Finite Group Databasebeta

← #27171086

Subgroups — #27171086

Order 12928.

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

normal subgroups only

This group has 620 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×#15692182C212yesmaximal, complemented
2#20265178C212yesmaximal, complemented
3C2×#15692172C211yesmaximal, characteristic, complemented
4#20265080C211yesmaximal, characteristic, complemented
5C2×H(5,𝔽2)×C101C211yesmaximal, characteristic
6C2×#13108058C2212yescomplemented
7C2×C4⋊Dic101C2211yescharacteristic, complemented
8C2×C22⋊Dic101—22no—
9C2×C4×Dic101C2211yescharacteristic, complemented
10#15692182—24no—
11#15692179—22no—
12#15692176—22no—
13#15692172—22no—
14H(5,𝔽2)×C101—24no—
15H(5,𝔽2)×C101—22no—
16H(5,𝔽2)×C101C414yescomplemented
17C2×#41×C101—22no—
18C2×#41×C101C2212yes—
19C22×D4×C101—22no—
20C22×D4×C101—22no—
21C22×D4×C101C2211yescharacteristic
22C22×Dic101—21no—
23C22×Dic101—21no—
24#13108058—22no—
25#13108058—22no—
26C2×#11717579—22no—
27C4⋊Dic101—22no—
28C22⋊Dic101—44no—
29C4×Dic101—22no—
30C24×C101—22no—
31C24×C101D412yes—
32#41×C101—28no—
33#41×C101—28no—
34#41×C101—22no—
35#41×C101—22no—
36#41×C101C2×C414yes—
37#41×C101D414yes—
38C2×Q8×C101D412yes—
39C2×D4×C101—24no—
40C2×D4×C101—28no—
41C2×D4×C101—28no—
42C2×D4×C101—28no—
43C2×D4×C101—22no—
44C2×D4×C101—28no—
45C2×D4×C101—22no—
46C2×D4×C101—22no—
47C2×D4×C101C2×C412yes—
48C2×D4×C101D412yes—
49C2×D4×C101D412yes—
50C22×C4×C101—22no—
51C22×C4×C101—22no—
52C22×C4×C101C2311yescharacteristic
53C23×C101—28no—
54C23×C101—22no—
55C23×C101—22no—
56C23×C101—22no—
57C23×C101—42no—
58C23×C101—42no—
59C23×C101—82no—
60C23×C101—82no—
61C23×C101C22⋊C414yes—
62C23×C101C2×D411yescharacteristic
63Q8×C101—22no—
64Q8×C101C22⋊C414yescomplemented
65D4×C101—216no—
66D4×C101—24no—
67D4×C101—24no—
68D4×C101—24no—
69D4×C101—24no—
70D4×C101—22no—
71D4×C101C22⋊C414yescomplemented
72C2×C4×C101—28no—
73C2×C4×C101—22no—
74C2×C4×C101—24no—
75C2×C4×C101—24no—
76C2×C4×C101C22×C411yescharacteristic
77C2×C4×C101C2×D412yes—
78C2×C4×C101C2×D412yes—
79C2×C4×C101C2×D411yescharacteristic
80C2×Dic101—22no—
81C2×Dic101—41no—
82C2×Dic101—41no—
83C2×Dic101—42no—
84C2×Dic101—41no—
85#11717579—42no—
86C22×C101—21no—
87C22×C101—21no—
88C22×C101—24no—
89C22×C101—24no—
90C22×C101—44no—
91C22×C101—42no—
92C22×C101—42no—
93C22×C101—42no—
94C22×C101—42no—
95C22×C101—82no—
96C22×C101—88no—
97C22×C101C2×C22⋊C411yescharacteristic
98C22×C101#12011yescharacteristic
99C22×C101#12011yescharacteristic
100C4×C101—24no—
101C4×C101C2×C22⋊C412yes—
102C4×C101#12012yes—
103Dic101—42no—
104Dic101—81no—
105C2×C101—21no—
106C2×C101—21no—
107C2×C101—44no—
108C2×C101—44no—
109C2×C101#37911yescharacteristic
110C2×C101#45311yescharacteristic
111C2×C101#45511yescharacteristic
112#1792—1011nomaximal, Sylow
113C101#179211yescharacteristic, complemented, Sylow
114C2×H(5,𝔽2)D10111yescharacteristic
115C2×#104—1012no—
116#417—1012no—
117C2×#100—1011no—
118#390—1011no—
119H(5,𝔽2)—24no—
120H(5,𝔽2)—22no—
121H(5,𝔽2)Dic10114yescomplemented
122C2×#41—22no—
123C2×#41C2×D10112yes—
124C22×D4—22no—
125C22×D4—22no—
126C22×D4C2×D10111yescharacteristic
127C2×M16—1012no—
128C2×C4⋊C4—1011no—
129C2×C22⋊C4—2022no—
130C2×C42—1011no—
131#104—2024no—
132#103—2022no—
133#102—2022no—
134#100—2022no—
135C24—22no—
136C24C22⋊D10112yes—
137#41—28no—
138#41—28no—
139#41—22no—
140#41—22no—
141#41C2×Dic10114yes—
142#41C22⋊D10114yes—
143C2×Q8C22⋊D10112yes—
144C2×D4—24no—
145C2×D4—28no—
146C2×D4—28no—
147C2×D4—28no—
148C2×D4—22no—
149C2×D4—28no—
150C2×D4—22no—
151C2×D4—22no—
152C2×D4C2×Dic10112yes—
153C2×D4C22⋊D10112yes—
154C2×D4C22⋊D10112yes—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4—2021no—
158C22×C4—2021no—
159C22×C4C22×D10111yescharacteristic
160M16—2022no—
161M16—2022no—
162C2×C8—2022no—
163C4⋊C4—2022no—
164C22⋊C4—4044no—
165C42—2022no—
166C23—28no—
167C23—22no—
168C23—22no—
169C23—22no—
170C23—42no—
171C23—42no—
172C23—82no—
173C23—82no—
174C23C22⋊Dic10114yes—
175C23C2×C22⋊D10111yescharacteristic
176Q8—22no—
177Q8C22⋊Dic10114yescomplemented
178D4—216no—
179D4—24no—
180D4—24no—
181D4—24no—
182D4—24no—
183D4—22no—
184D4C22⋊Dic10114yescomplemented
185C2×C4—28no—
186C2×C4—22no—
187C2×C4—24no—
188C2×C4—24no—
189C2×C4—4042no—
190C2×C4—4041no—
191C2×C4—4041no—
192C2×C4—4041no—
193C2×C4—2022no—
194C2×C4C22×Dic10111yescharacteristic
195C2×C4C2×C22⋊D10112yes—
196C2×C4C2×C22⋊D10112yes—
197C2×C4C2×C22⋊D10111yescharacteristic
198C8—4042no—
199C22—21no—
200C22—21no—
201C22—24no—
202C22—24no—
203C22—44no—
204C22—42no—
205C22—42no—
206C22—42no—
207C22—42no—
208C22—82no—
209C22—88no—
210C22C2×C22⋊Dic10111yescharacteristic
211C22#1569222411yescharacteristic
212C22#1569222411yescharacteristic
213C4—24no—
214C4—8081no—
215C4—4042no—
216C4C2×C22⋊Dic10112yes—
217C4#1569222412yes—
218C2—21no—
219C2—21no—
220C2—44no—
221C2—44no—
222C2#2026503311yescharacteristic
223C2#2026534511yescharacteristic
224C2#2026535511yescharacteristic
subgroupG/H
centerC22#15692224
commutatorC2×C4×C101C22×C4
FrattiniC22×C4C22×D101
Fitting, prime radicalC2×H(5,𝔽2)×C101C2
socleC22×C101#120
cosocleC22×C4×C101C23
hypercenterC2×H(5,𝔽2)D101
hypocenterC101#1792
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. C2×#15692182C2semidirect
2. #20265178C2semidirect
3. C2×#15692172C2semidirect
4. #20265080C2semidirect
6. C2×#13108058C22semidirect
7. C2×C4⋊Dic101C22semidirect
9. C2×C4×Dic101C22semidirect
16. H(5,𝔽2)×C101C4semidirect
64. Q8×C101C22⋊C4semidirect
71. D4×C101C22⋊C4semidirect
113. C101#1792semidirect
121. H(5,𝔽2)Dic101semidirect
177. Q8C22⋊Dic101semidirect
184. D4C22⋊Dic101semidirect

Maximal subgroups

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

subgroupindex
C2×#156921822
#202651782
C2×#156921722
#202650802
C2×H(5,𝔽2)×C1012
#1792101