Finite Group Databasebeta

← #1473×C223 #43250488

Subgroups — #1473×C223 #43250488

Order 28544.

214 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 88 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×#126×C223C211yesmaximal, characteristic
2#432×C223C211yesmaximal, characteristic, complemented
3#431×C223C211yesmaximal, characteristic, complemented
4#324×C223C212yesmaximal, complemented
5#323×C223C212yesmaximal
6#126×C223—21no—
7#126×C223—21no—
8#126×C223—21no—
9#126×C223—21no—
10C2×C4⋊C4×C223C411yescharacteristic
11C2×C4⋊C4×C223C411yescharacteristic
12C2×C4⋊C4×C223C2211yescharacteristic
13C2×C22⋊C4×C223C411yescharacteristic
14C2×C22⋊C4×C223C411yescharacteristic
15C2×C22⋊C4×C223C2211yescharacteristic
16C2×C42×C223C2211yescharacteristic
17C4⋊C8×C223—21no—
18C4⋊C8×C223—21no—
19C22⋊C8×C223C2212yes—
20C22⋊C8×C223C2212yes—
21#97×C223—21no—
22C4×C8×C223—21no—
23C24×C223C2×C411yescharacteristic
24C22×C4×C223C2×C411yescharacteristic
25C22×C4×C223C2×C411yescharacteristic
26C22×C4×C223C2×C411yescharacteristic
27C22×C4×C223C2×C411yescharacteristic
28C22×C4×C223C2×C411yescharacteristic
29C22×C4×C223C2311yescharacteristic
30C2×C8×C223—22no—
31C2×C8×C223—22no—
32C4⋊C4×C223—21no—
33C4⋊C4×C223—21no—
34C4⋊C4×C223—21no—
35C4⋊C4×C223—21no—
36C4⋊C4×C223—21no—
37C4⋊C4×C223—21no—
38C22⋊C4×C223—21no—
39C22⋊C4×C223—21no—
40C22⋊C4×C223—21no—
41C22⋊C4×C223—21no—
42C22⋊C4×C223—21no—
43C22⋊C4×C223—21no—
44C42×C223D411yescharacteristic
45C42×C223D411yescharacteristic
46C42×C223D411yescharacteristic
47C42×C223D411yescharacteristic
48C23×C223—21no—
49C23×C223—22no—
50C23×C223—41no—
51C23×C223—41no—
52C23×C223C22×C411yescharacteristic
53C2×C4×C223—22no—
54C2×C4×C223—21no—
55C2×C4×C223—21no—
56C2×C4×C223—21no—
57C2×C4×C223—22no—
58C2×C4×C223—41no—
59C2×C4×C223—41no—
60C2×C4×C223—41no—
61C2×C4×C223—41no—
62C2×C4×C223C22⋊C411yescharacteristic
63C2×C4×C223C22⋊C411yescharacteristic
64C2×C4×C223C22⋊C411yescharacteristic
65C2×C4×C223C22⋊C411yescharacteristic
66C2×C4×C223C2×D411yescharacteristic
67C2×C4×C223C2×D411yescharacteristic
68C8×C223—42no—
69C8×C223—42no—
70C22×C223—21no—
71C22×C223—21no—
72C22×C223—41no—
73C22×C223—41no—
74C22×C223—41no—
75C22×C223—82no—
76C22×C223C2×C22⋊C411yescharacteristic
77C22×C223#13112yes—
78C4×C223—21no—
79C4×C223—21no—
80C4×C223—21no—
81C4×C223—21no—
82C4×C223—21no—
83C4×C223—21no—
84C4×C223—41no—
85C4×C223—41no—
86C4×C223—41no—
87C2×C223—22no—
88C2×C223—81no—
89C2×C223#40811yescharacteristic
90C2×C223#41011yescharacteristic
91C2×C223#41311yescharacteristic
92C223#147311yescharacteristic, direct product factor, Sylow
93#1473C22311yesmaximal, characteristic, direct product factor, Sylow
94C2×#126C2×C22311yescharacteristic
95#432C2×C22311yescharacteristic, complemented
96#431C2×C22311yescharacteristic, complemented
97#324C2×C22312yescomplemented
98#323C2×C22312yes—
99#126—21no—
100#126—21no—
101#126—21no—
102#126—21no—
103C2×C4⋊C4C4×C22311yescharacteristic
104C2×C4⋊C4C4×C22311yescharacteristic
105C2×C4⋊C4C22×C22311yescharacteristic
106C2×C22⋊C4C4×C22311yescharacteristic
107C2×C22⋊C4C4×C22311yescharacteristic
108C2×C22⋊C4C22×C22311yescharacteristic
109C2×C42C22×C22311yescharacteristic
110C4⋊C8—21no—
111C4⋊C8—21no—
112C22⋊C8C22×C22312yes—
113C22⋊C8C22×C22312yes—
114#97—21no—
115C4×C8—21no—
116C24C2×C4×C22311yescharacteristic
117C22×C4C2×C4×C22311yescharacteristic
118C22×C4C2×C4×C22311yescharacteristic
119C22×C4C2×C4×C22311yescharacteristic
120C22×C4C2×C4×C22311yescharacteristic
121C22×C4C2×C4×C22311yescharacteristic
122C22×C4C23×C22311yescharacteristic
123C2×C8—22no—
124C2×C8—22no—
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—
137C42D4×C22311yescharacteristic
138C42D4×C22311yescharacteristic
139C42D4×C22311yescharacteristic
140C42D4×C22311yescharacteristic
141C23—21no—
142C23—22no—
143C23—41no—
144C23—41no—
145C23C22×C4×C22311yescharacteristic
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⋊C4×C22311yescharacteristic
156C2×C4C22⋊C4×C22311yescharacteristic
157C2×C4C22⋊C4×C22311yescharacteristic
158C2×C4C22⋊C4×C22311yescharacteristic
159C2×C4C2×D4×C22311yescharacteristic
160C2×C4C2×D4×C22311yescharacteristic
161C8—42no—
162C8—42no—
163C22—21no—
164C22—21no—
165C22—41no—
166C22—41no—
167C22—41no—
168C22—82no—
169C22C2×C22⋊C4×C22311yescharacteristic
170C22#131×C22312yes—
171C4—21no—
172C4—21no—
173C4—21no—
174C4—21no—
175C4—21no—
176C4—21no—
177C4—41no—
178C4—41no—
179C4—41no—
180C2—22no—
181C2—81no—
182C2#408×C22311yescharacteristic
183C2#410×C22311yescharacteristic
184C2#413×C22311yescharacteristic
subgroupG/H
center, socleC22×C223C2×C22⋊C4
commutatorC23C22×C4×C223
Frattini, cosocleC22×C4C23×C223
prime radicalC24×C223C2×C4
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C223 ◁ C2×C42×C223 ◁ G

Complements

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

NH ≅ G/Ntype
2. #432×C223C2semidirect
3. #431×C223C2semidirect
4. #324×C223C2semidirect
92. C223#1473direct (N × H)
93. #1473C223direct (N × H)
95. #432C2×C223semidirect
96. #431C2×C223semidirect
97. #324C2×C223semidirect

Maximal subgroups

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

subgroupindex
C2×#126×C2232
#432×C2232
#431×C2232
#324×C2232
#323×C2232
#1473223