Finite Group Databasebeta

← #38769040

Subgroups — #38769040

Order 25472.

360 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 129 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
1C22×C4⋊Dic199C211yesmaximal, characteristic
2C2×#20263889C212yesmaximal, complemented
3#27163878C211yesmaximal, characteristic
4#27163781C212yesmaximal, complemented
5C2×C22⋊C8×C199C211yesmaximal, characteristic
6C23×Dic199—21no—
7C2×C4⋊Dic199—24no—
8C2×C4⋊Dic199—21no—
9C2×C4⋊Dic199C2211yescharacteristic
10C2×C4⋊Dic199C2211yescharacteristic
11C2×C4⋊Dic199C2212yes—
12C2×C22⋊Dic199—21no—
13#20263889—24no—
14C199⋊#95—21no—
15C23×C4×C199C2211yescharacteristic
16C22×C8×C199C2212yes—
17C22⋊C8×C199C414yescomplemented
18C22×Dic199—24no—
19C22×Dic199—22no—
20C22×Dic199—21no—
21C22×Dic199—21no—
22C22×Dic199—42no—
23C22×Dic199—42no—
24C4⋊Dic199—24no—
25C4⋊Dic199—22no—
26C4⋊Dic199—22no—
27C4⋊Dic199—22no—
28C4⋊Dic199D414yes—
29C22⋊Dic199—42no—
30C24×C199D411yescharacteristic
31C22×C4×C199—22no—
32C22×C4×C199—22no—
33C22×C4×C199C2×C412yes—
34C22×C4×C199D411yescharacteristic
35C22×C4×C199Q812yes—
36C22×C4×C199C2311yescharacteristic
37C2×C8×C199—22no—
38C2×C8×C199—22no—
39C2×C8×C199C2×C414yescomplemented
40C2×Dic199—24no—
41C2×Dic199—48no—
42C2×Dic199—42no—
43C2×Dic199—42no—
44C2×Dic199—41no—
45C2×Dic199—41no—
46C2×Dic199—41no—
47C2×Dic199—41no—
48C2×Dic199—41no—
49C2×Dic199—41no—
50C23×C199—21no—
51C23×C199—21no—
52C23×C199—21no—
53C23×C199—21no—
54C23×C199C4⋊C412yes—
55C23×C199D812yes—
56C23×C199Q1612yes—
57C23×C199C2×D411yescharacteristic
58C2×C4×C199—21no—
59C2×C4×C199—21no—
60C2×C4×C199—24no—
61C2×C4×C199—24no—
62C2×C4×C199C4⋊C412yes—
63C2×C4×C199C22×C411yescharacteristic
64C2×C4×C199C2×D412yes—
65C2×C4×C199C2×Q811yescharacteristic
66C2×C4×C199#4112yes—
67C8×C199—24no—
68C22×C199—22no—
69C22×C199—22no—
70C22×C199—22no—
71C22×C199—22no—
72C22×C199—22no—
73C22×C199—22no—
74C22×C199#10714yescomplemented
75C22×C199C2×C4⋊C411yescharacteristic
76C22×C199#12011yescharacteristic
77C22×C199#12311yescharacteristic
78C22×C199C2×D811yescharacteristic
79C22×C199C2×Q1611yescharacteristic
80C22×C199AGL(1,ℤ/8)11yescharacteristic
81C22×C199#13711yescharacteristic
82C4×C199—22no—
83C4×C199C4×D412yes—
84C4×C199C22⋊Q812yes—
85Dic199—44no—
86Dic199—81no—
87Dic199—81no—
88C2×C199—24no—
89C2×C199#38511yescharacteristic
90C2×C199C2×#10711yescharacteristic
91C2×C199#42811yescharacteristic
92C2×C199#44711yescharacteristic
93C2×C199#45111yescharacteristic
94C2×C199#48011yescharacteristic
95C2×C199#48411yescharacteristic
96C199#185211yescharacteristic, complemented, Sylow
97#1852—1991nomaximal, Sylow
98C22×C4⋊C4—1991no—
99C2×#107—1992no—
100C2×C22⋊C8D19911yescharacteristic
101#380—1991no—
102#340—1992no—
103C23×C4—3981no—
104C23×C4C2×D19911yescharacteristic
105C22×C8C2×D19912yes—
106C2×C4⋊C4—3981no—
107C2×C4⋊C4—3984no—
108C2×C4⋊C4—1991no—
109C2×C4⋊C4—1991no—
110C2×C4⋊C4—1992no—
111C2×C22⋊C4—3981no—
112#107—3984no—
113C22⋊C8Dic19914yescomplemented
114#95—3981no—
115C24D79611yescharacteristic
116C22×C4—22no—
117C22×C4—22no—
118C22×C4—7962no—
119C22×C4—7962no—
120C22×C4—3981no—
121C22×C4—3981no—
122C22×C4—3984no—
123C22×C4—3982no—
124C22×C4C2×Dic19912yes—
125C22×C4D79611yescharacteristic
126C22×C4Dic39812yes—
127C22×C4C22×D19911yescharacteristic
128C2×C8—22no—
129C2×C8—22no—
130C2×C8C2×Dic19914yescomplemented
131C4⋊C4—3984no—
132C4⋊C4—3982no—
133C4⋊C4—3982no—
134C4⋊C4—3982no—
135C4⋊C4—1994no—
136C22⋊C4—7962no—
137C23—21no—
138C23—21no—
139C23—21no—
140C23—21no—
141C23C4⋊Dic19912yes—
142C23D159212yes—
143C23Dic79612yes—
144C23C2×D79611yescharacteristic
145C2×C4—21no—
146C2×C4—21no—
147C2×C4—24no—
148C2×C4—24no—
149C2×C4—7968no—
150C2×C4—7962no—
151C2×C4—7962no—
152C2×C4—7961no—
153C2×C4—7961no—
154C2×C4—7961no—
155C2×C4—7961no—
156C2×C4—7961no—
157C2×C4—7961no—
158C2×C4—3984no—
159C2×C4C4⋊Dic19912yes—
160C2×C4C22×Dic19911yescharacteristic
161C2×C4D4×D19912yes—
162C2×C4C2×Dic39811yescharacteristic
163C2×C4#1569188512yes—
164C8—24no—
165C22—22no—
166C22—22no—
167C22—22no—
168C22—22no—
169C22—22no—
170C22—22no—
171C22#2026388914yescomplemented
172C22C2×C4⋊Dic19911yescharacteristic
173C22#2026392511yescharacteristic
174C22#2026393911yescharacteristic
175C22C2×D159211yescharacteristic
176C22C2×Dic79611yescharacteristic
177C22#2026398311yescharacteristic
178C22#2026398811yescharacteristic
179C4—22no—
180C4—7964no—
181C4—15921no—
182C4—15921no—
183C4D4×Dic19912yes—
184C4#2026393212yes—
185C2—24no—
186C2#2716389311yescharacteristic
187C2C2×#2026388911yescharacteristic
188C2#2716406011yescharacteristic
189C2#2716415411yescharacteristic
190C2#2716417611yescharacteristic
191C2#2716433211yescharacteristic
192C2#2716435411yescharacteristic
subgroupG/H
centerC23C2×D796
commutatorC2×C4×C199C22×C4
FrattiniC22×C4C22×D199
FittingC2×C22⋊C8×C199C2
socleC23×C199C2×D4
prime radicalC24×C199D4
cosocleC22×C4×C199C23
hypercenterC2×C22⋊C8D199
hypocenterC199#1852
LCSG ▷ C2×C4×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C22⋊C8

Complements

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

NH ≅ G/Ntype
2. C2×#20263889C2semidirect
4. #27163781C2semidirect
17. C22⋊C8×C199C4semidirect
39. C2×C8×C199C2×C4semidirect
74. C22×C199#107semidirect
96. C199#1852semidirect
113. C22⋊C8Dic199semidirect
130. C2×C8C2×Dic199semidirect
171. C22#20263889semidirect

Maximal subgroups

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

subgroupindex
C22×C4⋊Dic1992
C2×#202638892
#271638782
#271637812
C2×C22⋊C8×C1992
#1852199