Finite Group Databasebeta

← #36369535

Subgroups — #36369535

Order 22144.

260 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 66 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#26012569C211yesmaximal, characteristic
2#26012497C212yesmaximal, complemented
3#26012464C211yesmaximal, characteristic
4#26012372C212yesmaximal, complemented
5#26011463C211yesmaximal, characteristic
6#19149766—22no—
7#19149766—22no—
8D173⋊M16—21no—
9C2×#14583835—21no—
10C2×#14583835C2211yescharacteristic
11C2×#14583835C2212yes—
12C22×#13063443C2211yescharacteristic
13#19149746—22no—
14C4⋊#13063443—22no—
15#19149736—21no—
16C22⋊#13063443—21no—
17C2×#14583825C2211yescharacteristic
18#19149635C2212yes—
19#19149594C411yescharacteristic
20#19149593C412yes—
21#99×C173C411yescharacteristic
22#14583836—21no—
23#14583836—21no—
24#14583835—22no—
25#14583835—22no—
26#14583835—22no—
27#14583835—22no—
28D173⋊C8—22no—
29C2×#13063443—21no—
30C2×#13063443—21no—
31C2×#13063443—21no—
32C2×#13063443—21no—
33C2×#13063443—22no—
34C2×#13063443D412yes—
35C2×#13063443D412yes—
36#14583825—22no—
37#14583825—22no—
38C2×Dic346Q811yescharacteristic
39C2×C22⋊D173C2×C411yescharacteristic
40C2×C22⋊D173Q811yescharacteristic
41C22×Dic173D411yescharacteristic
42C22×Dic173C2311yescharacteristic
43C2×C4×D173D411yescharacteristic
44#14583805—22no—
45C22⋊Dic173C2×C412yes—
46C4×Dic173—22no—
47C2×D4×C173C2×C411yescharacteristic
48C22⋊C4×C173C2×C412yes—
49#13063443—22no—
50#13063443—22no—
51#13063443—41no—
52#13063443—41no—
53C22×D173C4⋊C411yescharacteristic
54Dic346—22no—
55C22⋊D173—22no—
56C22⋊D173—22no—
57C2×Dic173—21no—
58C2×Dic173—21no—
59C2×Dic173—22no—
60C2×Dic173C4⋊C411yescharacteristic
61C2×Dic173C2×D412yes—
62C2×Dic173C2×D411yescharacteristic
63C2×Dic173C2×Q811yescharacteristic
64C2×Dic173#4112yes—
65C4×D173—21no—
66C4×D173—21no—
67C23×C173C4⋊C411yescharacteristic
68C23×C173C22×C411yescharacteristic
69D4×C173—22no—
70C2×C4×C173—22no—
71C2×C4×C173C4⋊C411yescharacteristic
72C22×C173—21no—
73C22×C173—41no—
74C22×C173—41no—
75C22×C173C2×C4⋊C411yescharacteristic
76C22×C173C4×D412yes—
77C2×D173—21no—
78C2×D173—41no—
79C4×C173—21no—
80C4×C173—42no—
81Dic173—21no—
82Dic173—42no—
83Dic173#12011yescharacteristic
84Dic173C22⋊Q812yes—
85Dic173#12311yescharacteristic
86C2×C173—22no—
87C2×C173—21no—
88C2×C173—41no—
89C2×C173#38511yescharacteristic
90D173—41no—
91C173#186811yescharacteristic, complemented, Sylow
92#1868—1731nomaximal, Sylow
93#568—1731no—
94#424—1732no—
95#410—1731no—
96#408—1731no—
97#344—1732no—
98C2×#41—1731no—
99#131—3462no—
100#131—3462no—
101C2×M16—3461no—
102C2×M16—3461no—
103C2×M16—1731no—
104C2×M16—1732no—
105C22×C8—1731no—
106#117—1732no—
107C4⋊C8—3462no—
108C4⋊C8—3462no—
109#99—1731no—
110#99—1732no—
111#99#1062192411yescharacteristic
112C22⋊C8—3461no—
113C22⋊C8—3461no—
114#41—3462no—
115#41—3462no—
116C2×Q8—1731no—
117C2×D4—1731no—
118C2×D4—1731no—
119C2×D4C2×#1062192411yescharacteristic
120C22×C4—1731no—
121C22×C4—1731no—
122C22×C4—1731no—
123M16—3462no—
124M16—3461no—
125M16—3461no—
126M16—3462no—
127M16—3462no—
128M16—3462no—
129C2×C8—3461no—
130C2×C8—3461no—
131C2×C8—3462no—
132C2×C8—3461no—
133C2×C8—3461no—
134C2×C8—3462no—
135C2×C8—1732no—
136C2×C8—1732no—
137C4⋊C4—3462no—
138C22⋊C4—1732no—
139C22⋊C4C2×#1062192412yes—
140C42—3462no—
141C23—1731no—
142C23C4⋊#1062192411yescharacteristic
143C23C22×#1062192411yescharacteristic
144Q8—3462no—
145D4—22no—
146D4—3462no—
147D4—3462no—
148C2×C4—22no—
149C2×C4—3461no—
150C2×C4—3461no—
151C2×C4—3461no—
152C2×C4—3461no—
153C2×C4—3462no—
154C2×C4—1731no—
155C2×C4—1731no—
156C2×C4—1731no—
157C2×C4—1732no—
158C2×C4—1732no—
159C2×C4C4⋊#1062192411yescharacteristic
160C8—3462no—
161C8—3462no—
162C8—6921no—
163C8—6921no—
164C22—21no—
165C22—41no—
166C22—41no—
167C22—3461no—
168C22—6921no—
169C22C2×C4⋊#1062192411yescharacteristic
170C22D4×#1062192412yes—
171C4—21no—
172C4—42no—
173C4—3461no—
174C4—1731no—
175C4—1731no—
176C4—1732no—
177C4—6922no—
178C2—22no—
179C2—21no—
180C2—41no—
181C2—6921no—
182C2#2601243811yescharacteristic
subgroupG/H
centerC2#26012438
commutatorC23×C173C22×C4
FrattiniC23C22×#10621924
Fitting#99×C173C4
socleC2×C173#385
prime radicalC2×#14583825C22
cosocleC22×Dic173C23
hypercenter#99#10621924
hypocenterC173#1868
LCSG ▷ C23×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C23 ◁ #99

Complements

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

NH ≅ G/Ntype
2. #26012497C2semidirect
4. #26012372C2semidirect
91. C173#1868semidirect

Maximal subgroups

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

subgroupindex
#260125692
#260124972
#260124642
#260123722
#260114632
#1868173