Finite Group Databasebeta

← #36354722

Subgroups — #36354722

Order 22144.

246 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 69 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#26012119C211yesmaximal, characteristic
2#26011500C211yesmaximal, characteristic
3#26011398C211yesmaximal, characteristic
4#26011259C211yesmaximal, characteristic
5#26011252C211yesmaximal, characteristic
6#26011250C211yesmaximal, characteristic
7#419×C173C211yesmaximal, characteristic
8C2×#14583810C2211yescharacteristic
9#19149679—21no—
10#19149678—21no—
11#19149671—21no—
12#19149670—21no—
13#19149656—22no—
14#19149638C2211yescharacteristic
15C2×C4⋊Dic173C2211yescharacteristic
16C2×C4×Dic173C2211yescharacteristic
17#19149603—21no—
18#19149603—21no—
19#19149586—22no—
20C22×Q8×C173C2211yescharacteristic
21C2×M16×C173C2211yescharacteristic
22#117×C173C2211yescharacteristic
23#103×C173—21no—
24#103×C173—21no—
25C2×Dic346—22no—
26C22×Dic173—21no—
27C22×Dic173D412yes—
28#14583810—21no—
29#14583810—21no—
30C2×#13063436—21no—
31C4⋊Dic173—22no—
32C4⋊Dic173—21no—
33C4⋊Dic173—21no—
34C4⋊Dic173D411yescharacteristic
35C4⋊Dic173D411yescharacteristic
36#14583805—22no—
37#14583805—22no—
38C22⋊Dic173—21no—
39C4×Dic173—21no—
40C4×Dic173—21no—
41C4×Dic173—21no—
42C2×Q8×C173—21no—
43C2×Q8×C173—21no—
44C2×Q8×C173—21no—
45C2×Q8×C173—21no—
46C2×Q8×C173—21no—
47C2×Q8×C173Q812yescomplemented
48C22×C4×C173—21no—
49C22×C4×C173C2311yescharacteristic
50M16×C173—21no—
51M16×C173—21no—
52C2×C8×C173—21no—
53C4⋊C4×C173D411yescharacteristic
54C4⋊C4×C173D411yescharacteristic
55C22⋊C4×C173—21no—
56C42×C173—21no—
57Dic346—82no—
58C2×Dic173—22no—
59C2×Dic173—22no—
60C2×Dic173—21no—
61C2×Dic173—21no—
62C2×Dic173—41no—
63C2×Dic173—41no—
64C2×Dic173—42no—
65#13063436—41no—
66C23×C173C2×D411yescharacteristic
67Q8×C173—22no—
68Q8×C173—22no—
69Q8×C173—41no—
70Q8×C173—41no—
71C2×C4×C173—21no—
72C2×C4×C173—21no—
73C2×C4×C173—22no—
74C2×C4×C173—41no—
75C2×C4×C173—41no—
76C2×C4×C173C2×D411yescharacteristic
77C2×C4×C173C2×D411yescharacteristic
78C2×C4×C173C2×Q811yescharacteristic
79C2×C4×C173#4111yescharacteristic
80C2×C4×C173#4111yescharacteristic
81C2×C4×C173#4111yescharacteristic
82C8×C173—41no—
83C22×C173—21no—
84C22×C173—21no—
85C22×C173#12011yescharacteristic
86C22×C173C22⋊Q811yescharacteristic
87C22×C173#12411yescharacteristic
88C4×C173—42no—
89C4×C173—41no—
90C4×C173—41no—
91C4×C173C22⋊Q811yescharacteristic
92C4×C173C22⋊Q811yescharacteristic
93C4×C173#12411yescharacteristic
94C4×C173#12411yescharacteristic
95Dic173—42no—
96Dic173—42no—
97Dic173—41no—
98Dic173—41no—
99C2×C173—21no—
100C2×C173—21no—
101C2×C173#39311yescharacteristic
102C2×C173#45511yescharacteristic
103C2×C173#45611yescharacteristic
104C173#202811yescharacteristic, complemented, Sylow
105#2028—1731nomaximal, Sylow
106#544—1731no—
107#419—1731no—
108#419D17311yescharacteristic
109#391—1731no—
110#343—1731no—
111#339—1731no—
112#339—1731no—
113C22×Q8C2×D17311yescharacteristic
114C2×M16—1731no—
115C2×M16C2×D17311yescharacteristic
116C4⋊Q8—3461no—
117C4⋊Q8—3461no—
118#125—3461no—
119#125—3461no—
120C22⋊Q8—3462no—
121#117—1731no—
122#117C2×D17311yescharacteristic
123C2×C4⋊C4—1731no—
124C2×C42—1731no—
125#103—21no—
126#103—21no—
127#103—3461no—
128#103—3461no—
129#95—3462no—
130C2×Q8—21no—
131C2×Q8—21no—
132C2×Q8—21no—
133C2×Q8—21no—
134C2×Q8—21no—
135C2×Q8—3462no—
136C2×Q8Dic34612yescomplemented
137C22×C4—21no—
138C22×C4—3461no—
139C22×C4—1732no—
140C22×C4C22×D17311yescharacteristic
141M16—21no—
142M16—21no—
143M16—3461no—
144M16—3461no—
145C2×C8—21no—
146C2×C8—3461no—
147C4⋊C4—3462no—
148C4⋊C4—3462no—
149C4⋊C4—3462no—
150C4⋊C4—3461no—
151C4⋊C4—3461no—
152C4⋊C4—1731no—
153C4⋊C4—1731no—
154C4⋊C4C22⋊D17311yescharacteristic
155C4⋊C4C22⋊D17311yescharacteristic
156C22⋊C4—21no—
157C22⋊C4—3461no—
158C42—21no—
159C42—3461no—
160C42—3461no—
161C42—3461no—
162C23D4×D17311yescharacteristic
163Q8—22no—
164Q8—22no—
165Q8—41no—
166Q8—41no—
167Q8—13842no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—22no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—3462no—
174C2×C4—3461no—
175C2×C4—3461no—
176C2×C4—3462no—
177C2×C4—6921no—
178C2×C4—6921no—
179C2×C4—6922no—
180C2×C4D4×D17311yescharacteristic
181C2×C4C2×C22⋊D17311yescharacteristic
182C2×C4C2×Dic34611yescharacteristic
183C2×C4#1458382511yescharacteristic
184C2×C4#1458382511yescharacteristic
185C2×C4#1458382611yescharacteristic
186C8—41no—
187C8—6921no—
188C22—21no—
189C22—21no—
190C22#1914964711yescharacteristic
191C22#1914965611yescharacteristic
192C22#1914966511yescharacteristic
193C4—42no—
194C4—41no—
195C4—41no—
196C4—6922no—
197C4—6922no—
198C4—6921no—
199C4—6921no—
200C4#1914965611yescharacteristic
201C4#1914965811yescharacteristic
202C4#1914966511yescharacteristic
203C4#1914966611yescharacteristic
204C2—21no—
205C2—21no—
206C2#2601140311yescharacteristic
207C2#2601166711yescharacteristic
208C2#2601167211yescharacteristic
subgroupG/H
centerC22#19149647
commutator, cosocleC22×C4×C173C23
FrattiniC22×C4C22×D173
Fitting#419×C173C2
socleC22×C173#120
prime radicalC23×C173C2×D4
hypercenter#419D173
hypocenterC173#2028
LCSG ▷ C22×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
47. C2×Q8×C173Q8semidirect
104. C173#2028semidirect
136. C2×Q8Dic346semidirect

Maximal subgroups

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

subgroupindex
#260121192
#260115002
#260113982
#260112592
#260112522
#260112502
#419×C1732
#2028173