Finite Group Databasebeta

← #38767569

Subgroups — #38767569

Order 25472.

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

normal subgroups only

This group has 246 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#27164040C211yesmaximal, characteristic, complemented
4#27163765C211yesmaximal, characteristic, complemented
5#27163763C211yesmaximal, characteristic, complemented
6#27163762C211yesmaximal, characteristic, complemented
7#331×C199C211yesmaximal, characteristic, complemented
8C2×#15691885—21no—
9C2×#15691869—21no—
10#20263955C2211yescharacteristic
11#20263942—21no—
12#20263926—21no—
13#20263926C412yes—
14D4×Dic199—21no—
15D4×Dic199C412yes—
16C42×D199C2211yescharacteristic
17C4⋊(C199⋊C8)C2211yescharacteristic
18C4⋊(C199⋊C8)C2211yescharacteristic
19Dic199⋊C8—21no—
20(C2×D199)⋊C8—21no—
21#20263865—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
33#15691869—42no—
34C2×C199⋊C8—21no—
35C2×C199⋊C8—21no—
36C4⋊Dic199—21no—
37C4⋊Dic199C2×C412yes—
38C22⋊Dic199—22no—
39C22⋊Dic199—22no—
40C4×Dic199C2×C411yescharacteristic
41C4×Dic199D411yescharacteristic
42C4×Dic199D411yescharacteristic
43C2×D4×C199—21no—
44C2×D4×C199—21no—
45C2×D4×C199C2×C412yes—
46C2×C8×C199—21no—
47C2×C8×C199—21no—
48C42×C199C2311yescharacteristic
49C22×D199C22⋊C411yescharacteristic
50Dic398—41no—
51Dic398—41no—
52C22⋊D199—44no—
53C2×Dic199—22no—
54C2×Dic199—44no—
55C2×Dic199C22⋊C411yescharacteristic
56C2×Dic199C22⋊C411yescharacteristic
57C2×Dic199C22⋊C411yescharacteristic
58C4×D199—21no—
59C4×D199—21no—
60C4×D199—21no—
61C4×D199—21no—
62C4×D199—41no—
63C199⋊C8—41no—
64C199⋊C8—41no—
65C23×C199—22no—
66D4×C199—42no—
67D4×C199—41no—
68D4×C199—41no—
69D4×C199—41no—
70D4×C199—41no—
71C2×C4×C199C22×C411yescharacteristic
72C2×C4×C199C2×D411yescharacteristic
73C2×C4×C199C2×D411yescharacteristic
74C8×C199—41no—
75C8×C199—41no—
76C22×C199—42no—
77C22×C199—42no—
78C22×C199—42no—
79C22×C199C2×C22⋊C411yescharacteristic
80C2×D199—21no—
81C2×D199—21no—
82C2×D199—21no—
83C4×C199—21no—
84C4×C199AGL(1,ℤ/8)11yescharacteristic
85C4×C199AGL(1,ℤ/8)11yescharacteristic
86C4×C199AGL(1,ℤ/8)11yescharacteristic
87C4×C199#13711yescharacteristic
88Dic199—21no—
89Dic199—21no—
90Dic199—21no—
91Dic199—42no—
92C2×C199—82no—
93C2×C199#41311yescharacteristic
94C2×C199#41711yescharacteristic
95C2×C199#41811yescharacteristic
96D199—41no—
97C199#155711yescharacteristic, complemented, Sylow
98#1557—1991nomaximal, Sylow
99#532—1991no—
100#432—1991no—
101#423—1991no—
102#332—1991no—
103#332—1991no—
104#331—1991no—
105#331D19911yescharacteristic, complemented
106C2×#41—3981no—
107C2×M16—3981no—
108C4⋊Q8—1991no—
109#127C2×D19911yescharacteristic
110#124—3981no—
111#121—3981no—
112#121—1992no—
113C4×D4—3981no—
114C4×D4—1992no—
115C2×C42—1991no—
116C4⋊C8—3981no—
117C4⋊C8—1991no—
118C4⋊C8—1991no—
119C4⋊C8C2×D19911yescharacteristic
120C4⋊C8C2×D19911yescharacteristic
121C22⋊C8—3981no—
122#97—3981no—
123#41—7962no—
124#41—7962no—
125C2×Q8—3981no—
126C2×D4—21no—
127C2×D4—21no—
128C2×D4—3982no—
129C2×D4C4×D19912yes—
130C22×C4—3982no—
131C22×C4—1991no—
132C22×C4—1991no—
133C22×C4—1991no—
134M16—7962no—
135C2×C8—21no—
136C2×C8—21no—
137C2×C8—3981no—
138C2×C8—3981no—
139C4⋊C4—3981no—
140C4⋊C4—1992no—
141C22⋊C4—3982no—
142C22⋊C4—3982no—
143C42—1991no—
144C42—1991no—
145C42—1991no—
146C42C22×D19911yescharacteristic
147C23—22no—
148C23—1991no—
149Q8—7961no—
150Q8—7961no—
151D4—42no—
152D4—41no—
153D4—41no—
154D4—41no—
155D4—41no—
156D4—7964no—
157C2×C4—7961no—
158C2×C4—7964no—
159C2×C4—3982no—
160C2×C4—3981no—
161C2×C4—3981no—
162C2×C4—3981no—
163C2×C4—3981no—
164C2×C4—1991no—
165C2×C4—1991no—
166C2×C4—1991no—
167C2×C4C2×C4×D19911yescharacteristic
168C2×C4D4×D19911yescharacteristic
169C2×C4D4×D19911yescharacteristic
170C8—41no—
171C8—41no—
172C8—7961no—
173C8—7961no—
174C22—42no—
175C22—42no—
176C22—42no—
177C22—3981no—
178C22—3981no—
179C22—3981no—
180C22C22⋊C4×D19911yescharacteristic
181C4—21no—
182C4—7962no—
183C4—3981no—
184C4—3981no—
185C4—3981no—
186C4#2026397911yescharacteristic
187C4#2026397911yescharacteristic
188C4#2026398211yescharacteristic
189C4#2026398411yescharacteristic
190C2—82no—
191C2—7961no—
192C2#2716399911yescharacteristic
193C2#2716401611yescharacteristic
194C2#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#1557
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. #27164040C2semidirect
4. #27163765C2semidirect
5. #27163763C2semidirect
6. #27163762C2semidirect
7. #331×C199C2semidirect
97. C199#1557semidirect
105. #331D199semidirect

Maximal subgroups

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

subgroupindex
#271645662
#271640742
#271640402
#271637652
#271637632
#271637622
#331×C1992
#1557199