Finite Group Databasebeta

← #38768175

Subgroups — #38768175

Order 25472.

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

normal subgroups only

This group has 196 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#27164762C211yesmaximal, characteristic
2#27164387C211yesmaximal, characteristic, complemented
3#27164378C211yesmaximal, characteristic, complemented
4#27164377C211yesmaximal, characteristic, complemented
5#27163759C211yesmaximal, characteristic, complemented
6#27163757C211yesmaximal, characteristic, complemented
7#329×C199C211yesmaximal, characteristic, complemented
8#20263955C2211yescharacteristic
9#20263947C2211yescharacteristic
10#20263943C2211yescharacteristic
11#20263939—21no—
12#20263937—21no—
13#20263932—21no—
14#20263931—21no—
15#20263928—21no—
16#20263927—21no—
17#20263880—21no—
18#20263880—21no—
19#20263880—21no—
20#20263879—21no—
21#20263877—21no—
22#20263876—21no—
23#20263866C2211yescharacteristic
24#124×C199C2211yescharacteristic
25#97×C199C2211yescharacteristic
26#97×C199C2211yescharacteristic
27C2×Dic398—21no—
28C2×Dic398D411yescharacteristic
29C2×D796D411yescharacteristic
30C2×C22⋊D199—21no—
31C2×C22⋊D199—21no—
32C22×Dic199—21no—
33C2×C4×D199—21no—
34C2×C199⋊C8—21no—
35C4⋊Dic199—21no—
36C4⋊Dic199—21no—
37C4⋊Dic199—21no—
38C4⋊Dic199D411yescharacteristic
39C4⋊Dic199D411yescharacteristic
40Dic199⋊C4—21no—
41Dic199⋊C4—21no—
42Dic199⋊C4—21no—
43C22⋊Dic199—21no—
44(C2×D199)⋊C4—21no—
45(C2×D199)⋊C4—21no—
46(C2×D199)⋊C4—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
54C22×D199—21no—
55Dic398—41no—
56Dic398—42no—
57D796—41no—
58C22⋊D199—81no—
59C22⋊D199—81no—
60C2×Dic199—21no—
61C2×Dic199—21no—
62C2×Dic199—21no—
63C2×Dic199—41no—
64C2×Dic199—81no—
65C4×D199—81no—
66C199⋊C8—41no—
67C23×C199—21no—
68Q8×C199—41no—
69D4×C199—41no—
70C2×C4×C199—21no—
71C2×C4×C199C2×D411yescharacteristic
72C2×C4×C199C2×D411yescharacteristic
73C2×C4×C199C2×D411yescharacteristic
74C8×C199—41no—
75C8×C199—41no—
76C22×C199—41no—
77C22×C199—41no—
78C22×C199—41no—
79C22×C199#12011yescharacteristic
80C2×D199—41no—
81C2×D199—41no—
82C2×D199—41no—
83C4×C199—21no—
84C4×C199—21no—
85C4×C199—21no—
86C4×C199—41no—
87Dic199—41no—
88Dic199—41no—
89Dic199—41no—
90Dic199—81no—
91C2×C199—81no—
92C2×C199#45411yescharacteristic
93C2×C199#45511yescharacteristic
94C2×C199#45511yescharacteristic
95D199—81no—
96C199#166211yescharacteristic, complemented, Sylow
97#1662—1991nomaximal, Sylow
98#562—1991no—
99#491—1991no—
100#489—1991no—
101#489—1991no—
102#330—1991no—
103#329—1991no—
104#329D19911yescharacteristic, complemented
105C4⋊Q8—1991no—
106#125—1991no—
107#124—1991no—
108#124C2×D19911yescharacteristic
109#123—3981no—
110#123—3981no—
111C22⋊Q8—3981no—
112C22⋊Q8—3981no—
113#121—3981no—
114#121—3981no—
115#103—3981no—
116#103—3981no—
117#103—3981no—
118#103—3981no—
119#102—3981no—
120#102—3981no—
121#97—1991no—
122#97C2×D19911yescharacteristic
123#97C2×D19911yescharacteristic
124C2×Q8—3981no—
125C2×Q8—1991no—
126C2×Q8D79611yescharacteristic
127C2×D4—3981no—
128C2×D4—3981no—
129C2×D4—1991no—
130C2×D4D79611yescharacteristic
131C22×C4—3981no—
132C22×C4—3981no—
133C2×C8—21no—
134C2×C8—21no—
135C2×C8—3981no—
136C4⋊C4—3981no—
137C4⋊C4—3981no—
138C4⋊C4—3981no—
139C4⋊C4—3981no—
140C4⋊C4—3981no—
141C4⋊C4—3981no—
142C4⋊C4—1991no—
143C4⋊C4—1991no—
144C22⋊C4—21no—
145C22⋊C4—21no—
146C22⋊C4—3981no—
147C22⋊C4—3981no—
148C22⋊C4—3981no—
149C22⋊C4—3981no—
150C42C22×D19911yescharacteristic
151C23—21no—
152C23—3981no—
153Q8—41no—
154Q8—7961no—
155Q8—7962no—
156D4—41no—
157D4—7961no—
158D4—15921no—
159D4—15921no—
160C2×C4—21no—
161C2×C4—7961no—
162C2×C4—15921no—
163C2×C4—15921no—
164C2×C4—3981no—
165C2×C4—3981no—
166C2×C4—3981no—
167C2×C4D4×D19911yescharacteristic
168C2×C4D4×D19911yescharacteristic
169C2×C4C2×D79611yescharacteristic
170C8—41no—
171C8—41no—
172C8—7961no—
173C22—41no—
174C22—41no—
175C22—41no—
176C22—7961no—
177C22—7961no—
178C22—7961no—
179C22#2026392511yescharacteristic
180C4—21no—
181C4—21no—
182C4—21no—
183C4—41no—
184C4—7961no—
185C4—7961no—
186C4—7961no—
187C4—15921no—
188C2—81no—
189C2—15921no—
190C2#2716419211yescharacteristic
191C2#2716419711yescharacteristic
192C2#2716419811yescharacteristic
subgroupG/H
centerC22#20263925
commutator, cosocleC42×C199C23
FrattiniC42C22×D199
Fitting#329×C199C2
socleC22×C199#120
prime radical#27164762C2
hypercenter#329D199
hypocenterC199#1662
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. #27164387C2semidirect
3. #27164378C2semidirect
4. #27164377C2semidirect
5. #27163759C2semidirect
6. #27163757C2semidirect
7. #329×C199C2semidirect
96. C199#1662semidirect
104. #329D199semidirect

Maximal subgroups

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

subgroupindex
#271647622
#271643872
#271643782
#271643772
#271637592
#271637572
#329×C1992
#1662199