Finite Group Databasebeta

← #38767825

Subgroups — #38767825

Order 25472.

238 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 51 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#27164647C211yesmaximal, characteristic
2#27164176C211yesmaximal, characteristic
3#27164173C211yesmaximal, characteristic
4#27163939C211yesmaximal, characteristic
5#27163754C211yesmaximal, characteristic
6#27163743C211yesmaximal, characteristic
7#328×C199C211yesmaximal, characteristic
8C22×Dic398C2211yescharacteristic
9C2×Dic796—21no—
10C2×#15691876—21no—
11#20263954—21no—
12#20263953—21no—
13#20263945—21no—
14#20263945—21no—
15#20263934C2211yescharacteristic
16#20263932—21no—
17#20263911—21no—
18C2×C4⋊Dic199—21no—
19C2×Dic199⋊C4C2211yescharacteristic
20#20263880—21no—
21#20263878—21no—
22C22⋊(C199⋊C8)C2211yescharacteristic
23C199⋊#95—21no—
24C22⋊Q8×C199C2211yescharacteristic
25C22⋊C8×C199C2211yescharacteristic
26#95×C199C2211yescharacteristic
27C2×Dic398—21no—
28C2×Dic398—21no—
29C2×Dic398—21no—
30C2×Dic398—42no—
31C2×Dic398D411yescharacteristic
32C22×Dic199—21no—
33C22×Dic199D411yescharacteristic
34C22×Dic199D411yescharacteristic
35Dic796—42no—
36#15691876—42no—
37C2×C199⋊C8—21no—
38C4⋊Dic199—21no—
39C4⋊Dic199—42no—
40C4⋊Dic199D411yescharacteristic
41Dic199⋊C4—21no—
42Dic199⋊C4—21no—
43Dic199⋊C4—21no—
44Dic199⋊C4—21no—
45Dic199⋊C4—21no—
46Dic199⋊C4—21no—
47Dic199⋊C4—21no—
48C22⋊Dic199—21no—
49C4×Dic199—21no—
50C4×Dic199—21no—
51C2×Q8×C199D411yescharacteristic
52C22×C4×C199—21no—
53C22×C4×C199C2311yescharacteristic
54C2×C8×C199—21no—
55C4⋊C4×C199—21no—
56C4⋊C4×C199D411yescharacteristic
57C22⋊C4×C199—21no—
58Dic398—22no—
59Dic398—22no—
60Dic398—41no—
61Dic398—41no—
62Dic398—41no—
63Dic398—81no—
64C2×Dic199—21no—
65C2×Dic199—21no—
66C2×Dic199—22no—
67C2×Dic199—21no—
68C2×Dic199—21no—
69C2×Dic199—21no—
70C2×Dic199—41no—
71C2×Dic199—41no—
72C2×Dic199—41no—
73C2×Dic199Q1612yescomplemented
74C199⋊C8—41no—
75C23×C199C2×D411yescharacteristic
76Q8×C199—41no—
77C2×C4×C199—22no—
78C2×C4×C199—21no—
79C2×C4×C199—21no—
80C2×C4×C199—41no—
81C2×C4×C199—41no—
82C2×C4×C199—41no—
83C2×C4×C199C2×D411yescharacteristic
84C2×C4×C199C2×D411yescharacteristic
85C8×C199—41no—
86C22×C199—21no—
87C22×C199—21no—
88C22×C199#12011yescharacteristic
89C22×C199C2×Q1611yescharacteristic
90C22×C199#13711yescharacteristic
91C4×C199—21no—
92C4×C199—21no—
93C4×C199—42no—
94C4×C199—41no—
95C4×C199—41no—
96Dic199—22no—
97Dic199—41no—
98Dic199—42no—
99Dic199—41no—
100Dic199—41no—
101C2×C199—21no—
102C2×C199—21no—
103C2×C199#45111yescharacteristic
104C2×C199#45611yescharacteristic
105C2×C199B(4,𝔽2)11yescharacteristic
106C199#160511yescharacteristic, complemented, Sylow
107#1605—1991nomaximal, Sylow
108#544—1991no—
109#451—1991no—
110#451—1991no—
111#395—1991no—
112#328—1991no—
113#328D19911yescharacteristic
114#324—1991no—
115C22×Q8—1991no—
116C2×Q16—3981no—
117C2×Q16—3981no—
118C4⋊Q8—3981no—
119C4⋊Q8—3981no—
120#125—3981no—
121#125—3981no—
122C22⋊Q8—3981no—
123C22⋊Q8—1991no—
124C22⋊Q8C2×D19911yescharacteristic
125#117—3981no—
126C2×C4⋊C4—3981no—
127C2×C4⋊C4—1991no—
128#103—3981no—
129#103—3981no—
130C22⋊C8—1991no—
131C22⋊C8C2×D19911yescharacteristic
132#95—3981no—
133#95C2×D19911yescharacteristic
134C2×Q8—7962no—
135C2×Q8—3981no—
136C2×Q8—3981no—
137C2×Q8—3981no—
138C2×Q8—1991no—
139C2×Q8D79611yescharacteristic
140C22×C4—21no—
141C22×C4—3981no—
142C22×C4—1991no—
143C22×C4—1991no—
144C22×C4C22×D19911yescharacteristic
145Q16—7962no—
146Q16—7962no—
147C2×C8—21no—
148C2×C8—3981no—
149C4⋊C4—21no—
150C4⋊C4—7962no—
151C4⋊C4—3981no—
152C4⋊C4—3981no—
153C4⋊C4—3981no—
154C4⋊C4—3981no—
155C4⋊C4—3981no—
156C4⋊C4—3981no—
157C4⋊C4—3981no—
158C4⋊C4—3981no—
159C4⋊C4—1991no—
160C4⋊C4D79611yescharacteristic
161C22⋊C4—21no—
162C22⋊C4—3981no—
163C42—3981no—
164C42—3981no—
165C23D4×D19911yescharacteristic
166Q8—41no—
167Q8—7961no—
168Q8—7961no—
169Q8—7961no—
170Q8—15921no—
171Q8—3982no—
172Q8—3982no—
173C2×C4—22no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—7961no—
180C2×C4—7961no—
181C2×C4—7961no—
182C2×C4—3982no—
183C2×C4—3981no—
184C2×C4—3981no—
185C2×C4—3981no—
186C2×C4—3981no—
187C2×C4—3981no—
188C2×C4—1992no—
189C2×C4D4×D19911yescharacteristic
190C2×C4C2×D79611yescharacteristic
191C8—41no—
192C8—7961no—
193C22—21no—
194C22—21no—
195C22#2026392511yescharacteristic
196C22Q16×D19911yescharacteristic
197C22#2026398411yescharacteristic
198C4—21no—
199C4—21no—
200C4—42no—
201C4—41no—
202C4—41no—
203C4—7961no—
204C4—7962no—
205C4—7961no—
206C4—7961no—
207C4—3982no—
208C2—21no—
209C2—21no—
210C2#2716417511yescharacteristic
211C2#2716420211yescharacteristic
212C2#2716420611yescharacteristic
subgroupG/H
centerC22#20263925
commutator, cosocleC22×C4×C199C23
FrattiniC22×C4C22×D199
Fitting#328×C199C2
socleC22×C199#120
prime radicalC23×C199C2×D4
hypercenter#328D199
hypocenterC199#1605
LCSG ▷ C22×C4×C199 ▷ C22×C199 ▷ C199
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
73. C2×Dic199Q16semidirect
106. C199#1605semidirect

Maximal subgroups

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

subgroupindex
#271646472
#271641762
#271641732
#271639392
#271637542
#271637432
#328×C1992
#1605199