Finite Group Databasebeta

← #1604×C223 #43250619

Subgroups — #1604×C223 #43250619

Order 28544.

282 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 68 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#537×C223C211yesmaximal, characteristic
2#448×C223C212yesmaximal, complemented
3#395×C223C211yesmaximal, characteristic, complemented
4#327×C223C212yesmaximal, complemented
5#324×C223C211yesmaximal, characteristic, complemented
6C2×#41×C223—21no—
7C22×Q8×C223C2211yescharacteristic
8C2×SD16×C223—22no—
9C22⋊Q8×C223—22no—
10#121×C223—22no—
11#121×C223—21no—
12#121×C223C2212yes—
13C2×C4⋊C4×C223—21no—
14C2×C4⋊C4×C223C2211yescharacteristic
15#103×C223—22no—
16C22⋊C8×C223C2212yes—
17#95×C223—21no—
18#95×C223C2211yescharacteristic
19#41×C223—42no—
20#41×C223—42no—
21C2×Q8×C223—21no—
22C2×Q8×C223—22no—
23C2×Q8×C223—42no—
24C2×D4×C223—21no—
25C2×D4×C223—22no—
26C2×D4×C223—21no—
27C2×D4×C223D412yes—
28C22×C4×C223—22no—
29C22×C4×C223—21no—
30C22×C4×C223—21no—
31C22×C4×C223D411yescharacteristic
32C22×C4×C223D411yescharacteristic
33C22×C4×C223C2311yescharacteristic
34SD16×C223—44no—
35C2×C8×C223—22no—
36C4⋊C4×C223—21no—
37C4⋊C4×C223—22no—
38C4⋊C4×C223—42no—
39C4⋊C4×C223D412yes—
40C22⋊C4×C223—22no—
41C22⋊C4×C223—22no—
42C23×C223—22no—
43C23×C223C2×D411yescharacteristic
44Q8×C223—24no—
45Q8×C223—42no—
46Q8×C223—42no—
47D4×C223—42no—
48D4×C223—42no—
49D4×C223—42no—
50D4×C223—42no—
51D4×C223—81no—
52C2×C4×C223—21no—
53C2×C4×C223—21no—
54C2×C4×C223—22no—
55C2×C4×C223—22no—
56C2×C4×C223—21no—
57C2×C4×C223—22no—
58C2×C4×C223—41no—
59C2×C4×C223—44no—
60C2×C4×C223—41no—
61C2×C4×C223—41no—
62C2×C4×C223—41no—
63C2×C4×C223—41no—
64C2×C4×C223—41no—
65C2×C4×C223SD1612yescomplemented
66C2×C4×C223C2×D412yes—
67C8×C223—42no—
68C22×C223—21no—
69C22×C223—21no—
70C22×C223—42no—
71C22×C223—42no—
72C22×C223—42no—
73C22×C223#12011yescharacteristic
74C22×C223C2×SD1611yescharacteristic
75C22×C223AGL(1,ℤ/8)11yescharacteristic
76C4×C223—22no—
77C4×C223—22no—
78C4×C223—42no—
79C4×C223—42no—
80C4×C223—42no—
81C4×C223—41no—
82C2×C223—21no—
83C2×C223—21no—
84C2×C223—82no—
85C2×C223#45011yescharacteristic
86C2×C223#45611yescharacteristic
87C2×C223B(4,𝔽2)11yescharacteristic
88C223#160411yescharacteristic, direct product factor, Sylow
89#1604C22311yesmaximal, characteristic, direct product factor, Sylow
90#537C2×C22311yescharacteristic
91#448C2×C22312yescomplemented
92#395C2×C22311yescharacteristic, complemented
93#327C2×C22312yescomplemented
94#324C2×C22311yescharacteristic, complemented
95C2×#41—21no—
96C22×Q8C22×C22311yescharacteristic
97C2×SD16—22no—
98C22⋊Q8—22no—
99#121—22no—
100#121—21no—
101#121C22×C22312yes—
102C2×C4⋊C4—21no—
103C2×C4⋊C4C22×C22311yescharacteristic
104#103—22no—
105C22⋊C8C22×C22312yes—
106#95—21no—
107#95C22×C22311yescharacteristic
108#41—42no—
109#41—42no—
110C2×Q8—21no—
111C2×Q8—22no—
112C2×Q8—42no—
113C2×D4—21no—
114C2×D4—22no—
115C2×D4—21no—
116C2×D4D4×C22312yes—
117C22×C4—22no—
118C22×C4—21no—
119C22×C4—21no—
120C22×C4D4×C22311yescharacteristic
121C22×C4D4×C22311yescharacteristic
122C22×C4C23×C22311yescharacteristic
123SD16—44no—
124C2×C8—22no—
125C4⋊C4—21no—
126C4⋊C4—22no—
127C4⋊C4—42no—
128C4⋊C4D4×C22312yes—
129C22⋊C4—22no—
130C22⋊C4—22no—
131C23—22no—
132C23C2×D4×C22311yescharacteristic
133Q8—24no—
134Q8—42no—
135Q8—42no—
136D4—42no—
137D4—42no—
138D4—42no—
139D4—42no—
140D4—81no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—22no—
144C2×C4—22no—
145C2×C4—21no—
146C2×C4—22no—
147C2×C4—41no—
148C2×C4—44no—
149C2×C4—41no—
150C2×C4—41no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—41no—
154C2×C4SD16×C22312yescomplemented
155C2×C4C2×D4×C22312yes—
156C8—42no—
157C22—21no—
158C22—21no—
159C22—42no—
160C22—42no—
161C22—42no—
162C22#120×C22311yescharacteristic
163C22C2×SD16×C22311yescharacteristic
164C22AGL(1,ℤ/8)×C22311yescharacteristic
165C4—22no—
166C4—22no—
167C4—42no—
168C4—42no—
169C4—42no—
170C4—41no—
171C2—21no—
172C2—21no—
173C2—82no—
174C2#450×C22311yescharacteristic
175C2#456×C22311yescharacteristic
176C2B(4,𝔽2)×C22311yescharacteristic
subgroupG/H
center, socleC22×C223#120
commutator, Frattini, cosocleC22×C4C23×C223
prime radical#537×C223C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C223 ◁ C22×C4×C223 ◁ G

Complements

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

NH ≅ G/Ntype
2. #448×C223C2semidirect
3. #395×C223C2semidirect
4. #327×C223C2semidirect
5. #324×C223C2semidirect
65. C2×C4×C223SD16semidirect
88. C223#1604direct (N × H)
89. #1604C223direct (N × H)
91. #448C2×C223semidirect
92. #395C2×C223semidirect
93. #327C2×C223semidirect
94. #324C2×C223semidirect
154. C2×C4SD16×C223semidirect

Maximal subgroups

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

subgroupindex
#537×C2232
#448×C2232
#395×C2232
#327×C2232
#324×C2232
#1604223