Finite Group Databasebeta

← #44550899

Subgroups — #44550899

Order 32128.

314 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 79 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#30632258C211yesmaximal, characteristic
2#30632164C211yesmaximal, characteristic, complemented
3#30631704C211yesmaximal, characteristic
4C2×#22548633C211yesmaximal, characteristic, complemented
5#30631444C211yesmaximal, characteristic, complemented
6#30631443C211yesmaximal, characteristic, complemented
7#421×C251C211yesmaximal, characteristic
8C22×Dic502C2211yescharacteristic
9C2×#16860794C2211yescharacteristic
10#22548713C412yescomplemented
11#22548702C412yescomplemented
12#22548696—22no—
13#22548694—22no—
14C4×Dic502—22no—
15#22548680—22no—
16#22548674C2211yescharacteristic
17#22548674C2211yescharacteristic
18#22548671—22no—
19#22548642—22no—
20#22548633—22no—
21C2×#41×C251C2211yescharacteristic
22C2×M16×C251C2211yescharacteristic
23#117×C251C2211yescharacteristic
24#104×C251—22no—
25C2×Dic502—21no—
26C2×Dic502—21no—
27C2×Dic502—21no—
28C2×Dic502—21no—
29C2×Dic502C2×C412yes—
30C2×Dic502D412yes—
31C22×Dic251D412yes—
32#16860794—21no—
33#16860794—21no—
34C2×C251⋊C8—21no—
35C4⋊Dic251—21no—
36C4⋊Dic251—21no—
37Dic251⋊C4—24no—
38Dic251⋊C4—24no—
39C22⋊Dic251—24no—
40C22⋊Dic251—21no—
41C22⋊Dic251—21no—
42C4×Dic251—24no—
43C4×Dic251C2×C412yes—
44C4×Dic251D412yes—
45#41×C251—21no—
46#41×C251—21no—
47#41×C251—21no—
48#41×C251—21no—
49C2×Q8×C251C2×C411yescharacteristic
50C2×D4×C251—21no—
51C2×D4×C251C2×C411yescharacteristic
52C22×C4×C251—21no—
53C22×C4×C251C2311yescharacteristic
54M16×C251—21no—
55M16×C251—21no—
56C2×C8×C251—21no—
57C4⋊C4×C251—21no—
58C22⋊C4×C251—21no—
59C42×C251D412yes—
60Dic502—22no—
61Dic502—22no—
62Dic502—22no—
63Dic502—22no—
64C2×Dic251—22no—
65C2×Dic251—22no—
66C2×Dic251—22no—
67C2×Dic251—22no—
68C2×Dic251C22⋊C414yes—
69C251⋊C8—41no—
70C23×C251—21no—
71C23×C251C2×D411yescharacteristic
72Q8×C251—21no—
73Q8×C251—21no—
74D4×C251—21no—
75D4×C251—21no—
76D4×C251—41no—
77D4×C251—41no—
78C2×C4×C251—21no—
79C2×C4×C251—22no—
80C2×C4×C251—41no—
81C2×C4×C251—41no—
82C2×C4×C251C22×C411yescharacteristic
83C2×C4×C251C2×D411yescharacteristic
84C2×C4×C251C2×D411yescharacteristic
85C2×C4×C251C2×D411yescharacteristic
86C2×C4×C251#4111yescharacteristic
87C2×C4×C251#4111yescharacteristic
88C8×C251—41no—
89C22×C251—41no—
90C22×C251—42no—
91C22×C251—41no—
92C22×C251C2×C22⋊C411yescharacteristic
93C22×C251#12011yescharacteristic
94C22×C251#12311yescharacteristic
95C4×C251—42no—
96C4×C251—41no—
97C4×C251C4×D411yescharacteristic
98C4×C251C4×D411yescharacteristic
99C4×C251#12111yescharacteristic
100C4×C251#12111yescharacteristic
101Dic251—24no—
102Dic251—42no—
103Dic251—42no—
104C2×C251—21no—
105C2×C251—21no—
106C2×C251—21no—
107C2×C251—81no—
108C2×C251#38611yescharacteristic
109C251#191211yescharacteristic, complemented, Sylow
110#1912—2511nomaximal, Sylow
111#536—2511no—
112#519—2511no—
113#421—2511no—
114#421D25111yescharacteristic
115C2×#101—2511no—
116H(3,ℤ/4)—2511no—
117H(3,ℤ/4)—2511no—
118C2×#41C2×D25111yescharacteristic
119C22×Q8—2511no—
120C2×M16—2511no—
121C2×M16C2×D25111yescharacteristic
122C4⋊Q8—2512no—
123#124—2512no—
124#123—5022no—
125C22⋊Q8—5022no—
126C4×Q8—5022no—
127C4×Q8—5022no—
128#117—5022no—
129#117—2511no—
130#117—2511no—
131#117C2×D25111yescharacteristic
132#104—22no—
133#104—5022no—
134#101—5022no—
135#41—21no—
136#41—21no—
137#41—21no—
138#41—21no—
139C2×Q8—5021no—
140C2×Q8—5021no—
141C2×Q8—5021no—
142C2×Q8—5021no—
143C2×Q8—2512no—
144C2×Q8—2512no—
145C2×Q8C4×D25111yescharacteristic
146C2×D4—21no—
147C2×D4C4×D25111yescharacteristic
148C22×C4—21no—
149C22×C4—2512no—
150C22×C4C22×D25111yescharacteristic
151M16—21no—
152M16—21no—
153M16—5021no—
154M16—5021no—
155C2×C8—21no—
156C2×C8—5021no—
157C4⋊C4—21no—
158C4⋊C4—5024no—
159C4⋊C4—5024no—
160C4⋊C4—5021no—
161C4⋊C4—5021no—
162C22⋊C4—21no—
163C22⋊C4—5021no—
164C22⋊C4—5021no—
165C22⋊C4—5024no—
166C42—5024no—
167C42—2512no—
168C42—2512no—
169C42C22⋊D25112yes—
170C23—21no—
171C23D4×D25111yescharacteristic
172Q8—21no—
173Q8—21no—
174Q8—5022no—
175Q8—5022no—
176Q8—5022no—
177Q8—5022no—
178D4—21no—
179D4—21no—
180D4—41no—
181D4—41no—
182C2×C4—21no—
183C2×C4—22no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—5022no—
187C2×C4—5022no—
188C2×C4—5022no—
189C2×C4—5022no—
190C2×C4—2514no—
191C2×C4C2×C4×D25111yescharacteristic
192C2×C4D4×D25111yescharacteristic
193C2×C4D4×D25111yescharacteristic
194C2×C4C2×C22⋊D25111yescharacteristic
195C2×C4#1686080911yescharacteristic
196C2×C4#1686081011yescharacteristic
197C8—41no—
198C8—10041no—
199C22—41no—
200C22—42no—
201C22—41no—
202C22C22⋊C4×D25111yescharacteristic
203C22#2254868311yescharacteristic
204C22#2254869711yescharacteristic
205C4—42no—
206C4—41no—
207C4—10042no—
208C4—10042no—
209C4—5024no—
210C4(C22⋊D251)⋊C411yescharacteristic
211C4C4×C22⋊D25111yescharacteristic
212C4#2254868611yescharacteristic
213C4#2254868811yescharacteristic
214C2—21no—
215C2—21no—
216C2—21no—
217C2—81no—
218C2#3063156311yescharacteristic
subgroupG/H
centerC2#30631563
commutatorC2×C4×C251C22×C4
FrattiniC22×C4C22×D251
Fitting#421×C251C2
socleC2×C251#386
prime radicalC2×#41×C251C22
cosocleC22×C4×C251C23
hypercenter#421D251
hypocenterC251#1912
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C2 ◁ C22×C4 ◁ #421

Complements

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

NH ≅ G/Ntype
2. #30632164C2semidirect
4. C2×#22548633C2semidirect
5. #30631444C2semidirect
6. #30631443C2semidirect
10. #22548713C4semidirect
11. #22548702C4semidirect
109. C251#1912semidirect

Maximal subgroups

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

subgroupindex
#306322582
#306321642
#306317042
C2×#225486332
#306314442
#306314432
#421×C2512
#1912251