Finite Group Databasebeta

← #27143020

Subgroups — #27143020

Order 12416.

266 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 67 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#20260758C211yesmaximal, characteristic, complemented
2C2×#15690737C212yesmaximal, complemented
3#20260719C212yesmaximal, complemented
4#20260493C211yesmaximal, characteristic, complemented
5#575×C97C211yesmaximal, characteristic
6C2×#13095495C2211yescharacteristic
7C2×#13095495C2212yes—
8C2×C4×Dic97C2211yescharacteristic
9#15690739—22no—
10#15690737—22no—
11#15690737—22no—
12#15690729—22no—
13#15690727—22no—
14C2×#41×C97C2212yes—
15#137×C97—22no—
16AGL(1,ℤ/8)×C97—22no—
17#135×C97—22no—
18C2×M16×C97C2211yescharacteristic
19C22×Dic97—21no—
20#13095495—22no—
21#13095495—22no—
22#13095495—22no—
23C2×#11715895—21no—
24C2×#11715895—22no—
25C4×Dic97—21no—
26C4×Dic97D411yescharacteristic
27C4×Dic97D411yescharacteristic
28#41×C97—22no—
29#41×C97—22no—
30#41×C97—22no—
31#41×C97—22no—
32C2×Q8×C97D412yes—
33C2×D4×C97—22no—
34C2×D4×C97D412yes—
35C22×C4×C97—22no—
36C22×C4×C97C2311yescharacteristic
37Q16×C97—22no—
38SD16×C97—24no—
39D8×C97—22no—
40M16×C97—22no—
41C2×C8×C97D412yes—
42C23×C97—22no—
43C23×C97#4111yescharacteristic
44Q8×C97—22no—
45Q8×C97—22no—
46D4×C97—22no—
47D4×C97—22no—
48D4×C97—42no—
49D4×C97—42no—
50C2×C4×C97—22no—
51C2×C4×C97—42no—
52C2×C4×C97—42no—
53C2×C4×C97C2×D411yescharacteristic
54C2×C4×C97C2×D412yes—
55C2×C4×C97C2×D411yescharacteristic
56C2×C4×C97#4112yes—
57C2×Dic97—21no—
58C2×Dic97—21no—
59C2×Dic97—42no—
60C8×C97—22no—
61#11715895—41no—
62#11715895—42no—
63C22×C97—41no—
64C22×C97—42no—
65C22×C97—82no—
66C22×C97#12111yescharacteristic
67C22×C97#12312yes—
68C4×C97—42no—
69C4×C97#12011yescharacteristic
70C4×C97#12112yes—
71C4×C97#12411yescharacteristic
72Dic97—22no—
73Dic97—41no—
74C2×C97—21no—
75C2×C97—22no—
76C2×C97—82no—
77C2×C97#39411yescharacteristic
78#2050—971nomaximal, Sylow
79C97#205011yescharacteristic, complemented, Sylow
80#575D9711yescharacteristic
81#423—971no—
82C2×#104—972no—
83#413—972no—
84#338—971no—
85C2×#41C2×D9712yes—
86#137—22no—
87AGL(1,ℤ/8)—22no—
88#135—22no—
89C2×M16—972no—
90C2×M16—971no—
91C2×M16C2×D9711yescharacteristic
92C2×C42—971no—
93C4⋊C8—1942no—
94#104—1942no—
95#104—1942no—
96#101—1942no—
97#100—1942no—
98#41—22no—
99#41—22no—
100#41—22no—
101#41—22no—
102C2×Q8C22⋊D9712yes—
103C2×D4—22no—
104C2×D4C22⋊D9712yes—
105C22×C4—22no—
106C22×C4—1941no—
107C22×C4C22×D9711yescharacteristic
108Q16—22no—
109SD16—24no—
110D8—22no—
111M16—22no—
112M16—1942no—
113M16—1942no—
114M16—1942no—
115C2×C8—1941no—
116C2×C8—1942no—
117C2×C8C22⋊D9712yes—
118C42—971no—
119C42—971no—
120C42—1941no—
121C23—22no—
122C23#1309551011yescharacteristic
123Q8—22no—
124Q8—22no—
125D4—22no—
126D4—22no—
127D4—42no—
128D4—42no—
129C2×C4—22no—
130C2×C4—42no—
131C2×C4—42no—
132C2×C4—3882no—
133C2×C4—1941no—
134C2×C4—1941no—
135C2×C4D4×D9711yescharacteristic
136C2×C4C2×C22⋊D9712yes—
137C2×C4C2×C22⋊D9711yescharacteristic
138C2×C4#1309551012yes—
139C8—22no—
140C8—3881no—
141C8—3882no—
142C22—41no—
143C22—42no—
144C22—82no—
145C22#1569078111yescharacteristic
146C22#1569079112yes—
147C4—42no—
148C4—3881no—
149C4—1942no—
150C4#1569077911yescharacteristic
151C4#1569078212yes—
152C4#1569079711yescharacteristic
153C2—21no—
154C2—22no—
155C2—82no—
156C2#2026064911yescharacteristic
subgroupG/H
centerC4#15690797
commutator, cosocleC22×C4×C97C23
FrattiniC22×C4C22×D97
Fitting, prime radical#575×C97C2
socleC2×C97#394
hypercenter#575D97
hypocenterC97#2050
LCSG ▷ C22×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C4 ◁ C22×C4 ◁ #575

Complements

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

NH ≅ G/Ntype
1. #20260758C2semidirect
2. C2×#15690737C2semidirect
3. #20260719C2semidirect
4. #20260493C2semidirect
79. C97#2050semidirect

Maximal subgroups

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

subgroupindex
#202607582
C2×#156907372
#202607192
#202604932
#575×C972
#205097