Finite Group Databasebeta

← #47985196

Subgroups — #47985196

Order 35456.

224 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 41 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#31800732C211yesmaximal, characteristic, complemented
2#31800185C212yesmaximal
3#31799801C211yesmaximal, characteristic, complemented
4#31799800C212yesmaximal, complemented
5#456×C277C211yesmaximal, characteristic
6#22579910—22no—
7#22579879—22no—
8#22579878—21no—
9#22579878C2212yes—
10C4×Dic554—21no—
11#22579845—22no—
12#22579807—22no—
13#22579799C2212yes—
142−1+4×C277C2211yescharacteristic
15#137×C277—21no—
16C4⋊Q8×C277C2211yescharacteristic
17#104×C277—21no—
18#101×C277C2211yescharacteristic
19C2×Dic554—21no—
20C2×Dic554—21no—
21C2×Dic554D412yes—
22#17984553—42no—
23#17984551—42no—
24#17984547—22no—
25C4⋊Dic277—21no—
26#17984542—22no—
27#17984542—22no—
28#17984542—22no—
29C4×Dic277—21no—
30C4×Dic277—22no—
31C4×Dic277D412yes—
32#41×C277—21no—
33#41×C277—22no—
34#41×C277—41no—
35C2×Q8×C277—41no—
36C2×Q8×C277D411yescharacteristic
37C2×Q8×C277C2311yescharacteristic
38Q16×C277—41no—
39SD16×C277—41no—
40M16×C277—21no—
41C4⋊C4×C277—22no—
42C42×C277D411yescharacteristic
43Dic554—24no—
44Dic554—42no—
45Dic554—81no—
46C2×Dic277—22no—
47C2×Dic277—22no—
48#14555660—42no—
49Q8×C277—21no—
50Q8×C277—22no—
51Q8×C277—41no—
52Q8×C277—41no—
53D4×C277—21no—
54D4×C277—22no—
55D4×C277—41no—
56C2×C4×C277—21no—
57C2×C4×C277—21no—
58C2×C4×C277—41no—
59C2×C4×C277—42no—
60C2×C4×C277C2×D412yes—
61C2×C4×C277C2×D411yescharacteristic
62C8×C277—41no—
63C22×C277—41no—
64C22×C277#12011yescharacteristic
65C4×C277—22no—
66C4×C277—21no—
67C4×C277—22no—
68C4×C277—41no—
69C4×C277—41no—
70Dic277—24no—
71Dic277—42no—
72C2×C277—21no—
73C2×C277—81no—
74C2×C277B(4,𝔽2)11yescharacteristic
75C277#220511yescharacteristic, complemented, Sylow
76#2205—2771nomaximal, Sylow
77Q82—2771no—
78#456—2772no—
79#456D27711yescharacteristic
80#356—2771no—
81#356—2772no—
822−1+4C2×D27711yescharacteristic
83#137—21no—
84#137—5542no—
85C4⋊Q8—2772no—
86C4⋊Q8—5542no—
87C4⋊Q8—5541no—
88C4⋊Q8C2×D27711yescharacteristic
89C4×Q8—5542no—
90C4×Q8—5541no—
91#104—21no—
92#104—5542no—
93#101—2772no—
94#101C2×D27711yescharacteristic
95#41—22no—
96#41—21no—
97#41—41no—
98C2×Q8—41no—
99C2×Q8—2772no—
100C2×Q8—5541no—
101C2×Q8—5541no—
102C2×Q8C22⋊D27711yescharacteristic
103C2×Q8C22×D27711yescharacteristic
104Q16—41no—
105Q16—11082no—
106SD16—41no—
107SD16—11082no—
108M16—21no—
109M16—5542no—
110C4⋊C4—22no—
111C4⋊C4—5542no—
112C4⋊C4—5541no—
113C4⋊C4—5542no—
114C4⋊C4—5542no—
115C42—2772no—
116C42—5542no—
117C42—5541no—
118C42C22⋊D27711yescharacteristic
119Q8—21no—
120Q8—22no—
121Q8—41no—
122Q8—41no—
123Q8—5544no—
124Q8—11082no—
125Q8—22161no—
126D4—21no—
127D4—22no—
128D4—41no—
129C2×C4—21no—
130C2×C4—21no—
131C2×C4—41no—
132C2×C4—42no—
133C2×C4—5542no—
134C2×C4—5542no—
135C2×C4D4×D27712yes—
136C2×C4C2×C22⋊D27711yescharacteristic
137C8—41no—
138C8—11082no—
139C22—41no—
140C22#2257984811yescharacteristic
141C4—22no—
142C4—21no—
143C4—22no—
144C4—41no—
145C4—41no—
146C4—5544no—
147C4—11082no—
148C2—21no—
149C2—81no—
150C2#3180018811yescharacteristic
subgroupG/H
centerC2#31800188
commutator, cosocleC2×Q8×C277C23
FrattiniC2×Q8C22×D277
Fitting#456×C277C2
socleC2×C277B(4,𝔽2)
prime radical2−1+4×C277C22
hypercenter#456D277
hypocenterC277#2205
LCSG ▷ C2×Q8×C277 ▷ C22×C277 ▷ C2×C277 ▷ C277
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #456

Complements

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

NH ≅ G/Ntype
1. #31800732C2semidirect
3. #31799801C2semidirect
4. #31799800C2semidirect
75. C277#2205semidirect

Maximal subgroups

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

subgroupindex
#318007322
#318001852
#317998012
#317998002
#456×C2772
#2205277