Finite Group Databasebeta

← #1603×C223 #43250618

Subgroups — #1603×C223 #43250618

Order 28544.

244 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 64 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#539×C223C211yesmaximal, characteristic
2#451×C223C211yesmaximal, characteristic, complemented
3#448×C223C211yesmaximal, characteristic
4#395×C223C211yesmaximal, characteristic, complemented
5#328×C223C211yesmaximal, characteristic, complemented
6#327×C223C211yesmaximal, characteristic
7#324×C223C211yesmaximal, characteristic, complemented
8C22×Q8×C223C2211yescharacteristic
9C2×Q16×C223—21no—
10C2×SD16×C223—21no—
11#126×C223—21no—
12#125×C223—21no—
13#123×C223—21no—
14#123×C223—21no—
15C22⋊Q8×C223—21no—
16C22⋊Q8×C223C2211yescharacteristic
17#121×C223C2211yescharacteristic
18C4×D4×C223—21no—
19C2×C4⋊C4×C223—21no—
20C2×C4⋊C4×C223C2211yescharacteristic
21#103×C223—21no—
22#103×C223—21no—
23C22⋊C8×C223C2211yescharacteristic
24C22⋊C8×C223C2211yescharacteristic
25#95×C223—21no—
26#95×C223C2211yescharacteristic
27C2×Q8×C223—21no—
28C2×Q8×C223—21no—
29C2×Q8×C223—42no—
30C2×Q8×C223D411yescharacteristic
31C2×D4×C223—21no—
32C2×D4×C223D411yescharacteristic
33C22×C4×C223—21no—
34C22×C4×C223—21no—
35C22×C4×C223—21no—
36C22×C4×C223D411yescharacteristic
37C22×C4×C223D411yescharacteristic
38C22×C4×C223C2311yescharacteristic
39Q16×C223—42no—
40SD16×C223—42no—
41C2×C8×C223—21no—
42C2×C8×C223—21no—
43C4⋊C4×C223—21no—
44C4⋊C4×C223—21no—
45C4⋊C4×C223—21no—
46C4⋊C4×C223—21no—
47C4⋊C4×C223—21no—
48C4⋊C4×C223—21no—
49C4⋊C4×C223—42no—
50C4⋊C4×C223D411yescharacteristic
51C4⋊C4×C223D411yescharacteristic
52C22⋊C4×C223—21no—
53C22⋊C4×C223—21no—
54C22⋊C4×C223—21no—
55C22⋊C4×C223—21no—
56C22⋊C4×C223—21no—
57C42×C223—21no—
58C23×C223—21no—
59C23×C223C2×D411yescharacteristic
60Q8×C223—22no—
61Q8×C223—22no—
62Q8×C223—41no—
63Q8×C223—41no—
64Q8×C223—41no—
65D4×C223—41no—
66D4×C223—41no—
67D4×C223—41no—
68C2×C4×C223—22no—
69C2×C4×C223—22no—
70C2×C4×C223—21no—
71C2×C4×C223—21no—
72C2×C4×C223—21no—
73C2×C4×C223—21no—
74C2×C4×C223—21no—
75C2×C4×C223—21no—
76C2×C4×C223—21no—
77C2×C4×C223—41no—
78C2×C4×C223—41no—
79C2×C4×C223—41no—
80C2×C4×C223—41no—
81C2×C4×C223—41no—
82C2×C4×C223—41no—
83C2×C4×C223—81no—
84C2×C4×C223C2×D411yescharacteristic
85C2×C4×C223C2×D411yescharacteristic
86C8×C223—41no—
87C8×C223—41no—
88C22×C223—21no—
89C22×C223—21no—
90C22×C223—41no—
91C22×C223—41no—
92C22×C223—41no—
93C22×C223#12011yescharacteristic
94C22×C223#13511yescharacteristic
95C22×C223#13711yescharacteristic
96C4×C223—21no—
97C4×C223—21no—
98C4×C223—42no—
99C4×C223—41no—
100C4×C223—41no—
101C4×C223—41no—
102C4×C223—42no—
103C4×C223—41no—
104C4×C223—41no—
105C2×C223—21no—
106C2×C223—21no—
107C2×C223—81no—
108C2×C223#45211yescharacteristic
109C2×C223#45611yescharacteristic
110C2×C223B(4,𝔽2)11yescharacteristic
111C223#160311yescharacteristic, direct product factor, Sylow
112#1603C22311yesmaximal, characteristic, direct product factor, Sylow
113#539C2×C22311yescharacteristic
114#451C2×C22311yescharacteristic, complemented
115#448C2×C22311yescharacteristic
116#395C2×C22311yescharacteristic, complemented
117#328C2×C22311yescharacteristic, complemented
118#327C2×C22311yescharacteristic
119#324C2×C22311yescharacteristic, complemented
120C22×Q8C22×C22311yescharacteristic
121C2×Q16—21no—
122C2×SD16—21no—
123#126—21no—
124#125—21no—
125#123—21no—
126#123—21no—
127C22⋊Q8—21no—
128C22⋊Q8C22×C22311yescharacteristic
129#121C22×C22311yescharacteristic
130C4×D4—21no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4C22×C22311yescharacteristic
133#103—21no—
134#103—21no—
135C22⋊C8C22×C22311yescharacteristic
136C22⋊C8C22×C22311yescharacteristic
137#95—21no—
138#95C22×C22311yescharacteristic
139C2×Q8—21no—
140C2×Q8—21no—
141C2×Q8—42no—
142C2×Q8D4×C22311yescharacteristic
143C2×D4—21no—
144C2×D4D4×C22311yescharacteristic
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4D4×C22311yescharacteristic
149C22×C4D4×C22311yescharacteristic
150C22×C4C23×C22311yescharacteristic
151Q16—42no—
152SD16—42no—
153C2×C8—21no—
154C2×C8—21no—
155C4⋊C4—21no—
156C4⋊C4—21no—
157C4⋊C4—21no—
158C4⋊C4—21no—
159C4⋊C4—21no—
160C4⋊C4—21no—
161C4⋊C4—42no—
162C4⋊C4D4×C22311yescharacteristic
163C4⋊C4D4×C22311yescharacteristic
164C22⋊C4—21no—
165C22⋊C4—21no—
166C22⋊C4—21no—
167C22⋊C4—21no—
168C22⋊C4—21no—
169C42—21no—
170C23—21no—
171C23C2×D4×C22311yescharacteristic
172Q8—22no—
173Q8—22no—
174Q8—41no—
175Q8—41no—
176Q8—41no—
177D4—41no—
178D4—41no—
179D4—41no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—81no—
196C2×C4C2×D4×C22311yescharacteristic
197C2×C4C2×D4×C22311yescharacteristic
198C8—41no—
199C8—41no—
200C22—21no—
201C22—21no—
202C22—41no—
203C22—41no—
204C22—41no—
205C22#120×C22311yescharacteristic
206C22#135×C22311yescharacteristic
207C22#137×C22311yescharacteristic
208C4—21no—
209C4—21no—
210C4—42no—
211C4—41no—
212C4—41no—
213C4—41no—
214C4—42no—
215C4—41no—
216C4—41no—
217C2—21no—
218C2—21no—
219C2—81no—
220C2#452×C22311yescharacteristic
221C2#456×C22311yescharacteristic
222C2B(4,𝔽2)×C22311yescharacteristic
subgroupG/H
center, socleC22×C223#120
commutator, Frattini, cosocleC22×C4C23×C223
prime radical#121×C223C22
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. #451×C223C2semidirect
4. #395×C223C2semidirect
5. #328×C223C2semidirect
7. #324×C223C2semidirect
111. C223#1603direct (N × H)
112. #1603C223direct (N × H)
114. #451C2×C223semidirect
116. #395C2×C223semidirect
117. #328C2×C223semidirect
119. #324C2×C223semidirect

Maximal subgroups

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

subgroupindex
#539×C2232
#451×C2232
#448×C2232
#395×C2232
#328×C2232
#327×C2232
#324×C2232
#1603223