Finite Group Databasebeta

← #44558553

Subgroups — #44558553

Order 32128.

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

normal subgroups only

This group has 380 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
1#30631629C212yesmaximal, complemented
2#30631628C212yesmaximal, complemented
3#30631625C212yesmaximal, complemented
4#30631615C212yesmaximal
5#30631612C211yesmaximal, characteristic
6#30631607C212yesmaximal, complemented
7#30631599C211yesmaximal, characteristic
8#30631560C212yesmaximal
9#385×C251C211yesmaximal, characteristic
10C22×Dic502C2211yescharacteristic
11C2×C4⋊Dic251C2211yescharacteristic
12C2×Dic251⋊C4C2212yes—
13C2×Dic251⋊C4C2212yes—
14C2×Dic251⋊C4C2212yes—
15C2×C22⋊Dic251C2212yes—
16C2×C22⋊Dic251C2212yes—
17C251⋊#95C2212yes—
18C251⋊#95C2212yes—
19C251⋊#95C2211yescharacteristic
20C251⋊#95C2212yes—
21C251⋊#95C2212yes—
22C251⋊#95C2212yes—
23C251⋊#95C2212yes—
24C251⋊#95C2211yescharacteristic
25C251⋊#95C2212yes—
26C23×C4×C251C2211yescharacteristic
27C2×C4⋊C4×C251C2212yes—
28C2×C22⋊C4×C251C2211yescharacteristic
29C2×C22⋊C4×C251C2211yescharacteristic
30#95×C251C2212yes—
31C2×Dic502—41no—
32C2×Dic502—42no—
33C22×Dic251C2312yes—
34C22×Dic251C2312yes—
35C22×Dic251C2312yes—
36C22×Dic251C2312yes—
37C4⋊Dic251—41no—
38Dic251⋊C4—42no—
39Dic251⋊C4—42no—
40Dic251⋊C4—42no—
41C22⋊Dic251—24no—
42C22⋊Dic251—42no—
43C24×C251C2311yescharacteristic
44C22×C4×C251—42no—
45C22×C4×C251—41no—
46C22×C4×C251C2311yescharacteristic
47C22×C4×C251C2311yescharacteristic
48C22×C4×C251C2312yes—
49C22×C4×C251C2312yes—
50C4⋊C4×C251—42no—
51C22⋊C4×C251—22no—
52C22⋊C4×C251—41no—
53Dic502—44no—
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×Dic251—22no—
65C2×Dic251—22no—
66C23×C251—21no—
67C23×C251—22no—
68C23×C251—22no—
69C23×C251—21no—
70C23×C251#4112yes—
71C23×C251C2411yescharacteristic
72C2×C4×C251—22no—
73C2×C4×C251—21no—
74C2×C4×C251—21no—
75C2×C4×C251—22no—
76C2×C4×C251—22no—
77C2×C4×C251—22no—
78C2×C4×C251—22no—
79C2×C4×C251—22no—
80C2×C4×C251—22no—
81C2×C4×C251—22no—
82C2×C4×C251—44no—
83C22×C251—24no—
84C22×C251—22no—
85C22×C251—41no—
86C22×C251—42no—
87C22×C251—41no—
88C22×C251C2×#4111yescharacteristic
89C22×C251H(5,𝔽2)12yes—
90C22×C251H(5,𝔽2)11yescharacteristic
91C22×C2512−1+412yes—
92C22×C2512−1+411yescharacteristic
93C4×C251—42no—
94C4×C251—42no—
95C4×C251—41no—
96C4×C251—41no—
97Dic251—42no—
98Dic251—42no—
99Dic251—42no—
100Dic251—42no—
101C2×C251—42no—
102C2×C251#53912yes—
103C2×C251#53911yescharacteristic
104C2×C251H(3,𝔽4)11yescharacteristic
105C2×C251#56312yes—
106C2×C251#56311yescharacteristic
107C251#283511yescharacteristic, complemented, Sylow
108#2835—2511nomaximal, Sylow
109#400—2512no—
110#400—2512no—
111#400—2512no—
112#397—2511no—
113#397—2512no—
114#395—2512no—
115#393—2511no—
116#385—2512no—
117#385D25111yescharacteristic
118C22×Q8—2511no—
119C23×C4C2×D25111yescharacteristic
120C2×C4⋊C4—2512no—
121C2×C4⋊C4—2512no—
122C2×C4⋊C4—2512no—
123C2×C4⋊C4—2511no—
124C2×C4⋊C4C2×D25112yes—
125C2×C22⋊C4—2512no—
126C2×C22⋊C4—2512no—
127C2×C22⋊C4C2×D25111yescharacteristic
128C2×C22⋊C4C2×D25111yescharacteristic
129#95—2512no—
130#95—2512no—
131#95—2511no—
132#95—2512no—
133#95—2512no—
134#95—2512no—
135#95—2512no—
136#95—2512no—
137#95—2511no—
138#95C2×D25112yes—
139C24C22×D25111yescharacteristic
140C2×Q8—10041no—
141C2×Q8—10042no—
142C22×C4—42no—
143C22×C4—41no—
144C22×C4—2512no—
145C22×C4—2512no—
146C22×C4—2512no—
147C22×C4—2512no—
148C22×C4C22×D25111yescharacteristic
149C22×C4C22×D25112yes—
150C22×C4C22×D25111yescharacteristic
151C22×C4C22×D25112yes—
152C4⋊C4—42no—
153C4⋊C4—10042no—
154C4⋊C4—10042no—
155C4⋊C4—10042no—
156C4⋊C4—10041no—
157C22⋊C4—22no—
158C22⋊C4—41no—
159C22⋊C4—10042no—
160C22⋊C4—5024no—
161C23—21no—
162C23—22no—
163C23—22no—
164C23—21no—
165C23#1686081012yes—
166C23C23×D25111yescharacteristic
167Q8—10044no—
168C2×C4—22no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—44no—
179C2×C4—5022no—
180C2×C4—5022no—
181C2×C4—5022no—
182C2×C4—5022no—
183C2×C4—5022no—
184C2×C4—5022no—
185C2×C4—5022no—
186C2×C4—5022no—
187C2×C4—5022no—
188C2×C4—5022no—
189C2×C4—5022no—
190C2×C4—5022no—
191C22—24no—
192C22—22no—
193C22—41no—
194C22—42no—
195C22—41no—
196C22C2×#1686081011yescharacteristic
197C22#2254876012yes—
198C22#2254876011yescharacteristic
199C22#2254876212yes—
200C22#2254876311yescharacteristic
201C4—42no—
202C4—42no—
203C4—41no—
204C4—41no—
205C4—10042no—
206C4—10042no—
207C4—10042no—
208C4—10042no—
209C2—42no—
210C2#3063227912yes—
211C2#3063228411yescharacteristic
212C2#3063243011yescharacteristic
213C2#3063243812yes—
214C2#3063244011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D251
commutator, socle, cosocleC23×C251C24
Fitting#385×C251C2
prime radicalC24×C251C23
hypercenter#385D251
hypocenterC251#2835
LCSG ▷ C23×C251 ▷ C251
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
1. #30631629C2semidirect
2. #30631628C2semidirect
3. #30631625C2semidirect
6. #30631607C2semidirect
107. C251#2835semidirect

Maximal subgroups

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

subgroupindex
#306316292
#306316282
#306316252
#306316152
#306316122
#306316072
#306315992
#306315602
#385×C2512
#2835251