Finite Group Databasebeta

← #45666014

Subgroups — #45666014

Order 32896.

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

normal subgroups only

This group has 230 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#30657308C211yesmaximal, characteristic
2#30656721C211yesmaximal, characteristic
3#30656535C214yesmaximal, complemented
4#413×C257C211yesmaximal, characteristic
5C22×Dic514C411yescharacteristic
6C2×#16861602—21no—
7#22551590C2212yes—
8#22551579C412yes—
9#22551573—21no—
10#22551573—21no—
11#22551571—21no—
12#22551571—21no—
13#22551551C411yescharacteristic
14#22551511C2212yes—
15#22551509—21no—
16C2×#41×C257C2211yescharacteristic
17C2×M16×C257—21no—
18#101×C257C2212yes—
19#100×C257—21no—
20C2×Dic514—21no—
21C2×Dic514—41no—
22C2×Dic514—41no—
23C2×Dic514C2×C412yes—
24C22×Dic257—21no—
25#16861602—22no—
26#16861602—41no—
27C2×#14523339—41no—
28C4⋊Dic257—21no—
29#16861597—22no—
30#16861597—22no—
31C22⋊Dic257—22no—
32C22⋊Dic257—22no—
33C22⋊Dic257—21no—
34C4×Dic257C2×C412yes—
35#41×C257—41no—
36#41×C257—41no—
37C2×Q8×C257C2311yescharacteristic
38C2×D4×C257C2×C411yescharacteristic
39C2×D4×C257D411yescharacteristic
40C2×D4×C257D411yescharacteristic
41C22×C4×C257C2×C411yescharacteristic
42C22×C4×C257D411yescharacteristic
43C22×C4×C257D411yescharacteristic
44M16×C257—22no—
45M16×C257—41no—
46C2×C8×C257—41no—
47Dic514—41no—
48Dic514—41no—
49Dic514—42no—
50C2×Dic257—22no—
51C2×Dic257—22no—
52C2×Dic257—42no—
53#14523339—42no—
54C23×C257C22⋊C411yescharacteristic
55C23×C257C22⋊C411yescharacteristic
56C23×C257C22⋊C411yescharacteristic
57Q8×C257—41no—
58D4×C257—41no—
59D4×C257—41no—
60D4×C257—41no—
61C2×C4×C257—21no—
62C2×C4×C257—21no—
63C2×C4×C257—21no—
64C2×C4×C257—21no—
65C2×C4×C257—21no—
66C2×C4×C257—21no—
67C2×C4×C257C22⋊C411yescharacteristic
68C2×C4×C257C22×C411yescharacteristic
69C2×C4×C257C2×D411yescharacteristic
70C2×C4×C257C2×D411yescharacteristic
71C8×C257—42no—
72C22×C257—21no—
73C22×C257—21no—
74C22×C257—21no—
75C22×C257—41no—
76C22×C257—41no—
77C22×C257—41no—
78C22×C257C2×C22⋊C411yescharacteristic
79C4×C257—21no—
80C4×C257—21no—
81C4×C257—21no—
82C4×C257#9911yescharacteristic
83C4×C257#9911yescharacteristic
84Dic257—42no—
85Dic257—42no—
86C2×C257—21no—
87C2×C257—41no—
88C2×C257—41no—
89C2×C257—41no—
90C2×C257C2×#9911yescharacteristic
91C257#213511yescharacteristic, complemented, Sylow
92#2135—2571nomaximal, Sylow
93#536—2571no—
94#413—2571no—
95#413D25711yescharacteristic
96#356—2574no—
97C2×#41C2×D25711yescharacteristic
98C22×Q8—2571no—
99C2×M16—21no—
100C2×M16—5141no—
101C4⋊Q8—2572no—
102#124—2572no—
103#123—5141no—
104#123—5141no—
105C22⋊Q8—5141no—
106C22⋊Q8—5141no—
107#117—2571no—
108#101—2572no—
109#101C2×D25712yes—
110#100—21no—
111#100—5141no—
112#41—41no—
113#41—41no—
114C2×Q8—2572no—
115C2×Q8—5141no—
116C2×Q8—10281no—
117C2×Q8—10281no—
118C2×Q8C22×D25711yescharacteristic
119C2×D4C4×D25711yescharacteristic
120C2×D4C22⋊D25711yescharacteristic
121C2×D4D102811yescharacteristic
122C22×C4—5141no—
123C22×C4C4×D25711yescharacteristic
124C22×C4C22⋊D25711yescharacteristic
125C22×C4D102811yescharacteristic
126M16—22no—
127M16—5142no—
128M16—41no—
129M16—10281no—
130C2×C8—41no—
131C2×C8—10281no—
132C4⋊C4—5141no—
133C4⋊C4—5142no—
134C4⋊C4—5142no—
135C22⋊C4—5141no—
136C22⋊C4—5142no—
137C22⋊C4—5142no—
138C42—2572no—
139C23#1686159511yescharacteristic
140C23#1686159511yescharacteristic
141C23#1686159511yescharacteristic
142Q8—41no—
143Q8—10282no—
144Q8—10281no—
145Q8—10281no—
146D4—41no—
147D4—41no—
148D4—41no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—5142no—
156C2×C4—5142no—
157C2×C4—10282no—
158C2×C4#1686159511yescharacteristic
159C2×C4C2×C4×D25711yescharacteristic
160C2×C4C2×C22⋊D25711yescharacteristic
161C2×C4C2×D102811yescharacteristic
162C8—42no—
163C8—10282no—
164C22—21no—
165C22—21no—
166C22—21no—
167C22—41no—
168C22—41no—
169C22—41no—
170C22C2×#1686159511yescharacteristic
171C4—21no—
172C4—21no—
173C4—21no—
174C4—10282no—
175C4—10282no—
176C4#2255150611yescharacteristic
177C4#2255150611yescharacteristic
178C2—21no—
179C2—41no—
180C2—41no—
181C2—41no—
182C2C2×#2255150611yescharacteristic
subgroupG/H
centerC2C2×#22551506
commutatorC2×C4×C257C22×C4
FrattiniC2×Q8C22×D257
Fitting#413×C257C2
socleC2×C257C2×#99
prime radicalC2×#41×C257C22
cosocleC2×Q8×C257C23
hypercenter#413D257
hypocenterC257#2135
LCSG ▷ C2×C4×C257 ▷ C22×C257 ▷ C2×C257 ▷ C257
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ #413

Complements

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

NH ≅ G/Ntype
3. #30656535C2semidirect
91. C257#2135semidirect

Maximal subgroups

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

subgroupindex
#306573082
#306567212
#306565352
#413×C2572
#2135257