Finite Group Databasebeta

← #27143035

Subgroups — #27143035

Order 12416.

262 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 67 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×#15690841C211yesmaximal, characteristic
2C2×#15690745C211yesmaximal, characteristic, complemented
3#20260737C211yesmaximal, characteristic
4C2×#15690734C211yesmaximal, characteristic, complemented
5#20260703C211yesmaximal, characteristic
6#20260503C211yesmaximal, characteristic, complemented
7C2×#101×C97C211yesmaximal, characteristic, complemented
8C2×#13095511C2211yescharacteristic
9#15690841—22no—
10#15690841—22no—
11C2×#13095501—21no—
12C2×#13095498—21no—
13C2×#13095495C2211yescharacteristic
14C22×#11715895C2211yescharacteristic
15C2×C4⋊Dic97C2211yescharacteristic
16#15690745—22no—
17#15690735—21no—
18#15690734—22no—
19#15690732—21no—
20#15690722—22no—
21C22×Q8×C97C2211yescharacteristic
22C2×M16×C97C2211yescharacteristic
23C2×M16×C97C2211yescharacteristic
24#101×C97—22no—
25#13095511—21no—
26#13095511—21no—
27#13095511—21no—
28#13095511—21no—
29C2×Dic194D411yescharacteristic
30C2×D388D411yescharacteristic
31C2×C22⋊D97—21no—
32C22×Dic97—21no—
33C2×C4×D97—21no—
34#13095501—24no—
35#13095498—24no—
36#13095495—21no—
37#13095495D412yes—
38C2×#11715895—21no—
39C2×#11715895—21no—
40C2×#11715895—21no—
41C2×#11715895D412yes—
42C4⋊Dic97—21no—
43C4⋊Dic97—21no—
44C2×Q8×C97—21no—
45C2×Q8×C97—21no—
46C2×Q8×C97—21no—
47C2×Q8×C97—21no—
48C2×Q8×C97—21no—
49C2×Q8×C97D412yes—
50C22×C4×C97—21no—
51C22×C4×C97C2311yescharacteristic
52M16×C97—21no—
53M16×C97—21no—
54M16×C97—21no—
55M16×C97—21no—
56C2×C8×C97—21no—
57C2×C8×C97—21no—
58C23×C97#4111yescharacteristic
59C22×D97—21no—
60Q8×C97—22no—
61Q8×C97—22no—
62Q8×C97—41no—
63Q8×C97—41no—
64C2×C4×C97—22no—
65C2×C4×C97—41no—
66C2×C4×C97—41no—
67C2×C4×C97C2×D411yescharacteristic
68C2×C4×C97C2×D411yescharacteristic
69C2×C4×C97C2×D411yescharacteristic
70C2×C4×C97C2×D411yescharacteristic
71C2×C4×C97#4111yescharacteristic
72C2×C4×C97#4111yescharacteristic
73C22⋊D97—41no—
74C22⋊D97—41no—
75C2×Dic97—21no—
76C2×Dic97—41no—
77C2×Dic97—81no—
78D388—21no—
79D388—21no—
80C4×D97—41no—
81C4×D97—41no—
82Dic194—21no—
83Dic194—21no—
84C8×C97—41no—
85C8×C97—41no—
86#11715895—22no—
87#11715895—22no—
88C22×C97—21no—
89C22×C97—21no—
90C22×C97#12111yescharacteristic
91C22×C97#12311yescharacteristic
92C22×C97#12411yescharacteristic
93C2×D97—41no—
94C2×D97—41no—
95C2×D97—41no—
96C4×C97—42no—
97C4×C97#12011yescharacteristic
98C4×C97#12111yescharacteristic
99C4×C97#12111yescharacteristic
100C4×C97#12311yescharacteristic
101Dic97—41no—
102Dic97—81no—
103C2×C97—21no—
104C2×C97—21no—
105C2×C97#39411yescharacteristic
106C2×C97#47111yescharacteristic
107C2×C97#47311yescharacteristic
108D97—81no—
109#2052—971nomaximal, Sylow
110C97#205211yescharacteristic, complemented, Sylow
111C2×#137—971no—
112C2×#108—971no—
113#417—971no—
114C2×#103—971no—
115C2×#101D9711yescharacteristic, complemented
116#408—971no—
117#343—971no—
118C2×#41—971no—
119C22×Q8C2×D9711yescharacteristic
120#137—1942no—
121#137—1942no—
122C2×Q16—1941no—
123C2×SD16—1941no—
124C2×M16—971no—
125C2×M16C2×D9711yescharacteristic
126C2×M16C2×D9711yescharacteristic
127C22×C8—971no—
128C2×C4⋊C4—971no—
129#108—1942no—
130#103—1941no—
131#103—1942no—
132#102—1941no—
133#101—22no—
134C22⋊C8—1942no—
135#41—1941no—
136#41—1941no—
137#41—1941no—
138#41—1941no—
139C2×Q8—21no—
140C2×Q8—21no—
141C2×Q8—21no—
142C2×Q8—21no—
143C2×Q8—21no—
144C2×Q8—971no—
145C2×Q8C22⋊D9712yes—
146C2×D4—971no—
147C2×D4—1941no—
148C22×C4—21no—
149C22×C4—1941no—
150C22×C4—1941no—
151C22×C4C22×D9711yescharacteristic
152Q16—1944no—
153SD16—1944no—
154M16—21no—
155M16—21no—
156M16—21no—
157M16—21no—
158M16—972no—
159M16—1941no—
160C2×C8—21no—
161C2×C8—21no—
162C2×C8—972no—
163C2×C8—1941no—
164C2×C8—1941no—
165C2×C8—1941no—
166C4⋊C4—1941no—
167C4⋊C4—1941no—
168C23—1941no—
169C23#1309551111yescharacteristic
170Q8—22no—
171Q8—22no—
172Q8—41no—
173Q8—41no—
174Q8—1941no—
175Q8—1941no—
176D4—3881no—
177D4—3881no—
178D4—1941no—
179D4—1941no—
180C2×C4—22no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—7761no—
184C2×C4—3881no—
185C2×C4—3881no—
186C2×C4—3881no—
187C2×C4—1941no—
188C2×C4D4×D9711yescharacteristic
189C2×C4D4×D9711yescharacteristic
190C2×C4D4×D9711yescharacteristic
191C2×C4C2×C22⋊D9711yescharacteristic
192C2×C4#1309551011yescharacteristic
193C2×C4#1309551111yescharacteristic
194C8—41no—
195C8—41no—
196C8—1942no—
197C8—1942no—
198C22—21no—
199C22—21no—
200C22—3881no—
201C22—3881no—
202C22—3881no—
203C22#1569078311yescharacteristic
204C22#1569079311yescharacteristic
205C22#1569079611yescharacteristic
206C4—42no—
207C4—7761no—
208C4—3881no—
209C4#1569077811yescharacteristic
210C4#1569078211yescharacteristic
211C4#1569078311yescharacteristic
212C4#1569079211yescharacteristic
213C2—21no—
214C2—21no—
215C2—7761no—
216C2#2026065011yescharacteristic
217C2#2026100111yescharacteristic
218C2#2026100911yescharacteristic
subgroupG/H
centerC22#15690783
commutator, cosocleC22×C4×C97C23
FrattiniC22×C4C22×D97
FittingC2×#101×C97C2
socleC22×C97#121
prime radicalC2×#13095511C22
hypercenterC2×#101D97
hypocenterC97#2052
LCSG ▷ C22×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#101

Complements

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

NH ≅ G/Ntype
2. C2×#15690745C2semidirect
4. C2×#15690734C2semidirect
6. #20260503C2semidirect
7. C2×#101×C97C2semidirect
110. C97#2052semidirect
115. C2×#101D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156908412
C2×#156907452
#202607372
C2×#156907342
#202607032
#202605032
C2×#101×C972
#205297