Finite Group Databasebeta

← #27141689

Subgroups — #27141689

Order 12416.

620 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 620 conjugacy classes of subgroups — too many to draw. Only its 101 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
1C2×#15690855C211yesmaximal, characteristic
2C2×#15690738C212yesmaximal, complemented
3#20260734C212yesmaximal, complemented
4C2×#15690727C211yesmaximal, characteristic, complemented
5#390×C97C211yesmaximal, characteristic, complemented
6#15690855—24no—
7#15690855—22no—
8#15690855C414yescomplemented
9C2×#13095511C2212yes—
10C2×#13095510—22no—
11C22×C22⋊D97—22no—
12C2×D4×D97—22no—
13C2×#13095495C2212yescomplemented
14#15690738—24no—
15#15690733—22no—
16#15690730—22no—
17#15690727—22no—
18C22×D4×C97C2211yescharacteristic
19C2×C4⋊C4×C97C2211yescharacteristic, complemented
20C2×C22⋊C4×C97—22no—
21C2×C42×C97C2211yescharacteristic, complemented
22C23×D97—22no—
23#13095511—22no—
24#13095511—22no—
25#13095511C2×C414yes—
26#13095511D414yes—
27#13095510—28no—
28#13095510—28no—
29C2×Dic194D412yes—
30C2×D388D412yes—
31C2×C22⋊D97—24no—
32C2×C22⋊D97—28no—
33C2×C22⋊D97—28no—
34C2×C22⋊D97—22no—
35D4×D97—28no—
36D4×D97—28no—
37C22×Dic97—22no—
38C2×C4×D97—22no—
39#13095495—22no—
40#13095495—22no—
41C2×#11715895—22no—
42C24×C97D412yes—
43C2×D4×C97—22no—
44C2×D4×C97—22no—
45C2×D4×C97C2×C412yes—
46C2×D4×C97D412yes—
47C22×C4×C97—21no—
48C22×C4×C97—21no—
49C22×C4×C97C2311yescharacteristic
50C4⋊C4×C97—22no—
51C22⋊C4×C97—44no—
52C42×C97—22no—
53C23×C97—22no—
54C23×C97—22no—
55C23×C97—42no—
56C23×C97—42no—
57C23×C97C22⋊C414yes—
58C23×C97C2×D411yescharacteristic
59C22×D97—22no—
60C22×D97—28no—
61C22×D97—82no—
62C22×D97—82no—
63D4×C97—24no—
64D4×C97—24no—
65C2×C4×C97—22no—
66C2×C4×C97—41no—
67C2×C4×C97—41no—
68C2×C4×C97—42no—
69C2×C4×C97—41no—
70C2×C4×C97C22×C411yescharacteristic
71C2×C4×C97C2×D412yes—
72C2×C4×C97C2×D412yes—
73C2×C4×C97C2×D411yescharacteristic
74C22⋊D97—216no—
75C22⋊D97—24no—
76C22⋊D97—24no—
77C2×Dic97—28no—
78C2×Dic97—22no—
79D388—22no—
80D388C22⋊C414yescomplemented
81C4×D97—24no—
82C4×D97—24no—
83Dic194—22no—
84Dic194C22⋊C414yescomplemented
85#11715895—42no—
86C22×C97—21no—
87C22×C97—21no—
88C22×C97—24no—
89C22×C97—44no—
90C22×C97—42no—
91C22×C97—42no—
92C22×C97—82no—
93C22×C97C2×C22⋊C411yescharacteristic
94C22×C97#12011yescharacteristic
95C22×C97#12011yescharacteristic
96C2×D97—24no—
97C2×D97—42no—
98C2×D97—42no—
99C2×D97—88no—
100C4×C97—42no—
101C4×C97—81no—
102C4×C97C2×C22⋊C412yes—
103C4×C97#12012yes—
104Dic97—24no—
105C2×C97—21no—
106C2×C97—21no—
107C2×C97—44no—
108C2×C97#37911yescharacteristic
109C2×C97#45311yescharacteristic
110C2×C97#45511yescharacteristic
111D97—44no—
112#1792—971nomaximal, Sylow
113C97#179211yescharacteristic, complemented, Sylow
114C2×H(5,𝔽2)—971no—
115C2×#104—972no—
116#417—972no—
117C2×#100—971no—
118#390D9711yescharacteristic, complemented
119H(5,𝔽2)—974no—
120H(5,𝔽2)—1944no—
121H(5,𝔽2)—1942no—
122C2×#41—972no—
123C2×#41—1942no—
124C22×D4—1942no—
125C22×D4—1942no—
126C22×D4C2×D9711yescharacteristic
127C2×M16—972no—
128C2×C4⋊C4C2×D9711yescharacteristic, complemented
129C2×C22⋊C4—22no—
130C2×C42C2×D9711yescharacteristic, complemented
131#104—1944no—
132#103—1942no—
133#102—1942no—
134#100—1942no—
135C24—1942no—
136C24C22⋊D9712yes—
137#41—974no—
138#41—974no—
139#41—1948no—
140#41—1948no—
141#41—1942no—
142#41—1942no—
143C2×Q8—972no—
144C2×D4—22no—
145C2×D4—22no—
146C2×D4—972no—
147C2×D4—1948no—
148C2×D4—1948no—
149C2×D4—1944no—
150C2×D4—1948no—
151C2×D4—1948no—
152C2×D4—1942no—
153C2×D4C4×D9712yes—
154C2×D4D38812yes—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—1942no—
158C22×C4—1942no—
159C22×C4C22×D9711yescharacteristic
160M16—1942no—
161M16—1942no—
162C2×C8—1942no—
163C4⋊C4—22no—
164C22⋊C4—44no—
165C42—22no—
166C23—22no—
167C23—22no—
168C23—42no—
169C23—42no—
170C23—7762no—
171C23—7762no—
172C23—1948no—
173C23—1942no—
174C23#1309548814yes—
175C23C2×C22⋊D9711yescharacteristic
176Q8—974no—
177Q8—1942no—
178D4—24no—
179D4—24no—
180D4—974no—
181D4—19416no—
182D4—1944no—
183D4—1944no—
184D4—1942no—
185C2×C4—22no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—42no—
189C2×C4—41no—
190C2×C4—1948no—
191C2×C4—1942no—
192C2×C4—1944no—
193C2×C4—1944no—
194C2×C4C2×C4×D9711yescharacteristic
195C2×C4D4×D9712yes—
196C2×C4D4×D9712yes—
197C2×C4C2×D38811yescharacteristic
198C8—3882no—
199C22—21no—
200C22—21no—
201C22—24no—
202C22—44no—
203C22—42no—
204C22—42no—
205C22—82no—
206C22—7768no—
207C22—3882no—
208C22—3882no—
209C22—1944no—
210C22C2×#1309548811yescharacteristic
211C22#1569077811yescharacteristic
212C22#1569077811yescharacteristic
213C4—42no—
214C4—81no—
215C4—1944no—
216C4C22⋊C4×D9712yes—
217C4#1569078012yes—
218C2—21no—
219C2—21no—
220C2—44no—
221C2—3884no—
222C2#2026059011yescharacteristic
223C2#2026090211yescharacteristic
224C2#2026091211yescharacteristic
subgroupG/H
centerC22#15690778
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
Fitting#390×C97C2
socleC22×C97#120
prime radicalC2×#15690855C2
cosocleC22×C4×C97C23
hypercenter#390D97
hypocenterC97#1792
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #390

Complements

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

NH ≅ G/Ntype
2. C2×#15690738C2semidirect
3. #20260734C2semidirect
4. C2×#15690727C2semidirect
5. #390×C97C2semidirect
8. #15690855C4semidirect
13. C2×#13095495C22semidirect
19. C2×C4⋊C4×C97C22semidirect
21. C2×C42×C97C22semidirect
80. D388C22⋊C4semidirect
84. Dic194C22⋊C4semidirect
113. C97#1792semidirect
118. #390D97semidirect
128. C2×C4⋊C4C2×D97semidirect
130. C2×C42C2×D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156908552
C2×#156907382
#202607342
C2×#156907272
#390×C972
#179297