Finite Group Databasebeta

← #44558627

Subgroups — #44558627

Order 32128.

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

normal subgroups only

This group has 396 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, complemented
2#30631620C212yesmaximal, complemented
3#30631617C212yesmaximal, complemented
4#30631610C212yesmaximal
5#30631607C212yesmaximal, complemented
6#30631599C211yesmaximal, characteristic
7#30631548C211yesmaximal, characteristic
8#30631545C212yesmaximal
9#399×C251C211yesmaximal, characteristic
10C22×Dic502C2211yescharacteristic
11C23×Dic251C2211yescharacteristic
12C2×C4⋊Dic251C2212yes—
13C2×Dic251⋊C4C2212yes—
14C2×Dic251⋊C4C2212yes—
15C2×Dic251⋊C4C2212yes—
16C2×Dic251⋊C4C2212yes—
17C2×C22⋊Dic251C2212yes—
18C2×C22⋊Dic251C2211yescharacteristic
19C251⋊#95C2212yes—
20C251⋊#95C2212yes—
21C251⋊#95C2212yes—
22C251⋊#95C2212yes—
23C251⋊#95C2211yescharacteristic
24C251⋊#95C2212yes—
25C251⋊#95C2212yes—
26C2×C4⋊C4×C251C2212yes—
27C2×C4⋊C4×C251C2211yescharacteristic
28C2×C22⋊C4×C251C2212yes—
29C2×C22⋊C4×C251C2211yescharacteristic
30#95×C251C2211yescharacteristic
31C2×Dic502—41no—
32C2×Dic502—42no—
33C22×Dic251—41no—
34C22×Dic251—41no—
35C22×Dic251Q814yescomplemented
36C22×Dic251C2312yes—
37C22×Dic251C2312yes—
38C22×Dic251C2312yes—
39C22×Dic251C2312yes—
40C4⋊Dic251—42no—
41Dic251⋊C4—42no—
42Dic251⋊C4—42no—
43Dic251⋊C4—42no—
44Dic251⋊C4—42no—
45C22⋊Dic251—42no—
46C22⋊Dic251—41no—
47C24×C251C2311yescharacteristic
48C22×C4×C251C2311yescharacteristic
49C22×C4×C251C2311yescharacteristic
50C22×C4×C251C2312yes—
51C22×C4×C251C2312yes—
52C4⋊C4×C251—41no—
53C4⋊C4×C251—42no—
54C22⋊C4×C251—42no—
55C22⋊C4×C251—41no—
56Dic502—44no—
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×Dic251—22no—
66C2×Dic251—22no—
67C2×Dic251—22no—
68C2×Dic251—44no—
69C2×Dic251C2×Q814yes—
70C23×C251—22no—
71C23×C251—21no—
72C23×C251—21no—
73C23×C251—22no—
74C23×C251C2×Q812yes—
75C23×C251C2411yescharacteristic
76C2×C4×C251—22no—
77C2×C4×C251—21no—
78C2×C4×C251—22no—
79C2×C4×C251—21no—
80C2×C4×C251—22no—
81C2×C4×C251—22no—
82C2×C4×C251—22no—
83C2×C4×C251—22no—
84C2×C4×C251—22no—
85C2×C4×C251—22no—
86C22×C251—22no—
87C22×C251—24no—
88C22×C251—41no—
89C22×C251—42no—
90C22×C251—41no—
91C22×C251C22×Q811yescharacteristic
92C22×C251H(5,𝔽2)11yescharacteristic
93C22×C251H(5,𝔽2)12yes—
94C22×C251H(5,𝔽2)11yescharacteristic
95C22×C2512−1+412yes—
96C4×C251—42no—
97C4×C251—41no—
98C4×C251—42no—
99C4×C251—41no—
100Dic251—24no—
101Dic251—42no—
102Dic251—42no—
103Dic251—42no—
104C2×C251—42no—
105C2×C251#54311yescharacteristic
106C2×C251#54412yes—
107C2×C251#56011yescharacteristic
108C2×C251#56212yes—
109C2×C251#56311yescharacteristic
110C251#284111yescharacteristic, complemented, Sylow
111#2841—2511nomaximal, Sylow
112#400—2512no—
113#399D25111yescharacteristic
114#398—2512no—
115#398—2512no—
116#396—2512no—
117#395—2512no—
118#393—2511no—
119#381—2511no—
120#380—2512no—
121C22×Q8—2511no—
122C23×C4—2511no—
123C2×C4⋊C4—2512no—
124C2×C4⋊C4—2512no—
125C2×C4⋊C4—2512no—
126C2×C4⋊C4—2512no—
127C2×C4⋊C4—2512no—
128C2×C4⋊C4C2×D25112yes—
129C2×C4⋊C4C2×D25111yescharacteristic
130C2×C22⋊C4—2512no—
131C2×C22⋊C4—2511no—
132C2×C22⋊C4C2×D25112yes—
133C2×C22⋊C4C2×D25111yescharacteristic
134#95—2512no—
135#95—2512no—
136#95—2512no—
137#95—2512no—
138#95—2511no—
139#95—2512no—
140#95—2512no—
141#95C2×D25111yescharacteristic
142C24C22×D25111yescharacteristic
143C2×Q8—10041no—
144C2×Q8—10042no—
145C22×C4—10041no—
146C22×C4—10041no—
147C22×C4—2514no—
148C22×C4—2512no—
149C22×C4—2512no—
150C22×C4—2512no—
151C22×C4—2512no—
152C22×C4C22×D25111yescharacteristic
153C22×C4C22×D25111yescharacteristic
154C22×C4C22×D25112yes—
155C22×C4C22×D25112yes—
156C4⋊C4—41no—
157C4⋊C4—42no—
158C4⋊C4—10042no—
159C4⋊C4—10042no—
160C4⋊C4—10042no—
161C4⋊C4—10042no—
162C4⋊C4—10042no—
163C22⋊C4—42no—
164C22⋊C4—41no—
165C22⋊C4—10042no—
166C22⋊C4—10041no—
167C23—22no—
168C23—21no—
169C23—21no—
170C23—22no—
171C23Q8×D25112yes—
172C23C23×D25111yescharacteristic
173Q8—10044no—
174C2×C4—22no—
175C2×C4—21no—
176C2×C4—22no—
177C2×C4—21no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—10044no—
185C2×C4—5022no—
186C2×C4—5022no—
187C2×C4—5022no—
188C2×C4—5022no—
189C2×C4—5022no—
190C2×C4—5022no—
191C2×C4—5022no—
192C2×C4—5022no—
193C2×C4—5022no—
194C2×C4—5022no—
195C2×C4—5022no—
196C2×C4—2514no—
197C22—22no—
198C22—24no—
199C22—41no—
200C22—42no—
201C22—41no—
202C22C2×Q8×D25111yescharacteristic
203C22#2254876011yescharacteristic
204C22#2254876012yes—
205C22#2254876111yescharacteristic
206C22#2254876212yes—
207C4—41no—
208C4—42no—
209C4—42no—
210C4—41no—
211C4—10042no—
212C4—10042no—
213C4—10042no—
214C4—5024no—
215C2—42no—
216C2#3063231211yescharacteristic
217C2#3063231612yes—
218C2#3063242411yescharacteristic
219C2#3063243412yes—
220C2#3063244011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D251
commutator, socle, cosocleC23×C251C24
Fitting#399×C251C2
prime radicalC24×C251C23
hypercenter#399D251
hypocenterC251#2841
LCSG ▷ C23×C251 ▷ C251
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. #30631628C2semidirect
2. #30631620C2semidirect
3. #30631617C2semidirect
5. #30631607C2semidirect
35. C22×Dic251Q8semidirect
110. C251#2841semidirect

Maximal subgroups

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

subgroupindex
#306316282
#306316202
#306316172
#306316102
#306316072
#306315992
#306315482
#306315452
#399×C2512
#2841251