Finite Group Databasebeta

← #36355236

Subgroups — #36355236

Order 22144.

238 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 55 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#26012061C211yesmaximal, characteristic
2#26011463C212yesmaximal
3#26011286C211yesmaximal, characteristic, complemented
4#26011285C212yesmaximal, complemented
5#354×C173C211yesmaximal, characteristic, complemented
6C2×#14583825C2211yescharacteristic
7C2×Q8×D173C411yescharacteristic, complemented
8#19149679C411yescharacteristic, complemented
9#19149677C411yescharacteristic, complemented
10#19149666C2211yescharacteristic
11#19149661—22no—
12#19149656—22no—
13#19149636C411yescharacteristic, complemented
14#19149635—22no—
15#19149594C2212yes—
16#19149593—22no—
17#124×C173C2211yescharacteristic
18#99×C173C2212yes—
19#14583825—42no—
20C2×Dic346—21no—
21C2×Dic346C2×C411yescharacteristic
22C2×Dic346C2×C411yescharacteristic
23Q8×D173—42no—
24C2×C22⋊D173D412yes—
25C22×Dic173D412yes—
26C2×C4×D173—21no—
27C2×C4×D173C2×C411yescharacteristic
28C4⋊Dic173—22no—
29#14583805—21no—
30#14583805—22no—
31#14583805—42no—
32C22⋊Dic173—22no—
33C22⋊Dic173—22no—
34#14583803—21no—
35C4×Dic173—42no—
36C4×Dic173C2×C411yescharacteristic
37C2×Q8×C173C2×C411yescharacteristic
38C2×D4×C173C2311yescharacteristic
39C22⋊C4×C173—22no—
40C22⋊C4×C173—22no—
41C42×C173C2×C411yescharacteristic
42C22×D173C22⋊C411yescharacteristic
43Dic346—41no—
44Dic346—42no—
45Dic346—41no—
46C22⋊D173—42no—
47C2×Dic173—22no—
48C2×Dic173—22no—
49C2×Dic173—21no—
50C2×Dic173—21no—
51C2×Dic173—42no—
52C2×Dic173C22⋊C412yes—
53C2×Dic173C22⋊C411yescharacteristic
54C4×D173—22no—
55C4×D173—42no—
56C23×C173C2×D412yes—
57Q8×C173—41no—
58D4×C173—41no—
59C2×C4×C173—21no—
60C2×C4×C173—21no—
61C2×C4×C173—42no—
62C2×C4×C173C22×C411yescharacteristic
63C22×C173—22no—
64C22×C173—42no—
65C22×C173C2×C22⋊C411yescharacteristic
66C2×D173—21no—
67C2×D173—41no—
68C4×C173—21no—
69C4×C173—41no—
70C4×C173—41no—
71C4×C173—82no—
72Dic173—22no—
73Dic173—41no—
74Dic173—41no—
75Dic173—82no—
76Dic173#9912yes—
77C2×C173—21no—
78C2×C173—42no—
79C2×C173C2×#9911yescharacteristic
80D173—41no—
81C173#212911yescharacteristic, complemented, Sylow
82#2129—1731nomaximal, Sylow
83#536—1731no—
84#410—1732no—
85#354—1731no—
86#354—1732no—
87#354D17311yescharacteristic, complemented
88C2×#41—1731no—
89C22×Q8—1731no—
90C4⋊Q8—1731no—
91C4⋊Q8—1731no—
92#124—1731no—
93#124C2×D17311yescharacteristic
94#123—3462no—
95C22⋊Q8—3462no—
96#117—3462no—
97#117—1731no—
98#99—3462no—
99#99—1732no—
100#99C2×D17312yes—
101#41—6922no—
102C2×Q8—3461no—
103C2×Q8—1731no—
104C2×Q8—1731no—
105C2×Q8—6922no—
106C2×Q8C4×D17311yescharacteristic
107C2×D4—1732no—
108C2×D4C22×D17311yescharacteristic
109C22×C4—3461no—
110C22×C4—1732no—
111C22×C4—1731no—
112C4⋊C4—3461no—
113C4⋊C4—3462no—
114C4⋊C4—3462no—
115C4⋊C4—6922no—
116C22⋊C4—22no—
117C22⋊C4—22no—
118C22⋊C4—3461no—
119C22⋊C4—3462no—
120C22⋊C4—3462no—
121C42—1731no—
122C42—6922no—
123C42C4×D17311yescharacteristic
124C23—1731no—
125C23D4×D17312yes—
126Q8—41no—
127Q8—6921no—
128Q8—6921no—
129Q8—6922no—
130D4—41no—
131D4—6922no—
132C2×C4—21no—
133C2×C4—21no—
134C2×C4—42no—
135C2×C4—3462no—
136C2×C4—3462no—
137C2×C4—3462no—
138C2×C4—3461no—
139C2×C4—3461no—
140C2×C4—1732no—
141C2×C4—1731no—
142C2×C4—6922no—
143C2×C4—6922no—
144C2×C4C2×C4×D17311yescharacteristic
145C22—22no—
146C22—42no—
147C22—3461no—
148C22—6921no—
149C22C22⋊C4×D17311yescharacteristic
150C4—21no—
151C4—41no—
152C4—41no—
153C4—82no—
154C4—3462no—
155C4—13842no—
156C4—1732no—
157C4—6921no—
158C4—6921no—
159C2—21no—
160C2—42no—
161C2—6921no—
162C2#99×D17311yescharacteristic
subgroupG/H
centerC2#99×D173
commutatorC2×C4×C173C22×C4
FrattiniC2×D4C22×D173
Fitting#354×C173C2
socleC2×C173C2×#99
prime radicalC2×#14583825C22
cosocleC2×D4×C173C23
hypercenter#354D173
hypocenterC173#2129
LCSG ▷ C2×C4×C173 ▷ C22×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #354

Complements

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

NH ≅ G/Ntype
3. #26011286C2semidirect
4. #26011285C2semidirect
5. #354×C173C2semidirect
7. C2×Q8×D173C4semidirect
8. #19149679C4semidirect
9. #19149677C4semidirect
13. #19149636C4semidirect
81. C173#2129semidirect
87. #354D173semidirect

Maximal subgroups

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

subgroupindex
#260120612
#260114632
#260112862
#260112852
#354×C1732
#2129173