Finite Group Databasebeta

← #38769897

Subgroups — #38769897

Order 25472.

370 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 75 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
1C2×#20263985C212yesmaximal, complemented
2C2×#20263882C212yesmaximal, complemented
3#27163923C211yesmaximal, characteristic, complemented
4#27163787C211yesmaximal, characteristic, complemented
5#537×C199C211yesmaximal, characteristic
6C22×Dic398C2211yescharacteristic
7#20263985—24no—
8#20263985—24no—
9C2×#15691876—22no—
10C2×#15691874—22no—
11C2×#15691870C2212yes—
12C2×C4×Dic199C2211yescharacteristic
13#20263882—24no—
14C199⋊#95—22no—
15C2×#41×C199C2212yes—
16C22⋊Q8×C199—22no—
17#121×C199—22no—
18#121×C199—22no—
19C2×C4⋊C4×C199C2211yescharacteristic
20C2×Dic398—22no—
21C2×Dic398—22no—
22C2×Dic398—21no—
23C2×Dic398D412yescomplemented
24C22×Dic199—21no—
25C22×Dic199—21no—
26#15691876—28no—
27#15691874—28no—
28#15691870—22no—
29#15691870D414yescomplemented
30C2×C199⋊C8—22no—
31C4×Dic199—22no—
32#41×C199—22no—
33#41×C199—22no—
34#41×C199—22no—
35#41×C199—22no—
36C2×Q8×C199D412yes—
37C2×D4×C199—22no—
38C2×D4×C199—22no—
39C2×D4×C199D412yes—
40C22×C4×C199—22no—
41C22×C4×C199D412yes—
42C22×C4×C199C2311yescharacteristic
43C4⋊C4×C199—22no—
44C4⋊C4×C199—22no—
45C22⋊C4×C199—24no—
46Dic398—24no—
47Dic398—42no—
48C2×Dic199—22no—
49C2×Dic199—22no—
50C2×Dic199—41no—
51C2×Dic199—41no—
52C2×Dic199—41no—
53C2×Dic199—41no—
54C199⋊C8—24no—
55C23×C199—22no—
56C23×C199C2×D411yescharacteristic
57Q8×C199—22no—
58Q8×C199—22no—
59D4×C199—22no—
60D4×C199—22no—
61D4×C199—42no—
62D4×C199—42no—
63D4×C199—82no—
64C2×C4×C199—22no—
65C2×C4×C199—22no—
66C2×C4×C199—42no—
67C2×C4×C199—42no—
68C2×C4×C199—42no—
69C2×C4×C199C2×D411yescharacteristic
70C2×C4×C199C2×D412yes—
71C2×C4×C199C2×D412yes—
72C2×C4×C199#4111yescharacteristic
73C22×C199—21no—
74C22×C199—21no—
75C22×C199—42no—
76C22×C199—42no—
77C22×C199—42no—
78C22×C199#12011yescharacteristic
79C22×C199#12111yescharacteristic
80C22×C199#12711yescharacteristic
81C4×C199—42no—
82C4×C199—42no—
83C4×C199#12012yes—
84C4×C199#12112yes—
85Dic199—42no—
86Dic199—42no—
87C2×C199—21no—
88C2×C199—21no—
89C2×C199—82no—
90C2×C199#39211yescharacteristic
91C2×C199#45511yescharacteristic
92C2×C199#45611yescharacteristic
93C199#200711yescharacteristic, complemented, Sylow
94#2007—1991nomaximal, Sylow
95C2×#137—1992no—
96#537D19911yescharacteristic
97C2×#104—1992no—
98#391—1991no—
99#343—1991no—
100C2×#41C2×D19912yes—
101C22×Q8—1991no—
102#137—3984no—
103#137—3984no—
104C2×Q16—3982no—
105C2×SD16—3982no—
106C2×M16—1992no—
107C22⋊Q8—22no—
108#121—22no—
109#121—22no—
110C2×C4⋊C4C2×D19911yescharacteristic
111C2×C42—1991no—
112#104—3984no—
113#95—3982no—
114#41—22no—
115#41—22no—
116#41—22no—
117#41—22no—
118C2×Q8—3982no—
119C2×Q8—3982no—
120C2×Q8—3981no—
121C2×Q8—1992no—
122C2×Q8C22⋊D19912yes—
123C2×D4—22no—
124C2×D4—22no—
125C2×D4C22⋊D19912yes—
126C22×C4—22no—
127C22×C4—3981no—
128C22×C4—3981no—
129C22×C4C22⋊D19912yes—
130C22×C4C22×D19911yescharacteristic
131Q16—3988no—
132SD16—3988no—
133M16—3982no—
134M16—1994no—
135C2×C8—3982no—
136C4⋊C4—22no—
137C4⋊C4—22no—
138C22⋊C4—24no—
139C42—3982no—
140C23—22no—
141C23C2×C22⋊D19911yescharacteristic
142Q8—22no—
143Q8—22no—
144Q8—7962no—
145Q8—3984no—
146D4—22no—
147D4—22no—
148D4—42no—
149D4—42no—
150D4—82no—
151C2×C4—22no—
152C2×C4—22no—
153C2×C4—42no—
154C2×C4—42no—
155C2×C4—42no—
156C2×C4—7961no—
157C2×C4—7961no—
158C2×C4—7961no—
159C2×C4—7961no—
160C2×C4—3982no—
161C2×C4—3982no—
162C2×C4D4×D19911yescharacteristic
163C2×C4D4×D19912yes—
164C2×C4C2×C22⋊D19912yes—
165C2×C4#1569188511yescharacteristic
166C8—3984no—
167C22—21no—
168C22—21no—
169C22—42no—
170C22—42no—
171C22—42no—
172C22#2026392411yescharacteristic
173C22#2026392611yescharacteristic
174C22#2026395111yescharacteristic
175C4—42no—
176C4—42no—
177C4—7962no—
178C4—7962no—
179C4#2026392312yes—
180C4#2026392712yes—
181C2—21no—
182C2—21no—
183C2—82no—
184C2#2716392611yescharacteristic
185C2#2716419611yescharacteristic
186C2#2716420111yescharacteristic
subgroupG/H
centerC22#20263924
commutator, cosocleC22×C4×C199C23
FrattiniC22×C4C22×D199
Fitting, prime radical#537×C199C2
socleC22×C199#120
hypercenter#537D199
hypocenterC199#2007
LCSG ▷ C22×C4×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C22 ◁ C22×C4 ◁ #537

Complements

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

NH ≅ G/Ntype
1. C2×#20263985C2semidirect
2. C2×#20263882C2semidirect
3. #27163923C2semidirect
4. #27163787C2semidirect
23. C2×Dic398D4semidirect
29. #15691870D4semidirect
93. C199#2007semidirect

Maximal subgroups

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

subgroupindex
C2×#202639852
C2×#202638822
#271639232
#271637872
#537×C1992
#2007199