Finite Group Databasebeta

← #27153138

Subgroups — #27153138

Order 12416.

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

normal subgroups only

This group has 354 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#20261271C212yesmaximal, complemented
2#20261040C212yesmaximal, complemented
3#20261034C212yesmaximal, complemented
4#20260988C211yesmaximal, characteristic
5#20260983C212yesmaximal, complemented
6#20260959C212yesmaximal, complemented
7#20260953C212yesmaximal, complemented
8#20260758C211yesmaximal, characteristic, complemented
9#470×C97C211yesmaximal, characteristic, complemented
10C2×#13095498C2212yes—
11C2×D776C2211yescharacteristic
12C2×#13095495C2211yescharacteristic
13#15690805—22no—
14#15690801C2212yes—
15#15690799C2212yes—
16#15690794—22no—
17#15690784C2212yes—
18Q8×Dic97C2212yes—
19#15690772C2212yes—
20#15690770—22no—
21#15690768—22no—
22C2×C4×Dic97C2211yescharacteristic
23#15690743C2212yes—
24#15690742C2211yescharacteristic
25#15690739C2212yes—
26#15690739C2212yes—
27#15690734C2212yes—
28#15690732C2212yes—
29#15690730C2212yes—
30C2×Q16×C97C2211yescharacteristic
31C2×M16×C97C2211yescharacteristic
32C22⋊Q8×C97C2212yes—
33#106×C97C2211yescharacteristic
34#103×C97C2212yes—
35C2×D388C2312yes—
36C2×C22⋊D97—22no—
37C22×Dic97D412yes—
38C2×C4×D97—22no—
39#13095498—42no—
40D776—41no—
41#13095495—41no—
42C2×#11715895C2312yes—
43C4⋊Dic97—22no—
44C4⋊Dic97—22no—
45C4⋊Dic97C2312yes—
46#13095490—22no—
47#13095488—22no—
48#13095488—22no—
49#13095488—22no—
50#13095488—22no—
51C4×Dic97—22no—
52C4×Dic97D411yescharacteristic
53C4×Dic97D411yescharacteristic
54C4×Dic97C2312yes—
55C2×Q8×C97C2312yes—
56C22×C4×C97C2311yescharacteristic
57Q16×C97—41no—
58M16×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—44no—
75C2×Dic97—22no—
76C2×Dic97—42no—
77C2×Dic97C2×D412yes—
78C2×Dic97C2×D412yes—
79D388—22no—
80D388—22no—
81C4×D97—44no—
82C8×C97—21no—
83C8×C97—21no—
84#11715895—22no—
85C22×C97—21no—
86C22×C97—22no—
87C22×C97C22×D411yescharacteristic
88C2×D97—42no—
89C2×D97—42no—
90C2×D97—42no—
91C4×C97—21no—
92C4×C97—42no—
93C4×C97—42no—
94C4×C97H(5,𝔽2)11yescharacteristic
95C4×C972−1+411yescharacteristic
96Dic97—22no—
97Dic97—22no—
98Dic97—42no—
99C2×C97—41no—
100C2×C97#53711yescharacteristic
101C2×C97#57512yes—
102D97—82no—
103#3256—971nomaximal, Sylow
104C97#325611yescharacteristic, complemented, Sylow
105#529—972no—
106#479—972no—
107#478—972no—
108#470D9711yescharacteristic, complemented
109#469—971no—
110#468—972no—
111#464—972no—
112#463—972no—
113#423—971no—
114C2×Q16C2×D9711yescharacteristic
115C2×SD16—972no—
116C2×D8—971no—
117C2×M16—971no—
118C2×M16C2×D9711yescharacteristic
119#126—1942no—
120#125—972no—
121#124—972no—
122#123—1942no—
123C22⋊Q8C2×D9712yes—
124#121—972no—
125C4×Q8—972no—
126C4×D4—972no—
127C4×D4—1942no—
128#117—1942no—
129C2×C42—971no—
130#107—972no—
131#106—971no—
132#106C2×D9711yescharacteristic
133C4⋊C8—972no—
134C4⋊C8—972no—
135#103—972no—
136#103C2×D9712yes—
137#102—972no—
138#102—972no—
139C2×Q8C22×D9712yes—
140C2×D4—972no—
141C2×D4—1942no—
142C22×C4—972no—
143C22×C4—1942no—
144C22×C4C22×D9711yescharacteristic
145Q16—41no—
146SD16—3882no—
147D8—3881no—
148M16—41no—
149M16—3881no—
150C2×C8—972no—
151C2×C8C22×D9711yescharacteristic
152C2×C8C22×D9711yescharacteristic
153C4⋊C4—22no—
154C4⋊C4—972no—
155C4⋊C4—1942no—
156C4⋊C4—1942no—
157C4⋊C4—1942no—
158C4⋊C4C22×D9712yes—
159C22⋊C4—22no—
160C22⋊C4—1942no—
161C22⋊C4—1942no—
162C22⋊C4—1942no—
163C22⋊C4—1942no—
164C42—971no—
165C42—971no—
166C42—972no—
167C42—1942no—
168C23—1942no—
169C23D4×D9711yescharacteristic
170Q8—22no—
171Q8—22no—
172D4—3884no—
173D4—1942no—
174D4—1942no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—972no—
180C2×C4—972no—
181C2×C4—3884no—
182C2×C4—3882no—
183C2×C4—1942no—
184C2×C4D4×D9711yescharacteristic
185C2×C4C23×D9711yescharacteristic
186C8—21no—
187C8—21no—
188C8—1942no—
189C22—22no—
190C22—21no—
191C22—3882no—
192C22—3882no—
193C22—3882no—
194C22C2×D4×D9711yescharacteristic
195C4—21no—
196C4—42no—
197C4—42no—
198C4—3882no—
199C4—1942no—
200C4—1942no—
201C4#1569085511yescharacteristic
202C4#1569085711yescharacteristic
203C2—41no—
204C2—7762no—
205C2#2026131211yescharacteristic
206C2#2026153412yes—
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#470×C97C2
socleC22×C97C22×D4
prime radical#20260988C2
hypercenter#470D97
hypocenterC97#3256
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

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

NH ≅ G/Ntype
1. #20261271C2semidirect
2. #20261040C2semidirect
3. #20261034C2semidirect
5. #20260983C2semidirect
6. #20260959C2semidirect
7. #20260953C2semidirect
8. #20260758C2semidirect
9. #470×C97C2semidirect
104. C97#3256semidirect
108. #470D97semidirect

Maximal subgroups

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

subgroupindex
#202612712
#202610402
#202610342
#202609882
#202609832
#202609592
#202609532
#202607582
#470×C972
#325697