Finite Group Databasebeta

← #45662949

Subgroups — #45662949

Order 32896.

200 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 65 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#30657271C211yesmaximal, characteristic
2#30656794C211yesmaximal, characteristic, complemented
3#30656478C212yesmaximal, complemented
4#30656476C212yesmaximal, complemented
5#432×C257C211yesmaximal, characteristic, complemented
6#22551586—21no—
7#22551584C2211yescharacteristic
8#22551580C2211yescharacteristic
9#22551575—21no—
10#22551571C411yescharacteristic
11#22551566C411yescharacteristic
12C4×Dic514C411yescharacteristic
13C4×D1028C411yescharacteristic
14C4×C22⋊D257—21no—
15#22551549—21no—
16C4⋊#14523339—21no—
17C22⋊#14523339—21no—
18#22551503C2212yes—
19C2×C42×C257C2211yescharacteristic
20C4⋊C8×C257—21no—
21C22⋊C8×C257—21no—
22#97×C257C2212yes—
23C2×Dic514C2×C411yescharacteristic
24C2×D1028C2×C411yescharacteristic
25C2×C22⋊D257—21no—
26C2×C4×D257—21no—
27C2×#14523339—22no—
28C4⋊Dic257C2×C411yescharacteristic
29C4⋊Dic257C2×C411yescharacteristic
30#16861597—21no—
31#16861597—21no—
32#16861597—21no—
33#16861597—21no—
34C22⋊Dic257—21no—
35#16861595—21no—
36#16861595—21no—
37#16861595—21no—
38#16861595—21no—
39C4×Dic257—21no—
40C22×C4×C257C2×C411yescharacteristic
41C22×C4×C257D411yescharacteristic
42C22×C4×C257D411yescharacteristic
43C2×C8×C257—22no—
44C42×C257C2×C411yescharacteristic
45C42×C257D411yescharacteristic
46C42×C257D411yescharacteristic
47C42×C257C2311yescharacteristic
48C22×D257—21no—
49Dic514—41no—
50D1028—41no—
51C22⋊D257—42no—
52C2×Dic257—21no—
53C2×Dic257—21no—
54C2×Dic257—21no—
55C4×D257—42no—
56#14523339—42no—
57C23×C257C22⋊C411yescharacteristic
58C2×C4×C257—41no—
59C2×C4×C257—41no—
60C2×C4×C257—41no—
61C2×C4×C257C22⋊C411yescharacteristic
62C2×C4×C257C22⋊C411yescharacteristic
63C2×C4×C257C22⋊C411yescharacteristic
64C2×C4×C257C22×C411yescharacteristic
65C2×C4×C257C2×D411yescharacteristic
66C2×C4×C257C2×D411yescharacteristic
67C8×C257—42no—
68C22×C257—21no—
69C22×C257—21no—
70C22×C257—21no—
71C22×C257C2×C22⋊C411yescharacteristic
72C2×D257—41no—
73C2×D257—41no—
74C2×D257—41no—
75C4×C257—21no—
76C4×C257—21no—
77C4×C257—21no—
78C4×C257—21no—
79C4×C257—21no—
80C4×C257—21no—
81Dic257—41no—
82Dic257—41no—
83Dic257—41no—
84C2×C257—41no—
85C2×C257#41311yescharacteristic
86C2×C257#42111yescharacteristic
87C2×C257#42111yescharacteristic
88D257—81no—
89C257#153511yescharacteristic, complemented, Sylow
90#1535—2571nomaximal, Sylow
91#529—2571no—
92#432—2571no—
93#432D25711yescharacteristic, complemented
94#330—2572no—
95#329—2572no—
96#126—5141no—
97#125—2571no—
98#124—2571no—
99#123—5141no—
100C22⋊Q8—2571no—
101#121—2571no—
102C4×Q8—2571no—
103C4×D4—2571no—
104C4×D4—5141no—
105#117—5141no—
106C2×C42C2×D25711yescharacteristic
107C4⋊C8—21no—
108C4⋊C8—5141no—
109C22⋊C8—21no—
110C22⋊C8—5141no—
111#97—2572no—
112#97C2×D25712yes—
113C2×Q8—2571no—
114C2×D4—2571no—
115C2×D4—5141no—
116C22×C4—5141no—
117C22×C4C4×D25711yescharacteristic
118C22×C4C22⋊D25711yescharacteristic
119C22×C4D102811yescharacteristic
120C2×C8—22no—
121C2×C8—5142no—
122C4⋊C4—2571no—
123C4⋊C4—2571no—
124C4⋊C4—5141no—
125C4⋊C4—5141no—
126C4⋊C4—5141no—
127C4⋊C4—5141no—
128C22⋊C4—5141no—
129C22⋊C4—5141no—
130C22⋊C4—5141no—
131C22⋊C4—5141no—
132C22⋊C4—5141no—
133C42—5141no—
134C42C4×D25711yescharacteristic
135C42C22⋊D25711yescharacteristic
136C42D102811yescharacteristic
137C42C22×D25711yescharacteristic
138C23—5141no—
139C23#1686159511yescharacteristic
140Q8—10281no—
141D4—10281no—
142D4—10282no—
143C2×C4—5141no—
144C2×C4—5141no—
145C2×C4—5141no—
146C2×C4—41no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—10282no—
150C2×C4#1686159511yescharacteristic
151C2×C4#1686159511yescharacteristic
152C2×C4#1686159511yescharacteristic
153C2×C4C2×C4×D25711yescharacteristic
154C2×C4C2×C22⋊D25711yescharacteristic
155C2×C4C2×D102811yescharacteristic
156C8—42no—
157C8—10282no—
158C22—21no—
159C22—21no—
160C22—21no—
161C22—10281no—
162C22—10281no—
163C22—10281no—
164C22C2×#1686159511yescharacteristic
165C4—21no—
166C4—21no—
167C4—21no—
168C4—21no—
169C4—21no—
170C4—21no—
171C4—10281no—
172C4—10281no—
173C4—10281no—
174C2—41no—
175C2—20561no—
176C2#3065671911yescharacteristic
177C2#3065675311yescharacteristic
178C2#3065675511yescharacteristic
subgroupG/H
centerC22C2×#16861595
commutatorC2×C4×C257C22×C4
FrattiniC42C22×D257
Fitting#432×C257C2
socleC22×C257C2×C22⋊C4
prime radical#22551566C4
cosocleC42×C257C23
hypercenter#432D257
hypocenterC257#1535
LCSG ▷ C2×C4×C257 ▷ C22×C257 ▷ C257
UCSC1 ◁ C22 ◁ C2×C42 ◁ #432

Complements

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

NH ≅ G/Ntype
2. #30656794C2semidirect
3. #30656478C2semidirect
4. #30656476C2semidirect
5. #432×C257C2semidirect
89. C257#1535semidirect
93. #432D257semidirect

Maximal subgroups

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

subgroupindex
#306572712
#306567942
#306564782
#306564762
#432×C2572
#1535257