Finite Group Databasebeta

← #27151266

Subgroups — #27151266

Order 12416.

458 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 149 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
2C22×#13095501C211yesmaximal, characteristic, complemented
3#20261311C211yesmaximal, characteristic
4#20260957C214yesmaximal, complemented
5#20260945C214yesmaximal, complemented
6#20260762C211yesmaximal, characteristic, complemented
7#20260743C211yesmaximal, characteristic, complemented
8#20260729C211yesmaximal, characteristic
9#519×C97C211yesmaximal, characteristic, complemented
10C2×#13095511C2211yescharacteristic
11C22×Dic194C2211yescharacteristic
12#15690841—24no—
13C2×#13095501—24no—
14C2×#13095501C2212yes—
15C2×#13095501C2214yes—
16C2×#13095498C2212yes—
17C2×#13095495C2211yescharacteristic
18C22×#11715895C2211yescharacteristic
19#15690810C2212yes—
20#15690798C2212yes—
21#15690794—22no—
22#15690790—22no—
23C4⋊#11715895C2214yes—
24#15690733C2212yes—
25#15690733C2214yes—
26#15690730C2212yes—
27C22×Q8×C97C2211yescharacteristic
28C4×Q8×C97—22no—
29C4×Q8×C97C2214yes—
30#117×C97—22no—
31#117×C97C2211yescharacteristic
32#117×C97C2211yescharacteristic
33#13095511—22no—
34#13095511—22no—
35C2×Dic194—22no—
36C2×Dic194—22no—
37C2×Dic194—21no—
38C2×Dic194C2312yes—
39C2×Dic194C2311yescharacteristic
40C2×D388C2311yescharacteristic
41C2×C22⋊D97—21no—
42C22×Dic97—21no—
43C2×C4×D97—21no—
44#13095501—28no—
45#13095501—24no—
46#13095498—24no—
47#13095495—22no—
48C2×#11715895—22no—
49C2×#11715895C2312yes—
50C2×#11715895C2312yes—
51C4⋊Dic97—24no—
52#13095488—24no—
53C2×Q8×C97—21no—
54C2×Q8×C97—22no—
55C2×Q8×C97D414yes—
56C2×Q8×C97C2312yes—
57C22×C4×C97—21no—
58C22×C4×C97C2311yescharacteristic
59C4⋊C4×C97—24no—
60C4⋊C4×C97D412yes—
61C4⋊C4×C97C2312yes—
62C22⋊C4×C97—21no—
63C22⋊C4×C97D412yes—
64C42×C97—24no—
65C42×C97—21no—
66C42×C97C2312yes—
67C23×C97C2×D411yescharacteristic
68C22×D97—21no—
69Q8×C97—24no—
70Q8×C97—22no—
71Q8×C97C2×D414yes—
72C2×C4×C97—24no—
73C2×C4×C97—22no—
74C2×C4×C97—22no—
75C2×C4×C97C2×D414yes—
76C2×C4×C97C2×D411yescharacteristic
77C2×C4×C97C2×D412yes—
78C2×C4×C97#4112yes—
79C2×C4×C97C2411yescharacteristic
80C22⋊D97—42no—
81C2×Dic97—22no—
82C2×Dic97—21no—
83C2×Dic97—42no—
84D388—21no—
85D388—21no—
86C4×D97—42no—
87Dic194—22no—
88Dic194—21no—
89Dic194—22no—
90Dic194—21no—
91Dic194—42no—
92#11715895—22no—
93#11715895—22no—
94C22×C97—22no—
95C22×C97#12112yes—
96C22×C97C22×D411yescharacteristic
97C2×D97—41no—
98C2×D97—42no—
99C4×C97—24no—
100C4×C97—24no—
101C4×C97—42no—
102C4×C97#12112yes—
103C4×C97C22×D411yescharacteristic
104C4×C97C2×#4111yescharacteristic
105Dic97—42no—
106Dic97—41no—
107C2×C97—22no—
108C2×C97C2×#12111yescharacteristic
109C2×C97#57812yes—
110D97—81no—
111#3044—971nomaximal, Sylow
112C97#304411yescharacteristic, complemented, Sylow
113C2×#137—971no—
114C22×Q16—971no—
115#536—971no—
116#519D9711yescharacteristic, complemented
117#464—974no—
118#462—974no—
119#424—971no—
120#419—971no—
121#416—971no—
122C2×#41—971no—
123C22×Q8—971no—
124C22×Q8C2×D9711yescharacteristic
125#137—1944no—
126C2×Q16—974no—
127C2×Q16—972no—
128C2×Q16—1944no—
129C2×SD16—972no—
130C2×M16—971no—
131C22×C8—971no—
132C4⋊Q8—972no—
133#124—972no—
134#123—1942no—
135C22⋊Q8—1942no—
136C4×Q8—22no—
137C4×Q8C2×D9714yes—
138#117—22no—
139#117C2×D9711yescharacteristic
140#117C2×D9711yescharacteristic
141C4⋊C8—974no—
142#103—972no—
143#103—974no—
144#102—972no—
145#41—1942no—
146#41—1942no—
147C2×Q8—21no—
148C2×Q8—22no—
149C2×Q8—972no—
150C2×Q8—971no—
151C2×Q8—1942no—
152C2×Q8—1942no—
153C2×Q8—1941no—
154C2×Q8D38814yes—
155C2×Q8C22×D9712yes—
156C2×D4—971no—
157C2×D4—1941no—
158C22×C4—21no—
159C22×C4—1941no—
160C22×C4—1941no—
161C22×C4C22×D9711yescharacteristic
162Q16—1948no—
163Q16—1944no—
164SD16—1944no—
165M16—1942no—
166C2×C8—972no—
167C2×C8—972no—
168C2×C8—1942no—
169C4⋊C4—24no—
170C4⋊C4—1944no—
171C4⋊C4C22⋊D9712yes—
172C4⋊C4C22×D9712yes—
173C22⋊C4—21no—
174C22⋊C4—1944no—
175C22⋊C4C22⋊D9712yes—
176C42—24no—
177C42—21no—
178C42C22×D9712yes—
179C23—1941no—
180C23C2×C22⋊D9711yescharacteristic
181Q8—24no—
182Q8—22no—
183Q8—3882no—
184Q8—1942no—
185Q8—1941no—
186Q8—1942no—
187Q8—1941no—
188Q8C2×D38814yes—
189D4—3882no—
190D4—1941no—
191D4—1941no—
192C2×C4—24no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—3882no—
196C2×C4—3882no—
197C2×C4—1942no—
198C2×C4—1941no—
199C2×C4C2×C22⋊D9714yes—
200C2×C4C2×C22⋊D9711yescharacteristic
201C2×C4C2×D38812yes—
202C2×C4#1309551112yes—
203C2×C4C23×D9711yescharacteristic
204C8—1942no—
205C8—1942no—
206C22—22no—
207C22—3881no—
208C22—3882no—
209C22#1569078412yes—
210C22C22×C22⋊D9711yescharacteristic
211C4—24no—
212C4—24no—
213C4—42no—
214C4—3881no—
215C4—3882no—
216C4#1569078412yes—
217C4C22×D38811yescharacteristic
218C4C2×#1309551111yescharacteristic
219C2—22no—
220C2—7761no—
221C2C2×#1569078411yescharacteristic
222C2#2026155012yes—
subgroupG/H
centerC22C22×C22⋊D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#519×C97C2
socleC22×C97C22×D4
prime radicalC2×#13095511C22
hypercenter#519D97
hypocenterC97#3044
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ #117 ◁ #519

Complements

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

NH ≅ G/Ntype
2. C22×#13095501C2semidirect
4. #20260957C2semidirect
5. #20260945C2semidirect
6. #20260762C2semidirect
7. #20260743C2semidirect
9. #519×C97C2semidirect
112. C97#3044semidirect
116. #519D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156908412
C22×#130955012
#202613112
#202609572
#202609452
#202607622
#202607432
#202607292
#519×C972
#304497