Finite Group Databasebeta

← #27157691

Subgroups — #27157691

Order 12416.

324 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 68 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#20261802C211yesmaximal, characteristic
2#20261746C212yesmaximal, complemented
3#20261687C211yesmaximal, characteristic
4#20261607C212yesmaximal, complemented
5#100×D97C211yesmaximal, characteristic
6D4×#87604—22no—
7D4×#87604—22no—
8#15690889C2212yes—
9C2×C4⋊#87604C2211yescharacteristic
10C2×C22⋊#87604—22no—
11C2×C22⋊#87604—22no—
12C2×C4×#87604C2211yescharacteristic
13#15690879—22no—
14#15690878—22no—
15C2×D4×D97C2211yescharacteristic
16M16×D97C2212yes—
17#15690727C411yescharacteristic, complemented
18#15690726C412yescomplemented
19#100×C97C411yescharacteristic, complemented
20C22×#87604—21no—
21C22×#87604—22no—
22C22×#87604—22no—
23C22×#87604—21no—
24C4⋊#87604—22no—
25C4⋊#87604D412yes—
26C4⋊#87604D412yes—
27C22⋊#87604—24no—
28C22⋊#87604—22no—
29C22⋊#87604—44no—
30C4×#87604—22no—
31C4×#87604—22no—
32C23×D97D412yes—
33C2×D388C2×C411yescharacteristic
34C2×C22⋊D97Q812yes—
35D4×D97—22no—
36D4×D97—22no—
37C2×C4×D97C2311yescharacteristic
38#13095495C2×C412yescomplemented
39#13095494—22no—
40C8×D97—22no—
41C2×D4×C97C2×C411yescharacteristic
42M16×C97C2×C412yescomplemented
43C23×C97C4⋊C412yes—
44C22×D97—22no—
45C22×D97—22no—
46C22×D97—42no—
47C22×D97—42no—
48C22×D97C4⋊C412yes—
49C22×D97C2×D411yescharacteristic
50C2×#87604—22no—
51C2×#87604—22no—
52C2×#87604—22no—
53C2×#87604—41no—
54C2×#87604—41no—
55C2×#87604—42no—
56C2×#87604—41no—
57C2×#87604—82no—
58C2×#87604—82no—
59D4×C97—22no—
60C2×C4×C97C22×C411yescharacteristic
61C22⋊D97—24no—
62C2×Dic97C2×Q811yescharacteristic
63D388—22no—
64C4×D97C2×D412yes—
65C4×D97#4112yes—
66C8×C97—22no—
67#11715895—22no—
68C22×C97—22no—
69C22×C97—42no—
70C22×C97C2×C4⋊C411yescharacteristic
71C2×D97—21no—
72C2×D97—21no—
73C2×D97—22no—
74C2×D97—42no—
75C2×D97—42no—
76C2×D97—42no—
77C2×D97—82no—
78C2×D97#12011yescharacteristic
79C2×D97#12311yescharacteristic
80#87604—42no—
81#87604—42no—
82#87604—42no—
83#87604—81no—
84C4×C97C4×D412yes—
85Dic97C22⋊Q812yes—
86C2×C97—21no—
87C2×C97—42no—
88C2×C97#38511yescharacteristic
89D97—21no—
90D97—42no—
91D97#45311yescharacteristic
92D97#45511yescharacteristic
93#1856—971nomaximal, Sylow
94C97#185611yescharacteristic, complemented, Sylow
95#518—971no—
96#428—972no—
97C2×#100—971no—
98#390—971no—
99#339—972no—
100C22×D4—971no—
101C2×M16—972no—
102C4×D4—1942no—
103C4×D4—1942no—
104#117—972no—
105C2×C4⋊C4—971no—
106C2×C22⋊C4—1942no—
107C2×C22⋊C4—1942no—
108C2×C42—971no—
109#107—1942no—
110#106—1942no—
111#100—971no—
112#100—972no—
113#100#8760411yescharacteristic, complemented
114C24—972no—
115C2×D4—972no—
116C2×D4—971no—
117C2×D4—1942no—
118C2×D4—1942no—
119C2×D4C2×#8760411yescharacteristic
120C22×C4—971no—
121C22×C4—1942no—
122C22×C4—1942no—
123C22×C4—1941no—
124C22×C4—1941no—
125M16—972no—
126M16—1942no—
127M16C2×#8760412yescomplemented
128C2×C8—1942no—
129C4⋊C4—972no—
130C4⋊C4—972no—
131C4⋊C4—1942no—
132C22⋊C4—3884no—
133C22⋊C4—1942no—
134C22⋊C4—1944no—
135C42—1942no—
136C42—1942no—
137C23—971no—
138C23—972no—
139C23—3882no—
140C23—3882no—
141C23—1942no—
142C23—1942no—
143C23C4⋊#8760412yes—
144D4—22no—
145D4—1944no—
146D4—1942no—
147C2×C4—7762no—
148C2×C4—7762no—
149C2×C4—971no—
150C2×C4—972no—
151C2×C4—972no—
152C2×C4—3881no—
153C2×C4—3882no—
154C2×C4—3881no—
155C2×C4—3881no—
156C2×C4—1942no—
157C2×C4—1942no—
158C2×C4—1942no—
159C2×C4C22×#8760411yescharacteristic
160C8—22no—
161C8—1942no—
162C22—22no—
163C22—42no—
164C22—7762no—
165C22—971no—
166C22—971no—
167C22—3882no—
168C22—3882no—
169C22—3882no—
170C22—1941no—
171C22—1941no—
172C22—1942no—
173C22C2×C4⋊#8760411yescharacteristic
174C4—7761no—
175C4—972no—
176C4—3882no—
177C4—3882no—
178C4—3882no—
179C4D4×#8760412yes—
180C2—21no—
181C2—42no—
182C2—971no—
183C2—971no—
184C2—3882no—
185C2—1941no—
186C2#2026168211yescharacteristic
subgroupG/H
centerC2#20261682
commutatorC2×C4×C97C22×C4
FrattiniC2×C4C22×#87604
Fitting#100×C97C4
socleC2×C97#385
prime radicalC2×D4×D97C22
cosocleC2×C4×D97C23
hypercenter#100#87604
hypocenterC97#1856
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C2×C4 ◁ #100

Complements

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

NH ≅ G/Ntype
2. #20261746C2semidirect
4. #20261607C2semidirect
17. #15690727C4semidirect
18. #15690726C4semidirect
19. #100×C97C4semidirect
38. #13095495C2×C4semidirect
42. M16×C97C2×C4semidirect
94. C97#1856semidirect
113. #100#87604semidirect
127. M16C2×#87604semidirect

Maximal subgroups

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

subgroupindex
#202618022
#202617462
#202616872
#202616072
#100×D972
#185697