Finite Group Databasebeta

← #47985180

Subgroups — #47985180

Order 35456.

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

normal subgroups only

This group has 226 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#31800191C212yesmaximal
2#31800179C211yesmaximal, characteristic
3#31799797C211yesmaximal, characteristic, complemented
4#31799792C212yesmaximal, complemented
5#563×C277C211yesmaximal, characteristic, complemented
6#22579925C2211yescharacteristic
7#22579911—21no—
8#22579863—22no—
9#22579808—21no—
10#22579797C2211yescharacteristic
11#22579795C2212yes—
12#22579794—22no—
13C4⋊Q8×C277—21no—
14#126×C277—22no—
15#125×C277—21no—
16#124×C277C2211yescharacteristic
17#123×C277C2212yes—
18C22⋊Q8×C277—22no—
19#17984563—21no—
20#17984562—41no—
21C2×C22⋊D277—22no—
22D4×D277—41no—
23#17984553—41no—
24#17984550—41no—
25#17984547—21no—
26#17984542—42no—
27C22⋊Dic277—22no—
28#17984540—42no—
29C2×Q8×C277—21no—
30C2×Q8×C277D411yescharacteristic
31C2×D4×C277C2311yescharacteristic
32C22×C4×C277D412yes—
33C4⋊C4×C277—21no—
34C4⋊C4×C277—22no—
35C4⋊C4×C277—22no—
36C4⋊C4×C277—21no—
37C4⋊C4×C277—22no—
38C22⋊C4×C277—22no—
39C22⋊C4×C277—22no—
40C22⋊C4×C277D412yes—
41C42×C277—21no—
42C42×C277D411yescharacteristic
43C22×D277—22no—
44Dic554—21no—
45D1108—21no—
46C22⋊D277—42no—
47C22⋊D277—41no—
48C2×Dic277—22no—
49C2×Dic277—42no—
50C4×D277—41no—
51#14555660—41no—
52C23×C277C2×D412yes—
53Q8×C277—41no—
54Q8×C277—81no—
55D4×C277—41no—
56C2×C4×C277—21no—
57C2×C4×C277—22no—
58C2×C4×C277—21no—
59C2×C4×C277—22no—
60C2×C4×C277—42no—
61C2×C4×C277C2×D411yescharacteristic
62C22×C277—22no—
63C22×C277—42no—
64C22×C277#12011yescharacteristic
65C2×D277—41no—
66C2×D277—82no—
67C4×C277—21no—
68C4×C277—42no—
69C4×C277—41no—
70C4×C277—42no—
71C4×C277—41no—
72Dic277—41no—
73Dic277—82no—
74C2×C277—21no—
75C2×C277—42no—
76C2×C277B(4,𝔽2)11yescharacteristic
77D277—81no—
78C277#220111yescharacteristic, complemented, Sylow
79#2201—2771nomaximal, Sylow
80#563D27711yescharacteristic, complemented
81#458—2772no—
82#455—2771no—
83#354—2771no—
84#352—2772no—
85H(5,𝔽2)—2771no—
86#137—5541no—
87C4⋊Q8—21no—
88#126—22no—
89#125—21no—
90#124C2×D27711yescharacteristic
91#123—5542no—
92#123C2×D27712yes—
93C22⋊Q8—22no—
94#104—5541no—
95#100—2771no—
96#99—2772no—
97#99—5542no—
98#41—5541no—
99#41—11081no—
100C2×Q8—21no—
101C2×Q8C22⋊D27711yescharacteristic
102C2×D4—5542no—
103C2×D4—11081no—
104C2×D4C22×D27711yescharacteristic
105C22×C4C22⋊D27712yes—
106Q16—11081no—
107SD16—11081no—
108M16—5541no—
109C4⋊C4—21no—
110C4⋊C4—22no—
111C4⋊C4—22no—
112C4⋊C4—21no—
113C4⋊C4—22no—
114C4⋊C4—11082no—
115C22⋊C4—22no—
116C22⋊C4—22no—
117C22⋊C4—5542no—
118C22⋊C4—11082no—
119C22⋊C4C22⋊D27712yes—
120C42—21no—
121C42C22⋊D27711yescharacteristic
122C23—5542no—
123C23C2×C22⋊D27712yes—
124Q8—41no—
125Q8—81no—
126Q8—5541no—
127D4—41no—
128D4—5541no—
129D4—11082no—
130D4—11081no—
131C2×C4—21no—
132C2×C4—22no—
133C2×C4—21no—
134C2×C4—22no—
135C2×C4—42no—
136C2×C4—5542no—
137C2×C4—11081no—
138C2×C4—11082no—
139C2×C4C2×C22⋊D27711yescharacteristic
140C8—11081no—
141C22—22no—
142C22—42no—
143C22—11081no—
144C22—22162no—
145C22#2257984911yescharacteristic
146C4—21no—
147C4—42no—
148C4—41no—
149C4—42no—
150C4—41no—
151C4—11081no—
152C4—22162no—
153C2—21no—
154C2—42no—
155C2—22161no—
156C2#3180018911yescharacteristic
subgroupG/H
centerC2#31800189
commutator, cosocleC2×D4×C277C23
FrattiniC2×D4C22×D277
Fitting#563×C277C2
socleC2×C277B(4,𝔽2)
prime radical#22579925C22
hypercenter#563D277
hypocenterC277#2201
LCSG ▷ C2×D4×C277 ▷ C22×C277 ▷ C2×C277 ▷ C277
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #563

Complements

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

NH ≅ G/Ntype
3. #31799797C2semidirect
4. #31799792C2semidirect
5. #563×C277C2semidirect
78. C277#2201semidirect
80. #563D277semidirect

Maximal subgroups

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

subgroupindex
#318001912
#318001792
#317997972
#317997922
#563×C2772
#2201277