Finite Group Databasebeta

← C2×#26011281 #36355199

Subgroups — C2×#26011281 #36355199

Order 22144.

278 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 77 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×#19149596C211yesmaximal, characteristic
2C2×#19149594C211yesmaximal, characteristic
3#26011281C214yesmaximal, direct product factor
4C2×#123×C173C211yesmaximal, characteristic
5C2×#14583810—21no—
6C2×C22⋊Dic173—21no—
7#19149596—21no—
8#19149596C2212yes—
9#19149594—21no—
10#19149594C2212yes—
11C22×D4×C173C2211yescharacteristic
12C23×C4×C173C411yescharacteristic, complemented
13#123×C173—21no—
14#123×C173—21no—
15#123×C173C412yescomplemented
16#123×C173C2212yes—
17C2×C4⋊C4×C173—21no—
18C2×C22⋊C4×C173—21no—
19C2×C22⋊C4×C173C411yescharacteristic, complemented
20C22×Dic173—41no—
21#14583810—22no—
22#14583810—41no—
23C2×#13063436—41no—
24C22⋊Dic173—22no—
25C22⋊Dic173—41no—
26C24×C173C2×C411yescharacteristic
27C24×C173D411yescharacteristic
28C2×D4×C173—41no—
29C2×D4×C173—41no—
30C2×D4×C173C2×C411yescharacteristic
31C2×D4×C173D411yescharacteristic
32C2×D4×C173D411yescharacteristic
33C2×D4×C173C2311yescharacteristic
34C22×C4×C173—21no—
35C22×C4×C173—21no—
36C22×C4×C173—21no—
37C22×C4×C173—41no—
38C22×C4×C173—41no—
39C22×C4×C173C2×C412yescomplemented
40C22×C4×C173D411yescharacteristic
41C4⋊C4×C173—22no—
42C4⋊C4×C173—22no—
43C22⋊C4×C173—22no—
44C22⋊C4×C173—21no—
45C22⋊C4×C173—22no—
46C22⋊C4×C173C2×C412yescomplemented
47C2×Dic173—42no—
48C2×Dic173—82no—
49#13063436—42no—
50C23×C173—21no—
51C23×C173—21no—
52C23×C173—21no—
53C23×C173—21no—
54C23×C173—42no—
55C23×C173—42no—
56C23×C173C22⋊C411yescharacteristic
57C23×C173C22⋊C411yescharacteristic
58C23×C173C22⋊C411yescharacteristic
59C23×C173C22×C411yescharacteristic
60C23×C173C2×D411yescharacteristic
61D4×C173—41no—
62D4×C173—41no—
63D4×C173—41no—
64D4×C173—41no—
65C2×C4×C173—21no—
66C2×C4×C173—21no—
67C2×C4×C173—22no—
68C2×C4×C173—22no—
69C2×C4×C173—42no—
70C2×C4×C173—42no—
71C2×C4×C173—41no—
72C2×C4×C173—41no—
73C2×C4×C173—42no—
74C2×C4×C173C22⋊C411yescharacteristic
75C2×C4×C173C2×D411yescharacteristic
76C22×C173—22no—
77C22×C173—21no—
78C22×C173—21no—
79C22×C173—21no—
80C22×C173—21no—
81C22×C173—42no—
82C22×C173—41no—
83C22×C173—41no—
84C22×C173—41no—
85C22×C173—41no—
86C22×C173—42no—
87C22×C173—42no—
88C22×C173—81no—
89C22×C173#9911yescharacteristic
90C22×C173#9911yescharacteristic
91C22×C173C2×C22⋊C411yescharacteristic
92C4×C173—21no—
93C4×C173—21no—
94C4×C173—42no—
95C4×C173—42no—
96Dic173—82no—
97C2×C173—21no—
98C2×C173—21no—
99C2×C173—41no—
100C2×C173—41no—
101C2×C173—41no—
102C2×C173—41no—
103C2×C173#35212yescomplemented
104C2×C173C2×#9911yescharacteristic
105C173C2×#35211yescharacteristic, complemented, Sylow
106C2×#352—1731nomaximal, Sylow
107C2×#123D17311yescharacteristic
108C2×#100—1731no—
109C2×#99—1731no—
110#352—1734no—
111C22×D4C2×D17311yescharacteristic
112C23×C4Dic17311yescharacteristic, complemented
113C2×M16—3461no—
114#123—21no—
115#123—21no—
116#123Dic17312yescomplemented
117#123C2×D17312yes—
118C2×C4⋊C4—21no—
119C2×C22⋊C4—21no—
120C2×C22⋊C4—3461no—
121C2×C22⋊C4Dic17311yescharacteristic, complemented
122#100—3461no—
123#100—1732no—
124#99—3461no—
125#99—1732no—
126C24C2×Dic17311yescharacteristic
127C24C22⋊D17311yescharacteristic
128C2×D4—41no—
129C2×D4—41no—
130C2×D4C2×Dic17311yescharacteristic
131C2×D4C22⋊D17311yescharacteristic
132C2×D4C22⋊D17311yescharacteristic
133C2×D4C22×D17311yescharacteristic
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—21no—
137C22×C4—41no—
138C22×C4—41no—
139C22×C4—6921no—
140C22×C4C2×Dic17312yescomplemented
141C22×C4C22⋊D17311yescharacteristic
142M16—3462no—
143M16—6921no—
144C2×C8—6921no—
145C4⋊C4—22no—
146C4⋊C4—22no—
147C22⋊C4—22no—
148C22⋊C4—22no—
149C22⋊C4—21no—
150C22⋊C4—3462no—
151C22⋊C4—6921no—
152C22⋊C4C2×Dic17312yescomplemented
153C23—21no—
154C23—21no—
155C23—21no—
156C23—21no—
157C23—42no—
158C23—42no—
159C23C22⋊Dic17311yescharacteristic
160C23C22⋊Dic17311yescharacteristic
161C23C22⋊Dic17311yescharacteristic
162C23C22×Dic17311yescharacteristic
163C23C2×C22⋊D17311yescharacteristic
164D4—41no—
165D4—41no—
166D4—41no—
167D4—41no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—42no—
173C2×C4—42no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—42no—
177C2×C4—13842no—
178C2×C4—6922no—
179C2×C4C22⋊Dic17311yescharacteristic
180C2×C4C2×C22⋊D17311yescharacteristic
181C8—6922no—
182C22—22no—
183C22—21no—
184C22—21no—
185C22—21no—
186C22—21no—
187C22—41no—
188C22—41no—
189C22—41no—
190C22—41no—
191C22—42no—
192C22—42no—
193C22—42no—
194C22—81no—
195C22#1914959411yescharacteristic
196C22#1914959411yescharacteristic
197C22C2×C22⋊Dic17311yescharacteristic
198C4—21no—
199C4—21no—
200C4—42no—
201C4—42no—
202C4—13842no—
203C2—21no—
204C2—21no—
205C2—41no—
206C2—41no—
207C2—41no—
208C2—41no—
209C2#2601128112yesdirect product factor
210C2C2×#1914959411yescharacteristic
subgroupG/H
centerC22#19149594
commutatorC23×C173C22×C4
FrattiniC2×D4C22×D173
FittingC2×#123×C173C2
socleC22×C173#99
prime radicalC22×D4×C173C22
cosocleC2×D4×C173C23
hypercenterC2×#123D173
hypocenterC173C2×#352
LCSG ▷ C23×C173 ▷ C22×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
3. #26011281C2direct (N × H)
12. C23×C4×C173C4semidirect
15. #123×C173C4semidirect
19. C2×C22⋊C4×C173C4semidirect
39. C22×C4×C173C2×C4semidirect
46. C22⋊C4×C173C2×C4semidirect
103. C2×C173#352semidirect
105. C173C2×#352semidirect
112. C23×C4Dic173semidirect
116. #123Dic173semidirect
121. C2×C22⋊C4Dic173semidirect
140. C22×C4C2×Dic173semidirect
152. C22⋊C4C2×Dic173semidirect
209. C2#26011281direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#191495962
C2×#191495942
#260112812
C2×#123×C1732
C2×#352173