Finite Group Databasebeta

← #44558638

Subgroups — #44558638

Order 32128.

356 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 125 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#30631628C212yesmaximal
2#30631627C212yesmaximal
3#30631626C211yesmaximal, characteristic
4#30631608C212yesmaximal
5#30631608C212yesmaximal
6#30631607C212yesmaximal
7#30631552C211yesmaximal, characteristic
8#30631551C212yesmaximal
9#400×C251C211yesmaximal, characteristic
10C22×Dic502C2212yes—
11C2×C4⋊Dic251C2211yescharacteristic
12C2×C4⋊Dic251C2211yescharacteristic
13C2×C4⋊Dic251C2212yes—
14C2×Dic251⋊C4C2212yes—
15C2×Dic251⋊C4C2212yes—
16C2×Dic251⋊C4C2212yes—
17C2×C4×Dic251C2211yescharacteristic
18C251⋊#95C2212yes—
19C251⋊#95C2212yes—
20C251⋊#95C2212yes—
21C251⋊#95C2212yes—
22C251⋊#95C2212yes—
23C251⋊#95C2212yes—
24C251⋊#95C2212yes—
25C251⋊#95C2211yescharacteristic
26C2×C4⋊C4×C251C2211yescharacteristic
27C2×C4⋊C4×C251C2211yescharacteristic
28#95×C251C2212yes—
29#95×C251C2212yes—
30#95×C251C2211yescharacteristic
31C2×Dic502—42no—
32C2×Dic502—42no—
33C2×Dic502—42no—
34C22×Dic251C2312yes—
35C22×Dic251C2312yes—
36C22×Dic251C2312yes—
37C22×Dic251C2312yes—
38C4⋊Dic251—41no—
39C4⋊Dic251—41no—
40C4⋊Dic251—42no—
41Dic251⋊C4—42no—
42Dic251⋊C4—42no—
43Dic251⋊C4—42no—
44C4×Dic251Q814yescomplemented
45C22×C4×C251C2312yes—
46C22×C4×C251C2312yes—
47C22×C4×C251C2311yescharacteristic
48C22×C4×C251C2311yescharacteristic
49C22×C4×C251C2311yescharacteristic
50C4⋊C4×C251—41no—
51C4⋊C4×C251—41no—
52Dic502—48no—
53C2×Dic251—22no—
54C2×Dic251—22no—
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×Dic251C2×Q814yes—
65C23×C251C2411yescharacteristic
66C2×C4×C251—21no—
67C2×C4×C251—21no—
68C2×C4×C251—21no—
69C2×C4×C251—21no—
70C2×C4×C251—22no—
71C2×C4×C251—22no—
72C2×C4×C251—22no—
73C2×C4×C251—22no—
74C2×C4×C251—22no—
75C2×C4×C251—22no—
76C2×C4×C251—22no—
77C2×C4×C251—22no—
78C2×C4×C251C2×Q812yes—
79C22×C251C22×Q811yescharacteristic
80C22×C251H(5,𝔽2)11yescharacteristic
81C22×C251H(5,𝔽2)12yes—
82C22×C2512−1+411yescharacteristic
83C22×C2512−1+412yes—
84C4×C251—22no—
85C4×C251—42no—
86C4×C251—42no—
87C4×C251—41no—
88C4×C251—41no—
89Dic251—24no—
90Dic251—42no—
91Dic251—42no—
92Dic251—42no—
93C2×C251#54411yescharacteristic
94C2×C251#54412yes—
95C2×C251H(3,𝔽4)11yescharacteristic
96C2×C251#56312yes—
97C2×C251#56311yescharacteristic
98C251#284411yescharacteristic, complemented, Sylow
99#2844—2511nomaximal, Sylow
100#400—2511no—
101#400—2512no—
102#400—2512no—
103#400D25111yescharacteristic
104#395—2512no—
105#395—2512no—
106#395—2512no—
107#382—2511no—
108#382—2512no—
109C22×Q8—2512no—
110C2×C4⋊C4—2512no—
111C2×C4⋊C4—2511no—
112C2×C4⋊C4—2512no—
113C2×C4⋊C4—2512no—
114C2×C4⋊C4—2511no—
115C2×C4⋊C4—2512no—
116C2×C4⋊C4C2×D25111yescharacteristic
117C2×C4⋊C4C2×D25111yescharacteristic
118C2×C42—2511no—
119#95—2512no—
120#95—2512no—
121#95—2512no—
122#95—2512no—
123#95—2512no—
124#95—2512no—
125#95—2512no—
126#95—2511no—
127#95C2×D25112yes—
128#95C2×D25112yes—
129#95C2×D25111yescharacteristic
130C2×Q8—10042no—
131C2×Q8—10042no—
132C2×Q8—10042no—
133C22×C4—2512no—
134C22×C4—2512no—
135C22×C4—2512no—
136C22×C4—2512no—
137C22×C4C22×D25111yescharacteristic
138C22×C4C22×D25111yescharacteristic
139C22×C4C22×D25112yes—
140C22×C4C22×D25112yes—
141C22×C4C22×D25111yescharacteristic
142C4⋊C4—41no—
143C4⋊C4—41no—
144C4⋊C4—10042no—
145C4⋊C4—10042no—
146C4⋊C4—10042no—
147C4⋊C4—10041no—
148C4⋊C4—10041no—
149C4⋊C4—10042no—
150C42—2514no—
151C23C23×D25111yescharacteristic
152Q8—10048no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—22no—
158C2×C4—22no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—22no—
165C2×C4—5022no—
166C2×C4—5022no—
167C2×C4—5022no—
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—2514no—
177C2×C4Q8×D25112yes—
178C22C2×Q8×D25111yescharacteristic
179C22#2254876011yescharacteristic
180C22#2254876112yes—
181C22#2254876211yescharacteristic
182C22#2254876212yes—
183C4—22no—
184C4—42no—
185C4—42no—
186C4—41no—
187C4—41no—
188C4—10042no—
189C4—10042no—
190C4—10042no—
191C4—5024no—
192C2#3063231611yescharacteristic
193C2#3063231712yes—
194C2#3063242911yescharacteristic
195C2#3063243912yes—
196C2#3063244011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D251
commutator, socle, prime radical, cosocleC23×C251C24
Fitting#400×C251C2
hypercenter#400D251
hypocenterC251#2844
LCSG ▷ C23×C251 ▷ C251
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
44. C4×Dic251Q8semidirect
98. C251#2844semidirect

Maximal subgroups

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

subgroupindex
#306316282
#306316272
#306316262
#306316082
#306316072
#306315522
#306315512
#400×C2512
#2844251