Finite Group Databasebeta

← #44550707

Subgroups — #44550707

Order 32128.

338 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 101 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
1C22×Dic1004C211yesmaximal, characteristic
2C2×#22548680C211yesmaximal, characteristic
3C2×#22548638C211yesmaximal, characteristic
4C2×#22548638C211yesmaximal, characteristic
5#30631592C211yesmaximal, characteristic
6#30631441C211yesmaximal, characteristic
7#340×C251C211yesmaximal, characteristic
8C22×Dic502C2211yescharacteristic
9C22×Dic502C2211yescharacteristic
10C2×Dic1004—22no—
11C2×Dic1004—22no—
12C2×Dic1004C414yescomplemented
13C22×C251⋊C8C2211yescharacteristic
14#22548680—22no—
15#22548680—22no—
16C2×Dic251⋊C4—21no—
17C2×C4×Dic251—21no—
18C2×C4×Dic251C2211yescharacteristic
19#22548638—22no—
20#22548638—22no—
21C251⋊#95—21no—
22C251⋊#95—21no—
23C22×C8×C251C2211yescharacteristic
24C2×C4⋊C4×C251C2211yescharacteristic
25C2×C4⋊C4×C251C2211yescharacteristic
26C2×Dic502—21no—
27C2×Dic502—21no—
28C2×Dic502—21no—
29C2×Dic502—21no—
30C2×Dic502—21no—
31C2×Dic502—21no—
32C2×Dic502—21no—
33C2×Dic502—21no—
34C2×Dic502C2×C412yes—
35C2×Dic502C2×C412yes—
36C2×Dic502D412yes—
37C2×Dic502D412yes—
38C22×Dic251—21no—
39C22×Dic251—21no—
40C22×Dic251D411yescharacteristic
41C22×Dic251D411yescharacteristic
42Dic1004—28no—
43C2×C251⋊C8—21no—
44C2×C251⋊C8—21no—
45C2×C251⋊C8—21no—
46Dic251⋊C4—24no—
47C4×Dic251—24no—
48C4×Dic251—21no—
49C4×Dic251—21no—
50C22×C4×C251—21no—
51C22×C4×C251—21no—
52C22×C4×C251C2311yescharacteristic
53C2×C8×C251—21no—
54C2×C8×C251—21no—
55C2×C8×C251C2×C412yes—
56C4⋊C4×C251—21no—
57C4⋊C4×C251—21no—
58C4⋊C4×C251—21no—
59C4⋊C4×C251D412yes—
60Dic502—22no—
61Dic502—22no—
62Dic502—22no—
63Dic502—22no—
64Dic502—22no—
65Dic502—41no—
66Dic502—41no—
67Dic502C22⋊C414yes—
68C2×Dic251—22no—
69C2×Dic251—21no—
70C2×Dic251—22no—
71C2×Dic251—22no—
72C2×Dic251—21no—
73C2×Dic251—22no—
74C2×Dic251—21no—
75C2×Dic251—21no—
76C2×Dic251—21no—
77C2×Dic251—41no—
78C2×Dic251—41no—
79C2×Dic251Q1612yescomplemented
80C251⋊C8—41no—
81C23×C251C2×D411yescharacteristic
82C2×C4×C251—22no—
83C2×C4×C251—41no—
84C2×C4×C251—41no—
85C2×C4×C251—41no—
86C2×C4×C251—41no—
87C2×C4×C251—41no—
88C2×C4×C251C22×C411yescharacteristic
89C2×C4×C251C2×D411yescharacteristic
90C2×C4×C251C2×D411yescharacteristic
91C2×C4×C251C2×D411yescharacteristic
92C2×C4×C251#4111yescharacteristic
93C2×C4×C251#4111yescharacteristic
94C8×C251—22no—
95C22×C251C4×D411yescharacteristic
96C22×C251#12011yescharacteristic
97C22×C251#12111yescharacteristic
98C22×C251C2×Q1611yescharacteristic
99C22×C251#13511yescharacteristic
100C22×C251#13711yescharacteristic
101C22×C251#13711yescharacteristic
102C4×C251—42no—
103C4×C251—81no—
104C4×C251C2×C22⋊C411yescharacteristic
105C4×C251C4×D411yescharacteristic
106C4×C251#12111yescharacteristic
107C4×C251#12311yescharacteristic
108Dic251—24no—
109Dic251—22no—
110Dic251—42no—
111Dic251—41no—
112C2×C251#38611yescharacteristic
113C2×C251C4×Q1611yescharacteristic
114C2×C251#44111yescharacteristic
115C2×C251#45111yescharacteristic
116C2×C251#45211yescharacteristic
117C2×C251#46211yescharacteristic
118C2×C251#46411yescharacteristic
119C251#187811yescharacteristic, complemented, Sylow
120#1878—2511nomaximal, Sylow
121C22×Q16—2511no—
122C2×C4×Q8—2511no—
123C2×#103—2511no—
124C2×#103—2511no—
125#391—2511no—
126#340D25111yescharacteristic
127#336—2511no—
128C22×Q8—2511no—
129C22×Q8—2511no—
130C2×Q16—5022no—
131C2×Q16—5022no—
132C2×Q16—2514no—
133C22×C8—2511no—
134C22×C8C2×D25111yescharacteristic
135C4×Q8—5022no—
136C4×Q8—5022no—
137C2×C4⋊C4—5021no—
138C2×C4⋊C4C2×D25111yescharacteristic
139C2×C4⋊C4C2×D25111yescharacteristic
140C2×C42—5021no—
141C2×C42—2511no—
142#103—5022no—
143#103—5022no—
144#95—5021no—
145#95—5021no—
146C2×Q8—5021no—
147C2×Q8—5021no—
148C2×Q8—5021no—
149C2×Q8—5021no—
150C2×Q8—5021no—
151C2×Q8—5021no—
152C2×Q8—5021no—
153C2×Q8—5021no—
154C2×Q8—2512no—
155C2×Q8—2512no—
156C2×Q8—2512no—
157C2×Q8—2512no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—5021no—
161C22×C4—5021no—
162C22×C4—2511no—
163C22×C4—2511no—
164C22×C4C22×D25111yescharacteristic
165Q16—5028no—
166C2×C8—21no—
167C2×C8—21no—
168C2×C8—5021no—
169C2×C8—5021no—
170C2×C8—5021no—
171C2×C8C4×D25112yes—
172C4⋊C4—21no—
173C4⋊C4—21no—
174C4⋊C4—21no—
175C4⋊C4—5024no—
176C4⋊C4D100412yes—
177C42—5024no—
178C42—5021no—
179C42—5021no—
180C23D4×D25111yescharacteristic
181Q8—10041no—
182Q8—10041no—
183Q8—5022no—
184Q8—5022no—
185Q8—5022no—
186Q8—5022no—
187Q8—5022no—
188Q8—2514no—
189C2×C4—22no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—10041no—
196C2×C4—10041no—
197C2×C4—5021no—
198C2×C4—5022no—
199C2×C4—5021no—
200C2×C4—5021no—
201C2×C4—5021no—
202C2×C4—5022no—
203C2×C4—5021no—
204C2×C4—5022no—
205C2×C4—5022no—
206C2×C4—2512no—
207C2×C4C2×C4×D25111yescharacteristic
208C2×C4D4×D25111yescharacteristic
209C2×C4D4×D25111yescharacteristic
210C2×C4C2×D100411yescharacteristic
211C2×C4#1686081011yescharacteristic
212C2×C4#1686081111yescharacteristic
213C8—22no—
214C8—10041no—
215C22#2254867711yescharacteristic
216C22#2254868511yescharacteristic
217C22#2254868811yescharacteristic
218C22Q16×D25111yescharacteristic
219C22#2254873411yescharacteristic
220C22#2254874411yescharacteristic
221C22#2254874411yescharacteristic
222C4—42no—
223C4—81no—
224C4—10042no—
225C4—10041no—
226C4—5024no—
227C4—5022no—
228C4C22⋊C4×D25111yescharacteristic
229C4C4×D100411yescharacteristic
230C4#2254869011yescharacteristic
231C4#2254870011yescharacteristic
232C2#3063156511yescharacteristic
233C2#3063178111yescharacteristic
234C2#3063179311yescharacteristic
235C2#3063184411yescharacteristic
236C2#3063185011yescharacteristic
237C2#3063189611yescharacteristic
238C2#3063190811yescharacteristic
subgroupG/H
centerC23D4×D251
commutatorC2×C4×C251C22×C4
FrattiniC22×C4C22×D251
Fitting#340×C251C2
socle, prime radicalC23×C251C2×D4
cosocleC22×C4×C251C23
hypercenter#340D251
hypocenterC251#1878
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C23 ◁ C22×C4 ◁ #340

Complements

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

NH ≅ G/Ntype
12. C2×Dic1004C4semidirect
79. C2×Dic251Q16semidirect
119. C251#1878semidirect

Maximal subgroups

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

subgroupindex
C22×Dic10042
C2×#225486802
C2×#225486382
#306315922
#306314412
#340×C2512
#1878251