Finite Group Databasebeta

← #34055830

Subgroups — #34055830

Order 19072.

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

normal subgroups only

This group has 370 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, complemented
2#23750176C211yesmaximal, characteristic
3#23750021C211yesmaximal, characteristic, complemented
4#23749973C211yesmaximal, characteristic, complemented
5#23749972C211yesmaximal, characteristic, complemented
6#23749952C214yesmaximal, complemented
7#23749928C214yesmaximal, complemented
8#23749676C211yesmaximal, characteristic, complemented
9#500×C149C211yesmaximal, characteristic, complemented
10C2×#14558737—21no—
11#18011911C2211yescharacteristic
12#18011908C2211yescharacteristic
13#18011902C2212yes—
14#18011900—21no—
15#18011891—21no—
16#18011887—21no—
17#18011886—21no—
18#18011886C2212yes—
19Q8×Dic149—21no—
20#18011876—21no—
21#18011873—21no—
22#18011873C2212yes—
23#18011869C2212yes—
24C42×D149C2211yescharacteristic
25#18011842C2214yes—
26#18011840C2212yes—
27#18011833C2214yes—
28#18011831C2214yes—
29#18011823C2212yes—
30C4×#11924746C2211yescharacteristic
31C4⋊Q8×C149C2211yescharacteristic
32C4⋊Q8×C149C2211yescharacteristic
33#107×C149C2214yes—
34C4×C8×C149C2211yescharacteristic
35#14558737—44no—
36C2×Dic298—21no—
37C2×D596—22no—
38C2×D596—21no—
39C2×D596—21no—
40C2×D596C2312yes—
41C2×C4×D149—22no—
42C2×C4×D149D411yescharacteristic
43C2×C4×D149D411yescharacteristic
44C2×C4×D149C2311yescharacteristic
45C2×#11924746C2312yes—
46C4⋊Dic149—22no—
47C4⋊Dic149—21no—
48C4⋊Dic149C2312yes—
49#14558715—22no—
50#14558713—22no—
51#14558713—22no—
52#14558713—22no—
53C4×Dic149—22no—
54C4×Dic149D411yescharacteristic
55C4×Dic149D411yescharacteristic
56C4×Dic149C2311yescharacteristic
57C2×Q8×C149—21no—
58C2×Q8×C149—21no—
59C2×C8×C149C2312yes—
60C4⋊C4×C149—21no—
61C4⋊C4×C149—21no—
62C4⋊C4×C149C2312yes—
63C4⋊C4×C149C2312yes—
64C42×C149C2311yescharacteristic
65C22×D149—22no—
66C22×D149C2×D411yescharacteristic
67Dic298—42no—
68D596—24no—
69D596—44no—
70D596—42no—
71D596—42no—
72C2×Dic149—22no—
73C2×Dic149C2×D411yescharacteristic
74C2×Dic149C2×D411yescharacteristic
75C2×Dic149C2×D411yescharacteristic
76C4×D149—22no—
77C4×D149—22no—
78C4×D149—22no—
79C4×D149—44no—
80#11924746—22no—
81Q8×C149—42no—
82Q8×C149—42no—
83C2×C4×C149—22no—
84C2×C4×C149—22no—
85C2×C4×C149C2×D411yescharacteristic
86C2×C4×C149C2×D411yescharacteristic
87C2×C4×C149C2411yescharacteristic
88C8×C149—22no—
89C22×C149C22×D411yescharacteristic
90C2×D149—21no—
91C2×D149—21no—
92C2×D149—21no—
93C2×D149—42no—
94C2×D149—42no—
95C2×D149—42no—
96C4×C149—42no—
97C4×C149—42no—
98C4×C149#13512yes—
99C4×C149AGL(1,ℤ/8)12yes—
100C4×C1492−1+412yes—
101Dic149—21no—
102Dic149—21no—
103Dic149—21no—
104Dic149—42no—
105C2×C149#53611yescharacteristic
106C2×C149C2×#13511yescharacteristic
107C2×C149C2×AGL(1,ℤ/8)11yescharacteristic
108D149—41no—
109D149—82no—
110C149#322811yescharacteristic, complemented, Sylow
111#3228—1491nomaximal, Sylow
112#533—1491no—
113#532—1491no—
114#500D14911yescharacteristic, complemented
115#498—1491no—
116#486—1491no—
117#486—1491no—
118#482—1494no—
119#478—1494no—
120#431—1491no—
121C2×#41—2981no—
122C4⋊Q8—1491no—
123C4⋊Q8C2×D14911yescharacteristic
124C4⋊Q8C2×D14911yescharacteristic
125#127—1491no—
126#125—1492no—
127#124—2981no—
128C22⋊Q8—2981no—
129C22⋊Q8—2981no—
130#121—2981no—
131#121—1492no—
132C4×Q8—2981no—
133C4×Q8—2981no—
134C4×D4—2981no—
135C4×D4—1492no—
136#117—1492no—
137C2×C42—1491no—
138#107C2×D14914yes—
139#106—1494no—
140C4⋊C8—1492no—
141#102—1494no—
142#102—1494no—
143C22⋊C8—1492no—
144C4×C8—1491no—
145C4×C8C2×D14911yescharacteristic
146#41—5964no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8—2981no—
150C2×D4—2982no—
151C2×D4—2981no—
152C2×D4—2981no—
153C2×D4—1492no—
154C22×C4—2982no—
155C22×C4—1491no—
156C22×C4—1491no—
157C22×C4—1491no—
158C2×C8—1492no—
159C2×C8C22×D14912yes—
160C4⋊C4—21no—
161C4⋊C4—21no—
162C4⋊C4—2982no—
163C4⋊C4—2982no—
164C4⋊C4—2981no—
165C4⋊C4—1492no—
166C4⋊C4C22×D14912yes—
167C4⋊C4C22×D14912yes—
168C22⋊C4—2982no—
169C22⋊C4—2982no—
170C22⋊C4—2982no—
171C42—2982no—
172C42—1491no—
173C42—1491no—
174C42—1491no—
175C42C22×D14911yescharacteristic
176C23—2982no—
177C23—1491no—
178Q8—42no—
179Q8—42no—
180Q8—5962no—
181D4—2984no—
182D4—5964no—
183D4—5962no—
184D4—5962no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—2982no—
188C2×C4—2982no—
189C2×C4—2982no—
190C2×C4—2982no—
191C2×C4—5964no—
192C2×C4—1491no—
193C2×C4—1491no—
194C2×C4—1491no—
195C2×C4D4×D14911yescharacteristic
196C2×C4D4×D14911yescharacteristic
197C2×C4C23×D14911yescharacteristic
198C8—22no—
199C8—2982no—
200C22—2981no—
201C22—2981no—
202C22—2981no—
203C22—5962no—
204C22—5962no—
205C22—5962no—
206C22C2×D4×D14911yescharacteristic
207C4—42no—
208C4—42no—
209C4—2981no—
210C4—2981no—
211C4—2981no—
212C4—5962no—
213C4#1801193412yes—
214C4#1801193512yes—
215C4#1801195912yes—
216C2—5961no—
217C2—11922no—
218C2#2375020011yescharacteristic
219C2C2×#1801193411yescharacteristic
220C2C2×#1801193511yescharacteristic
subgroupG/H
centerC22C2×D4×D149
commutator, cosocleC2×C4×C149C24
FrattiniC2×C4C23×D149
Fitting#500×C149C2
socleC22×C149C22×D4
prime radical#23750176C2
hypercenter#500D149
hypocenterC149#3228
LCSG ▷ C2×C4×C149 ▷ C2×C149 ▷ C149
UCSC1 ◁ C22 ◁ C42 ◁ #500

Complements

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

NH ≅ G/Ntype
1. #23750180C2semidirect
3. #23750021C2semidirect
4. #23749973C2semidirect
5. #23749972C2semidirect
6. #23749952C2semidirect
7. #23749928C2semidirect
8. #23749676C2semidirect
9. #500×C149C2semidirect
110. C149#3228semidirect
114. #500D149semidirect

Maximal subgroups

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

subgroupindex
#237501802
#237501762
#237500212
#237499732
#237499722
#237499522
#237499282
#237496762
#500×C1492
#3228149