Finite Group Databasebeta

← #27182533

Subgroups — #27182533

Order 12928.

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

normal subgroups only

This group has 354 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#20265711C212yesmaximal, complemented
2#20265485C212yesmaximal, complemented
3#20265478C212yesmaximal, complemented
4#20265435C211yesmaximal, characteristic, complemented
5#20265430C211yesmaximal, characteristic
6#20265423C212yesmaximal, complemented
7#20265400C212yesmaximal, complemented
8#20265393C212yesmaximal, complemented
9#423×C101C211yesmaximal, characteristic, complemented
10C2×Dic404C2211yescharacteristic
11C2×#13108063C2212yes—
12C2×D808C2211yescharacteristic
13#15692249—22no—
14#15692247C2212yes—
15#15692243C2212yes—
16#15692238—22no—
17#15692234C2212yes—
18#15692229C2212yes—
19C4×Dic202C2212yes—
20C4×D404C2212yes—
21C4×C22⋊D101—22no—
22#15692212—22no—
23#15692189C2212yes—
24#15692187C2211yescharacteristic
25#15692187C2211yescharacteristic
26#15692180C2212yes—
27#15692180C2212yes—
28#15692177C2212yes—
29#15692177C2212yes—
30C2×M16×C101C2211yescharacteristic
31C2×M16×C101C2211yescharacteristic
32C2×C42×C101C2211yescharacteristic
33C4⋊C8×C101C2212yes—
34C4⋊C8×C101C2212yes—
35C2×Dic202C2312yes—
36C2×D404C2312yes—
37C2×C22⋊D101—22no—
38C2×C4×D101—22no—
39Dic404—41no—
40#13108063—42no—
41D808—41no—
42C4⋊Dic101C2312yes—
43C4⋊Dic101C2312yes—
44#13108053—22no—
45#13108053—22no—
46#13108053—22no—
47#13108053—22no—
48C22⋊Dic101—22no—
49#13108051—22no—
50#13108051—22no—
51#13108051—22no—
52#13108051—22no—
53C4×Dic101—22no—
54C22×C4×C101D412yes—
55C22×C4×C101C2311yescharacteristic
56M16×C101—41no—
57M16×C101—41no—
58C2×C8×C101C2311yescharacteristic
59C2×C8×C101C2312yes—
60C2×C8×C101C2311yescharacteristic
61C42×C101D411yescharacteristic
62C42×C101D411yescharacteristic
63C42×C101C2312yes—
64C23×C101C2×D411yescharacteristic
65C22×D101—22no—
66C2×C4×C101—21no—
67C2×C4×C101—21no—
68C2×C4×C101—42no—
69C2×C4×C101C2×D411yescharacteristic
70C2×C4×C101C2×D412yes—
71C2×C4×C101C2×D412yes—
72C2×C4×C101C2411yescharacteristic
73C22⋊D101—44no—
74C2×Dic101—22no—
75C2×Dic101—22no—
76C2×Dic101—22no—
77D404—22no—
78D404—22no—
79C4×D101—44no—
80Dic202—22no—
81Dic202—22no—
82C8×C101—21no—
83C8×C101—22no—
84C8×C101—21no—
85C22×C101—21no—
86C22×C101—22no—
87C22×C101C22×D411yescharacteristic
88C2×D101—42no—
89C2×D101—42no—
90C2×D101—42no—
91C4×C101—21no—
92C4×C101—22no—
93C4×C101—22no—
94C4×C101H(5,𝔽2)11yescharacteristic
95C4×C1012−1+411yescharacteristic
96Dic101—42no—
97Dic101—42no—
98Dic101—42no—
99C2×C101—41no—
100C2×C101#53711yescharacteristic
101C2×C101#57512yes—
102D101—82no—
103#3256—1011nomaximal, Sylow
104C101#325611yescharacteristic, complemented, Sylow
105#529—1012no—
106#479—1012no—
107#478—1012no—
108#470—1011no—
109#469—1011no—
110#468—1012no—
111#464—1012no—
112#463—1012no—
113#423D10111yescharacteristic, complemented
114C2×Q16—1011no—
115C2×SD16—1012no—
116C2×D8—1011no—
117C2×M16C2×D10111yescharacteristic
118C2×M16C2×D10111yescharacteristic
119#126—2022no—
120#125—1012no—
121#124—1012no—
122#123—2022no—
123C22⋊Q8—1012no—
124#121—1012no—
125C4×Q8—1012no—
126C4×D4—1012no—
127C4×D4—2022no—
128#117—2022no—
129C2×C42C2×D10111yescharacteristic
130#107—1012no—
131#106—1011no—
132#106—1011no—
133C4⋊C8C2×D10112yes—
134C4⋊C8C2×D10112yes—
135#103—1012no—
136#103—1012no—
137#102—1012no—
138#102—1012no—
139C2×Q8—1012no—
140C2×D4—1012no—
141C2×D4—2022no—
142C22×C4—2022no—
143C22×C4D40412yes—
144C22×C4C22×D10111yescharacteristic
145Q16—4041no—
146SD16—4042no—
147D8—4041no—
148M16—41no—
149M16—41no—
150C2×C8C22×D10111yescharacteristic
151C2×C8C22×D10112yes—
152C2×C8C22×D10111yescharacteristic
153C4⋊C4—1012no—
154C4⋊C4—1012no—
155C4⋊C4—2022no—
156C4⋊C4—2022no—
157C4⋊C4—2022no—
158C4⋊C4—2022no—
159C22⋊C4—2022no—
160C22⋊C4—2022no—
161C22⋊C4—2022no—
162C22⋊C4—2022no—
163C22⋊C4—2022no—
164C42—2022no—
165C42D40411yescharacteristic
166C42D40411yescharacteristic
167C42C22×D10112yes—
168C23—2022no—
169C23C2×D40411yescharacteristic
170Q8—2022no—
171Q8—2022no—
172D4—4044no—
173D4—2022no—
174D4—2022no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—42no—
178C2×C4—4044no—
179C2×C4—2022no—
180C2×C4—2022no—
181C2×C4—2022no—
182C2×C4C2×D40412yes—
183C2×C4C2×D40411yescharacteristic
184C2×C4C2×D40412yes—
185C2×C4C23×D10111yescharacteristic
186C8—21no—
187C8—22no—
188C8—21no—
189C22—21no—
190C22—22no—
191C22—4042no—
192C22—4042no—
193C22—4042no—
194C22C22×D40411yescharacteristic
195C4—21no—
196C4—22no—
197C4—22no—
198C4—4042no—
199C4—4042no—
200C4—4042no—
201C4#1569230011yescharacteristic
202C4#1569230311yescharacteristic
203C2—41no—
204C2—8082no—
205C2#2026575811yescharacteristic
206C2#2026597712yes—
subgroupG/H
centerC22C22×D404
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#423×C101C2
socleC22×C101C22×D4
prime radical#20265430C2
hypercenter#423D101
hypocenterC101#3256
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C2×C42 ◁ #423

Complements

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

NH ≅ G/Ntype
1. #20265711C2semidirect
2. #20265485C2semidirect
3. #20265478C2semidirect
4. #20265435C2semidirect
6. #20265423C2semidirect
7. #20265400C2semidirect
8. #20265393C2semidirect
9. #423×C101C2semidirect
104. C101#3256semidirect
113. #423D101semidirect

Maximal subgroups

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

subgroupindex
#202657112
#202654852
#202654782
#202654352
#202654302
#202654232
#202654002
#202653932
#423×C1012
#3256101