Finite Group Databasebeta

← #36368896

Subgroups — #36368896

Order 22144.

246 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 64 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#26012512C211yesmaximal, characteristic, complemented
2#26012508C211yesmaximal, characteristic, complemented
3#26012341C212yesmaximal, complemented
4#26012339C212yesmaximal, complemented
5#26011959C211yesmaximal, characteristic
6#19149746—21no—
7C4⋊#13063443C2212yes—
8C22⋊#13063443—21no—
9C22⋊#13063443—21no—
10#19149734—21no—
11C4×#13063443C2212yes—
12C2×#14583826C411yescharacteristic
13#19149644C411yescharacteristic
14#19149644C2211yescharacteristic
15#19149641C411yescharacteristic
16#19149641C2211yescharacteristic
17#19149639—21no—
18#19149639—21no—
19#19149636—21no—
20C2×C4×Dic173C2211yescharacteristic
21C42×D173—21no—
22#117×C173C411yescharacteristic
23C2×#13063443—22no—
24C2×#13063443—22no—
25#14583826—22no—
26#14583826—22no—
27C2×Dic346C2×C411yescharacteristic
28C2×D692C2×C411yescharacteristic
29C2×C22⋊D173—21no—
30C22×Dic173D411yescharacteristic
31C22×Dic173D411yescharacteristic
32C2×C4×D173—21no—
33C2×C4×D173—21no—
34C2×C4×D173—21no—
35#14583805—21no—
36#14583805—21no—
37#14583803—21no—
38#14583803—21no—
39C4×Dic173—21no—
40C4×Dic173—21no—
41C4×Dic173C2×C411yescharacteristic
42C4×Dic173D411yescharacteristic
43C4×Dic173D411yescharacteristic
44C4×Dic173C2311yescharacteristic
45C22×C4×C173C2×C411yescharacteristic
46C4⋊C4×C173C2×C411yescharacteristic
47C4⋊C4×C173C2×C411yescharacteristic
48C22⋊C4×C173—21no—
49C42×C173—21no—
50#13063443—42no—
51#13063443—42no—
52C22×D173—21no—
53Dic346—21no—
54Dic346M1612yes—
55D692—21no—
56D692M1612yes—
57C22⋊D173—24no—
58C2×Dic173—21no—
59C2×Dic173—22no—
60C2×Dic173—41no—
61C2×Dic173C22⋊C411yescharacteristic
62C2×Dic173C22⋊C411yescharacteristic
63C2×Dic173C2×D411yescharacteristic
64C2×Dic173C2×D411yescharacteristic
65C4×D173—24no—
66C4×D173—42no—
67C4×D173—42no—
68C23×C173C22⋊C411yescharacteristic
69C2×C4×C173—21no—
70C2×C4×C173—21no—
71C2×C4×C173—21no—
72C2×C4×C173—21no—
73C2×C4×C173C22⋊C411yescharacteristic
74C2×C4×C173C22×C411yescharacteristic
75C22×C173—21no—
76C22×C173—21no—
77C22×C173—21no—
78C22×C173C2×C22⋊C411yescharacteristic
79C2×D173—22no—
80C2×D173—41no—
81C2×D173—41no—
82C4×C173—21no—
83C4×C173—41no—
84C4×C173—41no—
85C4×C173C2×M1611yescharacteristic
86C4×C173C2×M1611yescharacteristic
87Dic173—22no—
88Dic173—21no—
89Dic173—21no—
90Dic173—21no—
91Dic173#13512yes—
92C2×C173—41no—
93C2×C173#40711yescharacteristic
94C2×C173#41611yescharacteristic
95C2×C173#42111yescharacteristic
96D173—42no—
97C173#148811yescharacteristic, complemented, Sylow
98#1488—1731nomaximal, Sylow
99C4×#41—1731no—
100#432—1731no—
101#431—1731no—
102Q8⋊C8—1732no—
103D4⋊C8—1732no—
104C2×#41—1731no—
105C4×Q8—1731no—
106C4×Q8—1731no—
107C4×D4—3461no—
108C4×D4—3461no—
109C4×D4—1731no—
110C4×D4—1731no—
111#117—3461no—
112#117#1062192411yescharacteristic
113C2×C42—3461no—
114C2×C42—1731no—
115C4⋊C8—3461no—
116C4⋊C8—1732no—
117C22⋊C8—3461no—
118C22⋊C8—3461no—
119#97—3461no—
120C4×C8—1732no—
121#41—3462no—
122#41—3462no—
123C2×Q8—1731no—
124C2×D4—3461no—
125C2×D4—1731no—
126C22×C4—3461no—
127C22×C4—3461no—
128C22×C4—3461no—
129C22×C4—1731no—
130C22×C4—1731no—
131C22×C4C2×#1062192411yescharacteristic
132C2×C8—3462no—
133C2×C8—3462no—
134C4⋊C4—3461no—
135C4⋊C4—3461no—
136C4⋊C4C2×#1062192411yescharacteristic
137C4⋊C4C2×#1062192411yescharacteristic
138C22⋊C4—21no—
139C22⋊C4—3461no—
140C22⋊C4—3461no—
141C42—21no—
142C42—3461no—
143C42—3461no—
144C42—1731no—
145C42—1731no—
146C42—1731no—
147C42—1731no—
148C23—3461no—
149C23C22⋊#1062192411yescharacteristic
150Q8—3461no—
151Q8—1732no—
152D4—3464no—
153D4—3461no—
154D4—1732no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—21no—
158C2×C4—21no—
159C2×C4—3464no—
160C2×C4—3461no—
161C2×C4—3462no—
162C2×C4—1731no—
163C2×C4—1731no—
164C2×C4—1731no—
165C2×C4—1731no—
166C2×C4—6922no—
167C2×C4—6922no—
168C2×C4—6921no—
169C2×C4C22⋊#1062192411yescharacteristic
170C2×C4C22×#1062192411yescharacteristic
171C8—6922no—
172C8—6922no—
173C22—21no—
174C22—21no—
175C22—21no—
176C22—3462no—
177C22—6921no—
178C22—6921no—
179C22C2×C22⋊#1062192411yescharacteristic
180C4—21no—
181C4—41no—
182C4—41no—
183C4—3462no—
184C4—3461no—
185C4—3461no—
186C4—3461no—
187C4—1732no—
188C4D173⋊M1611yescharacteristic
189C4D173⋊M1611yescharacteristic
190C2—41no—
191C2—6922no—
192C2#2601246311yescharacteristic
193C2#2601248111yescharacteristic
194C2#2601248811yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10621924
commutatorC2×C4×C173C22×C4
FrattiniC2×C4C22×#10621924
Fitting#117×C173C4
socleC22×C173C2×C22⋊C4
prime radicalC2×#14583826C4
cosocleC4×Dic173C23
hypercenter#117#10621924
hypocenterC173#1488
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
1. #26012512C2semidirect
2. #26012508C2semidirect
3. #26012341C2semidirect
4. #26012339C2semidirect
97. C173#1488semidirect

Maximal subgroups

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

subgroupindex
#260125122
#260125082
#260123412
#260123392
#260119592
#1488173