—5961no—
181C4#1801188011yescharacteristic
182C4#1801188512yes—
183C4#1801189411yescharacteristic
184C2—21no—
185C2—22no—
186C2—42no—
187C2—5962no—
188C2#2374950411yescharacteristic
subgroupG/H
centerC4#18011880
commutatorC23×C149C22×C4
FrattiniC22×C4C22×D149
Fitting#568×C149C2
socleC2×C149#386
prime radical#23750132C2
cosocleC22×C4×C149C23
hypercenter#568D149
hypocenterC149#1891
LCSG ▷ C23×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C4 ◁ C22×C4 ◁ #568

Complements

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

NH ≅ G/Ntype
2. #23749609C2semidirect
3. #23749588C2semidirect
4. #23749396C2semidirect
5. #568×C149C2semidirect
48. C2×C8×C149D4semidirect
94. C149#1891semidirect
96. #568D149semidirect
137. C2×C8C22⋊D149semidirect

Maximal subgroups

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

subgroupindex
#237501322
#237496092
#237495882
#237493962
#568×C1492
#1891149