Finite Group Databasebeta

← #44558165

Subgroups — #44558165

Order 32128.

384 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 141 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#30631617C212yesmaximal
2#30631608C212yesmaximal
3#30631607C211yesmaximal, characteristic
4#30631586C212yesmaximal
5#30631584C212yesmaximal
6#30631569C212yesmaximal
7#30631568C212yesmaximal
8#30631556C211yesmaximal, characteristic
9#398×C251C211yesmaximal, characteristic
10C22×Dic502C2211yescharacteristic
11C2×C4⋊Dic251C2211yescharacteristic
12C2×C4⋊Dic251C2212yes—
13C2×C4⋊Dic251C2212yes—
14C2×Dic251⋊C4C2212yes—
15C2×Dic251⋊C4C2212yes—
16C2×Dic251⋊C4C2212yes—
17C2×Dic251⋊C4C2212yes—
18C2×C4×Dic251C2211yescharacteristic
19C2×C4×Dic251C2212yes—
20C251⋊#95C2212yes—
21C251⋊#95C2212yes—
22C251⋊#95C2212yes—
23C251⋊#95C2212yes—
24C251⋊#95C2211yescharacteristic
25C251⋊#95C2212yes—
26C2×C4⋊C4×C251C2212yes—
27C2×C4⋊C4×C251C2211yescharacteristic
28C2×C4⋊C4×C251C2211yescharacteristic
29#95×C251C2212yes—
30#95×C251C2211yescharacteristic
31C2×Dic502—41no—
32C2×Dic502—42no—
33C22×Dic251C2312yes—
34C22×Dic251C2312yes—
35C22×Dic251C2312yes—
36C22×Dic251C2312yes—
37C4⋊Dic251—41no—
38C4⋊Dic251—42no—
39C4⋊Dic251—42no—
40Dic251⋊C4—42no—
41Dic251⋊C4—42no—
42Dic251⋊C4—42no—
43Dic251⋊C4Q818yescomplemented
44C4×Dic251—22no—
45C4×Dic251—42no—
46C22×C4×C251C2311yescharacteristic
47C22×C4×C251C2312yes—
48C22×C4×C251C2311yescharacteristic
49C22×C4×C251C2312yes—
50C22×C4×C251C2311yescharacteristic
51C4⋊C4×C251—24no—
52C4⋊C4×C251—41no—
53C4⋊C4×C251—41no—
54Dic502—44no—
55C2×Dic251—22no—
56C2×Dic251—22no—
57C2×Dic251—22no—
58C2×Dic251—22no—
59C2×Dic251—22no—
60C2×Dic251—22no—
61C2×Dic251—22no—
62C2×Dic251—22no—
63C2×Dic251—22no—
64C2×Dic251—22no—
65C2×Dic251C2×Q814yes—
66C2×Dic251C2×Q814yes—
67C23×C251C2411yescharacteristic
68C2×C4×C251—22no—
69C2×C4×C251—21no—
70C2×C4×C251—22no—
71C2×C4×C251—21no—
72C2×C4×C251—22no—
73C2×C4×C251—21no—
74C2×C4×C251—22no—
75C2×C4×C251—21no—
76C2×C4×C251—22no—
77C2×C4×C251—22no—
78C2×C4×C251—22no—
79C2×C4×C251C2×Q814yes—
80C2×C4×C251#4112yes—
81C22×C251C22×Q812yes—
82C22×C251C2×#4111yescharacteristic
83C22×C251H(5,𝔽2)11yescharacteristic
84C22×C251H(5,𝔽2)12yes—
85C22×C2512−1+411yescharacteristic
86C4×C251—22no—
87C4×C251—24no—
88C4×C251—42no—
89C4×C251—41no—
90C4×C251—41no—
91Dic251—24no—
92Dic251—24no—
93Dic251—42no—
94Dic251—42no—
95C2×C251#53911yescharacteristic
96C2×C251#54412yes—
97C2×C251#55412yes—
98C2×C251Q8211yescharacteristic
99C2×C251#56011yescharacteristic
100C251#279211yescharacteristic, complemented, Sylow
101#2792—2511nomaximal, Sylow
102#398—2512no—
103#398D25111yescharacteristic
104#395—2511no—
105#395—2512no—
106#389—2512no—
107#389—2512no—
108#387—2512no—
109#387—2512no—
110#384—2511no—
111C22×Q8—2511no—
112C2×C4⋊C4—2512no—
113C2×C4⋊C4—2511no—
114C2×C4⋊C4—2512no—
115C2×C4⋊C4—2512no—
116C2×C4⋊C4—2512no—
117C2×C4⋊C4—2512no—
118C2×C4⋊C4—2512no—
119C2×C4⋊C4C2×D25112yes—
120C2×C4⋊C4C2×D25111yescharacteristic
121C2×C4⋊C4C2×D25111yescharacteristic
122C2×C42—2512no—
123C2×C42—2511no—
124#95—2511no—
125#95—2512no—
126#95—2512no—
127#95—2512no—
128#95—2512no—
129#95—2512no—
130#95C2×D25112yes—
131#95C2×D25111yescharacteristic
132C2×Q8—10041no—
133C2×Q8—10042no—
134C22×C4—2512no—
135C22×C4—2512no—
136C22×C4—2512no—
137C22×C4—2512no—
138C22×C4C22×D25112yes—
139C22×C4C22×D25111yescharacteristic
140C22×C4C22×D25111yescharacteristic
141C22×C4C22×D25112yes—
142C22×C4C22×D25111yescharacteristic
143C4⋊C4—24no—
144C4⋊C4—41no—
145C4⋊C4—41no—
146C4⋊C4—10042no—
147C4⋊C4—10042no—
148C4⋊C4—10042no—
149C4⋊C4—10041no—
150C4⋊C4—10042no—
151C4⋊C4—10042no—
152C4⋊C4—2518no—
153C42—10042no—
154C42—5022no—
155C23C23×D25111yescharacteristic
156Q8—10044no—
157C2×C4—22no—
158C2×C4—21no—
159C2×C4—22no—
160C2×C4—21no—
161C2×C4—22no—
162C2×C4—21no—
163C2×C4—22no—
164C2×C4—21no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—22no—
168C2×C4—5022no—
169C2×C4—5022no—
170C2×C4—5022no—
171C2×C4—5022no—
172C2×C4—5022no—
173C2×C4—5022no—
174C2×C4—5022no—
175C2×C4—5022no—
176C2×C4—5022no—
177C2×C4—5022no—
178C2×C4—2514no—
179C2×C4—2514no—
180C2×C4Q8×D25114yes—
181C2×C4#1686080912yes—
182C22C2×Q8×D25112yes—
183C22C2×#1686080911yescharacteristic
184C22#2254876011yescharacteristic
185C22#2254876112yes—
186C22#2254876211yescharacteristic
187C4—22no—
188C4—24no—
189C4—42no—
190C4—41no—
191C4—41no—
192C4—10042no—
193C4—10042no—
194C4—5024no—
195C4—5024no—
196C2#3063228211yescharacteristic
197C2#3063231712yes—
198C2#3063239512yes—
199C2#3063241711yescharacteristic
200C2#3063242711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D251
commutator, socle, prime radical, cosocleC23×C251C24
Fitting#398×C251C2
hypercenter#398D251
hypocenterC251#2792
LCSG ▷ C23×C251 ▷ C251
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
43. Dic251⋊C4Q8semidirect
100. C251#2792semidirect

Maximal subgroups

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

subgroupindex
#306316172
#306316082
#306316072
#306315862
#306315842
#306315692
#306315682
#306315562
#398×C2512
#2792251