Finite Group Databasebeta

← #38767313

Subgroups — #38767313

Order 25472.

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

normal subgroups only

This group has 216 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
2#27164071C211yesmaximal, characteristic, complemented
3#27163748C212yesmaximal, complemented
4#27163745C212yesmaximal, complemented
5#423×C199C211yesmaximal, characteristic, complemented
6#20263949—21no—
7#20263947C411yescharacteristic
8#20263943C411yescharacteristic
9#20263938—21no—
10#20263934C411yescharacteristic
11#20263929C411yescharacteristic
12C4×Dic398C2211yescharacteristic
13C4×D796C2211yescharacteristic
14C4×C22⋊D199—21no—
15#20263912—21no—
16C4⋊(C199⋊C8)—21no—
17C22⋊(C199⋊C8)—21no—
18C4×C199⋊C8C2212yes—
19C2×M16×C199—21no—
20C2×C42×C199C2211yescharacteristic
21C4⋊C8×C199—21no—
22C4⋊C8×C199C2212yes—
23C2×Dic398C2×C411yescharacteristic
24C2×D796C2×C411yescharacteristic
25C2×C22⋊D199—21no—
26C2×C4×D199—21no—
27C2×C199⋊C8—22no—
28C4⋊Dic199C2×C411yescharacteristic
29C4⋊Dic199C2×C411yescharacteristic
30Dic199⋊C4—21no—
31Dic199⋊C4—21no—
32Dic199⋊C4—21no—
33Dic199⋊C4—21no—
34C22⋊Dic199—21no—
35(C2×D199)⋊C4—21no—
36(C2×D199)⋊C4—21no—
37(C2×D199)⋊C4—21no—
38(C2×D199)⋊C4—21no—
39C4×Dic199—21no—
40C22×C4×C199C2×C411yescharacteristic
41C22×C4×C199D411yescharacteristic
42C22×C4×C199D411yescharacteristic
43M16×C199—42no—
44C2×C8×C199—22no—
45C42×C199C2×C411yescharacteristic
46C42×C199D411yescharacteristic
47C42×C199D411yescharacteristic
48C42×C199C2311yescharacteristic
49C22×D199—21no—
50Dic398—22no—
51D796—22no—
52C22⋊D199—42no—
53C2×Dic199—21no—
54C2×Dic199—21no—
55C2×Dic199—21no—
56C4×D199—41no—
57C4×D199—41no—
58C199⋊C8—42no—
59C23×C199C22⋊C411yescharacteristic
60C2×C4×C199—22no—
61C2×C4×C199—21no—
62C2×C4×C199—21no—
63C2×C4×C199—41no—
64C2×C4×C199C22⋊C411yescharacteristic
65C2×C4×C199C22⋊C411yescharacteristic
66C2×C4×C199C22⋊C411yescharacteristic
67C2×C4×C199C22×C411yescharacteristic
68C2×C4×C199C2×D411yescharacteristic
69C2×C4×C199C2×D411yescharacteristic
70C8×C199—42no—
71C22×C199—21no—
72C22×C199—21no—
73C22×C199—21no—
74C22×C199C2×C22⋊C411yescharacteristic
75C2×D199—41no—
76C2×D199—41no—
77C2×D199—41no—
78C4×C199—21no—
79C4×C199—21no—
80C4×C199—21no—
81C4×C199—21no—
82C4×C199#13112yes—
83C4×C199AGL(1,ℤ/8)11yescharacteristic
84C4×C199#13711yescharacteristic
85Dic199—41no—
86Dic199—41no—
87Dic199—41no—
88C2×C199—41no—
89C2×C199#40811yescharacteristic
90C2×C199#41711yescharacteristic
91C2×C199#42111yescharacteristic
92D199—81no—
93C199#149511yescharacteristic, complemented, Sylow
94#1495—1991nomaximal, Sylow
95#529—1991no—
96#431—1991no—
97#423D19911yescharacteristic, complemented
98Q8⋊C8—1992no—
99D4⋊C8—1992no—
100C2×M16—21no—
101#126—3981no—
102#125—1991no—
103#124—1991no—
104#123—3981no—
105C22⋊Q8—1991no—
106#121—1991no—
107C4×Q8—1991no—
108C4×D4—3981no—
109C4×D4—1991no—
110#117—3981no—
111C2×C42C2×D19911yescharacteristic
112C4⋊C8—21no—
113C4⋊C8—3981no—
114C4⋊C8C2×D19912yes—
115C22⋊C8—3981no—
116C4×C8—1992no—
117C2×Q8—1991no—
118C2×D4—3981no—
119C2×D4—1991no—
120C22×C4—3981no—
121C22×C4C4×D19911yescharacteristic
122C22×C4C22⋊D19911yescharacteristic
123C22×C4D79611yescharacteristic
124M16—42no—
125C2×C8—22no—
126C2×C8—3982no—
127C4⋊C4—3981no—
128C4⋊C4—3981no—
129C4⋊C4—3981no—
130C4⋊C4—3981no—
131C4⋊C4—1991no—
132C4⋊C4—1991no—
133C22⋊C4—3981no—
134C22⋊C4—3981no—
135C22⋊C4—3981no—
136C22⋊C4—3981no—
137C22⋊C4—3981no—
138C42—3981no—
139C42C4×D19911yescharacteristic
140C42C22⋊D19911yescharacteristic
141C42D79611yescharacteristic
142C42C22×D19911yescharacteristic
143C23—3981no—
144C23(C2×D199)⋊C411yescharacteristic
145Q8—3982no—
146D4—7962no—
147D4—3982no—
148C2×C4—22no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—41no—
152C2×C4—7961no—
153C2×C4—7961no—
154C2×C4—3981no—
155C2×C4—3981no—
156C2×C4—3981no—
157C2×C4(C2×D199)⋊C411yescharacteristic
158C2×C4(C2×D199)⋊C411yescharacteristic
159C2×C4(C2×D199)⋊C411yescharacteristic
160C2×C4C2×C4×D19911yescharacteristic
161C2×C4C2×C22⋊D19911yescharacteristic
162C2×C4C2×D79611yescharacteristic
163C8—42no—
164C8—7962no—
165C22—21no—
166C22—21no—
167C22—21no—
168C22—7961no—
169C22—7961no—
170C22—7961no—
171C22C2×(C2×D199)⋊C411yescharacteristic
172C4—21no—
173C4—21no—
174C4—21no—
175C4—21no—
176C4—7961no—
177C4—7961no—
178C4—7961no—
179C4#2026396312yes—
180C4#2026398111yescharacteristic
181C4#2026398611yescharacteristic
182C2—41no—
183C2—15921no—
184C2#2716398611yescharacteristic
185C2#2716401711yescharacteristic
186C2#2716403411yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊D199
commutatorC2×C4×C199C22×C4
FrattiniC42C22×D199
Fitting#423×C199C2
socleC22×C199C2×C22⋊C4
prime radical#20263929C4
cosocleC42×C199C23
hypercenter#423D199
hypocenterC199#1495
LCSG ▷ C2×C4×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #423

Complements

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

NH ≅ G/Ntype
2. #27164071C2semidirect
3. #27163748C2semidirect
4. #27163745C2semidirect
5. #423×C199C2semidirect
93. C199#1495semidirect
97. #423D199semidirect

Maximal subgroups

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

subgroupindex
#271645522
#271640712
#271637482
#271637452
#423×C1992
#1495199