Finite Group Databasebeta

← #38768187

Subgroups — #38768187

Order 25472.

184 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 45 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#27164768C211yesmaximal, characteristic
2#27164427C211yesmaximal, characteristic, complemented
3#27164387C211yesmaximal, characteristic, complemented
4#27164377C211yesmaximal, characteristic
5#27163759C211yesmaximal, characteristic, complemented
6#27163759C211yesmaximal, characteristic, complemented
7#329×C199C211yesmaximal, characteristic
8#20263955C2211yescharacteristic
9#20263953—21no—
10#20263948—21no—
11#20263948—21no—
12#20263947C2211yescharacteristic
13#20263947C2211yescharacteristic
14#20263936—21no—
15#20263932—21no—
16#20263932—21no—
17#20263889—21no—
18#20263887—21no—
19#20263880—21no—
20#20263880—21no—
21#20263879—21no—
22#20263876—21no—
23#20263866C2211yescharacteristic
24#124×C199C2211yescharacteristic
25#97×C199C2211yescharacteristic
26#97×C199C2211yescharacteristic
27C2×Dic398—21no—
28C2×Dic398—21no—
29C22×Dic199—21no—
30C2×C199⋊C8—21no—
31C4⋊Dic199—21no—
32C4⋊Dic199—21no—
33C4⋊Dic199D411yescharacteristic
34C4⋊Dic199D411yescharacteristic
35C4⋊Dic199D411yescharacteristic
36C4⋊Dic199D411yescharacteristic
37Dic199⋊C4—21no—
38Dic199⋊C4—21no—
39Dic199⋊C4—21no—
40Dic199⋊C4—21no—
41Dic199⋊C4—21no—
42Dic199⋊C4—21no—
43C22⋊Dic199—21no—
44C22⋊Dic199—21no—
45C22⋊Dic199—21no—
46C4×Dic199—21no—
47C2×Q8×C199D411yescharacteristic
48C2×D4×C199D411yescharacteristic
49C2×C8×C199—21no—
50C2×C8×C199—21no—
51C22⋊C4×C199—21no—
52C22⋊C4×C199—21no—
53C42×C199C2311yescharacteristic
54Dic398—42no—
55Dic398—81no—
56C2×Dic199—21no—
57C2×Dic199—21no—
58C2×Dic199—21no—
59C2×Dic199—21no—
60C2×Dic199—41no—
61C2×Dic199—81no—
62C199⋊C8—41no—
63C23×C199—21no—
64Q8×C199—41no—
65D4×C199—41no—
66C2×C4×C199—21no—
67C2×C4×C199C2×D411yescharacteristic
68C2×C4×C199C2×D411yescharacteristic
69C2×C4×C199C2×D411yescharacteristic
70C8×C199—22no—
71C8×C199—41no—
72C22×C199—41no—
73C22×C199—41no—
74C22×C199—41no—
75C22×C199#12011yescharacteristic
76C4×C199—21no—
77C4×C199—21no—
78C4×C199—21no—
79C4×C199—41no—
80Dic199—41no—
81Dic199—41no—
82Dic199—41no—
83Dic199—41no—
84Dic199—81no—
85C2×C199—81no—
86C2×C199#45411yescharacteristic
87C2×C199#45511yescharacteristic
88C2×C199#45611yescharacteristic
89C199#166411yescharacteristic, complemented, Sylow
90#1664—1991nomaximal, Sylow
91#563—1991no—
92#501—1991no—
93#491—1991no—
94#489—1991no—
95#330—1991no—
96#330—1991no—
97#329D19911yescharacteristic
98C4⋊Q8—3981no—
99C4⋊Q8—1991no—
100#126—3981no—
101#126—3981no—
102#125—1991no—
103#125—1991no—
104#124C2×D19911yescharacteristic
105#123—3981no—
106C22⋊Q8—3981no—
107C22⋊Q8—3981no—
108#107—3981no—
109#106—3981no—
110#103—3981no—
111#103—3981no—
112#103—3981no—
113#102—3981no—
114#97—1991no—
115#97C2×D19911yescharacteristic
116#97C2×D19911yescharacteristic
117C2×Q8—3981no—
118C2×Q8—3981no—
119C2×Q8D79611yescharacteristic
120C2×D4D79611yescharacteristic
121C22×C4—3981no—
122C2×C8—21no—
123C2×C8—21no—
124C2×C8—3981no—
125C4⋊C4—3981no—
126C4⋊C4—3981no—
127C4⋊C4—3981no—
128C4⋊C4—3981no—
129C4⋊C4—3981no—
130C4⋊C4—3981no—
131C4⋊C4—3981no—
132C4⋊C4—3981no—
133C4⋊C4—1991no—
134C4⋊C4—1991no—
135C4⋊C4—1991no—
136C4⋊C4—1991no—
137C22⋊C4—21no—
138C22⋊C4—21no—
139C22⋊C4—3981no—
140C22⋊C4—3981no—
141C22⋊C4—3981no—
142C42—3981no—
143C42C22×D19911yescharacteristic
144C23—21no—
145Q8—41no—
146Q8—7962no—
147Q8—15921no—
148D4—41no—
149C2×C4—21no—
150C2×C4—7961no—
151C2×C4—15921no—
152C2×C4—3981no—
153C2×C4—3981no—
154C2×C4—3981no—
155C2×C4—3981no—
156C2×C4D4×D19911yescharacteristic
157C2×C4D4×D19911yescharacteristic
158C2×C4C2×D79611yescharacteristic
159C8—22no—
160C8—41no—
161C8—7961no—
162C22—41no—
163C22—41no—
164C22—41no—
165C22#2026392511yescharacteristic
166C4—21no—
167C4—21no—
168C4—21no—
169C4—41no—
170C4—7961no—
171C4—7961no—
172C4—7961no—
173C4—7961no—
174C4—15921no—
175C2—81no—
176C2#2716419211yescharacteristic
177C2#2716419811yescharacteristic
178C2#2716420211yescharacteristic
subgroupG/H
centerC22#20263925
commutator, cosocleC42×C199C23
FrattiniC42C22×D199
Fitting#329×C199C2
socleC22×C199#120
prime radicalC2×D4×C199D4
hypercenter#329D199
hypocenterC199#1664
LCSG ▷ C42×C199 ▷ C22×C199 ▷ C199
UCSC1 ◁ C22 ◁ C42 ◁ #329

Complements

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

NH ≅ G/Ntype
2. #27164427C2semidirect
3. #27164387C2semidirect
5. #27163759C2semidirect
6. #27163759C2semidirect
89. C199#1664semidirect

Maximal subgroups

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

subgroupindex
#271647682
#271644272
#271643872
#271643772
#271637592
#329×C1992
#1664199