Finite Group Databasebeta

← #38767372

Subgroups — #38767372

Order 25472.

242 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 63 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#27164549C211yesmaximal, characteristic
2#27164072C211yesmaximal, characteristic, complemented
3#27163869C211yesmaximal, characteristic, complemented
4#27163756C211yesmaximal, characteristic, complemented
5#27163754C211yesmaximal, characteristic, complemented
6#27163752C211yesmaximal, characteristic
7#327×C199C211yesmaximal, characteristic
8#20263948—21no—
9#20263946C411yescharacteristic, complemented
10#20263942C411yescharacteristic, complemented
11#20263936—21no—
12#20263934C2211yescharacteristic
13#20263920C411yescharacteristic, complemented
14D4×Dic199—21no—
15D4×Dic199C411yescharacteristic, complemented
16#20263911—21no—
17C2×Dic199⋊C4—21no—
18C2×C4×Dic199—21no—
19C2×C4×Dic199C2211yescharacteristic
20Dic199⋊C8—21no—
21C22⋊(C199⋊C8)C2211yescharacteristic
22#20263865—21no—
23C199⋊#95C2211yescharacteristic
24#121×C199C2211yescharacteristic
25C22⋊C8×C199C2211yescharacteristic
26#95×C199C2211yescharacteristic
27C2×Dic398C2×C411yescharacteristic
28C22×Dic199—21no—
29C22×Dic199—21no—
30C22×Dic199D411yescharacteristic
31C22×Dic199D411yescharacteristic
32C2×C199⋊C8—21no—
33C4⋊Dic199—21no—
34C4⋊Dic199C2×C411yescharacteristic
35Dic199⋊C4—24no—
36Dic199⋊C4—21no—
37Dic199⋊C4—21no—
38Dic199⋊C4—21no—
39C22⋊Dic199—21no—
40C22⋊Dic199—21no—
41C22⋊Dic199—21no—
42C22⋊Dic199—21no—
43C4×Dic199—21no—
44C4×Dic199—42no—
45C4×Dic199C2×C411yescharacteristic
46C4×Dic199C2×C411yescharacteristic
47C4×Dic199D411yescharacteristic
48C4×Dic199D411yescharacteristic
49C2×D4×C199—21no—
50C2×D4×C199C2×C411yescharacteristic
51C22×C4×C199—21no—
52C22×C4×C199C2311yescharacteristic
53C2×C8×C199—21no—
54C4⋊C4×C199C2×C411yescharacteristic
55C22⋊C4×C199—21no—
56Dic398—41no—
57C2×Dic199—21no—
58C2×Dic199—21no—
59C2×Dic199—22no—
60C2×Dic199—22no—
61C2×Dic199—21no—
62C2×Dic199—21no—
63C2×Dic199—41no—
64C2×Dic199—41no—
65C2×Dic199—42no—
66C2×Dic199C22⋊C411yescharacteristic
67C2×Dic199C22⋊C411yescharacteristic
68C2×Dic199C22⋊C411yescharacteristic
69C2×Dic199C22⋊C411yescharacteristic
70C199⋊C8—41no—
71C23×C199—21no—
72C23×C199C2×D411yescharacteristic
73D4×C199—41no—
74D4×C199—41no—
75D4×C199—41no—
76C2×C4×C199—22no—
77C2×C4×C199—21no—
78C2×C4×C199—41no—
79C2×C4×C199—41no—
80C2×C4×C199—41no—
81C2×C4×C199C22×C411yescharacteristic
82C2×C4×C199C2×D411yescharacteristic
83C8×C199—41no—
84C22×C199—21no—
85C22×C199—21no—
86C22×C199—41no—
87C22×C199—41no—
88C22×C199—41no—
89C22×C199C2×C22⋊C411yescharacteristic
90C22×C199AGL(1,ℤ/8)11yescharacteristic
91C22×C199#13711yescharacteristic
92C4×C199—21no—
93C4×C199—21no—
94C4×C199—42no—
95C4×C199—41no—
96Dic199—21no—
97Dic199—21no—
98Dic199—21no—
99Dic199—42no—
100Dic199—41no—
101Dic199—41no—
102Dic199#10412yes—
103C2×C199—21no—
104C2×C199—21no—
105C2×C199—81no—
106C2×C199#41011yescharacteristic
107C2×C199#41711yescharacteristic
108C2×C199C2×#10411yescharacteristic
109C199#150911yescharacteristic, complemented, Sylow
110#1509—1991nomaximal, Sylow
111#529—1991no—
112#432—1991no—
113C4×C4⋊C4—1991no—
114#328—1991no—
115#328—1991no—
116#327—1991no—
117#327D19911yescharacteristic
118#126—3981no—
119#125—1991no—
120#124—1991no—
121#123—3981no—
122C22⋊Q8—1991no—
123#121C2×D19911yescharacteristic
124C4×Q8—1991no—
125C4×D4—3981no—
126C4×D4—1991no—
127#117—3981no—
128C2×C4⋊C4—3981no—
129C2×C42—3981no—
130C2×C42—1991no—
131C4⋊C8—3981no—
132C22⋊C8—1991no—
133C22⋊C8C2×D19911yescharacteristic
134#97—3981no—
135#95—1991no—
136#95C2×D19911yescharacteristic
137C2×Q8—1991no—
138C2×D4—21no—
139C2×D4C4×D19911yescharacteristic
140C22×C4—21no—
141C22×C4—3981no—
142C22×C4—3981no—
143C22×C4—1991no—
144C22×C4—1991no—
145C22×C4C22×D19911yescharacteristic
146C2×C8—21no—
147C2×C8—3981no—
148C4⋊C4—3984no—
149C4⋊C4—3981no—
150C4⋊C4—3981no—
151C4⋊C4—3981no—
152C4⋊C4—3981no—
153C4⋊C4—1991no—
154C4⋊C4C4×D19911yescharacteristic
155C22⋊C4—21no—
156C22⋊C4—3981no—
157C22⋊C4—3981no—
158C22⋊C4—3981no—
159C22⋊C4—3981no—
160C42—7962no—
161C42—3981no—
162C42—1991no—
163C42—1991no—
164C42—1991no—
165C42—1991no—
166C23—21no—
167C23D4×D19911yescharacteristic
168Q8—7961no—
169D4—41no—
170D4—41no—
171D4—41no—
172C2×C4—22no—
173C2×C4—21no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—7961no—
178C2×C4—7961no—
179C2×C4—7962no—
180C2×C4—3982no—
181C2×C4—3981no—
182C2×C4—3981no—
183C2×C4—3981no—
184C2×C4—3981no—
185C2×C4—3982no—
186C2×C4—1991no—
187C2×C4—1991no—
188C2×C4—1991no—
189C2×C4—1991no—
190C2×C4C2×C4×D19911yescharacteristic
191C2×C4D4×D19911yescharacteristic
192C8—41no—
193C8—7961no—
194C22—21no—
195C22—21no—
196C22—41no—
197C22—41no—
198C22—41no—
199C22C22⋊C4×D19911yescharacteristic
200C22#2026397911yescharacteristic
201C22#2026398411yescharacteristic
202C4—21no—
203C4—21no—
204C4—42no—
205C4—41no—
206C4—7962no—
207C4—7961no—
208C4—7961no—
209C4—3981no—
210C4—3981no—
211C4—3981no—
212C4—1992no—
213C2—21no—
214C2—21no—
215C2—81no—
216C2#2716399011yescharacteristic
217C2#2716401611yescharacteristic
218C2#104×D19911yescharacteristic
subgroupG/H
centerC22C22⋊C4×D199
commutatorC2×C4×C199C22×C4
FrattiniC22×C4C22×D199
Fitting#327×C199C2
socleC22×C199C2×C22⋊C4
prime radical#121×C199C22
cosocleC22×C4×C199C23
hypercenter#327D199
hypocenterC199#1509
LCSG ▷ C2×C4×C199 ▷ C22×C199 ▷ C199
UCSC1 ◁ C22 ◁ C22×C4 ◁ #327

Complements

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

NH ≅ G/Ntype
2. #27164072C2semidirect
3. #27163869C2semidirect
4. #27163756C2semidirect
5. #27163754C2semidirect
9. #20263946C4semidirect
10. #20263942C4semidirect
13. #20263920C4semidirect
15. D4×Dic199C4semidirect
109. C199#1509semidirect

Maximal subgroups

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

subgroupindex
#271645492
#271640722
#271638692
#271637562
#271637542
#271637522
#327×C1992
#1509199