Finite Group Databasebeta

← #44558435

Subgroups — #44558435

Order 32128.

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

normal subgroups only

This group has 380 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
1C2×#22548706C211yesmaximal, characteristic
2C2×#22548705C211yesmaximal, characteristic
3#30631629C212yesmaximal
4#30631627C211yesmaximal, characteristic
5#30631626C211yesmaximal, characteristic
6#30631621C212yesmaximal
7#30631620C212yesmaximal
8#30631618C211yesmaximal, characteristic
9#30631618C212yesmaximal
10#30631551C211yesmaximal, characteristic
11#398×C251C211yesmaximal, characteristic
12#22548706—24no—
13#22548705—24no—
14C2×C4⋊Dic251C2211yescharacteristic
15C2×C4⋊Dic251C2212yes—
16C2×C4⋊Dic251C2211yescharacteristic
17C2×Dic251⋊C4C2211yescharacteristic
18C2×Dic251⋊C4C2211yescharacteristic
19C2×Dic251⋊C4C2211yescharacteristic
20C2×Dic251⋊C4C2211yescharacteristic
21C2×Dic251⋊C4C2212yes—
22C2×Dic251⋊C4C2212yes—
23C2×Dic251⋊C4C2212yes—
24C2×C4×Dic251C2211yescharacteristic
25C251⋊#95C2211yescharacteristic
26C251⋊#95C2212yes—
27C251⋊#95C2212yes—
28C251⋊#95C2212yes—
29C251⋊#95C2212yes—
30C251⋊#95C2212yes—
31C251⋊#95C2212yes—
32C2×C4⋊C4×C251C2211yescharacteristic
33C2×C4⋊C4×C251C2211yescharacteristic
34C2×C4⋊C4×C251C2212yes—
35#95×C251C2211yescharacteristic
36#95×C251C2212yes—
37C22×Dic251C2312yes—
38C22×Dic251C2311yescharacteristic
39C22×Dic251C2312yes—
40C22×Dic251C2311yescharacteristic
41C22×Dic251C2311yescharacteristic
42C22×Dic251C2311yescharacteristic
43C4⋊Dic251—22no—
44C4⋊Dic251—24no—
45C4⋊Dic251—22no—
46Dic251⋊C4—22no—
47Dic251⋊C4—22no—
48Dic251⋊C4—22no—
49Dic251⋊C4—24no—
50Dic251⋊C4—22no—
51Dic251⋊C4—42no—
52Dic251⋊C4—42no—
53C4×Dic251D414yes—
54C22×C4×C251C2312yes—
55C22×C4×C251C2311yescharacteristic
56C22×C4×C251C2312yes—
57C22×C4×C251C2311yescharacteristic
58C22×C4×C251C2311yescharacteristic
59C4⋊C4×C251—22no—
60C4⋊C4×C251—22no—
61C4⋊C4×C251—42no—
62C2×Dic251—21no—
63C2×Dic251—21no—
64C2×Dic251—22no—
65C2×Dic251—21no—
66C2×Dic251—21no—
67C2×Dic251—22no—
68C2×Dic251—21no—
69C2×Dic251—22no—
70C2×Dic251—22no—
71C2×Dic251—21no—
72C2×Dic251—22no—
73C2×Dic251—24no—
74C2×Dic251—22no—
75C2×Dic251C2×D412yes—
76C2×Dic251C2×D412yes—
77C23×C251C2411yescharacteristic
78C2×C4×C251—22no—
79C2×C4×C251—21no—
80C2×C4×C251—21no—
81C2×C4×C251—22no—
82C2×C4×C251—21no—
83C2×C4×C251—22no—
84C2×C4×C251—22no—
85C2×C4×C251—22no—
86C2×C4×C251—22no—
87C2×C4×C251—21no—
88C2×C4×C251—22no—
89C2×C4×C251—22no—
90C2×C4×C251C2×D412yes—
91C22×C251C22×D411yescharacteristic
92C22×C251H(5,𝔽2)11yescharacteristic
93C22×C2512−1+412yes—
94C22×C2512−1+411yescharacteristic
95C22×C2512−1+412yes—
96C4×C251—22no—
97C4×C251—41no—
98C4×C251—41no—
99C4×C251—42no—
100C4×C251—42no—
101Dic251—22no—
102Dic251—22no—
103Dic251—41no—
104Dic251—42no—
105Dic251—41no—
106Dic251—42no—
107C2×C251#53611yescharacteristic
108C2×C251#53611yescharacteristic
109C2×C251#53711yescharacteristic
110C2×C251#56312yes—
111C2×C251#56412yes—
112C251#282211yescharacteristic, complemented, Sylow
113#2822—2511nomaximal, Sylow
114C2×#125—2511no—
115C2×#125—2511no—
116#400—2511no—
117#400—2511no—
118#400—2512no—
119#398—2511no—
120#398—2512no—
121#398—2512no—
122#398—2512no—
123#398D25111yescharacteristic
124#382—2511no—
125#125—5024no—
126#125—5024no—
127C2×C4⋊C4—2512no—
128C2×C4⋊C4—2512no—
129C2×C4⋊C4—2511no—
130C2×C4⋊C4—2512no—
131C2×C4⋊C4—2511no—
132C2×C4⋊C4—2511no—
133C2×C4⋊C4—2511no—
134C2×C4⋊C4—2512no—
135C2×C4⋊C4—2511no—
136C2×C4⋊C4—2511no—
137C2×C4⋊C4C2×D25111yescharacteristic
138C2×C4⋊C4C2×D25111yescharacteristic
139C2×C4⋊C4C2×D25112yes—
140C2×C42—2511no—
141#95—2511no—
142#95—2512no—
143#95—2512no—
144#95—2512no—
145#95—2512no—
146#95—2512no—
147#95—2512no—
148#95C2×D25111yescharacteristic
149#95C2×D25112yes—
150C22×C4—2512no—
151C22×C4—2511no—
152C22×C4—2512no—
153C22×C4—2511no—
154C22×C4—2511no—
155C22×C4—2511no—
156C22×C4C22×D25112yes—
157C22×C4C22×D25111yescharacteristic
158C22×C4C22×D25111yescharacteristic
159C22×C4C22×D25112yes—
160C22×C4C22×D25111yescharacteristic
161C4⋊C4—22no—
162C4⋊C4—22no—
163C4⋊C4—42no—
164C4⋊C4—10042no—
165C4⋊C4—10042no—
166C4⋊C4—5022no—
167C4⋊C4—5022no—
168C4⋊C4—5024no—
169C4⋊C4—5022no—
170C4⋊C4—5022no—
171C4⋊C4—5024no—
172C4⋊C4—5022no—
173C4⋊C4—5022no—
174C42—2514no—
175C23C23×D25111yescharacteristic
176C2×C4—22no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—22no—
180C2×C4—21no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—21no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—5021no—
189C2×C4—5022no—
190C2×C4—5021no—
191C2×C4—5021no—
192C2×C4—5022no—
193C2×C4—5022no—
194C2×C4—5021no—
195C2×C4—5022no—
196C2×C4—5024no—
197C2×C4—5022no—
198C2×C4—5021no—
199C2×C4—5021no—
200C2×C4—5022no—
201C2×C4—2512no—
202C2×C4—2512no—
203C2×C4D4×D25112yes—
204C22C2×D4×D25111yescharacteristic
205C22#2254876011yescharacteristic
206C22#2254876212yes—
207C22#2254876211yescharacteristic
208C22#2254876312yes—
209C4—22no—
210C4—41no—
211C4—41no—
212C4—42no—
213C4—42no—
214C4—10041no—
215C4—10042no—
216C4—10041no—
217C4—10042no—
218C4—5022no—
219C4—5022no—
220C2#3063225911yescharacteristic
221C2#3063226211yescharacteristic
222C2#3063226611yescharacteristic
223C2#3063243812yes—
224C2C251⋊#56412yes—
subgroupG/H
center, FrattiniC23C23×D251
commutator, socle, prime radical, cosocleC23×C251C24
Fitting#398×C251C2
hypercenter#398D251
hypocenterC251#2822
LCSG ▷ C23×C251 ▷ C251
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
112. C251#2822semidirect

Maximal subgroups

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

subgroupindex
C2×#225487062
C2×#225487052
#306316292
#306316272
#306316262
#306316212
#306316202
#306316182
#306315512
#398×C2512
#2822251