Finite Group Databasebeta

← #27142990

Subgroups — #27142990

Order 12416.

296 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 67 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
1C22×#13095496C211yesmaximal, characteristic
2#20260763C211yesmaximal, characteristic, complemented
3#20260741C211yesmaximal, characteristic, complemented
4#20260740C211yesmaximal, characteristic, complemented
5C2×#15690726C211yesmaximal, characteristic, complemented
6#20260499C211yesmaximal, characteristic, complemented
7C2×#100×C97C211yesmaximal, characteristic, complemented
8C22×D388C2211yescharacteristic
9C2×#13095496—22no—
10C2×#13095496—22no—
11C2×#13095496—22no—
12C2×#13095495C2211yescharacteristic
13C22×#11715895C2211yescharacteristic
14#15690769C2211yescharacteristic
15#15690739—22no—
16#15690732—22no—
17#15690731—22no—
18#15690726—22no—
19C22×D4×C97C2211yescharacteristic
20C2×M16×C97C2211yescharacteristic
21C2×M16×C97C2211yescharacteristic
22#100×C97—22no—
23C23×D97—21no—
24C2×D388—21no—
25C2×D388—21no—
26C2×D388—21no—
27C2×D388—21no—
28C2×D388—21no—
29C2×D388D412yes—
30#13095496—28no—
31#13095495—21no—
32#13095495—21no—
33C2×#11715895—21no—
34C2×#11715895—21no—
35C2×#11715895—21no—
36C2×#11715895D412yes—
37C4⋊Dic97D412yes—
38C22⋊Dic97—21no—
39C4×Dic97—21no—
40C24×C97—21no—
41C2×D4×C97—21no—
42C2×D4×C97—21no—
43C2×D4×C97—21no—
44C2×D4×C97—21no—
45C2×D4×C97—21no—
46C2×D4×C97D412yes—
47C22×C4×C97C2311yescharacteristic
48M16×C97—21no—
49M16×C97—21no—
50M16×C97—21no—
51M16×C97—21no—
52C2×C8×C97—21no—
53C2×C8×C97—21no—
54C23×C97—22no—
55C23×C97—41no—
56C23×C97—41no—
57C23×C97—82no—
58C23×C97#4111yescharacteristic
59C22×D97—22no—
60C22×D97—41no—
61C22×D97—41no—
62C22×D97—82no—
63D4×C97—22no—
64D4×C97—22no—
65D4×C97—41no—
66D4×C97—41no—
67C2×C4×C97C2×D411yescharacteristic
68C2×C4×C97C2×D411yescharacteristic
69C2×C4×C97C2×D411yescharacteristic
70C2×C4×C97C2×D411yescharacteristic
71C2×C4×C97#4111yescharacteristic
72C2×C4×C97#4111yescharacteristic
73C2×Dic97—22no—
74D388—22no—
75D388—22no—
76D388—41no—
77D388—41no—
78C8×C97—41no—
79C8×C97—41no—
80#11715895—22no—
81#11715895—41no—
82C22×C97—22no—
83C22×C97—42no—
84C22×C97—82no—
85C22×C97—82no—
86C22×C97—82no—
87C22×C97#12111yescharacteristic
88C22×C97#12311yescharacteristic
89C22×C97#12311yescharacteristic
90C2×D97—42no—
91C2×D97—82no—
92C2×D97—82no—
93C2×D97—82no—
94C4×C97#12011yescharacteristic
95C4×C97#12111yescharacteristic
96C4×C97#12111yescharacteristic
97C4×C97#12411yescharacteristic
98Dic97—42no—
99C2×C97—21no—
100C2×C97—21no—
101C2×C97—82no—
102C2×C97#39411yescharacteristic
103C2×C97#47212yes—
104D97—82no—
105#2045—971nomaximal, Sylow
106C97#204511yescharacteristic, complemented, Sylow
107C22×D8—971no—
108#424—971no—
109#418—971no—
110#418—971no—
111C2×#100—971no—
112C2×#100D9711yescharacteristic, complemented
113#341—971no—
114C22×D4—971no—
115C22×D4C2×D9711yescharacteristic
116C2×D8—1942no—
117C2×D8—1942no—
118C2×D8—1942no—
119C2×M16—971no—
120C2×M16C2×D9711yescharacteristic
121C2×M16C2×D9711yescharacteristic
122C22×C8—971no—
123#117—971no—
124C4⋊C8—1942no—
125#102—1942no—
126#102—1942no—
127#100—22no—
128#100—1942no—
129C24—21no—
130C24—1941no—
131C2×D4—21no—
132C2×D4—21no—
133C2×D4—21no—
134C2×D4—21no—
135C2×D4—21no—
136C2×D4—972no—
137C2×D4—1941no—
138C2×D4—1941no—
139C2×D4—1941no—
140C2×D4—1941no—
141C2×D4—1941no—
142C2×D4C22⋊D9712yes—
143C22×C4C22×D9711yescharacteristic
144D8—1948no—
145M16—21no—
146M16—21no—
147M16—21no—
148M16—21no—
149M16—1941no—
150M16—1941no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—972no—
154C2×C8—1941no—
155C2×C8—1941no—
156C2×C8—1941no—
157C4⋊C4—972no—
158C22⋊C4—1941no—
159C42—1941no—
160C23—22no—
161C23—41no—
162C23—41no—
163C23—82no—
164C23—7762no—
165C23—3881no—
166C23—3881no—
167C23—1942no—
168C23#1309551111yescharacteristic
169D4—22no—
170D4—22no—
171D4—41no—
172D4—41no—
173D4—3881no—
174D4—3881no—
175D4—1942no—
176D4—1942no—
177C2×C4—1942no—
178C2×C4D4×D9711yescharacteristic
179C2×C4D4×D9711yescharacteristic
180C2×C4D4×D9711yescharacteristic
181C2×C4C2×C22⋊D9711yescharacteristic
182C2×C4#1309551011yescharacteristic
183C2×C4#1309551111yescharacteristic
184C8—41no—
185C8—41no—
186C8—3881no—
187C8—1942no—
188C22—22no—
189C22—42no—
190C22—82no—
191C22—82no—
192C22—82no—
193C22—7762no—
194C22—7762no—
195C22—7762no—
196C22—3882no—
197C22#1569078311yescharacteristic
198C22#1569079211yescharacteristic
199C22#1569079311yescharacteristic
200C4—3882no—
201C4#1569077811yescharacteristic
202C4#1569078211yescharacteristic
203C4#1569078311yescharacteristic
204C4#1569079611yescharacteristic
205C2—21no—
206C2—21no—
207C2—82no—
208C2—7762no—
209C2#2026065011yescharacteristic
210C2#2026100412yes—
subgroupG/H
centerC22#15690783
commutator, cosocleC22×C4×C97C23
FrattiniC22×C4C22×D97
FittingC2×#100×C97C2
socleC22×C97#121
prime radicalC22×#13095496C2
hypercenterC2×#100D97
hypocenterC97#2045
LCSG ▷ C22×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#100

Complements

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

NH ≅ G/Ntype
2. #20260763C2semidirect
3. #20260741C2semidirect
4. #20260740C2semidirect
5. C2×#15690726C2semidirect
6. #20260499C2semidirect
7. C2×#100×C97C2semidirect
106. C97#2045semidirect
112. C2×#100D97semidirect

Maximal subgroups

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

subgroupindex
C22×#130954962
#202607632
#202607412
#202607402
C2×#156907262
#202604992
C2×#100×C972
#204597