Finite Group Databasebeta

← #38767552

Subgroups — #38767552

Order 25472.

242 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 65 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#27164566C211yesmaximal, characteristic
2#27164074C211yesmaximal, characteristic, complemented
3#27164070C211yesmaximal, characteristic, complemented
4#27163765C211yesmaximal, characteristic, complemented
5#27163763C211yesmaximal, characteristic, complemented
6#27163762C211yesmaximal, characteristic, complemented
7#331×C199C211yesmaximal, characteristic, complemented
8C2×#15691885—21no—
9#20263955C2211yescharacteristic
10#20263942—21no—
11#20263926—21no—
12#20263926C412yes—
13D4×Dic199—21no—
14D4×Dic199C412yes—
15C42×D199C2211yescharacteristic
16C4⋊(C199⋊C8)C2211yescharacteristic
17C4⋊(C199⋊C8)C2211yescharacteristic
18(C2×D199)⋊C8—21no—
19(C2×D199)⋊C8—21no—
20#20263865—21no—
21C8×Dic199—21no—
22#127×C199C2211yescharacteristic
23C4⋊C8×C199C2211yescharacteristic
24C4⋊C8×C199C2211yescharacteristic
25#15691885—42no—
26#15691885—42no—
27C2×Dic398—21no—
28C2×C22⋊D199—22no—
29C22×Dic199—22no—
30C2×C4×D199C2×C411yescharacteristic
31C2×C4×D199D411yescharacteristic
32C2×C4×D199D411yescharacteristic
33C2×C199⋊C8—21no—
34C2×C199⋊C8—21no—
35C4⋊Dic199—21no—
36C4⋊Dic199C2×C412yes—
37C22⋊Dic199—22no—
38C22⋊Dic199—22no—
39C4×Dic199C2×C411yescharacteristic
40C4×Dic199D411yescharacteristic
41C4×Dic199D411yescharacteristic
42C2×D4×C199—21no—
43C2×D4×C199—21no—
44C2×D4×C199C2×C412yes—
45C2×C8×C199—21no—
46C2×C8×C199—21no—
47C42×C199C2311yescharacteristic
48C22×D199C22⋊C411yescharacteristic
49Dic398—41no—
50Dic398—41no—
51C22⋊D199—44no—
52C2×Dic199—22no—
53C2×Dic199—44no—
54C2×Dic199C22⋊C411yescharacteristic
55C2×Dic199C22⋊C411yescharacteristic
56C2×Dic199C22⋊C411yescharacteristic
57C4×D199—21no—
58C4×D199—21no—
59C4×D199—21no—
60C4×D199—21no—
61C4×D199—41no—
62C199⋊C8—41no—
63C199⋊C8—41no—
64C23×C199—22no—
65D4×C199—42no—
66D4×C199—41no—
67D4×C199—41no—
68D4×C199—41no—
69D4×C199—41no—
70C2×C4×C199C22×C411yescharacteristic
71C2×C4×C199C2×D411yescharacteristic
72C2×C4×C199C2×D411yescharacteristic
73C8×C199—41no—
74C8×C199—41no—
75C22×C199—42no—
76C22×C199—42no—
77C22×C199—42no—
78C22×C199C2×C22⋊C411yescharacteristic
79C2×D199—21no—
80C2×D199—21no—
81C2×D199—21no—
82C4×C199—21no—
83C4×C199#13511yescharacteristic
84C4×C199#13511yescharacteristic
85C4×C199AGL(1,ℤ/8)11yescharacteristic
86C4×C199AGL(1,ℤ/8)11yescharacteristic
87Dic199—21no—
88Dic199—21no—
89Dic199—21no—
90Dic199—42no—
91C2×C199—82no—
92C2×C199#41311yescharacteristic
93C2×C199#41611yescharacteristic
94C2×C199#41811yescharacteristic
95D199—41no—
96C199#155311yescharacteristic, complemented, Sylow
97#1553—1991nomaximal, Sylow
98#532—1991no—
99#432—1991no—
100#431—1991no—
101#332—1991no—
102#332—1991no—
103#331—1991no—
104#331D19911yescharacteristic, complemented
105C2×#41—3981no—
106C4⋊Q8—1991no—
107#127C2×D19911yescharacteristic
108#124—3981no—
109#121—3981no—
110#121—1992no—
111C4×D4—3981no—
112C4×D4—1992no—
113C2×C42—1991no—
114C4⋊C8—1991no—
115C4⋊C8—1991no—
116C4⋊C8C2×D19911yescharacteristic
117C4⋊C8C2×D19911yescharacteristic
118C22⋊C8—3981no—
119C22⋊C8—3981no—
120#97—3981no—
121C4×C8—3981no—
122#41—7962no—
123#41—7962no—
124C2×Q8—3981no—
125C2×D4—21no—
126C2×D4—21no—
127C2×D4—3982no—
128C2×D4C4×D19912yes—
129C22×C4—3982no—
130C22×C4—1991no—
131C22×C4—1991no—
132C22×C4—1991no—
133C2×C8—21no—
134C2×C8—21no—
135C2×C8—3981no—
136C2×C8—3981no—
137C4⋊C4—3981no—
138C4⋊C4—1992no—
139C22⋊C4—3982no—
140C22⋊C4—3982no—
141C42—1991no—
142C42—1991no—
143C42—1991no—
144C42C22×D19911yescharacteristic
145C23—22no—
146C23—1991no—
147Q8—7961no—
148Q8—7961no—
149D4—42no—
150D4—41no—
151D4—41no—
152D4—41no—
153D4—41no—
154D4—7964no—
155C2×C4—7961no—
156C2×C4—7964no—
157C2×C4—3982no—
158C2×C4—3981no—
159C2×C4—3981no—
160C2×C4—3981no—
161C2×C4—3981no—
162C2×C4—1991no—
163C2×C4—1991no—
164C2×C4—1991no—
165C2×C4C2×C4×D19911yescharacteristic
166C2×C4D4×D19911yescharacteristic
167C2×C4D4×D19911yescharacteristic
168C8—41no—
169C8—41no—
170C8—7961no—
171C8—7961no—
172C22—42no—
173C22—42no—
174C22—42no—
175C22—3981no—
176C22—3981no—
177C22—3981no—
178C22C22⋊C4×D19911yescharacteristic
179C4—21no—
180C4—7962no—
181C4—3981no—
182C4—3981no—
183C4—3981no—
184C4#2026397411yescharacteristic
185C4#2026397611yescharacteristic
186C4#2026397911yescharacteristic
187C4#2026398211yescharacteristic
188C2—82no—
189C2—7961no—
190C2#2716399911yescharacteristic
191C2#2716401111yescharacteristic
192C2#2716402111yescharacteristic
subgroupG/H
centerC22C22⋊C4×D199
commutatorC2×C4×C199C22×C4
FrattiniC42C22×D199
Fitting#331×C199C2
socleC22×C199C2×C22⋊C4
prime radical#27164566C2
cosocleC42×C199C23
hypercenter#331D199
hypocenterC199#1553
LCSG ▷ C2×C4×C199 ▷ C22×C199 ▷ C199
UCSC1 ◁ C22 ◁ C42 ◁ #331

Complements

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

NH ≅ G/Ntype
2. #27164074C2semidirect
3. #27164070C2semidirect
4. #27163765C2semidirect
5. #27163763C2semidirect
6. #27163762C2semidirect
7. #331×C199C2semidirect
96. C199#1553semidirect
104. #331D199semidirect

Maximal subgroups

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

subgroupindex
#271645662
#271640742
#271640702
#271637652
#271637632
#271637622
#331×C1992
#1553199