Finite Group Databasebeta

← #1804×C251 #44546391

Subgroups — #1804×C251 #44546391

Order 32128.

356 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 136 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
1C2×C4×D4×C251C211yesmaximal, characteristic
2C2×C4⋊C8×C251C211yesmaximal, characteristic, complemented
3C2×#102×C251C212yesmaximal
4#382×C251C211yesmaximal, characteristic, complemented
5#340×C251C212yesmaximal, complemented
6C22×D4×C251C2211yescharacteristic
7C23×C4×C251—21no—
8C22×C8×C251C2212yes—
9C4×D4×C251—22no—
10C4×D4×C251C414yescomplemented
11C2×C4⋊C4×C251C2212yes—
12C2×C4⋊C4×C251C2211yescharacteristic
13C2×C22⋊C4×C251—21no—
14C2×C42×C251C2211yescharacteristic
15C4⋊C8×C251—22no—
16#102×C251—24no—
17#95×C251—21no—
18#95×C251—21no—
19C24×C251—21no—
20C2×D4×C251—21no—
21C2×D4×C251—21no—
22C2×D4×C251—21no—
23C2×D4×C251C2×C412yes—
24C2×D4×C251D412yes—
25C2×D4×C251Q812yes—
26C22×C4×C251—21no—
27C22×C4×C251—24no—
28C22×C4×C251—22no—
29C22×C4×C251—44no—
30C22×C4×C251D411yescharacteristic
31C22×C4×C251D411yescharacteristic
32C22×C4×C251C2311yescharacteristic
33C2×C8×C251—22no—
34C2×C8×C251—22no—
35C2×C8×C251—22no—
36C4⋊C4×C251—21no—
37C4⋊C4×C251—22no—
38C4⋊C4×C251—22no—
39C4⋊C4×C251C2×C412yes—
40C22⋊C4×C251—24no—
41C42×C251C2×C412yes—
42C42×C251D412yes—
43C23×C251—22no—
44C23×C251—22no—
45C23×C251—22no—
46C23×C251—42no—
47C23×C251—42no—
48C23×C251C2×D411yescharacteristic
49D4×C251—22no—
50D4×C251—22no—
51D4×C251—22no—
52D4×C251C4⋊C414yes—
53C2×C4×C251—22no—
54C2×C4×C251—22no—
55C2×C4×C251—22no—
56C2×C4×C251—48no—
57C2×C4×C251—42no—
58C2×C4×C251—42no—
59C2×C4×C251—42no—
60C2×C4×C251—42no—
61C2×C4×C251C22⋊C412yes—
62C2×C4×C251C22⋊C412yes—
63C2×C4×C251C22×C411yescharacteristic
64C2×C4×C251C2×D411yescharacteristic
65C2×C4×C251C2×D411yescharacteristic
66C2×C4×C251C2×Q811yescharacteristic
67C2×C4×C251#4111yescharacteristic
68C2×C4×C251#4111yescharacteristic
69C8×C251—42no—
70C22×C251—24no—
71C22×C251—42no—
72C22×C251—42no—
73C22×C251—44no—
74C22×C251—44no—
75C22×C251C2×C22⋊C411yescharacteristic
76C22×C251#12111yescharacteristic
77C22×C251C22⋊Q811yescharacteristic
78C22×C251#13512yes—
79C22×C251AGL(1,ℤ/8)12yes—
80C4×C251—22no—
81C4×C251—22no—
82C4×C251—42no—
83C4×C251—82no—
84C4×C251C2×C4⋊C411yescharacteristic
85C4×C251#11711yescharacteristic
86C4×C251#12111yescharacteristic
87C4×C251C22⋊Q811yescharacteristic
88C2×C251—44no—
89C2×C251#38011yescharacteristic
90C2×C251#41611yescharacteristic
91C2×C251#41811yescharacteristic
92C2×C251#46312yes—
93C2×C251#47812yes—
94C251#180411yescharacteristic, direct product factor, Sylow
95#1804C25111yesmaximal, characteristic, direct product factor, Sylow
96C2×C4×D4C2×C25111yescharacteristic
97C2×C4⋊C8C2×C25111yescharacteristic, complemented
98C2×#102C2×C25112yes—
99#382C2×C25111yescharacteristic, complemented
100#340C2×C25112yescomplemented
101C22×D4C22×C25111yescharacteristic
102C23×C4—21no—
103C22×C8C22×C25112yes—
104C4×D4—22no—
105C4×D4C4×C25114yescomplemented
106C2×C4⋊C4C22×C25112yes—
107C2×C4⋊C4C22×C25111yescharacteristic
108C2×C22⋊C4—21no—
109C2×C42C22×C25111yescharacteristic
110C4⋊C8—22no—
111#102—24no—
112#95—21no—
113#95—21no—
114C24—21no—
115C2×D4—21no—
116C2×D4—21no—
117C2×D4—21no—
118C2×D4C2×C4×C25112yes—
119C2×D4D4×C25112yes—
120C2×D4Q8×C25112yes—
121C22×C4—21no—
122C22×C4—24no—
123C22×C4—22no—
124C22×C4—44no—
125C22×C4D4×C25111yescharacteristic
126C22×C4D4×C25111yescharacteristic
127C22×C4C23×C25111yescharacteristic
128C2×C8—22no—
129C2×C8—22no—
130C2×C8—22no—
131C4⋊C4—21no—
132C4⋊C4—22no—
133C4⋊C4—22no—
134C4⋊C4C2×C4×C25112yes—
135C22⋊C4—24no—
136C42C2×C4×C25112yes—
137C42D4×C25112yes—
138C23—22no—
139C23—22no—
140C23—22no—
141C23—42no—
142C23—42no—
143C23C2×D4×C25111yescharacteristic
144D4—22no—
145D4—22no—
146D4—22no—
147D4C4⋊C4×C25114yes—
148C2×C4—22no—
149C2×C4—22no—
150C2×C4—22no—
151C2×C4—48no—
152C2×C4—42no—
153C2×C4—42no—
154C2×C4—42no—
155C2×C4—42no—
156C2×C4C22⋊C4×C25112yes—
157C2×C4C22⋊C4×C25112yes—
158C2×C4C22×C4×C25111yescharacteristic
159C2×C4C2×D4×C25111yescharacteristic
160C2×C4C2×D4×C25111yescharacteristic
161C2×C4C2×Q8×C25111yescharacteristic
162C2×C4#41×C25111yescharacteristic
163C2×C4#41×C25111yescharacteristic
164C8—42no—
165C22—24no—
166C22—42no—
167C22—42no—
168C22—44no—
169C22—44no—
170C22C2×C22⋊C4×C25111yescharacteristic
171C22#121×C25111yescharacteristic
172C22C22⋊Q8×C25111yescharacteristic
173C22#135×C25112yes—
174C22AGL(1,ℤ/8)×C25112yes—
175C4—22no—
176C4—22no—
177C4—42no—
178C4—82no—
179C4C2×C4⋊C4×C25111yescharacteristic
180C4#117×C25111yescharacteristic
181C4#121×C25111yescharacteristic
182C4C22⋊Q8×C25111yescharacteristic
183C2—44no—
184C2#380×C25111yescharacteristic
185C2#416×C25111yescharacteristic
186C2#418×C25111yescharacteristic
187C2#463×C25112yes—
188C2#478×C25112yes—
subgroupG/H
center, socleC23×C251C2×D4
commutatorC2×C4C22×C4×C251
Frattini, cosocleC22×C4C23×C251
prime radicalC22×D4×C251C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C251 ◁ C2×C42×C251 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊C8×C251C2semidirect
4. #382×C251C2semidirect
5. #340×C251C2semidirect
10. C4×D4×C251C4semidirect
94. C251#1804direct (N × H)
95. #1804C251direct (N × H)
97. C2×C4⋊C8C2×C251semidirect
99. #382C2×C251semidirect
100. #340C2×C251semidirect
105. C4×D4C4×C251semidirect

Maximal subgroups

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

subgroupindex
C2×C4×D4×C2512
C2×C4⋊C8×C2512
C2×#102×C2512
#382×C2512
#340×C2512
#1804251