Finite Group Databasebeta

← #38767334

Subgroups — #38767334

Order 25472.

238 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 73 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#27164552C211yesmaximal, characteristic
2#27164071C211yesmaximal, characteristic, complemented
3#27163754C212yesmaximal, complemented
4#27163751C212yesmaximal
5#376×C199C211yesmaximal, characteristic, complemented
6#20263949—21no—
7#20263947C411yescharacteristic, complemented
8#20263943C411yescharacteristic, complemented
9#20263938—21no—
10#20263934C2211yescharacteristic
11#20263929C2211yescharacteristic
12C4×Dic398C411yescharacteristic, complemented
13C4×D796C411yescharacteristic, complemented
14C4×C22⋊D199—21no—
15#20263912—21no—
16C4⋊(C199⋊C8)—21no—
17C22⋊(C199⋊C8)C2212yes—
18C4×C199⋊C8—21no—
19C2×C42×C199—21no—
20C2×C42×C199C2211yescharacteristic
21#95×C199—21no—
22#95×C199C2212yes—
23C2×Dic398C2×C411yescharacteristic
24C2×D796C2×C411yescharacteristic
25C2×C22⋊D199—21no—
26C2×C4×D199—21no—
27C2×C199⋊C8—22no—
28C4⋊Dic199C2×C411yescharacteristic
29C4⋊Dic199C2×C411yescharacteristic
30Dic199⋊C4—21no—
31Dic199⋊C4—21no—
32Dic199⋊C4—21no—
33Dic199⋊C4—21no—
34C22⋊Dic199—21no—
35(C2×D199)⋊C4—21no—
36(C2×D199)⋊C4—21no—
37(C2×D199)⋊C4—21no—
38(C2×D199)⋊C4—21no—
39C4×Dic199—21no—
40C22×C4×C199—22no—
41C22×C4×C199D411yescharacteristic
42C22×C4×C199D411yescharacteristic
43C22×C4×C199C2311yescharacteristic
44C42×C199—24no—
45C42×C199C2×C411yescharacteristic
46C42×C199C2×C411yescharacteristic
47C42×C199D411yescharacteristic
48C42×C199D411yescharacteristic
49C22×D199—21no—
50Dic398—41no—
51D796—41no—
52C22⋊D199—42no—
53C2×Dic199—21no—
54C2×Dic199—21no—
55C2×Dic199—21no—
56C4×D199—42no—
57C199⋊C8—42no—
58C23×C199C2×D411yescharacteristic
59C2×C4×C199—22no—
60C2×C4×C199—22no—
61C2×C4×C199—24no—
62C2×C4×C199—41no—
63C2×C4×C199—42no—
64C2×C4×C199—42no—
65C2×C4×C199C22⋊C411yescharacteristic
66C2×C4×C199C22⋊C411yescharacteristic
67C2×C4×C199C22⋊C411yescharacteristic
68C2×C4×C199C22⋊C411yescharacteristic
69C2×C4×C199C22×C411yescharacteristic
70C2×C4×C199C2×D411yescharacteristic
71C22×C199—21no—
72C22×C199—21no—
73C22×C199C2×C22⋊C411yescharacteristic
74C22×C199#13512yes—
75C2×D199—41no—
76C2×D199—41no—
77C2×D199—41no—
78C4×C199—21no—
79C4×C199—21no—
80C4×C199—21no—
81C4×C199—21no—
82C4×C199—21no—
83C4×C199—44no—
84C4×C199#10412yes—
85Dic199—41no—
86Dic199—41no—
87Dic199—41no—
88C2×C199—22no—
89C2×C199#41011yescharacteristic
90C2×C199#41611yescharacteristic
91C2×C199C2×#10411yescharacteristic
92D199—81no—
93C199#150011yescharacteristic, complemented, Sylow
94#1500—1991nomaximal, Sylow
95#529—1991no—
96#431—1991no—
97#376D19911yescharacteristic, complemented
98#328—1992no—
99#327—1992no—
100#126—3981no—
101#125—1991no—
102#124—1991no—
103#123—3981no—
104C22⋊Q8—1991no—
105#121—1991no—
106C4×Q8—1991no—
107C4×D4—3981no—
108C4×D4—1991no—
109#117—3981no—
110C2×C42—21no—
111C2×C42C2×D19911yescharacteristic
112C4⋊C8—3981no—
113C22⋊C8—1992no—
114C4×C8—3981no—
115#95—21no—
116#95C2×D19912yes—
117C2×Q8—1991no—
118C2×D4—3981no—
119C2×D4—1991no—
120C22×C4—22no—
121C22×C4—3981no—
122C22×C4C22⋊D19911yescharacteristic
123C22×C4C22⋊D19911yescharacteristic
124C22×C4C22×D19911yescharacteristic
125C2×C8—3982no—
126C4⋊C4—3981no—
127C4⋊C4—3981no—
128C4⋊C4—3981no—
129C4⋊C4—3981no—
130C4⋊C4—1991no—
131C4⋊C4—1991no—
132C22⋊C4—3981no—
133C22⋊C4—3981no—
134C22⋊C4—3981no—
135C22⋊C4—3981no—
136C22⋊C4—3981no—
137C42—24no—
138C42—3981no—
139C42C4×D19911yescharacteristic
140C42C4×D19911yescharacteristic
141C42D79611yescharacteristic
142C42D79611yescharacteristic
143C23—3981no—
144C23C2×C22⋊D19911yescharacteristic
145Q8—7961no—
146D4—7961no—
147D4—7962no—
148C2×C4—22no—
149C2×C4—22no—
150C2×C4—24no—
151C2×C4—41no—
152C2×C4—42no—
153C2×C4—42no—
154C2×C4—7962no—
155C2×C4—3981no—
156C2×C4—3981no—
157C2×C4—3981no—
158C2×C4(C2×D199)⋊C411yescharacteristic
159C2×C4(C2×D199)⋊C411yescharacteristic
160C2×C4(C2×D199)⋊C411yescharacteristic
161C2×C4(C2×D199)⋊C411yescharacteristic
162C2×C4C2×C4×D19911yescharacteristic
163C2×C4C2×D79611yescharacteristic
164C8—7962no—
165C22—21no—
166C22—21no—
167C22—7961no—
168C22—7961no—
169C22—7961no—
170C22C2×(C2×D199)⋊C411yescharacteristic
171C22#2026397512yes—
172C4—21no—
173C4—21no—
174C4—21no—
175C4—21no—
176C4—21no—
177C4—44no—
178C4—7961no—
179C4—7961no—
180C4—7961no—
181C4#2026388312yes—
182C2—22no—
183C2—15921no—
184C2#2716399111yescharacteristic
185C2#2716401211yescharacteristic
186C2C2×#2026388311yescharacteristic
subgroupG/H
centerC2×C4(C2×D199)⋊C4
commutatorC2×C4×C199C22×C4
FrattiniC22×C4C22×D199
Fitting#376×C199C2
socleC22×C199C2×C22⋊C4
prime radical#20263929C22
cosocleC22×C4×C199C23
hypercenter#376D199
hypocenterC199#1500
LCSG ▷ C2×C4×C199 ▷ C22×C199 ▷ C199
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #376

Complements

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

NH ≅ G/Ntype
2. #27164071C2semidirect
3. #27163754C2semidirect
5. #376×C199C2semidirect
7. #20263947C4semidirect
8. #20263943C4semidirect
12. C4×Dic398C4semidirect
13. C4×D796C4semidirect
93. C199#1500semidirect
97. #376D199semidirect

Maximal subgroups

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

subgroupindex
#271645522
#271640712
#271637542
#271637512
#376×C1992
#1500199