Finite Group Databasebeta

← #27140671

Subgroups — #27140671

Order 12416.

310 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 57 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#20260868C212yesmaximal, complemented
2#20260652C211yesmaximal, characteristic, complemented
3#20260469C212yesmaximal, complemented
4#20260461C211yesmaximal, characteristic, complemented
5#537×C97C211yesmaximal, characteristic, complemented
6C22×D388C2211yescharacteristic
7C2×#13095496—22no—
8C2×#13095488—21no—
9C2×#13095488—21no—
10#15690730—22no—
11C22⋊#11715895C2212yes—
12#15690717C2211yescharacteristic
13C2×#41×C97—21no—
14C22⋊Q8×C97—22no—
15#121×C97—22no—
16#121×C97—21no—
17#121×C97C2212yes—
18C2×C4⋊C4×C97C2211yescharacteristic
19C23×D97—21no—
20C2×D388—22no—
21C2×D388—42no—
22C22×Dic97—21no—
23#13095496—44no—
24C2×#11715895—22no—
25#13095488—42no—
26#13095488—42no—
27#41×C97—42no—
28#41×C97—42no—
29C2×Q8×C97—21no—
30C2×D4×C97—21no—
31C2×D4×C97—22no—
32C2×D4×C97—21no—
33C2×D4×C97D412yes—
34C22×C4×C97—22no—
35C22×C4×C97D411yescharacteristic
36C22×C4×C97D411yescharacteristic
37C22×C4×C97C2311yescharacteristic
38C4⋊C4×C97—21no—
39C4⋊C4×C97—22no—
40C4⋊C4×C97D412yes—
41C22⋊C4×C97—22no—
42C22⋊C4×C97—22no—
43C23×C97—22no—
44C23×C97C2×D411yescharacteristic
45C22×D97—22no—
46C22×D97—41no—
47C22×D97—41no—
48C22×D97—41no—
49C22×D97—41no—
50C22×D97—41no—
51C22×D97—41no—
52Q8×C97—42no—
53D4×C97—42no—
54D4×C97—42no—
55D4×C97—42no—
56D4×C97—42no—
57D4×C97—81no—
58C2×C4×C97—21no—
59C2×C4×C97—21no—
60C2×C4×C97—21no—
61C2×C4×C97—22no—
62C2×C4×C97—41no—
63C2×C4×C97—44no—
64C2×C4×C97—41no—
65C2×C4×C97D812yescomplemented
66C2×C4×C97C2×D412yes—
67C2×Dic97—41no—
68C2×Dic97—41no—
69C2×Dic97—41no—
70D388—24no—
71D388—42no—
72#11715895—42no—
73C22×C97—21no—
74C22×C97—21no—
75C22×C97—42no—
76C22×C97—42no—
77C22×C97—42no—
78C22×C97#12011yescharacteristic
79C22×C97C2×D811yescharacteristic
80C22×C97AGL(1,ℤ/8)11yescharacteristic
81C2×D97—42no—
82C2×D97—42no—
83C2×D97—42no—
84C2×D97—81no—
85C2×D97—81no—
86C2×D97—81no—
87C2×D97—81no—
88C4×C97—22no—
89C4×C97—22no—
90C4×C97—42no—
91C4×C97—41no—
92Dic97—81no—
93C2×C97—21no—
94C2×C97—21no—
95C2×C97—82no—
96C2×C97#44711yescharacteristic
97C2×C97#45411yescharacteristic
98C2×C97B(4,𝔽2)11yescharacteristic
99D97—82no—
100#1598—971nomaximal, Sylow
101C97#159811yescharacteristic, complemented, Sylow
102#537D9711yescharacteristic, complemented
103#447—972no—
104#394—971no—
105#327—972no—
106#324—971no—
107C2×#41—21no—
108C22×D4—971no—
109C2×D8—1942no—
110C22⋊Q8—22no—
111#121—22no—
112#121—21no—
113#121C2×D9712yes—
114C2×C4⋊C4C2×D9711yescharacteristic
115C2×C22⋊C4—1941no—
116C2×C22⋊C4—1941no—
117#102—1942no—
118C22⋊C8—972no—
119#95—971no—
120C24—1941no—
121#41—42no—
122#41—42no—
123C2×Q8—21no—
124C2×D4—21no—
125C2×D4—22no—
126C2×D4—21no—
127C2×D4—3882no—
128C2×D4—1942no—
129C2×D4C22⋊D9712yes—
130C22×C4—22no—
131C22×C4—1941no—
132C22×C4C22⋊D9711yescharacteristic
133C22×C4C22⋊D9711yescharacteristic
134C22×C4C22×D9711yescharacteristic
135D8—3884no—
136C2×C8—1942no—
137C4⋊C4—21no—
138C4⋊C4—22no—
139C4⋊C4C22⋊D9712yes—
140C22⋊C4—22no—
141C22⋊C4—22no—
142C22⋊C4—3882no—
143C22⋊C4—3882no—
144C23—22no—
145C23—3881no—
146C23—3881no—
147C23—3881no—
148C23—3881no—
149C23—3881no—
150C23—3881no—
151C23—1942no—
152C23C2×C22⋊D9711yescharacteristic
153Q8—42no—
154D4—42no—
155D4—42no—
156D4—42no—
157D4—42no—
158D4—81no—
159D4—3882no—
160D4—1944no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—22no—
165C2×C4—41no—
166C2×C4—44no—
167C2×C4—41no—
168C2×C4—3881no—
169C2×C4—3881no—
170C2×C4—3881no—
171C2×C4#1309549612yescomplemented
172C2×C4C2×C22⋊D9712yes—
173C8—3882no—
174C22—21no—
175C22—21no—
176C22—42no—
177C22—42no—
178C22—42no—
179C22—7761no—
180C22—7761no—
181C22—7761no—
182C22—7761no—
183C22—3882no—
184C22—3882no—
185C22—3882no—
186C22#1569077911yescharacteristic
187C22C2×#1309549611yescharacteristic
188C22#1569083611yescharacteristic
189C4—22no—
190C4—22no—
191C4—42no—
192C4—41no—
193C4—7761no—
194C2—21no—
195C2—21no—
196C2—82no—
197C2—7762no—
198C2#2026086911yescharacteristic
199C2#2026090811yescharacteristic
200C2#2026092211yescharacteristic
subgroupG/H
centerC22#15690779
commutator, cosocleC22×C4×C97C23
FrattiniC22×C4C22×D97
Fitting#537×C97C2
socleC22×C97#120
hypercenter#537D97
hypocenterC97#1598
LCSG ▷ C22×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #537

Complements

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

NH ≅ G/Ntype
1. #20260868C2semidirect
2. #20260652C2semidirect
3. #20260469C2semidirect
4. #20260461C2semidirect
5. #537×C97C2semidirect
65. C2×C4×C97D8semidirect
101. C97#1598semidirect
102. #537D97semidirect
171. C2×C4#13095496semidirect

Maximal subgroups

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

subgroupindex
#202608682
#202606522
#202604692
#202604612
#537×C972
#159897