Finite Group Databasebeta

← #47985188

Subgroups — #47985188

Order 35456.

216 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 45 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#31800184C212yesmaximal, complemented
2#31800174C211yesmaximal, characteristic
3#31799801C212yesmaximal, complemented
4#31799799C211yesmaximal, characteristic, complemented
5#563×C277C211yesmaximal, characteristic, complemented
6#22579928C2211yescharacteristic
7#22579911—22no—
8#22579906—21no—
9#22579808—21no—
10#22579808—22no—
11#22579799C2211yescharacteristic
12#22579799C2212yes—
13C4⋊Q8×C277C2212yes—
14#126×C277—22no—
15#125×C277—21no—
16#124×C277C2211yescharacteristic
17#123×C277—21no—
18C22⋊Q8×C277—22no—
19#17984564—41no—
20#17984563—21no—
21#17984563—22no—
22Q8×D277—41no—
23#17984553—42no—
24#17984551—41no—
25#17984550—42no—
26#17984548—41no—
27#17984547—21no—
28#17984547—22no—
29C2×Q8×C277D412yes—
30C2×Q8×C277C2311yescharacteristic
31C2×D4×C277D411yescharacteristic
32C22×C4×C277—21no—
33C4⋊C4×C277—21no—
34C4⋊C4×C277—22no—
35C4⋊C4×C277—21no—
36C4⋊C4×C277—22no—
37C4⋊C4×C277—22no—
38C22⋊C4×C277—22no—
39C22⋊C4×C277—21no—
40C22⋊C4×C277—22no—
41C42×C277D411yescharacteristic
42C42×C277D412yes—
43Dic554—21no—
44Dic554—22no—
45D1108—21no—
46D1108—22no—
47C22⋊D277—41no—
48C4×D277—41no—
49C4×D277—42no—
50#14555660—41no—
51#14555660—42no—
52C23×C277—21no—
53Q8×C277—41no—
54Q8×C277—42no—
55D4×C277—41no—
56C2×C4×C277—21no—
57C2×C4×C277—22no—
58C2×C4×C277—22no—
59C2×C4×C277—81no—
60C2×C4×C277C2×D412yes—
61C2×C4×C277C2×D411yescharacteristic
62C22×C277—42no—
63C22×C277—41no—
64C22×C277#12011yescharacteristic
65C2×D277—41no—
66C4×C277—21no—
67C4×C277—22no—
68C4×C277—41no—
69C4×C277—42no—
70C4×C277—42no—
71Dic277—41no—
72C2×C277—21no—
73C2×C277—81no—
74C2×C277B(4,𝔽2)11yescharacteristic
75D277—81no—
76C277#220311yescharacteristic, complemented, Sylow
77#2203—2771nomaximal, Sylow
78#563D27711yescharacteristic, complemented
79#456—2772no—
80#454—2771no—
81#356—2772no—
82#355—2771no—
832−1+4—2771no—
84#137—5542no—
85AGL(1,ℤ/8)—5541no—
86C4⋊Q8C2×D27712yes—
87#126—22no—
88#125—21no—
89#124C2×D27711yescharacteristic
90#123—21no—
91C22⋊Q8—22no—
92#104—5541no—
93#104—5542no—
94#101—2772no—
95#101—2771no—
96#41—5541no—
97#41—5542no—
98#41—11081no—
99C2×Q8—11081no—
100C2×Q8C22⋊D27712yes—
101C2×Q8C22×D27711yescharacteristic
102C2×D4C22⋊D27711yescharacteristic
103C22×C4—21no—
104Q16—11082no—
105SD16—11081no—
106SD16—11082no—
107D8—11081no—
108M16—5541no—
109M16—5542no—
110C4⋊C4—21no—
111C4⋊C4—22no—
112C4⋊C4—21no—
113C4⋊C4—22no—
114C4⋊C4—22no—
115C22⋊C4—22no—
116C22⋊C4—21no—
117C22⋊C4—22no—
118C42C22⋊D27711yescharacteristic
119C42C22⋊D27712yes—
120C23—21no—
121Q8—41no—
122Q8—42no—
123Q8—5541no—
124Q8—5542no—
125D4—41no—
126D4—5541no—
127D4—5542no—
128D4—11081no—
129C2×C4—21no—
130C2×C4—22no—
131C2×C4—22no—
132C2×C4—81no—
133C2×C4—11081no—
134C2×C4—11082no—
135C2×C4C2×C22⋊D27712yes—
136C2×C4C2×C22⋊D27711yescharacteristic
137C8—11081no—
138C8—11082no—
139C22—42no—
140C22—41no—
141C22—11081no—
142C22#2257984911yescharacteristic
143C4—21no—
144C4—22no—
145C4—41no—
146C4—42no—
147C4—42no—
148C4—11081no—
149C2—21no—
150C2—81no—
151C2—22161no—
152C2#3180018911yescharacteristic
subgroupG/H
centerC2#31800189
commutator, cosocleC2×Q8×C277C23
FrattiniC2×Q8C22×D277
Fitting#563×C277C2
socleC2×C277B(4,𝔽2)
prime radical#31800174C2
hypercenter#563D277
hypocenterC277#2203
LCSG ▷ C2×Q8×C277 ▷ C22×C277 ▷ C2×C277 ▷ C277
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #563

Complements

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

NH ≅ G/Ntype
1. #31800184C2semidirect
3. #31799801C2semidirect
4. #31799799C2semidirect
5. #563×C277C2semidirect
76. C277#2203semidirect
78. #563D277semidirect

Maximal subgroups

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

subgroupindex
#318001842
#318001742
#317998012
#317997992
#563×C2772
#2203277