Finite Group Databasebeta

← #38767899

Subgroups — #38767899

Order 25472.

240 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 49 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#27164605C211yesmaximal, characteristic
2#27164354C211yesmaximal, characteristic, complemented
3#27164334C211yesmaximal, characteristic, complemented
4#27163955C211yesmaximal, characteristic, complemented
5#27163754C211yesmaximal, characteristic, complemented
6#27163741C211yesmaximal, characteristic, complemented
7#327×C199C211yesmaximal, characteristic, complemented
8#20263950—21no—
9#20263941—21no—
10#20263939—21no—
11#20263934C2211yescharacteristic
12#20263927—21no—
13#20263923—21no—
14D4×Dic199—21no—
15C2×C4⋊Dic199—21no—
16C2×C4⋊Dic199C2211yescharacteristic
17C2×(C2×D199)⋊C4—21no—
18C2×(C2×D199)⋊C4C2211yescharacteristic
19#20263889—21no—
20#20263886—21no—
21#20263880—21no—
22#20263876—21no—
23C22⋊(C199⋊C8)C2211yescharacteristic
24#121×C199C2211yescharacteristic
25C22⋊C8×C199C2211yescharacteristic
26#95×C199C2211yescharacteristic
27C23×D199D411yescharacteristic
28C2×Dic398D411yescharacteristic
29C2×C22⋊D199—21no—
30C2×C22⋊D199—21no—
31C22×Dic199—21no—
32C22×Dic199—21no—
33C22×Dic199D411yescharacteristic
34C2×C199⋊C8—21no—
35C4⋊Dic199—21no—
36C4⋊Dic199—21no—
37C4⋊Dic199—21no—
38C4⋊Dic199—42no—
39C4⋊Dic199D411yescharacteristic
40Dic199⋊C4—21no—
41C22⋊Dic199—21no—
42C22⋊Dic199—21no—
43(C2×D199)⋊C4—21no—
44(C2×D199)⋊C4—21no—
45(C2×D199)⋊C4—21no—
46(C2×D199)⋊C4—21no—
47(C2×D199)⋊C4—42no—
48C4×Dic199—21no—
49C2×D4×C199—21no—
50C2×D4×C199D411yescharacteristic
51C22×C4×C199—21no—
52C22×C4×C199C2311yescharacteristic
53C2×C8×C199—21no—
54C4⋊C4×C199D411yescharacteristic
55C22⋊C4×C199—21no—
56C22×D199—21no—
57C22×D199—21no—
58C22×D199—21no—
59C22×D199—41no—
60C22×D199—41no—
61Dic398—41no—
62C22⋊D199—81no—
63C22⋊D199—81no—
64C2×Dic199—21no—
65C2×Dic199—21no—
66C2×Dic199—21no—
67C2×Dic199—41no—
68C2×Dic199—41no—
69C2×Dic199—41no—
70C2×Dic199—41no—
71C2×Dic199—81no—
72C199⋊C8—41no—
73C23×C199—21no—
74C23×C199C2×D411yescharacteristic
75D4×C199—41no—
76D4×C199—41no—
77D4×C199—41no—
78C2×C4×C199—22no—
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—41no—
89C22×C199—41no—
90C22×C199—41no—
91C22×C199#12011yescharacteristic
92C22×C199#13511yescharacteristic
93C22×C199#13711yescharacteristic
94C2×D199—42no—
95C2×D199—41no—
96C2×D199—41no—
97C2×D199—41no—
98C2×D199—81no—
99C4×C199—21no—
100C4×C199—21no—
101C4×C199—42no—
102C4×C199—41no—
103Dic199—41no—
104Dic199—41no—
105Dic199—41no—
106Dic199—81no—
107C2×C199—21no—
108C2×C199—21no—
109C2×C199—81no—
110C2×C199#45211yescharacteristic
111C2×C199#45311yescharacteristic
112C2×C199#45811yescharacteristic
113D199—81no—
114C199#161811yescharacteristic, complemented, Sylow
115#1618—1991nomaximal, Sylow
116#538—1991no—
117#484—1991no—
118#481—1991no—
119#399—1991no—
120#328—1991no—
121#327D19911yescharacteristic, complemented
122#323—1991no—
123#126—3981no—
124#124—3981no—
125#123—3981no—
126C22⋊Q8—1991no—
127#121—3981no—
128#121C2×D19911yescharacteristic
129#120—3981no—
130C4×D4—3981no—
131C2×C4⋊C4—3981no—
132C2×C4⋊C4—1991no—
133C2×C22⋊C4—3981no—
134C2×C22⋊C4—1991no—
135#107—3981no—
136#106—3981no—
137#103—3981no—
138#102—3981no—
139C22⋊C8—1991no—
140C22⋊C8C2×D19911yescharacteristic
141#95C2×D19911yescharacteristic
142C24—1991no—
143C2×Q8—1991no—
144C2×D4—21no—
145C2×D4—3981no—
146C2×D4—3981no—
147C2×D4D79611yescharacteristic
148C22×C4—21no—
149C22×C4—3981no—
150C22×C4—3981no—
151C22×C4—1991no—
152C22×C4C22×D19911yescharacteristic
153C2×C8—21no—
154C2×C8—3981no—
155C4⋊C4—7962no—
156C4⋊C4—3981no—
157C4⋊C4—3981no—
158C4⋊C4—3981no—
159C4⋊C4—3981no—
160C4⋊C4—1991no—
161C4⋊C4D79611yescharacteristic
162C22⋊C4—21no—
163C22⋊C4—7962no—
164C22⋊C4—3981no—
165C22⋊C4—3981no—
166C22⋊C4—3981no—
167C22⋊C4—3981no—
168C22⋊C4—3981no—
169C22⋊C4—3981no—
170C42—3981no—
171C23—21no—
172C23—7961no—
173C23—7961no—
174C23—3981no—
175C23—3981no—
176C23—3981no—
177C23D4×D19911yescharacteristic
178Q8—7961no—
179D4—41no—
180D4—41no—
181D4—41no—
182D4—15921no—
183D4—15921no—
184C2×C4—22no—
185C2×C4—21no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—7961no—
190C2×C4—7961no—
191C2×C4—7961no—
192C2×C4—7961no—
193C2×C4—15921no—
194C2×C4—3981no—
195C2×C4—3981no—
196C2×C4—3981no—
197C2×C4D4×D19911yescharacteristic
198C2×C4C2×D79611yescharacteristic
199C8—41no—
200C8—7961no—
201C22—21no—
202C22—21no—
203C22—41no—
204C22—41no—
205C22—41no—
206C22—7962no—
207C22—7961no—
208C22—7961no—
209C22—7961no—
210C22—15921no—
211C22#2026392511yescharacteristic
212C22#2026397411yescharacteristic
213C22#2026398411yescharacteristic
214C4—21no—
215C4—21no—
216C4—42no—
217C4—41no—
218C4—7961no—
219C4—7961no—
220C4—7961no—
221C4—15921no—
222C2—21no—
223C2—21no—
224C2—81no—
225C2—15921no—
226C2#2716418111yescharacteristic
227C2#2716418711yescharacteristic
228C2#2716420911yescharacteristic
subgroupG/H
centerC22#20263925
commutator, cosocleC22×C4×C199C23
FrattiniC22×C4C22×D199
Fitting#327×C199C2
socleC22×C199#120
prime radical#27164605C2
hypercenter#327D199
hypocenterC199#1618
LCSG ▷ C22×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. #27164354C2semidirect
3. #27164334C2semidirect
4. #27163955C2semidirect
5. #27163754C2semidirect
6. #27163741C2semidirect
7. #327×C199C2semidirect
114. C199#1618semidirect
121. #327D199semidirect

Maximal subgroups

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

subgroupindex
#271646052
#271643542
#271643342
#271639552
#271637542
#271637412
#327×C1992
#1618199