Finite Group Databasebeta

← #36355245

Subgroups — #36355245

Order 22144.

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

normal subgroups only

This group has 250 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#26012040C211yesmaximal, characteristic
2#26011474C212yesmaximal, complemented
3#26011289C212yesmaximal, complemented
4#26011288C211yesmaximal, characteristic, complemented
5#355×C173C211yesmaximal, characteristic, complemented
6C2×#14583827C2211yescharacteristic
7C2×#14583825C411yescharacteristic
8M16×D173—22no—
9#19149677C2211yescharacteristic
10#19149675C411yescharacteristic
11#19149667C411yescharacteristic
12#19149652—22no—
13#19149639—22no—
14C42×D173C411yescharacteristic
15#19149598C2212yes—
16#19149595—22no—
17#124×C173C2211yescharacteristic
18#101×C173C2212yes—
19#14583827—41no—
20#14583827—41no—
21#14583825—41no—
22#14583825—41no—
23C2×Dic346C2×C411yescharacteristic
24C2×D692C2×C411yescharacteristic
25C2×D692D412yes—
26C2×C22⋊D173—22no—
27C2×C22⋊D173—21no—
28C22×Dic173—21no—
29C2×C4×D173—21no—
30C2×C4×D173C2×C411yescharacteristic
31C2×C4×D173D412yes—
32#14583810—22no—
33#14583809—42no—
34C8×D173—42no—
35#14583805—22no—
36#14583803—22no—
37C4×Dic173—21no—
38C4×Dic173C2×C411yescharacteristic
39C2×Q8×C173C2311yescharacteristic
40C2×D4×C173C2×C411yescharacteristic
41M16×C173—22no—
42C22⋊C4×C173—22no—
43C42×C173C2×C411yescharacteristic
44C22×D173C22⋊C411yescharacteristic
45C22×D173C22⋊C412yes—
46Dic346—41no—
47D692—41no—
48D692—42no—
49C22⋊D173—44no—
50C22⋊D173—42no—
51C2×Dic173—21no—
52C2×Dic173—21no—
53C2×Dic173—41no—
54C2×Dic173—41no—
55C2×Dic173C22⋊C411yescharacteristic
56C4×D173—22no—
57C4×D173—21no—
58C4×D173—21no—
59C4×D173—22no—
60C4×D173—42no—
61#13063436—42no—
62C23×C173—21no—
63Q8×C173—41no—
64D4×C173—41no—
65C2×C4×C173—21no—
66C2×C4×C173C22×C411yescharacteristic
67C2×C4×C173C2×D412yes—
68C8×C173—42no—
69C22×C173—42no—
70C22×C173—41no—
71C22×C173C2×C22⋊C411yescharacteristic
72C2×D173—21no—
73C2×D173—22no—
74C2×D173—41no—
75C2×D173—42no—
76C4×C173—22no—
77C4×C173—21no—
78C4×C173—41no—
79Dic173—22no—
80Dic173—41no—
81Dic173#9911yescharacteristic
82Dic173#9911yescharacteristic
83C2×C173—21no—
84C2×C173—81no—
85C2×C173C2×#9911yescharacteristic
86D173—41no—
87D173—42no—
88C173#213211yescharacteristic, complemented, Sylow
89#2132—1731nomaximal, Sylow
90#532—1731no—
91#413—1732no—
92#356—1732no—
93#355—1731no—
94#355D17311yescharacteristic, complemented
95C2×#41—1731no—
96C2×#41—1731no—
97C2×M16—3462no—
98C4⋊Q8—1731no—
99#127—1731no—
100#124—1731no—
101#124C2×D17311yescharacteristic
102#121—3462no—
103C4×D4—3462no—
104C2×C42—1731no—
105#101—1732no—
106#101C2×D17312yes—
107#100—3462no—
108#41—6921no—
109#41—6921no—
110#41—6921no—
111#41—6921no—
112C2×Q8—1731no—
113C2×Q8C22×D17311yescharacteristic
114C2×D4—3462no—
115C2×D4—3461no—
116C2×D4—1731no—
117C2×D4—1732no—
118C2×D4C4×D17311yescharacteristic
119C22×C4—3461no—
120C22×C4—3461no—
121C22×C4—1731no—
122C22×C4—1732no—
123M16—22no—
124M16—3462no—
125M16—6922no—
126C2×C8—6922no—
127C4⋊C4—3462no—
128C22⋊C4—22no—
129C22⋊C4—3462no—
130C42—3461no—
131C42—1731no—
132C42C4×D17311yescharacteristic
133C23—21no—
134C23—1732no—
135C23—1731no—
136Q8—41no—
137Q8—6921no—
138D4—41no—
139D4—6922no—
140D4—6921no—
141D4—6924no—
142D4—6922no—
143C2×C4—21no—
144C2×C4—3462no—
145C2×C4—3462no—
146C2×C4—3461no—
147C2×C4—3461no—
148C2×C4—3461no—
149C2×C4—3461no—
150C2×C4—1731no—
151C2×C4—6922no—
152C2×C4—6921no—
153C2×C4—6921no—
154C2×C4C2×C4×D17311yescharacteristic
155C2×C4D4×D17312yes—
156C8—42no—
157C8—6922no—
158C22—42no—
159C22—41no—
160C22—3461no—
161C22—3462no—
162C22—6921no—
163C22—6922no—
164C22C22⋊C4×D17311yescharacteristic
165C4—22no—
166C4—21no—
167C4—41no—
168C4—3462no—
169C4—1731no—
170C4—1731no—
171C4—6921no—
172C2—21no—
173C2—81no—
174C2—6921no—
175C2—6922no—
176C2#99×D17311yescharacteristic
subgroupG/H
centerC2#99×D173
commutatorC2×C4×C173C22×C4
FrattiniC2×Q8C22×D173
Fitting#355×C173C2
socleC2×C173C2×#99
prime radical#26012040C2
cosocleC2×Q8×C173C23
hypercenter#355D173
hypocenterC173#2132
LCSG ▷ C2×C4×C173 ▷ C22×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #355

Complements

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

NH ≅ G/Ntype
2. #26011474C2semidirect
3. #26011289C2semidirect
4. #26011288C2semidirect
5. #355×C173C2semidirect
88. C173#2132semidirect
94. #355D173semidirect

Maximal subgroups

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

subgroupindex
#260120402
#260114742
#260112892
#260112882
#355×C1732
#2132173