Finite Group Databasebeta

← #27143714

Subgroups — #27143714

Order 12416.

206 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 53 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#20261175C211yesmaximal, characteristic, complemented
2C2×#15690756C211yesmaximal, characteristic, complemented
3#20261010C211yesmaximal, characteristic, complemented
4#20260551C211yesmaximal, characteristic, complemented
5#20260542C211yesmaximal, characteristic, complemented
6#20260519C211yesmaximal, characteristic, complemented
7#578×C97C211yesmaximal, characteristic
8#15690840—21no—
9C2×Dic388C2211yescharacteristic
10#15690756—22no—
11#15690756—21no—
12#15690754—21no—
13#15690750C2211yescharacteristic
14C2×#13095486C2211yescharacteristic
15#15690746C2211yescharacteristic
16#15690728—21no—
172−1+4×C97—21no—
18#137×C97—22no—
19#137×C97—21no—
20#135×C97—21no—
21#135×C97C2211yescharacteristic
22C2×Q16×C97—21no—
23C2×Q16×C97C2211yescharacteristic
24#131×C97C2211yescharacteristic
25C2×Dic194—21no—
26Dic388—21no—
27Dic388—21no—
28#13095501—41no—
29#13095499—41no—
30#13095495—21no—
31#13095486—21no—
32#13095486—21no—
33#41×C97—22no—
34#41×C97—21no—
35#41×C97—42no—
36#41×C97—41no—
37#41×C97D411yescharacteristic
38C2×Q8×C97—22no—
39C2×Q8×C97—21no—
40C2×Q8×C97—41no—
41Q16×C97—22no—
42Q16×C97—21no—
43Q16×C97D411yescharacteristic
44Q16×C97D412yes—
45SD16×C97—22no—
46SD16×C97—21no—
47D8×C97D411yescharacteristic
48M16×C97—21no—
49M16×C97D411yescharacteristic
50C2×C8×C97—21no—
51C2×C8×C97C2311yescharacteristic
52Q8×C97—22no—
53Q8×C97—21no—
54Q8×C97—42no—
55Q8×C97—41no—
56Q8×C97D811yescharacteristic
57D4×C97—21no—
58D4×C97—21no—
59D4×C97—42no—
60D4×C97—41no—
61D4×C97D811yescharacteristic
62C2×C4×C97—21no—
63C2×C4×C97—42no—
64C2×C4×C97—42no—
65C2×C4×C97—41no—
66C2×C4×C97—41no—
67C2×C4×C97C2×D411yescharacteristic
68C2×Dic97—41no—
69Dic194—41no—
70Dic194—41no—
71C8×C97—21no—
72C8×C97C2×D411yescharacteristic
73C8×C97C2×D411yescharacteristic
74#11715895—41no—
75C22×C97—21no—
76C22×C97—41no—
77C22×C97AGL(1,ℤ/8)11yescharacteristic
78C4×C97—21no—
79C4×C97—42no—
80C4×C97—41no—
81C4×C97#12011yescharacteristic
82C4×C97C2×D811yescharacteristic
83Dic97—81no—
84C2×C97—21no—
85C2×C97—41no—
86C2×C97—81no—
87C2×C97#44711yescharacteristic
88#2195—971nomaximal, Sylow
89C97#219511yescharacteristic, complemented, Sylow
90#578D9711yescharacteristic
91#510—971no—
92C2×Q32—971no—
93#473—971no—
94#362—971no—
95#359—971no—
96#350—971no—
972−1+4—21no—
98#137—22no—
99#137—21no—
100#137—1941no—
101#135—21no—
102#135C2×D9711yescharacteristic
103C2×Q16—21no—
104C2×Q16—971no—
105C2×Q16C2×D9711yescharacteristic
106#131C2×D9711yescharacteristic
107Q32—1942no—
108Q32—1941no—
109SD32—1941no—
110M32—971no—
111C2×C16—971no—
112#108—971no—
113#101—1941no—
114#41—22no—
115#41—21no—
116#41—42no—
117#41—41no—
118#41C22⋊D9711yescharacteristic
119C2×Q8—22no—
120C2×Q8—21no—
121C2×Q8—41no—
122C2×Q8—1941no—
123Q16—22no—
124Q16—21no—
125Q16—3881no—
126Q16—1941no—
127Q16—1941no—
128Q16C22⋊D9711yescharacteristic
129Q16C22⋊D9712yes—
130SD16—22no—
131SD16—21no—
132SD16—3881no—
133D8C22⋊D9711yescharacteristic
134M16—21no—
135M16—1941no—
136M16C22⋊D9711yescharacteristic
137C2×C8—21no—
138C2×C8C22×D9711yescharacteristic
139C16—1941no—
140C16—1941no—
141Q8—22no—
142Q8—21no—
143Q8—42no—
144Q8—41no—
145Q8—3881no—
146Q8—3881no—
147Q8#1309549611yescharacteristic
148D4—21no—
149D4—21no—
150D4—42no—
151D4—41no—
152D4#1309549611yescharacteristic
153C2×C4—21no—
154C2×C4—42no—
155C2×C4—42no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—3881no—
159C2×C4C2×C22⋊D9711yescharacteristic
160C8—21no—
161C8—3881no—
162C8C2×C22⋊D9711yescharacteristic
163C8C2×C22⋊D9711yescharacteristic
164C22—21no—
165C22—41no—
166C22#1569083611yescharacteristic
167C4—21no—
168C4—42no—
169C4—41no—
170C4—7761no—
171C4#1569077911yescharacteristic
172C4C2×#1309549611yescharacteristic
173C2—21no—
174C2—41no—
175C2—81no—
176C2#2026086911yescharacteristic
subgroupG/H
centerC2#20260869
commutator, cosocleC2×C8×C97C23
FrattiniC2×C8C22×D97
Fitting, prime radical#578×C97C2
socleC2×C97#447
hypercenter#578D97
hypocenterC97#2195
LCSG ▷ C2×C8×C97 ▷ C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C2×C4 ◁ #131 ◁ #578

Complements

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

NH ≅ G/Ntype
1. #20261175C2semidirect
2. C2×#15690756C2semidirect
3. #20261010C2semidirect
4. #20260551C2semidirect
5. #20260542C2semidirect
6. #20260519C2semidirect
89. C97#2195semidirect

Maximal subgroups

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

subgroupindex
#202611752
C2×#156907562
#202610102
#202605512
#202605422
#202605192
#578×C972
#219597