Finite Group Databasebeta

← #44549337

Subgroups — #44549337

Order 32128.

238 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 51 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#30632316C211yesmaximal, characteristic
2#30631843C211yesmaximal, characteristic
3#30631606C211yesmaximal, characteristic
4#30631425C211yesmaximal, characteristic
5#30631424C211yesmaximal, characteristic
6#30631412C211yesmaximal, characteristic
7#451×C251C211yesmaximal, characteristic
8C2×#16860800—21no—
9#22548714—21no—
10#22548713—21no—
11#22548705—21no—
12#22548705—21no—
13#22548694C2211yescharacteristic
14#22548692—21no—
15#22548671—21no—
16C2×Dic251⋊C4—21no—
17C2×Dic251⋊C4C2211yescharacteristic
18#22548639—21no—
19C22⋊(C251⋊C8)C2211yescharacteristic
20C251⋊#95—21no—
21C251⋊#95C2211yescharacteristic
22C22×Q8×C251C2211yescharacteristic
23C2×Q16×C251—21no—
24C22⋊Q8×C251C2211yescharacteristic
25#103×C251—21no—
26C22⋊C8×C251C2211yescharacteristic
27C2×Dic502—21no—
28C2×Dic502D411yescharacteristic
29C22×Dic251—21no—
30C22×Dic251D411yescharacteristic
31C22×Dic251D411yescharacteristic
32#16860800—42no—
33C2×C251⋊C8—21no—
34C4⋊Dic251—21no—
35C4⋊Dic251D411yescharacteristic
36Dic251⋊C4—21no—
37Dic251⋊C4—21no—
38Dic251⋊C4—21no—
39Dic251⋊C4—21no—
40Dic251⋊C4—21no—
41Dic251⋊C4—21no—
42Dic251⋊C4—21no—
43Dic251⋊C4—42no—
44C22⋊Dic251—21no—
45C4×Dic251—21no—
46C4×Dic251—21no—
47C2×Q8×C251—21no—
48C2×Q8×C251—21no—
49C2×Q8×C251—42no—
50C2×Q8×C251D411yescharacteristic
51C22×C4×C251—21no—
52C22×C4×C251C2311yescharacteristic
53Q16×C251—42no—
54C2×C8×C251—21no—
55C4⋊C4×C251—21no—
56C4⋊C4×C251D411yescharacteristic
57C22⋊C4×C251—21no—
58Dic502—41no—
59Dic502—81no—
60C2×Dic251—21no—
61C2×Dic251—21no—
62C2×Dic251—22no—
63C2×Dic251—21no—
64C2×Dic251—21no—
65C2×Dic251—21no—
66C2×Dic251—41no—
67C2×Dic251—41no—
68C2×Dic251—41no—
69C2×Dic251Q1612yescomplemented
70C251⋊C8—41no—
71C23×C251C2×D411yescharacteristic
72Q8×C251—22no—
73Q8×C251—22no—
74Q8×C251—41no—
75Q8×C251—41no—
76Q8×C251—41no—
77C2×C4×C251—22no—
78C2×C4×C251—21no—
79C2×C4×C251—21no—
80C2×C4×C251—41no—
81C2×C4×C251—41no—
82C2×C4×C251—41no—
83C2×C4×C251C2×D411yescharacteristic
84C2×C4×C251C2×D411yescharacteristic
85C8×C251—41no—
86C22×C251—21no—
87C22×C251—21no—
88C22×C251#12011yescharacteristic
89C22×C251C2×Q1611yescharacteristic
90C22×C251#13711yescharacteristic
91C4×C251—21no—
92C4×C251—21no—
93C4×C251—41no—
94C4×C251—42no—
95C4×C251—41no—
96Dic251—22no—
97Dic251—42no—
98Dic251—41no—
99Dic251—41no—
100Dic251—41no—
101C2×C251—21no—
102C2×C251—21no—
103C2×C251#45111yescharacteristic
104C2×C251#45611yescharacteristic
105C2×C251B(4,𝔽2)11yescharacteristic
106C251#160511yescharacteristic, complemented, Sylow
107#1605—2511nomaximal, Sylow
108#544—2511no—
109#451—2511no—
110#451D25111yescharacteristic
111#395—2511no—
112#328—2511no—
113#328—2511no—
114#324—2511no—
115C22×Q8C2×D25111yescharacteristic
116C2×Q16—21no—
117C2×Q16—5021no—
118C4⋊Q8—5021no—
119C4⋊Q8—5021no—
120#125—5021no—
121#125—5021no—
122C22⋊Q8—5021no—
123C22⋊Q8—2511no—
124C22⋊Q8C2×D25111yescharacteristic
125#117—5021no—
126C2×C4⋊C4—5021no—
127C2×C4⋊C4—2511no—
128#103—21no—
129#103—5021no—
130C22⋊C8—2511no—
131C22⋊C8C2×D25111yescharacteristic
132#95—5021no—
133#95—2511no—
134C2×Q8—21no—
135C2×Q8—21no—
136C2×Q8—42no—
137C2×Q8—5021no—
138C2×Q8—2511no—
139C2×Q8C22⋊D25111yescharacteristic
140C22×C4—21no—
141C22×C4—5021no—
142C22×C4—2511no—
143C22×C4—2511no—
144C22×C4C22×D25111yescharacteristic
145Q16—42no—
146Q16—10042no—
147C2×C8—21no—
148C2×C8—5021no—
149C4⋊C4—21no—
150C4⋊C4—10042no—
151C4⋊C4—5021no—
152C4⋊C4—5021no—
153C4⋊C4—5021no—
154C4⋊C4—5021no—
155C4⋊C4—5021no—
156C4⋊C4—5021no—
157C4⋊C4—5021no—
158C4⋊C4—5021no—
159C4⋊C4—2511no—
160C4⋊C4C22⋊D25111yescharacteristic
161C22⋊C4—21no—
162C22⋊C4—5021no—
163C42—5021no—
164C42—5021no—
165C23D4×D25111yescharacteristic
166Q8—22no—
167Q8—22no—
168Q8—41no—
169Q8—41no—
170Q8—41no—
171Q8—20081no—
172Q8—10041no—
173C2×C4—22no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—10041no—
180C2×C4—10041no—
181C2×C4—10041no—
182C2×C4—5022no—
183C2×C4—5021no—
184C2×C4—5021no—
185C2×C4—5021no—
186C2×C4—5021no—
187C2×C4—5021no—
188C2×C4—2512no—
189C2×C4D4×D25111yescharacteristic
190C2×C4C2×C22⋊D25111yescharacteristic
191C8—41no—
192C8—10041no—
193C22—21no—
194C22—21no—
195C22#2254868311yescharacteristic
196C22Q16×D25111yescharacteristic
197C22#2254874711yescharacteristic
198C4—21no—
199C4—21no—
200C4—41no—
201C4—42no—
202C4—41no—
203C4—10042no—
204C4—10041no—
205C4—10041no—
206C4—10041no—
207C4—5022no—
208C2—21no—
209C2—21no—
210C2#3063184111yescharacteristic
211C2#3063186811yescharacteristic
212C2#3063187311yescharacteristic
subgroupG/H
centerC22#22548683
commutator, cosocleC22×C4×C251C23
FrattiniC22×C4C22×D251
Fitting#451×C251C2
socleC22×C251#120
prime radicalC23×C251C2×D4
hypercenter#451D251
hypocenterC251#1605
LCSG ▷ C22×C4×C251 ▷ C22×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
69. C2×Dic251Q16semidirect
106. C251#1605semidirect

Maximal subgroups

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

subgroupindex
#306323162
#306318432
#306316062
#306314252
#306314242
#306314122
#451×C2512
#1605251