Finite Group Databasebeta

← #27143749

Subgroups — #27143749

Order 12416.

226 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 45 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#20260924C212yesmaximal
2#20260912C211yesmaximal, characteristic
3#20260530C211yesmaximal, characteristic, complemented
4#20260525C212yesmaximal, complemented
5#563×C97C211yesmaximal, characteristic, complemented
6#15690855C2211yescharacteristic
7#15690841—21no—
8#15690793—22no—
9#15690738—21no—
10#15690727C2211yescharacteristic
11#15690725C2212yes—
12#15690724—22no—
13C4⋊Q8×C97—21no—
14#126×C97—22no—
15#125×C97—21no—
16#124×C97C2211yescharacteristic
17#123×C97C2212yes—
18C22⋊Q8×C97—22no—
19#13095511—21no—
20#13095510—41no—
21C2×C22⋊D97—22no—
22D4×D97—41no—
23#13095501—41no—
24#13095498—41no—
25#13095495—21no—
26#13095490—42no—
27C22⋊Dic97—22no—
28#13095488—42no—
29C2×Q8×C97—21no—
30C2×Q8×C97D411yescharacteristic
31C2×D4×C97C2311yescharacteristic
32C22×C4×C97D412yes—
33C4⋊C4×C97—21no—
34C4⋊C4×C97—22no—
35C4⋊C4×C97—22no—
36C4⋊C4×C97—21no—
37C4⋊C4×C97—22no—
38C22⋊C4×C97—22no—
39C22⋊C4×C97—22no—
40C22⋊C4×C97D412yes—
41C42×C97—21no—
42C42×C97D411yescharacteristic
43C23×C97C2×D412yes—
44C22×D97—22no—
45Q8×C97—41no—
46Q8×C97—81no—
47D4×C97—41no—
48C2×C4×C97—21no—
49C2×C4×C97—22no—
50C2×C4×C97—21no—
51C2×C4×C97—22no—
52C2×C4×C97—42no—
53C2×C4×C97C2×D411yescharacteristic
54C22⋊D97—42no—
55C22⋊D97—41no—
56C2×Dic97—22no—
57C2×Dic97—42no—
58D388—21no—
59C4×D97—41no—
60Dic194—21no—
61#11715895—41no—
62C22×C97—22no—
63C22×C97—42no—
64C22×C97#12011yescharacteristic
65C2×D97—41no—
66C2×D97—82no—
67C4×C97—21no—
68C4×C97—42no—
69C4×C97—41no—
70C4×C97—42no—
71C4×C97—41no—
72Dic97—41no—
73Dic97—82no—
74C2×C97—21no—
75C2×C97—42no—
76C2×C97B(4,𝔽2)11yescharacteristic
77D97—81no—
78#2201—971nomaximal, Sylow
79C97#220111yescharacteristic, complemented, Sylow
80#563D9711yescharacteristic, complemented
81#458—972no—
82#455—971no—
83#354—971no—
84#352—972no—
85H(5,𝔽2)—971no—
86#137—1941no—
87C4⋊Q8—21no—
88#126—22no—
89#125—21no—
90#124C2×D9711yescharacteristic
91#123—1942no—
92#123C2×D9712yes—
93C22⋊Q8—22no—
94#104—1941no—
95#100—971no—
96#99—972no—
97#99—1942no—
98#41—3881no—
99#41—1941no—
100C2×Q8—21no—
101C2×Q8C22⋊D9711yescharacteristic
102C2×D4—3881no—
103C2×D4—1942no—
104C2×D4C22×D9711yescharacteristic
105C22×C4C22⋊D9712yes—
106Q16—3881no—
107SD16—3881no—
108M16—1941no—
109C4⋊C4—21no—
110C4⋊C4—22no—
111C4⋊C4—22no—
112C4⋊C4—21no—
113C4⋊C4—22no—
114C4⋊C4—3882no—
115C22⋊C4—22no—
116C22⋊C4—22no—
117C22⋊C4—3882no—
118C22⋊C4—1942no—
119C22⋊C4C22⋊D9712yes—
120C42—21no—
121C42C22⋊D9711yescharacteristic
122C23—1942no—
123C23C2×C22⋊D9712yes—
124Q8—41no—
125Q8—81no—
126Q8—1941no—
127D4—41no—
128D4—3882no—
129D4—3881no—
130D4—1941no—
131C2×C4—21no—
132C2×C4—22no—
133C2×C4—21no—
134C2×C4—22no—
135C2×C4—42no—
136C2×C4—3881no—
137C2×C4—3882no—
138C2×C4—1942no—
139C2×C4C2×C22⋊D9711yescharacteristic
140C8—3881no—
141C22—22no—
142C22—42no—
143C22—7762no—
144C22—3881no—
145C22#1569077911yescharacteristic
146C4—21no—
147C4—42no—
148C4—41no—
149C4—42no—
150C4—41no—
151C4—7762no—
152C4—3881no—
153C2—21no—
154C2—42no—
155C2—7761no—
156C2#2026092211yescharacteristic
subgroupG/H
centerC2#20260922
commutator, cosocleC2×D4×C97C23
FrattiniC2×D4C22×D97
Fitting#563×C97C2
socleC2×C97B(4,𝔽2)
prime radical#15690855C22
hypercenter#563D97
hypocenterC97#2201
LCSG ▷ C2×D4×C97 ▷ C22×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #563

Complements

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

NH ≅ G/Ntype
3. #20260530C2semidirect
4. #20260525C2semidirect
5. #563×C97C2semidirect
79. C97#2201semidirect
80. #563D97semidirect

Maximal subgroups

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

subgroupindex
#202609242
#202609122
#202605302
#202605252
#563×C972
#220197