Finite Group Databasebeta

← #36364335

Subgroups — #36364335

Order 22144.

416 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 137 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×#19149658C211yesmaximal, characteristic
2#26011826C212yesmaximal, complemented
3#26011824C212yesmaximal, complemented
4#26011730C214yesmaximal, complemented
5#26011647C212yesmaximal, complemented
6#26011646C212yesmaximal, complemented
7C2×C22⋊#13063436C211yesmaximal, characteristic
8C2×C22⋊Q8×C173C211yesmaximal, characteristic
9C22×Dic346C2211yescharacteristic
10C2×#14583816C2214yescomplemented
11C22×#13063436C2212yes—
12#19149658—21no—
13#19149658—21no—
14#19149658C2212yes—
15#19149658C2212yes—
16C2×C4⋊Dic173C2211yescharacteristic
17C2×#14583805—21no—
18C2×C22⋊Dic173—21no—
19#19149612C2214yescomplemented
20#19149603C2214yescomplemented
21#19149602C2214yescomplemented
22C22⋊#13063436C2212yes—
23C22⋊#13063436C2212yes—
24C22×Q8×C173C2211yescharacteristic
25C23×C4×C173C2211yescharacteristic
26C22⋊Q8×C173—21no—
27C22⋊Q8×C173—21no—
28C22⋊Q8×C173C2212yes—
29C22⋊Q8×C173C2212yes—
30C2×C4⋊C4×C173—21no—
31C2×C4⋊C4×C173C2211yescharacteristic
32C2×C22⋊C4×C173—21no—
33C2×Dic346—21no—
34C2×Dic346—42no—
35C2×Dic346C2312yes—
36C22×Dic173—21no—
37C22×Dic173—21no—
38#14583816—44no—
39C2×#13063436—42no—
40C2×#13063436C2314yes—
41C4⋊Dic173—21no—
42C4⋊Dic173C2312yes—
43#14583805—22no—
44#14583805—22no—
45C22⋊Dic173—22no—
46C22⋊Dic173—22no—
47C24×C173D411yescharacteristic
48C2×Q8×C173—21no—
49C2×Q8×C173—42no—
50C2×Q8×C173C2312yes—
51C22×C4×C173—21no—
52C22×C4×C173—21no—
53C22×C4×C173—42no—
54C22×C4×C173D411yescharacteristic
55C22×C4×C173D411yescharacteristic
56C22×C4×C173D411yescharacteristic
57C22×C4×C173C2311yescharacteristic
58C22×C4×C173C2311yescharacteristic
59C22×C4×C173C2311yescharacteristic
60C4⋊C4×C173—22no—
61C4⋊C4×C173—22no—
62C4⋊C4×C173—21no—
63C4⋊C4×C173C2312yes—
64C22⋊C4×C173—22no—
65C22⋊C4×C173—22no—
66Dic346—24no—
67Dic346—42no—
68C2×Dic173—22no—
69C2×Dic173—22no—
70C2×Dic173—42no—
71C2×Dic173—42no—
72#13063436—24no—
73C23×C173—41no—
74C23×C173—41no—
75C23×C173Q1614yescomplemented
76C23×C173C2×D411yescharacteristic
77C23×C173C2×D411yescharacteristic
78C23×C173C2×D411yescharacteristic
79Q8×C173—24no—
80Q8×C173—42no—
81C2×C4×C173—22no—
82C2×C4×C173—22no—
83C2×C4×C173—22no—
84C2×C4×C173—22no—
85C2×C4×C173—22no—
86C2×C4×C173—41no—
87C2×C4×C173—41no—
88C2×C4×C173—41no—
89C2×C4×C173—42no—
90C2×C4×C173—42no—
91C2×C4×C173C2×D411yescharacteristic
92C2×C4×C173C2×D411yescharacteristic
93C2×C4×C173C2×D411yescharacteristic
94C2×C4×C173C2411yescharacteristic
95C22×C173—21no—
96C22×C173—21no—
97C22×C173—21no—
98C22×C173—21no—
99C22×C173—21no—
100C22×C173—21no—
101C22×C173—44no—
102C22×C173C2×Q1612yes—
103C22×C173C2×Q1612yes—
104C22×C173C2×Q1612yes—
105C22×C173C22×D411yescharacteristic
106C4×C173—21no—
107C4×C173—21no—
108C4×C173—21no—
109C4×C173—42no—
110C4×C173—42no—
111C4×C173H(5,𝔽2)12yes—
112Dic173—42no—
113Dic173—42no—
114C2×C173—22no—
115C2×C173—22no—
116C2×C173—22no—
117C2×C173#53411yescharacteristic
118C2×C173C22×Q1611yescharacteristic
119C2×C173#57511yescharacteristic
120C173#318911yescharacteristic, complemented, Sylow
121#3189—1731nomaximal, Sylow
122C2×C22⋊Q8—1731no—
123C2×C22⋊Q8D17311yescharacteristic
124#484—1732no—
125#484—1732no—
126#467—1734no—
127#451—1732no—
128#451—1732no—
129C2×C22⋊C8—1731no—
130C22×Q8—1731no—
131C22×Q8C2×D17311yescharacteristic
132C23×C4C2×D17311yescharacteristic
133C2×Q16—1734no—
134C22×C8—1732no—
135C22⋊Q8—21no—
136C22⋊Q8—21no—
137C22⋊Q8—3461no—
138C22⋊Q8—3461no—
139C22⋊Q8—1732no—
140C22⋊Q8—1732no—
141C22⋊Q8C2×D17312yes—
142C22⋊Q8C2×D17312yes—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—3461no—
145C2×C4⋊C4—1731no—
146C2×C4⋊C4C2×D17311yescharacteristic
147C2×C22⋊C4—21no—
148C2×C22⋊C4—3461no—
149#107—1734no—
150#103—1734no—
151#103—1734no—
152C22⋊C8—1732no—
153C22⋊C8—1732no—
154C24C22⋊D17311yescharacteristic
155C2×Q8—21no—
156C2×Q8—42no—
157C2×Q8—3461no—
158C2×Q8—1732no—
159C2×Q8—6922no—
160C2×Q8C22×D17312yes—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—42no—
164C22×C4—3461no—
165C22×C4—3461no—
166C22×C4C22⋊D17311yescharacteristic
167C22×C4C22⋊D17311yescharacteristic
168C22×C4C22⋊D17311yescharacteristic
169C22×C4C22×D17311yescharacteristic
170C22×C4C22×D17311yescharacteristic
171C22×C4C22×D17311yescharacteristic
172Q16—6924no—
173C2×C8—1734no—
174C2×C8—6922no—
175C4⋊C4—22no—
176C4⋊C4—22no—
177C4⋊C4—21no—
178C4⋊C4—3462no—
179C4⋊C4—3462no—
180C4⋊C4—3461no—
181C4⋊C4—1732no—
182C4⋊C4C22×D17312yes—
183C22⋊C4—22no—
184C22⋊C4—22no—
185C22⋊C4—3462no—
186C22⋊C4—3462no—
187C23—41no—
188C23—41no—
189C23#1458381614yescomplemented
190C23C2×C22⋊D17311yescharacteristic
191C23C2×C22⋊D17311yescharacteristic
192C23C2×C22⋊D17311yescharacteristic
193Q8—24no—
194Q8—42no—
195Q8—3464no—
196Q8—6922no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—3462no—
208C2×C4—3462no—
209C2×C4—6922no—
210C2×C4—6922no—
211C2×C4C2×C22⋊D17311yescharacteristic
212C2×C4C2×C22⋊D17311yescharacteristic
213C2×C4C2×C22⋊D17311yescharacteristic
214C2×C4C23×D17311yescharacteristic
215C8—3464no—
216C22—21no—
217C22—21no—
218C22—21no—
219C22—21no—
220C22—21no—
221C22—21no—
222C22—44no—
223C22C2×#1458381612yes—
224C22C2×#1458381612yes—
225C22C2×#1458381612yes—
226C22C22×C22⋊D17311yescharacteristic
227C4—21no—
228C4—21no—
229C4—21no—
230C4—42no—
231C4—42no—
232C4—6922no—
233C4—6922no—
234C4#1914972412yes—
235C2—22no—
236C2—22no—
237C2—22no—
238C2#2601205011yescharacteristic
239C2C22×#1458381611yescharacteristic
240C2#2601228911yescharacteristic
subgroupG/H
centerC22C22×C22⋊D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
FittingC2×C22⋊Q8×C173C2
socleC22×C173C22×D4
prime radicalC24×C173D4
hypercenterC2×C22⋊Q8D173
hypocenterC173#3189
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. #26011826C2semidirect
3. #26011824C2semidirect
4. #26011730C2semidirect
5. #26011647C2semidirect
6. #26011646C2semidirect
10. C2×#14583816C22semidirect
19. #19149612C22semidirect
20. #19149603C22semidirect
21. #19149602C22semidirect
75. C23×C173Q16semidirect
120. C173#3189semidirect
189. C23#14583816semidirect

Maximal subgroups

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

subgroupindex
C2×#191496582
#260118262
#260118242
#260117302
#260116472
#260116462
C2×C22⋊#130634362
C2×C22⋊Q8×C1732
#3189173