Finite Group Databasebeta

← #27142154

Subgroups — #27142154

Order 12416.

342 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 89 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#20261242C211yesmaximal, characteristic
2#20260719C211yesmaximal, characteristic, complemented
3#20260698C212yesmaximal, complemented
4#20260506C212yesmaximal, complemented
5#568×C97C211yesmaximal, characteristic, complemented
6C22×C4×D97C2211yescharacteristic
7C2×#13095495C2212yes—
8#15690791C412yes—
9#15690786—21no—
10#15690781—21no—
11#15690778C412yes—
12C4×C22⋊D97—22no—
13#15690769C2211yescharacteristic
14#15690729—21no—
15#15690727—21no—
16#15690722—24no—
17C2×#41×C97C2211yescharacteristic
18#131×C97—24no—
19C2×M16×C97—22no—
20C2×M16×C97C2212yes—
21C23×D97C2×C411yescharacteristic
22C2×C22⋊D97—22no—
23C22×Dic97C2×C411yescharacteristic
24C2×C4×D97—21no—
25C2×C4×D97—21no—
26C2×C4×D97—22no—
27C2×C4×D97—22no—
28#13095495—22no—
29#13095495—22no—
30C2×#11715895—22no—
31C4⋊Dic97—21no—
32#13095490—22no—
33C22⋊Dic97—22no—
34C22⋊Dic97C2×C412yes—
35#13095488—22no—
36C4×Dic97—21no—
37#41×C97—22no—
38#41×C97—22no—
39C2×Q8×C97D411yescharacteristic
40C2×D4×C97C2×C412yes—
41C2×D4×C97D411yescharacteristic
42C22×C4×C97D412yes—
43C22×C4×C97C2311yescharacteristic
44M16×C97—24no—
45M16×C97—22no—
46M16×C97—22no—
47C2×C8×C97—24no—
48C2×C8×C97D414yescomplemented
49C23×C97C22⋊C412yes—
50C23×C97C22×C411yescharacteristic
51C22×D97—21no—
52C22×D97—22no—
53C22×D97—42no—
54Q8×C97—22no—
55D4×C97—24no—
56D4×C97—22no—
57C2×C4×C97—22no—
58C2×C4×C97—22no—
59C2×C4×C97C22⋊C412yes—
60C2×C4×C97C2×D412yes—
61C2×C4×C97C2×D411yescharacteristic
62C2×C4×C97C2×D411yescharacteristic
63C2×C4×C97#4112yes—
64C22⋊D97—44no—
65C2×Dic97—22no—
66C2×Dic97—21no—
67C2×Dic97—22no—
68C4×D97—41no—
69C4×D97—42no—
70C4×D97—41no—
71C8×C97—24no—
72#11715895—42no—
73C22×C97—22no—
74C22×C97—41no—
75C22×C97—42no—
76C22×C97C2×C22⋊C411yescharacteristic
77C22×C97C4×D412yes—
78C2×D97—22no—
79C2×D97—24no—
80C2×D97—41no—
81C2×D97—42no—
82C2×D97—41no—
83C4×C97—22no—
84C4×C97#12011yescharacteristic
85C4×C97#12112yes—
86C4×C97#12311yescharacteristic
87Dic97—42no—
88Dic97—41no—
89C2×C97—21no—
90C2×C97—22no—
91C2×C97—42no—
92C2×C97#38611yescharacteristic
93D97—42no—
94#1891—971nomaximal, Sylow
95C97#189111yescharacteristic, complemented, Sylow
96#568D9711yescharacteristic, complemented
97#525—971no—
98#413—971no—
99#407—972no—
100#344—972no—
101C2×#41C2×D9711yescharacteristic
102C23×C4—971no—
103#131—24no—
104C2×M16—22no—
105C2×M16—972no—
106C2×M16C2×D9712yes—
107#123—972no—
108C22⋊Q8—1941no—
109#121—1941no—
110#120—972no—
111C4×D4—1942no—
112#117—971no—
113#101—1941no—
114#100—1941no—
115C22⋊C8—1944no—
116C24—971no—
117#41—22no—
118#41—22no—
119C2×Q8D38811yescharacteristic
120C2×D4—1942no—
121C2×D4C4×D9712yes—
122C2×D4D38811yescharacteristic
123C22×C4—971no—
124C22×C4—1942no—
125C22×C4—1941no—
126C22×C4—1941no—
127C22×C4—1942no—
128C22×C4C22⋊D9712yes—
129C22×C4C22×D9711yescharacteristic
130M16—24no—
131M16—22no—
132M16—22no—
133M16—1942no—
134M16—1942no—
135C2×C8—24no—
136C2×C8—1942no—
137C2×C8C22⋊D9714yescomplemented
138C4⋊C4—1942no—
139C4⋊C4—1941no—
140C22⋊C4—972no—
141C22⋊C4—1942no—
142C22⋊C4—1942no—
143C42—1941no—
144C23—3882no—
145C23—1941no—
146C23—1942no—
147C23#1309548812yes—
148C23C2×C4×D9711yescharacteristic
149Q8—22no—
150D4—24no—
151D4—22no—
152D4—3884no—
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—3881no—
156C2×C4—3882no—
157C2×C4—3881no—
158C2×C4—1942no—
159C2×C4—1941no—
160C2×C4—1942no—
161C2×C4#1309548812yes—
162C2×C4C2×C22⋊D9712yes—
163C2×C4C2×C22⋊D9711yescharacteristic
164C2×C4C2×D38811yescharacteristic
165C2×C4#1309551112yes—
166C8—24no—
167C8—3882no—
168C22—22no—
169C22—41no—
170C22—42no—
171C22—3881no—
172C22—3882no—
173C22—3881no—
174C22—1942no—
175C22—1944no—
176C22C2×#1309548811yescharacteristic
177C22C4×C22⋊D9712yes—
178C4—22no—
179C4—3882no—
180C4—3881no—
181C4#1569077911yescharacteristic
182C4#1569078412yes—
183C4#1569079311yescharacteristic
184C2—21no—
185C2—22no—
186C2—42no—
187C2—3882no—
188C2#2026061411yescharacteristic
subgroupG/H
centerC4#15690779
commutatorC23×C97C22×C4
FrattiniC22×C4C22×D97
Fitting#568×C97C2
socleC2×C97#386
prime radical#20261242C2
cosocleC22×C4×C97C23
hypercenter#568D97
hypocenterC97#1891
LCSG ▷ C23×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C4 ◁ C22×C4 ◁ #568

Complements

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

NH ≅ G/Ntype
2. #20260719C2semidirect
3. #20260698C2semidirect
4. #20260506C2semidirect
5. #568×C97C2semidirect
48. C2×C8×C97D4semidirect
95. C97#1891semidirect
96. #568D97semidirect
137. C2×C8C22⋊D97semidirect

Maximal subgroups

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

subgroupindex
#202612422
#202607192
#202606982
#202605062
#568×C972
#189197