Finite Group Databasebeta

← #44558513

Subgroups — #44558513

Order 32128.

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

normal subgroups only

This group has 408 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#30631617C211yesmaximal, characteristic, complemented
2#30631616C212yesmaximal, complemented
3#30631613C212yesmaximal, complemented
4#30631609C212yesmaximal, complemented
5#30631605C212yesmaximal
6#30631598C212yesmaximal, complemented
7#30631588C211yesmaximal, characteristic
8#30631569C212yesmaximal, complemented
9#395×C251C211yesmaximal, characteristic, complemented
10C22×D1004C2211yescharacteristic
11C2×C4⋊Dic251C2211yescharacteristic
12C2×Dic251⋊C4C2212yes—
13C2×Dic251⋊C4C2212yes—
14C2×(C2×D251)⋊C4C2212yes—
15C2×(C2×D251)⋊C4C2212yes—
16C2×(C2×D251)⋊C4C2212yes—
17C2×(C2×D251)⋊C4C2212yes—
18C2×(C2×D251)⋊C4C2212yes—
19C2×(C2×D251)⋊C4C2212yes—
20C2×C4×Dic251C2211yescharacteristic
21C251⋊#95C2212yes—
22C251⋊#95C2212yes—
23C251⋊#95C2212yes—
24C251⋊#95C2212yes—
25C251⋊#95C2211yescharacteristic
26C22×Q8×C251C2211yescharacteristic
27C2×C4⋊C4×C251C2211yescharacteristic
28C2×C4⋊C4×C251C2212yes—
29#95×C251C2212yes—
30#95×C251C2211yescharacteristic
31C23×D251C2312yes—
32C2×D1004—22no—
33C2×D1004—41no—
34C2×D1004—41no—
35C22×Dic251C2312yes—
36C22×Dic251C2312yes—
37C22×Dic251C2312yes—
38C4⋊Dic251—41no—
39Dic251⋊C4—24no—
40Dic251⋊C4—42no—
41(C2×D251)⋊C4—42no—
42(C2×D251)⋊C4—42no—
43(C2×D251)⋊C4—42no—
44(C2×D251)⋊C4—42no—
45(C2×D251)⋊C4—42no—
46(C2×D251)⋊C4—42no—
47C4×Dic251—41no—
48C2×Q8×C251—41no—
49C2×Q8×C251—42no—
50C22×C4×C251C2312yes—
51C22×C4×C251C2312yes—
52C22×C4×C251C2311yescharacteristic
53C22×C4×C251C2311yescharacteristic
54C22×C4×C251C2311yescharacteristic
55C4⋊C4×C251—41no—
56C4⋊C4×C251—42no—
57C22×D251—22no—
58C22×D251—22no—
59C22×D251—22no—
60C22×D251—22no—
61C22×D251—22no—
62C22×D251—22no—
63C22×D251—22no—
64D1004—44no—
65C2×Dic251—22no—
66C2×Dic251—22no—
67C2×Dic251—22no—
68C2×Dic251—22no—
69C2×Dic251—22no—
70C2×Dic251—22no—
71C2×Dic251—22no—
72C2×Dic251—22no—
73C2×Dic251—22no—
74C23×C251C2411yescharacteristic
75Q8×C251—44no—
76C2×C4×C251—21no—
77C2×C4×C251—21no—
78C2×C4×C251—21no—
79C2×C4×C251—22no—
80C2×C4×C251—22no—
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—22no—
88C2×C4×C251#4112yes—
89C22×C251C2×#4111yescharacteristic
90C22×C251H(5,𝔽2)11yescharacteristic
91C22×C251H(5,𝔽2)12yes—
92C22×C251H(5,𝔽2)11yescharacteristic
93C22×C2512−1+412yes—
94C2×D251—42no—
95C2×D251—42no—
96C2×D251—42no—
97C2×D251—42no—
98C2×D251—42no—
99C2×D251—42no—
100C2×D251—42no—
101C4×C251—22no—
102C4×C251—41no—
103C4×C251—41no—
104C4×C251—42no—
105C4×C251—42no—
106Dic251—42no—
107Dic251—42no—
108Dic251—42no—
109C2×C251#53811yescharacteristic
110C2×C251#54212yes—
111C2×C251#56011yescharacteristic
112C2×C251#56212yes—
113C2×C251#56311yescharacteristic
114D251—82no—
115C251#283011yescharacteristic, complemented, Sylow
116#2830—2511nomaximal, Sylow
117#398—2511no—
118#397—2512no—
119#397—2512no—
120#396—2512no—
121#395D25111yescharacteristic, complemented
122#394—2512no—
123#393—2512no—
124#390—2511no—
125#387—2512no—
126C22×Q8C2×D25111yescharacteristic
127C22×D4—2511no—
128C2×C4⋊C4—2512no—
129C2×C4⋊C4—2511no—
130C2×C4⋊C4—2512no—
131C2×C4⋊C4C2×D25111yescharacteristic
132C2×C4⋊C4C2×D25112yes—
133C2×C22⋊C4—2512no—
134C2×C22⋊C4—2512no—
135C2×C22⋊C4—2512no—
136C2×C22⋊C4—2512no—
137C2×C22⋊C4—2512no—
138C2×C22⋊C4—2512no—
139C2×C42—2511no—
140#95—2512no—
141#95—2512no—
142#95—2512no—
143#95—2512no—
144#95—2511no—
145#95C2×D25112yes—
146#95C2×D25111yescharacteristic
147C24—2512no—
148C2×Q8—41no—
149C2×Q8—42no—
150C2×D4—10041no—
151C2×D4—10041no—
152C2×D4—5022no—
153C22×C4—2512no—
154C22×C4—2512no—
155C22×C4—2512no—
156C22×C4C22×D25112yes—
157C22×C4C22×D25111yescharacteristic
158C22×C4C22×D25112yes—
159C22×C4C22×D25111yescharacteristic
160C22×C4C22×D25111yescharacteristic
161C4⋊C4—41no—
162C4⋊C4—42no—
163C4⋊C4—10042no—
164C4⋊C4—10041no—
165C4⋊C4—5024no—
166C22⋊C4—10042no—
167C22⋊C4—10042no—
168C22⋊C4—10042no—
169C22⋊C4—10042no—
170C22⋊C4—10042no—
171C22⋊C4—10042no—
172C42—10041no—
173C23—5022no—
174C23—5022no—
175C23—5022no—
176C23—5022no—
177C23—5022no—
178C23—5022no—
179C23—5022no—
180C23C23×D25111yescharacteristic
181Q8—44no—
182D4—10044no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—5022no—
196C2×C4—5022no—
197C2×C4—5022no—
198C2×C4—5022no—
199C2×C4—5022no—
200C2×C4—5022no—
201C2×C4—5022no—
202C2×C4—5022no—
203C2×C4—5022no—
204C2×C4#1686081112yes—
205C22—10042no—
206C22—10042no—
207C22—10042no—
208C22—10042no—
209C22—10042no—
210C22—10042no—
211C22—10042no—
212C22C2×#1686081111yescharacteristic
213C22#2254876011yescharacteristic
214C22#2254876012yes—
215C22#2254876111yescharacteristic
216C22#2254876312yes—
217C4—22no—
218C4—42no—
219C4—41no—
220C4—41no—
221C4—42no—
222C4—10042no—
223C4—10042no—
224C4—10042no—
225C2—20082no—
226C2#3063227711yescharacteristic
227C2#3063231112yes—
228C2#3063242611yescharacteristic
229C2#3063243312yes—
230C2#3063243611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D251
commutator, socle, cosocleC23×C251C24
Fitting#395×C251C2
prime radicalC22×D1004C22
hypercenter#395D251
hypocenterC251#2830
LCSG ▷ C23×C251 ▷ C251
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
1. #30631617C2semidirect
2. #30631616C2semidirect
3. #30631613C2semidirect
4. #30631609C2semidirect
6. #30631598C2semidirect
8. #30631569C2semidirect
9. #395×C251C2semidirect
115. C251#2830semidirect
121. #395D251semidirect

Maximal subgroups

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

subgroupindex
#306316172
#306316162
#306316132
#306316092
#306316052
#306315982
#306315882
#306315692
#395×C2512
#2830251