Finite Group Databasebeta

← #44558127

Subgroups — #44558127

Order 32128.

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

normal subgroups only

This group has 352 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
1C251⋊#401C212yesmaximal
2#30631626C212yesmaximal
3#30631617C211yesmaximal, characteristic
4#30631594C212yesmaximal
5#30631574C212yesmaximal
6#30631573C212yesmaximal
7#30631572C212yesmaximal
8#30631552C211yesmaximal, characteristic
9#395×C251C211yesmaximal, characteristic
10C2×C4⋊Dic251C2212yes—
11C2×C4⋊Dic251C2211yescharacteristic
12C2×C4⋊Dic251C2211yescharacteristic
13C2×Dic251⋊C4C2212yes—
14C2×C4×Dic251C2211yescharacteristic
15C2×C4×Dic251C2212yes—
16C251⋊#95C2212yes—
17C251⋊#95C2212yes—
18C251⋊#95C2212yes—
19C251⋊#95C2212yes—
20C251⋊#95C2212yes—
21C251⋊#95C2212yes—
22C251⋊#95C2212yes—
23C251⋊#95C2212yes—
24C251⋊#95C2212yes—
25C251⋊#95C2211yescharacteristic
26C22×Q8×C251C2211yescharacteristic
27C2×C4⋊C4×C251C2212yes—
28C2×C4⋊C4×C251C2211yescharacteristic
29#95×C251C2212yes—
30#95×C251C2211yescharacteristic
31C22×Dic251C2312yes—
32C22×Dic251C2312yes—
33C22×Dic251C2312yes—
34C22×Dic251C2312yes—
35C4⋊Dic251—24no—
36C4⋊Dic251—41no—
37C4⋊Dic251—41no—
38Dic251⋊C4—24no—
39C4×Dic251—22no—
40C4×Dic251—42no—
41C2×Q8×C251—24no—
42C2×Q8×C251—41no—
43C22×C4×C251C2312yes—
44C22×C4×C251C2311yescharacteristic
45C22×C4×C251C2311yescharacteristic
46C22×C4×C251C2311yescharacteristic
47C22×C4×C251C2312yes—
48C4⋊C4×C251—24no—
49C4⋊C4×C251—41no—
50C2×Dic251—22no—
51C2×Dic251—22no—
52C2×Dic251—22no—
53C2×Dic251—22no—
54C2×Dic251—22no—
55C2×Dic251—22no—
56C2×Dic251—22no—
57C2×Dic251—22no—
58C2×Dic251—22no—
59C2×Dic251—22no—
60C2×Dic251—24no—
61C23×C251C2411yescharacteristic
62Q8×C251—44no—
63C2×C4×C251—22no—
64C2×C4×C251—21no—
65C2×C4×C251—22no—
66C2×C4×C251—21no—
67C2×C4×C251—22no—
68C2×C4×C251—22no—
69C2×C4×C251—22no—
70C2×C4×C251—21no—
71C2×C4×C251—21no—
72C2×C4×C251—22no—
73C2×C4×C251—22no—
74C2×C4×C251#4114yes—
75C2×C4×C251#4112yes—
76C22×C251C2×#4112yes—
77C22×C251C2×#4111yescharacteristic
78C22×C251H(5,𝔽2)11yescharacteristic
79C22×C2512−1+411yescharacteristic
80C22×C2512−1+412yes—
81C4×C251—22no—
82C4×C251—24no—
83C4×C251—42no—
84C4×C251—41no—
85C4×C251—41no—
86Dic251—42no—
87Dic251—42no—
88Dic251—42no—
89Dic251—42no—
90C2×C251#53911yescharacteristic
91C2×C251#54112yes—
92C2×C251#55111yescharacteristic
93C2×C251#55612yes—
94C2×C251#56311yescharacteristic
95C251#278911yescharacteristic, complemented, Sylow
96#2789—2511nomaximal, Sylow
97#401—2512no—
98#400—2512no—
99#398—2511no—
100#395D25111yescharacteristic
101#391—2512no—
102#387—2512no—
103#387—2512no—
104#387—2512no—
105#382—2511no—
106C22×Q8C2×D25111yescharacteristic
107C2×C4⋊C4—2511no—
108C2×C4⋊C4—2511no—
109C2×C4⋊C4—2512no—
110C2×C4⋊C4—2512no—
111C2×C4⋊C4C2×D25112yes—
112C2×C4⋊C4C2×D25111yescharacteristic
113C2×C42—2512no—
114C2×C42—2511no—
115#95—2512no—
116#95—2512no—
117#95—2512no—
118#95—2512no—
119#95—2512no—
120#95—2512no—
121#95—2512no—
122#95—2512no—
123#95—2512no—
124#95—2511no—
125#95C2×D25112yes—
126#95C2×D25111yescharacteristic
127C2×Q8—24no—
128C2×Q8—41no—
129C22×C4—2512no—
130C22×C4—2512no—
131C22×C4—2512no—
132C22×C4—2512no—
133C22×C4C22×D25112yes—
134C22×C4C22×D25111yescharacteristic
135C22×C4C22×D25111yescharacteristic
136C22×C4C22×D25112yes—
137C22×C4C22×D25111yescharacteristic
138C4⋊C4—24no—
139C4⋊C4—41no—
140C4⋊C4—10041no—
141C4⋊C4—10041no—
142C4⋊C4—5024no—
143C4⋊C4—5024no—
144C42—10042no—
145C42—5022no—
146C23C23×D25111yescharacteristic
147Q8—44no—
148C2×C4—22no—
149C2×C4—22no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—22no—
158C2×C4—22no—
159C2×C4—5022no—
160C2×C4—5022no—
161C2×C4—5022no—
162C2×C4—5022no—
163C2×C4—5022no—
164C2×C4—5022no—
165C2×C4—5022no—
166C2×C4—5022no—
167C2×C4—5022no—
168C2×C4—5022no—
169C2×C4—5024no—
170C2×C4#1686080914yes—
171C2×C4#1686081112yes—
172C22C2×#1686080912yes—
173C22C2×#1686081111yescharacteristic
174C22#2254876011yescharacteristic
175C22#2254876211yescharacteristic
176C22#2254876312yes—
177C4—22no—
178C4—24no—
179C4—42no—
180C4—41no—
181C4—41no—
182C4—10042no—
183C4—10042no—
184C4—10042no—
185C4—10042no—
186C2#3063228611yescharacteristic
187C2#3063229712yes—
188C2#3063236811yescharacteristic
189C2#3063241112yes—
190C2#3063243611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D251
commutator, socle, prime radical, cosocleC23×C251C24
Fitting#395×C251C2
hypercenter#395D251
hypocenterC251#2789
LCSG ▷ C23×C251 ▷ C251
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
95. C251#2789semidirect

Maximal subgroups

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

subgroupindex
C251⋊#4012
#306316262
#306316172
#306315942
#306315742
#306315732
#306315722
#306315522
#395×C2512
#2789251