Finite Group Databasebeta

← #47983714

Subgroups — #47983714

Order 35456.

340 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 89 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#31800566C211yesmaximal, characteristic
2#31799986C211yesmaximal, characteristic, complemented
3C2×#22579796C212yesmaximal, complemented
4#31799774C212yesmaximal, complemented
5#568×C277C211yesmaximal, characteristic, complemented
6C22×D1108C2211yescharacteristic
7C2×#17984547C2212yes—
8#22579876—21no—
9#22579869—21no—
10#22579854—22no—
11#22579848C414yes—
12#22579839C2211yescharacteristic
13#22579799—21no—
14#22579797—21no—
15#22579796—24no—
16C2×#41×C277C2211yescharacteristic
17#131×C277—24no—
18C2×M16×C277—22no—
19C2×M16×C277C2212yes—
20C23×D277C2×C412yes—
21C2×D1108—22no—
22C2×D1108—21no—
23C2×D1108—21no—
24C2×D1108—22no—
25C2×C22⋊D277—24no—
26#17984547—22no—
27#17984547—22no—
28C2×#14555660—22no—
29C4⋊Dic277—21no—
30C22⋊Dic277C2×C412yes—
31#17984540—24no—
32C4×Dic277—21no—
33#41×C277—22no—
34#41×C277—22no—
35C2×Q8×C277D411yescharacteristic
36C2×D4×C277C2×C412yes—
37C2×D4×C277D411yescharacteristic
38C22×C4×C277D412yes—
39C22×C4×C277C2311yescharacteristic
40M16×C277—24no—
41M16×C277—22no—
42M16×C277D414yescomplemented
43C2×C8×C277—24no—
44C2×C8×C277—22no—
45C22×D277—22no—
46C22×D277—24no—
47C22×D277—82no—
48D1108—41no—
49D1108—42no—
50D1108—41no—
51C22⋊D277—84no—
52C2×Dic277—22no—
53#14555660—42no—
54C23×C277C22⋊C412yes—
55C23×C277C22×C411yescharacteristic
56Q8×C277—22no—
57D4×C277—24no—
58D4×C277—22no—
59C2×C4×C277—22no—
60C2×C4×C277—22no—
61C2×C4×C277C22⋊C412yes—
62C2×C4×C277C2×D412yes—
63C2×C4×C277C2×D411yescharacteristic
64C2×C4×C277C2×D411yescharacteristic
65C2×C4×C277#4112yes—
66C8×C277—24no—
67C22×C277—22no—
68C22×C277—41no—
69C22×C277—42no—
70C22×C277C2×C22⋊C411yescharacteristic
71C22×C277C4×D412yes—
72C2×D277—44no—
73C2×D277—42no—
74C2×D277—84no—
75C4×C277—22no—
76C4×C277#12011yescharacteristic
77C4×C277#12112yes—
78C4×C277#12311yescharacteristic
79Dic277—42no—
80C2×C277—21no—
81C2×C277—22no—
82C2×C277—42no—
83C2×C277#38611yescharacteristic
84D277—82no—
85C277#191411yescharacteristic, complemented, Sylow
86#1914—2771nomaximal, Sylow
87#568D27711yescharacteristic, complemented
88#535—2771no—
89#413—2771no—
90C2×#100—2772no—
91#344—2772no—
92C2×#41C2×D27711yescharacteristic
93C22×D4—2771no—
94#131—24no—
95C2×M16—22no—
96C2×M16—2772no—
97C2×M16C2×D27712yes—
98#127—5541no—
99#124—5541no—
100#121—5542no—
101#120—2774no—
102#117—2771no—
103#101—5541no—
104#100—5541no—
105#100—5544no—
106C24—2772no—
107#41—22no—
108#41—22no—
109C2×Q8D110811yescharacteristic
110C2×D4—5544no—
111C2×D4—5542no—
112C2×D4—5541no—
113C2×D4—5541no—
114C2×D4—5542no—
115C2×D4C4×D27712yes—
116C2×D4D110811yescharacteristic
117C22×C4C22⋊D27712yes—
118C22×C4C22×D27711yescharacteristic
119M16—24no—
120M16—22no—
121M16—5542no—
122M16—5542no—
123M16C22⋊D27714yescomplemented
124C2×C8—24no—
125C2×C8—22no—
126C2×C8—5542no—
127C4⋊C4—5541no—
128C22⋊C4—2772no—
129C22⋊C4—5544no—
130C42—5541no—
131C23—5542no—
132C23—5544no—
133C23—22162no—
134C23#1798454012yes—
135C23C2×C4×D27711yescharacteristic
136Q8—22no—
137D4—24no—
138D4—22no—
139D4—11081no—
140D4—11082no—
141D4—11081no—
142D4—22164no—
143C2×C4—22no—
144C2×C4—22no—
145C2×C4—5542no—
146C2×C4#1798454012yes—
147C2×C4C2×C22⋊D27712yes—
148C2×C4C2×C22⋊D27711yescharacteristic
149C2×C4C2×D110811yescharacteristic
150C2×C4#1798456312yes—
151C8—24no—
152C8—11082no—
153C22—22no—
154C22—41no—
155C22—42no—
156C22—11084no—
157C22—11082no—
158C22—22164no—
159C22C2×#1798454011yescharacteristic
160C22C4×C22⋊D27712yes—
161C4—22no—
162C4—11082no—
163C4#2257984911yescharacteristic
164C4#2257985412yes—
165C4#2257986311yescharacteristic
166C2—21no—
167C2—22no—
168C2—42no—
169C2—22162no—
170C2#3179988111yescharacteristic
subgroupG/H
centerC2#31799881
commutatorC23×C277C22×C4
FrattiniC22×C4C22×D277
Fitting#568×C277C2
socleC2×C277#386
prime radical#31800566C2
cosocleC22×C4×C277C23
hypercenter#568D277
hypocenterC277#1914
LCSG ▷ C23×C277 ▷ C2×C277 ▷ C277
UCSC1 ◁ C2 ◁ C22×C4 ◁ #568

Complements

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

NH ≅ G/Ntype
2. #31799986C2semidirect
3. C2×#22579796C2semidirect
4. #31799774C2semidirect
5. #568×C277C2semidirect
42. M16×C277D4semidirect
85. C277#1914semidirect
87. #568D277semidirect
123. M16C22⋊D277semidirect

Maximal subgroups

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

subgroupindex
#318005662
#317999862
C2×#225797962
#317997742
#568×C2772
#1914277