Finite Group Databasebeta

← #27140724

Subgroups — #27140724

Order 12416.

280 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 53 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#20261297C211yesmaximal, characteristic
2#20261049C212yesmaximal, complemented
3#20260660C211yesmaximal, characteristic, complemented
4#20260470C212yesmaximal, complemented
5#323×C97C211yesmaximal, characteristic, complemented
6C22×C22⋊D97—21no—
7#15690794—22no—
8#15690784—21no—
9#15690784C2212yes—
10#15690780—22no—
11C2×C4⋊Dic97C2211yescharacteristic, complemented
12C2×C22⋊Dic97—21no—
13#15690744—22no—
14#15690732—22no—
15#15690717—21no—
16#15690717C2211yescharacteristic, complemented
17C2×C22⋊C4×C97C2211yescharacteristic
18C22⋊C8×C97C2212yescomplemented
19C23×D97—21no—
20C2×D388—21no—
21C2×D388D412yes—
22C2×C22⋊D97—22no—
23C2×C22⋊D97—22no—
24C2×C22⋊D97—42no—
25C2×C22⋊D97—42no—
26C22×Dic97—21no—
27C22×Dic97—21no—
28C22×Dic97—21no—
29C4⋊Dic97—21no—
30C4⋊Dic97—22no—
31C4⋊Dic97D412yes—
32C22⋊Dic97—42no—
33#13095488—22no—
34#13095488—22no—
35C24×C97D411yescharacteristic
36C22×C4×C97D411yescharacteristic
37C22×C4×C97C2311yescharacteristic
38C2×C8×C97—22no—
39C22⋊C4×C97—22no—
40C23×C97—21no—
41C23×C97—21no—
42C23×C97—41no—
43C23×C97—41no—
44C23×C97D812yes—
45C23×C97C2×D411yescharacteristic
46C22×D97—22no—
47C22×D97—42no—
48C22×D97—81no—
49C22×D97—81no—
50C2×C4×C97—21no—
51C2×C4×C97—41no—
52C2×C4×C97—41no—
53C2×C4×C97C2×D412yes—
54C22⋊D97—44no—
55C22⋊D97—42no—
56C22⋊D97—42no—
57C2×Dic97—22no—
58C2×Dic97—42no—
59C2×Dic97—41no—
60C2×Dic97—41no—
61C2×Dic97—41no—
62C2×Dic97—41no—
63C2×Dic97—41no—
64C2×Dic97—41no—
65D388—42no—
66D388—81no—
67C8×C97—42no—
68C22×C97—21no—
69C22×C97—21no—
70C22×C97—22no—
71C22×C97—42no—
72C22×C97—41no—
73C22×C97—41no—
74C22×C97—81no—
75C22×C97#12011yescharacteristic
76C22×C97C2×D811yescharacteristic
77C22×C97AGL(1,ℤ/8)11yescharacteristic
78C2×D97—42no—
79C2×D97—42no—
80C2×D97—42no—
81C2×D97—84no—
82C4×C97—22no—
83C4×C97—41no—
84Dic97—42no—
85Dic97—81no—
86Dic97—81no—
87C2×C97—21no—
88C2×C97—21no—
89C2×C97—42no—
90C2×C97#44711yescharacteristic
91C2×C97#45511yescharacteristic
92C2×C97#45811yescharacteristic
93D97—82no—
94#1607—971nomaximal, Sylow
95C97#160711yescharacteristic, complemented, Sylow
96#534—971no—
97#480—972no—
98#397—971no—
99#327—972no—
100#323D9711yescharacteristic, complemented
101C22×D4—1941no—
102#123—1942no—
103#121—972no—
104#121—1941no—
105#120—1942no—
106C2×C4⋊C4—971no—
107C2×C22⋊C4—1941no—
108C2×C22⋊C4C2×D9711yescharacteristic
109#107—1942no—
110#102—1942no—
111C22⋊C8C2×D9712yescomplemented
112#95—971no—
113#95—1941no—
114C24—1941no—
115C24D38811yescharacteristic
116C2×D4—972no—
117C2×D4—3882no—
118C2×D4—3882no—
119C2×D4—1941no—
120C2×D4—1942no—
121C2×D4—1942no—
122C22×C4—1941no—
123C22×C4—1941no—
124C22×C4—1941no—
125C22×C4D38811yescharacteristic
126C22×C4C22×D9711yescharacteristic
127C2×C8—22no—
128C4⋊C4—972no—
129C4⋊C4—1941no—
130C4⋊C4—1942no—
131C22⋊C4—22no—
132C22⋊C4—3882no—
133C22⋊C4—1942no—
134C22⋊C4—1942no—
135C23—21no—
136C23—21no—
137C23—41no—
138C23—41no—
139C23—7761no—
140C23—7761no—
141C23—3882no—
142C23—1942no—
143C23D77612yes—
144C23C2×D38811yescharacteristic
145D4—7761no—
146D4—3882no—
147D4—3884no—
148D4—3882no—
149D4—3882no—
150C2×C4—21no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—3881no—
154C2×C4—3881no—
155C2×C4—3881no—
156C2×C4—3881no—
157C2×C4—3881no—
158C2×C4—3881no—
159C2×C4—3882no—
160C2×C4—1942no—
161C2×C4D4×D9712yes—
162C8—42no—
163C22—21no—
164C22—21no—
165C22—22no—
166C22—42no—
167C22—41no—
168C22—41no—
169C22—81no—
170C22—7764no—
171C22—3882no—
172C22—3882no—
173C22—3882no—
174C22#1569078011yescharacteristic
175C22C2×D77611yescharacteristic
176C22#1569083811yescharacteristic
177C4—22no—
178C4—41no—
179C4—7761no—
180C4—7761no—
181C4—3882no—
182C2—21no—
183C2—21no—
184C2—42no—
185C2—7762no—
186C2#2026087111yescharacteristic
187C2#2026091511yescharacteristic
188C2#2026092611yescharacteristic
subgroupG/H
centerC22#15690780
commutator, cosocleC22×C4×C97C23
FrattiniC22×C4C22×D97
Fitting#323×C97C2
socleC22×C97#120
prime radical#20261297C2
hypercenter#323D97
hypocenterC97#1607
LCSG ▷ C22×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #323

Complements

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

NH ≅ G/Ntype
2. #20261049C2semidirect
3. #20260660C2semidirect
4. #20260470C2semidirect
5. #323×C97C2semidirect
11. C2×C4⋊Dic97C22semidirect
16. #15690717C22semidirect
18. C22⋊C8×C97C22semidirect
95. C97#1607semidirect
100. #323D97semidirect
111. C22⋊C8C2×D97semidirect

Maximal subgroups

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

subgroupindex
#202612972
#202610492
#202606602
#202604702
#323×C972
#160797