Finite Group Databasebeta

← #44552017

Subgroups — #44552017

Order 32128.

234 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 55 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#30632259C211yesmaximal, characteristic
2#30631668C211yesmaximal, characteristic
3#30631659C211yesmaximal, characteristic
4#30631478C211yesmaximal, characteristic, complemented
5#30631477C211yesmaximal, characteristic, complemented
6#30631476C211yesmaximal, characteristic, complemented
7#352×C251C211yesmaximal, characteristic, complemented
8C2×#16860809C2211yescharacteristic
9C22×Dic502C411yescharacteristic, complemented
10M16×D251—21no—
11#22548714—21no—
12#22548701—21no—
13#22548699C411yescharacteristic, complemented
14#22548698C411yescharacteristic, complemented
15#22548696C2211yescharacteristic
16#22548693—21no—
17#22548692—21no—
18#22548671—21no—
19#22548671C411yescharacteristic, complemented
20#22548633—21no—
21#22548632C2211yescharacteristic
22#22548630C2211yescharacteristic
23#22548629—21no—
24#123×C251C2211yescharacteristic
25#100×C251C2211yescharacteristic
26#99×C251C2211yescharacteristic
27#16860809—41no—
28#16860809—41no—
29C2×Dic502—21no—
30C2×Dic502—21no—
31C2×Dic502—41no—
32C2×Dic502—41no—
33C2×Dic502D411yescharacteristic
34C2×C22⋊D251C2×C411yescharacteristic
35C2×C22⋊D251D411yescharacteristic
36C22×Dic251C2×C411yescharacteristic
37C22×Dic251C2×C411yescharacteristic
38C22×Dic251D411yescharacteristic
39C2×C4×D251D411yescharacteristic
40#16860794—21no—
41#16860793—41no—
42C8×D251—41no—
43C4⋊Dic251—21no—
44Dic251⋊C4—21no—
45Dic251⋊C4—21no—
46Dic251⋊C4—21no—
47Dic251⋊C4—41no—
48C22⋊Dic251—21no—
49C22⋊Dic251—21no—
50C22⋊Dic251C2×C411yescharacteristic
51(C2×D251)⋊C4—21no—
52(C2×D251)⋊C4—21no—
53C4×Dic251—21no—
54C4×Dic251—41no—
55C2×D4×C251C2311yescharacteristic
56C22×C4×C251C2×C411yescharacteristic
57M16×C251—21no—
58C4⋊C4×C251—21no—
59C22⋊C4×C251—21no—
60C22⋊C4×C251—21no—
61C22⋊C4×C251C2×C411yescharacteristic
62C22×D251C22⋊C411yescharacteristic
63Dic502—41no—
64Dic502—42no—
65Dic502—41no—
66Dic502—41no—
67C22⋊D251—41no—
68C22⋊D251—41no—
69C2×Dic251—21no—
70C2×Dic251—21no—
71C2×Dic251—21no—
72C2×Dic251—21no—
73C2×Dic251—21no—
74C2×Dic251—21no—
75C2×Dic251—41no—
76C2×Dic251—41no—
77C2×Dic251C22⋊C411yescharacteristic
78C2×Dic251C22⋊C411yescharacteristic
79C2×Dic251C22⋊C411yescharacteristic
80C4×D251—21no—
81C4×D251—21no—
82C251⋊C8—41no—
83C23×C251C22×C411yescharacteristic
84C23×C251C2×D411yescharacteristic
85D4×C251—41no—
86C2×C4×C251—21no—
87C2×C4×C251—21no—
88C2×C4×C251—41no—
89C2×C4×C251—41no—
90C2×C4×C251C2×D411yescharacteristic
91C8×C251—41no—
92C22×C251—21no—
93C22×C251—21no—
94C22×C251—41no—
95C22×C251—41no—
96C22×C251C2×C22⋊C411yescharacteristic
97C2×D251—21no—
98C2×D251—41no—
99C4×C251—21no—
100C4×C251—41no—
101C4×C251—41no—
102C4×C251—81no—
103Dic251—21no—
104Dic251—21no—
105Dic251—41no—
106Dic251—41no—
107Dic251—81no—
108Dic251#9911yescharacteristic
109Dic251#9911yescharacteristic
110C2×C251—21no—
111C2×C251—41no—
112C2×C251—41no—
113C2×C251C2×#9911yescharacteristic
114D251—41no—
115C251#212311yescharacteristic, complemented, Sylow
116#2123—2511nomaximal, Sylow
117#536—2511no—
118#413—2511no—
119#410—2511no—
120#352—2511no—
121#352—2511no—
122#352—2511no—
123#352D25111yescharacteristic, complemented
124C2×#41—2511no—
125C22×Q8—2511no—
126C2×M16—5021no—
127C4⋊Q8—5021no—
128#124—5021no—
129#123—2511no—
130#123—2511no—
131#123—2511no—
132#123C2×D25111yescharacteristic
133C22⋊Q8—5021no—
134C22⋊Q8—5021no—
135#117—5021no—
136#117—2511no—
137#101—5021no—
138#100—2511no—
139#100C2×D25111yescharacteristic
140#99—5021no—
141#99—2511no—
142#99C2×D25111yescharacteristic
143#41—10041no—
144#41—10041no—
145C2×Q8—10041no—
146C2×Q8—10041no—
147C2×Q8—5021no—
148C2×Q8—5021no—
149C2×Q8—2511no—
150C2×D4—2511no—
151C2×D4—2511no—
152C2×D4C22×D25111yescharacteristic
153C22×C4—2511no—
154C22×C4—2511no—
155C22×C4—2511no—
156C22×C4—2511no—
157C22×C4C4×D25111yescharacteristic
158M16—21no—
159M16—10041no—
160M16—5021no—
161C2×C8—10041no—
162C4⋊C4—21no—
163C4⋊C4—10041no—
164C4⋊C4—5021no—
165C4⋊C4—5021no—
166C4⋊C4—5021no—
167C4⋊C4—5021no—
168C22⋊C4—21no—
169C22⋊C4—21no—
170C22⋊C4—5021no—
171C22⋊C4—5021no—
172C22⋊C4—5021no—
173C22⋊C4—5021no—
174C22⋊C4—2511no—
175C22⋊C4C4×D25111yescharacteristic
176C42—10041no—
177C42—5021no—
178C23—2511no—
179C23C2×C4×D25111yescharacteristic
180C23D4×D25111yescharacteristic
181Q8—10041no—
182Q8—10042no—
183Q8—10041no—
184Q8—10041no—
185D4—41no—
186D4—10041no—
187D4—10041no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—10041no—
193C2×C4—10041no—
194C2×C4—5021no—
195C2×C4—5021no—
196C2×C4—5021no—
197C2×C4—5021no—
198C2×C4—5021no—
199C2×C4—5021no—
200C2×C4—5021no—
201C2×C4—5021no—
202C2×C4—2511no—
203C2×C4—2511no—
204C2×C4—2511no—
205C2×C4D4×D25111yescharacteristic
206C8—41no—
207C8—10041no—
208C22—21no—
209C22—21no—
210C22—41no—
211C22—41no—
212C22—10041no—
213C22—5021no—
214C22C22⋊C4×D25111yescharacteristic
215C4—21no—
216C4—41no—
217C4—41no—
218C4—81no—
219C4—20081no—
220C4—10041no—
221C4—10041no—
222C4—5021no—
223C4—5021no—
224C4—2511no—
225C4—2511no—
226C2—21no—
227C2—41no—
228C2—41no—
229C2—10041no—
230C2#99×D25111yescharacteristic
subgroupG/H
centerC2#99×D251
commutatorC23×C251C22×C4
FrattiniC2×D4C22×D251
Fitting#352×C251C2
socleC2×C251C2×#99
prime radicalC2×#16860809C22
cosocleC2×D4×C251C23
hypercenter#352D251
hypocenterC251#2123
LCSG ▷ C23×C251 ▷ C22×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #352

Complements

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

NH ≅ G/Ntype
4. #30631478C2semidirect
5. #30631477C2semidirect
6. #30631476C2semidirect
7. #352×C251C2semidirect
9. C22×Dic502C4semidirect
13. #22548699C4semidirect
14. #22548698C4semidirect
19. #22548671C4semidirect
115. C251#2123semidirect
123. #352D251semidirect

Maximal subgroups

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

subgroupindex
#306322592
#306316682
#306316592
#306314782
#306314772
#306314762
#352×C2512
#2123251