Finite Group Databasebeta

← D173⋊#324 #36368792

Subgroups — D173⋊#324 #36368792

Order 22144.

328 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 328 conjugacy classes of subgroups — too many to draw. Only its 84 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⋊(C22⋊C8)C212yesmaximal
2#26012337C212yesmaximal, complemented
3#26012336C212yesmaximal, complemented
4C2×C4⋊C4×D173C211yesmaximal, characteristic
5C2×D173⋊C8—22no—
6#19149736—22no—
7C22⋊#13063443C2214yes—
8C22×C4×D173C411yescharacteristic
9C22×C4×D173C411yescharacteristic
10C22×C4×D173C2211yescharacteristic
11C2×C4⋊Dic173C2211yescharacteristic
12C2×#14583805C411yescharacteristic
13C2×#14583805C2211yescharacteristic
14C4⋊C4×D173—22no—
15C4⋊C4×D173—22no—
16C2×C4⋊C4×C173C411yescharacteristic
17D173⋊C8—44no—
18C2×#13063443—24no—
19C23×D173C2×C411yescharacteristic
20C22×Dic173C2×C411yescharacteristic
21C22×Dic173C2×C411yescharacteristic
22C22×Dic173C2311yescharacteristic
23C2×C4×D173—21no—
24C2×C4×D173—21no—
25C2×C4×D173—22no—
26C2×C4×D173—21no—
27C2×C4×D173—21no—
28C2×C4×D173—42no—
29C2×C4×D173—42no—
30C2×C4×D173C814yescomplemented
31C2×C4×D173D412yes—
32C2×C4×D173D412yes—
33C4⋊Dic173—22no—
34#14583805—22no—
35#14583805—22no—
36C22×C4×C173C2×C411yescharacteristic
37C22×C4×C173C2×C411yescharacteristic
38C22×C4×C173C2×C411yescharacteristic
39C4⋊C4×C173—22no—
40#13063443—44no—
41C22×D173—22no—
42C22×D173—22no—
43C22×D173C22⋊C411yescharacteristic
44C22×D173C22⋊C411yescharacteristic
45C22×D173C2×C812yes—
46C22×D173M1612yes—
47C2×Dic173—21no—
48C2×Dic173—21no—
49C2×Dic173—21no—
50C2×Dic173—41no—
51C2×Dic173—41no—
52C2×Dic173C2×C812yescomplemented
53C2×Dic173C2×D412yes—
54C4×D173—24no—
55C4×D173—24no—
56C4×D173—24no—
57C4×D173—42no—
58C4×D173—42no—
59C4×D173—42no—
60C23×C173C22×C411yescharacteristic
61C2×C4×C173—21no—
62C2×C4×C173—21no—
63C2×C4×C173—21no—
64C2×C4×C173—41no—
65C2×C4×C173—41no—
66C2×C4×C173C22⋊C412yes—
67C2×C4×C173C2×C812yescomplemented
68C22×C173—21no—
69C22×C173—21no—
70C22×C173C2×C22⋊C411yescharacteristic
71C22×C173C22×C811yescharacteristic
72C22×C173C2×M1611yescharacteristic
73C2×D173—21no—
74C2×D173—21no—
75C2×D173—24no—
76C2×D173—24no—
77C2×D173C22⋊C812yes—
78C2×D173C22⋊C812yes—
79C2×D173#9912yes—
80C2×D173#10112yes—
81C4×C173—22no—
82C4×C173—22no—
83C4×C173—41no—
84Dic173—22no—
85Dic173—22no—
86Dic173—41no—
87C2×C173—21no—
88C2×C173—21no—
89C2×C173C2×C22⋊C811yescharacteristic
90C2×C173C2×#9911yescharacteristic
91C2×C173C2×#10111yescharacteristic
92D173—22no—
93D173#32414yescomplemented
94C173C2×#32411yescharacteristic, complemented, Sylow
95C2×#324—1731nomaximal, Sylow
96C22×C4⋊C4—1731no—
97C2×C22⋊C8—1732no—
98#324—1732no—
99#324—1732no—
100C23×C4—1731no—
101C23×C4—1731no—
102C23×C4—1731no—
103C22×C8—3462no—
104C2×C4⋊C4—3462no—
105C2×C4⋊C4—3462no—
106C2×C4⋊C4—1731no—
107C2×C4⋊C4—1731no—
108C2×C4⋊C4—1731no—
109C2×C4⋊C4#1062192411yescharacteristic
110C22⋊C8—3462no—
111C22⋊C8—1734no—
112C24—1731no—
113C22×C4—3461no—
114C22×C4—3461no—
115C22×C4—3462no—
116C22×C4—3461no—
117C22×C4—3461no—
118C22×C4—1731no—
119C22×C4—1732no—
120C22×C4—1734no—
121C22×C4—1731no—
122C22×C4—1731no—
123C22×C4—1732no—
124C22×C4—6922no—
125C22×C4—6922no—
126C22×C4C2×#1062192411yescharacteristic
127C22×C4C2×#1062192411yescharacteristic
128C22×C4C2×#1062192411yescharacteristic
129C2×C8—3464no—
130C2×C8—6924no—
131C4⋊C4—22no—
132C4⋊C4—3462no—
133C4⋊C4—3462no—
134C4⋊C4—3462no—
135C23—3462no—
136C23—3462no—
137C23—1731no—
138C23—1732no—
139C23—1732no—
140C23—1731no—
141C23C22×#1062192411yescharacteristic
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—41no—
146C2×C4—41no—
147C2×C4—3464no—
148C2×C4—3464no—
149C2×C4—3464no—
150C2×C4—3461no—
151C2×C4—3461no—
152C2×C4—3461no—
153C2×C4—1732no—
154C2×C4—1732no—
155C2×C4—6922no—
156C2×C4—6922no—
157C2×C4—6922no—
158C2×C4—6921no—
159C2×C4—6921no—
160C2×C4C22⋊#1062192412yes—
161C2×C4D173⋊C812yescomplemented
162C8—6924no—
163C22—21no—
164C22—21no—
165C22—3464no—
166C22—3464no—
167C22—3461no—
168C22—3461no—
169C22—1732no—
170C22—1732no—
171C22—1732no—
172C22—1732no—
173C22C2×C22⋊#1062192411yescharacteristic
174C22C2×D173⋊C811yescharacteristic
175C22D173⋊M1611yescharacteristic
176C4—22no—
177C4—22no—
178C4—41no—
179C4—3462no—
180C4—3462no—
181C4—6921no—
182C2—21no—
183C2—21no—
184C2—3462no—
185C2—1734no—
186C2D173⋊(C22⋊C8)11yescharacteristic
187C2D173⋊#9911yescharacteristic
188C2D173⋊#10111yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10621924
commutatorC22×C173C22×C8
FrattiniC23C22×#10621924
FittingC2×C4⋊C4×C173C4
socleC22×C173C2×C22⋊C4
prime radicalC23×D173C2×C4
cosocleC22×Dic173C23
hypercenterC2×C4⋊C4#10621924
hypocenterC173C2×#324
LCSG ▷ C22×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #26012337C2semidirect
3. #26012336C2semidirect
30. C2×C4×D173C8semidirect
52. C2×Dic173C2×C8semidirect
67. C2×C4×C173C2×C8semidirect
93. D173#324semidirect
94. C173C2×#324semidirect
161. C2×C4D173⋊C8semidirect

Maximal subgroups

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

subgroupindex
D173⋊(C22⋊C8)2
#260123372
#260123362
C2×C4⋊C4×D1732
C2×#324173