Finite Group Databasebeta

← #20290877

Subgroups — #20290877

Order 6784.

276 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 68 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
1D53⋊#108C211yesmaximal, characteristic
2#16811616C212yesmaximal
3D53⋊#101C212yesmaximal
4#16811586C211yesmaximal, characteristic
5C2×Q16×D53C211yesmaximal, characteristic
6D53⋊M16C2212yes—
7D53⋊M16C2211yescharacteristic
8#13133296C2211yescharacteristic
9#13133288—21no—
10#13133287—21no—
11#13133280—24no—
12#13133278—24no—
13#13133271—21no—
14C8×#6157—21no—
15C2×Q8×D53C2212yes—
16C2×Dic212C411yescharacteristic, complemented
17C2×#11719925C412yescomplemented
18Q16×D53—21no—
19Q16×D53—21no—
20Q16×D53—21no—
21Q16×D53—21no—
22C2×C8×D53C2211yescharacteristic
23C2×Q16×C53C411yescharacteristic, complemented
24C2×Q8×C53C2×C412yes—
25Q8×D53—22no—
26Q8×D53—22no—
27Q8×D53—22no—
28Q8×D53—22no—
29C2×C4×D53—22no—
30C2×C4×D53C2311yescharacteristic
31C2×Dic106—22no—
32C2×Dic106C2×C412yes—
33C22⋊#6157—21no—
34#11719940—22no—
35#11719940D412yes—
36C4⋊#6157D412yes—
37C4×#6157—21no—
38#11719936—21no—
39#11719936—22no—
40D53⋊C8—21no—
41D53⋊C8—22no—
42Q16×C53—21no—
43Q16×C53—21no—
44C2×C8×C53C2×C411yescharacteristic
45#11719925—22no—
46#11719925—22no—
47C2×#88322C2×C411yescharacteristic
48Dic212—21no—
49Dic212—21no—
50C8×D53D411yescharacteristic
51C8×D53D411yescharacteristic
52C8×D53D411yescharacteristic
53C8×D53D411yescharacteristic
54C22×D53#4111yescharacteristic
55C2×#6157—22no—
56Q8×C53—22no—
57Q8×C53—22no—
58C2×C4×C53—22no—
59C2×C4×C53C22×C411yescharacteristic
60C2×Dic53—22no—
61C2×Dic53C2×D411yescharacteristic
62C4×D53—42no—
63C4×D53—42no—
64C4×D53C2×D411yescharacteristic
65C4×D53C2×D411yescharacteristic
66C4×D53C2×D411yescharacteristic
67C4×D53#4111yescharacteristic
68Dic106—22no—
69Dic106—22no—
70Dic106—42no—
71Dic106—42no—
72#88324—22no—
73#88324—42no—
74C8×C53C22⋊C411yescharacteristic
75C8×C53C22⋊C411yescharacteristic
76#88322C22⋊C411yescharacteristic
77#88322C22⋊C411yescharacteristic
78C22×C53C4×D411yescharacteristic
79C2×D53—22no—
80C2×D53#12111yescharacteristic
81C2×D53#12411yescharacteristic
82#6157—42no—
83C4×C53—42no—
84C4×C53C2×C22⋊C411yescharacteristic
85C4×C53C4×D411yescharacteristic
86Dic53—42no—
87Dic53#12111yescharacteristic
88Dic53#12711yescharacteristic
89#1979—531nomaximal, Sylow
90C2×C53—21no—
91C2×C53#39011yescharacteristic
92D53—21no—
93D53#47312yes—
94C22×Q16—531no—
95C2×#108—531no—
96#419—532no—
97C2×#101—532no—
98#405—531no—
99C53#197911yescharacteristic, complemented, Sylow
100C22×Q8—532no—
101C2×Q16—532no—
102C2×Q16—531no—
103C2×Q16—1061no—
104C2×Q16—1061no—
105C2×Q16—1061no—
106C2×Q16—1061no—
107C2×Q16#615711yescharacteristic, complemented
108C2×M16—531no—
109C2×M16—532no—
110C22×C8—531no—
111#117—531no—
112#108—1061no—
113#108—1061no—
114#103—1064no—
115#101—1064no—
116#97—1061no—
117C4×C8—1061no—
118C2×Q8—532no—
119C2×Q8—1062no—
120C2×Q8—1062no—
121C2×Q8—1062no—
122C2×Q8—1062no—
123C2×Q8—1062no—
124C2×Q8C2×#615712yes—
125C22×C4—531no—
126C22×C4—1062no—
127Q16—21no—
128Q16—21no—
129Q16—1062no—
130Q16—1062no—
131Q16—1061no—
132Q16—1061no—
133M16—532no—
134M16—1061no—
135M16—1062no—
136M16—1062no—
137C2×C8—531no—
138C2×C8—531no—
139C2×C8—531no—
140C2×C8—531no—
141C2×C8—531no—
142C2×C8—1061no—
143C2×C8—1062no—
144C2×C8C2×#615711yescharacteristic
145C4⋊C4—532no—
146C22⋊C4—1061no—
147C42—1061no—
148C23—531no—
149Q8—22no—
150Q8—22no—
151Q8—1062no—
152Q8—1062no—
153Q8—2122no—
154Q8—2122no—
155C2×C4—22no—
156C2×C4—531no—
157C2×C4—531no—
158C2×C4—531no—
159C2×C4—531no—
160C2×C4—531no—
161C2×C4—1062no—
162C2×C4—1062no—
163C2×C4—2122no—
164C2×C4—2122no—
165C2×C4C22×#615711yescharacteristic
166C8—531no—
167C8—531no—
168C8—1062no—
169C8—2122no—
170C8C22⋊#615711yescharacteristic
171C8C22⋊#615711yescharacteristic
172C22—531no—
173C22—531no—
174C22—1062no—
175C22D4×#615711yescharacteristic
176C4—42no—
177C4—531no—
178C4—531no—
179C4—2122no—
180C4—2122no—
181C4C2×C22⋊#615711yescharacteristic
182C4D4×#615711yescharacteristic
183C2—21no—
184C2—532no—
185C2—1061no—
186C2#1681157511yescharacteristic
subgroupG/H
centerC2#16811575
commutatorC2×C4×C53C22×C4
FrattiniC2×C4C22×#6157
FittingC2×Q16×C53C4
socleC2×C53#390
hypercenterC2×Q16#6157
hypocenterC53#1979
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×Q16

Complements

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

NH ≅ G/Ntype
16. C2×Dic212C4semidirect
17. C2×#11719925C4semidirect
23. C2×Q16×C53C4semidirect
99. C53#1979semidirect
107. C2×Q16#6157semidirect