Finite Group Databasebeta

← C2×#26012394 #36369813

Subgroups — C2×#26012394 #36369813

Order 22144.

278 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 58 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
1C2×#19149739C211yesmaximal, characteristic
2C2×#19149737C211yesmaximal, characteristic
3#26012394C214yesmaximal, direct product factor
4C2×#19149663C211yesmaximal, characteristic
5C2×#14583835—21no—
6C2×C22⋊#10621924—21no—
7#19149739—21no—
8#19149739C2212yes—
9#19149737—21no—
10#19149737C2212yes—
11C22×C22⋊D173C2211yescharacteristic
12C23×Dic173C411yescharacteristic, complemented
13#19149663—21no—
14#19149663—21no—
15#19149663C412yescomplemented
16#19149663C2212yes—
17C2×C4⋊Dic173—21no—
18C2×#14583803—21no—
19C2×C22⋊C4×C173C411yescharacteristic, complemented
20C22×#10621924—41no—
21#14583835—22no—
22#14583835—41no—
23C2×#13063443—41no—
24C22⋊#10621924—22no—
25C22⋊#10621924—41no—
26C23×D173D411yescharacteristic
27C2×C22⋊D173—41no—
28C2×C22⋊D173—41no—
29C2×C22⋊D173C2×C411yescharacteristic
30C2×C22⋊D173D411yescharacteristic
31C2×C22⋊D173D411yescharacteristic
32C2×C22⋊D173C2311yescharacteristic
33C22×Dic173—21no—
34C22×Dic173—21no—
35C22×Dic173—41no—
36C22×Dic173—41no—
37C22×Dic173C2×C412yescomplemented
38C22×Dic173D411yescharacteristic
39C4⋊Dic173—22no—
40C4⋊Dic173—22no—
41#14583803—22no—
42#14583803—22no—
43C24×C173C2×C411yescharacteristic
44C22×C4×C173—21no—
45C22⋊C4×C173—21no—
46C22⋊C4×C173C2×C412yescomplemented
47C2×#10621924—42no—
48C2×#10621924—82no—
49#13063443—42no—
50C22×D173—21no—
51C22×D173—21no—
52C22×D173—42no—
53C22×D173C22⋊C411yescharacteristic
54C22×D173C2×D411yescharacteristic
55C22⋊D173—41no—
56C22⋊D173—41no—
57C22⋊D173—41no—
58C22⋊D173—41no—
59C2×Dic173—21no—
60C2×Dic173—21no—
61C2×Dic173—22no—
62C2×Dic173—42no—
63C2×Dic173—41no—
64C2×Dic173—41no—
65C2×Dic173—42no—
66C2×Dic173C22⋊C411yescharacteristic
67C2×Dic173C2×D411yescharacteristic
68C23×C173—21no—
69C23×C173—21no—
70C23×C173—42no—
71C23×C173C22⋊C411yescharacteristic
72C23×C173C22⋊C411yescharacteristic
73C23×C173C22×C411yescharacteristic
74C2×C4×C173—22no—
75C2×C4×C173—42no—
76C22×C173—22no—
77C22×C173—21no—
78C22×C173—21no—
79C22×C173—42no—
80C22×C173—41no—
81C22×C173—41no—
82C22×C173—42no—
83C22×C173#9911yescharacteristic
84C22×C173#9911yescharacteristic
85C22×C173C2×C22⋊C411yescharacteristic
86C2×D173—21no—
87C2×D173—21no—
88C2×D173—42no—
89C2×D173—41no—
90C2×D173—41no—
91C2×D173—81no—
92#10621924—82no—
93C4×C173—42no—
94Dic173—21no—
95Dic173—21no—
96Dic173—42no—
97C2×C173—21no—
98C2×C173—21no—
99C2×C173—41no—
100C2×C173—41no—
101C2×C173#35212yescomplemented
102C2×C173C2×#9911yescharacteristic
103D173—41no—
104D173—41no—
105C173C2×#35211yescharacteristic, complemented, Sylow
106C2×#352—1731nomaximal, Sylow
107C2×#123—1731no—
108C2×#100—1731no—
109C2×#99—1731no—
110#352—1734no—
111C22×D4—1731no—
112C23×C4—1731no—
113C2×M16—3461no—
114#123—3461no—
115#123—3461no—
116#123—1732no—
117#123—1732no—
118C2×C4⋊C4—3461no—
119C2×C22⋊C4—3461no—
120C2×C22⋊C4—3461no—
121C2×C22⋊C4#1062192411yescharacteristic, complemented
122#100—3461no—
123#100—1732no—
124#99—3461no—
125#99—1732no—
126C24—1731no—
127C24C2×#1062192411yescharacteristic
128C2×D4—1731no—
129C2×D4—1731no—
130C2×D4—1731no—
131C2×D4—1731no—
132C2×D4—6921no—
133C2×D4—6921no—
134C22×C4—21no—
135C22×C4—3461no—
136C22×C4—3461no—
137C22×C4—1731no—
138C22×C4—1732no—
139C22×C4—6921no—
140C22×C4—6921no—
141C22×C4—6921no—
142M16—3462no—
143M16—6921no—
144C2×C8—6921no—
145C4⋊C4—3462no—
146C4⋊C4—3462no—
147C22⋊C4—21no—
148C22⋊C4—3462no—
149C22⋊C4—3462no—
150C22⋊C4—3462no—
151C22⋊C4—6921no—
152C22⋊C4C2×#1062192412yescomplemented
153C23—21no—
154C23—21no—
155C23—42no—
156C23—3461no—
157C23—3461no—
158C23—1731no—
159C23—1731no—
160C23—6922no—
161C23C22⋊#1062192411yescharacteristic
162C23C22⋊#1062192411yescharacteristic
163C23C22×#1062192411yescharacteristic
164D4—6921no—
165D4—6921no—
166D4—6921no—
167D4—6921no—
168C2×C4—22no—
169C2×C4—42no—
170C2×C4—3461no—
171C2×C4—3461no—
172C2×C4—3462no—
173C2×C4—13842no—
174C2×C4—1731no—
175C2×C4—1731no—
176C2×C4—6922no—
177C2×C4—6922no—
178C2×C4—6921no—
179C2×C4—6921no—
180C2×C4—6922no—
181C8—6922no—
182C22—22no—
183C22—21no—
184C22—21no—
185C22—41no—
186C22—41no—
187C22—42no—
188C22—42no—
189C22—3461no—
190C22—3461no—
191C22—13841no—
192C22—6921no—
193C22—6921no—
194C22—6922no—
195C22#1914973811yescharacteristic
196C22#1914973811yescharacteristic
197C22C2×C22⋊#1062192411yescharacteristic
198C4—42no—
199C4—3461no—
200C4—3461no—
201C4—13842no—
202C4—6922no—
203C2—21no—
204C2—21no—
205C2—41no—
206C2—41no—
207C2—6921no—
208C2—6921no—
209C2#2601239412yesdirect product factor
210C2C2×#1914973811yescharacteristic
subgroupG/H
centerC22#19149738
commutatorC23×C173C22×C4
FrattiniC23C22×#10621924
FittingC2×C22⋊C4×C173C4
socleC22×C173#99
prime radicalC22×C22⋊D173C22
cosocleC2×C22⋊D173C23
hypercenterC2×C22⋊C4#10621924
hypocenterC173C2×#352
LCSG ▷ C23×C173 ▷ C22×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C23 ◁ C24 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
3. #26012394C2direct (N × H)
12. C23×Dic173C4semidirect
15. #19149663C4semidirect
19. C2×C22⋊C4×C173C4semidirect
37. C22×Dic173C2×C4semidirect
46. C22⋊C4×C173C2×C4semidirect
101. C2×C173#352semidirect
105. C173C2×#352semidirect
121. C2×C22⋊C4#10621924semidirect
152. C22⋊C4C2×#10621924semidirect
209. C2#26012394direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#191497392
C2×#191497372
#260123942
C2×#191496632
C2×#352173