Finite Group Databasebeta

← #38767439

Subgroups — #38767439

Order 25472.

234 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 73 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#27164552C211yesmaximal, characteristic
2C2×#20263866C211yesmaximal, characteristic, complemented
3#27163759C212yesmaximal, complemented
4#27163757C212yesmaximal, complemented
5C4×M16×C199C211yesmaximal, characteristic, complemented
6C22×C199⋊C8—21no—
7#20263949—21no—
8#20263947C2211yescharacteristic
9#20263943C2211yescharacteristic
10#20263938—21no—
11#20263934C411yescharacteristic
12#20263929C411yescharacteristic
13C4×Dic398C411yescharacteristic
14C4×D796C411yescharacteristic
15C4×C22⋊D199—21no—
16#20263912—21no—
17#20263866—21no—
18#20263866C2212yes—
19C2×M16×C199—21no—
20C2×C42×C199C2211yescharacteristic
21#97×C199C2212yes—
22C4×C8×C199—21no—
23C2×Dic398C2×C411yescharacteristic
24C2×D796C2×C411yescharacteristic
25C2×C22⋊D199—21no—
26C2×C4×D199—21no—
27C2×C199⋊C8—24no—
28C2×C199⋊C8—22no—
29C4⋊Dic199C2×C411yescharacteristic
30C4⋊Dic199C2×C411yescharacteristic
31Dic199⋊C4—21no—
32Dic199⋊C4—21no—
33Dic199⋊C4—21no—
34Dic199⋊C4—21no—
35C22⋊Dic199—21no—
36(C2×D199)⋊C4—21no—
37(C2×D199)⋊C4—21no—
38(C2×D199)⋊C4—21no—
39(C2×D199)⋊C4—21no—
40C4×Dic199—21no—
41C22×C4×C199C2×C411yescharacteristic
42C22×C4×C199D411yescharacteristic
43C22×C4×C199D411yescharacteristic
44M16×C199—24no—
45C2×C8×C199—22no—
46C42×C199C2×C411yescharacteristic
47C42×C199D411yescharacteristic
48C42×C199D411yescharacteristic
49C42×C199C2311yescharacteristic
50C22×D199—21no—
51Dic398—41no—
52D796—41no—
53C22⋊D199—42no—
54C2×Dic199—21no—
55C2×Dic199—21no—
56C2×Dic199—21no—
57C4×D199—42no—
58C199⋊C8—24no—
59C23×C199C22⋊C411yescharacteristic
60C2×C4×C199—22no—
61C2×C4×C199—22no—
62C2×C4×C199—41no—
63C2×C4×C199C22⋊C411yescharacteristic
64C2×C4×C199C22⋊C411yescharacteristic
65C2×C4×C199C22⋊C411yescharacteristic
66C2×C4×C199C22×C411yescharacteristic
67C2×C4×C199C2×D411yescharacteristic
68C2×C4×C199C2×D411yescharacteristic
69C8×C199—24no—
70C22×C199—21no—
71C22×C199—21no—
72C22×C199#10412yes—
73C22×C199C2×C22⋊C411yescharacteristic
74C2×D199—41no—
75C2×D199—41no—
76C2×D199—41no—
77C4×C199—21no—
78C4×C199—21no—
79C4×C199—21no—
80C4×C199—21no—
81C4×C199—21no—
82C4×C199#10412yes—
83Dic199—41no—
84Dic199—41no—
85Dic199—41no—
86C2×C199—22no—
87C2×C199#41311yescharacteristic
88C2×C199C2×#10411yescharacteristic
89C2×C199C2×#10411yescharacteristic
90D199—81no—
91C199#152411yescharacteristic, complemented, Sylow
92#1524—1991nomaximal, Sylow
93#529—1991no—
94C4×M16D19911yescharacteristic, complemented
95C2×#97—1991no—
96#330—1992no—
97#329—1992no—
98C2×M16—21no—
99C22×C8—3981no—
100#126—3981no—
101#125—1991no—
102#124—1991no—
103#123—3981no—
104C22⋊Q8—1991no—
105#121—1991no—
106C4×Q8—1991no—
107C4×D4—3981no—
108C4×D4—1991no—
109#117—3981no—
110C2×C42C2×D19911yescharacteristic
111#97—3981no—
112#97—1992no—
113#97C2×D19912yes—
114C4×C8—21no—
115C2×Q8—1991no—
116C2×D4—3981no—
117C2×D4—1991no—
118C22×C4—3981no—
119C22×C4C4×D19911yescharacteristic
120C22×C4C22⋊D19911yescharacteristic
121C22×C4D79611yescharacteristic
122M16—24no—
123C2×C8—22no—
124C2×C8—3984no—
125C2×C8—3982no—
126C4⋊C4—3981no—
127C4⋊C4—3981no—
128C4⋊C4—3981no—
129C4⋊C4—3981no—
130C4⋊C4—1991no—
131C4⋊C4—1991no—
132C22⋊C4—3981no—
133C22⋊C4—3981no—
134C22⋊C4—3981no—
135C22⋊C4—3981no—
136C22⋊C4—3981no—
137C42—3981no—
138C42C4×D19911yescharacteristic
139C42C22⋊D19911yescharacteristic
140C42D79611yescharacteristic
141C42C22×D19911yescharacteristic
142C23—3981no—
143C23(C2×D199)⋊C411yescharacteristic
144Q8—7961no—
145D4—7961no—
146D4—7962no—
147C2×C4—22no—
148C2×C4—22no—
149C2×C4—41no—
150C2×C4—7962no—
151C2×C4—3981no—
152C2×C4—3981no—
153C2×C4—3981no—
154C2×C4(C2×D199)⋊C411yescharacteristic
155C2×C4(C2×D199)⋊C411yescharacteristic
156C2×C4(C2×D199)⋊C411yescharacteristic
157C2×C4C2×C4×D19911yescharacteristic
158C2×C4C2×C22⋊D19911yescharacteristic
159C2×C4C2×D79611yescharacteristic
160C8—24no—
161C8—3984no—
162C22—21no—
163C22—21no—
164C22—7961no—
165C22—7961no—
166C22—7961no—
167C22#2026388312yes—
168C22C2×(C2×D199)⋊C411yescharacteristic
169C4—21no—
170C4—21no—
171C4—21no—
172C4—21no—
173C4—21no—
174C4—7961no—
175C4—7961no—
176C4—7961no—
177C4#2026388112yes—
178C2—22no—
179C2—15921no—
180C2#2716400011yescharacteristic
181C2C2×#2026388111yescharacteristic
182C2C2×#2026388311yescharacteristic
subgroupG/H
centerC2×C4(C2×D199)⋊C4
commutatorC2×C4×C199C22×C4
FrattiniC42C22×D199
FittingC4×M16×C199C2
socleC22×C199C2×C22⋊C4
prime radical#20263929C4
cosocleC42×C199C23
hypercenterC4×M16D199
hypocenterC199#1524
LCSG ▷ C2×C4×C199 ▷ C22×C199 ▷ C199
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C4×M16

Complements

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

NH ≅ G/Ntype
2. C2×#20263866C2semidirect
3. #27163759C2semidirect
4. #27163757C2semidirect
5. C4×M16×C199C2semidirect
91. C199#1524semidirect
94. C4×M16D199semidirect

Maximal subgroups

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

subgroupindex
#271645522
C2×#202638662
#271637592
#271637572
C4×M16×C1992
#1524199