Finite Group Databasebeta

← #13056436

Subgroups — #13056436

Order 1344.

192 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 61 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#10621430C211yesmaximal, characteristic
2#10621373C211yesmaximal, characteristic, complemented
3C3×C7×#103C211yesmaximal, characteristic, complemented
4#10620674C211yesmaximal, characteristic, complemented
5#10620660C211yesmaximal, characteristic
6C8×Dic21C211yesmaximal, characteristic, complemented
7#10620648C211yesmaximal, characteristic
8#89572—31nomaximal, Hall
9C2×C3×C7×Q8C2211yescharacteristic
10C2×D84C2211yescharacteristic
11C2×C3×C7×C8C2211yescharacteristic
12C3×C7×C4⋊C4C2211yescharacteristic
13#66631—21no—
14#66631—21no—
15C4⋊Dic21—21no—
16C4⋊Dic21C2211yescharacteristic
17#66629—21no—
18C4×Dic21C2211yescharacteristic
19C2×C3⋊(C7⋊C8)C2211yescharacteristic
20#6510—31no—
21#6453—31no—
22C7×#103D311yescharacteristic, complemented
23#6409—31no—
24#6395—31no—
25C8×Dic7—31no—
26#6383—31no—
27#4800—71nomaximal, Hall
28C22×D21—21no—
29C3×C7×Q8—21no—
30C3×C7×Q8—21no—
31C2×C3×C4×C7—21no—
32C2×C3×C4×C7—21no—
33C2×C3×C4×C7C2311yescharacteristic
34C2×Dic21—21no—
35C2×Dic21D411yescharacteristic
36C2×Dic21D411yescharacteristic
37D84—21no—
38D84—21no—
39C3×C7×C8—21no—
40C3⋊(C7⋊C8)—21no—
41C2×C7×Q8C2×D311yescharacteristic
42C2×D28—31no—
43C2×C7×C8C2×D311yescharacteristic
44C7×C4⋊C4C2×D311yescharacteristic
45(C2×D7)⋊C4—61no—
46(C2×D7)⋊C4—61no—
47C4⋊Dic7—31no—
48C4⋊Dic7—61no—
49Dic7⋊C4—61no—
50C4×Dic7—31no—
51C2×C7⋊C8—31no—
52#947—71no—
53#890—71no—
54C3×#103D711yescharacteristic, complemented
55#835—71no—
56#821—71no—
57C8×Dic3—71no—
58#809—71no—
59C22×C3×C7C2×D411yescharacteristic
60C2×D21—41no—
61C2×D21—41no—
62C2×D21—41no—
63C3×C4×C7—41no—
64C3×C4×C7—41no—
65C3×C4×C7#4111yescharacteristic
66C3×C4×C7#4111yescharacteristic
67Dic21—21no—
68Dic21—21no—
69Dic21—41no—
70#488—211noSylow
71C22×D7—61no—
72C7×Q8—21no—
73C7×Q8—21no—
74C2×C4×C7—21no—
75C2×C4×C7—21no—
76C2×C4×C7C22×D311yescharacteristic
77C2×Dic7—31no—
78C2×Dic7—31no—
79C2×Dic7—61no—
80D28—61no—
81D28—61no—
82C7×C8—21no—
83C7⋊C8—61no—
84C2×C3×Q8C2×D711yescharacteristic
85C2×D12—71no—
86C2×C3×C8C2×D711yescharacteristic
87C3×C4⋊C4C2×D711yescharacteristic
88(C2×D3)⋊C4—141no—
89(C2×D3)⋊C4—141no—
90C4⋊Dic3—71no—
91C4⋊Dic3—141no—
92Dic3⋊C4—141no—
93C4×Dic3—71no—
94C2×C3⋊C8—71no—
95C2×C3×C7#12411yescharacteristic
96C2×C3×C7#13511yescharacteristic
97C2×C3×C7#13511yescharacteristic
98D21—81no—
99#125—211no—
100#124—211no—
101#103—211no—
102#103D2111yescharacteristic, complemented
103#102—211no—
104#102—211no—
105C4×C8—211no—
106C22×C7D3×D411yescharacteristic
107C2×D7—121no—
108C2×D7—121no—
109C2×D7—121no—
110C4×C7—41no—
111C4×C7—41no—
112C4×C7#23511yescharacteristic
113C4×C7#23711yescharacteristic
114Dic7—61no—
115Dic7—61no—
116Dic7—121no—
117C22×D3—141no—
118C3×Q8—21no—
119C3×Q8—21no—
120C2×C3×C4—21no—
121C2×C3×C4—21no—
122C2×C3×C4C22×D711yescharacteristic
123C2×Dic3—71no—
124C2×Dic3—71no—
125C2×Dic3—141no—
126D12—141no—
127D12—141no—
128C3×C8—21no—
129C3⋊C8—141no—
130C3×C7#48811yescharacteristic, complemented, Hall
131C2×Q8C2×D2111yescharacteristic
132C2×D4—211no—
133C2×C8—211no—
134C2×C8C2×D2111yescharacteristic
135C4⋊C4—211no—
136C4⋊C4—421no—
137C4⋊C4—421no—
138C4⋊C4C2×D2111yescharacteristic
139C22⋊C4—421no—
140C22⋊C4—421no—
141C42—211no—
142C2×C7#88511yescharacteristic
143C2×C7#91611yescharacteristic
144C2×C7#91911yescharacteristic
145D7—241no—
146C22×C3D4×D711yescharacteristic
147C2×D3—281no—
148C2×D3—281no—
149C2×D3—281no—
150C3×C4—41no—
151C3×C4—41no—
152C3×C4#115711yescharacteristic
153C3×C4#115911yescharacteristic
154Dic3—141no—
155Dic3—141no—
156Dic3—281no—
157C23—421no—
158Q8—21no—
159Q8—21no—
160D4—421no—
161D4—421no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—211no—
165C2×C4—211no—
166C2×C4—421no—
167C2×C4C22×D2111yescharacteristic
168C8—21no—
169C8—421no—
170C7#480011yescharacteristic, complemented, Sylow
171C2×C3#644811yescharacteristic
172C2×C3#647911yescharacteristic
173C2×C3#648211yescharacteristic
174D3—561no—
175C22—841no—
176C22—841no—
177C22—841no—
178C22D4×D2111yescharacteristic
179C4—41no—
180C4—41no—
181C4—421no—
182C4—421no—
183C4—841no—
184C4#6672811yescharacteristic
185C4#6673011yescharacteristic
186C3#8957211yescharacteristic, complemented, Sylow
187C2—1681no—
188C2#1062136811yescharacteristic
189C2#1062139911yescharacteristic
190C2#1062140211yescharacteristic
subgroupG/H
centerC22D4×D21
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C22×D21
FittingC3×C7×#103C2
socleC22×C3×C7C2×D4
prime radicalC2×D84C22
hypercenter#103D21
hypocenterC3×C7#488
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C2×C4 ◁ #103

Complements

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

NH ≅ G/Ntype
2. #10621373C2semidirect
3. C3×C7×#103C2semidirect
4. #10620674C2semidirect
6. C8×Dic21C2semidirect
22. C7×#103D3semidirect
54. C3×#103D7semidirect
102. #103D21semidirect
130. C3×C7#488semidirect
170. C7#4800semidirect
186. C3#89572semidirect

Maximal subgroups

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

subgroupindex
#106214302
#106213732
C3×C7×#1032
#106206742
#106206602
C8×Dic212
#106206482
#895723
#48007