Finite Group Databasebeta

← #36364489

Subgroups — #36364489

Order 22144.

428 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 125 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#26012042C211yesmaximal, characteristic
2C4⋊Q8×D173C211yesmaximal, characteristic, complemented
3#26011865C211yesmaximal, characteristic, complemented
4#26011864C211yesmaximal, characteristic, complemented
5#26011843C211yesmaximal, characteristic, complemented
6#26011695C214yesmaximal, complemented
7#26011631C214yesmaximal, complemented
8#26011542C211yesmaximal, characteristic, complemented
9#487×C173C211yesmaximal, characteristic, complemented
10C2×#14583827—21no—
11C2×Q8×D173C2212yes—
12C2×#14583815C2214yes—
13C2×#14583813C2214yes—
14#19149679C2211yescharacteristic
15#19149678—21no—
16#19149678—21no—
17#19149677C2212yes—
18#19149676C2211yescharacteristic
19#19149668—21no—
20#19149654—21no—
21#19149654C2212yes—
22#19149641—21no—
23#19149641C2212yes—
24C4⋊C4×D173—21no—
25C4⋊C4×D173—21no—
26C42×D173C2211yescharacteristic
27#19149608C2212yes—
28#19149602C2214yes—
29#19149591C2212yes—
30C4×#13063436C2211yescharacteristic
31C4⋊Q8×C173C2211yescharacteristic
32C4⋊Q8×C173C2211yescharacteristic
33#103×C173C2214yes—
34C4×C8×C173C2211yescharacteristic
35#14583827—44no—
36C2×Dic346—22no—
37C2×Dic346C2312yes—
38Q8×D173—44no—
39C2×D692—22no—
40C2×D692—21no—
41C2×D692—21no—
42C2×D692C2312yes—
43C2×C4×D173—22no—
44C2×C4×D173—22no—
45C2×C4×D173D411yescharacteristic
46C2×C4×D173D411yescharacteristic
47C2×C4×D173C2311yescharacteristic
48#14583815—44no—
49#14583813—44no—
50C2×#13063436C2312yes—
51C4⋊Dic173—21no—
52C4⋊Dic173—21no—
53#14583805—22no—
54#14583805—22no—
55#14583803—22no—
56#14583803—22no—
57C4×Dic173D411yescharacteristic
58C4×Dic173D411yescharacteristic
59C4×Dic173C2311yescharacteristic
60C2×Q8×C173—21no—
61C2×Q8×C173C2312yes—
62C2×C8×C173C2312yes—
63C4⋊C4×C173—21no—
64C4⋊C4×C173—21no—
65C4⋊C4×C173—21no—
66C4⋊C4×C173C2312yes—
67C42×C173C2311yescharacteristic
68C22×D173—22no—
69C22×D173C2×D411yescharacteristic
70Dic346—24no—
71Dic346—44no—
72D692—24no—
73D692—44no—
74D692—42no—
75D692—42no—
76C2×Dic173—22no—
77C2×Dic173C2×D411yescharacteristic
78C2×Dic173C2×D411yescharacteristic
79C2×Dic173C2×D411yescharacteristic
80C4×D173—22no—
81C4×D173—22no—
82C4×D173—44no—
83C4×D173—44no—
84C4×D173SD1614yescomplemented
85#13063436—22no—
86Q8×C173—24no—
87Q8×C173—42no—
88C2×C4×C173—22no—
89C2×C4×C173—22no—
90C2×C4×C173C2×D411yescharacteristic
91C2×C4×C173C2×D411yescharacteristic
92C2×C4×C173C2411yescharacteristic
93C8×C173—22no—
94C22×C173C22×D411yescharacteristic
95C2×D173—21no—
96C2×D173—21no—
97C2×D173—42no—
98C2×D173—42no—
99C2×D173—42no—
100C2×D173C2×SD1612yes—
101C4×C173—42no—
102C4×C173—42no—
103C4×C173C2×SD1612yes—
104C4×C173#13712yes—
105C4×C173H(5,𝔽2)12yes—
106Dic173—21no—
107Dic173—21no—
108Dic173—42no—
109Dic173C2×SD1612yes—
110C2×C173#53511yescharacteristic
111C2×C173C22×SD1611yescharacteristic
112C2×C173C2×#13711yescharacteristic
113D173—22no—
114D173—82no—
115C173#320611yescharacteristic, complemented, Sylow
116#3206—1731nomaximal, Sylow
117#532—1731no—
118C2×C4⋊Q8—1731no—
119#492—1731no—
120#492—1731no—
121#487—1731no—
122#487D17311yescharacteristic, complemented
123#461—1734no—
124#448—1734no—
125#431—1731no—
126C2×#41—3461no—
127C22×Q8—1732no—
128C2×SD16—1734no—
129C2×SD16—1734no—
130C4⋊Q8—3461no—
131C4⋊Q8—3461no—
132C4⋊Q8—1732no—
133C4⋊Q8—1731no—
134C4⋊Q8C2×D17311yescharacteristic
135C4⋊Q8C2×D17311yescharacteristic
136#127—1731no—
137#124—3461no—
138#121—3461no—
139#121—1732no—
140C4×D4—3461no—
141C4×D4—1732no—
142C2×C4⋊C4—3461no—
143C2×C4⋊C4—3461no—
144C2×C42—1731no—
145C4⋊C8—1732no—
146#103—1734no—
147#103C2×D17314yes—
148C22⋊C8—1732no—
149C4×C8—1731no—
150C4×C8C2×D17311yescharacteristic
151#41—6924no—
152C2×Q8—21no—
153C2×Q8—3462no—
154C2×Q8—1732no—
155C2×Q8—6924no—
156C2×Q8C22×D17312yes—
157C2×D4—3462no—
158C2×D4—3461no—
159C2×D4—3461no—
160C2×D4—1732no—
161C22×C4—3462no—
162C22×C4—3462no—
163C22×C4—1731no—
164C22×C4—1731no—
165C22×C4—1731no—
166SD16—6924no—
167SD16—6924no—
168C2×C8—1732no—
169C2×C8C22×D17312yes—
170C4⋊C4—21no—
171C4⋊C4—21no—
172C4⋊C4—21no—
173C4⋊C4—3462no—
174C4⋊C4—3462no—
175C4⋊C4—3461no—
176C4⋊C4—3461no—
177C4⋊C4C22×D17312yes—
178C22⋊C4—3462no—
179C22⋊C4—3462no—
180C42—1731no—
181C42—1731no—
182C42—1731no—
183C42C22×D17311yescharacteristic
184C23—3462no—
185C23—1731no—
186Q8—24no—
187Q8—42no—
188Q8—3464no—
189Q8—6924no—
190D4—3464no—
191D4—6924no—
192D4—6922no—
193D4—6922no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—3462no—
197C2×C4—3462no—
198C2×C4—3462no—
199C2×C4—1731no—
200C2×C4—1731no—
201C2×C4—1731no—
202C2×C4—1734no—
203C2×C4—6924no—
204C2×C4—6924no—
205C2×C4D4×D17311yescharacteristic
206C2×C4D4×D17311yescharacteristic
207C2×C4C23×D17311yescharacteristic
208C8—22no—
209C8—3462no—
210C22—3461no—
211C22—3461no—
212C22—1732no—
213C22—6922no—
214C22—6922no—
215C22—6922no—
216C22C2×D4×D17311yescharacteristic
217C4—42no—
218C4—42no—
219C4—3461no—
220C4—3461no—
221C4—1732no—
222C4—6922no—
223C4SD16×D17312yes—
224C4#1914971112yes—
225C4#1914972512yes—
226C2—3462no—
227C2—13842no—
228C2#2601205911yescharacteristic
229C2C2×SD16×D17311yescharacteristic
230C2C2×#1914971111yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#487×C173C2
socleC22×C173C22×D4
prime radical#26012042C2
hypercenter#487D173
hypocenterC173#3206
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
2. C4⋊Q8×D173C2semidirect
3. #26011865C2semidirect
4. #26011864C2semidirect
5. #26011843C2semidirect
6. #26011695C2semidirect
7. #26011631C2semidirect
8. #26011542C2semidirect
9. #487×C173C2semidirect
84. C4×D173SD16semidirect
115. C173#3206semidirect
122. #487D173semidirect

Maximal subgroups

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

subgroupindex
#260120422
C4⋊Q8×D1732
#260118652
#260118642
#260118432
#260116952
#260116312
#260115422
#487×C1732
#3206173