Finite Group Databasebeta

← C2×H(5,𝔽2)×D109 #28377832

Subgroups — C2×H(5,𝔽2)×D109 #28377832

Order 13952.

6560 conjugacy classes of subgroups; only the 3213 normal subgroups are shown.

normal subgroups only

This group has 6560 conjugacy classes of subgroups — too many to draw. Only its 3213 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×#16812745C216yesmaximal, complemented
2C2×#16812744C219yesmaximal, complemented
3H(5,𝔽2)×D109C2132yesmaximal, direct product factor
4C2×#41×D109C216yesmaximal, complemented
5C22×D4×D109C219yesmaximal, complemented
6C2×H(5,𝔽2)×C109C211yesmaximal, characteristic, complemented
7C24×D109C2216yescomplemented
8#16812745C22196yescomplemented
9#16812744C221144yescomplemented
10C2×#13182030C2216yescomplemented
11C2×#13182029C22118yescomplemented
12C2×#13182028C22118yescomplemented
13#41×D109C22196yescomplemented
14C2×Q8×D109C2212yescomplemented
15C22×D436C2219yescomplemented
16C22×C22⋊D109C22118yescomplemented
17C2×D4×D109C22118yescomplemented
18C2×D4×D109C221144yescomplemented
19C2×D4×D109C22136yescomplemented
20C22×C4×D109C2219yescomplemented
21H(5,𝔽2)×C109C22116yescomplemented
22C2×#41×C109C2216yescomplemented
23C22×D4×C109C2219yescomplemented
24C23×D109—424no—
25C23×D109C2319yes—
26C23×D109C23148yescomplemented
27C23×D109C23118yescomplemented
28#13182030C23148yescomplemented
29#13182029C231144yescomplemented
30#13182028C231144yescomplemented
31C2×Dic218C2316yescomplemented
32Q8×D109C23116yescomplemented
33C2×D436C23118yescomplemented
34C2×D436C23172yescomplemented
35C2×C22⋊D109C23136yescomplemented
36C2×C22⋊D109C231144yescomplemented
37D4×D109C231144yescomplemented
38D4×D109C231288yescomplemented
39C22×Dic109C2319yescomplemented
40C2×C4×D109C2316yes—
41C2×C4×D109C23172yescomplemented
42C2×C4×D109C23118yescomplemented
43C24×C109C2316yescomplemented
44#41×C109C23148yescomplemented
45C2×Q8×C109C2312yescomplemented
46C2×D4×C109C23118yescomplemented
47C2×D4×C109C23172yescomplemented
48C22×C4×C109C2319yescomplemented
49C23×C109—412no—
50C23×C109C2419yes—
51C23×C109C24124yescomplemented
52C22×D109—236no—
53C22×D109—496no—
54C22×D109—436no—
55C22×D109C24136yes—
56C22×D109C2419yes—
57C22×D109C2411yescharacteristic
58C22×D109C24172yescomplemented
59Q8×C109C2418yescomplemented
60D4×C109C24172yescomplemented
61C2×C4×C109C2416yes—
62C2×C4×C109C24136yescomplemented
63C22⋊D109C241144yescomplemented
64C2×Dic109C2416yes—
65C2×Dic109C24136yescomplemented
66D436C24172yescomplemented
67C4×D109C24124yes—
68C4×D109C24172yescomplemented
69Dic218C24124yescomplemented
70C22×C109—218no—
71C22×C109—424no—
72C22×C109C25118yes—
73C22×C109C2511yescharacteristic
74C2×D109—218no—
75C2×D109—272no—
76C2×D109—472no—
77C2×D109H(5,𝔽2)14yesdirect product factor
78C2×D109C25118yes—
79C2×D109C2512yes—
80C4×C109C25112yes—
81Dic109C25112yes—
82C2×C109—218no—
83C2×C109C2×H(5,𝔽2)12yescomplemented
84C2×C109C2611yescharacteristic
85D109—218no—
86D109C2×H(5,𝔽2)14yesdirect product factor
87C22×H(5,𝔽2)—1091nomaximal, Sylow
88C109C22×H(5,𝔽2)11yescharacteristic, complemented, Sylow
89C2×H(5,𝔽2)—10932no—
90C2×H(5,𝔽2)—1096no—
91C2×H(5,𝔽2)—1099no—
92C2×H(5,𝔽2)D10911yescharacteristic, direct product factor
93C22×#41—1096no—
94C23×D4—1099no—
95C25—1096no—
96H(5,𝔽2)—10996no—
97H(5,𝔽2)—109144no—
98H(5,𝔽2)C2×D109116yesdirect product factor
99C2×#41—10996no—
100C2×#41—1096no—
101C2×#41—10918no—
102C2×#41—10918no—
103C2×#41C2×D10916yescomplemented
104C22×Q8—1092no—
105C22×D4—10918no—
106C22×D4—10918no—
107C22×D4—10936no—
108C22×D4—109144no—
109C22×D4—1099no—
110C22×D4C2×D10919yescomplemented
111C23×C4—1099no—
112C24—10918no—
113C24—10948no—
114C24—1099no—
115C24—43624no—
116C24C22×D10916yescomplemented
117#41—10948no—
118#41—109144no—
119#41—109144no—
120#41C22×D109148yescomplemented
121C2×Q8—10916no—
122C2×Q8—1096no—
123C2×Q8C22×D10912yescomplemented
124C2×D4—10936no—
125C2×D4—109144no—
126C2×D4—109144no—
127C2×D4—109288no—
128C2×D4—10918no—
129C2×D4—10972no—
130C2×D4C22×D109118yescomplemented
131C2×D4C22×D109172yescomplemented
132C22×C4—1099no—
133C22×C4—1096no—
134C22×C4—10918no—
135C22×C4—10972no—
136C22×C4C22×D10919yescomplemented
137C23—412no—
138C23—1091no—
139C23—10972no—
140C23—10936no—
141C23—1099no—
142C23—43636no—
143C23—43696no—
144C23—21836no—
145C23C23×D10919yes—
146C23C23×D109124yescomplemented
147Q8—10924no—
148Q8C23×D10918yescomplemented
149D4—109144no—
150D4—10972no—
151D4C23×D109172yescomplemented
152C2×C4—10936no—
153C2×C4—1096no—
154C2×C4—10924no—
155C2×C4—10972no—
156C2×C4C23×D10916yes—
157C2×C4C23×D109136yescomplemented
158C22—218no—
159C22—424no—
160C22—1094no—
161C22—1092no—
162C22—10918no—
163C22—43672no—
164C22—21872no—
165C22—21818no—
166C22C24×D109118yes—
167C22C24×D10911yescharacteristic
168C4—10912no—
169C4C24×D109112yes—
170C2—218no—
171C2—1094no—
172C2—21818no—
173C2H(5,𝔽2)×D10912yesdirect product factor
174C2C25×D10911yescharacteristic
subgroupG/H
centerC22C24×D109
commutator, cosocleC2×C109C26
FrattiniC2C25×D109
FittingC2×H(5,𝔽2)×C109C2
socleC22×C109C25
hypercenterC2×H(5,𝔽2)D109
hypocenterC109C22×H(5,𝔽2)
LCSG ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. C2×#16812745C2semidirect
2. C2×#16812744C2semidirect
3. H(5,𝔽2)×D109C2direct (N × H)
4. C2×#41×D109C2semidirect
5. C22×D4×D109C2semidirect
6. C2×H(5,𝔽2)×C109C2semidirect
7. C24×D109C22semidirect
8. #16812745C22semidirect
9. #16812744C22semidirect
10. C2×#13182030C22semidirect
11. C2×#13182029C22semidirect
12. C2×#13182028C22semidirect
13. #41×D109C22semidirect
14. C2×Q8×D109C22semidirect
15. C22×D436C22semidirect
16. C22×C22⋊D109C22semidirect
17. C2×D4×D109C22semidirect
18. C2×D4×D109C22semidirect
19. C2×D4×D109C22semidirect
20. C22×C4×D109C22semidirect
21. H(5,𝔽2)×C109C22semidirect
22. C2×#41×C109C22semidirect
23. C22×D4×C109C22semidirect
26. C23×D109C23semidirect
27. C23×D109C23semidirect
28. #13182030C23semidirect
29. #13182029C23semidirect
30. #13182028C23semidirect
31. C2×Dic218C23semidirect
32. Q8×D109C23semidirect
33. C2×D436C23semidirect
34. C2×D436C23semidirect
35. C2×C22⋊D109C23semidirect
36. C2×C22⋊D109C23semidirect
37. D4×D109C23semidirect
38. D4×D109C23semidirect
39. C22×Dic109C23semidirect
41. C2×C4×D109C23semidirect
42. C2×C4×D109C23semidirect
43. C24×C109C23semidirect
44. #41×C109C23semidirect
45. C2×Q8×C109C23semidirect
46. C2×D4×C109C23semidirect
47. C2×D4×C109C23semidirect
48. C22×C4×C109C23semidirect
51. C23×C109C24semidirect
58. C22×D109C24semidirect
59. Q8×C109C24semidirect
60. D4×C109C24semidirect
62. C2×C4×C109C24semidirect
63. C22⋊D109C24semidirect
65. C2×Dic109C24semidirect
66. D436C24semidirect
68. C4×D109C24semidirect
69. Dic218C24semidirect
77. C2×D109H(5,𝔽2)direct (N × H)
83. C2×C109C2×H(5,𝔽2)semidirect
86. D109C2×H(5,𝔽2)direct (N × H)
88. C109C22×H(5,𝔽2)semidirect
92. C2×H(5,𝔽2)D109direct (N × H)
98. H(5,𝔽2)C2×D109direct (N × H)
103. C2×#41C2×D109semidirect
110. C22×D4C2×D109semidirect
116. C24C22×D109semidirect
120. #41C22×D109semidirect
123. C2×Q8C22×D109semidirect
130. C2×D4C22×D109semidirect
131. C2×D4C22×D109semidirect
136. C22×C4C22×D109semidirect
146. C23C23×D109semidirect
148. Q8C23×D109semidirect
151. D4C23×D109semidirect
157. C2×C4C23×D109semidirect
173. C2H(5,𝔽2)×D109direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168127452
C2×#168127442
H(5,𝔽2)×D1092
C2×#41×D1092
C22×D4×D1092
C2×H(5,𝔽2)×C1092
C22×H(5,𝔽2)109