Finite Group Databasebeta

← #36355561

Subgroups — #36355561

Order 22144.

246 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 41 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#26012227C211yesmaximal, characteristic, complemented
2#26011682C212yesmaximal, complemented
3#26011658C211yesmaximal, characteristic
4#26011280C212yesmaximal, complemented
5#353×C173C211yesmaximal, characteristic, complemented
6#19149724C2211yescharacteristic
7#19149705—21no—
8#19149666—22no—
9#19149663—22no—
10#19149660—21no—
11#19149660C2212yes—
12C4×Dic346—21no—
13D4×Dic173—22no—
14#19149607—21no—
15#19149596C2211yescharacteristic
16#19149593—22no—
17#127×C173C2211yescharacteristic
18#99×C173C2212yes—
19#14583826—21no—
20#14583825—41no—
21C2×Dic346—21no—
22C2×C22⋊D173—22no—
23D4×D173—41no—
24C22×Dic173—21no—
25C22×Dic173D412yes—
26#14583814—41no—
27#14583811—41no—
28#14583810—21no—
29C4⋊Dic173—21no—
30C4⋊Dic173—42no—
31#14583805—22no—
32C22⋊Dic173—22no—
33C22⋊Dic173—21no—
34C22⋊Dic173—22no—
35C22⋊Dic173D412yes—
36#14583803—42no—
37C4×Dic173—22no—
38C2×D4×C173—22no—
39C2×D4×C173D411yescharacteristic
40C2×D4×C173C2311yescharacteristic
41C22⋊C4×C173—22no—
42C42×C173D411yescharacteristic
43C22×D173—22no—
44Dic346—21no—
45Dic346—42no—
46D692—21no—
47C22⋊D173—42no—
48C22⋊D173—41no—
49C2×Dic173—22no—
50C2×Dic173—22no—
51C2×Dic173—22no—
52C2×Dic173—42no—
53C2×Dic173—81no—
54C4×D173—41no—
55#13063436—41no—
56C23×C173—21no—
57C23×C173C2×D412yes—
58D4×C173—41no—
59D4×C173—41no—
60D4×C173—44no—
61C2×C4×C173—21no—
62C2×C4×C173—42no—
63C2×C4×C173C2×D411yescharacteristic
64C22×C173—22no—
65C22×C173—42no—
66C22×C173—42no—
67C22×C173—41no—
68C22×C173#12011yescharacteristic
69C2×D173—41no—
70C2×D173—82no—
71C4×C173—22no—
72C4×C173—21no—
73C4×C173—82no—
74Dic173—42no—
75Dic173—41no—
76Dic173—42no—
77C2×C173—21no—
78C2×C173—42no—
79C2×C173—81no—
80C2×C173B(4,𝔽2)11yescharacteristic
81D173—81no—
82C173#219811yescharacteristic, complemented, Sylow
83#2198—1731nomaximal, Sylow
84#559—1731no—
85#458—1732no—
86#453—1731no—
87#353D17311yescharacteristic, complemented
88#352—1732no—
89H(5,𝔽2)—1731no—
90AGL(1,ℤ/8)—3461no—
91#127C2×D17311yescharacteristic
92#124—3462no—
93#123—3462no—
94#123—3461no—
95#123—1732no—
96C4×Q8—3461no—
97C4×D4—3462no—
98#104—3461no—
99#100—1731no—
100#99—3462no—
101#99C2×D17312yes—
102#41—3461no—
103#41—6921no—
104C2×Q8—3461no—
105C2×D4—22no—
106C2×D4—3462no—
107C2×D4—6921no—
108C2×D4D69211yescharacteristic
109C2×D4C22×D17311yescharacteristic
110C22×C4—3461no—
111C22×C4—1732no—
112SD16—6921no—
113D8—6921no—
114M16—3461no—
115C4⋊C4—3461no—
116C4⋊C4—3462no—
117C4⋊C4—6922no—
118C22⋊C4—22no—
119C22⋊C4—3462no—
120C22⋊C4—3461no—
121C22⋊C4—3462no—
122C22⋊C4—1732no—
123C22⋊C4—6922no—
124C42—3462no—
125C42D69211yescharacteristic
126C23—21no—
127C23—3462no—
128C23D4×D17312yes—
129Q8—3461no—
130Q8—6922no—
131D4—41no—
132D4—41no—
133D4—44no—
134D4—3461no—
135D4—6922no—
136D4—6921no—
137C2×C4—21no—
138C2×C4—42no—
139C2×C4—3462no—
140C2×C4—3462no—
141C2×C4—3462no—
142C2×C4—13841no—
143C2×C4—6921no—
144C2×C4—6922no—
145C2×C4C2×D69211yescharacteristic
146C8—6921no—
147C22—22no—
148C22—42no—
149C22—42no—
150C22—41no—
151C22—13842no—
152C22—6921no—
153C22#1914964911yescharacteristic
154C4—22no—
155C4—21no—
156C4—82no—
157C4—6922no—
158C4—6921no—
159C4—6922no—
160C2—21no—
161C2—42no—
162C2—81no—
163C2—13841no—
164C2#2601167911yescharacteristic
subgroupG/H
centerC2#26011679
commutator, cosocleC2×D4×C173C23
FrattiniC2×D4C22×D173
Fitting#353×C173C2
socleC2×C173B(4,𝔽2)
prime radical#26011658C2
hypercenter#353D173
hypocenterC173#2198
LCSG ▷ C2×D4×C173 ▷ C22×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #353

Complements

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

NH ≅ G/Ntype
1. #26012227C2semidirect
2. #26011682C2semidirect
4. #26011280C2semidirect
5. #353×C173C2semidirect
82. C173#2198semidirect
87. #353D173semidirect

Maximal subgroups

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

subgroupindex
#260122272
#260116822
#260116582
#260112802
#353×C1732
#2198173