Finite Group Databasebeta

← #36352196

Subgroups — #36352196

Order 22144.

238 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 59 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#26012040C211yesmaximal, characteristic
2#26011546C212yesmaximal, complemented
3#26011231C211yesmaximal, characteristic, complemented
4#26011230C212yesmaximal, complemented
5#329×C173C211yesmaximal, characteristic, complemented
6C2×#14583827C411yescharacteristic
7C2×#14583825C411yescharacteristic
8#19149677C411yescharacteristic
9#19149675C411yescharacteristic
10#19149667C2211yescharacteristic
11#19149652—22no—
12#19149639—22no—
13C42×D173C2211yescharacteristic
14#19149608—22no—
15#19149591—22no—
16#19149590C2212yes—
17#124×C173C2211yescharacteristic
18#97×C173C2212yes—
19#14583827—42no—
20#14583825—42no—
21C2×Dic346C2×C411yescharacteristic
22C2×D692—21no—
23C2×D692C2×C411yescharacteristic
24C2×C22⋊D173—21no—
25C2×C22⋊D173—22no—
26C22×Dic173—21no—
27C2×C4×D173—21no—
28C2×C4×D173C2×C411yescharacteristic
29C2×C4×D173D412yes—
30C2×#13063436—22no—
31#14583805—22no—
32#14583803—22no—
33C4×Dic173C2×C411yescharacteristic
34C4×Dic173D412yes—
35C2×Q8×C173C2×C411yescharacteristic
36C2×D4×C173C2×C411yescharacteristic
37C2×C8×C173—22no—
38C22⋊C4×C173—22no—
39C42×C173C2311yescharacteristic
40C22×D173—21no—
41C22×D173C22⋊C411yescharacteristic
42Dic346—41no—
43D692—41no—
44D692—42no—
45C22⋊D173—44no—
46C22⋊D173—42no—
47C2×Dic173—21no—
48C2×Dic173—42no—
49C2×Dic173C22⋊C412yes—
50C2×Dic173C22⋊C411yescharacteristic
51C4×D173—22no—
52C4×D173—42no—
53C4×D173—42no—
54#13063436—42no—
55C23×C173—21no—
56Q8×C173—41no—
57D4×C173—41no—
58C2×C4×C173—21no—
59C2×C4×C173C22×C411yescharacteristic
60C2×C4×C173C2×D412yes—
61C8×C173—42no—
62C22×C173—41no—
63C22×C173—42no—
64C22×C173C2×C22⋊C411yescharacteristic
65C2×D173—21no—
66C2×D173—22no—
67C2×D173—41no—
68C2×D173—42no—
69C4×C173—21no—
70C4×C173—22no—
71C4×C173—41no—
72Dic173—22no—
73Dic173—41no—
74Dic173#10012yes—
75C2×C173—81no—
76C2×C173C2×#10011yescharacteristic
77C2×C173#42112yes—
78D173—41no—
79D173—81no—
80C173#153611yescharacteristic, complemented, Sylow
81#1536—1731nomaximal, Sylow
82#532—1731no—
83#432—1732no—
84#329—1731no—
85#329—1732no—
86#329D17311yescharacteristic, complemented
87C2×#41—1731no—
88C2×#41—1731no—
89C4⋊Q8—1731no—
90#127—1731no—
91#124—1731no—
92#124C2×D17311yescharacteristic
93#121—3462no—
94C4×D4—3462no—
95C2×C42—1731no—
96C4⋊C8—3462no—
97C22⋊C8—3462no—
98#97—1732no—
99#97C2×D17312yes—
100#41—6922no—
101#41—6922no—
102C2×Q8—1731no—
103C2×Q8C4×D17311yescharacteristic
104C2×D4—3461no—
105C2×D4—3461no—
106C2×D4—3462no—
107C2×D4—1731no—
108C2×D4C4×D17311yescharacteristic
109C22×C4—3461no—
110C22×C4—3461no—
111C22×C4—1731no—
112C22×C4—1732no—
113C2×C8—22no—
114C2×C8—3462no—
115C4⋊C4—3462no—
116C22⋊C4—22no—
117C22⋊C4—3462no—
118C42—1731no—
119C42—1732no—
120C42C22×D17311yescharacteristic
121C23—21no—
122C23—3461no—
123C23—1731no—
124Q8—41no—
125Q8—6921no—
126D4—41no—
127D4—6921no—
128D4—6922no—
129D4—6924no—
130D4—6922no—
131C2×C4—21no—
132C2×C4—3462no—
133C2×C4—3461no—
134C2×C4—1732no—
135C2×C4—1731no—
136C2×C4—6922no—
137C2×C4—6922no—
138C2×C4—6922no—
139C2×C4C2×C4×D17311yescharacteristic
140C2×C4D4×D17312yes—
141C8—42no—
142C8—6922no—
143C22—41no—
144C22—42no—
145C22—3461no—
146C22—3462no—
147C22—6921no—
148C22—6922no—
149C22C22⋊C4×D17311yescharacteristic
150C4—21no—
151C4—22no—
152C4—41no—
153C4—3462no—
154C4—1732no—
155C4—6921no—
156C2—81no—
157C2—13841no—
158C2—6921no—
159C2#100×D17311yescharacteristic
160C2#2601150612yes—
subgroupG/H
centerC22C22⋊C4×D173
commutatorC2×C4×C173C22×C4
FrattiniC42C22×D173
Fitting#329×C173C2
socleC22×C173C2×C22⋊C4
prime radical#26012040C2
cosocleC42×C173C23
hypercenter#329D173
hypocenterC173#1536
LCSG ▷ C2×C4×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #329

Complements

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

NH ≅ G/Ntype
2. #26011546C2semidirect
3. #26011231C2semidirect
4. #26011230C2semidirect
5. #329×C173C2semidirect
80. C173#1536semidirect
86. #329D173semidirect

Maximal subgroups

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

subgroupindex
#260120402
#260115462
#260112312
#260112302
#329×C1732
#1536173