Finite Group Databasebeta

← #27140105

Subgroups — #27140105

Order 12416.

214 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 77 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#20260792C211yesmaximal, characteristic, complemented
2#20260788C211yesmaximal, characteristic, complemented
3#20260461C212yesmaximal, complemented
4#20260459C212yesmaximal
5C2×#126×C97C211yesmaximal, characteristic
6C4⋊#11715895—21no—
7C4⋊#11715895—21no—
8C22⋊#11715895C2212yes—
9C22⋊#11715895C2212yes—
10#15690721—21no—
11C4×#11715895—21no—
12#126×C97—21no—
13#126×C97—21no—
14#126×C97—21no—
15#126×C97—21no—
16C2×C4⋊C4×C97C411yescharacteristic
17C2×C4⋊C4×C97C411yescharacteristic
18C2×C4⋊C4×C97C2211yescharacteristic
19C2×C22⋊C4×C97C411yescharacteristic
20C2×C22⋊C4×C97C411yescharacteristic
21C2×C22⋊C4×C97C2211yescharacteristic
22C2×C42×C97C2211yescharacteristic
23C2×#11715895—22no—
24C2×#11715895—22no—
25C24×C97C2×C411yescharacteristic
26C22×C4×C97C2×C411yescharacteristic
27C22×C4×C97C2×C411yescharacteristic
28C22×C4×C97C2×C411yescharacteristic
29C22×C4×C97C2×C411yescharacteristic
30C22×C4×C97C2×C411yescharacteristic
31C22×C4×C97C2311yescharacteristic
32C4⋊C4×C97—21no—
33C4⋊C4×C97—21no—
34C4⋊C4×C97—21no—
35C4⋊C4×C97—21no—
36C4⋊C4×C97—21no—
37C4⋊C4×C97—21no—
38C22⋊C4×C97—21no—
39C22⋊C4×C97—21no—
40C22⋊C4×C97—21no—
41C22⋊C4×C97—21no—
42C22⋊C4×C97—21no—
43C22⋊C4×C97—21no—
44C42×C97D411yescharacteristic
45C42×C97D411yescharacteristic
46C42×C97D411yescharacteristic
47C42×C97D411yescharacteristic
48C23×C97—21no—
49C23×C97—22no—
50C23×C97—41no—
51C23×C97—41no—
52C23×C97C22×C411yescharacteristic
53C2×C4×C97—22no—
54C2×C4×C97—21no—
55C2×C4×C97—21no—
56C2×C4×C97—21no—
57C2×C4×C97—22no—
58C2×C4×C97—41no—
59C2×C4×C97—41no—
60C2×C4×C97—41no—
61C2×C4×C97—41no—
62C2×C4×C97C22⋊C411yescharacteristic
63C2×C4×C97C22⋊C411yescharacteristic
64C2×C4×C97C22⋊C411yescharacteristic
65C2×C4×C97C22⋊C411yescharacteristic
66C2×C4×C97C2×D411yescharacteristic
67C2×C4×C97C2×D411yescharacteristic
68#11715895—42no—
69#11715895—42no—
70C22×C97—21no—
71C22×C97—21no—
72C22×C97—41no—
73C22×C97—41no—
74C22×C97—41no—
75C22×C97—82no—
76C22×C97C2×C22⋊C411yescharacteristic
77C22×C97#13112yes—
78C4×C97—21no—
79C4×C97—21no—
80C4×C97—21no—
81C4×C97—21no—
82C4×C97—21no—
83C4×C97—21no—
84C4×C97—41no—
85C4×C97—41no—
86C4×C97—41no—
87C2×C97—22no—
88C2×C97—81no—
89C2×C97#40811yescharacteristic
90C2×C97#41011yescharacteristic
91C2×C97#41311yescharacteristic
92#1473—971nomaximal, Sylow
93C97#147311yescharacteristic, complemented, Sylow
94C2×#126D9711yescharacteristic
95#432—971no—
96#431—971no—
97#324—972no—
98#323—972no—
99#126—21no—
100#126—21no—
101#126—21no—
102#126—21no—
103C2×C4⋊C4Dic9711yescharacteristic
104C2×C4⋊C4Dic9711yescharacteristic
105C2×C4⋊C4C2×D9711yescharacteristic
106C2×C22⋊C4Dic9711yescharacteristic
107C2×C22⋊C4Dic9711yescharacteristic
108C2×C22⋊C4C2×D9711yescharacteristic
109C2×C42C2×D9711yescharacteristic
110C4⋊C8—1941no—
111C4⋊C8—1941no—
112C22⋊C8—972no—
113C22⋊C8—972no—
114#97—1941no—
115C4×C8—1941no—
116C24C2×Dic9711yescharacteristic
117C22×C4C2×Dic9711yescharacteristic
118C22×C4C2×Dic9711yescharacteristic
119C22×C4C2×Dic9711yescharacteristic
120C22×C4C2×Dic9711yescharacteristic
121C22×C4C2×Dic9711yescharacteristic
122C22×C4C22×D9711yescharacteristic
123C2×C8—1942no—
124C2×C8—1942no—
125C4⋊C4—21no—
126C4⋊C4—21no—
127C4⋊C4—21no—
128C4⋊C4—21no—
129C4⋊C4—21no—
130C4⋊C4—21no—
131C22⋊C4—21no—
132C22⋊C4—21no—
133C22⋊C4—21no—
134C22⋊C4—21no—
135C22⋊C4—21no—
136C22⋊C4—21no—
137C42C22⋊D9711yescharacteristic
138C42C22⋊D9711yescharacteristic
139C42C22⋊D9711yescharacteristic
140C42C22⋊D9711yescharacteristic
141C23—21no—
142C23—22no—
143C23—41no—
144C23—41no—
145C23C22×Dic9711yescharacteristic
146C2×C4—22no—
147C2×C4—21no—
148C2×C4—21no—
149C2×C4—21no—
150C2×C4—22no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4—41no—
155C2×C4C22⋊Dic9711yescharacteristic
156C2×C4C22⋊Dic9711yescharacteristic
157C2×C4C22⋊Dic9711yescharacteristic
158C2×C4C22⋊Dic9711yescharacteristic
159C2×C4C2×C22⋊D9711yescharacteristic
160C2×C4C2×C22⋊D9711yescharacteristic
161C8—3882no—
162C8—3882no—
163C22—21no—
164C22—21no—
165C22—41no—
166C22—41no—
167C22—41no—
168C22—82no—
169C22C2×C22⋊Dic9711yescharacteristic
170C22#1569081712yes—
171C4—21no—
172C4—21no—
173C4—21no—
174C4—21no—
175C4—21no—
176C4—21no—
177C4—41no—
178C4—41no—
179C4—41no—
180C2—22no—
181C2—81no—
182C2#2026070211yescharacteristic
183C2#2026070911yescharacteristic
184C2#2026071911yescharacteristic
subgroupG/H
centerC22C2×C22⋊Dic97
commutatorC23×C97C22×C4
FrattiniC22×C4C22×D97
FittingC2×#126×C97C2
socleC22×C97C2×C22⋊C4
prime radicalC24×C97C2×C4
cosocleC22×C4×C97C23
hypercenterC2×#126D97
hypocenterC97#1473
LCSG ▷ C23×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#126

Complements

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

NH ≅ G/Ntype
1. #20260792C2semidirect
2. #20260788C2semidirect
3. #20260461C2semidirect
93. C97#1473semidirect

Maximal subgroups

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

subgroupindex
#202607922
#202607882
#202604612
#202604592
C2×#126×C972
#147397