Finite Group Databasebeta

← D173⋊#353 #36369829

Subgroups — D173⋊#353 #36369829

Order 22144.

368 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 54 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
1D173⋊#99C212yesmaximal, complemented
2#26012396C212yesmaximal, complemented
3#26012395C212yesmaximal, complemented
4#127×D173C211yesmaximal, characteristic
5C2×C22⋊#10621924—22no—
6#19149738C2214yescomplemented
7#19149737—22no—
8C2×D4×D173—22no—
9C2×D4×D173C411yescharacteristic, complemented
10C2×D4×D173C2211yescharacteristic
11#19149676C2211yescharacteristic
12#19149675—22no—
13#19149675C411yescharacteristic, complemented
14#19149675C2211yescharacteristic
15C42×D173C411yescharacteristic, complemented
16#127×C173C411yescharacteristic, complemented
17C22×#10621924—42no—
18C22⋊#10621924—24no—
19C22⋊#10621924—42no—
20C23×D173—21no—
21C23×D173D412yes—
22C2×D692—22no—
23C2×D692C2×C411yescharacteristic, complemented
24C2×D692C2311yescharacteristic
25C2×C22⋊D173—22no—
26C2×C22⋊D173—21no—
27C2×C22⋊D173—22no—
28C2×C22⋊D173D412yes—
29D4×D173—41no—
30D4×D173—41no—
31D4×D173—48no—
32D4×D173—41no—
33D4×D173—41no—
34C2×C4×D173—21no—
35C2×C4×D173C2×C411yescharacteristic
36C4×Dic173—21no—
37C4×Dic173C2×C411yescharacteristic, complemented
38C2×D4×C173—22no—
39C2×D4×C173C2×C411yescharacteristic
40C2×D4×C173C2×C411yescharacteristic, complemented
41C42×C173C2×C411yescharacteristic, complemented
42C2×#10621924—44no—
43C2×#10621924—84no—
44C22×D173—22no—
45C22×D173—22no—
46C22×D173—21no—
47C22×D173—44no—
48C22×D173—44no—
49C22×D173—41no—
50C22×D173—41no—
51C22×D173C22⋊C411yescharacteristic
52C22×D173C2×D412yes—
53D692—41no—
54D692—41no—
55D692—44no—
56C22⋊D173—42no—
57C22⋊D173—44no—
58C22⋊D173—44no—
59C22⋊D173—42no—
60C2×Dic173—21no—
61C2×Dic173C22⋊C411yescharacteristic
62C4×D173—24no—
63C4×D173—21no—
64C4×D173—21no—
65C23×C173—21no—
66C23×C173C22⋊C412yes—
67D4×C173—41no—
68D4×C173—41no—
69D4×C173—44no—
70C2×C4×C173—21no—
71C2×C4×C173C22×C411yescharacteristic
72C22×C173—22no—
73C22×C173—42no—
74C22×C173—42no—
75C22×C173—41no—
76C22×C173C2×C22⋊C411yescharacteristic
77C2×D173—22no—
78C2×D173—22no—
79C2×D173—44no—
80C2×D173—42no—
81C2×D173—42no—
82C2×D173—41no—
83C2×D173—82no—
84C2×D173—82no—
85C2×D173—82no—
86C2×D173#9911yescharacteristic
87C2×D173#9911yescharacteristic
88#10621924—84no—
89C4×C173—22no—
90C4×C173—21no—
91Dic173—22no—
92Dic173—21no—
93C2×C173—21no—
94C2×C173—42no—
95C2×C173—81no—
96C2×C173C2×#9911yescharacteristic
97D173—21no—
98D173—42no—
99D173—81no—
100D173#35312yescomplemented
101C173C2×#35311yescharacteristic, complemented, Sylow
102C2×#353—1731nomaximal, Sylow
103C2×#127—1731no—
104C2×#99—1732no—
105#353—1732no—
106#353—1732no—
107C22×D4—3462no—
108C22×D4—1731no—
109C22×D4—1731no—
110#127—3462no—
111#127—1731no—
112#127—1731no—
113#127—1731no—
114#127#1062192411yescharacteristic, complemented
115C2×C22⋊C4—3462no—
116C2×C42—1731no—
117#99—3462no—
118#99—1734no—
119C24—3461no—
120C24—1732no—
121C2×D4—22no—
122C2×D4—3462no—
123C2×D4—3462no—
124C2×D4—3462no—
125C2×D4—3461no—
126C2×D4—1732no—
127C2×D4—1731no—
128C2×D4—1731no—
129C2×D4—6921no—
130C2×D4—6921no—
131C2×D4—6921no—
132C2×D4—6921no—
133C2×D4—6928no—
134C2×D4C2×#1062192411yescharacteristic
135C2×D4C2×#1062192411yescharacteristic, complemented
136C22×C4—3461no—
137C22×C4—1731no—
138C22×C4—6922no—
139C22⋊C4—3464no—
140C22⋊C4—6922no—
141C42—3461no—
142C42—1731no—
143C42C2×#1062192411yescharacteristic, complemented
144C23—21no—
145C23—3462no—
146C23—3462no—
147C23—3461no—
148C23—1731no—
149C23—1732no—
150C23—6924no—
151C23—6924no—
152C23—6921no—
153C23—6921no—
154C23C22⋊#1062192412yes—
155D4—41no—
156D4—41no—
157D4—44no—
158D4—6921no—
159D4—6921no—
160D4—6924no—
161D4—6922no—
162D4—6924no—
163D4—6924no—
164D4—6922no—
165C2×C4—21no—
166C2×C4—3464no—
167C2×C4—3461no—
168C2×C4—3461no—
169C2×C4—3461no—
170C2×C4—13844no—
171C2×C4—1731no—
172C2×C4—6924no—
173C2×C4C22×#1062192411yescharacteristic
174C22—22no—
175C22—42no—
176C22—42no—
177C22—41no—
178C22—3462no—
179C22—3462no—
180C22—13842no—
181C22—13842no—
182C22—13842no—
183C22—1731no—
184C22—1731no—
185C22—6922no—
186C22—6922no—
187C22—6924no—
188C22—6921no—
189C22C2×C22⋊#1062192411yescharacteristic
190C4—22no—
191C4—21no—
192C4—3462no—
193C4—3461no—
194C4—13844no—
195C2—21no—
196C2—42no—
197C2—81no—
198C2—3461no—
199C2—13841no—
200C2—1732no—
201C2—6922no—
202C2D173⋊#9911yescharacteristic
subgroupG/H
centerC2D173⋊#99
commutatorC2×C4×C173C22×C4
FrattiniC2×C4C22×#10621924
Fitting#127×C173C4
socleC2×C173C2×#99
prime radical#127×D173C2
cosocleC2×D692C23
hypercenter#127#10621924
hypocenterC173C2×#353
LCSG ▷ C2×C4×C173 ▷ C22×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #127

Complements

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

NH ≅ G/Ntype
1. D173⋊#99C2semidirect
2. #26012396C2semidirect
3. #26012395C2semidirect
6. #19149738C22semidirect
9. C2×D4×D173C4semidirect
13. #19149675C4semidirect
15. C42×D173C4semidirect
16. #127×C173C4semidirect
23. C2×D692C2×C4semidirect
37. C4×Dic173C2×C4semidirect
40. C2×D4×C173C2×C4semidirect
41. C42×C173C2×C4semidirect
100. D173#353semidirect
101. C173C2×#353semidirect
114. #127#10621924semidirect
135. C2×D4C2×#10621924semidirect
143. C42C2×#10621924semidirect

Maximal subgroups

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

subgroupindex
D173⋊#992
#260123962
#260123952
#127×D1732
C2×#353173