Finite Group Databasebeta

← #44558426

Subgroups — #44558426

Order 32128.

430 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 133 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
1C2×#22548715C211yesmaximal, characteristic, complemented
2C2×#22548709C211yesmaximal, characteristic
3#30631627C212yesmaximal, complemented
4#30631624C212yesmaximal
5#30631619C212yesmaximal, complemented
6#30631614C212yesmaximal
7#30631608C212yesmaximal, complemented
8#30631600C212yesmaximal
9#383×C251C211yesmaximal, characteristic, complemented
10C22×Dic502C2212yes—
11#22548715—24no—
12#22548709—24no—
13C2×C4⋊Dic251C2212yes—
14C2×C4⋊Dic251C2212yes—
15C2×C4⋊Dic251C2214yes—
16C2×Dic251⋊C4C2211yescharacteristic
17C2×Dic251⋊C4C2211yescharacteristic
18C2×Dic251⋊C4C2211yescharacteristic
19C2×(C2×D251)⋊C4C2211yescharacteristic
20C2×(C2×D251)⋊C4C2211yescharacteristic
21C2×(C2×D251)⋊C4C2211yescharacteristic
22C2×(C2×D251)⋊C4C2214yes—
23C251⋊#95C2214yes—
24C251⋊#95C2214yes—
25C2×C42×C251C2211yescharacteristic
26#95×C251C2212yes—
27#95×C251C2212yes—
28#95×C251C2212yes—
29C23×D251C2311yescharacteristic
30C2×Dic502—24no—
31C2×Dic502—44no—
32C22×Dic251C2314yes—
33C22×Dic251C2311yescharacteristic
34C22×Dic251C2311yescharacteristic
35C22×Dic251C2311yescharacteristic
36C4⋊Dic251—24no—
37C4⋊Dic251—24no—
38C4⋊Dic251—44no—
39Dic251⋊C4—22no—
40Dic251⋊C4—22no—
41Dic251⋊C4—22no—
42(C2×D251)⋊C4—22no—
43(C2×D251)⋊C4—22no—
44(C2×D251)⋊C4—22no—
45(C2×D251)⋊C4—44no—
46C22×C4×C251C2314yes—
47C22×C4×C251C2311yescharacteristic
48C22×C4×C251C2311yescharacteristic
49C22×C4×C251C2311yescharacteristic
50C42×C251D414yes—
51C22×D251—21no—
52C22×D251—22no—
53C22×D251—22no—
54C22×D251—22no—
55Dic502—48no—
56C2×Dic251—21no—
57C2×Dic251—21no—
58C2×Dic251—21no—
59C2×Dic251—24no—
60C2×Dic251—24no—
61C2×Dic251—22no—
62C2×Dic251—24no—
63C2×Dic251—22no—
64C2×Dic251—22no—
65C23×C251C2411yescharacteristic
66C2×C4×C251—22no—
67C2×C4×C251—22no—
68C2×C4×C251—22no—
69C2×C4×C251—24no—
70C2×C4×C251—24no—
71C2×C4×C251—24no—
72C2×C4×C251C2×D412yes—
73C2×C4×C251C2×D412yes—
74C2×C4×C251C2×D412yes—
75C22×C251C22×D411yescharacteristic
76C22×C251H(5,𝔽2)12yes—
77C22×C2512−1+412yes—
78C22×C2512−1+412yes—
79C2×D251—41no—
80C2×D251—41no—
81C2×D251—41no—
82C2×D251—44no—
83C4×C251—22no—
84C4×C251—22no—
85C4×C251—22no—
86C4×C251—44no—
87Dic251—41no—
88Dic251—44no—
89Dic251—41no—
90Dic251—41no—
91C2×C251#53511yescharacteristic
92C2×C251#53611yescharacteristic
93C2×C251#53611yescharacteristic
94C2×C251#56314yes—
95D251—81no—
96C251#282111yescharacteristic, complemented, Sylow
97#2821—2511nomaximal, Sylow
98C2×C4⋊Q8—2511no—
99C2×#126—2511no—
100#400—2512no—
101#399—2512no—
102#398—2512no—
103#397—2512no—
104#395—2512no—
105#393—2512no—
106#383D25111yescharacteristic, complemented
107C22×Q8—2512no—
108C4⋊Q8—5024no—
109#126—5024no—
110C2×C4⋊C4—2514no—
111C2×C4⋊C4—2511no—
112C2×C4⋊C4—2511no—
113C2×C4⋊C4—2511no—
114C2×C4⋊C4—2512no—
115C2×C4⋊C4—2512no—
116C2×C22⋊C4—2514no—
117C2×C22⋊C4—2511no—
118C2×C22⋊C4—2511no—
119C2×C22⋊C4—2511no—
120C2×C42C2×D25111yescharacteristic
121#95—2514no—
122#95—2514no—
123#95C2×D25112yes—
124#95C2×D25112yes—
125#95C2×D25112yes—
126C24—2511no—
127C2×Q8—10044no—
128C2×Q8—5024no—
129C22×C4—2511no—
130C22×C4—2511no—
131C22×C4—2514no—
132C22×C4—2511no—
133C22×C4C22×D25114yes—
134C22×C4C22×D25111yescharacteristic
135C22×C4C22×D25111yescharacteristic
136C22×C4C22×D25111yescharacteristic
137C4⋊C4—10044no—
138C4⋊C4—5022no—
139C4⋊C4—5022no—
140C4⋊C4—5022no—
141C4⋊C4—5024no—
142C4⋊C4—5024no—
143C22⋊C4—10044no—
144C22⋊C4—5022no—
145C22⋊C4—5022no—
146C22⋊C4—5022no—
147C42D100414yes—
148C23—5021no—
149C23—5022no—
150C23—5022no—
151C23—5022no—
152C23C23×D25111yescharacteristic
153Q8—10048no—
154C2×C4—22no—
155C2×C4—22no—
156C2×C4—22no—
157C2×C4—24no—
158C2×C4—24no—
159C2×C4—24no—
160C2×C4—5021no—
161C2×C4—5021no—
162C2×C4—5021no—
163C2×C4—5024no—
164C2×C4—5024no—
165C2×C4—5022no—
166C2×C4—5024no—
167C2×C4—5022no—
168C2×C4—5022no—
169C2×C4C2×D100412yes—
170C2×C4C2×D100412yes—
171C2×C4C2×D100412yes—
172C22—10041no—
173C22—10041no—
174C22—10041no—
175C22—10044no—
176C22C22×D100411yescharacteristic
177C22#2254876112yes—
178C22#2254876212yes—
179C22#2254876212yes—
180C4—22no—
181C4—22no—
182C4—22no—
183C4—44no—
184C4—10044no—
185C4—10041no—
186C4—10041no—
187C4—10041no—
188C2—20081no—
189C2#3063225611yescharacteristic
190C2#3063226311yescharacteristic
191C2#3063226311yescharacteristic
192C2#3063243914yes—
subgroupG/H
center, FrattiniC23C23×D251
commutator, socle, cosocleC23×C251C24
Fitting#383×C251C2
prime radicalC23×D251C23
hypercenter#383D251
hypocenterC251#2821
LCSG ▷ C23×C251 ▷ C251
UCSC1 ◁ C23 ◁ #383

Complements

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

NH ≅ G/Ntype
1. C2×#22548715C2semidirect
3. #30631627C2semidirect
5. #30631619C2semidirect
7. #30631608C2semidirect
9. #383×C251C2semidirect
96. C251#2821semidirect
106. #383D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487152
C2×#225487092
#306316272
#306316242
#306316192
#306316142
#306316082
#306316002
#383×C2512
#2821251