Finite Group Databasebeta

← #27170136

Subgroups — #27170136

Order 12928.

254 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 51 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#20265754C211yesmaximal, characteristic
2#20265491C211yesmaximal, characteristic, complemented
3#20265488C211yesmaximal, characteristic, complemented
4#20265109C211yesmaximal, characteristic, complemented
5#20264910C211yesmaximal, characteristic, complemented
6#20264902C211yesmaximal, characteristic, complemented
7#327×C101C211yesmaximal, characteristic, complemented
8C2×#13108073—21no—
9#15692233—21no—
10#15692232—21no—
11#15692229C2211yescharacteristic
12#15692228—21no—
13#15692227—21no—
14#15692227—21no—
15C2×C4⋊Dic101—21no—
16C2×C4⋊Dic101C2211yescharacteristic
17C2×#13108053C2211yescharacteristic
18#15692189—21no—
19#15692188—21no—
20#15692177—21no—
21#15692176—21no—
22C22⋊#11717579C2211yescharacteristic
23#15692162—21no—
24#121×C101C2211yescharacteristic
25C22⋊C8×C101C2211yescharacteristic
26#95×C101C2211yescharacteristic
27#13108073—42no—
28#13108073—42no—
29C2×Dic202—21no—
30C2×D404D411yescharacteristic
31C2×C22⋊D101—21no—
32C2×C22⋊D101—21no—
33C2×C22⋊D101—21no—
34C22×Dic101—21no—
35C22×Dic101—21no—
36C22×Dic101D411yescharacteristic
37C22×Dic101D411yescharacteristic
38C2×C4×D101—21no—
39C2×#11717579—21no—
40C4⋊Dic101—21no—
41C4⋊Dic101—21no—
42C4⋊Dic101—42no—
43C4⋊Dic101D411yescharacteristic
44#13108053—21no—
45#13108053—21no—
46#13108053—21no—
47C22⋊Dic101—21no—
48#13108051—21no—
49#13108051—21no—
50C2×D4×C101—21no—
51C2×D4×C101D411yescharacteristic
52C22×C4×C101—21no—
53C22×C4×C101C2311yescharacteristic
54C2×C8×C101—21no—
55C4⋊C4×C101D411yescharacteristic
56C22⋊C4×C101—21no—
57C23×C101—21no—
58C23×C101C2×D411yescharacteristic
59C22×D101—21no—
60D4×C101—41no—
61D4×C101—41no—
62D4×C101—41no—
63C2×C4×C101—22no—
64C2×C4×C101—21no—
65C2×C4×C101—41no—
66C2×C4×C101—41no—
67C2×C4×C101—41no—
68C2×C4×C101C2×D411yescharacteristic
69C2×C4×C101C2×D411yescharacteristic
70C22⋊D101—42no—
71C22⋊D101—41no—
72C22⋊D101—41no—
73C22⋊D101—81no—
74C2×Dic101—21no—
75C2×Dic101—21no—
76C2×Dic101—21no—
77C2×Dic101—21no—
78C2×Dic101—42no—
79C2×Dic101—41no—
80C2×Dic101—41no—
81C2×Dic101—41no—
82C2×Dic101—41no—
83C2×Dic101D812yes—
84D404—41no—
85C4×D101—42no—
86Dic202—42no—
87C8×C101—41no—
88#11717579—41no—
89C22×C101—21no—
90C22×C101—21no—
91C22×C101—41no—
92C22×C101—41no—
93C22×C101—41no—
94C22×C101#12011yescharacteristic
95C22×C101C2×D811yescharacteristic
96C22×C101AGL(1,ℤ/8)11yescharacteristic
97C2×D101—41no—
98C2×D101—41no—
99C2×D101—41no—
100C4×C101—21no—
101C4×C101—21no—
102C4×C101—42no—
103C4×C101—41no—
104Dic101—22no—
105Dic101—41no—
106Dic101—41no—
107Dic101—81no—
108C2×C101—21no—
109C2×C101—21no—
110C2×C101—81no—
111C2×C101#44711yescharacteristic
112C2×C101#45611yescharacteristic
113C2×C101#45811yescharacteristic
114D101—81no—
115#1610—1011nomaximal, Sylow
116C101#161011yescharacteristic, complemented, Sylow
117#537—1011no—
118#480—1011no—
119#480—1011no—
120#398—1011no—
121#327—1011no—
122#327D10111yescharacteristic, complemented
123#324—1011no—
124C2×#41—2021no—
125C22⋊Q8—2021no—
126C22⋊Q8—2021no—
127#121—1011no—
128#121—2021no—
129#121—2021no—
130#121—2021no—
131#121C2×D10111yescharacteristic
132C2×C4⋊C4—1011no—
133C2×C4⋊C4—1011no—
134C2×C4⋊C4—2021no—
135#107—2021no—
136#107—2021no—
137#102—2021no—
138#102—2021no—
139C22⋊C8—1011no—
140C22⋊C8C2×D10111yescharacteristic
141#95—2021no—
142#95C2×D10111yescharacteristic
143#41—4042no—
144#41—4042no—
145C2×Q8—2021no—
146C2×D4—21no—
147C2×D4—1011no—
148C2×D4—2021no—
149C2×D4—2021no—
150C2×D4—2021no—
151C2×D4D40411yescharacteristic
152C22×C4—21no—
153C22×C4—1011no—
154C22×C4—1011no—
155C22×C4—2021no—
156C22×C4—2021no—
157C22×C4—2021no—
158C22×C4C22×D10111yescharacteristic
159C2×C8—21no—
160C2×C8—2021no—
161C4⋊C4—1011no—
162C4⋊C4—4042no—
163C4⋊C4—2021no—
164C4⋊C4—2021no—
165C4⋊C4—2021no—
166C4⋊C4—2021no—
167C4⋊C4—2021no—
168C4⋊C4D40411yescharacteristic
169C22⋊C4—21no—
170C22⋊C4—2021no—
171C22⋊C4—2021no—
172C22⋊C4—2021no—
173C23—21no—
174C23—2021no—
175C23D4×D10111yescharacteristic
176Q8—4042no—
177D4—41no—
178D4—41no—
179D4—41no—
180D4—8081no—
181D4—4041no—
182D4—4041no—
183D4—4041no—
184D4—4042no—
185C2×C4—22no—
186C2×C4—21no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—1012no—
191C2×C4—4042no—
192C2×C4—4041no—
193C2×C4—4041no—
194C2×C4—4041no—
195C2×C4—4042no—
196C2×C4—4041no—
197C2×C4—2021no—
198C2×C4—2021no—
199C2×C4—2021no—
200C2×C4—2021no—
201C2×C4D4×D10111yescharacteristic
202C2×C4C2×D40411yescharacteristic
203C8—41no—
204C8—4041no—
205C22—21no—
206C22—21no—
207C22—41no—
208C22—41no—
209C22—41no—
210C22—4041no—
211C22—4041no—
212C22—4041no—
213C22#1569222511yescharacteristic
214C22D8×D10111yescharacteristic
215C22#1569228211yescharacteristic
216C4—21no—
217C4—21no—
218C4—42no—
219C4—41no—
220C4—8081no—
221C4—4041no—
222C4—4041no—
223C4—2022no—
224C2—21no—
225C2—21no—
226C2—81no—
227C2—8081no—
228C2#2026531211yescharacteristic
229C2#2026536111yescharacteristic
230C2#2026536811yescharacteristic
subgroupG/H
centerC22#15692225
commutator, cosocleC22×C4×C101C23
FrattiniC22×C4C22×D101
Fitting#327×C101C2
socleC22×C101#120
prime radical#20265754C2
hypercenter#327D101
hypocenterC101#1610
LCSG ▷ C22×C4×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #327

Complements

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

NH ≅ G/Ntype
2. #20265491C2semidirect
3. #20265488C2semidirect
4. #20265109C2semidirect
5. #20264910C2semidirect
6. #20264902C2semidirect
7. #327×C101C2semidirect
116. C101#1610semidirect
122. #327D101semidirect

Maximal subgroups

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

subgroupindex
#202657542
#202654912
#202654882
#202651092
#202649102
#202649022
#327×C1012
#1610101