Finite Group Databasebeta

← #36364641

Subgroups — #36364641

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#26012102C211yesmaximal, characteristic, complemented
2#26012061C211yesmaximal, characteristic
3#26011870C211yesmaximal, characteristic, complemented
4#26011860C211yesmaximal, characteristic, complemented
5#26011845C211yesmaximal, characteristic, complemented
6#26011711C212yesmaximal, complemented
7#26011699C212yesmaximal, complemented
8#26011650C212yesmaximal
9#26011645C212yesmaximal, complemented
10#26011553C211yesmaximal, characteristic, complemented
11#497×C173C211yesmaximal, characteristic, complemented
12C2×#14583825C2211yescharacteristic
13C2×Q8×D173C2211yescharacteristic
14C2×#14583816C2212yes—
15C2×#14583816C2212yes—
16#19149679C2211yescharacteristic
17#19149677C2211yescharacteristic
18#19149674—22no—
19#19149671C2211yescharacteristic
20#19149669—22no—
21#19149666C2211yescharacteristic
22#19149661—22no—
23#19149656—22no—
24#19149655C2212yes—
25Q8×Dic173C2212yes—
26#19149636C2211yescharacteristic
27#19149608C2212yes—
28#19149604C2212yes—
29#19149604C2212yes—
30#19149603C2212yes—
31#19149600C2212yes—
32#19149591C2212yes—
33C4×#13063436C2211yescharacteristic
34C2×Q16×C173C2212yes—
35C2×SD16×C173C2212yes—
36C4⋊Q8×C173C2211yescharacteristic
37#124×C173C2211yescharacteristic
38#97×C173C2211yescharacteristic
39#14583825—42no—
40C2×Dic346—21no—
41C2×Dic346C2311yescharacteristic
42C2×Dic346C2311yescharacteristic
43Q8×D173—42no—
44C2×C22⋊D173—21no—
45C22×Dic173—21no—
46C2×C4×D173—21no—
47C2×C4×D173C2311yescharacteristic
48#14583816—42no—
49#14583816—42no—
50C2×#13063436C2312yes—
51C4⋊Dic173—22no—
52C4⋊Dic173—22no—
53C4⋊Dic173C2312yes—
54#14583805—22no—
55#14583805—22no—
56#14583805—22no—
57#14583805D412yes—
58C22⋊Dic173—22no—
59#14583803—22no—
60#14583803D412yes—
61C4×Dic173—22no—
62C4×Dic173C2311yescharacteristic
63C2×Q8×C173C2312yes—
64C2×Q8×C173C2311yescharacteristic
65C2×D4×C173C2311yescharacteristic
66Q16×C173—42no—
67SD16×C173—42no—
68C2×C8×C173C2312yes—
69C4⋊C4×C173—22no—
70C22⋊C4×C173—22no—
71C42×C173C2311yescharacteristic
72C22×D173C2×D411yescharacteristic
73Dic346—22no—
74Dic346—22no—
75Dic346—42no—
76C22⋊D173—42no—
77C2×Dic173—21no—
78C2×Dic173—21no—
79C2×Dic173—22no—
80C2×Dic173—42no—
81C2×Dic173C2×D411yescharacteristic
82C2×Dic173C2×D412yes—
83C4×D173—22no—
84C4×D173—42no—
85#13063436—22no—
86C23×C173—21no—
87Q8×C173—22no—
88Q8×C173—22no—
89Q8×C173—22no—
90D4×C173—22no—
91C2×C4×C173—21no—
92C2×C4×C173—22no—
93C2×C4×C173C2×D412yes—
94C2×C4×C173C2411yescharacteristic
95C8×C173—22no—
96C22×C173—41no—
97C22×C173—42no—
98C22×C173C22×D411yescharacteristic
99C2×D173—21no—
100C2×D173—22no—
101C4×C173—22no—
102C4×C173—42no—
103C4×C173—41no—
104C4×C173H(5,𝔽2)12yes—
105Dic173—21no—
106Dic173—22no—
107Dic173—41no—
108Dic173—41no—
109Dic173—42no—
110C2×C173—81no—
111C2×C173#53511yescharacteristic
112C2×C173#57812yes—
113D173—41no—
114C173#322311yescharacteristic, complemented, Sylow
115#3223—1731nomaximal, Sylow
116#541—1731no—
117#536—1731no—
118#497D17311yescharacteristic, complemented
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×Q16C2×D17312yes—
132C2×SD16C2×D17312yes—
133C4⋊Q8—1731no—
134C4⋊Q8—1731no—
135C4⋊Q8C2×D17311yescharacteristic
136#126—3462no—
137#125—3462no—
138#125—1731no—
139#124—1731no—
140#124C2×D17311yescharacteristic
141#123—3462no—
142C22⋊Q8—3462no—
143C22⋊Q8—1732no—
144C4×Q8—1732no—
145#117—1731no—
146C4⋊C8—1732no—
147#103—1732no—
148#103—1732no—
149#103—1732no—
150#102—1732no—
151C22⋊C8—1732no—
152#97C2×D17311yescharacteristic
153C4×C8—1731no—
154#41—6922no—
155C2×Q8—3461no—
156C2×Q8—1731no—
157C2×Q8—1731no—
158C2×Q8—6922no—
159C2×Q8C22×D17311yescharacteristic
160C2×Q8C22×D17312yes—
161C2×D4—3461no—
162C2×D4C22×D17311yescharacteristic
163C22×C4—3461no—
164C22×C4—3461no—
165C22×C4—1731no—
166Q16—42no—
167Q16—6922no—
168Q16—6922no—
169SD16—42no—
170C2×C8—1732no—
171C2×C8C22×D17312yes—
172C4⋊C4—22no—
173C4⋊C4—3462no—
174C4⋊C4—3462no—
175C4⋊C4—3462no—
176C4⋊C4—3462no—
177C4⋊C4—3462no—
178C4⋊C4—1732no—
179C4⋊C4—1732no—
180C22⋊C4—22no—
181C22⋊C4—3462no—
182C22⋊C4—3462no—
183C22⋊C4—1732no—
184C42—3462no—
185C42—1731no—
186C42C22×D17311yescharacteristic
187C23—21no—
188C23—1731no—
189Q8—22no—
190Q8—22no—
191Q8—22no—
192Q8—3462no—
193Q8—3462no—
194Q8—6922no—
195D4—22no—
196D4—6922no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—3462no—
200C2×C4—3462no—
201C2×C4—3461no—
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—42no—
212C22—41no—
213C22—3462no—
214C22—3461no—
215C22C2×D4×D17311yescharacteristic
216C4—22no—
217C4—41no—
218C4—42no—
219C4—3462no—
220C4—3461no—
221C4—6922no—
222C4—6921no—
223C4—6921no—
224C4#1914972412yes—
225C2—81no—
226C2—6921no—
227C2#2601205411yescharacteristic
228C2#2601230712yes—
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#497×C173C2
socleC22×C173C22×D4
prime radicalC2×#14583825C22
hypercenter#497D173
hypocenterC173#3223
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
1. #26012102C2semidirect
3. #26011870C2semidirect
4. #26011860C2semidirect
5. #26011845C2semidirect
6. #26011711C2semidirect
7. #26011699C2semidirect
9. #26011645C2semidirect
10. #26011553C2semidirect
11. #497×C173C2semidirect
114. C173#3223semidirect
118. #497D173semidirect

Maximal subgroups

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

subgroupindex
#260121022
#260120612
#260118702
#260118602
#260118452
#260117112
#260116992
#260116502
#260116452
#260115532
#497×C1732
#3223173