Finite Group Databasebeta

← #38770015

Subgroups — #38770015

Order 25472.

466 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 67 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, complemented
2#27163873C212yesmaximal
3#27163873C211yesmaximal, characteristic
4#27163784C212yesmaximal, complemented
5#342×C199C211yesmaximal, characteristic, complemented
6C24×D199C2211yescharacteristic
7#20263934—22no—
8#20263932—24no—
9C2×C22⋊Dic199C2211yescharacteristic
10C2×C22⋊Dic199C2212yes—
11C2×(C2×D199)⋊C4—24no—
12C2×(C2×D199)⋊C4—22no—
13C2×C22⋊C4×C199C2212yes—
14C2×C22⋊C4×C199C2211yescharacteristic
15C23×D199—22no—
16C23×D199—21no—
17C23×D199—21no—
18C23×D199—22no—
19C23×D199—44no—
20C23×D199Q812yescomplemented
21C2×Dic398—22no—
22C22×Dic199—21no—
23C22×Dic199D414yes—
24C4⋊Dic199—22no—
25Dic199⋊C4—24no—
26C22⋊Dic199—22no—
27C22⋊Dic199—22no—
28C22⋊Dic199—22no—
29(C2×D199)⋊C4—44no—
30(C2×D199)⋊C4—48no—
31C24×C199C2311yescharacteristic
32C22×C4×C199—22no—
33C22×C4×C199D412yes—
34C22⋊C4×C199—22no—
35C22⋊C4×C199—22no—
36C22⋊C4×C199—22no—
37C22×D199—28no—
38C22×D199—24no—
39C22×D199—24no—
40C22×D199—22no—
41C22×D199—44no—
42C22×D199—44no—
43C22×D199—48no—
44C22×D199—44no—
45C22×D199—42no—
46C22×D199—41no—
47C22×D199—42no—
48C22×D199—41no—
49Dic398—82no—
50C2×Dic199—24no—
51C2×Dic199—44no—
52C2×Dic199—42no—
53C2×Dic199—41no—
54C23×C199—42no—
55C23×C199C2×D412yes—
56C23×C199C2×D411yescharacteristic
57C23×C199C2×Q811yescharacteristic
58C23×C199#4111yescharacteristic
59C23×C199#4112yes—
60C2×C4×C199—22no—
61C2×C4×C199—42no—
62C2×C4×C199—44no—
63C2×C4×C199—42no—
64C22×C199—24no—
65C22×C199—22no—
66C22×C199—41no—
67C22×C199—41no—
68C22×C199—42no—
69C22×C199#12011yescharacteristic
70C22×C199C22⋊Q812yes—
71C22×C199C22⋊Q811yescharacteristic
72C22×C199#12412yes—
73C22×C199#12411yescharacteristic
74C2×D199—28no—
75C2×D199—24no—
76C2×D199—44no—
77C2×D199—48no—
78C2×D199—44no—
79C2×D199—42no—
80C2×D199—42no—
81C2×D199—44no—
82C4×C199—42no—
83C4×C199—82no—
84Dic199—44no—
85Dic199—81no—
86C2×C199—21no—
87C2×C199—22no—
88C2×C199—22no—
89C2×C199—21no—
90C2×C199#39311yescharacteristic
91C2×C199B(4,𝔽2)12yes—
92D199—44no—
93C199#202911yescharacteristic, complemented, Sylow
94#2029—1991nomaximal, Sylow
95#543—1991no—
96#379—1992no—
97#379—1991no—
98#342—1992no—
99#342D19911yescharacteristic, complemented
100C25—1991no—
101C22⋊Q8—3984no—
102C22⋊Q8—3982no—
103C2×C22⋊C4—3984no—
104C2×C22⋊C4—3982no—
105C2×C22⋊C4—1991no—
106C2×C22⋊C4—1992no—
107C2×C22⋊C4C2×D19912yes—
108C2×C22⋊C4C2×D19911yescharacteristic
109C24—7964no—
110C24—3981no—
111C24—3982no—
112C24—3982no—
113C24—3981no—
114C24—1992no—
115C24C22×D19911yescharacteristic
116C2×Q8—3982no—
117C22×C4—22no—
118C22×C4—3981no—
119C22×C4—1994no—
120C22×C4D79612yes—
121C4⋊C4—3984no—
122C4⋊C4—3982no—
123C22⋊C4—22no—
124C22⋊C4—22no—
125C22⋊C4—22no—
126C22⋊C4—7964no—
127C22⋊C4—7968no—
128C22⋊C4—3982no—
129C22⋊C4—3982no—
130C22⋊C4—3982no—
131C23—42no—
132C23—7964no—
133C23—7962no—
134C23—7964no—
135C23—7968no—
136C23—7964no—
137C23—7962no—
138C23—7961no—
139C23—7961no—
140C23—3988no—
141C23—3984no—
142C23—3984no—
143C23—3982no—
144C23D4×D19912yes—
145C23C2×D79611yescharacteristic
146C23Q8×D19911yescharacteristic
147C23#1569188511yescharacteristic
148C23#1569188612yes—
149Q8—15922no—
150C2×C4—22no—
151C2×C4—42no—
152C2×C4—44no—
153C2×C4—42no—
154C2×C4—7964no—
155C2×C4—7962no—
156C2×C4—7961no—
157C2×C4—3984no—
158C22—24no—
159C22—22no—
160C22—41no—
161C22—41no—
162C22—42no—
163C22—7968no—
164C22—7964no—
165C22—7964no—
166C22—7962no—
167C22—7962no—
168C22—7964no—
169C22—3988no—
170C22—3984no—
171C22#2026392511yescharacteristic
172C22#2026393312yes—
173C22#2026393511yescharacteristic
174C22#2026394112yes—
175C22#2026394311yescharacteristic
176C4—42no—
177C4—82no—
178C4—7964no—
179C4—15921no—
180C2—22no—
181C2—21no—
182C2—22no—
183C2—21no—
184C2—7964no—
185C2#2716393111yescharacteristic
186C2#2716420612yes—
subgroupG/H
centerC22#20263925
commutator, cosocleC24×C199C23
FrattiniC24C22×D199
Fitting#342×C199C2
socleC22×C199#120
prime radicalC24×D199C22
hypercenter#342D199
hypocenterC199#2029
LCSG ▷ C24×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C22 ◁ C24 ◁ #342

Complements

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

NH ≅ G/Ntype
1. #27164644C2semidirect
4. #27163784C2semidirect
5. #342×C199C2semidirect
20. C23×D199Q8semidirect
93. C199#2029semidirect
99. #342D199semidirect

Maximal subgroups

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

subgroupindex
#271646442
#271638732
#271637842
#342×C1992
#2029199