Finite Group Databasebeta

← #27133462

Subgroups — #27133462

Order 12224.

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×#15690288C211yesmaximal, characteristic, complemented
2#41×D191C211yesmaximal, characteristic, complemented
3#20259354C212yesmaximal, complemented
4#20259353C211yesmaximal, characteristic, complemented
5#20259350C211yesmaximal, characteristic, complemented
6#20259349C212yesmaximal, complemented
7#20259347C211yesmaximal, characteristic, complemented
8#20259345C212yesmaximal, complemented
9#20259343C212yesmaximal, complemented
10M16×D191C211yesmaximal, characteristic, complemented
11#137×C191C211yesmaximal, characteristic, complemented
12#15690288—21no—
13#15690288—21no—
14#15690288C2212yes—
15#15690288C2212yescomplemented
16#15690288C2211yescharacteristic, complemented
17#15690287—21no—
18#15690287C2211yescharacteristic
19#15690286—21no—
20#15690286C2211yescharacteristic, complemented
21Q8×D191C2211yescharacteristic, complemented
22C2×D764—21no—
23C2×D764C2211yescharacteristic, complemented
24D4×D191—21no—
25D4×D191C2211yescharacteristic, complemented
26C2×C4×D191—21no—
27C2×C4×D191—21no—
28C2×C4×D191C2211yescharacteristic
29#15690277C2212yescomplemented
30#15690276C2212yescomplemented
31#15690274C2212yescomplemented
32#15690274C2212yescomplemented
33D1528C2212yescomplemented
34#15690272C2212yescomplemented
35#15690271C2211yescharacteristic, complemented
36#15690270C2212yescomplemented
37C8×D191C2212yescomplemented
38#41×C191C2211yescharacteristic
39C2×Q8×C191C2211yescharacteristic, complemented
40Q16×C191C2212yescomplemented
41SD16×C191C2212yescomplemented
42M16×C191C2211yescharacteristic, complemented
43C22×D191—21no—
44C22×D191—21no—
45C22×D191D411yescharacteristic
46Dic382—21no—
47Dic382C2311yescharacteristic
48D764—21no—
49D764—22no—
50D764—22no—
51D764—21no—
52D764C2312yes—
53D764C2311yescharacteristic
54C22⋊D191—21no—
55C22⋊D191—21no—
56C22⋊D191—21no—
57C2×Dic191—21no—
58C2×Dic191D411yescharacteristic
59C4×D191—21no—
60C4×D191—21no—
61C4×D191—21no—
62C4×D191—22no—
63C4×D191—22no—
64C4×D191D411yescharacteristic
65C4×D191D411yescharacteristic
66C4×D191C2311yescharacteristic
67C4×D191C2311yescharacteristic
68C191⋊C8C2312yes—
69Q8×C191—21no—
70Q8×C191C2312yes—
71Q8×C191C2311yescharacteristic
72D4×C191—21no—
73D4×C191C2311yescharacteristic
74C2×C4×C191—21no—
75C2×C4×C191—21no—
76C2×C4×C191C2311yescharacteristic
77C8×C191C2312yes—
78C22×C191—21no—
79C22×C191C2×D411yescharacteristic
80C2×D191—21no—
81C2×D191—22no—
82C2×D191—41no—
83C2×D191—42no—
84C2×D191—42no—
85C2×D191C2×D411yescharacteristic
86C2×D191C2×D411yescharacteristic
87C4×C191—21no—
88C4×C191—22no—
89C4×C191C2×D411yescharacteristic
90C4×C191C2411yescharacteristic
91Dic191—21no—
92Dic191C2×D411yescharacteristic
93Dic191C2×D411yescharacteristic
94C2×C191—21no—
95C2×C191—41no—
96C2×C191C22×D411yescharacteristic
97D191—21no—
98D191—21no—
99D191—41no—
100D191—42no—
101C191#57511yescharacteristic, complemented, Sylow
102#575—1911nomaximal, Sylow
103C2×#41—1911no—
104C2×#41—1911no—
105#137—1911no—
106#137—1912no—
107#137D19111yescharacteristic, complemented
108AGL(1,ℤ/8)—1911no—
109AGL(1,ℤ/8)—1912no—
110AGL(1,ℤ/8)—1911no—
111#135—1912no—
112#135—1912no—
113C2×M16—1911no—
114#41—3821no—
115#41—3821no—
116#41—3821no—
117#41—3821no—
118#41—1912no—
119#41—1911no—
120#41—1912no—
121#41—1911no—
122#41—1911no—
123#41C2×D19111yescharacteristic
124C2×Q8—1911no—
125C2×Q8C2×D19111yescharacteristic, complemented
126C2×D4—3821no—
127C2×D4—3821no—
128C2×D4—1911no—
129C2×D4—1911no—
130C22×C4—3821no—
131C22×C4—3821no—
132C22×C4—1911no—
133Q16—1912no—
134Q16C2×D19112yescomplemented
135SD16—1912no—
136SD16—1912no—
137SD16—1912no—
138SD16C2×D19112yescomplemented
139D8—1912no—
140D8—1912no—
141M16—1911no—
142M16—1912no—
143M16C2×D19111yescharacteristic, complemented
144C2×C8—1912no—
145C23—3821no—
146C23—3821no—
147C23—1911no—
148Q8—21no—
149Q8—3821no—
150Q8—1911no—
151Q8C22×D19111yescharacteristic
152Q8C22×D19112yes—
153D4—21no—
154D4—3821no—
155D4—3822no—
156D4—3822no—
157D4—3821no—
158D4—3821no—
159D4—3821no—
160D4—3821no—
161D4—1912no—
162D4—1911no—
163D4C22×D19111yescharacteristic
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—3821no—
167C2×C4—3821no—
168C2×C4—3822no—
169C2×C4—3822no—
170C2×C4—3821no—
171C2×C4—3821no—
172C2×C4—1911no—
173C2×C4—1911no—
174C2×C4—1911no—
175C2×C4—1911no—
176C2×C4—1911no—
177C2×C4C22×D19111yescharacteristic
178C8—1912no—
179C8C22×D19112yes—
180C22—21no—
181C22—3822no—
182C22—3821no—
183C22—1911no—
184C22—1911no—
185C22—7641no—
186C22—7642no—
187C22—7642no—
188C22D4×D19111yescharacteristic
189C4—21no—
190C4—22no—
191C4—3821no—
192C4—1911no—
193C4—1911no—
194C4D4×D19111yescharacteristic
195C4C23×D19111yescharacteristic
196C2—21no—
197C2—41no—
198C2—3821no—
199C2—3821no—
200C2—7642no—
201C2—7641no—
202C2C2×D4×D19111yescharacteristic
subgroupG/H
centerC2C2×D4×D191
commutator, cosocleC4×C191C24
FrattiniC4C23×D191
Fitting#137×C191C2
socleC2×C191C22×D4
hypercenter#137D191
hypocenterC191#575
LCSG ▷ C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C2 ◁ C2×C4 ◁ #137

Complements

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

NH ≅ G/Ntype
1. C2×#15690288C2semidirect
2. #41×D191C2semidirect
3. #20259354C2semidirect
4. #20259353C2semidirect
5. #20259350C2semidirect
6. #20259349C2semidirect
7. #20259347C2semidirect
8. #20259345C2semidirect
9. #20259343C2semidirect
10. M16×D191C2semidirect
11. #137×C191C2semidirect
15. #15690288C22semidirect
16. #15690288C22semidirect
20. #15690286C22semidirect
21. Q8×D191C22semidirect
23. C2×D764C22semidirect
25. D4×D191C22semidirect
29. #15690277C22semidirect
30. #15690276C22semidirect
31. #15690274C22semidirect
32. #15690274C22semidirect
33. D1528C22semidirect
34. #15690272C22semidirect
35. #15690271C22semidirect
36. #15690270C22semidirect
37. C8×D191C22semidirect
39. C2×Q8×C191C22semidirect
40. Q16×C191C22semidirect
41. SD16×C191C22semidirect
42. M16×C191C22semidirect
101. C191#575semidirect
107. #137D191semidirect
125. C2×Q8C2×D191semidirect
134. Q16C2×D191semidirect
138. SD16C2×D191semidirect
143. M16C2×D191semidirect

Maximal subgroups

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

subgroupindex
C2×#156902882
#41×D1912
#202593542
#202593532
#202593502
#202593492
#202593472
#202593452
#202593432
M16×D1912
#137×C1912
#575191