Finite Group Databasebeta

← #44558545

Subgroups — #44558545

Order 32128.

332 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 117 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#30631629C212yesmaximal
3#30631627C212yesmaximal
4#30631626C211yesmaximal, characteristic
5#30631620C212yesmaximal
6#30631607C212yesmaximal
7#30631592C211yesmaximal, characteristic
8#30631573C212yesmaximal
9#398×C251C211yesmaximal, characteristic
10C22×Dic502C2211yescharacteristic
11C2×C4⋊Dic251C2212yes—
12C2×C4⋊Dic251C2211yescharacteristic
13C2×Dic251⋊C4C2212yes—
14C2×Dic251⋊C4C2212yes—
15C2×C4×Dic251C2211yescharacteristic
16C251⋊#95C2212yes—
17C251⋊#95C2212yes—
18C251⋊#95C2212yes—
19C251⋊#95C2212yes—
20C251⋊#95C2212yes—
21C251⋊#95C2212yes—
22C251⋊#95C2212yes—
23C251⋊#95C2212yes—
24C251⋊#95C2212yes—
25C251⋊#95C2211yescharacteristic
26C2×C4⋊C4×C251C2212yes—
27C2×C4⋊C4×C251C2211yescharacteristic
28C2×C4⋊C4×C251C2211yescharacteristic
29#95×C251C2212yes—
30#95×C251C2211yescharacteristic
31C2×Dic502—22no—
32C2×Dic502—41no—
33C2×Dic502—41no—
34C22×Dic251C2312yes—
35C22×Dic251C2312yes—
36C22×Dic251C2312yes—
37C22×Dic251C2312yes—
38C4⋊Dic251—41no—
39C4⋊Dic251—42no—
40Dic251⋊C4—42no—
41Dic251⋊C4—42no—
42C4×Dic251—41no—
43C22×C4×C251C2312yes—
44C22×C4×C251C2311yescharacteristic
45C22×C4×C251C2311yescharacteristic
46C22×C4×C251C2312yes—
47C22×C4×C251C2311yescharacteristic
48C4⋊C4×C251—24no—
49C4⋊C4×C251—41no—
50C4⋊C4×C251—41no—
51Dic502—44no—
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×Dic251—22no—
63C23×C251C2411yescharacteristic
64C2×C4×C251—21no—
65C2×C4×C251—21no—
66C2×C4×C251—22no—
67C2×C4×C251—22no—
68C2×C4×C251—21no—
69C2×C4×C251—22no—
70C2×C4×C251—21no—
71C2×C4×C251—22no—
72C2×C4×C251—22no—
73C2×C4×C251—22no—
74C2×C4×C251—22no—
75C2×C4×C251—22no—
76C2×C4×C251#4112yes—
77C22×C251C2×#4111yescharacteristic
78C22×C251H(5,𝔽2)11yescharacteristic
79C22×C251H(5,𝔽2)11yescharacteristic
80C22×C2512−1+412yes—
81C22×C2512−1+412yes—
82C4×C251—22no—
83C4×C251—42no—
84C4×C251—42no—
85C4×C251—41no—
86C4×C251—41no—
87Dic251—42no—
88Dic251—42no—
89Dic251—42no—
90Dic251—42no—
91C2×C251#53811yescharacteristic
92C2×C251#54112yes—
93C2×C251#56311yescharacteristic
94C2×C251#56311yescharacteristic
95C2×C251#56312yes—
96C251#283311yescharacteristic, complemented, Sylow
97#2833—2511nomaximal, Sylow
98#401—2512no—
99#400—2511no—
100#400—2512no—
101#400—2512no—
102#398—2512no—
103#398D25111yescharacteristic
104#395—2512no—
105#391—2511no—
106#387—2512no—
107C22×Q8—2511no—
108C2×C4⋊C4—2512no—
109C2×C4⋊C4—2512no—
110C2×C4⋊C4—2512no—
111C2×C4⋊C4—2511no—
112C2×C4⋊C4C2×D25112yes—
113C2×C4⋊C4C2×D25111yescharacteristic
114C2×C4⋊C4C2×D25111yescharacteristic
115C2×C42—2511no—
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—2512no—
125#95—2511no—
126#95C2×D25112yes—
127#95C2×D25111yescharacteristic
128C2×Q8—10041no—
129C2×Q8—10041no—
130C2×Q8—5022no—
131C22×C4—2512no—
132C22×C4—2512no—
133C22×C4—2512no—
134C22×C4—2512no—
135C22×C4C22×D25112yes—
136C22×C4C22×D25111yescharacteristic
137C22×C4C22×D25111yescharacteristic
138C22×C4C22×D25112yes—
139C22×C4C22×D25111yescharacteristic
140C4⋊C4—24no—
141C4⋊C4—41no—
142C4⋊C4—41no—
143C4⋊C4—10042no—
144C4⋊C4—10042no—
145C4⋊C4—10041no—
146C4⋊C4—10042no—
147C42—10041no—
148C23C23×D25111yescharacteristic
149Q8—10044no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—21no—
155C2×C4—22no—
156C2×C4—21no—
157C2×C4—22no—
158C2×C4—22no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—5022no—
163C2×C4—5022no—
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#1686080912yes—
174C22C2×#1686080911yescharacteristic
175C22#2254876011yescharacteristic
176C22#2254876111yescharacteristic
177C22#2254876212yes—
178C22#2254876312yes—
179C4—22no—
180C4—42no—
181C4—42no—
182C4—41no—
183C4—41no—
184C4—10042no—
185C4—10042no—
186C4—10042no—
187C4—10042no—
188C2#3063227411yescharacteristic
189C2#3063229712yes—
190C2#3063243611yescharacteristic
191C2#3063244011yescharacteristic
192C2#3063244112yes—
subgroupG/H
center, FrattiniC23C23×D251
commutator, socle, prime radical, cosocleC23×C251C24
Fitting#398×C251C2
hypercenter#398D251
hypocenterC251#2833
LCSG ▷ C23×C251 ▷ C251
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
96. C251#2833semidirect

Maximal subgroups

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

subgroupindex
C251⋊#4012
#306316292
#306316272
#306316262
#306316202
#306316072
#306315922
#306315732
#398×C2512
#2833251