Finite Group Databasebeta

← #356×D53 #20276394

Subgroups — #356×D53 #20276394

Order 6784.

240 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 57 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
1C4⋊Q8×D53C211yesmaximal, characteristic
2#101×D53C212yesmaximal
3#16810422C211yesmaximal, characteristic, complemented
4#16810421C212yesmaximal, complemented
5C53×#356C211yesmaximal, characteristic, complemented
6C2×Q8×D53C411yescharacteristic
7C2×Q8×D53C2211yescharacteristic
8M16×D53—22no—
9#13133217C411yescharacteristic
10#13133216—22no—
11#13133215C411yescharacteristic
12#13133215C2211yescharacteristic
13C4⋊C4×D53—22no—
14C42×D53C411yescharacteristic
15#13133136C2212yes—
16#13133135—22no—
17C4⋊Q8×C53C2211yescharacteristic
18#101×C53C2212yes—
19C2×Q8×C53C2×C411yescharacteristic
20C2×Q8×C53C2311yescharacteristic
21Q8×D53—41no—
22Q8×D53—41no—
23Q8×D53—41no—
24Q8×D53—41no—
25C2×C4×D53—21no—
26C2×C4×D53—21no—
27C2×C4×D53C2×C411yescharacteristic
28C2×C4×D53D412yes—
29C2×Dic106—21no—
30C2×Dic106C2×C411yescharacteristic
31C2×Dic106C2×C411yescharacteristic
32C2×Dic106D412yes—
33M16×C53—22no—
34C4⋊C4×C53—22no—
35C42×C53C2×C411yescharacteristic
36C4⋊Dic53—22no—
37#11719919—22no—
38#11719919—22no—
39C4×Dic53—21no—
40C4×Dic53C2×C411yescharacteristic
41#11719917—22no—
42#11719912—42no—
43C8×D53—42no—
44C22×D53C22⋊C411yescharacteristic
45Q8×C53—41no—
46Q8×C53—41no—
47C2×C4×C53—21no—
48C2×C4×C53—21no—
49C2×C4×C53C22×C411yescharacteristic
50C2×C4×C53C2×D412yes—
51C2×Dic53—21no—
52C2×Dic53—21no—
53C2×Dic53C22⋊C411yescharacteristic
54C2×Dic53C22⋊C412yes—
55C4×D53—24no—
56C4×D53—21no—
57C4×D53—21no—
58C4×D53—22no—
59C4×D53—22no—
60C4×D53—41no—
61C4×D53—41no—
62Dic106—41no—
63Dic106—42no—
64Dic106—42no—
65Dic106—41no—
66C8×C53—42no—
67#88322—42no—
68C22×C53C2×C22⋊C411yescharacteristic
69C2×D53—22no—
70C2×D53#9911yescharacteristic
71C2×D53#9911yescharacteristic
72C4×C53—22no—
73C4×C53—22no—
74C4×C53—21no—
75C4×C53—41no—
76Dic53—22no—
77Dic53—21no—
78Dic53—22no—
79Dic53—41no—
80C2×#356—531nomaximal, Sylow
81C2×C53—21no—
82C2×C53C2×#9911yescharacteristic
83D53—21no—
84D53#35612yesdirect product factor
85C2×C4⋊Q8—531no—
86C2×#101—532no—
87#356—531no—
88#356—532no—
89#356D5311yescharacteristic, direct product factor
90C53C2×#35611yescharacteristic, complemented, Sylow
91C22×Q8—531no—
92C22×Q8—531no—
93C2×M16—1062no—
94C4⋊Q8—531no—
95C4⋊Q8—531no—
96C4⋊Q8—531no—
97C4⋊Q8—1062no—
98C4⋊Q8C2×D5311yescharacteristic
99C2×C4⋊C4—1062no—
100C2×C42—531no—
101#101—532no—
102#101—1062no—
103#101C2×D5312yes—
104C2×Q8—531no—
105C2×Q8—532no—
106C2×Q8—531no—
107C2×Q8—1061no—
108C2×Q8—2121no—
109C2×Q8—2121no—
110C2×Q8—2121no—
111C2×Q8—2121no—
112C2×Q8C4×D5311yescharacteristic
113C2×Q8C22×D5311yescharacteristic
114C22×C4—531no—
115C22×C4—532no—
116C22×C4—1061no—
117C22×C4—1061no—
118M16—22no—
119M16—1062no—
120M16—2122no—
121C2×C8—2122no—
122C4⋊C4—22no—
123C4⋊C4—1062no—
124C4⋊C4—1062no—
125C4⋊C4—1062no—
126C42—531no—
127C42—1061no—
128C42C4×D5311yescharacteristic
129C23—531no—
130Q8—41no—
131Q8—41no—
132Q8—2122no—
133Q8—2121no—
134Q8—2121no—
135Q8—2122no—
136C2×C4—21no—
137C2×C4—21no—
138C2×C4—531no—
139C2×C4—532no—
140C2×C4—1062no—
141C2×C4—1062no—
142C2×C4—1061no—
143C2×C4—1061no—
144C2×C4—1064no—
145C2×C4—1061no—
146C2×C4—1061no—
147C2×C4—2121no—
148C2×C4—2121no—
149C2×C4C2×C4×D5311yescharacteristic
150C2×C4D4×D5312yes—
151C8—42no—
152C8—2122no—
153C22—531no—
154C22—531no—
155C22—1062no—
156C22C22⋊C4×D5311yescharacteristic
157C4—22no—
158C4—22no—
159C4—21no—
160C4—41no—
161C4—1062no—
162C4—1062no—
163C4—1061no—
164C4—2121no—
165C2—21no—
166C2—532no—
167C2—1061no—
168C2#99×D5311yescharacteristic
subgroupG/H
centerC2#99×D53
commutatorC2×C4×C53C22×C4
FrattiniC2×Q8C22×D53
FittingC53×#356C2
socleC2×C53C2×#99
hypercenter#356D53
hypocenterC53C2×#356
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #356

Complements

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

NH ≅ G/Ntype
3. #16810422C2semidirect
4. #16810421C2semidirect
5. C53×#356C2semidirect
84. D53#356direct (N × H)
89. #356D53direct (N × H)
90. C53C2×#356semidirect