Finite Group Databasebeta

← #36364647

Subgroups — #36364647

Order 22144.

364 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 109 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#26012104C211yesmaximal, characteristic, complemented
2#26012066C211yesmaximal, characteristic
3#26011885C211yesmaximal, characteristic, complemented
4#26011870C211yesmaximal, characteristic, complemented
5#26011848C211yesmaximal, characteristic, complemented
6#26011716C212yesmaximal, complemented
7#26011700C212yesmaximal, complemented
8#26011656C212yesmaximal
9#26011648C212yesmaximal, complemented
10#26011553C211yesmaximal, characteristic, complemented
11#491×C173C211yesmaximal, characteristic, complemented
12C2×#14583827C2211yescharacteristic
13C2×Q8×D173C2211yescharacteristic
14C2×Dic692C2212yes—
15C2×#14583816C2212yes—
16C2×#14583816C2212yes—
17C2×#14583815C2212yes—
18#19149679C2211yescharacteristic
19#19149677C2211yescharacteristic
20#19149674—22no—
21#19149670—22no—
22#19149668C2211yescharacteristic
23#19149667C2211yescharacteristic
24#19149664—22no—
25#19149659C2212yes—
26#19149657—22no—
27#19149644C2212yes—
28#19149636C2211yescharacteristic
29#19149608C2212yes—
30#19149602C2212yes—
31#19149599C2212yes—
32#19149591C2212yes—
33C4×#13063436C2211yescharacteristic
34C4⋊Q8×C173C2211yescharacteristic
35#125×C173C2211yescharacteristic
36#103×C173C2212yes—
37#103×C173C2212yes—
38#97×C173C2211yescharacteristic
39#14583827—42no—
40C2×Dic346—21no—
41C2×Dic346C2311yescharacteristic
42C2×Dic346C2312yes—
43Q8×D173—42no—
44C2×D692—21no—
45C2×D692C2311yescharacteristic
46C2×C4×D173—21no—
47C2×C4×D173—21no—
48C2×C4×D173C2311yescharacteristic
49Dic692—42no—
50#14583816—42no—
51#14583816—42no—
52#14583815—42no—
53C2×#13063436C2312yes—
54C4⋊Dic173—22no—
55C4⋊Dic173—22no—
56#14583805—22no—
57#14583805—22no—
58#14583805D412yes—
59#14583803—22no—
60#14583803—22no—
61#14583803—22no—
62#14583803D412yes—
63C4×Dic173—22no—
64C4×Dic173C2311yescharacteristic
65C2×Q8×C173C2311yescharacteristic
66C2×Q8×C173C2311yescharacteristic
67C2×C8×C173C2312yes—
68C4⋊C4×C173—22no—
69C4⋊C4×C173—22no—
70C4⋊C4×C173C2312yes—
71C42×C173C2311yescharacteristic
72C22×D173—21no—
73C22×D173C2×D411yescharacteristic
74Dic346—22no—
75Dic346—22no—
76Dic346—22no—
77Dic346—42no—
78D692—22no—
79D692—42no—
80C2×Dic173—21no—
81C2×Dic173—22no—
82C2×Dic173C2×D411yescharacteristic
83C2×Dic173C2×D412yes—
84C4×D173—22no—
85C4×D173—42no—
86C4×D173—42no—
87#13063436—22no—
88Q8×C173—22no—
89Q8×C173—22no—
90C2×C4×C173—21no—
91C2×C4×C173—21no—
92C2×C4×C173—22no—
93C2×C4×C173C2×D412yes—
94C2×C4×C173C2411yescharacteristic
95C8×C173—22no—
96C22×C173C22×D411yescharacteristic
97C2×D173—21no—
98C2×D173—22no—
99C2×D173—41no—
100C2×D173—42no—
101C4×C173—22no—
102C4×C173—41no—
103C4×C173—41no—
104C4×C173—42no—
105C4×C173H(5,𝔽2)12yes—
106Dic173—21no—
107Dic173—22no—
108Dic173—41no—
109Dic173—42no—
110C2×C173#53511yescharacteristic
111C2×C173#57812yes—
112D173—41no—
113D173—81no—
114C173#322311yescharacteristic, complemented, Sylow
115#3223—1731nomaximal, Sylow
116#541—1731no—
117#536—1731no—
118#497—1731no—
119#494—1731no—
120#491D17311yescharacteristic, complemented
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—3462no—
137#125—3462no—
138#125C2×D17311yescharacteristic
139#124—1731no—
140#124—1731no—
141#123—3462no—
142C22⋊Q8—3462no—
143C22⋊Q8—1732no—
144C4×Q8—1732no—
145#117—1731no—
146C4⋊C8—1732no—
147#103—1732no—
148#103C2×D17312yes—
149#103C2×D17312yes—
150#102—1732no—
151C22⋊C8—1732no—
152#97C2×D17311yescharacteristic
153C4×C8—1731no—
154#41—6922no—
155C2×Q8—3461no—
156C2×Q8—1731no—
157C2×Q8—1732no—
158C2×Q8—6922no—
159C2×Q8C22×D17311yescharacteristic
160C2×Q8C22×D17311yescharacteristic
161C2×D4—3461no—
162C2×D4—1731no—
163C22×C4—3461no—
164C22×C4—3461no—
165C22×C4—1731no—
166Q16—6922no—
167Q16—6922no—
168Q16—6922no—
169SD16—6922no—
170C2×C8—1732no—
171C2×C8C22×D17312yes—
172C4⋊C4—22no—
173C4⋊C4—22no—
174C4⋊C4—3462no—
175C4⋊C4—3462no—
176C4⋊C4—3462no—
177C4⋊C4—3462no—
178C4⋊C4—1732no—
179C4⋊C4C22×D17312yes—
180C22⋊C4—3462no—
181C22⋊C4—3462no—
182C22⋊C4—3462no—
183C22⋊C4—1732no—
184C42—3462no—
185C42—1731no—
186C42C22×D17311yescharacteristic
187C23—3461no—
188C23—1731no—
189Q8—22no—
190Q8—22no—
191Q8—3462no—
192Q8—3462no—
193Q8—3462no—
194Q8—6922no—
195D4—3462no—
196D4—6922no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—3462no—
201C2×C4—3462no—
202C2×C4—3461no—
203C2×C4—1732no—
204C2×C4—1731no—
205C2×C4—6922no—
206C2×C4—6922no—
207C2×C4D4×D17312yes—
208C2×C4C23×D17311yescharacteristic
209C8—22no—
210C8—3462no—
211C22—3462no—
212C22—3461no—
213C22—6922no—
214C22—6921no—
215C22C2×D4×D17311yescharacteristic
216C4—22no—
217C4—42no—
218C4—41no—
219C4—41no—
220C4—3462no—
221C4—3461no—
222C4—6921no—
223C4—6922no—
224C4#1914972512yes—
225C2—13841no—
226C2—6921no—
227C2#2601205911yescharacteristic
228C2#2601230712yes—
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#491×C173C2
socleC22×C173C22×D4
prime radicalC2×#14583827C22
hypercenter#491D173
hypocenterC173#3223
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #491

Complements

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

NH ≅ G/Ntype
1. #26012104C2semidirect
3. #26011885C2semidirect
4. #26011870C2semidirect
5. #26011848C2semidirect
6. #26011716C2semidirect
7. #26011700C2semidirect
9. #26011648C2semidirect
10. #26011553C2semidirect
11. #491×C173C2semidirect
114. C173#3223semidirect
120. #491D173semidirect

Maximal subgroups

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

subgroupindex
#260121042
#260120662
#260118852
#260118702
#260118482
#260117162
#260117002
#260116562
#260116482
#260115532
#491×C1732
#3223173