Finite Group Databasebeta

← #13056652

Subgroups — #13056652

Order 1344.

280 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 83 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
1C3×C4×C7×D4C211yesmaximal, characteristic, complemented
2C2×#66632C212yesmaximal, complemented
3#10621357C211yesmaximal, characteristic, complemented
4#10620648C212yesmaximal, complemented
5C4⋊(C3⋊(C7⋊C8))C211yesmaximal, characteristic, complemented
6#89787—31nomaximal, Hall
7C2×C3×C7×D4C2211yescharacteristic
8C22×C3×C4×C7—21no—
9C2×D84—21no—
10C2×D84—21no—
11C2×D84C2212yes—
12C3×C7×C4⋊C4C2211yescharacteristic
13C3×C7×C22⋊C4—21no—
14C3×C42×C7C2211yescharacteristic
15#66632—24no—
16C2×C3⋊(C7⋊C8)C2212yes—
17C4×C7×D4D311yescharacteristic, complemented
18C2×#1141—32no—
19#6437—31no—
20#6383—32no—
21C4⋊(C7⋊C8)—31no—
22#5015—71nomaximal, Hall
23C23×C3×C7—21no—
24C22×D21—22no—
25C3×C7×D4—21no—
26C3×C7×D4D412yescomplemented
27C2×C3×C4×C7—21no—
28C2×C3×C4×C7—42no—
29C2×C3×C4×C7D411yescharacteristic
30C2×C3×C4×C7D411yescharacteristic
31C2×C3×C4×C7C2311yescharacteristic
32D84—22no—
33D84—22no—
34D84—42no—
35D84—42no—
36C3⋊(C7⋊C8)—22no—
37C2×C7×D4C2×D311yescharacteristic
38C22×C4×C7—21no—
39C2×D28—32no—
40C2×D28—61no—
41C2×D28—61no—
42C7×C4⋊C4C2×D311yescharacteristic
43C7×C22⋊C4—21no—
44C42×C7C2×D311yescharacteristic
45#1141—64no—
46C2×C7⋊C8—32no—
47C3×C4×D4D711yescharacteristic, complemented
48C2×#213—72no—
49#874—71no—
50#809—72no—
51C4⋊(C3⋊C8)—71no—
52C22×C3×C7—22no—
53C22×C3×C7—41no—
54C22×C3×C7—41no—
55C22×C3×C7C2×D411yescharacteristic
56C2×D21—42no—
57C2×D21—42no—
58C2×D21—42no—
59C3×C4×C7—21no—
60C3×C4×C7—41no—
61C3×C4×C7D812yescomplemented
62C3×C4×C7C2×D411yescharacteristic
63C3×C4×C7#4111yescharacteristic
64#459—211noSylow
65C23×C7—21no—
66C22×D7—62no—
67C7×D4—21no—
68C7×D4D1212yescomplemented
69C2×C4×C7—21no—
70C2×C4×C7—42no—
71C2×C4×C7#6711yescharacteristic
72C2×C4×C7#6711yescharacteristic
73C2×C4×C7C22×D311yescharacteristic
74D28—62no—
75D28—62no—
76D28—122no—
77D28—122no—
78C7⋊C8—62no—
79C2×C3×D4C2×D711yescharacteristic
80C22×C3×C4—21no—
81C2×D12—72no—
82C2×D12—141no—
83C2×D12—141no—
84C3×C4⋊C4C2×D711yescharacteristic
85C3×C22⋊C4—21no—
86C3×C42C2×D711yescharacteristic
87#213—144no—
88C2×C3⋊C8—72no—
89C2×C3×C7—42no—
90C2×C3×C7#12111yescharacteristic
91C2×C3×C7C2×D811yescharacteristic
92C2×C3×C7AGL(1,ℤ/8)11yescharacteristic
93D21—82no—
94C2×D8—212no—
95#127—211no—
96C4×D4D2111yescharacteristic, complemented
97C4⋊C8—211no—
98#102—212no—
99C22×C7—22no—
100C22×C7—41no—
101C22×C7—41no—
102C22×C7C2×#6711yescharacteristic
103C2×D7—122no—
104C2×D7—122no—
105C2×D7—122no—
106C4×C7—21no—
107C4×C7—41no—
108C4×C7#21312yescomplemented
109C4×C7C2×D1211yescharacteristic
110C4×C7#23511yescharacteristic
111C23×C3—21no—
112C22×D3—142no—
113C3×D4—21no—
114C3×D4D2812yescomplemented
115C2×C3×C4—21no—
116C2×C3×C4—42no—
117C2×C3×C4C22⋊D711yescharacteristic
118C2×C3×C4C22⋊D711yescharacteristic
119C2×C3×C4C22×D711yescharacteristic
120D12—142no—
121D12—142no—
122D12—282no—
123D12—282no—
124C3⋊C8—142no—
125C3×C7#45911yescharacteristic, complemented, Hall
126C2×D4—212no—
127C2×D4—421no—
128C2×D4—421no—
129C2×D4C2×D2111yescharacteristic
130C22×C4—21no—
131D8—424no—
132C2×C8—212no—
133C4⋊C4C2×D2111yescharacteristic
134C22⋊C4—21no—
135C42C2×D2111yescharacteristic
136C2×C7—42no—
137C2×C7#93011yescharacteristic
138C2×C7C2×#21311yescharacteristic
139C2×C7#94911yescharacteristic
140D7—242no—
141C22×C3—22no—
142C22×C3—41no—
143C22×C3—41no—
144C22×C3C2×C22⋊D711yescharacteristic
145C2×D3—282no—
146C2×D3—282no—
147C2×D3—282no—
148C3×C4—21no—
149C3×C4—41no—
150C3×C4#114112yescomplemented
151C3×C4C2×D2811yescharacteristic
152C3×C4#115711yescharacteristic
153C23—21no—
154C23—422no—
155D4—21no—
156D4—422no—
157D4—422no—
158D4—842no—
159D4—842no—
160D4D8412yescomplemented
161C2×C4—21no—
162C2×C4—42no—
163C2×C4#426811yescharacteristic
164C2×C4#426811yescharacteristic
165C2×C4C22×D2111yescharacteristic
166C8—422no—
167C7#501511yescharacteristic, complemented, Sylow
168C2×C3—42no—
169C2×C3#649311yescharacteristic
170C2×C3C2×#114111yescharacteristic
171C2×C3#651211yescharacteristic
172D3—562no—
173C22—22no—
174C22—41no—
175C22—41no—
176C22—842no—
177C22—842no—
178C22—842no—
179C22C2×#426811yescharacteristic
180C4—21no—
181C4—41no—
182C4#6663212yescomplemented
183C4C2×D8411yescharacteristic
184C4#6672811yescharacteristic
185C3#8978711yescharacteristic, complemented, Sylow
186C2—42no—
187C2—1682no—
188C2#1062141311yescharacteristic
189C2C2×#6663211yescharacteristic
190C2#1062143211yescharacteristic
subgroupG/H
centerC22C2×#4268
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C22×D21
FittingC3×C4×C7×D4C2
socleC22×C3×C7C2×D4
hypercenterC4×D4D21
hypocenterC3×C7#459
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C42 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. C3×C4×C7×D4C2semidirect
2. C2×#66632C2semidirect
3. #10621357C2semidirect
4. #10620648C2semidirect
5. C4⋊(C3⋊(C7⋊C8))C2semidirect
17. C4×C7×D4D3semidirect
26. C3×C7×D4D4semidirect
47. C3×C4×D4D7semidirect
61. C3×C4×C7D8semidirect
68. C7×D4D12semidirect
96. C4×D4D21semidirect
108. C4×C7#213semidirect
114. C3×D4D28semidirect
125. C3×C7#459semidirect
150. C3×C4#1141semidirect
160. D4D84semidirect
167. C7#5015semidirect
182. C4#66632semidirect
185. C3#89787semidirect

Maximal subgroups

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

subgroupindex
C3×C4×C7×D42
C2×#666322
#106213572
#106206482
C4⋊(C3⋊(C7⋊C8))2
#897873
#50157