Finite Group Databasebeta

← #38770102

Subgroups — #38770102

Order 25472.

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

normal subgroups only

This group has 382 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#27164580C211yesmaximal, characteristic
2C2×#20263869C212yesmaximal, complemented
3#27163895C212yesmaximal, complemented
4#27163890C211yesmaximal, characteristic, complemented
5#342×C199C211yesmaximal, characteristic, complemented
6C22×C22⋊D199C2212yes—
7C23×Dic199C2211yescharacteristic
8#20263939—22no—
9#20263929—21no—
10#20263929—21no—
11#20263925—22no—
12C2×Dic199⋊C4—21no—
13C2×C22⋊Dic199C2211yescharacteristic
14C2×(C2×D199)⋊C4—22no—
15#20263869—24no—
16C199⋊#95—22no—
17C199⋊#95—21no—
18C2×C22⋊C4×C199C2212yes—
19C2×C22⋊C4×C199C2211yescharacteristic
20C23×D199D412yes—
21C2×D796—21no—
22C2×D796—21no—
23C2×C22⋊D199—22no—
24C2×C22⋊D199—22no—
25C2×C22⋊D199—22no—
26C2×C22⋊D199—22no—
27C2×C22⋊D199—22no—
28C2×C22⋊D199—22no—
29C22×Dic199—22no—
30C22×Dic199—21no—
31C22×Dic199—21no—
32C22×Dic199—42no—
33C22×Dic199D412yes—
34C4⋊Dic199—21no—
35C4⋊Dic199—21no—
36Dic199⋊C4—42no—
37C22⋊Dic199—21no—
38C22⋊Dic199D412yescomplemented
39(C2×D199)⋊C4—22no—
40(C2×D199)⋊C4—22no—
41(C2×D199)⋊C4—44no—
42C24×C199C2311yescharacteristic
43C22×C4×C199—22no—
44C22×C4×C199D411yescharacteristic
45C22×C4×C199D411yescharacteristic
46C22⋊C4×C199—22no—
47C22⋊C4×C199—22no—
48C22⋊C4×C199—22no—
49C22×D199—22no—
50C22×D199—22no—
51C22×D199—22no—
52C22×D199—42no—
53C22×D199—42no—
54D796—81no—
55D796—81no—
56C22⋊D199—42no—
57C22⋊D199—42no—
58C22⋊D199—42no—
59C22⋊D199—42no—
60C2×Dic199—24no—
61C2×Dic199—22no—
62C2×Dic199—22no—
63C2×Dic199—22no—
64C2×Dic199—22no—
65C2×Dic199—22no—
66C2×Dic199—41no—
67C2×Dic199—41no—
68C2×Dic199—41no—
69C2×Dic199—42no—
70C2×Dic199—41no—
71C23×C199—41no—
72C23×C199—41no—
73C23×C199C2×D412yes—
74C23×C199C2×D411yescharacteristic
75C23×C199C2×D411yescharacteristic
76C23×C199#4112yes—
77C23×C199#4111yescharacteristic
78C2×C4×C199—21no—
79C2×C4×C199—21no—
80C2×C4×C199—41no—
81C2×C4×C199—41no—
82C2×C4×C199—42no—
83C2×C4×C199—42no—
84C2×C4×C199—42no—
85C22×C199—21no—
86C22×C199—22no—
87C22×C199—21no—
88C22×C199—22no—
89C22×C199—41no—
90C22×C199—42no—
91C22×C199—41no—
92C22×C199#12011yescharacteristic
93C22×C199#12112yes—
94C22×C199#12111yescharacteristic
95C22×C199#12312yes—
96C22×C199#12411yescharacteristic
97C2×D199—44no—
98C2×D199—42no—
99C2×D199—42no—
100C2×D199—42no—
101C2×D199—82no—
102C4×C199—41no—
103C4×C199—41no—
104C4×C199—82no—
105Dic199—42no—
106Dic199—42no—
107Dic199—42no—
108C2×C199—21no—
109C2×C199—22no—
110C2×C199—22no—
111C2×C199—21no—
112C2×C199#39411yescharacteristic
113C2×C199B(4,𝔽2)11yescharacteristic
114C2×C199#45811yescharacteristic
115D199—82no—
116C199#204411yescharacteristic, complemented, Sylow
117#2044—1991nomaximal, Sylow
118#534—1991no—
119C2×#99—1992no—
120#386—1992no—
121#385—1991no—
122#342D19911yescharacteristic, complemented
123C22×D4—1992no—
124C23×C4—1991no—
125#123—3982no—
126#121—3981no—
127#121—3981no—
128#120—3982no—
129C2×C4⋊C4—3981no—
130C2×C22⋊C4—3982no—
131C2×C22⋊C4—1991no—
132C2×C22⋊C4C2×D19912yes—
133C2×C22⋊C4C2×D19911yescharacteristic
134#99—3984no—
135#95—3982no—
136#95—3981no—
137C24—1992no—
138C24C22×D19911yescharacteristic
139C2×D4—3981no—
140C2×D4—3981no—
141C2×D4—3982no—
142C2×D4—3982no—
143C2×D4—3982no—
144C2×D4—3982no—
145C2×D4—3982no—
146C2×D4—3982no—
147C22×C4—22no—
148C22×C4—7962no—
149C22×C4—3982no—
150C22×C4—3981no—
151C22×C4—3981no—
152C22×C4—1992no—
153C22×C4D79611yescharacteristic
154C22×C4D79611yescharacteristic
155C4⋊C4—7962no—
156C4⋊C4—3981no—
157C4⋊C4—3981no—
158C22⋊C4—22no—
159C22⋊C4—22no—
160C22⋊C4—22no—
161C22⋊C4—7964no—
162C22⋊C4—3982no—
163C22⋊C4—3982no—
164C22⋊C4—3981no—
165C22⋊C4—1992no—
166C23—41no—
167C23—41no—
168C23—7962no—
169C23—7962no—
170C23—3982no—
171C23—3982no—
172C23—3982no—
173C23D4×D19912yes—
174C23D4×D19911yescharacteristic
175C23C2×D79611yescharacteristic
176C23#1569188612yes—
177C23#1569188711yescharacteristic
178D4—7962no—
179D4—7962no—
180D4—7962no—
181D4—7962no—
182D4—15921no—
183D4—15921no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—42no—
191C2×C4—7961no—
192C2×C4—7961no—
193C2×C4—7962no—
194C2×C4—7961no—
195C2×C4—7961no—
196C2×C4—3984no—
197C2×C4—3982no—
198C2×C4—3982no—
199C2×C4—3982no—
200C2×C4—3982no—
201C2×C4—3982no—
202C22—21no—
203C22—22no—
204C22—21no—
205C22—22no—
206C22—41no—
207C22—42no—
208C22—41no—
209C22—7964no—
210C22—7962no—
211C22—7962no—
212C22—7962no—
213C22—15922no—
214C22#2026392511yescharacteristic
215C22#2026392812yes—
216C22#2026393011yescharacteristic
217C22#2026394012yes—
218C22#2026394311yescharacteristic
219C4—41no—
220C4—41no—
221C4—82no—
222C4—7962no—
223C4—7962no—
224C4—7962no—
225C2—22no—
226C2—21no—
227C2—22no—
228C2—21no—
229C2—15922no—
230C2#2716393611yescharacteristic
231C2#2716420611yescharacteristic
232C2#2716420911yescharacteristic
subgroupG/H
centerC22#20263925
commutator, cosocleC24×C199C23
FrattiniC24C22×D199
Fitting#342×C199C2
socleC22×C199#120
prime radical#27164580C2
hypercenter#342D199
hypocenterC199#2044
LCSG ▷ C24×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C22 ◁ C24 ◁ #342

Complements

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

NH ≅ G/Ntype
2. C2×#20263869C2semidirect
3. #27163895C2semidirect
4. #27163890C2semidirect
5. #342×C199C2semidirect
38. C22⋊Dic199D4semidirect
116. C199#2044semidirect
122. #342D199semidirect

Maximal subgroups

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

subgroupindex
#271645802
C2×#202638692
#271638952
#271638902
#342×C1992
#2044199