Finite Group Databasebeta

← #2191×C173 #36349976

Subgroups — #2191×C173 #36349976

Order 22144.

216 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 64 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
1#575×C173C211yesmaximal, characteristic
2#510×C173C212yesmaximal, complemented
3#470×C173C211yesmaximal, characteristic, complemented
4#360×C173C212yesmaximal, complemented
5#349×C173C211yesmaximal, characteristic, complemented
6C2×#41×C173—21no—
7#137×C173—22no—
8AGL(1,ℤ/8)×C173—22no—
9#135×C173—21no—
10#135×C173C2212yes—
11C2×Q16×C173C2211yescharacteristic, complemented
12C2×M16×C173C2211yescharacteristic
13C22⋊Q8×C173—21no—
14Q32×C173—22no—
15SD32×C173—22no—
16M32×C173C2212yescomplemented
17#106×C173C2211yescharacteristic, complemented
18#103×C173—21no—
19#41×C173—22no—
20#41×C173—22no—
21#41×C173—41no—
22#41×C173—41no—
23C2×Q8×C173—21no—
24C2×Q8×C173—21no—
25C2×D4×C173—21no—
26C2×D4×C173—41no—
27C22×C4×C173—41no—
28C22×C4×C173D411yescharacteristic
29Q16×C173—21no—
30Q16×C173—22no—
31Q16×C173D412yes—
32SD16×C173—22no—
33SD16×C173—22no—
34D8×C173—21no—
35D8×C173D412yes—
36M16×C173—22no—
37C2×C8×C173D411yescharacteristic
38C2×C8×C173C2311yescharacteristic
39C4⋊C4×C173—21no—
40C4⋊C4×C173—41no—
41C22⋊C4×C173—41no—
42C16×C173—22no—
43C23×C173—41no—
44C23×C173D811yescharacteristic
45Q8×C173—22no—
46Q8×C173—41no—
47Q8×C173—41no—
48Q8×C173—41no—
49D4×C173—22no—
50D4×C173—41no—
51D4×C173—42no—
52D4×C173—42no—
53C2×C4×C173—21no—
54C2×C4×C173—21no—
55C2×C4×C173—42no—
56C2×C4×C173—41no—
57C2×C4×C173—42no—
58C2×C4×C173—41no—
59C2×C4×C173—41no—
60C2×C4×C173D811yescharacteristic
61C2×C4×C173C2×D411yescharacteristic
62C8×C173—21no—
63C8×C173C2×D412yes—
64C22×C173—21no—
65C22×C173—41no—
66C22×C173—42no—
67C22×C173—82no—
68C22×C173C2×D811yescharacteristic
69C4×C173—21no—
70C4×C173—42no—
71C4×C173—81no—
72C4×C173—81no—
73C4×C173#12011yescharacteristic
74C4×C173AGL(1,ℤ/8)11yescharacteristic
75C2×C173—21no—
76C2×C173—41no—
77C2×C173—82no—
78C2×C173#44711yescharacteristic
79C173#219111yescharacteristic, direct product factor, Sylow
80#2191C17311yesmaximal, characteristic, direct product factor, Sylow
81#575C2×C17311yescharacteristic
82#510C2×C17312yescomplemented
83#470C2×C17311yescharacteristic, complemented
84#360C2×C17312yescomplemented
85#349C2×C17311yescharacteristic, complemented
86C2×#41—21no—
87#137—22no—
88AGL(1,ℤ/8)—22no—
89#135—21no—
90#135C22×C17312yes—
91C2×Q16C22×C17311yescharacteristic, complemented
92C2×M16C22×C17311yescharacteristic
93C22⋊Q8—21no—
94Q32—22no—
95SD32—22no—
96M32C22×C17312yescomplemented
97#106C22×C17311yescharacteristic, complemented
98#103—21no—
99#41—22no—
100#41—22no—
101#41—41no—
102#41—41no—
103C2×Q8—21no—
104C2×Q8—21no—
105C2×D4—21no—
106C2×D4—41no—
107C22×C4—41no—
108C22×C4D4×C17311yescharacteristic
109Q16—21no—
110Q16—22no—
111Q16D4×C17312yes—
112SD16—22no—
113SD16—22no—
114D8—21no—
115D8D4×C17312yes—
116M16—22no—
117C2×C8D4×C17311yescharacteristic
118C2×C8C23×C17311yescharacteristic
119C4⋊C4—21no—
120C4⋊C4—41no—
121C22⋊C4—41no—
122C16—22no—
123C23—41no—
124C23D8×C17311yescharacteristic
125Q8—22no—
126Q8—41no—
127Q8—41no—
128Q8—41no—
129D4—22no—
130D4—41no—
131D4—42no—
132D4—42no—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—42no—
136C2×C4—41no—
137C2×C4—42no—
138C2×C4—41no—
139C2×C4—41no—
140C2×C4D8×C17311yescharacteristic
141C2×C4C2×D4×C17311yescharacteristic
142C8—21no—
143C8C2×D4×C17312yes—
144C22—21no—
145C22—41no—
146C22—42no—
147C22—82no—
148C22C2×D8×C17311yescharacteristic
149C4—21no—
150C4—42no—
151C4—81no—
152C4—81no—
153C4#120×C17311yescharacteristic
154C4AGL(1,ℤ/8)×C17311yescharacteristic
155C2—21no—
156C2—41no—
157C2—82no—
158C2#447×C17311yescharacteristic
subgroupG/H
center, socleC2×C173#447
commutator, Frattini, cosocleC2×C8C23×C173
prime radical#575×C173C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C173 ◁ C2×C4×C173 ◁ C2×M16×C173 ◁ G

Complements

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

NH ≅ G/Ntype
2. #510×C173C2semidirect
3. #470×C173C2semidirect
4. #360×C173C2semidirect
5. #349×C173C2semidirect
11. C2×Q16×C173C22semidirect
16. M32×C173C22semidirect
17. #106×C173C22semidirect
79. C173#2191direct (N × H)
80. #2191C173direct (N × H)
82. #510C2×C173semidirect
83. #470C2×C173semidirect
84. #360C2×C173semidirect
85. #349C2×C173semidirect
91. C2×Q16C22×C173semidirect
96. M32C22×C173semidirect
97. #106C22×C173semidirect

Maximal subgroups

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

subgroupindex
#575×C1732
#510×C1732
#470×C1732
#360×C1732
#349×C1732
#2191173