Finite Group Databasebeta

← #27140102

Subgroups — #27140102

Order 12416.

214 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 61 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
1C2×#15690803C211yesmaximal, characteristic
2#20260789C211yesmaximal, characteristic, complemented
3#20260785C211yesmaximal, characteristic, complemented
4#20260460C211yesmaximal, characteristic, complemented
5#20260460C211yesmaximal, characteristic, complemented
6#20260459C211yesmaximal, characteristic
7#323×C97C211yesmaximal, characteristic
8#15690803—21no—
9#15690803—21no—
10#15690803—21no—
11#15690803—21no—
12C2×C4⋊Dic97C411yescharacteristic
13C2×#13095490C411yescharacteristic
14C2×#13095490C2211yescharacteristic
15C2×C22⋊Dic97C411yescharacteristic
16C2×C22⋊Dic97C411yescharacteristic
17C2×C4×Dic97C2211yescharacteristic
18#15690739—21no—
19#15690739—21no—
20C22⋊#11715895C2211yescharacteristic
21C22⋊#11715895C2211yescharacteristic
22#15690720—21no—
23C8×Dic97—21no—
24C2×C22⋊C4×C97C2211yescharacteristic
25C22⋊C8×C97C2211yescharacteristic
26C22⋊C8×C97C2211yescharacteristic
27C22×Dic97C2×C411yescharacteristic
28C22×Dic97C2×C411yescharacteristic
29C22×Dic97C2×C411yescharacteristic
30C22×Dic97C2×C411yescharacteristic
31C2×#11715895—21no—
32C2×#11715895—21no—
33C4⋊Dic97—21no—
34C4⋊Dic97—21no—
35#13095490—21no—
36#13095490—21no—
37#13095490—21no—
38#13095490—21no—
39C22⋊Dic97—21no—
40C22⋊Dic97—21no—
41C22⋊Dic97—21no—
42C22⋊Dic97—21no—
43C4×Dic97D411yescharacteristic
44C4×Dic97D411yescharacteristic
45C4×Dic97D411yescharacteristic
46C4×Dic97D411yescharacteristic
47C24×C97C2×C411yescharacteristic
48C22×C4×C97C2×C411yescharacteristic
49C22×C4×C97C2311yescharacteristic
50C2×C8×C97—21no—
51C2×C8×C97—21no—
52C22⋊C4×C97—21no—
53C22⋊C4×C97—21no—
54C23×C97—21no—
55C23×C97—21no—
56C23×C97—21no—
57C23×C97—41no—
58C23×C97—41no—
59C23×C97C22×C411yescharacteristic
60C2×C4×C97—21no—
61C2×C4×C97—41no—
62C2×C4×C97—41no—
63C2×C4×C97C2×D411yescharacteristic
64C2×C4×C97C2×D411yescharacteristic
65C2×Dic97—21no—
66C2×Dic97—21no—
67C2×Dic97—21no—
68C2×Dic97—21no—
69C2×Dic97—21no—
70C2×Dic97—21no—
71C2×Dic97—41no—
72C2×Dic97—41no—
73C2×Dic97C22⋊C411yescharacteristic
74C2×Dic97C22⋊C411yescharacteristic
75C2×Dic97C22⋊C411yescharacteristic
76C2×Dic97C22⋊C411yescharacteristic
77C8×C97—41no—
78C8×C97—41no—
79#11715895—41no—
80#11715895—41no—
81C22×C97—21no—
82C22×C97—21no—
83C22×C97—41no—
84C22×C97—41no—
85C22×C97—41no—
86C22×C97—81no—
87C22×C97—81no—
88C22×C97C2×C22⋊C411yescharacteristic
89C22×C97#13111yescharacteristic
90C22×C97#13111yescharacteristic
91C4×C97—21no—
92C4×C97—21no—
93C4×C97—41no—
94Dic97—21no—
95Dic97—21no—
96Dic97—21no—
97Dic97—21no—
98Dic97—41no—
99Dic97—41no—
100C2×C97—21no—
101C2×C97—21no—
102C2×C97—81no—
103C2×C97#40811yescharacteristic
104C2×C97#41011yescharacteristic
105C2×C97#41311yescharacteristic
106#1473—971nomaximal, Sylow
107C97#147311yescharacteristic, complemented, Sylow
108C2×#126—971no—
109#432—971no—
110#431—971no—
111#324—971no—
112#324—971no—
113#323—971no—
114#323D9711yescharacteristic
115#126—1941no—
116#126—1941no—
117#126—1941no—
118#126—1941no—
119C2×C4⋊C4—971no—
120C2×C4⋊C4—971no—
121C2×C4⋊C4—971no—
122C2×C22⋊C4—971no—
123C2×C22⋊C4—971no—
124C2×C22⋊C4C2×D9711yescharacteristic
125C2×C42—971no—
126C4⋊C8—1941no—
127C4⋊C8—1941no—
128C22⋊C8—971no—
129C22⋊C8—971no—
130C22⋊C8C2×D9711yescharacteristic
131C22⋊C8C2×D9711yescharacteristic
132#97—1941no—
133C4×C8—1941no—
134C24C4×D9711yescharacteristic
135C22×C4—971no—
136C22×C4—971no—
137C22×C4—971no—
138C22×C4—971no—
139C22×C4C4×D9711yescharacteristic
140C22×C4C22×D9711yescharacteristic
141C2×C8—21no—
142C2×C8—21no—
143C2×C8—1941no—
144C2×C8—1941no—
145C4⋊C4—1941no—
146C4⋊C4—1941no—
147C4⋊C4—1941no—
148C4⋊C4—1941no—
149C4⋊C4—1941no—
150C4⋊C4—1941no—
151C22⋊C4—21no—
152C22⋊C4—21no—
153C22⋊C4—1941no—
154C22⋊C4—1941no—
155C22⋊C4—1941no—
156C22⋊C4—1941no—
157C42—971no—
158C42—971no—
159C42—971no—
160C42—971no—
161C23—21no—
162C23—21no—
163C23—21no—
164C23—41no—
165C23—41no—
166C23C2×C4×D9711yescharacteristic
167C2×C4—21no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—971no—
171C2×C4—971no—
172C2×C4—971no—
173C2×C4—971no—
174C2×C4—3881no—
175C2×C4—3881no—
176C2×C4—1941no—
177C2×C4—1941no—
178C2×C4—1941no—
179C2×C4—1941no—
180C2×C4—1941no—
181C2×C4—1941no—
182C2×C4D4×D9711yescharacteristic
183C2×C4D4×D9711yescharacteristic
184C8—41no—
185C8—41no—
186C8—3881no—
187C8—3881no—
188C22—21no—
189C22—21no—
190C22—41no—
191C22—41no—
192C22—41no—
193C22—81no—
194C22—81no—
195C22C22⋊C4×D9711yescharacteristic
196C22#1569081611yescharacteristic
197C22#1569081811yescharacteristic
198C4—21no—
199C4—21no—
200C4—41no—
201C4—3881no—
202C4—3881no—
203C4—1941no—
204C4—1941no—
205C4—1941no—
206C4—1941no—
207C2—21no—
208C2—21no—
209C2—81no—
210C2#2026070011yescharacteristic
211C2#2026070711yescharacteristic
212C2#2026071611yescharacteristic
subgroupG/H
centerC22C22⋊C4×D97
commutatorC23×C97C22×C4
FrattiniC22×C4C22×D97
Fitting#323×C97C2
socleC22×C97C2×C22⋊C4
prime radicalC24×C97C2×C4
cosocleC22×C4×C97C23
hypercenter#323D97
hypocenterC97#1473
LCSG ▷ C23×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #323

Complements

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

NH ≅ G/Ntype
2. #20260789C2semidirect
3. #20260785C2semidirect
4. #20260460C2semidirect
5. #20260460C2semidirect
107. C97#1473semidirect

Maximal subgroups

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

subgroupindex
C2×#156908032
#202607892
#202607852
#202604602
#202604592
#323×C972
#147397