Finite Group Databasebeta

← #36369846

Subgroups — #36369846

Order 22144.

238 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 52 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#26012472C212yesmaximal
2#26012398C212yesmaximal, complemented
3#26012397C212yesmaximal, complemented
4#26012066C211yesmaximal, characteristic
5#19149758—22no—
6#19149738—22no—
7#19149738C2214yes—
8C2×#14583827C2211yescharacteristic
9C2×Q8×D173C411yescharacteristic, complemented
10#19149677C411yescharacteristic, complemented
11#19149668C2211yescharacteristic
12#19149667C2211yescharacteristic
13#19149664—22no—
14#19149657—22no—
15#19149636C411yescharacteristic, complemented
16C4⋊Q8×C173C411yescharacteristic, complemented
17C4⋊#10621924—42no—
18C22⋊#10621924—24no—
19C4×#10621924—42no—
20#14583827—42no—
21C2×Dic346—21no—
22C2×Dic346C2×C411yescharacteristic
23Q8×D173—42no—
24C2×D692D412yes—
25C2×D692C2311yescharacteristic
26C2×C4×D173—21no—
27C2×C4×D173C2×C411yescharacteristic
28C2×C4×D173D412yes—
29#14583805—21no—
30#14583805—22no—
31#14583803—21no—
32#14583803—22no—
33#14583803—22no—
34C4×Dic173C2×C411yescharacteristic
35C2×Q8×C173C2×C411yescharacteristic
36C2×Q8×C173C2×C411yescharacteristic
37C4⋊C4×C173—22no—
38C42×C173C2×C411yescharacteristic
39C2×#10621924—44no—
40C22×D173C22⋊C411yescharacteristic
41C22×D173C2×D412yes—
42Dic346—42no—
43Dic346—41no—
44D692—42no—
45D692—41no—
46C2×Dic173—21no—
47C2×Dic173—21no—
48C2×Dic173C22⋊C411yescharacteristic
49C4×D173—22no—
50C4×D173—22no—
51C4×D173—22no—
52C4×D173—42no—
53Q8×C173—41no—
54Q8×C173—41no—
55C2×C4×C173—21no—
56C2×C4×C173—21no—
57C2×C4×C173C22⋊C412yes—
58C2×C4×C173C22×C411yescharacteristic
59C22×C173C2×C22⋊C411yescharacteristic
60C2×D173—21no—
61C2×D173—22no—
62C2×D173—41no—
63C2×D173—42no—
64#10621924—84no—
65C4×C173—22no—
66C4×C173—21no—
67C4×C173—41no—
68C4×C173—41no—
69Dic173—41no—
70Dic173—41no—
71Dic173#9912yes—
72C2×C173—21no—
73C2×C173C2×#9911yescharacteristic
74D173—42no—
75D173—41no—
76C173#212911yescharacteristic, complemented, Sylow
77#2129—1731nomaximal, Sylow
78#536—1731no—
79#410—1732no—
80#354—1732no—
81#354—1732no—
82C2×#41—1731no—
83C22×Q8—1731no—
84C4⋊Q8—1731no—
85C4⋊Q8#1062192411yescharacteristic, complemented
86#124—1731no—
87#124—1731no—
88#123—3462no—
89C22⋊Q8—3462no—
90#117—3462no—
91#117—1731no—
92#99—3462no—
93#99—1734no—
94#41—6922no—
95C2×Q8—3461no—
96C2×Q8—1731no—
97C2×Q8—6922no—
98C2×Q8C2×#1062192411yescharacteristic
99C2×Q8C2×#1062192411yescharacteristic
100C2×D4—1732no—
101C2×D4—1731no—
102C22×C4—3461no—
103C22×C4—1732no—
104C22×C4—1731no—
105C4⋊C4—22no—
106C4⋊C4—3461no—
107C4⋊C4—3462no—
108C4⋊C4—6922no—
109C22⋊C4—3461no—
110C22⋊C4—3462no—
111C22⋊C4—3462no—
112C22⋊C4—3464no—
113C42—1731no—
114C42—6922no—
115C42C2×#1062192411yescharacteristic
116C23—1731no—
117C23—1732no—
118Q8—41no—
119Q8—41no—
120Q8—6921no—
121Q8—6922no—
122D4—6922no—
123D4—6921no—
124C2×C4—21no—
125C2×C4—21no—
126C2×C4—3462no—
127C2×C4—3462no—
128C2×C4—3462no—
129C2×C4—3461no—
130C2×C4—3461no—
131C2×C4—1731no—
132C2×C4—6922no—
133C2×C4—6924no—
134C2×C4C22⋊#1062192412yes—
135C2×C4C22×#1062192411yescharacteristic
136C22—3462no—
137C22—3461no—
138C22—6922no—
139C22—6921no—
140C22C2×C22⋊#1062192411yescharacteristic
141C4—22no—
142C4—21no—
143C4—41no—
144C4—41no—
145C4—13844no—
146C4—1732no—
147C4—6921no—
148C4—6921no—
149C2—21no—
150C2—6922no—
151C2—6921no—
152C2D173⋊#9911yescharacteristic
subgroupG/H
centerC2D173⋊#99
commutatorC2×C4×C173C22×C4
FrattiniC2×C4C22×#10621924
FittingC4⋊Q8×C173C4
socleC2×C173C2×#99
prime radicalC2×#14583827C22
cosocleC2×D692C23
hypercenterC4⋊Q8#10621924
hypocenterC173#2129
LCSG ▷ C2×C4×C173 ▷ C22×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
2. #26012398C2semidirect
3. #26012397C2semidirect
9. C2×Q8×D173C4semidirect
10. #19149677C4semidirect
15. #19149636C4semidirect
16. C4⋊Q8×C173C4semidirect
76. C173#2129semidirect
85. C4⋊Q8#10621924semidirect

Maximal subgroups

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

subgroupindex
#260124722
#260123982
#260123972
#260120662
#2129173