Finite Group Databasebeta

← #47985121

Subgroups — #47985121

Order 35456.

220 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 47 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#31800804C211yesmaximal, characteristic, complemented
2C2×#22579821C211yesmaximal, characteristic, complemented
3#31800269C211yesmaximal, characteristic, complemented
4#31799816C211yesmaximal, characteristic, complemented
5#31799809C211yesmaximal, characteristic, complemented
6#31799787C211yesmaximal, characteristic, complemented
7#509×C277C211yesmaximal, characteristic, complemented
8#22579926—21no—
9#22579907—22no—
10#22579906—21no—
11#22579905—21no—
12#22579903—21no—
13C2×D2216C2211yescharacteristic
14D8×D277—21no—
15#22579886C2211yescharacteristic
16#22579821—22no—
17C2×#17984538C2211yescharacteristic, complemented
18#22579816C2211yescharacteristic, complemented
19#22579797—21no—
20#135×C277C2211yescharacteristic
21C2×D8×C277C2211yescharacteristic, complemented
22SD32×C277—21no—
23D16×C277—21no—
24M32×C277C2211yescharacteristic, complemented
25#17984564—22no—
26#17984563—41no—
27#17984563D411yescharacteristic
28C2×D1108—21no—
29C2×D1108—41no—
30D4×D277—22no—
31D4×D277—42no—
32#17984551—41no—
33#17984550—22no—
34D2216—21no—
35D2216D412yes—
36#17984548—22no—
37#17984548—41no—
38#17984547—21no—
39#17984547D411yescharacteristic
40#17984546—21no—
41C8×D277—21no—
42#17984538—21no—
43#41×C277—21no—
44C2×D4×C277—21no—
45Q16×C277D411yescharacteristic
46SD16×C277—21no—
47D8×C277—21no—
48D8×C277—21no—
49D8×C277D411yescharacteristic
50C2×C8×C277C2311yescharacteristic
51C16×C277—21no—
52C22×D277—42no—
53C22×D277—41no—
54Dic554D811yescharacteristic
55D1108—22no—
56D1108—42no—
57D1108—41no—
58D1108D811yescharacteristic
59C22⋊D277—21no—
60C22⋊D277—42no—
61C4×D277—21no—
62C4×D277—42no—
63#14555660—21no—
64#14555660—41no—
65C23×C277—41no—
66Q8×C277—21no—
67D4×C277—21no—
68D4×C277—41no—
69D4×C277—41no—
70D4×C277—41no—
71C2×C4×C277—41no—
72C2×C4×C277C2×D411yescharacteristic
73C8×C277C2×D411yescharacteristic
74C8×C277C2×D411yescharacteristic
75C22×C277—41no—
76C22×C277—81no—
77C22×C277—161no—
78C22×C277AGL(1,ℤ/8)11yescharacteristic
79C2×D277—21no—
80C2×D277—42no—
81C2×D277—162no—
82C2×D277—161no—
83C4×C277—41no—
84C4×C277#12011yescharacteristic
85C4×C277C2×D811yescharacteristic
86Dic277—21no—
87C2×C277—21no—
88C2×C277—81no—
89C2×C277—161no—
90C2×C277#44711yescharacteristic
91D277—41no—
92D277—82no—
93C277#219211yescharacteristic, complemented, Sylow
94#2192—2771nomaximal, Sylow
95#576—2771no—
96#509D27711yescharacteristic, complemented
97C2×D16—2771no—
98#472—2771no—
99#361—2771no—
100#359—2771no—
101#350—2771no—
102H(5,𝔽2)—5541no—
103AGL(1,ℤ/8)—5541no—
104AGL(1,ℤ/8)—5542no—
105AGL(1,ℤ/8)—5541no—
106#135—5541no—
107#135C2×D27711yescharacteristic
108C2×D8—2771no—
109C2×D8—5541no—
110C2×D8C2×D27711yescharacteristic, complemented
111#131—2771no—
112SD32—21no—
113D16—21no—
114D16—5542no—
115M32C2×D27711yescharacteristic, complemented
116C2×C16—2771no—
117#108—2771no—
118#100—5541no—
119#41—21no—
120#41—2771no—
121#41—5542no—
122#41—11081no—
123C2×D4—21no—
124C2×D4—5542no—
125C2×D4—5541no—
126C2×D4—11081no—
127C2×D4—11082no—
128Q16C22⋊D27711yescharacteristic
129SD16—21no—
130SD16—5542no—
131SD16—11081no—
132D8—21no—
133D8—21no—
134D8—2772no—
135D8—5542no—
136D8—5541no—
137D8—11081no—
138D8C22⋊D27711yescharacteristic
139M16—2771no—
140M16—5541no—
141M16—5541no—
142C2×C8—5541no—
143C2×C8C22×D27711yescharacteristic
144C16—21no—
145C16—5541no—
146C23—41no—
147C23—11082no—
148C23—11081no—
149Q8—21no—
150Q8—2771no—
151D4—21no—
152D4—41no—
153D4—41no—
154D4—41no—
155D4—2771no—
156D4—5541no—
157D4—5542no—
158D4—11082no—
159D4—11082no—
160D4—11081no—
161C2×C4—41no—
162C2×C4—5541no—
163C2×C4—11082no—
164C2×C4D4×D27711yescharacteristic
165C8—5541no—
166C8—11081no—
167C8D4×D27711yescharacteristic
168C8C2×C22⋊D27711yescharacteristic
169C22—41no—
170C22—81no—
171C22—161no—
172C22—5541no—
173C22—44321no—
174C22—44322no—
175C22—11082no—
176C22#2257990411yescharacteristic
177C4—41no—
178C4—5541no—
179C4#2257984811yescharacteristic
180C4D8×D27711yescharacteristic
181C2—21no—
182C2—81no—
183C2—161no—
184C2—11081no—
185C2—22162no—
186C2#3180013411yescharacteristic
subgroupG/H
centerC2#31800134
commutator, cosocleC2×C8×C277C23
FrattiniC2×C8C22×D277
Fitting#509×C277C2
socleC2×C277#447
hypercenter#509D277
hypocenterC277#2192
LCSG ▷ C2×C8×C277 ▷ C4×C277 ▷ C2×C277 ▷ C277
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #509

Complements

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

NH ≅ G/Ntype
1. #31800804C2semidirect
2. C2×#22579821C2semidirect
3. #31800269C2semidirect
4. #31799816C2semidirect
5. #31799809C2semidirect
6. #31799787C2semidirect
7. #509×C277C2semidirect
17. C2×#17984538C22semidirect
18. #22579816C22semidirect
21. C2×D8×C277C22semidirect
24. M32×C277C22semidirect
93. C277#2192semidirect
96. #509D277semidirect
110. C2×D8C2×D277semidirect
115. M32C2×D277semidirect

Maximal subgroups

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

subgroupindex
#318008042
C2×#225798212
#318002692
#317998162
#317998092
#317997872
#509×C2772
#2192277