Finite Group Databasebeta

← #36364640

Subgroups — #36364640

Order 22144.

364 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 117 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#26012067C211yesmaximal, characteristic
2#26011883C211yesmaximal, characteristic, complemented
3#26011870C211yesmaximal, characteristic, complemented
4#26011861C211yesmaximal, characteristic, complemented
5#26011848C211yesmaximal, characteristic, complemented
6#26011713C212yesmaximal, complemented
7#26011701C212yesmaximal, complemented
8#26011651C212yesmaximal
9#26011646C212yesmaximal, complemented
10#26011552C211yesmaximal, characteristic, complemented
11#541×C173C211yesmaximal, characteristic, complemented
12C2×#14583826C2211yescharacteristic
13C22×Dic346C2211yescharacteristic
14C2×#14583816C2212yes—
15C2×#14583816C2212yes—
16C2×#14583816C2212yes—
17C2×#14583813C2212yes—
18#19149679C2211yescharacteristic
19#19149679C2211yescharacteristic
20#19149667C2211yescharacteristic
21#19149667C2211yescharacteristic
22#19149663—22no—
23#19149659—22no—
24C4⋊#13063436C2212yes—
25#19149602C2212yes—
26#19149602C2212yes—
27#19149602C2212yes—
28#19149599C2212yes—
29C22⋊#13063436C2212yes—
30#19149590C2211yescharacteristic
31C4×#13063436C2211yescharacteristic
32C4⋊Q8×C173C2211yescharacteristic
33#126×C173—22no—
34#125×C173—22no—
35#125×C173C2211yescharacteristic
36C22⋊Q8×C173C2212yes—
37C4×Q8×C173C2212yes—
38#117×C173C2211yescharacteristic
39#14583826—42no—
40C2×Dic346—21no—
41C2×Dic346—42no—
42C2×Dic346C2311yescharacteristic
43C2×Dic346C2311yescharacteristic
44C2×Dic346C2311yescharacteristic
45C2×D692C2311yescharacteristic
46C2×C22⋊D173—21no—
47C22×Dic173—21no—
48C2×C4×D173—21no—
49#14583816—42no—
50#14583816—42no—
51#14583816—42no—
52#14583813—42no—
53C2×#13063436C2312yes—
54C2×#13063436C2312yes—
55C4⋊Dic173—22no—
56C4⋊Dic173—22no—
57#14583803—22no—
58#14583803—22no—
59C2×Q8×C173C2312yes—
60C22×C4×C173C2311yescharacteristic
61C4⋊C4×C173—22no—
62C4⋊C4×C173—22no—
63C4⋊C4×C173—22no—
64C4⋊C4×C173—22no—
65C4⋊C4×C173D412yes—
66C4⋊C4×C173C2312yes—
67C22⋊C4×C173—22no—
68C22⋊C4×C173D412yes—
69C42×C173—22no—
70C42×C173C2311yescharacteristic
71C42×C173C2311yescharacteristic
72C22×D173—21no—
73Dic346—22no—
74Dic346—22no—
75Dic346—22no—
76Dic346—42no—
77D692—22no—
78C22⋊D173—42no—
79C2×Dic173—21no—
80C2×Dic173—21no—
81C2×Dic173—21no—
82C2×Dic173—42no—
83C4×D173—42no—
84#13063436—22no—
85#13063436—22no—
86C23×C173C2×D411yescharacteristic
87Q8×C173—22no—
88Q8×C173—22no—
89C2×C4×C173—22no—
90C2×C4×C173—22no—
91C2×C4×C173—22no—
92C2×C4×C173C2×D411yescharacteristic
93C2×C4×C173C2×D412yes—
94C2×C4×C173C2×D412yes—
95C2×C4×C173C2411yescharacteristic
96C22×C173—21no—
97C22×C173—22no—
98C22×C173C22×D411yescharacteristic
99C2×D173—41no—
100C2×D173—42no—
101C4×C173—21no—
102C4×C173—22no—
103C4×C173—22no—
104C4×C173—42no—
105C4×C173—42no—
106C4×C173H(5,𝔽2)12yes—
107Dic173—41no—
108Dic173—41no—
109Dic173—41no—
110C2×C173—41no—
111C2×C173#53511yescharacteristic
112C2×C173#57812yes—
113D173—81no—
114C173#322311yescharacteristic, complemented, Sylow
115#3223—1731nomaximal, Sylow
116#541D17311yescharacteristic, complemented
117#536—1731no—
118#497—1731no—
119#494—1731no—
120#491—1731no—
121#488—1731no—
122#464—1732no—
123#462—1732no—
124#452—1732no—
125#451—1732no—
126#433—1731no—
127C2×#41—1731no—
128C22×Q8—1731no—
129C2×Q16—1732no—
130C2×Q16—1732no—
131C2×Q16—1732no—
132C2×SD16—1732no—
133C4⋊Q8—1731no—
134C4⋊Q8—1731no—
135C4⋊Q8C2×D17311yescharacteristic
136#126—22no—
137#125—22no—
138#125C2×D17311yescharacteristic
139#124—1731no—
140#124—1731no—
141#123—3462no—
142C22⋊Q8—3462no—
143C22⋊Q8C2×D17312yes—
144C4×Q8C2×D17312yes—
145#117C2×D17311yescharacteristic
146C4⋊C8—1732no—
147#103—1732no—
148#103—1732no—
149#103—1732no—
150#102—1732no—
151C22⋊C8—1732no—
152#97—1731no—
153C4×C8—1731no—
154#41—6922no—
155C2×Q8—3461no—
156C2×Q8—1731no—
157C2×Q8—1731no—
158C2×Q8—1731no—
159C2×Q8—6922no—
160C2×Q8C22×D17312yes—
161C2×D4—3461no—
162C2×D4—1731no—
163C22×C4—3461no—
164C22×C4—3461no—
165C22×C4C22×D17311yescharacteristic
166Q16—6922no—
167Q16—6922no—
168Q16—6922no—
169SD16—6922no—
170C2×C8—1732no—
171C2×C8—1732no—
172C4⋊C4—22no—
173C4⋊C4—22no—
174C4⋊C4—22no—
175C4⋊C4—22no—
176C4⋊C4—3462no—
177C4⋊C4—3462no—
178C4⋊C4C22⋊D17312yes—
179C4⋊C4C22×D17312yes—
180C22⋊C4—22no—
181C22⋊C4—3462no—
182C22⋊C4—3462no—
183C22⋊C4C22⋊D17312yes—
184C42—22no—
185C42C22×D17311yescharacteristic
186C42C22×D17311yescharacteristic
187C23—3461no—
188C23C2×C22⋊D17311yescharacteristic
189Q8—22no—
190Q8—22no—
191Q8—3462no—
192Q8—3462no—
193Q8—3462no—
194Q8—6922no—
195D4—3462no—
196D4—6922no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—3461no—
201C2×C4—3461no—
202C2×C4—3461no—
203C2×C4—6922no—
204C2×C4—6922no—
205C2×C4C2×C22⋊D17312yes—
206C2×C4C2×C22⋊D17312yes—
207C2×C4C2×C22⋊D17311yescharacteristic
208C2×C4C23×D17311yescharacteristic
209C8—3462no—
210C8—3462no—
211C22—22no—
212C22—21no—
213C22—6922no—
214C22—6921no—
215C22C22×C22⋊D17311yescharacteristic
216C4—22no—
217C4—22no—
218C4—21no—
219C4—42no—
220C4—42no—
221C4—6921no—
222C4—6921no—
223C4—6921no—
224C4#1914972512yes—
225C2—41no—
226C2—13841no—
227C2#2601205511yescharacteristic
228C2#2601230612yes—
subgroupG/H
centerC22C22×C22⋊D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#541×C173C2
socleC22×C173C22×D4
prime radicalC2×#14583826C22
hypercenter#541D173
hypocenterC173#3223
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ #117 ◁ #541

Complements

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

NH ≅ G/Ntype
2. #26011883C2semidirect
3. #26011870C2semidirect
4. #26011861C2semidirect
5. #26011848C2semidirect
6. #26011713C2semidirect
7. #26011701C2semidirect
9. #26011646C2semidirect
10. #26011552C2semidirect
11. #541×C173C2semidirect
114. C173#3223semidirect
116. #541D173semidirect

Maximal subgroups

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

subgroupindex
#260120672
#260118832
#260118702
#260118612
#260118482
#260117132
#260117012
#260116512
#260116462
#260115522
#541×C1732
#3223173