Finite Group Databasebeta

← #45665993

Subgroups — #45665993

Order 32896.

238 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 55 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#30657307C211yesmaximal, characteristic
2#30656709C212yesmaximal
3#30656532C211yesmaximal, characteristic, complemented
4#30656531C212yesmaximal, complemented
5#354×C257C211yesmaximal, characteristic, complemented
6C2×#16861617C2211yescharacteristic
7C2×Q8×D257C411yescharacteristic, complemented
8#22551592C411yescharacteristic, complemented
9#22551590C411yescharacteristic, complemented
10#22551579C2211yescharacteristic
11#22551574—22no—
12#22551569—22no—
13#22551549C411yescharacteristic, complemented
14#22551548—22no—
15#22551507C2212yes—
16#22551506—22no—
17#124×C257C2211yescharacteristic
18#99×C257C2212yes—
19#16861617—42no—
20C2×Dic514—21no—
21C2×Dic514C2×C411yescharacteristic
22C2×Dic514C2×C411yescharacteristic
23Q8×D257—42no—
24C2×C22⋊D257D412yes—
25C22×Dic257D412yes—
26C2×C4×D257—21no—
27C2×C4×D257C2×C411yescharacteristic
28C4⋊Dic257—22no—
29#16861597—21no—
30#16861597—22no—
31#16861597—42no—
32C22⋊Dic257—22no—
33C22⋊Dic257—22no—
34#16861595—21no—
35C4×Dic257—42no—
36C4×Dic257C2×C411yescharacteristic
37C2×Q8×C257C2×C411yescharacteristic
38C2×D4×C257C2311yescharacteristic
39C22⋊C4×C257—22no—
40C22⋊C4×C257—22no—
41C42×C257C2×C411yescharacteristic
42C22×D257C22⋊C411yescharacteristic
43Dic514—41no—
44Dic514—42no—
45Dic514—41no—
46C22⋊D257—42no—
47C2×Dic257—22no—
48C2×Dic257—22no—
49C2×Dic257—21no—
50C2×Dic257—21no—
51C2×Dic257—42no—
52C2×Dic257C22⋊C412yes—
53C2×Dic257C22⋊C411yescharacteristic
54C4×D257—22no—
55C4×D257—42no—
56C23×C257C2×D412yes—
57Q8×C257—41no—
58D4×C257—41no—
59C2×C4×C257—21no—
60C2×C4×C257—21no—
61C2×C4×C257—42no—
62C2×C4×C257C22×C411yescharacteristic
63C22×C257—22no—
64C22×C257—42no—
65C22×C257C2×C22⋊C411yescharacteristic
66C2×D257—21no—
67C2×D257—41no—
68C4×C257—21no—
69C4×C257—41no—
70C4×C257—41no—
71C4×C257—82no—
72Dic257—22no—
73Dic257—41no—
74Dic257—41no—
75Dic257—82no—
76Dic257#9912yes—
77C2×C257—21no—
78C2×C257—42no—
79C2×C257C2×#9911yescharacteristic
80D257—41no—
81C257#212911yescharacteristic, complemented, Sylow
82#2129—2571nomaximal, Sylow
83#536—2571no—
84#410—2572no—
85#354—2571no—
86#354—2572no—
87#354D25711yescharacteristic, complemented
88C2×#41—2571no—
89C22×Q8—2571no—
90C4⋊Q8—2571no—
91C4⋊Q8—2571no—
92#124—2571no—
93#124C2×D25711yescharacteristic
94#123—5142no—
95C22⋊Q8—5142no—
96#117—2571no—
97#117—5142no—
98#99—2572no—
99#99—5142no—
100#99C2×D25712yes—
101#41—10282no—
102C2×Q8—2571no—
103C2×Q8—2571no—
104C2×Q8—5141no—
105C2×Q8—10282no—
106C2×Q8C4×D25711yescharacteristic
107C2×D4—2572no—
108C2×D4C22×D25711yescharacteristic
109C22×C4—2572no—
110C22×C4—2571no—
111C22×C4—5141no—
112C4⋊C4—5141no—
113C4⋊C4—5142no—
114C4⋊C4—5142no—
115C4⋊C4—10282no—
116C22⋊C4—22no—
117C22⋊C4—22no—
118C22⋊C4—5141no—
119C22⋊C4—5142no—
120C22⋊C4—5142no—
121C42—2571no—
122C42—10282no—
123C42C4×D25711yescharacteristic
124C23—2571no—
125C23D4×D25712yes—
126Q8—41no—
127Q8—10282no—
128Q8—10281no—
129Q8—10281no—
130D4—41no—
131D4—10282no—
132C2×C4—2571no—
133C2×C4—2572no—
134C2×C4—21no—
135C2×C4—21no—
136C2×C4—5142no—
137C2×C4—5142no—
138C2×C4—5142no—
139C2×C4—5141no—
140C2×C4—5141no—
141C2×C4—42no—
142C2×C4—10282no—
143C2×C4—10282no—
144C2×C4C2×C4×D25711yescharacteristic
145C22—22no—
146C22—5141no—
147C22—42no—
148C22—10281no—
149C22C22⋊C4×D25711yescharacteristic
150C4—2572no—
151C4—21no—
152C4—5142no—
153C4—41no—
154C4—41no—
155C4—10281no—
156C4—10281no—
157C4—82no—
158C4—20562no—
159C2—21no—
160C2—42no—
161C2—10281no—
162C2#99×D25711yescharacteristic
subgroupG/H
centerC2#99×D257
commutatorC2×C4×C257C22×C4
FrattiniC2×D4C22×D257
Fitting#354×C257C2
socleC2×C257C2×#99
prime radicalC2×#16861617C22
cosocleC2×D4×C257C23
hypercenter#354D257
hypocenterC257#2129
LCSG ▷ C2×C4×C257 ▷ C22×C257 ▷ C2×C257 ▷ C257
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #354

Complements

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

NH ≅ G/Ntype
3. #30656532C2semidirect
4. #30656531C2semidirect
5. #354×C257C2semidirect
7. C2×Q8×D257C4semidirect
8. #22551592C4semidirect
9. #22551590C4semidirect
13. #22551549C4semidirect
81. C257#2129semidirect
87. #354D257semidirect

Maximal subgroups

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

subgroupindex
#306573072
#306567092
#306565322
#306565312
#354×C2572
#2129257