Finite Group Databasebeta

← #27143712

Subgroups — #27143712

Order 12416.

206 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 49 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#20261552C211yesmaximal, characteristic
2#20261176C211yesmaximal, characteristic, complemented
3C2×#15690756C211yesmaximal, characteristic, complemented
4#20260550C211yesmaximal, characteristic, complemented
5#20260541C211yesmaximal, characteristic, complemented
6#20260519C211yesmaximal, characteristic, complemented
7#473×C97C211yesmaximal, characteristic, complemented
8#15690857—21no—
9#15690843—22no—
10#15690843—21no—
11#15690832—21no—
12#15690832C2211yescharacteristic
13C2×Dic388—21no—
14C2×Dic388C2211yescharacteristic
15#15690756—22no—
16#15690756—21no—
17#15690753—21no—
18#15690750C2211yescharacteristic
19C2×#13095486C2211yescharacteristic
20#137×C97—21no—
21C2×Q16×C97C2211yescharacteristic
22#131×C97C2211yescharacteristic
23#108×C97C2211yescharacteristic
24#101×C97—21no—
25#13095511—22no—
26#13095511—21no—
27#13095510—42no—
28#13095510—41no—
29C2×Dic194—22no—
30C2×Dic194—21no—
31Q8×D97—41no—
32Dic388—22no—
33Dic388—21no—
34Dic388D411yescharacteristic
35Dic388D412yes—
36#13095500—22no—
37#13095500—21no—
38D776D411yescharacteristic
39#13095486—21no—
40#13095486—21no—
41#41×C97D411yescharacteristic
42C2×Q8×C97—21no—
43Q16×C97—21no—
44Q16×C97—21no—
45Q16×C97—41no—
46SD16×C97—41no—
47M16×C97—21no—
48M16×C97—21no—
49M16×C97D411yescharacteristic
50C2×C8×C97—21no—
51C2×C8×C97C2311yescharacteristic
52Q8×C97—41no—
53Q8×C97—41no—
54Q8×C97D811yescharacteristic
55D4×C97—21no—
56D4×C97D811yescharacteristic
57C2×C4×C97—21no—
58C2×C4×C97—41no—
59C2×C4×C97C2×D411yescharacteristic
60C22⋊D97—42no—
61C22⋊D97—41no—
62C2×Dic97—42no—
63C2×Dic97—41no—
64D388—21no—
65C4×D97—42no—
66C4×D97—41no—
67Dic194—22no—
68Dic194—21no—
69Dic194—42no—
70Dic194—41no—
71C8×C97—21no—
72C8×C97—41no—
73C8×C97C2×D411yescharacteristic
74C8×C97C2×D411yescharacteristic
75C22×C97—21no—
76C22×C97AGL(1,ℤ/8)11yescharacteristic
77C2×D97—41no—
78C4×C97—21no—
79C4×C97—81no—
80C4×C97#12011yescharacteristic
81C4×C97C2×D811yescharacteristic
82Dic97—41no—
83Dic97—42no—
84C2×C97—21no—
85C2×C97—41no—
86C2×C97#44711yescharacteristic
87D97—81no—
88#2195—971nomaximal, Sylow
89C97#219511yescharacteristic, complemented, Sylow
90#578—971no—
91#510—971no—
92C2×Q32—971no—
93#473D9711yescharacteristic, complemented
94#362—971no—
95#359—971no—
96#350—971no—
972−1+4—1941no—
98#137—21no—
99#137—1942no—
100#137—1941no—
101#135—971no—
102#135—1941no—
103C2×Q16—971no—
104C2×Q16—1941no—
105C2×Q16C2×D9711yescharacteristic
106#131C2×D9711yescharacteristic
107Q32—1942no—
108Q32—1941no—
109SD32—1941no—
110M32—971no—
111C2×C16—971no—
112#108C2×D9711yescharacteristic
113#101—21no—
114#41—3882no—
115#41—3881no—
116#41—1942no—
117#41—1941no—
118#41C22⋊D9711yescharacteristic
119C2×Q8—21no—
120C2×Q8—3881no—
121C2×Q8—1942no—
122C2×Q8—1941no—
123Q16—21no—
124Q16—21no—
125Q16—41no—
126Q16—972no—
127Q16—971no—
128Q16—1942no—
129Q16—1941no—
130SD16—41no—
131SD16—1942no—
132SD16—1941no—
133D8—971no—
134M16—21no—
135M16—21no—
136M16C22⋊D9711yescharacteristic
137C2×C8—21no—
138C2×C8C22×D9711yescharacteristic
139C16—1941no—
140C16—1941no—
141Q8—41no—
142Q8—41no—
143Q8—3882no—
144Q8—3881no—
145Q8—1942no—
146Q8—1941no—
147Q8#1309549611yescharacteristic
148D4—21no—
149D4—3882no—
150D4—3881no—
151D4—1941no—
152D4#1309549611yescharacteristic
153C2×C4—21no—
154C2×C4—41no—
155C2×C4—3882no—
156C2×C4—3882no—
157C2×C4—3881no—
158C2×C4—3881no—
159C2×C4C2×C22⋊D9711yescharacteristic
160C8—21no—
161C8—41no—
162C8D4×D9711yescharacteristic
163C8D4×D9711yescharacteristic
164C22—21no—
165C22—3881no—
166C22#1569083511yescharacteristic
167C4—21no—
168C4—81no—
169C4—3882no—
170C4—3881no—
171C4#1569077811yescharacteristic
172C4C2×#1309549611yescharacteristic
173C2—21no—
174C2—41no—
175C2—7761no—
176C2#2026086811yescharacteristic
subgroupG/H
centerC2#20260868
commutator, cosocleC2×C8×C97C23
FrattiniC2×C8C22×D97
Fitting#473×C97C2
socleC2×C97#447
prime radical#20261552C2
hypercenter#473D97
hypocenterC97#2195
LCSG ▷ C2×C8×C97 ▷ C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C2×C4 ◁ #131 ◁ #473

Complements

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

NH ≅ G/Ntype
2. #20261176C2semidirect
3. C2×#15690756C2semidirect
4. #20260550C2semidirect
5. #20260541C2semidirect
6. #20260519C2semidirect
7. #473×C97C2semidirect
89. C97#2195semidirect
93. #473D97semidirect

Maximal subgroups

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

subgroupindex
#202615522
#202611762
C2×#156907562
#202605502
#202605412
#202605192
#473×C972
#219597