Finite Group Databasebeta

← #47982474

Subgroups — #47982474

Order 35456.

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

normal subgroups only

This group has 228 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#31800744C211yesmaximal, characteristic
2#31800397C211yesmaximal, characteristic, complemented
3#31800362C212yesmaximal, complemented
4#31799741C212yesmaximal, complemented
5#329×C277C211yesmaximal, characteristic, complemented
6C2×#17984553—21no—
7C2×#17984551—21no—
8#22579880C2211yescharacteristic, complemented
9#22579869—21no—
10#22579868C2212yes—
11#22579866—22no—
12#22579850—22no—
13#22579848—21no—
14#22579805—22no—
15#22579802—22no—
16#22579791C2211yescharacteristic, complemented
17#124×C277C2211yescharacteristic
18#97×C277C2212yescomplemented
19C23×D277—21no—
20C2×Dic554—21no—
21C2×Dic554D412yes—
22C2×D1108—21no—
23C2×D1108D412yes—
24C2×C22⋊D277—22no—
25#17984553—42no—
26#17984551—42no—
27C2×#14555660—21no—
28C4⋊Dic277—22no—
29C22⋊Dic277—21no—
30#17984540—22no—
31#17984540—22no—
32#17984540—22no—
33C4×Dic277—21no—
34C2×Q8×C277D411yescharacteristic
35C2×D4×C277D411yescharacteristic
36C2×C8×C277—22no—
37C22⋊C4×C277—22no—
38C42×C277C2311yescharacteristic
39C22×D277—22no—
40C22×D277—81no—
41C22×D277—82no—
42Dic554—42no—
43Dic554—41no—
44Dic554—41no—
45D1108—42no—
46D1108—81no—
47C22⋊D277—82no—
48C2×Dic277—22no—
49C2×Dic277—41no—
50#14555660—22no—
51C23×C277—21no—
52Q8×C277—41no—
53D4×C277—41no—
54C2×C4×C277—21no—
55C2×C4×C277C2×D412yes—
56C2×C4×C277C2×D411yescharacteristic
57C8×C277—42no—
58C22×C277—41no—
59C22×C277—42no—
60C22×C277#12011yescharacteristic
61C2×D277—42no—
62C2×D277—42no—
63C2×D277—42no—
64C2×D277—84no—
65C4×C277—21no—
66C4×C277—22no—
67C4×C277—41no—
68Dic277—42no—
69Dic277—81no—
70C2×C277—81no—
71C2×C277#45311yescharacteristic
72C2×C277#45512yes—
73D277—82no—
74C277#165911yescharacteristic, complemented, Sylow
75#1659—2771nomaximal, Sylow
76H(3,𝔽4)—2771no—
77#497—2771no—
78#489—2772no—
79#329—2772no—
80#329D27711yescharacteristic, complemented
81C2×Q16—5541no—
82C2×SD16—5541no—
83C4⋊Q8—2771no—
84#124—2772no—
85#124—5542no—
86#124—5541no—
87#124C2×D27711yescharacteristic
88#120—5542no—
89#120—5541no—
90#103—5542no—
91#102—5542no—
92#97—2771no—
93#97C2×D27712yescomplemented
94C24—5541no—
95C2×Q8—2772no—
96C2×Q8—5541no—
97C2×Q8D110811yescharacteristic
98C2×D4—2772no—
99C2×D4—5541no—
100C2×D4—5542no—
101C2×D4D110811yescharacteristic
102Q16—11082no—
103SD16—11082no—
104C2×C8—22no—
105C2×C8—5541no—
106C4⋊C4—5542no—
107C22⋊C4—22no—
108C22⋊C4—5542no—
109C22⋊C4—5541no—
110C22⋊C4—5542no—
111C22⋊C4—5542no—
112C42—5541no—
113C42C22×D27711yescharacteristic
114C23—21no—
115C23—5542no—
116C23—22161no—
117C23—22162no—
118Q8—41no—
119Q8—11082no—
120Q8—11081no—
121Q8—11081no—
122D4—41no—
123D4—11082no—
124D4—22161no—
125D4—22162no—
126C2×C4—21no—
127C2×C4—5542no—
128C2×C4—11081no—
129C2×C4D4×D27712yes—
130C2×C4C2×D110811yescharacteristic
131C8—42no—
132C8—5542no—
133C22—41no—
134C22—42no—
135C22—11082no—
136C22—11082no—
137C22—11082no—
138C22—22164no—
139C22#2257985011yescharacteristic
140C4—21no—
141C4—22no—
142C4—41no—
143C4—11082no—
144C4—22161no—
145C2—81no—
146C2—22162no—
147C2#3180017211yescharacteristic
148C2#3180018112yes—
subgroupG/H
centerC22#22579850
commutator, cosocleC42×C277C23
FrattiniC42C22×D277
Fitting#329×C277C2
socleC22×C277#120
prime radical#31800744C2
hypercenter#329D277
hypocenterC277#1659
LCSG ▷ C42×C277 ▷ C22×C277 ▷ C277
UCSC1 ◁ C22 ◁ C42 ◁ #329

Complements

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

NH ≅ G/Ntype
2. #31800397C2semidirect
3. #31800362C2semidirect
4. #31799741C2semidirect
5. #329×C277C2semidirect
8. #22579880C22semidirect
16. #22579791C22semidirect
18. #97×C277C22semidirect
74. C277#1659semidirect
80. #329D277semidirect
93. #97C2×D277semidirect

Maximal subgroups

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

subgroupindex
#318007442
#318003972
#318003622
#317997412
#329×C2772
#1659277