Finite Group Databasebeta

← #36369098

Subgroups — #36369098

Order 22144.

202 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 202 conjugacy classes of subgroups — too many to draw. Only its 58 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#26012514C211yesmaximal, characteristic
2#26012496C211yesmaximal, characteristic
3#26012351C212yesmaximal, complemented
4#26012350C212yesmaximal, complemented
5#26012045C211yesmaximal, characteristic
6C2×#14583836—21no—
7#19149746—21no—
8#19149746C2212yes—
9#19149746C2212yes—
10#19149736—21no—
11#19149734—21no—
12#19149678C2211yescharacteristic
13#19149677C2211yescharacteristic
14#19149669—21no—
15#19149659C411yescharacteristic
16#19149657—21no—
17#19149655C411yescharacteristic
18C4×Dic346—21no—
19C4×Dic346C411yescharacteristic
20#19149636—21no—
21C42×D173C2211yescharacteristic
22C4×Q8×C173C411yescharacteristic
23#14583836—42no—
24C2×#13063443—22no—
25C2×#13063443—22no—
26C2×Dic346—21no—
27C2×Dic346C2×C411yescharacteristic
28C2×C4×D173C2×C411yescharacteristic
29C2×C4×D173D411yescharacteristic
30C2×C4×D173D411yescharacteristic
31C4⋊Dic173—21no—
32C4⋊Dic173C2×C411yescharacteristic
33#14583805—21no—
34#14583805—21no—
35#14583805—21no—
36#14583805—21no—
37#14583805—21no—
38#14583803—21no—
39#14583803—21no—
40C4×Dic173—21no—
41C4×Dic173D411yescharacteristic
42C4×Dic173D411yescharacteristic
43C4×Dic173C2311yescharacteristic
44C2×Q8×C173C2×C411yescharacteristic
45C4⋊C4×C173—21no—
46C4⋊C4×C173C2×C411yescharacteristic
47C42×C173—21no—
48C42×C173C2×C411yescharacteristic
49#13063443—42no—
50#13063443—42no—
51C22×D173C22⋊C411yescharacteristic
52Dic346—41no—
53Dic346—41no—
54Dic346—41no—
55C2×Dic173—21no—
56C2×Dic173—21no—
57C2×Dic173C22⋊C411yescharacteristic
58C2×Dic173C2×D411yescharacteristic
59C2×Dic173C2×D411yescharacteristic
60C4×D173—21no—
61C4×D173—21no—
62C4×D173—22no—
63C4×D173—41no—
64Q8×C173—41no—
65C2×C4×C173—21no—
66C2×C4×C173—21no—
67C2×C4×C173C22⋊C411yescharacteristic
68C2×C4×C173C22⋊C411yescharacteristic
69C2×C4×C173C22×C411yescharacteristic
70C22×C173C2×C22⋊C411yescharacteristic
71C2×D173—21no—
72C2×D173—21no—
73C2×D173—21no—
74C4×C173—21no—
75C4×C173—21no—
76C4×C173—21no—
77C4×C173—41no—
78C4×C173—41no—
79Dic173—21no—
80Dic173—41no—
81Dic173—41no—
82Dic173AGL(1,ℤ/8)11yescharacteristic
83Dic173#13711yescharacteristic
84Dic173#13712yes—
85C2×C173#41311yescharacteristic
86C2×C173#41711yescharacteristic
87C2×C173#41911yescharacteristic
88D173—41no—
89C173#155911yescharacteristic, complemented, Sylow
90#1559—1731nomaximal, Sylow
91#533—1731no—
92#432—1731no—
93#423—1731no—
94#333—1732no—
95#332—1732no—
96C2×M16—3461no—
97C4⋊Q8—1731no—
98C4⋊Q8—1731no—
99#125—3461no—
100C22⋊Q8—3461no—
101C22⋊Q8—1731no—
102C22⋊Q8—1731no—
103C4×Q8—3461no—
104C4×Q8—1731no—
105C4×Q8#1062192411yescharacteristic
106#117—3461no—
107C2×C42—1731no—
108C4⋊C8—3461no—
109C4⋊C8—1732no—
110C4⋊C8—1732no—
111C22⋊C8—3461no—
112#97—3461no—
113C2×Q8—3461no—
114C2×Q8—1731no—
115C2×Q8C2×#1062192411yescharacteristic
116C22×C4—1731no—
117C22×C4—1731no—
118C22×C4—1731no—
119M16—6922no—
120C2×C8—3462no—
121C2×C8—3462no—
122C4⋊C4—21no—
123C4⋊C4—3461no—
124C4⋊C4—3461no—
125C4⋊C4—3461no—
126C4⋊C4—3461no—
127C4⋊C4—3461no—
128C4⋊C4—3461no—
129C4⋊C4—1731no—
130C4⋊C4C2×#1062192411yescharacteristic
131C22⋊C4—3461no—
132C22⋊C4—3461no—
133C42—21no—
134C42—3461no—
135C42—1731no—
136C42—1731no—
137C42—1731no—
138C42C2×#1062192411yescharacteristic
139C23—1731no—
140Q8—41no—
141Q8—6921no—
142Q8—6921no—
143Q8—6921no—
144C2×C4—21no—
145C2×C4—21no—
146C2×C4—3461no—
147C2×C4—3461no—
148C2×C4—3462no—
149C2×C4—3461no—
150C2×C4—3461no—
151C2×C4—1731no—
152C2×C4—1731no—
153C2×C4—1731no—
154C2×C4—6921no—
155C2×C4C22⋊#1062192411yescharacteristic
156C2×C4C22⋊#1062192411yescharacteristic
157C2×C4C22×#1062192411yescharacteristic
158C8—6922no—
159C8—6922no—
160C22—3461no—
161C22—3461no—
162C22—3461no—
163C22C2×C22⋊#1062192411yescharacteristic
164C4—21no—
165C4—21no—
166C4—21no—
167C4—41no—
168C4—41no—
169C4—3461no—
170C4—1731no—
171C4—1731no—
172C4—1732no—
173C4—6921no—
174C4—6921no—
175C2—6921no—
176C2#2601247711yescharacteristic
177C2#2601248211yescharacteristic
178C2#2601248411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10621924
commutatorC2×C4×C173C22×C4
FrattiniC2×C4C22×#10621924
FittingC4×Q8×C173C4
socleC22×C173C2×C22⋊C4
prime radicalC22×D173C22⋊C4
cosocleC4×Dic173C23
hypercenterC4×Q8#10621924
hypocenterC173#1559
LCSG ▷ C2×C4×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ C4×Q8

Complements

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

NH ≅ G/Ntype
3. #26012351C2semidirect
4. #26012350C2semidirect
89. C173#1559semidirect

Maximal subgroups

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

subgroupindex
#260125142
#260124962
#260123512
#260123502
#260120452
#1559173