Finite Group Databasebeta

← #44558336

Subgroups — #44558336

Order 32128.

408 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 133 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#30631625C212yesmaximal, complemented
2#30631623C211yesmaximal, characteristic
3#30631617C212yesmaximal, complemented
4#30631586C212yesmaximal, complemented
5#30631584C212yesmaximal, complemented
6#30631581C212yesmaximal
7#30631562C212yesmaximal
8#30631559C211yesmaximal, characteristic
9#399×C251C211yesmaximal, characteristic
10C23×Dic251C2211yescharacteristic
11C2×C4⋊Dic251C2212yes—
12C2×C4⋊Dic251C2212yes—
13C2×C4⋊Dic251C2211yescharacteristic
14C2×Dic251⋊C4C2212yes—
15C2×Dic251⋊C4C2212yes—
16C2×Dic251⋊C4C2212yes—
17C2×C22⋊Dic251C2212yes—
18C2×C22⋊Dic251C2211yescharacteristic
19C2×C4×Dic251C2212yes—
20C251⋊#95C2212yes—
21C251⋊#95C2212yes—
22C251⋊#95C2212yes—
23C251⋊#95C2212yes—
24C251⋊#95C2212yes—
25C251⋊#95C2211yescharacteristic
26C2×C4⋊C4×C251C2212yes—
27C2×C4⋊C4×C251C2211yescharacteristic
28C2×C22⋊C4×C251C2212yes—
29C2×C22⋊C4×C251C2211yescharacteristic
30#95×C251C2211yescharacteristic
31C22×Dic251—41no—
32C22×Dic251—42no—
33C22×Dic251C2312yes—
34C22×Dic251C2312yes—
35C22×Dic251C2312yes—
36C22×Dic251C2312yes—
37C4⋊Dic251—42no—
38C4⋊Dic251—41no—
39C4⋊Dic251—42no—
40Dic251⋊C4—24no—
41Dic251⋊C4—42no—
42Dic251⋊C4—42no—
43C22⋊Dic251—42no—
44C22⋊Dic251Q814yes—
45C4×Dic251—42no—
46C24×C251C2311yescharacteristic
47C22×C4×C251C2311yescharacteristic
48C22×C4×C251C2312yes—
49C22×C4×C251C2311yescharacteristic
50C22×C4×C251C2312yes—
51C4⋊C4×C251—24no—
52C4⋊C4×C251—41no—
53C22⋊C4×C251—24no—
54C22⋊C4×C251—41no—
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—
66C2×Dic251—44no—
67C2×Dic251C2×Q814yes—
68C23×C251—22no—
69C23×C251—21no—
70C23×C251—22no—
71C23×C251—21no—
72C23×C251C2×Q812yes—
73C23×C251C2411yescharacteristic
74C2×C4×C251—22no—
75C2×C4×C251—22no—
76C2×C4×C251—22no—
77C2×C4×C251—22no—
78C2×C4×C251—22no—
79C2×C4×C251—21no—
80C2×C4×C251—21no—
81C2×C4×C251—22no—
82C2×C4×C251—22no—
83C2×C4×C251#4114yes—
84C22×C251—22no—
85C22×C251—24no—
86C22×C251—42no—
87C22×C251—41no—
88C22×C251—41no—
89C22×C251C22×Q811yescharacteristic
90C22×C251C2×#4112yes—
91C22×C251H(5,𝔽2)11yescharacteristic
92C22×C251H(5,𝔽2)12yes—
93C22×C2512−1+411yescharacteristic
94C4×C251—24no—
95C4×C251—42no—
96C4×C251—41no—
97C4×C251—41no—
98Dic251—24no—
99Dic251—42no—
100Dic251—42no—
101Dic251—42no—
102C2×C251—42no—
103C2×C251#53912yes—
104C2×C251#54411yescharacteristic
105C2×C251#55412yes—
106C2×C251#55911yescharacteristic
107C2×C251#56011yescharacteristic
108C251#280711yescharacteristic, complemented, Sylow
109#2807—2511nomaximal, Sylow
110#400—2512no—
111#399—2511no—
112#399D25111yescharacteristic
113#398—2512no—
114#389—2512no—
115#389—2512no—
116#388—2512no—
117#385—2511no—
118#385—2512no—
119C23×C4—2511no—
120C2×C4⋊C4—2512no—
121C2×C4⋊C4—2512no—
122C2×C4⋊C4—2512no—
123C2×C4⋊C4—2511no—
124C2×C4⋊C4—2512no—
125C2×C4⋊C4—2512no—
126C2×C4⋊C4C2×D25112yes—
127C2×C4⋊C4C2×D25111yescharacteristic
128C2×C22⋊C4—2512no—
129C2×C22⋊C4—2511no—
130C2×C22⋊C4C2×D25112yes—
131C2×C22⋊C4C2×D25111yescharacteristic
132C2×C42—2512no—
133#95—2512no—
134#95—2512no—
135#95—2512no—
136#95—2512no—
137#95—2512no—
138#95—2511no—
139#95C2×D25111yescharacteristic
140C24C22×D25111yescharacteristic
141C22×C4—10041no—
142C22×C4—10042no—
143C22×C4—2512no—
144C22×C4—2512no—
145C22×C4—2512no—
146C22×C4—2512no—
147C22×C4C22×D25112yes—
148C22×C4C22×D25111yescharacteristic
149C22×C4C22×D25111yescharacteristic
150C22×C4C22×D25112yes—
151C4⋊C4—24no—
152C4⋊C4—41no—
153C4⋊C4—10042no—
154C4⋊C4—10042no—
155C4⋊C4—10042no—
156C4⋊C4—10041no—
157C4⋊C4—10042no—
158C4⋊C4—5024no—
159C22⋊C4—24no—
160C22⋊C4—41no—
161C22⋊C4—10042no—
162C22⋊C4—2514no—
163C42—10042no—
164C23—22no—
165C23—21no—
166C23—22no—
167C23—21no—
168C23Q8×D25112yes—
169C23C23×D25111yescharacteristic
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—10044no—
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—
191C2×C4—2514no—
192C2×C4#1686080914yes—
193C22—22no—
194C22—24no—
195C22—42no—
196C22—41no—
197C22—41no—
198C22C2×Q8×D25111yescharacteristic
199C22C2×#1686080912yes—
200C22#2254876011yescharacteristic
201C22#2254876112yes—
202C22#2254876211yescharacteristic
203C4—24no—
204C4—42no—
205C4—41no—
206C4—41no—
207C4—10042no—
208C4—10042no—
209C4—10042no—
210C4—5024no—
211C2—42no—
212C2#3063228112yes—
213C2#3063231611yescharacteristic
214C2#3063239512yes—
215C2#3063242311yescharacteristic
216C2#3063242711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D251
commutator, socle, cosocleC23×C251C24
Fitting#399×C251C2
prime radicalC24×C251C23
hypercenter#399D251
hypocenterC251#2807
LCSG ▷ C23×C251 ▷ C251
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. #30631625C2semidirect
3. #30631617C2semidirect
4. #30631586C2semidirect
5. #30631584C2semidirect
108. C251#2807semidirect

Maximal subgroups

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

subgroupindex
#306316252
#306316232
#306316172
#306315862
#306315842
#306315812
#306315622
#306315592
#399×C2512
#2807251