Finite Group Databasebeta

← #27169706

Subgroups — #27169706

Order 12928.

218 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 61 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#20265708C211yesmaximal, characteristic
2#20265227C211yesmaximal, characteristic, complemented
3#20265044C211yesmaximal, characteristic, complemented
4#20264915C211yesmaximal, characteristic, complemented
5#20264913C211yesmaximal, characteristic, complemented
6#20264911C211yesmaximal, characteristic
7#327×C101C211yesmaximal, characteristic
8#15692248—21no—
9#15692246C411yescharacteristic, complemented
10#15692242C411yescharacteristic, complemented
11#15692236—21no—
12#15692234C2211yescharacteristic
13#15692220C411yescharacteristic, complemented
14D4×Dic101—21no—
15D4×Dic101C411yescharacteristic, complemented
16#15692211—21no—
17C2×C4×Dic101C2211yescharacteristic
18#15692184—21no—
19C22⋊#11717579C2211yescharacteristic
20C8×Dic101—21no—
21#15692162—21no—
22#15692162—21no—
23#15692162C2211yescharacteristic
24#121×C101C2211yescharacteristic
25C22⋊C8×C101C2211yescharacteristic
26#95×C101C2211yescharacteristic
27C2×Dic202C2×C411yescharacteristic
28C22×Dic101—21no—
29C22×Dic101—21no—
30C22×Dic101D411yescharacteristic
31C22×Dic101D411yescharacteristic
32C2×#11717579—21no—
33C4⋊Dic101—21no—
34C4⋊Dic101C2×C411yescharacteristic
35#13108053—21no—
36#13108053—21no—
37#13108053—21no—
38C22⋊Dic101—21no—
39C22⋊Dic101—21no—
40C22⋊Dic101—21no—
41C22⋊Dic101—21no—
42C4×Dic101—21no—
43C4×Dic101C2×C411yescharacteristic
44C4×Dic101C2×C411yescharacteristic
45C4×Dic101D411yescharacteristic
46C4×Dic101D411yescharacteristic
47C2×D4×C101—21no—
48C2×D4×C101C2×C411yescharacteristic
49C22×C4×C101—21no—
50C22×C4×C101C2311yescharacteristic
51C2×C8×C101—21no—
52C4⋊C4×C101C2×C411yescharacteristic
53C22⋊C4×C101—21no—
54C23×C101—21no—
55C23×C101C2×D411yescharacteristic
56D4×C101—41no—
57D4×C101—41no—
58D4×C101—41no—
59C2×C4×C101—21no—
60C2×C4×C101—41no—
61C2×C4×C101—41no—
62C2×C4×C101—41no—
63C2×C4×C101—41no—
64C2×C4×C101C22×C411yescharacteristic
65C2×C4×C101C2×D411yescharacteristic
66C2×Dic101—21no—
67C2×Dic101—21no—
68C2×Dic101—21no—
69C2×Dic101—21no—
70C2×Dic101—41no—
71C2×Dic101—42no—
72C2×Dic101—41no—
73C2×Dic101—41no—
74C2×Dic101—41no—
75C2×Dic101C22⋊C411yescharacteristic
76C2×Dic101C22⋊C411yescharacteristic
77C2×Dic101C22⋊C411yescharacteristic
78C2×Dic101C22⋊C411yescharacteristic
79Dic202—41no—
80C8×C101—41no—
81#11717579—41no—
82C22×C101—21no—
83C22×C101—21no—
84C22×C101—41no—
85C22×C101—41no—
86C22×C101—41no—
87C22×C101C2×C22⋊C411yescharacteristic
88C22×C101#13511yescharacteristic
89C22×C101#13511yescharacteristic
90C4×C101—21no—
91C4×C101—21no—
92C4×C101—41no—
93C4×C101—81no—
94Dic101—21no—
95Dic101—21no—
96Dic101—21no—
97Dic101—21no—
98Dic101—41no—
99Dic101—41no—
100Dic101—81no—
101C2×C101—21no—
102C2×C101—21no—
103C2×C101—81no—
104C2×C101#41011yescharacteristic
105C2×C101#41611yescharacteristic
106C2×C101#42111yescharacteristic
107#1521—1011nomaximal, Sylow
108C101#152111yescharacteristic, complemented, Sylow
109#529—1011no—
110#431—1011no—
111#383—1011no—
112#328—1011no—
113#328—1011no—
114#327—1011no—
115#327D10111yescharacteristic
116#126—2021no—
117#125—1011no—
118#124—1011no—
119#123—2021no—
120C22⋊Q8—1011no—
121#121C2×D10111yescharacteristic
122C4×Q8—1011no—
123C4×D4—1011no—
124C4×D4—2021no—
125#117—2021no—
126C2×C42—1011no—
127C4⋊C8—2021no—
128C22⋊C8—1011no—
129C22⋊C8C2×D10111yescharacteristic
130C4×C8—2021no—
131#95—1011no—
132#95—2021no—
133#95—2021no—
134#95C2×D10111yescharacteristic
135C2×Q8—1011no—
136C2×D4—21no—
137C2×D4C4×D10111yescharacteristic
138C22×C4—21no—
139C22×C4—1011no—
140C22×C4—1011no—
141C22×C4—2021no—
142C22×C4—2021no—
143C22×C4C22×D10111yescharacteristic
144C2×C8—21no—
145C2×C8—2021no—
146C4⋊C4—1011no—
147C4⋊C4—2021no—
148C4⋊C4—2021no—
149C4⋊C4—2021no—
150C4⋊C4—2021no—
151C4⋊C4C4×D10111yescharacteristic
152C22⋊C4—21no—
153C22⋊C4—2021no—
154C22⋊C4—2021no—
155C22⋊C4—2021no—
156C22⋊C4—2021no—
157C42—1011no—
158C42—1011no—
159C42—1011no—
160C42—1011no—
161C42—2021no—
162C23—21no—
163C23D4×D10111yescharacteristic
164Q8—4041no—
165D4—41no—
166D4—41no—
167D4—41no—
168C2×C4—21no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—1011no—
174C2×C4—1011no—
175C2×C4—1011no—
176C2×C4—1011no—
177C2×C4—4041no—
178C2×C4—4041no—
179C2×C4—4041no—
180C2×C4—4042no—
181C2×C4—4041no—
182C2×C4—2021no—
183C2×C4—2021no—
184C2×C4—2021no—
185C2×C4—2021no—
186C2×C4C2×C4×D10111yescharacteristic
187C2×C4D4×D10111yescharacteristic
188C8—41no—
189C8—4041no—
190C22—21no—
191C22—21no—
192C22—41no—
193C22—41no—
194C22—41no—
195C22C22⋊C4×D10111yescharacteristic
196C22#1569227411yescharacteristic
197C22#1569227611yescharacteristic
198C4—21no—
199C4—21no—
200C4—41no—
201C4—81no—
202C4—8081no—
203C4—4041no—
204C4—4041no—
205C4—2021no—
206C4—2021no—
207C4—2021no—
208C4—2021no—
209C2—21no—
210C2—21no—
211C2—81no—
212C2#2026514911yescharacteristic
213C2#2026517011yescharacteristic
214C2#2026519211yescharacteristic
subgroupG/H
centerC22C22⋊C4×D101
commutatorC2×C4×C101C22×C4
FrattiniC22×C4C22×D101
Fitting#327×C101C2
socleC22×C101C2×C22⋊C4
prime radical#121×C101C22
cosocleC22×C4×C101C23
hypercenter#327D101
hypocenterC101#1521
LCSG ▷ C2×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. #20265227C2semidirect
3. #20265044C2semidirect
4. #20264915C2semidirect
5. #20264913C2semidirect
9. #15692246C4semidirect
10. #15692242C4semidirect
13. #15692220C4semidirect
15. D4×Dic101C4semidirect
108. C101#1521semidirect

Maximal subgroups

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

subgroupindex
#202657082
#202652272
#202650442
#202649152
#202649132
#202649112
#327×C1012
#1521101