Finite Group Databasebeta

← #27140733

Subgroups — #27140733

Order 12416.

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

normal subgroups only

This group has 216 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#20261063C211yesmaximal, characteristic, complemented
2#20261046C211yesmaximal, characteristic
3#20260665C211yesmaximal, characteristic, complemented
4#20260472C211yesmaximal, characteristic, complemented
5#20260469C211yesmaximal, characteristic
6#20260461C211yesmaximal, characteristic, complemented
7#539×C97C211yesmaximal, characteristic
8C2×C4⋊Dic97C2211yescharacteristic
9C2×#13095490—21no—
10#15690743—21no—
11#15690741—21no—
12#15690734—21no—
13#15690731—21no—
14C22⋊#11715895C2211yescharacteristic
15C22⋊#11715895C2211yescharacteristic
16#15690717—21no—
17#15690717C2211yescharacteristic
18#126×C97—21no—
19#125×C97—21no—
20#123×C97—21no—
21#123×C97—21no—
22C22⋊Q8×C97—21no—
23C22⋊Q8×C97C2211yescharacteristic
24#121×C97C2211yescharacteristic
25C4×D4×C97—21no—
26C2×C4⋊C4×C97C2211yescharacteristic
27C22×Dic97—21no—
28C22×Dic97—21no—
29C2×#11715895—21no—
30C2×#11715895—21no—
31C4⋊Dic97—21no—
32C4⋊Dic97—21no—
33#13095490—42no—
34C2×Q8×C97D411yescharacteristic
35C2×D4×C97—21no—
36C2×D4×C97D411yescharacteristic
37C22×C4×C97—21no—
38C22×C4×C97D411yescharacteristic
39C22×C4×C97D411yescharacteristic
40C22×C4×C97C2311yescharacteristic
41C4⋊C4×C97—21no—
42C4⋊C4×C97—21no—
43C4⋊C4×C97—21no—
44C4⋊C4×C97—21no—
45C4⋊C4×C97—21no—
46C4⋊C4×C97—21no—
47C4⋊C4×C97D411yescharacteristic
48C4⋊C4×C97D411yescharacteristic
49C22⋊C4×C97—21no—
50C22⋊C4×C97—21no—
51C22⋊C4×C97—21no—
52C22⋊C4×C97—21no—
53C22⋊C4×C97—21no—
54C42×C97—21no—
55C23×C97—21no—
56C23×C97C2×D411yescharacteristic
57Q8×C97—41no—
58D4×C97—41no—
59D4×C97—41no—
60D4×C97—41no—
61C2×C4×C97—21no—
62C2×C4×C97—21no—
63C2×C4×C97—21no—
64C2×C4×C97—21no—
65C2×C4×C97—21no—
66C2×C4×C97—21no—
67C2×C4×C97—21no—
68C2×C4×C97—41no—
69C2×C4×C97—41no—
70C2×C4×C97—81no—
71C2×C4×C97C2×D411yescharacteristic
72C2×C4×C97C2×D411yescharacteristic
73C2×Dic97—22no—
74C2×Dic97—41no—
75C2×Dic97—41no—
76C2×Dic97—41no—
77C2×Dic97—41no—
78C2×Dic97—41no—
79#11715895—41no—
80#11715895—41no—
81C22×C97—21no—
82C22×C97—21no—
83C22×C97—41no—
84C22×C97—41no—
85C22×C97—41no—
86C22×C97#12011yescharacteristic
87C22×C97#13511yescharacteristic
88C22×C97AGL(1,ℤ/8)11yescharacteristic
89C4×C97—21no—
90C4×C97—21no—
91C4×C97—41no—
92C4×C97—41no—
93C4×C97—41no—
94C4×C97—41no—
95C4×C97—41no—
96Dic97—42no—
97Dic97—81no—
98C2×C97—21no—
99C2×C97—21no—
100C2×C97—81no—
101C2×C97#44911yescharacteristic
102C2×C97#45611yescharacteristic
103C2×C97#45811yescharacteristic
104#1609—971nomaximal, Sylow
105C97#160911yescharacteristic, complemented, Sylow
106#539D9711yescharacteristic
107#483—971no—
108#480—971no—
109#398—971no—
110#328—971no—
111#327—971no—
112#324—971no—
113#126—21no—
114#125—21no—
115#123—21no—
116#123—21no—
117C22⋊Q8—21no—
118C22⋊Q8C2×D9711yescharacteristic
119#121C2×D9711yescharacteristic
120C4×D4—21no—
121C2×C4⋊C4—971no—
122C2×C4⋊C4—1941no—
123C2×C4⋊C4C2×D9711yescharacteristic
124#107—1941no—
125#106—1941no—
126#103—1941no—
127#102—1941no—
128C22⋊C8—971no—
129C22⋊C8—971no—
130#95—971no—
131#95—1941no—
132C2×Q8C22⋊D9711yescharacteristic
133C2×D4—21no—
134C2×D4C22⋊D9711yescharacteristic
135C22×C4—21no—
136C22×C4—1941no—
137C22×C4—1941no—
138C22×C4C22⋊D9711yescharacteristic
139C22×C4C22⋊D9711yescharacteristic
140C22×C4C22×D9711yescharacteristic
141C2×C8—1941no—
142C2×C8—1941no—
143C4⋊C4—21no—
144C4⋊C4—21no—
145C4⋊C4—21no—
146C4⋊C4—21no—
147C4⋊C4—21no—
148C4⋊C4—21no—
149C4⋊C4—3882no—
150C4⋊C4—1941no—
151C4⋊C4—1941no—
152C4⋊C4C22⋊D9711yescharacteristic
153C4⋊C4C22⋊D9711yescharacteristic
154C22⋊C4—21no—
155C22⋊C4—21no—
156C22⋊C4—21no—
157C22⋊C4—21no—
158C22⋊C4—21no—
159C42—21no—
160C23—21no—
161C23C2×C22⋊D9711yescharacteristic
162Q8—41no—
163D4—41no—
164D4—41no—
165D4—41no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—81no—
176C2×C4—3881no—
177C2×C4—3881no—
178C2×C4—3881no—
179C2×C4—3881no—
180C2×C4—3881no—
181C2×C4—1942no—
182C2×C4C2×C22⋊D9711yescharacteristic
183C2×C4C2×C22⋊D9711yescharacteristic
184C8—3881no—
185C8—3881no—
186C22—21no—
187C22—21no—
188C22—41no—
189C22—41no—
190C22—41no—
191C22#1569077911yescharacteristic
192C22#1569083011yescharacteristic
193C22#1569083511yescharacteristic
194C4—21no—
195C4—21no—
196C4—41no—
197C4—41no—
198C4—41no—
199C4—41no—
200C4—41no—
201C4—7761no—
202C4—3882no—
203C2—21no—
204C2—21no—
205C2—81no—
206C2#2026088011yescharacteristic
207C2#2026091811yescharacteristic
208C2#2026092511yescharacteristic
subgroupG/H
centerC22#15690779
commutator, cosocleC22×C4×C97C23
FrattiniC22×C4C22×D97
Fitting#539×C97C2
socleC22×C97#120
prime radical#121×C97C22
hypercenter#539D97
hypocenterC97#1609
LCSG ▷ C22×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #539

Complements

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

NH ≅ G/Ntype
1. #20261063C2semidirect
3. #20260665C2semidirect
4. #20260472C2semidirect
6. #20260461C2semidirect
105. C97#1609semidirect

Maximal subgroups

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

subgroupindex
#202610632
#202610462
#202606652
#202604722
#202604692
#202604612
#539×C972
#160997