Finite Group Databasebeta

← #20295822

Subgroups — #20295822

Order 6976.

262 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 99 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×#13182028C211yesmaximal, characteristic, complemented
2#41×D109C211yesmaximal, characteristic, complemented
3#16812732C211yesmaximal, characteristic, complemented
4#16812729C211yesmaximal, characteristic, complemented
5#16812728C212yesmaximal, complemented
6#16812725C211yesmaximal, characteristic, complemented
7#16812723C212yesmaximal, complemented
8#16812720C212yesmaximal, complemented
9#16812718C212yesmaximal, complemented
10M16×D109C211yesmaximal, characteristic, complemented
11AGL(1,ℤ/8)×C109C211yesmaximal, characteristic, complemented
12#13182030C2211yescharacteristic, complemented
13#13182029—21no—
14#13182029C2211yescharacteristic
15#13182028—21no—
16#13182028—21no—
17#13182028—21no—
18#13182028C2212yes—
19#13182028C2211yescharacteristic, complemented
20#13182028C2212yescomplemented
21C2×Dic218C2211yescharacteristic, complemented
22Q8×D109C2211yescharacteristic, complemented
23C2×C22⋊D109—21no—
24D4×D109—21no—
25D4×D109C2211yescharacteristic, complemented
26C22×Dic109—21no—
27C2×C4×D109—21no—
28C2×C4×D109C2211yescharacteristic
29Dic436C2212yescomplemented
30#13182019C2212yescomplemented
31#13182018C2212yescomplemented
32#13182017C2212yescomplemented
33#13182017C2212yescomplemented
34#13182014C2212yescomplemented
35#13182013C2211yescharacteristic, complemented
36#13182012C2212yescomplemented
37C8×D109C2212yescomplemented
38#41×C109C2211yescharacteristic
39C2×D4×C109C2211yescharacteristic, complemented
40SD16×C109C2212yescomplemented
41D8×C109C2212yescomplemented
42M16×C109C2211yescharacteristic, complemented
43C23×C109—21no—
44C22×D109—21no—
45C22×D109D411yescharacteristic
46Q8×C109C2311yescharacteristic
47D4×C109—21no—
48D4×C109—21no—
49D4×C109C2311yescharacteristic
50D4×C109C2312yes—
51C2×C4×C109—21no—
52C2×C4×C109C2311yescharacteristic
53C22⋊D109—21no—
54C22⋊D109—21no—
55C22⋊D109—22no—
56C22⋊D109—22no—
57C22⋊D109—21no—
58C2×Dic109—21no—
59C2×Dic109—21no—
60C2×Dic109—22no—
61C2×Dic109—22no—
62C2×Dic109D411yescharacteristic
63D436—21no—
64D436C2311yescharacteristic
65C4×D109—21no—
66C4×D109—21no—
67C4×D109—21no—
68C4×D109D411yescharacteristic
69C4×D109D411yescharacteristic
70C4×D109C2311yescharacteristic
71C4×D109C2311yescharacteristic
72Dic218—21no—
73Dic218—21no—
74Dic218C2311yescharacteristic
75Dic218C2312yes—
76C8×C109C2312yes—
77#11724784C2312yes—
78C22×C109—22no—
79C22×C109—21no—
80C22×C109—42no—
81C22×C109C2×D411yescharacteristic
82C2×D109—21no—
83C2×D109—41no—
84C2×D109—42no—
85C2×D109C2×D411yescharacteristic
86C2×D109C2×D411yescharacteristic
87C4×C109—21no—
88C4×C109C2×D411yescharacteristic
89C4×C109C2411yescharacteristic
90Dic109—21no—
91Dic109—22no—
92Dic109C2×D411yescharacteristic
93Dic109C2×D411yescharacteristic
94C2×C109—21no—
95C2×C109—41no—
96C2×C109—42no—
97C2×C109C22×D411yescharacteristic
98D109—21no—
99D109—21no—
100D109—41no—
101C109#57511yescharacteristic, complemented, Sylow
102#575—1091nomaximal, Sylow
103C2×#41—1091no—
104C2×#41—1091no—
105#137—1091no—
106#137—1092no—
107#137—1091no—
108AGL(1,ℤ/8)—1091no—
109AGL(1,ℤ/8)—1092no—
110AGL(1,ℤ/8)D10911yescharacteristic, complemented
111#135—1092no—
112#135—1092no—
113C2×M16—1091no—
114#41—1091no—
115#41—1092no—
116#41—1092no—
117#41—1091no—
118#41—1091no—
119#41—2181no—
120#41—2181no—
121#41—2181no—
122#41—2181no—
123#41C2×D10911yescharacteristic
124C2×Q8—1091no—
125C2×Q8—1091no—
126C2×D4—1091no—
127C2×D4—2181no—
128C2×D4—2181no—
129C2×D4C2×D10911yescharacteristic, complemented
130C22×C4—1091no—
131C22×C4—2181no—
132C22×C4—2181no—
133Q16—1092no—
134Q16—1092no—
135SD16—1092no—
136SD16—1092no—
137SD16—1092no—
138SD16C2×D10912yescomplemented
139D8—1092no—
140D8C2×D10912yescomplemented
141M16—1091no—
142M16—1092no—
143M16C2×D10911yescharacteristic, complemented
144C2×C8—1092no—
145C23—21no—
146C23—1091no—
147C23—2181no—
148Q8—1091no—
149Q8—1092no—
150Q8—2181no—
151Q8—2181no—
152Q8C22×D10911yescharacteristic
153D4—21no—
154D4—21no—
155D4—1091no—
156D4—2181no—
157D4—2181no—
158D4—2181no—
159D4—2182no—
160D4—2182no—
161D4—2181no—
162D4C22×D10912yes—
163D4C22×D10911yescharacteristic
164C2×C4—21no—
165C2×C4—1091no—
166C2×C4—1091no—
167C2×C4—1091no—
168C2×C4—1091no—
169C2×C4—1091no—
170C2×C4—2181no—
171C2×C4—2181no—
172C2×C4—2181no—
173C2×C4—2182no—
174C2×C4—2182no—
175C2×C4—2181no—
176C2×C4—2181no—
177C2×C4C22×D10911yescharacteristic
178C8—1092no—
179C8C22×D10912yes—
180C22—21no—
181C22—22no—
182C22—42no—
183C22—1091no—
184C22—1091no—
185C22—4361no—
186C22—4362no—
187C22—2181no—
188C22D4×D10911yescharacteristic
189C4—21no—
190C4—1091no—
191C4—1091no—
192C4—2181no—
193C4—2182no—
194C4D4×D10911yescharacteristic
195C4C23×D10911yescharacteristic
196C2—21no—
197C2—42no—
198C2—41no—
199C2—4361no—
200C2—2181no—
201C2—2181no—
202C2C2×D4×D10911yescharacteristic
subgroupG/H
centerC2C2×D4×D109
commutatorC4×C109C24
FrattiniC4C23×D109
FittingAGL(1,ℤ/8)×C109C2
socleC2×C109C22×D4
hypercenterAGL(1,ℤ/8)D109
hypocenterC109#575
LCSG ▷ C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#13182028C2semidirect
2. #41×D109C2semidirect
3. #16812732C2semidirect
4. #16812729C2semidirect
5. #16812728C2semidirect
6. #16812725C2semidirect
7. #16812723C2semidirect
8. #16812720C2semidirect
9. #16812718C2semidirect
10. M16×D109C2semidirect
11. AGL(1,ℤ/8)×C109C2semidirect
12. #13182030C22semidirect
19. #13182028C22semidirect
20. #13182028C22semidirect
21. C2×Dic218C22semidirect
22. Q8×D109C22semidirect
25. D4×D109C22semidirect
29. Dic436C22semidirect
30. #13182019C22semidirect
31. #13182018C22semidirect
32. #13182017C22semidirect
33. #13182017C22semidirect
34. #13182014C22semidirect
35. #13182013C22semidirect
36. #13182012C22semidirect
37. C8×D109C22semidirect
39. C2×D4×C109C22semidirect
40. SD16×C109C22semidirect
41. D8×C109C22semidirect
42. M16×C109C22semidirect
101. C109#575semidirect
110. AGL(1,ℤ/8)D109semidirect
129. C2×D4C2×D109semidirect
138. SD16C2×D109semidirect
140. D8C2×D109semidirect
143. M16C2×D109semidirect