Finite Group Databasebeta

← #27142214

Subgroups — #27142214

Order 12416.

324 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 85 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
1C2×#15690835C211yesmaximal, characteristic
2#20261502C211yesmaximal, characteristic, complemented
3#20260739C211yesmaximal, characteristic, complemented
4C2×#15690726C211yesmaximal, characteristic, complemented
5#20260505C211yesmaximal, characteristic, complemented
6#20260493C211yesmaximal, characteristic, complemented
7#421×C97C211yesmaximal, characteristic, complemented
8C22×D388C2211yescharacteristic
9#15690835—22no—
10#15690835—22no—
11C2×#13095498C412yescomplemented
12C2×#13095496C412yescomplemented
13#15690817—22no—
14#15690817—22no—
15C2×#13095495—22no—
16C2×#13095495C2211yescharacteristic
17C2×#13095495C2211yescharacteristic
18C2×#13095495C2211yescharacteristic
19#15690730—22no—
20#15690726—22no—
21C2×#41×C97C2211yescharacteristic
22C2×M16×C97C2211yescharacteristic
23#117×C97C2211yescharacteristic
24#104×C97—22no—
25C23×D97—21no—
26C2×D388—21no—
27C2×D388—21no—
28C2×D388—21no—
29C2×D388—21no—
30C2×D388—21no—
31C2×D388C2×C412yes—
32#13095498—24no—
33#13095496—24no—
34#13095495—24no—
35#13095495—21no—
36#13095495—21no—
37#13095495—21no—
38#13095495—21no—
39#13095495—21no—
40#13095495D412yes—
41C2×#11715895—24no—
42C2×#11715895—21no—
43C2×#11715895C2×C412yes—
44C2×#11715895D412yes—
45#41×C97—21no—
46#41×C97—21no—
47#41×C97D412yes—
48#41×C97D412yes—
49C2×Q8×C97C2×C411yescharacteristic
50C2×D4×C97—21no—
51C2×D4×C97C2×C411yescharacteristic
52C22×C4×C97—21no—
53C22×C4×C97C2311yescharacteristic
54M16×C97—21no—
55M16×C97—21no—
56C2×C8×C97—21no—
57C4⋊C4×C97—21no—
58C22⋊C4×C97—21no—
59C42×C97—21no—
60C23×C97—21no—
61C23×C97#4111yescharacteristic
62C22×D97—22no—
63C22×D97—41no—
64C22×D97—41no—
65C22×D97—82no—
66Q8×C97—21no—
67Q8×C97C22⋊C412yes—
68D4×C97—22no—
69D4×C97—22no—
70D4×C97—21no—
71D4×C97C22⋊C412yes—
72C2×C4×C97—21no—
73C2×C4×C97—22no—
74C2×C4×C97—22no—
75C2×C4×C97—41no—
76C2×C4×C97C22×C411yescharacteristic
77C2×C4×C97C2×D411yescharacteristic
78C2×C4×C97C2×D411yescharacteristic
79C2×C4×C97C2×D411yescharacteristic
80C2×C4×C97C2×D411yescharacteristic
81C2×C4×C97#4111yescharacteristic
82D388—22no—
83D388—22no—
84D388—41no—
85D388—41no—
86C8×C97—41no—
87#11715895—22no—
88#11715895—22no—
89#11715895—22no—
90C22×C97—22no—
91C22×C97—41no—
92C22×C97—81no—
93C22×C97C4×D411yescharacteristic
94C22×C97#12111yescharacteristic
95C22×C97#12311yescharacteristic
96C2×D97—42no—
97C2×D97—82no—
98C2×D97—82no—
99C2×D97—82no—
100C4×C97—22no—
101C4×C97—81no—
102C4×C97C2×C22⋊C411yescharacteristic
103C4×C97C4×D411yescharacteristic
104C4×C97#12011yescharacteristic
105C4×C97#12111yescharacteristic
106C2×C97—21no—
107C2×C97—21no—
108C2×C97—21no—
109C2×C97—42no—
110C2×C97#38611yescharacteristic
111D97—82no—
112#1903—971nomaximal, Sylow
113C97#190311yescharacteristic, complemented, Sylow
114C2×AGL(1,ℤ/8)—971no—
115#568—971no—
116#421D9711yescharacteristic, complemented
117#418—971no—
118C2×#100—971no—
119#344—971no—
120#338—971no—
121C2×#41C2×D9711yescharacteristic
122C22×D4—971no—
123AGL(1,ℤ/8)—1942no—
124AGL(1,ℤ/8)—1942no—
125C2×SD16—972no—
126C2×D8—972no—
127#131—1942no—
128#131—1942no—
129C2×M16—971no—
130C2×M16—971no—
131C2×M16—971no—
132C2×M16—1942no—
133C2×M16C2×D9711yescharacteristic
134#117C2×D9711yescharacteristic
135#104—22no—
136#102—1942no—
137#100—1942no—
138C24—1941no—
139#41—21no—
140#41—21no—
141#41D38812yes—
142#41C22⋊D9712yes—
143C2×Q8C4×D9711yescharacteristic
144C2×D4—21no—
145C2×D4—972no—
146C2×D4—1941no—
147C2×D4—1941no—
148C2×D4—1941no—
149C2×D4—1941no—
150C2×D4—1941no—
151C2×D4C4×D9711yescharacteristic
152C22×C4—21no—
153C22×C4C22×D9711yescharacteristic
154SD16—1944no—
155D8—1944no—
156M16—21no—
157M16—21no—
158M16—972no—
159M16—1944no—
160M16—1941no—
161M16—1941no—
162M16—1941no—
163M16—1941no—
164M16—1941no—
165C2×C8—21no—
166C2×C8—972no—
167C2×C8—972no—
168C2×C8—1944no—
169C2×C8—1941no—
170C4⋊C4—21no—
171C22⋊C4—21no—
172C42—21no—
173C23—21no—
174C23—7762no—
175C23—3881no—
176C23—3881no—
177C23—1942no—
178C23#1309551011yescharacteristic
179Q8—21no—
180Q8#1309548812yes—
181D4—22no—
182D4—22no—
183D4—21no—
184D4—3881no—
185D4—3881no—
186D4—1942no—
187D4—1942no—
188D4#1309548812yes—
189C2×C4—21no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—41no—
193C2×C4C2×C4×D9711yescharacteristic
194C2×C4D4×D9711yescharacteristic
195C2×C4D4×D9711yescharacteristic
196C2×C4C2×C22⋊D9711yescharacteristic
197C2×C4C2×D38811yescharacteristic
198C2×C4#1309551011yescharacteristic
199C8—41no—
200C8—1942no—
201C8—1942no—
202C8—1942no—
203C22—22no—
204C22—41no—
205C22—81no—
206C22—7762no—
207C22—7762no—
208C22—7762no—
209C22—3882no—
210C22#1569077011yescharacteristic
211C22#1569078211yescharacteristic
212C22#1569079411yescharacteristic
213C4—22no—
214C4—81no—
215C4C2×#1309548811yescharacteristic
216C4#1569077011yescharacteristic
217C4#1569078011yescharacteristic
218C4#1569078211yescharacteristic
219C2—21no—
220C2—21no—
221C2—21no—
222C2—42no—
223C2—7762no—
224C2#2026061211yescharacteristic
subgroupG/H
centerC2#20260612
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
Fitting#421×C97C2
socleC2×C97#386
prime radicalC2×#15690835C2
cosocleC22×C4×C97C23
hypercenter#421D97
hypocenterC97#1903
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C22×C4 ◁ #421

Complements

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

NH ≅ G/Ntype
2. #20261502C2semidirect
3. #20260739C2semidirect
4. C2×#15690726C2semidirect
5. #20260505C2semidirect
6. #20260493C2semidirect
7. #421×C97C2semidirect
11. C2×#13095498C4semidirect
12. C2×#13095496C4semidirect
113. C97#1903semidirect
116. #421D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156908352
#202615022
#202607392
C2×#156907262
#202605052
#202604932
#421×C972
#190397