Finite Group Databasebeta

← #27170096

Subgroups — #27170096

Order 12928.

244 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 49 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#20265772C211yesmaximal, characteristic
2#20265335C211yesmaximal, characteristic, complemented
3#20265315C211yesmaximal, characteristic
4#20265098C211yesmaximal, characteristic, complemented
5#20264913C211yesmaximal, characteristic, complemented
6#20264902C211yesmaximal, characteristic, complemented
7#327×C101C211yesmaximal, characteristic
8C22×Dic202C2211yescharacteristic
9C2×Dic404—21no—
10C2×#13108062—21no—
11#15692248—21no—
12#15692245—21no—
13#15692236—21no—
14#15692236—21no—
15#15692234C2211yescharacteristic
16#15692232—21no—
17D4×Dic101—21no—
18C2×C4⋊Dic101—21no—
19C2×#13108053C2211yescharacteristic
20#15692180—21no—
21#15692178—21no—
22C22⋊#11717579C2211yescharacteristic
23#15692162—21no—
24#121×C101C2211yescharacteristic
25C22⋊C8×C101C2211yescharacteristic
26#95×C101C2211yescharacteristic
27C2×Dic202—21no—
28C2×Dic202—21no—
29C2×Dic202—42no—
30C2×Dic202D411yescharacteristic
31C22×Dic101—21no—
32C22×Dic101—21no—
33C22×Dic101D411yescharacteristic
34C22×Dic101D411yescharacteristic
35Dic404—42no—
36#13108062—42no—
37C2×#11717579—21no—
38C4⋊Dic101—21no—
39C4⋊Dic101—42no—
40C4⋊Dic101D411yescharacteristic
41#13108053—21no—
42#13108053—21no—
43#13108053—21no—
44#13108053—21no—
45#13108053—21no—
46C22⋊Dic101—21no—
47C22⋊Dic101—21no—
48C22⋊Dic101—21no—
49C22⋊Dic101—21no—
50C4×Dic101—21no—
51C2×D4×C101—21no—
52C2×D4×C101D411yescharacteristic
53C22×C4×C101—21no—
54C22×C4×C101C2311yescharacteristic
55C2×C8×C101—21no—
56C4⋊C4×C101D411yescharacteristic
57C22⋊C4×C101—21no—
58C23×C101—21no—
59C23×C101C2×D411yescharacteristic
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
70C2×Dic101—22no—
71C2×Dic101—21no—
72C2×Dic101—21no—
73C2×Dic101—21no—
74C2×Dic101—21no—
75C2×Dic101—21no—
76C2×Dic101—21no—
77C2×Dic101—41no—
78C2×Dic101—41no—
79C2×Dic101—41no—
80C2×Dic101—81no—
81Dic202—22no—
82Dic202—22no—
83Dic202—41no—
84Dic202—41no—
85Dic202—41no—
86C8×C101—41no—
87#11717579—41no—
88C22×C101—21no—
89C22×C101—21no—
90C22×C101—41no—
91C22×C101—41no—
92C22×C101—41no—
93C22×C101#12011yescharacteristic
94C22×C101#13511yescharacteristic
95C22×C101#13711yescharacteristic
96C4×C101—21no—
97C4×C101—21no—
98C4×C101—42no—
99C4×C101—41no—
100Dic101—41no—
101Dic101—41no—
102Dic101—42no—
103Dic101—41no—
104Dic101—41no—
105C2×C101—21no—
106C2×C101—21no—
107C2×C101—81no—
108C2×C101#45211yescharacteristic
109C2×C101#45611yescharacteristic
110C2×C101B(4,𝔽2)11yescharacteristic
111#1603—1011nomaximal, Sylow
112C101#160311yescharacteristic, complemented, Sylow
113#539—1011no—
114#451—1011no—
115#448—1011no—
116#395—1011no—
117#328—1011no—
118#327D10111yescharacteristic
119#324—1011no—
120C22×Q8—1011no—
121C2×Q16—2021no—
122C2×SD16—2021no—
123#126—2021no—
124#125—2021no—
125#123—2021no—
126#123—2021no—
127C22⋊Q8—1011no—
128C22⋊Q8—2021no—
129#121C2×D10111yescharacteristic
130C4×D4—2021no—
131C2×C4⋊C4—1011no—
132C2×C4⋊C4—2021no—
133#103—2021no—
134#103—2021no—
135C22⋊C8—1011no—
136C22⋊C8C2×D10111yescharacteristic
137#95—2021no—
138#95C2×D10111yescharacteristic
139C2×Q8—1011no—
140C2×Q8—4042no—
141C2×Q8—2021no—
142C2×Q8—2021no—
143C2×D4—21no—
144C2×D4D40411yescharacteristic
145C22×C4—21no—
146C22×C4—1011no—
147C22×C4—1011no—
148C22×C4—2021no—
149C22×C4—2021no—
150C22×C4C22×D10111yescharacteristic
151Q16—4042no—
152SD16—4042no—
153C2×C8—21no—
154C2×C8—2021no—
155C4⋊C4—1011no—
156C4⋊C4—4042no—
157C4⋊C4—2021no—
158C4⋊C4—2021no—
159C4⋊C4—2021no—
160C4⋊C4—2021no—
161C4⋊C4—2021no—
162C4⋊C4—2021no—
163C4⋊C4D40411yescharacteristic
164C22⋊C4—21no—
165C22⋊C4—2021no—
166C22⋊C4—2021no—
167C22⋊C4—2021no—
168C22⋊C4—2021no—
169C42—2021no—
170C23—21no—
171C23D4×D10111yescharacteristic
172Q8—4041no—
173Q8—4041no—
174Q8—4041no—
175Q8—2022no—
176Q8—2022no—
177D4—41no—
178D4—41no—
179D4—41no—
180C2×C4—22no—
181C2×C4—21no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—8081no—
186C2×C4—4041no—
187C2×C4—4041no—
188C2×C4—4041no—
189C2×C4—2022no—
190C2×C4—2021no—
191C2×C4—2021no—
192C2×C4—2021no—
193C2×C4—2021no—
194C2×C4—2021no—
195C2×C4—2021no—
196C2×C4D4×D10111yescharacteristic
197C2×C4C2×D40411yescharacteristic
198C8—41no—
199C8—4041no—
200C22—21no—
201C22—21no—
202C22—41no—
203C22—41no—
204C22—41no—
205C22#1569222511yescharacteristic
206C22#1569227411yescharacteristic
207C22#1569228411yescharacteristic
208C4—21no—
209C4—21no—
210C4—42no—
211C4—41no—
212C4—4041no—
213C4—4041no—
214C4—4041no—
215C4—4041no—
216C4—4042no—
217C2—21no—
218C2—21no—
219C2—81no—
220C2#2026534011yescharacteristic
221C2#2026536111yescharacteristic
222C2#2026536511yescharacteristic
subgroupG/H
centerC22#15692225
commutator, cosocleC22×C4×C101C23
FrattiniC22×C4C22×D101
Fitting#327×C101C2
socleC22×C101#120
prime radical#121×C101C22
hypercenter#327D101
hypocenterC101#1603
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. #20265335C2semidirect
4. #20265098C2semidirect
5. #20264913C2semidirect
6. #20264902C2semidirect
112. C101#1603semidirect

Maximal subgroups

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

subgroupindex
#202657722
#202653352
#202653152
#202650982
#202649132
#202649022
#327×C1012
#1603101