Finite Group Databasebeta

← #36355253

Subgroups — #36355253

Order 22144.

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

normal subgroups only

This group has 262 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#26012059C211yesmaximal, characteristic
2#26011474C212yesmaximal, complemented
3#26011288C211yesmaximal, characteristic, complemented
4#26011287C212yesmaximal, complemented
5#355×C173C211yesmaximal, characteristic, complemented
6C2×#14583827C2211yescharacteristic
7C2×D4×D173C411yescharacteristic
8M16×D173—22no—
9#19149676C411yescharacteristic
10#19149675C411yescharacteristic
11#19149668C2211yescharacteristic
12#19149652—22no—
13#19149649—22no—
14#19149636C411yescharacteristic
15#19149598C2212yescomplemented
16#19149595—22no—
17#124×C173C2211yescharacteristic
18#101×C173C2212yescomplemented
19C23×D173—21no—
20#14583827—42no—
21C2×D692—22no—
22C2×D692C2×C411yescharacteristic
23C2×D692C2×C411yescharacteristic
24C2×D692D412yes—
25C2×C22⋊D173—22no—
26C2×C22⋊D173—21no—
27D4×D173—42no—
28C2×C4×D173C2×C411yescharacteristic
29C2×C4×D173D412yes—
30#14583810—22no—
31#14583809—42no—
32C8×D173—42no—
33#14583805—21no—
34#14583803—21no—
35#14583803—22no—
36C4×Dic173C2×C411yescharacteristic
37C2×Q8×C173C2311yescharacteristic
38C2×D4×C173C2×C411yescharacteristic
39M16×C173—22no—
40C22⋊C4×C173—22no—
41C42×C173C2×C411yescharacteristic
42C22×D173—21no—
43C22×D173—42no—
44C22×D173—82no—
45C22×D173C22⋊C411yescharacteristic
46C22×D173C22⋊C412yes—
47D692—41no—
48D692—42no—
49D692—41no—
50D692—82no—
51C22⋊D173—42no—
52C22⋊D173—82no—
53C2×Dic173—21no—
54C2×Dic173C22⋊C411yescharacteristic
55C4×D173—22no—
56C4×D173—22no—
57C4×D173—22no—
58#13063436—42no—
59C23×C173—21no—
60Q8×C173—41no—
61D4×C173—41no—
62C2×C4×C173—21no—
63C2×C4×C173C22×C411yescharacteristic
64C2×C4×C173C2×D412yes—
65C8×C173—42no—
66C22×C173—42no—
67C22×C173—41no—
68C22×C173C2×C22⋊C411yescharacteristic
69C2×D173—21no—
70C2×D173—22no—
71C2×D173—42no—
72C2×D173—41no—
73C2×D173—42no—
74C2×D173—41no—
75C2×D173—84no—
76C4×C173—22no—
77C4×C173—21no—
78C4×C173—41no—
79Dic173—41no—
80Dic173#9912yes—
81C2×C173—21no—
82C2×C173—81no—
83C2×C173C2×#9911yescharacteristic
84D173—42no—
85D173—41no—
86D173—81no—
87C173#213411yescharacteristic, complemented, Sylow
88#2134—1731nomaximal, Sylow
89#535—1731no—
90#413—1732no—
91#355—1731no—
92#355—1732no—
93#355D17311yescharacteristic, complemented
94C2×#41—1731no—
95C22×D4—1731no—
96C2×M16—3462no—
97#127—1731no—
98#127—1731no—
99#124—1731no—
100#124C2×D17311yescharacteristic
101#121—3462no—
102#120—3462no—
103#117—1731no—
104#101—1732no—
105#101C2×D17312yescomplemented
106#100—3462no—
107C24—3461no—
108#41—6922no—
109C2×Q8C22×D17311yescharacteristic
110C2×D4—3462no—
111C2×D4—3462no—
112C2×D4—3461no—
113C2×D4—1731no—
114C2×D4—1731no—
115C2×D4—1732no—
116C2×D4—6922no—
117C2×D4C4×D17311yescharacteristic
118C22×C4—1731no—
119C22×C4—1732no—
120M16—22no—
121M16—3462no—
122M16—6922no—
123C2×C8—6922no—
124C4⋊C4—3461no—
125C22⋊C4—22no—
126C22⋊C4—3461no—
127C22⋊C4—3462no—
128C42—1731no—
129C42C4×D17311yescharacteristic
130C23—21no—
131C23—3461no—
132C23—13842no—
133C23—1731no—
134C23—1732no—
135C23—6922no—
136Q8—41no—
137D4—41no—
138D4—13842no—
139D4—13842no—
140D4—6922no—
141D4—6921no—
142D4—6921no—
143D4—6922no—
144C2×C4—21no—
145C2×C4—3462no—
146C2×C4—3462no—
147C2×C4—3462no—
148C2×C4—3461no—
149C2×C4—1731no—
150C2×C4C2×C4×D17311yescharacteristic
151C2×C4D4×D17312yes—
152C8—42no—
153C8—6922no—
154C22—42no—
155C22—41no—
156C22—3461no—
157C22—3462no—
158C22—13844no—
159C22—6922no—
160C22—6922no—
161C22—6921no—
162C22—6921no—
163C22C22⋊C4×D17311yescharacteristic
164C4—22no—
165C4—21no—
166C4—41no—
167C4—1732no—
168C4—6921no—
169C2—21no—
170C2—81no—
171C2—13841no—
172C2—6921no—
173C2—6922no—
174C2#99×D17311yescharacteristic
subgroupG/H
centerC2#99×D173
commutatorC2×C4×C173C22×C4
FrattiniC2×Q8C22×D173
Fitting#355×C173C2
socleC2×C173C2×#99
prime radical#26012059C2
cosocleC2×Q8×C173C23
hypercenter#355D173
hypocenterC173#2134
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. #26011288C2semidirect
4. #26011287C2semidirect
5. #355×C173C2semidirect
15. #19149598C22semidirect
18. #101×C173C22semidirect
87. C173#2134semidirect
93. #355D173semidirect
105. #101C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260120592
#260114742
#260112882
#260112872
#355×C1732
#2134173