Finite Group Databasebeta

← #27173122

Subgroups — #27173122

Order 12928.

258 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 45 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#20265366C211yesmaximal, characteristic, complemented
2#20265363C211yesmaximal, characteristic
3#20265354C211yesmaximal, characteristic, complemented
4#20264972C211yesmaximal, characteristic, complemented
5#20264967C211yesmaximal, characteristic, complemented
6#20264964C211yesmaximal, characteristic, complemented
7#551×C101C211yesmaximal, characteristic, complemented
8#15692300C2211yescharacteristic
9#15692286—21no—
10#15692238—21no—
11#15692224—21no—
12#15692183—21no—
13#15692172C2211yescharacteristic
14#15692170C2211yescharacteristic
15#15692170C2211yescharacteristic
16#15692169—21no—
17#15692169—21no—
18#126×C101—21no—
19#124×C101C2211yescharacteristic
20#123×C101—21no—
21#123×C101C2211yescharacteristic
22C22⋊Q8×C101—21no—
23#120×C101C2211yescharacteristic
24C4×D4×C101—21no—
25#117×C101—21no—
26C2×C22⋊C4×C101—21no—
27#13108074—21no—
28#13108073—41no—
29C2×C22⋊D101—21no—
30C2×C22⋊D101—21no—
31C2×C22⋊D101—41no—
32D4×D101—41no—
33#13108064—41no—
34#13108061—41no—
35#13108058—21no—
36#13108053—41no—
37C22⋊Dic101—21no—
38C22⋊Dic101—21no—
39C22⋊Dic101—41no—
40#13108051—41no—
41C24×C101D411yescharacteristic
42C2×Q8×C101D411yescharacteristic, complemented
43C2×D4×C101—21no—
44C2×D4×C101C2311yescharacteristic
45C22×C4×C101—21no—
46C22×C4×C101—21no—
47C22×C4×C101D411yescharacteristic, complemented
48C4⋊C4×C101—21no—
49C4⋊C4×C101—21no—
50C4⋊C4×C101—21no—
51C4⋊C4×C101—21no—
52C22⋊C4×C101—21no—
53C22⋊C4×C101—21no—
54C22⋊C4×C101—21no—
55C22⋊C4×C101—21no—
56C22⋊C4×C101—21no—
57C22⋊C4×C101—21no—
58C22⋊C4×C101D411yescharacteristic
59C22⋊C4×C101D411yescharacteristic, complemented
60C42×C101—21no—
61C42×C101D411yescharacteristic, complemented
62C23×C101—21no—
63C23×C101—42no—
64C23×C101—41no—
65C23×C101C2×D411yescharacteristic
66C23×C101C2×D411yescharacteristic
67C22×D101—21no—
68C22×D101—21no—
69Q8×C101—41no—
70D4×C101—41no—
71D4×C101—42no—
72C2×C4×C101—21no—
73C2×C4×C101—21no—
74C2×C4×C101—21no—
75C2×C4×C101—21no—
76C2×C4×C101—21no—
77C2×C4×C101—42no—
78C2×C4×C101—41no—
79C2×C4×C101—81no—
80C2×C4×C101C2×D411yescharacteristic
81C22⋊D101—41no—
82C22⋊D101—41no—
83C22⋊D101—41no—
84C22⋊D101—82no—
85C2×Dic101—21no—
86C2×Dic101—21no—
87C2×Dic101—41no—
88C2×Dic101—41no—
89D404—21no—
90C4×D101—41no—
91Dic202—21no—
92#11717579—41no—
93C22×C101—22no—
94C22×C101—21no—
95C22×C101—21no—
96C22×C101—41no—
97C22×C101—41no—
98C22×C101—41no—
99C22×C101—42no—
100C22×C101—41no—
101C22×C101—81no—
102C22×C101#12011yescharacteristic
103C2×D101—41no—
104C2×D101—81no—
105C2×D101—81no—
106C4×C101—21no—
107C4×C101—41no—
108C4×C101—41no—
109C4×C101—41no—
110C4×C101—41no—
111C4×C101—41no—
112Dic101—41no—
113Dic101—81no—
114Dic101—81no—
115C2×C101—21no—
116C2×C101—41no—
117C2×C101—41no—
118C2×C101—42no—
119C2×C101B(4,𝔽2)11yescharacteristic
120D101—81no—
121#2197—1011nomaximal, Sylow
122C101#219711yescharacteristic, complemented, Sylow
123#551D10111yescharacteristic, complemented
124#458—1011no—
125B(4,𝔽2)—1011no—
126#455—1011no—
127#354—1011no—
128#352—1011no—
129AΣL(1,𝔽16)—1011no—
130H(5,𝔽2)—1011no—
131#137—2021no—
132#126—21no—
133#124C2×D10111yescharacteristic
134#123—21no—
135#123—2021no—
136#123C2×D10111yescharacteristic
137C22⋊Q8—21no—
138#120—2021no—
139#120C2×D10111yescharacteristic
140C4×D4—21no—
141#117—21no—
142C2×C22⋊C4—21no—
143#104—2021no—
144#100—1011no—
145#99—1011no—
146#99—1011no—
147#99—2021no—
148#99—2021no—
149C24C22⋊D10111yescharacteristic
150#41—4041no—
151#41—2021no—
152C2×Q8C22⋊D10111yescharacteristic, complemented
153C2×D4—21no—
154C2×D4—4041no—
155C2×D4—4041no—
156C2×D4—2021no—
157C2×D4—2021no—
158C2×D4C22×D10111yescharacteristic
159C22×C4—21no—
160C22×C4—21no—
161C22×C4C22⋊D10111yescharacteristic, complemented
162Q16—4041no—
163SD16—4041no—
164M16—2021no—
165C4⋊C4—21no—
166C4⋊C4—21no—
167C4⋊C4—21no—
168C4⋊C4—21no—
169C4⋊C4—4041no—
170C22⋊C4—21no—
171C22⋊C4—21no—
172C22⋊C4—21no—
173C22⋊C4—21no—
174C22⋊C4—21no—
175C22⋊C4—21no—
176C22⋊C4—4041no—
177C22⋊C4—4041no—
178C22⋊C4—2021no—
179C22⋊C4—2021no—
180C22⋊C4C22⋊D10111yescharacteristic
181C22⋊C4C22⋊D10111yescharacteristic, complemented
182C42—21no—
183C42C22⋊D10111yescharacteristic, complemented
184C23—21no—
185C23—41no—
186C23—42no—
187C23—2021no—
188C23—2021no—
189C23C2×C22⋊D10111yescharacteristic
190C23C2×C22⋊D10111yescharacteristic
191Q8—41no—
192Q8—2021no—
193D4—41no—
194D4—42no—
195D4—8082no—
196D4—4041no—
197D4—4041no—
198D4—4041no—
199D4—2021no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—42no—
206C2×C4—41no—
207C2×C4—81no—
208C2×C4—4041no—
209C2×C4—4041no—
210C2×C4—4041no—
211C2×C4—2021no—
212C2×C4—2021no—
213C2×C4C2×C22⋊D10111yescharacteristic
214C8—4041no—
215C22—22no—
216C22—21no—
217C22—21no—
218C22—41no—
219C22—41no—
220C22—41no—
221C22—42no—
222C22—41no—
223C22—81no—
224C22—8081no—
225C22—8081no—
226C22—4041no—
227C22#1569222411yescharacteristic
228C4—21no—
229C4—41no—
230C4—41no—
231C4—41no—
232C4—41no—
233C4—41no—
234C4—8081no—
235C4—8081no—
236C4—4041no—
237C2—21no—
238C2—41no—
239C2—41no—
240C2—42no—
241C2—8081no—
242C2#2026536411yescharacteristic
subgroupG/H
centerC2#20265364
commutator, cosocleC2×D4×C101C23
FrattiniC2×D4C22×D101
Fitting#551×C101C2
socleC2×C101B(4,𝔽2)
prime radical#20265363C2
hypercenter#551D101
hypocenterC101#2197
LCSG ▷ C2×D4×C101 ▷ C22×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #551

Complements

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

NH ≅ G/Ntype
1. #20265366C2semidirect
3. #20265354C2semidirect
4. #20264972C2semidirect
5. #20264967C2semidirect
6. #20264964C2semidirect
7. #551×C101C2semidirect
42. C2×Q8×C101D4semidirect
47. C22×C4×C101D4semidirect
59. C22⋊C4×C101D4semidirect
61. C42×C101D4semidirect
122. C101#2197semidirect
123. #551D101semidirect
152. C2×Q8C22⋊D101semidirect
161. C22×C4C22⋊D101semidirect
181. C22⋊C4C22⋊D101semidirect
183. C42C22⋊D101semidirect

Maximal subgroups

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

subgroupindex
#202653662
#202653632
#202653542
#202649722
#202649672
#202649642
#551×C1012
#2197101