Finite Group Databasebeta

← #38767877

Subgroups — #38767877

Order 25472.

236 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 53 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#27164644C211yesmaximal, characteristic
2#27164354C212yesmaximal, complemented
3#27163954C211yesmaximal, characteristic
4#27163756C212yesmaximal, complemented
5#323×C199C211yesmaximal, characteristic
6#20263934—21no—
7#20263934C2212yes—
8#20263932—22no—
9#20263932—22no—
10C2×C4⋊Dic199C2211yescharacteristic
11C2×Dic199⋊C4—21no—
12C2×C22⋊Dic199—21no—
13C2×C22⋊Dic199—21no—
14#20263889—22no—
15#20263880—22no—
16C199⋊#95C2211yescharacteristic
17C2×C22⋊C4×C199C2211yescharacteristic
18C22⋊C8×C199C2212yes—
19C2×Dic398—21no—
20C2×Dic398D412yes—
21C22×Dic199—21no—
22C22×Dic199—21no—
23C22×Dic199—21no—
24C22×Dic199—21no—
25C4⋊Dic199—21no—
26C4⋊Dic199—22no—
27C4⋊Dic199D412yes—
28Dic199⋊C4—22no—
29Dic199⋊C4—22no—
30Dic199⋊C4—42no—
31C22⋊Dic199—22no—
32C22⋊Dic199—22no—
33C22⋊Dic199—42no—
34C24×C199D411yescharacteristic
35C22×C4×C199D411yescharacteristic
36C22×C4×C199C2311yescharacteristic
37C2×C8×C199—22no—
38C22⋊C4×C199—22no—
39Dic398—42no—
40Dic398—81no—
41C2×Dic199—22no—
42C2×Dic199—22no—
43C2×Dic199—22no—
44C2×Dic199—41no—
45C2×Dic199—41no—
46C2×Dic199—41no—
47C2×Dic199—41no—
48C2×Dic199—41no—
49C2×Dic199—81no—
50C2×Dic199—81no—
51C23×C199—21no—
52C23×C199—21no—
53C23×C199—41no—
54C23×C199—41no—
55C23×C199Q1612yes—
56C23×C199C2×D411yescharacteristic
57C2×C4×C199—21no—
58C2×C4×C199—41no—
59C2×C4×C199—41no—
60C2×C4×C199C2×D412yes—
61C8×C199—42no—
62C22×C199—21no—
63C22×C199—21no—
64C22×C199—22no—
65C22×C199—42no—
66C22×C199—41no—
67C22×C199—41no—
68C22×C199—81no—
69C22×C199#12011yescharacteristic
70C22×C199C2×Q1611yescharacteristic
71C22×C199#13711yescharacteristic
72C4×C199—22no—
73C4×C199—41no—
74Dic199—42no—
75Dic199—42no—
76Dic199—42no—
77Dic199—81no—
78C2×C199—21no—
79C2×C199—21no—
80C2×C199—42no—
81C2×C199#45111yescharacteristic
82C2×C199#45311yescharacteristic
83C2×C199#45811yescharacteristic
84C199#161411yescharacteristic, complemented, Sylow
85#1614—1991nomaximal, Sylow
86#543—1991no—
87#484—1992no—
88#399—1991no—
89#328—1992no—
90#323D19911yescharacteristic
91C22⋊Q8—3982no—
92C22⋊Q8—3982no—
93C22⋊Q8—3981no—
94C22⋊Q8—1992no—
95C2×C4⋊C4—3981no—
96C2×C4⋊C4—1991no—
97C2×C22⋊C4—3981no—
98C2×C22⋊C4—3981no—
99C2×C22⋊C4C2×D19911yescharacteristic
100#107—3982no—
101#103—3982no—
102C22⋊C8C2×D19912yes—
103#95—1991no—
104C24D79611yescharacteristic
105C2×Q8—3981no—
106C2×Q8—1992no—
107C22×C4—3981no—
108C22×C4—3981no—
109C22×C4—3981no—
110C22×C4—3981no—
111C22×C4D79611yescharacteristic
112C22×C4C22×D19911yescharacteristic
113C2×C8—22no—
114C4⋊C4—7962no—
115C4⋊C4—3982no—
116C4⋊C4—3981no—
117C4⋊C4—3982no—
118C4⋊C4—3982no—
119C4⋊C4—1992no—
120C22⋊C4—22no—
121C22⋊C4—7962no—
122C22⋊C4—3982no—
123C22⋊C4—3982no—
124C23—21no—
125C23—21no—
126C23—41no—
127C23—41no—
128C23Dic79612yes—
129C23C2×D79611yescharacteristic
130Q8—7962no—
131Q8—15921no—
132C2×C4—21no—
133C2×C4—41no—
134C2×C4—41no—
135C2×C4—7961no—
136C2×C4—7961no—
137C2×C4—7961no—
138C2×C4—7961no—
139C2×C4—7961no—
140C2×C4—15921no—
141C2×C4—15921no—
142C2×C4—3982no—
143C2×C4—3982no—
144C2×C4—3982no—
145C2×C4D4×D19912yes—
146C8—42no—
147C22—21no—
148C22—21no—
149C22—22no—
150C22—42no—
151C22—41no—
152C22—41no—
153C22—81no—
154C22#2026392511yescharacteristic
155C22C2×Dic79611yescharacteristic
156C22#2026398811yescharacteristic
157C4—22no—
158C4—41no—
159C4—7962no—
160C4—7962no—
161C4—7962no—
162C4—15921no—
163C2—21no—
164C2—21no—
165C2—42no—
166C2#2716417611yescharacteristic
167C2#2716418811yescharacteristic
168C2#2716420911yescharacteristic
subgroupG/H
centerC22#20263925
commutator, cosocleC22×C4×C199C23
FrattiniC22×C4C22×D199
Fitting#323×C199C2
socleC22×C199#120
prime radicalC24×C199D4
hypercenter#323D199
hypocenterC199#1614
LCSG ▷ C22×C4×C199 ▷ C22×C199 ▷ C199
UCSC1 ◁ C22 ◁ C22×C4 ◁ #323

Complements

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

NH ≅ G/Ntype
2. #27164354C2semidirect
4. #27163756C2semidirect
84. C199#1614semidirect

Maximal subgroups

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

subgroupindex
#271646442
#271643542
#271639542
#271637562
#323×C1992
#1614199