Finite Group Databasebeta

← #27182594

Subgroups — #27182594

Order 12928.

346 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 109 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#20265800C212yesmaximal, complemented
2#20265470C212yesmaximal, complemented
3#20265456C212yesmaximal, complemented
4#20265425C212yesmaximal, complemented
5#20265413C211yesmaximal, characteristic
6#20265401C212yesmaximal, complemented
7#20265395C212yesmaximal, complemented
8#20265203C211yesmaximal, characteristic, complemented
9#467×C101C211yesmaximal, characteristic, complemented
10C2×#13108061C2212yes—
11C2×D808C2211yescharacteristic
12C2×#13108058C2211yescharacteristic
13#15692254C2212yes—
14#15692250—22no—
15#15692244C2212yes—
16#15692241—22no—
17#15692237—22no—
18#15692235—22no—
19#15692229C2212yes—
20Q8×Dic101C2212yes—
21#15692217C2212yes—
22#15692214C2211yescharacteristic
23#15692189C2211yescharacteristic
24#15692188C2212yes—
25#15692184C2212yes—
26#15692184C2212yes—
27#15692179C2212yes—
28#15692177C2212yes—
29#15692175C2212yes—
30C2×Q16×C101C2211yescharacteristic
31C22×C8×C101C2211yescharacteristic
32C22⋊Q8×C101C2212yes—
33#107×C101C2211yescharacteristic
34#103×C101C2212yes—
35C2×Dic202—22no—
36C2×D404C2312yes—
37C2×C22⋊D101—22no—
38C2×C4×D101—22no—
39#13108061—42no—
40D808—41no—
41#13108058—41no—
42C2×#11717579C2312yes—
43C4⋊Dic101—22no—
44C4⋊Dic101D411yescharacteristic
45C4⋊Dic101D411yescharacteristic
46C4⋊Dic101C2312yes—
47#13108053—22no—
48C22⋊Dic101D412yes—
49#13108051—22no—
50#13108051—22no—
51#13108051—22no—
52#13108051—22no—
53C4×Dic101—22no—
54C4×Dic101C2312yes—
55C2×Q8×C101C2312yes—
56C22×C4×C101C2311yescharacteristic
57Q16×C101—41no—
58C2×C8×C101—41no—
59C2×C8×C101C2311yescharacteristic
60C2×C8×C101C2311yescharacteristic
61C4⋊C4×C101—22no—
62C4⋊C4×C101C2312yes—
63C22⋊C4×C101—22no—
64C23×C101C2×D411yescharacteristic
65C22×D101—22no—
66Q8×C101—22no—
67Q8×C101—22no—
68C2×C4×C101—21no—
69C2×C4×C101—21no—
70C2×C4×C101—22no—
71C2×C4×C101—22no—
72C2×C4×C101C2×D411yescharacteristic
73C2×C4×C101C2411yescharacteristic
74C22⋊D101—82no—
75C2×Dic101—22no—
76C2×Dic101C2×D412yes—
77C2×Dic101C2×D412yes—
78D404—22no—
79D404—22no—
80C4×D101—82no—
81Dic202—82no—
82C8×C101—21no—
83C8×C101—21no—
84#11717579—22no—
85C22×C101—21no—
86C22×C101—21no—
87C22×C101—21no—
88C22×C101C22×D411yescharacteristic
89C2×D101—42no—
90C2×D101—42no—
91C2×D101—42no—
92C4×C101—21no—
93C4×C101—42no—
94C4×C101—42no—
95C4×C101H(5,𝔽2)11yescharacteristic
96C4×C1012−1+411yescharacteristic
97Dic101—22no—
98Dic101—22no—
99Dic101—42no—
100C2×C101—41no—
101C2×C101#53711yescharacteristic
102C2×C101#57611yescharacteristic
103C2×C101#57811yescharacteristic
104D101—82no—
105#3262—1011nomaximal, Sylow
106C101#326211yescharacteristic, complemented, Sylow
107#542—1012no—
108#477—1012no—
109#474—1012no—
110#468—1012no—
111#467D10111yescharacteristic, complemented
112#466—1011no—
113#464—1012no—
114#463—1012no—
115#424—1011no—
116C2×Q16C2×D10111yescharacteristic
117C2×SD16—1012no—
118C2×D8—1011no—
119C2×M16—1011no—
120C22×C8C2×D10111yescharacteristic
121C4⋊Q8—1012no—
122#126—2022no—
123#124—1012no—
124#124—2022no—
125#123—2022no—
126C22⋊Q8—2022no—
127C22⋊Q8C2×D10112yes—
128#121—1012no—
129C4×Q8—1012no—
130C4×D4—1012no—
131#117—1011no—
132#107—1011no—
133#107—1012no—
134#107C2×D10111yescharacteristic
135C4⋊C8—1012no—
136C4⋊C8—1012no—
137#103—1012no—
138#103C2×D10112yes—
139#102—1012no—
140#102—1012no—
141C2×Q8—2022no—
142C2×Q8C22×D10112yes—
143C2×D4—1012no—
144C2×D4—2022no—
145C22×C4—2022no—
146C22×C4C22×D10111yescharacteristic
147Q16—41no—
148SD16—4042no—
149D8—4041no—
150M16—4041no—
151C2×C8—41no—
152C2×C8—1012no—
153C2×C8C22×D10111yescharacteristic
154C2×C8C22×D10111yescharacteristic
155C4⋊C4—22no—
156C4⋊C4—1011no—
157C4⋊C4—1011no—
158C4⋊C4—1012no—
159C4⋊C4—2022no—
160C4⋊C4—2022no—
161C4⋊C4C22×D10112yes—
162C22⋊C4—22no—
163C22⋊C4—1012no—
164C22⋊C4—2022no—
165C22⋊C4—2022no—
166C22⋊C4—2022no—
167C22⋊C4—2022no—
168C42—1012no—
169C42—2022no—
170C23—2022no—
171C23D4×D10111yescharacteristic
172Q8—22no—
173Q8—22no—
174Q8—8082no—
175D4—8082no—
176D4—2022no—
177D4—2022no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—8082no—
183C2×C4—1012no—
184C2×C4—1012no—
185C2×C4—2022no—
186C2×C4D4×D10111yescharacteristic
187C2×C4C23×D10111yescharacteristic
188C8—21no—
189C8—21no—
190C8—2022no—
191C22—21no—
192C22—21no—
193C22—21no—
194C22—4042no—
195C22—4042no—
196C22—4042no—
197C22C2×D4×D10111yescharacteristic
198C4—21no—
199C4—42no—
200C4—42no—
201C4—4042no—
202C4—2022no—
203C4—2022no—
204C4#1569230011yescharacteristic
205C4#1569230211yescharacteristic
206C2—41no—
207C2—8082no—
208C2#2026575411yescharacteristic
209C2#2026597911yescharacteristic
210C2#2026599011yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#467×C101C2
socleC22×C101C22×D4
prime radical#20265413C2
hypercenter#467D101
hypocenterC101#3262
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #467

Complements

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

NH ≅ G/Ntype
1. #20265800C2semidirect
2. #20265470C2semidirect
3. #20265456C2semidirect
4. #20265425C2semidirect
6. #20265401C2semidirect
7. #20265395C2semidirect
8. #20265203C2semidirect
9. #467×C101C2semidirect
106. C101#3262semidirect
111. #467D101semidirect

Maximal subgroups

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

subgroupindex
#202658002
#202654702
#202654562
#202654252
#202654132
#202654012
#202653952
#202652032
#467×C1012
#3262101