Finite Group Databasebeta

← #27304455

Subgroups — #27304455

Order 13568.

312 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 79 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×#16810422C212yesmaximal
2#20272888C214yesmaximal, complemented
3C53×#3139C211yesmaximal, characteristic
4C2×#13133136—22no—
5#16810422—22no—
6#16810422C2214yes—
7C53×#533C411yescharacteristic
8C2×C4⋊Q8×C53C2211yescharacteristic
9C53×#491C412yes—
10C53×#491C2212yes—
11C53×#487—22no—
12C53×#419—22no—
13C4×M16×C53C411yescharacteristic
14C2×#11719917—42no—
15#13133136—24no—
16#13133136—42no—
17C22×Q8×C53D412yes—
18C2×M16×C53—21no—
19C4⋊Q8×C53—21no—
20C4⋊Q8×C53—41no—
21C4⋊Q8×C53C2×C411yescharacteristic
22C4⋊Q8×C53D412yes—
23C4⋊Q8×C53C2311yescharacteristic
24#125×C53C2×C412yes—
25C22⋊Q8×C53—41no—
26C4×Q8×C53—41no—
27#117×C53—21no—
28C2×C4⋊C4×C53—41no—
29C2×C42×C53C2×C411yescharacteristic
30#103×C53—24no—
31#103×C53—24no—
32#97×C53C2×C412yes—
33C4×C8×C53—21no—
34C2×Q8×C53—22no—
35C2×Q8×C53—41no—
36C2×Q8×C53—42no—
37C2×Q8×C53—42no—
38C2×Q8×C53C22⋊C412yes—
39C2×Q8×C53C2×D412yes—
40C22×C4×C53—21no—
41C22×C4×C53—22no—
42C22×C4×C53C22⋊C411yescharacteristic
43M16×C53—42no—
44C2×C8×C53—22no—
45C4⋊C4×C53—22no—
46C4⋊C4×C53—41no—
47C4⋊C4×C53—42no—
48C4⋊C4×C53—41no—
49C4⋊C4×C53—41no—
50C4⋊C4×C53—42no—
51C22⋊C4×C53—41no—
52C42×C53—21no—
53C42×C53—41no—
54C42×C53C22⋊C411yescharacteristic
55C42×C53C22×C411yescharacteristic
56#11719917—44no—
57#11719917—82no—
58C2×#88322—82no—
59C23×C53#9911yescharacteristic
60Q8×C53—42no—
61Q8×C53—42no—
62Q8×C53—82no—
63Q8×C53—82no—
64C2×C4×C53—22no—
65C2×C4×C53—21no—
66C2×C4×C53—22no—
67C2×C4×C53—22no—
68C2×C4×C53—21no—
69C2×C4×C53—42no—
70C2×C4×C53—42no—
71C2×C4×C53—42no—
72C2×C4×C53—42no—
73C2×C4×C53#9911yescharacteristic
74C2×C4×C53C2×C22⋊C411yescharacteristic
75C8×C53—42no—
76#88322—84no—
77#13599—531nomaximal, Sylow
78C22×C53—41no—
79C22×C53#35412yes—
80C22×C53C2×#9911yescharacteristic
81C4×C53—22no—
82C4×C53—21no—
83C4×C53—21no—
84C4×C53—41no—
85C4×C53—42no—
86C4×C53—42no—
87C4×C53—82no—
88#3139D5311yescharacteristic
89C2×#356—532no—
90#1413—534no—
91C2×C53—21no—
92C2×C53—22no—
93C2×C53C2×#35411yescharacteristic
94#533Dic5311yescharacteristic
95C2×C4⋊Q8C2×D5311yescharacteristic
96#491Dic5312yes—
97#491C2×D5312yes—
98#487—22no—
99#419—22no—
100C2×#101—1062no—
101C4×M16Dic5311yescharacteristic
102#356—534no—
103#356—1062no—
104C53#1359911yescharacteristic, complemented, Sylow
105C22×Q8C22⋊D5312yes—
106C2×M16—21no—
107C2×M16—2122no—
108C4⋊Q8—21no—
109C4⋊Q8—41no—
110C4⋊Q8C2×Dic5311yescharacteristic
111C4⋊Q8C22⋊D5312yes—
112C4⋊Q8C22×D5311yescharacteristic
113#125C2×Dic5312yes—
114C22⋊Q8—41no—
115C4×Q8—41no—
116#117—21no—
117C2×C4⋊C4—41no—
118C2×C42C2×Dic5311yescharacteristic
119#103—24no—
120#103—24no—
121#101—1064no—
122#101—2122no—
123#97C2×Dic5312yes—
124C4×C8—21no—
125C2×Q8—22no—
126C2×Q8—41no—
127C2×Q8—42no—
128C2×Q8—42no—
129C2×Q8C22⋊Dic5312yes—
130C2×Q8C2×C22⋊D5312yes—
131C22×C4—21no—
132C22×C4—22no—
133C22×C4C22⋊Dic5311yescharacteristic
134M16—42no—
135M16—4242no—
136M16—2124no—
137C2×C8—22no—
138C2×C8—4242no—
139C4⋊C4—22no—
140C4⋊C4—41no—
141C4⋊C4—42no—
142C4⋊C4—41no—
143C4⋊C4—42no—
144C4⋊C4—41no—
145C22⋊C4—41no—
146C42—21no—
147C42—41no—
148C42C22⋊Dic5311yescharacteristic
149C42C22×Dic5311yescharacteristic
150C23#1313313211yescharacteristic
151Q8—42no—
152Q8—42no—
153Q8—82no—
154Q8—82no—
155C2×C4—21no—
156C2×C4—22no—
157C2×C4—21no—
158C2×C4—22no—
159C2×C4—22no—
160C2×C4—42no—
161C2×C4—42no—
162C2×C4—42no—
163C2×C4—42no—
164C2×C4#1313313211yescharacteristic
165C2×C4C2×C22⋊Dic5311yescharacteristic
166C8—42no—
167C8—4244no—
168C22—41no—
169C22#1681041812yes—
170C22C2×#1313313211yescharacteristic
171C4—22no—
172C4—21no—
173C4—21no—
174C4—41no—
175C4—42no—
176C4—42no—
177C4—82no—
178C2—21no—
179C2—22no—
180C2C2×#1681041811yescharacteristic
subgroupG/H
centerC2C2×#16810418
commutatorC42×C53C22×C4
FrattiniC4⋊Q8C22×D53
FittingC53×#3139C2
socleC2×C53C2×#354
prime radicalC23×C53#99
cosocleC4⋊Q8×C53C23
hypercenter#3139D53
hypocenterC53#13599
LCSG ▷ C42×C53 ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C23 ◁ C22×C4 ◁ C2×C4⋊Q8 ◁ #3139

Complements

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

NH ≅ G/Ntype
2. #20272888C2semidirect
104. C53#13599semidirect

Maximal subgroups

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

subgroupindex
C2×#168104222
#202728882
C53×#31392
#1359953