Finite Group Databasebeta

← #36354712

Subgroups — #36354712

Order 22144.

258 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 85 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
1C2×#19149670C211yesmaximal, characteristic
2C2×#19149603C211yesmaximal, characteristic
3#26011398C211yesmaximal, characteristic
4#26011254C211yesmaximal, characteristic
5#26011253C211yesmaximal, characteristic
6#26011245C211yesmaximal, characteristic
7C2×#103×C173C211yesmaximal, characteristic
8C22×#13063436C2211yescharacteristic
9#19149670—22no—
10#19149670—22no—
11C2×C4⋊Dic173—21no—
12C2×C4⋊Dic173C2211yescharacteristic
13C2×C4⋊Dic173C2211yescharacteristic
14C2×#14583805—21no—
15C2×C4×Dic173C2211yescharacteristic
16#19149603—22no—
17#19149586—21no—
18#19149586—21no—
19C22×Q8×C173C2211yescharacteristic
20C22×C8×C173C2211yescharacteristic
21C2×C4⋊C4×C173C2211yescharacteristic
22#103×C173—22no—
23C22×Dic173—21no—
24C22×Dic173—21no—
25C22×Dic173D411yescharacteristic
26C22×Dic173D411yescharacteristic
27C2×#13063436—21no—
28C2×#13063436—21no—
29C2×#13063436—21no—
30C4⋊Dic173—24no—
31C4⋊Dic173—21no—
32C4⋊Dic173—21no—
33C4⋊Dic173—21no—
34C4⋊Dic173D412yes—
35#14583805—24no—
36C4×Dic173—21no—
37C4×Dic173—21no—
38C2×Q8×C173—21no—
39C2×Q8×C173—21no—
40C2×Q8×C173—21no—
41C2×Q8×C173—21no—
42C2×Q8×C173—21no—
43C2×Q8×C173Q812yes—
44C22×C4×C173—21no—
45C22×C4×C173—21no—
46C22×C4×C173C2311yescharacteristic
47C2×C8×C173—21no—
48C2×C8×C173—21no—
49C2×C8×C173—21no—
50C4⋊C4×C173—21no—
51C4⋊C4×C173D412yes—
52C2×Dic173—22no—
53C2×Dic173—21no—
54C2×Dic173—21no—
55C2×Dic173—21no—
56C2×Dic173—21no—
57C2×Dic173—21no—
58C2×Dic173—21no—
59C2×Dic173—41no—
60C2×Dic173—41no—
61C2×Dic173—41no—
62C2×Dic173—41no—
63C2×Dic173—41no—
64#13063436—41no—
65C23×C173C2×D411yescharacteristic
66Q8×C173—22no—
67Q8×C173—22no—
68Q8×C173—41no—
69Q8×C173—41no—
70C2×C4×C173—22no—
71C2×C4×C173—22no—
72C2×C4×C173—41no—
73C2×C4×C173—41no—
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#12011yescharacteristic
84C22×C173C22⋊Q811yescharacteristic
85C22×C173#12411yescharacteristic
86C22×C173#13511yescharacteristic
87C22×C173#13511yescharacteristic
88C22×C173#13711yescharacteristic
89C22×C173#13711yescharacteristic
90C4×C173—42no—
91C4×C173—42no—
92C4×C173C22⋊Q811yescharacteristic
93C4×C173C22⋊Q811yescharacteristic
94C4×C173#12411yescharacteristic
95C4×C173#12411yescharacteristic
96Dic173—41no—
97Dic173—41no—
98Dic173—42no—
99Dic173—81no—
100C2×C173#39311yescharacteristic
101C2×C173#45211yescharacteristic
102C2×C173#45211yescharacteristic
103C2×C173#47911yescharacteristic
104C2×C173#47911yescharacteristic
105C2×C173#48811yescharacteristic
106C2×C173#49111yescharacteristic
107C173#202611yescharacteristic, complemented, Sylow
108#2026—1731nomaximal, Sylow
109C2×#125—1731no—
110C2×#103—1731no—
111C2×#103D17311yescharacteristic
112#391—1731no—
113#340—1731no—
114#340—1731no—
115#336—1731no—
116C22×Q8C2×D17311yescharacteristic
117C22×C8—1731no—
118C22×C8C2×D17311yescharacteristic
119#125—3462no—
120#125—3462no—
121C2×C4⋊C4—3461no—
122C2×C4⋊C4—3461no—
123C2×C4⋊C4—1731no—
124C2×C4⋊C4—1731no—
125C2×C4⋊C4C2×D17311yescharacteristic
126C2×C42—1731no—
127#103—22no—
128#103—3462no—
129#95—3461no—
130#95—3461no—
131C2×Q8—21no—
132C2×Q8—21no—
133C2×Q8—21no—
134C2×Q8—21no—
135C2×Q8—21no—
136C2×Q8Dic34612yes—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—3461no—
140C22×C4—3461no—
141C22×C4—1731no—
142C22×C4—1731no—
143C22×C4C22×D17311yescharacteristic
144C2×C8—21no—
145C2×C8—21no—
146C2×C8—21no—
147C2×C8—3461no—
148C2×C8—3461no—
149C2×C8—3461no—
150C4⋊C4—21no—
151C4⋊C4—3464no—
152C4⋊C4—3464no—
153C4⋊C4—3461no—
154C4⋊C4—3461no—
155C4⋊C4—3461no—
156C4⋊C4—1732no—
157C4⋊C4C22⋊D17312yes—
158C42—3461no—
159C42—3461no—
160C23D4×D17311yescharacteristic
161Q8—22no—
162Q8—22no—
163Q8—41no—
164Q8—41no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—3461no—
172C2×C4—3461no—
173C2×C4—3462no—
174C2×C4—3461no—
175C2×C4—3461no—
176C2×C4—3461no—
177C2×C4—3461no—
178C2×C4—6921no—
179C2×C4—6921no—
180C2×C4—6921no—
181C2×C4—6921no—
182C2×C4—6921no—
183C2×C4D4×D17311yescharacteristic
184C2×C4C2×C22⋊D17311yescharacteristic
185C2×C4C2×Dic34611yescharacteristic
186C2×C4#1458382511yescharacteristic
187C2×C4#1458382511yescharacteristic
188C2×C4#1458382611yescharacteristic
189C8—41no—
190C8—6921no—
191C22#1914964711yescharacteristic
192C22#1914965611yescharacteristic
193C22#1914966511yescharacteristic
194C22#1914970011yescharacteristic
195C22#1914970211yescharacteristic
196C22#1914970811yescharacteristic
197C22#1914971111yescharacteristic
198C4—42no—
199C4—42no—
200C4—13841no—
201C4—6921no—
202C4—6921no—
203C4—6922no—
204C4#1914965611yescharacteristic
205C4#1914965811yescharacteristic
206C4#1914966511yescharacteristic
207C4#1914966611yescharacteristic
208C2#2601140311yescharacteristic
209C2#2601165011yescharacteristic
210C2#2601165211yescharacteristic
211C2#2601179611yescharacteristic
212C2#2601180011yescharacteristic
213C2#2601184711yescharacteristic
214C2#2601185811yescharacteristic
subgroupG/H
centerC23D4×D173
commutator, cosocleC22×C4×C173C23
FrattiniC22×C4C22×D173
FittingC2×#103×C173C2
socle, prime radicalC23×C173C2×D4
hypercenterC2×#103D173
hypocenterC173#2026
LCSG ▷ C22×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#103

Complements

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

NH ≅ G/Ntype
107. C173#2026semidirect

Maximal subgroups

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

subgroupindex
C2×#191496702
C2×#191496032
#260113982
#260112542
#260112532
#260112452
C2×#103×C1732
#2026173