Finite Group Databasebeta

← #38767218

Subgroups — #38767218

Order 25472.

214 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 61 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×#20263950C211yesmaximal, characteristic
2#27164072C211yesmaximal, characteristic, complemented
3#27164068C211yesmaximal, characteristic, complemented
4#27163744C211yesmaximal, characteristic, complemented
5#27163741C211yesmaximal, characteristic
6#27163741C211yesmaximal, characteristic
7#324×C199C211yesmaximal, characteristic, complemented
8#20263950—21no—
9#20263950—21no—
10#20263950—21no—
11#20263950—21no—
12C2×C4⋊Dic199C411yescharacteristic
13C2×Dic199⋊C4C411yescharacteristic
14C2×(C2×D199)⋊C4C411yescharacteristic
15C2×(C2×D199)⋊C4C411yescharacteristic
16C2×(C2×D199)⋊C4C2211yescharacteristic
17C2×C4×Dic199C2211yescharacteristic
18Dic199⋊C8—21no—
19Dic199⋊C8—21no—
20C22⋊(C199⋊C8)C2211yescharacteristic
21C22⋊(C199⋊C8)C2211yescharacteristic
22#20263865—21no—
23C8×Dic199—21no—
24C2×C4⋊C4×C199C2211yescharacteristic
25C22⋊C8×C199C2211yescharacteristic
26C22⋊C8×C199C2211yescharacteristic
27C23×D199C2×C411yescharacteristic
28C22×Dic199C2×C411yescharacteristic
29C22×Dic199C2×C411yescharacteristic
30C22×Dic199C2×C411yescharacteristic
31C2×C199⋊C8—21no—
32C2×C199⋊C8—21no—
33C4⋊Dic199—21no—
34C4⋊Dic199—21no—
35Dic199⋊C4—21no—
36Dic199⋊C4—21no—
37(C2×D199)⋊C4—21no—
38(C2×D199)⋊C4—21no—
39(C2×D199)⋊C4—21no—
40(C2×D199)⋊C4—21no—
41(C2×D199)⋊C4—21no—
42(C2×D199)⋊C4—21no—
43C4×Dic199D411yescharacteristic
44C4×Dic199D411yescharacteristic
45C4×Dic199D411yescharacteristic
46C4×Dic199D411yescharacteristic
47C22×C4×C199C2×C411yescharacteristic
48C22×C4×C199C2×C411yescharacteristic
49C22×C4×C199C2311yescharacteristic
50C2×C8×C199—21no—
51C2×C8×C199—21no—
52C4⋊C4×C199—21no—
53C4⋊C4×C199—21no—
54C22×D199—21no—
55C22×D199—21no—
56C22×D199—21no—
57C22×D199—41no—
58C22×D199—41no—
59C2×Dic199—21no—
60C2×Dic199—21no—
61C2×Dic199—21no—
62C2×Dic199—41no—
63C2×Dic199—41no—
64C2×Dic199C22⋊C411yescharacteristic
65C2×Dic199C22⋊C411yescharacteristic
66C2×Dic199C22⋊C411yescharacteristic
67C2×Dic199C22⋊C411yescharacteristic
68C199⋊C8—41no—
69C199⋊C8—41no—
70C23×C199C22×C411yescharacteristic
71C2×C4×C199—21no—
72C2×C4×C199—21no—
73C2×C4×C199—21no—
74C2×C4×C199—21no—
75C2×C4×C199—41no—
76C2×C4×C199—41no—
77C2×C4×C199C2×D411yescharacteristic
78C2×C4×C199C2×D411yescharacteristic
79C8×C199—41no—
80C8×C199—41no—
81C22×C199—21no—
82C22×C199—21no—
83C22×C199C2×C22⋊C411yescharacteristic
84C22×C199#13111yescharacteristic
85C22×C199#13111yescharacteristic
86C2×D199—41no—
87C2×D199—41no—
88C2×D199—41no—
89C2×D199—81no—
90C2×D199—81no—
91C4×C199—21no—
92C4×C199—21no—
93C4×C199—41no—
94C4×C199—41no—
95Dic199—21no—
96Dic199—21no—
97Dic199—21no—
98Dic199—21no—
99Dic199—41no—
100C2×C199—21no—
101C2×C199—21no—
102C2×C199#40811yescharacteristic
103C2×C199#41011yescharacteristic
104C2×C199#41311yescharacteristic
105D199—81no—
106C199#147311yescharacteristic, complemented, Sylow
107#1473—1991nomaximal, Sylow
108C2×#126—1991no—
109#432—1991no—
110#431—1991no—
111#324—1991no—
112#324D19911yescharacteristic, complemented
113#323—1991no—
114#323—1991no—
115#126—3981no—
116#126—3981no—
117#126—3981no—
118#126—3981no—
119C2×C4⋊C4—1991no—
120C2×C4⋊C4—1991no—
121C2×C4⋊C4C2×D19911yescharacteristic
122C2×C22⋊C4—1991no—
123C2×C22⋊C4—1991no—
124C2×C22⋊C4—1991no—
125C2×C42—1991no—
126C4⋊C8—3981no—
127C4⋊C8—3981no—
128C22⋊C8—1991no—
129C22⋊C8—1991no—
130C22⋊C8C2×D19911yescharacteristic
131C22⋊C8C2×D19911yescharacteristic
132#97—3981no—
133C4×C8—3981no—
134C24—1991no—
135C22×C4—1991no—
136C22×C4—1991no—
137C22×C4—1991no—
138C22×C4C4×D19911yescharacteristic
139C22×C4C4×D19911yescharacteristic
140C22×C4C22×D19911yescharacteristic
141C2×C8—21no—
142C2×C8—21no—
143C2×C8—3981no—
144C2×C8—3981no—
145C4⋊C4—21no—
146C4⋊C4—21no—
147C4⋊C4—3981no—
148C4⋊C4—3981no—
149C4⋊C4—3981no—
150C4⋊C4—3981no—
151C22⋊C4—3981no—
152C22⋊C4—3981no—
153C22⋊C4—3981no—
154C22⋊C4—3981no—
155C22⋊C4—3981no—
156C22⋊C4—3981no—
157C42—1991no—
158C42—1991no—
159C42—1991no—
160C42—1991no—
161C23—7961no—
162C23—7961no—
163C23—3981no—
164C23—3981no—
165C23—3981no—
166C23C2×C4×D19911yescharacteristic
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—7961no—
174C2×C4—7961no—
175C2×C4—3981no—
176C2×C4—3981no—
177C2×C4—3981no—
178C2×C4—1991no—
179C2×C4—1991no—
180C2×C4—1991no—
181C2×C4—1991no—
182C2×C4D4×D19911yescharacteristic
183C2×C4D4×D19911yescharacteristic
184C8—41no—
185C8—41no—
186C8—7961no—
187C8—7961no—
188C22—21no—
189C22—21no—
190C22—7961no—
191C22—7961no—
192C22—7961no—
193C22—15921no—
194C22—15921no—
195C22C22⋊C4×D19911yescharacteristic
196C22#2026396111yescharacteristic
197C22#2026396311yescharacteristic
198C4—21no—
199C4—21no—
200C4—41no—
201C4—41no—
202C4—7961no—
203C4—3981no—
204C4—3981no—
205C4—3981no—
206C4—3981no—
207C2—21no—
208C2—21no—
209C2—15921no—
210C2#2716398311yescharacteristic
211C2#2716399011yescharacteristic
212C2#2716400111yescharacteristic
subgroupG/H
centerC22C22⋊C4×D199
commutatorC23×C199C22×C4
FrattiniC22×C4C22×D199
Fitting#324×C199C2
socleC22×C199C2×C22⋊C4
prime radicalC23×D199C2×C4
cosocleC22×C4×C199C23
hypercenter#324D199
hypocenterC199#1473
LCSG ▷ C23×C199 ▷ C2×C199 ▷ C199
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
2. #27164072C2semidirect
3. #27164068C2semidirect
4. #27163744C2semidirect
7. #324×C199C2semidirect
106. C199#1473semidirect
112. #324D199semidirect

Maximal subgroups

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

subgroupindex
C2×#202639502
#271640722
#271640682
#271637442
#271637412
#324×C1992
#1473199