Finite Group Databasebeta

← #27153198

Subgroups — #27153198

Order 12416.

346 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 109 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#20261358C212yesmaximal, complemented
2#20261028C212yesmaximal, complemented
3#20261014C212yesmaximal, complemented
4#20260983C212yesmaximal, complemented
5#20260971C211yesmaximal, characteristic
6#20260959C212yesmaximal, complemented
7#20260953C212yesmaximal, complemented
8#20260761C211yesmaximal, characteristic, complemented
9#467×C97C211yesmaximal, characteristic, complemented
10C2×#13095498C2212yes—
11C2×D776C2211yescharacteristic
12C2×#13095495C2211yescharacteristic
13#15690809C2212yes—
14#15690805—22no—
15#15690799C2212yes—
16#15690796—22no—
17#15690792—22no—
18#15690790—22no—
19#15690784C2212yes—
20Q8×Dic97C2212yes—
21#15690772C2212yes—
22#15690769C2211yescharacteristic
23#15690744C2211yescharacteristic
24#15690743C2212yes—
25#15690739C2212yes—
26#15690739C2212yes—
27#15690734C2212yes—
28#15690732C2212yes—
29#15690730C2212yes—
30C2×Q16×C97C2211yescharacteristic
31C22×C8×C97C2211yescharacteristic
32C22⋊Q8×C97C2212yes—
33#107×C97C2211yescharacteristic
34#103×C97C2212yes—
35C2×Dic194—22no—
36C2×D388C2312yes—
37C2×C22⋊D97—22no—
38C2×C4×D97—22no—
39#13095498—42no—
40D776—41no—
41#13095495—41no—
42C2×#11715895C2312yes—
43C4⋊Dic97—22no—
44C4⋊Dic97D411yescharacteristic
45C4⋊Dic97D411yescharacteristic
46C4⋊Dic97C2312yes—
47#13095490—22no—
48C22⋊Dic97D412yes—
49#13095488—22no—
50#13095488—22no—
51#13095488—22no—
52#13095488—22no—
53C4×Dic97—22no—
54C4×Dic97C2312yes—
55C2×Q8×C97C2312yes—
56C22×C4×C97C2311yescharacteristic
57Q16×C97—41no—
58C2×C8×C97—41no—
59C2×C8×C97C2311yescharacteristic
60C2×C8×C97C2311yescharacteristic
61C4⋊C4×C97—22no—
62C4⋊C4×C97C2312yes—
63C22⋊C4×C97—22no—
64C23×C97C2×D411yescharacteristic
65C22×D97—22no—
66Q8×C97—22no—
67Q8×C97—22no—
68C2×C4×C97—21no—
69C2×C4×C97—21no—
70C2×C4×C97—22no—
71C2×C4×C97—22no—
72C2×C4×C97C2×D411yescharacteristic
73C2×C4×C97C2411yescharacteristic
74C22⋊D97—82no—
75C2×Dic97—22no—
76C2×Dic97C2×D412yes—
77C2×Dic97C2×D412yes—
78D388—22no—
79D388—22no—
80C4×D97—82no—
81Dic194—82no—
82C8×C97—21no—
83C8×C97—21no—
84#11715895—22no—
85C22×C97—21no—
86C22×C97—21no—
87C22×C97—21no—
88C22×C97C22×D411yescharacteristic
89C2×D97—42no—
90C2×D97—42no—
91C2×D97—42no—
92C4×C97—21no—
93C4×C97—42no—
94C4×C97—42no—
95C4×C97H(5,𝔽2)11yescharacteristic
96C4×C972−1+411yescharacteristic
97Dic97—22no—
98Dic97—22no—
99Dic97—42no—
100C2×C97—41no—
101C2×C97#53711yescharacteristic
102C2×C97#57611yescharacteristic
103C2×C97#57811yescharacteristic
104D97—82no—
105#3262—971nomaximal, Sylow
106C97#326211yescharacteristic, complemented, Sylow
107#542—972no—
108#477—972no—
109#474—972no—
110#468—972no—
111#467D9711yescharacteristic, complemented
112#466—971no—
113#464—972no—
114#463—972no—
115#424—971no—
116C2×Q16C2×D9711yescharacteristic
117C2×SD16—972no—
118C2×D8—971no—
119C2×M16—971no—
120C22×C8C2×D9711yescharacteristic
121C4⋊Q8—972no—
122#126—1942no—
123#124—972no—
124#124—1942no—
125#123—1942no—
126C22⋊Q8—1942no—
127C22⋊Q8C2×D9712yes—
128#121—972no—
129C4×Q8—972no—
130C4×D4—972no—
131#117—971no—
132#107—971no—
133#107—972no—
134#107C2×D9711yescharacteristic
135C4⋊C8—972no—
136C4⋊C8—972no—
137#103—972no—
138#103C2×D9712yes—
139#102—972no—
140#102—972no—
141C2×Q8—1942no—
142C2×Q8C22×D9712yes—
143C2×D4—972no—
144C2×D4—1942no—
145C22×C4—1942no—
146C22×C4C22×D9711yescharacteristic
147Q16—41no—
148SD16—3882no—
149D8—3881no—
150M16—3881no—
151C2×C8—41no—
152C2×C8—972no—
153C2×C8C22×D9711yescharacteristic
154C2×C8C22×D9711yescharacteristic
155C4⋊C4—22no—
156C4⋊C4—971no—
157C4⋊C4—971no—
158C4⋊C4—972no—
159C4⋊C4—1942no—
160C4⋊C4—1942no—
161C4⋊C4C22×D9712yes—
162C22⋊C4—22no—
163C22⋊C4—972no—
164C22⋊C4—1942no—
165C22⋊C4—1942no—
166C22⋊C4—1942no—
167C22⋊C4—1942no—
168C42—972no—
169C42—1942no—
170C23—1942no—
171C23D4×D9711yescharacteristic
172Q8—22no—
173Q8—22no—
174Q8—7762no—
175D4—7762no—
176D4—1942no—
177D4—1942no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—7762no—
183C2×C4—972no—
184C2×C4—972no—
185C2×C4—1942no—
186C2×C4D4×D9711yescharacteristic
187C2×C4C23×D9711yescharacteristic
188C8—21no—
189C8—21no—
190C8—1942no—
191C22—21no—
192C22—21no—
193C22—21no—
194C22—3882no—
195C22—3882no—
196C22—3882no—
197C22C2×D4×D9711yescharacteristic
198C4—21no—
199C4—42no—
200C4—42no—
201C4—3882no—
202C4—1942no—
203C4—1942no—
204C4#1569085511yescharacteristic
205C4#1569085711yescharacteristic
206C2—41no—
207C2—7762no—
208C2#2026131211yescharacteristic
209C2#2026153711yescharacteristic
210C2#2026154811yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#467×C97C2
socleC22×C97C22×D4
prime radical#20260971C2
hypercenter#467D97
hypocenterC97#3262
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #467

Complements

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

NH ≅ G/Ntype
1. #20261358C2semidirect
2. #20261028C2semidirect
3. #20261014C2semidirect
4. #20260983C2semidirect
6. #20260959C2semidirect
7. #20260953C2semidirect
8. #20260761C2semidirect
9. #467×C97C2semidirect
106. C97#3262semidirect
111. #467D97semidirect

Maximal subgroups

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

subgroupindex
#202613582
#202610282
#202610142
#202609832
#202609712
#202609592
#202609532
#202607612
#467×C972
#326297