Finite Group Databasebeta

← #38768002

Subgroups — #38768002

Order 25472.

234 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 55 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#27164671C211yesmaximal, characteristic
2#27164370C211yesmaximal, characteristic, complemented
3#27164309C211yesmaximal, characteristic, complemented
4#27164309C211yesmaximal, characteristic, complemented
5#27163745C211yesmaximal, characteristic, complemented
6#27163745C211yesmaximal, characteristic, complemented
7#332×C199C211yesmaximal, characteristic, complemented
8C2×#15691887—21no—
9#20263955C2211yescharacteristic
10#20263940—22no—
11#20263931—21no—
12#20263930—21no—
13C4×D796C2211yescharacteristic
14C4×D796C2211yescharacteristic
15C4⋊C4×D199—21no—
16#20263887—21no—
17#20263887—21no—
18#20263878—22no—
19#20263877—21no—
20#20263877—21no—
21C4×C199⋊C8C2211yescharacteristic
22C4⋊Q8×C199C2211yescharacteristic
23C4⋊C8×C199C2211yescharacteristic
24C4⋊C8×C199C2211yescharacteristic
25#15691887—42no—
26#15691887—42no—
27C2×Dic398—21no—
28C2×D796—22no—
29C2×D796D411yescharacteristic
30C2×D796D411yescharacteristic
31C2×C4×D199—22no—
32C2×C4×D199—21no—
33C2×C4×D199—21no—
34C2×C199⋊C8—21no—
35C4⋊Dic199—21no—
36C4⋊Dic199—21no—
37C4⋊Dic199D411yescharacteristic
38C4⋊Dic199D411yescharacteristic
39Dic199⋊C4—22no—
40(C2×D199)⋊C4—22no—
41(C2×D199)⋊C4—21no—
42(C2×D199)⋊C4—21no—
43C2×Q8×C199—21no—
44C2×C8×C199—21no—
45C2×C8×C199—21no—
46C4⋊C4×C199—21no—
47C4⋊C4×C199D412yes—
48C42×C199C2311yescharacteristic
49C22×D199—21no—
50C22×D199—21no—
51Dic398—42no—
52D796—21no—
53D796—21no—
54D796—21no—
55D796—44no—
56D796SD1612yes—
57C2×Dic199—21no—
58C2×Dic199—21no—
59C2×Dic199—41no—
60C4×D199—44no—
61C4×D199—41no—
62C4×D199—41no—
63C4×D199—41no—
64C4×D199—41no—
65C199⋊C8—41no—
66Q8×C199—41no—
67Q8×C199—41no—
68C2×C4×C199—22no—
69C2×C4×C199C2×D411yescharacteristic
70C2×C4×C199C2×D411yescharacteristic
71C2×C4×C199C2×D411yescharacteristic
72C8×C199—41no—
73C8×C199—41no—
74C22×C199#12011yescharacteristic
75C2×D199—22no—
76C2×D199—41no—
77C2×D199—41no—
78C2×D199—41no—
79C2×D199—41no—
80C2×D199—41no—
81C4×C199—21no—
82C4×C199—42no—
83C4×C199C2×SD1611yescharacteristic
84C4×C199#13511yescharacteristic
85C4×C199AGL(1,ℤ/8)11yescharacteristic
86C4×C199#13711yescharacteristic
87Dic199—41no—
88Dic199—41no—
89Dic199—81no—
90C2×C199#45011yescharacteristic
91C2×C199#45211yescharacteristic
92C2×C199#45411yescharacteristic
93D199—42no—
94D199—81no—
95C199#163511yescharacteristic, complemented, Sylow
96#1635—1991nomaximal, Sylow
97#547—1991no—
98#487—1991no—
99#476—1991no—
100#476—1991no—
101#332D19911yescharacteristic, complemented
102D4⋊C8—1991no—
103D4⋊C8—1991no—
104C2×#41—3981no—
105C4⋊Q8—1991no—
106C4⋊Q8C2×D19911yescharacteristic
107#123—3982no—
108C22⋊Q8—3981no—
109#121—3981no—
110C4×D4—1991no—
111C4×D4—1991no—
112C2×C4⋊C4—3981no—
113#106—3981no—
114#106—3981no—
115C4⋊C8C2×D19911yescharacteristic
116C4⋊C8C2×D19911yescharacteristic
117#103—3982no—
118#102—3981no—
119#102—3981no—
120C4×C8—1991no—
121#41—7962no—
122#41—7962no—
123C2×Q8—21no—
124C2×Q8—3981no—
125C2×D4—3982no—
126C2×D4—1991no—
127C2×D4—1991no—
128C22×C4—3982no—
129C22×C4—3981no—
130C22×C4—3981no—
131C2×C8—21no—
132C2×C8—21no—
133C2×C8—3981no—
134C4⋊C4—21no—
135C4⋊C4—3982no—
136C4⋊C4—3981no—
137C4⋊C4—3981no—
138C4⋊C4—1991no—
139C4⋊C4—1991no—
140C4⋊C4D79612yes—
141C22⋊C4—3982no—
142C22⋊C4—3981no—
143C22⋊C4—3981no—
144C42C22×D19911yescharacteristic
145C23—3981no—
146C23—3981no—
147Q8—41no—
148Q8—41no—
149Q8—7962no—
150D4—7964no—
151D4—3981no—
152D4—3981no—
153D4—3981no—
154D4—1992no—
155C2×C4—22no—
156C2×C4—7964no—
157C2×C4—7961no—
158C2×C4—7961no—
159C2×C4—7961no—
160C2×C4—7961no—
161C2×C4—7961no—
162C2×C4—3981no—
163C2×C4—3981no—
164C2×C4D4×D19911yescharacteristic
165C2×C4D4×D19911yescharacteristic
166C2×C4C2×D79611yescharacteristic
167C8—41no—
168C8—41no—
169C8—7961no—
170C22—7961no—
171C22—7961no—
172C22—7961no—
173C22—7961no—
174C22—7961no—
175C22—3982no—
176C22#2026392511yescharacteristic
177C4—21no—
178C4—42no—
179C4—7961no—
180C4—7961no—
181C4—15921no—
182C4SD16×D19911yescharacteristic
183C4#2026397611yescharacteristic
184C4#2026397911yescharacteristic
185C4#2026398711yescharacteristic
186C2—7962no—
187C2—15921no—
188C2#2716417011yescharacteristic
189C2#2716418311yescharacteristic
190C2#2716419311yescharacteristic
subgroupG/H
centerC22#20263925
commutator, cosocleC42×C199C23
FrattiniC42C22×D199
Fitting#332×C199C2
socleC22×C199#120
prime radical#27164671C2
hypercenter#332D199
hypocenterC199#1635
LCSG ▷ C42×C199 ▷ C22×C199 ▷ C199
UCSC1 ◁ C22 ◁ C42 ◁ #332

Complements

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

NH ≅ G/Ntype
2. #27164370C2semidirect
3. #27164309C2semidirect
4. #27164309C2semidirect
5. #27163745C2semidirect
6. #27163745C2semidirect
7. #332×C199C2semidirect
95. C199#1635semidirect
101. #332D199semidirect

Maximal subgroups

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

subgroupindex
#271646712
#271643702
#271643092
#271637452
#332×C1992
#1635199