Finite Group Databasebeta

← #44558631

Subgroups — #44558631

Order 32128.

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

normal subgroups only

This group has 348 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#30631626C211yesmaximal, characteristic
3#30631621C212yesmaximal
4#30631620C212yesmaximal
5#30631618C212yesmaximal
6#30631606C212yesmaximal
7#30631556C211yesmaximal, characteristic
8#30631554C212yesmaximal
9#395×C251C211yesmaximal, characteristic
10C2×C4⋊Dic251C2211yescharacteristic
11C2×C4⋊Dic251C2211yescharacteristic
12C2×Dic251⋊C4C2212yes—
13C2×Dic251⋊C4C2212yes—
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⋊#95C2211yescharacteristic
25C251⋊#95C2212yes—
26C22×Q8×C251C2211yescharacteristic
27C2×C4⋊C4×C251C2211yescharacteristic
28C2×C4⋊C4×C251C2212yes—
29#95×C251C2212yes—
30#95×C251C2211yescharacteristic
31C22×Dic251C2312yes—
32C22×Dic251C2312yes—
33C22×Dic251C2312yes—
34C22×Dic251C2312yes—
35C4⋊Dic251—41no—
36C4⋊Dic251—41no—
37Dic251⋊C4—42no—
38Dic251⋊C4—42no—
39Dic251⋊C4—42no—
40Dic251⋊C4—42no—
41Dic251⋊C4—42no—
42C4×Dic251Q814yescomplemented
43C2×Q8×C251—42no—
44C2×Q8×C251—41no—
45C22×C4×C251C2312yes—
46C22×C4×C251C2312yes—
47C22×C4×C251C2311yescharacteristic
48C22×C4×C251C2311yescharacteristic
49C22×C4×C251C2311yescharacteristic
50C4⋊C4×C251—41no—
51C4⋊C4×C251—42no—
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—22no—
61C2×Dic251—24no—
62C2×Dic251C2×Q814yes—
63C23×C251C2411yescharacteristic
64Q8×C251—44no—
65C2×C4×C251—21no—
66C2×C4×C251—21no—
67C2×C4×C251—22no—
68C2×C4×C251—21no—
69C2×C4×C251—22no—
70C2×C4×C251—22no—
71C2×C4×C251—21no—
72C2×C4×C251—22no—
73C2×C4×C251—22no—
74C2×C4×C251—22no—
75C2×C4×C251—22no—
76C2×C4×C251—22no—
77C2×C4×C251C2×Q812yes—
78C22×C251C22×Q811yescharacteristic
79C22×C251H(5,𝔽2)12yes—
80C22×C251H(5,𝔽2)11yescharacteristic
81C22×C2512−1+411yescharacteristic
82C22×C2512−1+412yes—
83C4×C251—22no—
84C4×C251—41no—
85C4×C251—42no—
86C4×C251—41no—
87C4×C251—42no—
88Dic251—24no—
89Dic251—42no—
90Dic251—42no—
91Dic251—42no—
92C2×C251#54412yes—
93C2×C251#54411yescharacteristic
94C2×C251#56011yescharacteristic
95C2×C251#56311yescharacteristic
96C2×C251#56312yes—
97C251#284311yescharacteristic, complemented, Sylow
98#2843—2511nomaximal, Sylow
99#400—2511no—
100#400—2512no—
101#398—2512no—
102#398—2512no—
103#398—2512no—
104#395—2512no—
105#395D25111yescharacteristic
106#384—2511no—
107#383—2512no—
108C22×Q8C2×D25111yescharacteristic
109C2×C4⋊C4—2512no—
110C2×C4⋊C4—2512no—
111C2×C4⋊C4—2511no—
112C2×C4⋊C4—2512no—
113C2×C4⋊C4—2512no—
114C2×C4⋊C4—2512no—
115C2×C4⋊C4—2511no—
116C2×C4⋊C4C2×D25111yescharacteristic
117C2×C4⋊C4C2×D25112yes—
118C2×C42—2511no—
119#95—2512no—
120#95—2512no—
121#95—2512no—
122#95—2512no—
123#95—2512no—
124#95—2512no—
125#95—2511no—
126#95—2512no—
127#95C2×D25112yes—
128#95C2×D25111yescharacteristic
129C2×Q8—42no—
130C2×Q8—41no—
131C22×C4—2512no—
132C22×C4—2512no—
133C22×C4—2512no—
134C22×C4—2512no—
135C22×C4C22×D25112yes—
136C22×C4C22×D25112yes—
137C22×C4C22×D25111yescharacteristic
138C22×C4C22×D25111yescharacteristic
139C22×C4C22×D25111yescharacteristic
140C4⋊C4—41no—
141C4⋊C4—42no—
142C4⋊C4—10042no—
143C4⋊C4—10042no—
144C4⋊C4—10042no—
145C4⋊C4—10042no—
146C4⋊C4—10042no—
147C4⋊C4—10041no—
148C4⋊C4—10041no—
149C42—2514no—
150C23C23×D25111yescharacteristic
151Q8—44no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—22no—
155C2×C4—21no—
156C2×C4—22no—
157C2×C4—22no—
158C2×C4—21no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—5022no—
165C2×C4—5022no—
166C2×C4—5022no—
167C2×C4—5022no—
168C2×C4—5022no—
169C2×C4—5022no—
170C2×C4—5024no—
171C2×C4—5022no—
172C2×C4—5022no—
173C2×C4—5022no—
174C2×C4—2514no—
175C2×C4Q8×D25112yes—
176C22C2×Q8×D25111yescharacteristic
177C22#2254876012yes—
178C22#2254876011yescharacteristic
179C22#2254876211yescharacteristic
180C22#2254876312yes—
181C4—22no—
182C4—42no—
183C4—41no—
184C4—41no—
185C4—42no—
186C4—10042no—
187C4—10042no—
188C4—10042no—
189C4—5024no—
190C2#3063231412yes—
191C2#3063231611yescharacteristic
192C2#3063242611yescharacteristic
193C2#3063243611yescharacteristic
194C2#3063243812yes—
subgroupG/H
center, FrattiniC23C23×D251
commutator, socle, prime radical, cosocleC23×C251C24
Fitting#395×C251C2
hypercenter#395D251
hypocenterC251#2843
LCSG ▷ C23×C251 ▷ C251
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
42. C4×Dic251Q8semidirect
97. C251#2843semidirect

Maximal subgroups

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

subgroupindex
#306316282
#306316262
#306316212
#306316202
#306316182
#306316062
#306315562
#306315542
#395×C2512
#2843251