Finite Group Databasebeta

← #27142957

Subgroups — #27142957

Order 12416.

306 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 71 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
1C2×#15690737C212yesmaximal, complemented
2#20260736C212yesmaximal, complemented
3#20260604C211yesmaximal, characteristic, complemented
4#20260504C211yesmaximal, characteristic, complemented
5#537×C97C211yesmaximal, characteristic
6C2×#13095495C2212yes—
7C2×C4⋊Dic97C2211yescharacteristic
8C2×#13095490—22no—
9C2×C4×Dic97C2211yescharacteristic
10#15690737—24no—
11#15690734—22no—
12#15690731—22no—
13C2×#41×C97C2212yes—
14C22⋊Q8×C97—22no—
15#121×C97—22no—
16#121×C97—22no—
17C2×C4⋊C4×C97C2211yescharacteristic
18C22×Dic97—21no—
19C22×Dic97—21no—
20#13095495—22no—
21#13095495—22no—
22C2×#11715895—22no—
23C4⋊Dic97—22no—
24#13095490—44no—
25C4×Dic97—21no—
26C4×Dic97D412yescomplemented
27#41×C97—22no—
28#41×C97—22no—
29#41×C97—22no—
30#41×C97—22no—
31C2×Q8×C97D412yes—
32C2×D4×C97—22no—
33C2×D4×C97—22no—
34C2×D4×C97D412yes—
35C22×C4×C97—22no—
36C22×C4×C97D412yes—
37C22×C4×C97C2311yescharacteristic
38C4⋊C4×C97—22no—
39C4⋊C4×C97—22no—
40C22⋊C4×C97—24no—
41C23×C97—22no—
42C23×C97C2×D411yescharacteristic
43Q8×C97—22no—
44Q8×C97—22no—
45D4×C97—22no—
46D4×C97—22no—
47D4×C97—42no—
48D4×C97—42no—
49D4×C97—82no—
50C2×C4×C97—22no—
51C2×C4×C97—22no—
52C2×C4×C97—42no—
53C2×C4×C97—42no—
54C2×C4×C97—42no—
55C2×C4×C97C2×D411yescharacteristic
56C2×C4×C97C2×D412yes—
57C2×C4×C97#4112yes—
58C2×C4×C97#4111yescharacteristic
59C2×Dic97—22no—
60C2×Dic97—22no—
61C2×Dic97—22no—
62C2×Dic97—42no—
63C2×Dic97—41no—
64#11715895—42no—
65C22×C97—21no—
66C22×C97—21no—
67C22×C97—42no—
68C22×C97—42no—
69C22×C97—42no—
70C22×C97#12011yescharacteristic
71C22×C97#12111yescharacteristic
72C22×C97#12411yescharacteristic
73C4×C97—42no—
74C4×C97—42no—
75C4×C97#12112yes—
76C4×C97#12312yes—
77Dic97—24no—
78Dic97—81no—
79C2×C97—21no—
80C2×C97—21no—
81C2×C97—82no—
82C2×C97#39411yescharacteristic
83C2×C97#45311yescharacteristic
84C2×C97#45611yescharacteristic
85#2039—971nomaximal, Sylow
86C97#203911yescharacteristic, complemented, Sylow
87#537D9711yescharacteristic
88C2×#104—972no—
89#417—972no—
90#384—971no—
91#343—971no—
92C2×#41C2×D9712yes—
93C2×M16—972no—
94C22⋊Q8—22no—
95#121—22no—
96#121—22no—
97C2×C4⋊C4—971no—
98C2×C4⋊C4—1942no—
99C2×C4⋊C4C2×D9711yescharacteristic
100C2×C42—971no—
101#104—1944no—
102#103—1942no—
103#102—1942no—
104#41—22no—
105#41—22no—
106#41—22no—
107#41—22no—
108C2×Q8C22⋊D9712yes—
109C2×D4—22no—
110C2×D4—22no—
111C2×D4C22⋊D9712yes—
112C22×C4—22no—
113C22×C4—1941no—
114C22×C4—1941no—
115C22×C4C22⋊D9712yes—
116C22×C4C22×D9711yescharacteristic
117M16—1942no—
118M16—1942no—
119C2×C8—1942no—
120C4⋊C4—22no—
121C4⋊C4—22no—
122C4⋊C4—3884no—
123C4⋊C4—1942no—
124C22⋊C4—24no—
125C42—972no—
126C42—1941no—
127C23—22no—
128C23C2×C22⋊D9711yescharacteristic
129Q8—22no—
130Q8—22no—
131D4—22no—
132D4—22no—
133D4—42no—
134D4—42no—
135D4—82no—
136C2×C4—22no—
137C2×C4—22no—
138C2×C4—42no—
139C2×C4—42no—
140C2×C4—42no—
141C2×C4—3882no—
142C2×C4—3881no—
143C2×C4—1942no—
144C2×C4—1942no—
145C2×C4—1942no—
146C2×C4D4×D9711yescharacteristic
147C2×C4C2×C22⋊D9712yes—
148C2×C4#1309551012yes—
149C2×C4#1309551011yescharacteristic
150C8—3882no—
151C22—21no—
152C22—21no—
153C22—42no—
154C22—42no—
155C22—42no—
156C22#1569077911yescharacteristic
157C22#1569078111yescharacteristic
158C22#1569079711yescharacteristic
159C4—42no—
160C4—42no—
161C4—7761no—
162C4—1944no—
163C4#1569078212yes—
164C4#1569079112yes—
165C2—21no—
166C2—21no—
167C2—82no—
168C2#2026064911yescharacteristic
169C2#2026090311yescharacteristic
170C2#2026091811yescharacteristic
subgroupG/H
centerC22#15690779
commutator, cosocleC22×C4×C97C23
FrattiniC22×C4C22×D97
Fitting, prime radical#537×C97C2
socleC22×C97#120
hypercenter#537D97
hypocenterC97#2039
LCSG ▷ C22×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #537

Complements

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

NH ≅ G/Ntype
1. C2×#15690737C2semidirect
2. #20260736C2semidirect
3. #20260604C2semidirect
4. #20260504C2semidirect
26. C4×Dic97D4semidirect
86. C97#2039semidirect

Maximal subgroups

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

subgroupindex
C2×#156907372
#202607362
#202606042
#202605042
#537×C972
#203997