Finite Group Databasebeta

← #1613×C223 #43250628

Subgroups — #1613×C223 #43250628

Order 28544.

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

normal subgroups only

This group has 288 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#534×C223C211yesmaximal, characteristic
2#481×C223C212yesmaximal, complemented
3#399×C223C211yesmaximal, characteristic, complemented
4#327×C223C212yesmaximal, complemented
5#323×C223C211yesmaximal, characteristic, complemented
6C22×D4×C223—21no—
7#123×C223—22no—
8#121×C223—21no—
9#121×C223C2212yes—
10#120×C223—22no—
11C2×C4⋊C4×C223—21no—
12C2×C4⋊C4×C223C2211yescharacteristic, complemented
13C2×C22⋊C4×C223—21no—
14C2×C22⋊C4×C223C2211yescharacteristic
15#106×C223—22no—
16#102×C223—22no—
17C22⋊C8×C223C2212yescomplemented
18#95×C223C2211yescharacteristic, complemented
19C24×C223—21no—
20C24×C223D411yescharacteristic
21C2×D4×C223—22no—
22C2×D4×C223—22no—
23C2×D4×C223—21no—
24C2×D4×C223—42no—
25C2×D4×C223—42no—
26C2×D4×C223D412yes—
27C22×C4×C223—21no—
28C22×C4×C223—21no—
29C22×C4×C223—21no—
30C22×C4×C223D411yescharacteristic
31C22×C4×C223C2311yescharacteristic
32C2×C8×C223—22no—
33C4⋊C4×C223—21no—
34C4⋊C4×C223—22no—
35C4⋊C4×C223—42no—
36C4⋊C4×C223D412yes—
37C22⋊C4×C223—22no—
38C22⋊C4×C223—22no—
39C22⋊C4×C223—22no—
40C22⋊C4×C223—42no—
41C23×C223—21no—
42C23×C223—21no—
43C23×C223—22no—
44C23×C223—41no—
45C23×C223—41no—
46C23×C223—42no—
47C23×C223—81no—
48C23×C223—81no—
49C23×C223SD1612yes—
50C23×C223C2×D411yescharacteristic
51D4×C223—42no—
52D4×C223—44no—
53D4×C223—42no—
54D4×C223—42no—
55D4×C223—81no—
56C2×C4×C223—22no—
57C2×C4×C223—21no—
58C2×C4×C223—22no—
59C2×C4×C223—41no—
60C2×C4×C223—41no—
61C2×C4×C223—42no—
62C2×C4×C223—41no—
63C2×C4×C223—41no—
64C2×C4×C223—41no—
65C2×C4×C223—41no—
66C2×C4×C223—41no—
67C2×C4×C223C2×D412yes—
68C8×C223—42no—
69C22×C223—21no—
70C22×C223—21no—
71C22×C223—22no—
72C22×C223—42no—
73C22×C223—41no—
74C22×C223—41no—
75C22×C223—42no—
76C22×C223—42no—
77C22×C223—42no—
78C22×C223—81no—
79C22×C223—84no—
80C22×C223#12011yescharacteristic
81C22×C223C2×SD1611yescharacteristic
82C22×C223AGL(1,ℤ/8)11yescharacteristic
83C4×C223—22no—
84C4×C223—42no—
85C4×C223—42no—
86C4×C223—41no—
87C4×C223—81no—
88C2×C223—21no—
89C2×C223—21no—
90C2×C223—42no—
91C2×C223—82no—
92C2×C223#45011yescharacteristic
93C2×C223#45311yescharacteristic
94C2×C223#45811yescharacteristic
95C223#161311yescharacteristic, direct product factor, Sylow
96#1613C22311yesmaximal, characteristic, direct product factor, Sylow
97#534C2×C22311yescharacteristic
98#481C2×C22312yescomplemented
99#399C2×C22311yescharacteristic, complemented
100#327C2×C22312yescomplemented
101#323C2×C22311yescharacteristic, complemented
102C22×D4—21no—
103#123—22no—
104#121—21no—
105#121C22×C22312yes—
106#120—22no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4C22×C22311yescharacteristic, complemented
109C2×C22⋊C4—21no—
110C2×C22⋊C4C22×C22311yescharacteristic
111#106—22no—
112#102—22no—
113C22⋊C8C22×C22312yescomplemented
114#95C22×C22311yescharacteristic, complemented
115C24—21no—
116C24D4×C22311yescharacteristic
117C2×D4—22no—
118C2×D4—22no—
119C2×D4—21no—
120C2×D4—42no—
121C2×D4—42no—
122C2×D4D4×C22312yes—
123C22×C4—21no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4D4×C22311yescharacteristic
127C22×C4C23×C22311yescharacteristic
128C2×C8—22no—
129C4⋊C4—21no—
130C4⋊C4—22no—
131C4⋊C4—42no—
132C4⋊C4D4×C22312yes—
133C22⋊C4—22no—
134C22⋊C4—22no—
135C22⋊C4—22no—
136C22⋊C4—42no—
137C23—21no—
138C23—21no—
139C23—22no—
140C23—41no—
141C23—41no—
142C23—42no—
143C23—81no—
144C23—81no—
145C23SD16×C22312yes—
146C23C2×D4×C22311yescharacteristic
147D4—42no—
148D4—44no—
149D4—42no—
150D4—42no—
151D4—81no—
152C2×C4—22no—
153C2×C4—21no—
154C2×C4—22no—
155C2×C4—41no—
156C2×C4—42no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4C2×D4×C22312yes—
164C8—42no—
165C22—21no—
166C22—21no—
167C22—22no—
168C22—42no—
169C22—41no—
170C22—41no—
171C22—42no—
172C22—42no—
173C22—42no—
174C22—81no—
175C22—84no—
176C22#120×C22311yescharacteristic
177C22C2×SD16×C22311yescharacteristic
178C22AGL(1,ℤ/8)×C22311yescharacteristic
179C4—22no—
180C4—42no—
181C4—42no—
182C4—41no—
183C4—81no—
184C2—21no—
185C2—21no—
186C2—42no—
187C2—82no—
188C2#450×C22311yescharacteristic
189C2#453×C22311yescharacteristic
190C2#458×C22311yescharacteristic
subgroupG/H
center, socleC22×C223#120
commutator, Frattini, cosocleC22×C4C23×C223
prime radical#534×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. #481×C223C2semidirect
3. #399×C223C2semidirect
4. #327×C223C2semidirect
5. #323×C223C2semidirect
12. C2×C4⋊C4×C223C22semidirect
17. C22⋊C8×C223C22semidirect
18. #95×C223C22semidirect
95. C223#1613direct (N × H)
96. #1613C223direct (N × H)
98. #481C2×C223semidirect
99. #399C2×C223semidirect
100. #327C2×C223semidirect
101. #323C2×C223semidirect
108. C2×C4⋊C4C22×C223semidirect
113. C22⋊C8C22×C223semidirect
114. #95C22×C223semidirect

Maximal subgroups

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

subgroupindex
#534×C2232
#481×C2232
#399×C2232
#327×C2232
#323×C2232
#1613223