Finite Group Databasebeta

← #27140739

Subgroups — #27140739

Order 12416.

254 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 57 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#20261046C212yesmaximal, complemented
2#20260665C211yesmaximal, characteristic, complemented
3#20260469C212yesmaximal, complemented
4#20260461C211yesmaximal, characteristic, complemented
5#537×C97C211yesmaximal, characteristic
6C2×C4⋊Dic97C2211yescharacteristic
7C2×#13095490—21no—
8#15690743—22no—
9#15690731—22no—
10C22⋊#11715895C2212yes—
11#15690717—21no—
12#15690717C2211yescharacteristic
13C2×#41×C97—21no—
14C22⋊Q8×C97—22no—
15#121×C97—22no—
16#121×C97—21no—
17#121×C97C2212yes—
18C2×C4⋊C4×C97C2211yescharacteristic
19C22×Dic97—21no—
20C22×Dic97—21no—
21C2×#11715895—22no—
22C4⋊Dic97—22no—
23#13095490—42no—
24#41×C97—42no—
25#41×C97—42no—
26C2×Q8×C97—21no—
27C2×D4×C97—21no—
28C2×D4×C97—22no—
29C2×D4×C97—21no—
30C2×D4×C97D412yes—
31C22×C4×C97—22no—
32C22×C4×C97D411yescharacteristic
33C22×C4×C97D411yescharacteristic
34C22×C4×C97C2311yescharacteristic
35C4⋊C4×C97—21no—
36C4⋊C4×C97—22no—
37C4⋊C4×C97D412yes—
38C22⋊C4×C97—22no—
39C22⋊C4×C97—22no—
40C23×C97—22no—
41C23×C97C2×D411yescharacteristic
42Q8×C97—42no—
43D4×C97—42no—
44D4×C97—42no—
45D4×C97—42no—
46D4×C97—42no—
47D4×C97—81no—
48C2×C4×C97—21no—
49C2×C4×C97—21no—
50C2×C4×C97—21no—
51C2×C4×C97—22no—
52C2×C4×C97—41no—
53C2×C4×C97—44no—
54C2×C4×C97—41no—
55C2×C4×C97D812yes—
56C2×C4×C97C2×D412yes—
57C2×Dic97—22no—
58C2×Dic97—41no—
59C2×Dic97—41no—
60C2×Dic97—41no—
61C2×Dic97—41no—
62C2×Dic97—41no—
63#11715895—42no—
64C22×C97—21no—
65C22×C97—21no—
66C22×C97—42no—
67C22×C97—42no—
68C22×C97—42no—
69C22×C97#12011yescharacteristic
70C22×C97C2×D811yescharacteristic
71C22×C97AGL(1,ℤ/8)11yescharacteristic
72C4×C97—22no—
73C4×C97—22no—
74C4×C97—42no—
75C4×C97—41no—
76Dic97—42no—
77Dic97—81no—
78C2×C97—21no—
79C2×C97—21no—
80C2×C97—82no—
81C2×C97#44711yescharacteristic
82C2×C97#45611yescharacteristic
83C2×C97#45811yescharacteristic
84#1610—971nomaximal, Sylow
85C97#161011yescharacteristic, complemented, Sylow
86#537D9711yescharacteristic
87#480—972no—
88#398—971no—
89#327—972no—
90#324—971no—
91C2×#41—21no—
92C22⋊Q8—22no—
93#121—22no—
94#121—21no—
95#121C2×D9712yes—
96C2×C4⋊C4—971no—
97C2×C4⋊C4—1941no—
98C2×C4⋊C4C2×D9711yescharacteristic
99#107—1942no—
100#102—1942no—
101C22⋊C8—972no—
102#95—971no—
103#95—1941no—
104#41—42no—
105#41—42no—
106C2×Q8—21no—
107C2×D4—21no—
108C2×D4—22no—
109C2×D4—21no—
110C2×D4C22⋊D9712yes—
111C22×C4—22no—
112C22×C4—1941no—
113C22×C4—1941no—
114C22×C4C22⋊D9711yescharacteristic
115C22×C4C22⋊D9711yescharacteristic
116C22×C4C22×D9711yescharacteristic
117C2×C8—1942no—
118C4⋊C4—21no—
119C4⋊C4—22no—
120C4⋊C4—3882no—
121C4⋊C4—1942no—
122C4⋊C4C22⋊D9712yes—
123C22⋊C4—22no—
124C22⋊C4—22no—
125C23—22no—
126C23C2×C22⋊D9711yescharacteristic
127Q8—42no—
128D4—42no—
129D4—42no—
130D4—42no—
131D4—42no—
132D4—81no—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—21no—
136C2×C4—22no—
137C2×C4—41no—
138C2×C4—44no—
139C2×C4—41no—
140C2×C4—3881no—
141C2×C4—3881no—
142C2×C4—3881no—
143C2×C4—3881no—
144C2×C4—3881no—
145C2×C4—1942no—
146C2×C4#1309549612yes—
147C2×C4C2×C22⋊D9712yes—
148C8—3882no—
149C22—21no—
150C22—21no—
151C22—42no—
152C22—42no—
153C22—42no—
154C22#1569077911yescharacteristic
155C22C2×#1309549611yescharacteristic
156C22#1569083611yescharacteristic
157C4—22no—
158C4—22no—
159C4—42no—
160C4—41no—
161C4—7761no—
162C4—3882no—
163C2—21no—
164C2—21no—
165C2—82no—
166C2#2026086911yescharacteristic
167C2#2026091811yescharacteristic
168C2#2026092511yescharacteristic
subgroupG/H
centerC22#15690779
commutator, cosocleC22×C4×C97C23
FrattiniC22×C4C22×D97
Fitting, prime radical#537×C97C2
socleC22×C97#120
hypercenter#537D97
hypocenterC97#1610
LCSG ▷ C22×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #537

Complements

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

NH ≅ G/Ntype
1. #20261046C2semidirect
2. #20260665C2semidirect
3. #20260469C2semidirect
4. #20260461C2semidirect
85. C97#1610semidirect

Maximal subgroups

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

subgroupindex
#202610462
#202606652
#202604692
#202604612
#537×C972
#161097