Finite Group Databasebeta

← #38770824

Subgroups — #38770824

Order 25472.

216 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 47 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#27164819C211yesmaximal, characteristic
2#27164461C211yesmaximal, characteristic, complemented
3C2×#20263899C211yesmaximal, characteristic, complemented
4#27164294C211yesmaximal, characteristic, complemented
5#27163824C211yesmaximal, characteristic, complemented
6#27163803C211yesmaximal, characteristic, complemented
7#361×C199C211yesmaximal, characteristic, complemented
8#20264000—21no—
9#20263988—21no—
10#20263981—21no—
11#20263979—22no—
12#20263977C2211yescharacteristic
13#20263974—21no—
14C2×Dic796C2211yescharacteristic, complemented
15D8×D199—21no—
16#20263961C2211yescharacteristic
17Dic1592—21no—
18#20263900—21no—
19#20263899—22no—
20C2×C199⋊C16C2211yescharacteristic, complemented
21#20263873—21no—
22C2×D8×C199C2211yescharacteristic
23M32×C199C2211yescharacteristic, complemented
24#108×C199C2211yescharacteristic, complemented
25#15691886—21no—
26#15691886D411yescharacteristic
27#15691885—22no—
28#15691885—41no—
29C2×Dic398—21no—
30C2×C22⋊D199—42no—
31D4×D199—22no—
32D4×D199—41no—
33Dic796—21no—
34Dic796—21no—
35Dic796—41no—
36Dic796D411yescharacteristic
37#15691875—21no—
38#15691875—41no—
39#15691874—22no—
40D1592D411yescharacteristic
41#15691871—22no—
42#15691870D411yescharacteristic
43#15691869—21no—
44C8×D199—21no—
45C199⋊C16—21no—
46C2×D4×C199—21no—
47D8×C199—21no—
48D8×C199D412yes—
49M16×C199—21no—
50C2×C8×C199C2311yescharacteristic
51C16×C199—21no—
52C22×D199—42no—
53Dic398—21no—
54Dic398—41no—
55Dic398—41no—
56Dic398D811yescharacteristic
57D796—21no—
58D796D811yescharacteristic
59C22⋊D199—21no—
60C22⋊D199—42no—
61C22⋊D199—42no—
62C22⋊D199—41no—
63C2×Dic199—42no—
64C2×Dic199—41no—
65C4×D199—21no—
66C4×D199—41no—
67C199⋊C8—21no—
68C23×C199—41no—
69D4×C199—22no—
70D4×C199—41no—
71C2×C4×C199C2×D411yescharacteristic
72C8×C199—41no—
73C8×C199C2×D411yescharacteristic
74C8×C199C2×D411yescharacteristic
75C22×C199—42no—
76C22×C199—161no—
77C22×C199AGL(1,ℤ/8)11yescharacteristic
78C2×D199—21no—
79C2×D199—41no—
80C2×D199—162no—
81C4×C199#12011yescharacteristic
82C4×C199C2×D811yescharacteristic
83Dic199—21no—
84Dic199—41no—
85Dic199—81no—
86C2×C199—21no—
87C2×C199—82no—
88C2×C199#44711yescharacteristic
89D199—41no—
90D199—81no—
91C199#219411yescharacteristic, complemented, Sylow
92#2194—1991nomaximal, Sylow
93#576—1991no—
94#510—1991no—
95C2×SD32—1991no—
96#473—1991no—
97#361D19911yescharacteristic, complemented
98#359—1991no—
99#350—1991no—
100H(5,𝔽2)—3981no—
101#137—3981no—
102AGL(1,ℤ/8)—3982no—
103AGL(1,ℤ/8)—3981no—
104#135—3981no—
105#135—1991no—
106C2×Q16—1991no—
107C2×D8—3981no—
108C2×D8C2×D19911yescharacteristic
109#131—1991no—
110Q32—3981no—
111SD32—3981no—
112SD32—3982no—
113M32C2×D19911yescharacteristic, complemented
114C2×C16—1991no—
115#108C2×D19911yescharacteristic, complemented
116#101—3981no—
117#41—7961no—
118#41—3982no—
119#41—3981no—
120#41—1991no—
121C2×Q8—3981no—
122C2×D4—21no—
123C2×D4—7961no—
124C2×D4—7962no—
125C2×D4—3982no—
126Q16—7961no—
127Q16—3981no—
128Q16—3981no—
129Q16—1991no—
130SD16—7961no—
131SD16—3982no—
132SD16—3981no—
133D8—21no—
134D8—3982no—
135D8—1991no—
136D8D79612yes—
137M16—21no—
138M16—3981no—
139M16—1991no—
140C2×C8—3981no—
141C2×C8C22×D19911yescharacteristic
142C16—21no—
143C16—3981no—
144C23—41no—
145C23—7962no—
146Q8—7961no—
147Q8—7961no—
148Q8—3981no—
149Q8—1991no—
150D4—22no—
151D4—41no—
152D4—7962no—
153D4—7961no—
154D4—7962no—
155D4—3981no—
156D4—3981no—
157D4—1991no—
158C2×C4—7962no—
159C2×C4—7961no—
160C2×C4—7961no—
161C2×C4—3981no—
162C2×C4D4×D19911yescharacteristic
163C8—41no—
164C8—3981no—
165C8D4×D19911yescharacteristic
166C8C2×D79611yescharacteristic
167C22—42no—
168C22—161no—
169C22—7961no—
170C22—31842no—
171C22—3981no—
172C22#2026398211yescharacteristic
173C4—7961no—
174C4—15921no—
175C4—3981no—
176C4#2026392511yescharacteristic
177C4D8×D19911yescharacteristic
178C2—21no—
179C2—82no—
180C2—7961no—
181C2—15921no—
182C2#2716415311yescharacteristic
subgroupG/H
centerC2#27164153
commutator, cosocleC2×C8×C199C23
FrattiniC2×C8C22×D199
Fitting#361×C199C2
socleC2×C199#447
prime radical#27164819C2
hypercenter#361D199
hypocenterC199#2194
LCSG ▷ C2×C8×C199 ▷ C4×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #361

Complements

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

NH ≅ G/Ntype
2. #27164461C2semidirect
3. C2×#20263899C2semidirect
4. #27164294C2semidirect
5. #27163824C2semidirect
6. #27163803C2semidirect
7. #361×C199C2semidirect
14. C2×Dic796C22semidirect
20. C2×C199⋊C16C22semidirect
23. M32×C199C22semidirect
24. #108×C199C22semidirect
91. C199#2194semidirect
97. #361D199semidirect
113. M32C2×D199semidirect
115. #108C2×D199semidirect

Maximal subgroups

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

subgroupindex
#271648192
#271644612
C2×#202638992
#271642942
#271638242
#271638032
#361×C1992
#2194199