Finite Group Databasebeta

← #44549288

Subgroups — #44549288

Order 32128.

316 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 57 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#30631837C212yesmaximal, complemented
2#30631596C211yesmaximal, characteristic
3#30631424C212yesmaximal, complemented
4#30631411C211yesmaximal, characteristic, complemented
5#543×C251C211yesmaximal, characteristic, complemented
6C22×D1004C2211yescharacteristic
7C22×C22⋊D251—21no—
8C2×#16860797—22no—
9C2×(C2×D251)⋊C4—21no—
10#22548635—22no—
11C22⋊(C251⋊C8)C2212yes—
12C251⋊#95C2211yescharacteristic
13C22⋊Q8×C251—22no—
14C22⋊Q8×C251—21no—
15C22⋊Q8×C251—22no—
16C22⋊Q8×C251C2212yes—
17C2×C22⋊C4×C251—21no—
18C2×C22⋊C4×C251C2211yescharacteristic
19C23×D251—21no—
20C2×D1004—22no—
21C2×D1004—42no—
22C2×C22⋊D251—42no—
23C2×C22⋊D251—42no—
24C2×C22⋊D251—42no—
25C22×Dic251—21no—
26#16860797—44no—
27C2×C251⋊C8—22no—
28(C2×D251)⋊C4—42no—
29C24×C251D411yescharacteristic
30C2×Q8×C251—21no—
31C2×Q8×C251D412yescomplemented
32C22×C4×C251—21no—
33C22×C4×C251—21no—
34C22×C4×C251D411yescharacteristic
35C22×C4×C251C2311yescharacteristic
36C4⋊C4×C251—21no—
37C4⋊C4×C251—22no—
38C4⋊C4×C251—22no—
39C4⋊C4×C251D412yescomplemented
40C22⋊C4×C251—22no—
41C22⋊C4×C251—22no—
42C22⋊C4×C251—22no—
43C22×D251—22no—
44C22×D251—41no—
45C22×D251—41no—
46C22×D251—41no—
47C22×D251—41no—
48C22×D251—41no—
49C22×D251—41no—
50D1004—24no—
51D1004—42no—
52C22⋊D251—48no—
53C2×Dic251—22no—
54C2×Dic251—41no—
55C2×Dic251—41no—
56C251⋊C8—42no—
57C23×C251—21no—
58C23×C251—21no—
59C23×C251—41no—
60C23×C251—41no—
61C23×C251SD1612yescomplemented
62C23×C251C2×D411yescharacteristic
63Q8×C251—42no—
64Q8×C251—81no—
65C2×C4×C251—22no—
66C2×C4×C251—21no—
67C2×C4×C251—22no—
68C2×C4×C251—41no—
69C2×C4×C251—41no—
70C2×C4×C251—81no—
71C2×C4×C251—81no—
72C2×C4×C251C2×D412yes—
73C22×C251—21no—
74C22×C251—21no—
75C22×C251—22no—
76C22×C251—42no—
77C22×C251—41no—
78C22×C251—41no—
79C22×C251—81no—
80C22×C251#12011yescharacteristic
81C22×C251C2×SD1611yescharacteristic
82C22×C251#13711yescharacteristic
83C2×D251—42no—
84C2×D251—42no—
85C2×D251—42no—
86C2×D251—81no—
87C2×D251—81no—
88C2×D251—81no—
89C2×D251—81no—
90C4×C251—22no—
91C4×C251—41no—
92C4×C251—42no—
93C4×C251—42no—
94Dic251—42no—
95C2×C251—21no—
96C2×C251—21no—
97C2×C251—42no—
98C2×C251#44811yescharacteristic
99C2×C251#45311yescharacteristic
100C2×C251B(4,𝔽2)11yescharacteristic
101D251—82no—
102C251#159611yescharacteristic, complemented, Sylow
103#1596—2511nomaximal, Sylow
104#543D25111yescharacteristic, complemented
105#450—2512no—
106#392—2511no—
107#328—2512no—
108#323—2511no—
109C22×D4—5021no—
110C22×D4—2511no—
111C2×SD16—5022no—
112C22⋊Q8—22no—
113C22⋊Q8—21no—
114C22⋊Q8—22no—
115C22⋊Q8C2×D25112yes—
116C2×C22⋊C4—21no—
117C2×C22⋊C4—5021no—
118C2×C22⋊C4C2×D25111yescharacteristic
119#102—5022no—
120C22⋊C8—2512no—
121#95—2511no—
122C24—5021no—
123C24C22⋊D25111yescharacteristic
124C2×Q8—21no—
125C2×Q8C22⋊D25112yescomplemented
126C2×D4—10042no—
127C2×D4—10042no—
128C2×D4—10042no—
129C2×D4—10042no—
130C2×D4—5022no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—5021no—
134C22×C4C22⋊D25111yescharacteristic
135C22×C4C22×D25111yescharacteristic
136SD16—10044no—
137C2×C8—5022no—
138C4⋊C4—21no—
139C4⋊C4—22no—
140C4⋊C4—22no—
141C4⋊C4C22⋊D25112yescomplemented
142C22⋊C4—22no—
143C22⋊C4—22no—
144C22⋊C4—22no—
145C22⋊C4—10042no—
146C23—21no—
147C23—21no—
148C23—41no—
149C23—41no—
150C23—10041no—
151C23—10041no—
152C23—10041no—
153C23—10041no—
154C23—10041no—
155C23—10041no—
156C23—5022no—
157C23#1686079712yescomplemented
158C23C2×C22⋊D25111yescharacteristic
159Q8—42no—
160Q8—81no—
161D4—10042no—
162D4—10048no—
163D4—5024no—
164C2×C4—22no—
165C2×C4—21no—
166C2×C4—22no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—81no—
170C2×C4—81no—
171C2×C4—10041no—
172C2×C4—10041no—
173C2×C4—5022no—
174C2×C4C2×C22⋊D25112yes—
175C8—10042no—
176C22—21no—
177C22—21no—
178C22—22no—
179C22—42no—
180C22—41no—
181C22—41no—
182C22—81no—
183C22—20081no—
184C22—20081no—
185C22—20081no—
186C22—20081no—
187C22—10042no—
188C22—10042no—
189C22—10042no—
190C22#2254868411yescharacteristic
191C22C2×#1686079711yescharacteristic
192C22#2254874611yescharacteristic
193C4—22no—
194C4—41no—
195C4—42no—
196C4—42no—
197C4—10042no—
198C2—21no—
199C2—21no—
200C2—42no—
201C2—20082no—
202C2#3063182611yescharacteristic
203C2#3063185511yescharacteristic
204C2#3063187411yescharacteristic
subgroupG/H
centerC22#22548684
commutator, cosocleC22×C4×C251C23
FrattiniC22×C4C22×D251
Fitting#543×C251C2
socleC22×C251#120
prime radical#30631596C2
hypercenter#543D251
hypocenterC251#1596
LCSG ▷ C22×C4×C251 ▷ C22×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #543

Complements

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

NH ≅ G/Ntype
1. #30631837C2semidirect
3. #30631424C2semidirect
4. #30631411C2semidirect
5. #543×C251C2semidirect
31. C2×Q8×C251D4semidirect
39. C4⋊C4×C251D4semidirect
61. C23×C251SD16semidirect
102. C251#1596semidirect
104. #543D251semidirect
125. C2×Q8C22⋊D251semidirect
141. C4⋊C4C22⋊D251semidirect
157. C23#16860797semidirect

Maximal subgroups

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

subgroupindex
#306318372
#306315962
#306314242
#306314112
#543×C2512
#1596251