Finite Group Databasebeta

← #13056682

Subgroups — #13056682

Order 1344.

192 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 67 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×C7×C22⋊Q8C211yesmaximal, characteristic
2#10621409C211yesmaximal, characteristic
3#10620674C211yesmaximal, characteristic, complemented
4#10620674C211yesmaximal, characteristic, complemented
5#10620669C211yesmaximal, characteristic
6#10620647C211yesmaximal, characteristic, complemented
7#10620646C211yesmaximal, characteristic, complemented
8#89817—31nomaximal, Hall
9C2×C3×C7×Q8C2211yescharacteristic
10C22×C3×C4×C7C2211yescharacteristic
11C3×C7×C4⋊C4—21no—
12C3×C7×C4⋊C4C2211yescharacteristic
13C3×C7×C22⋊C4—21no—
14#66636—21no—
15C4⋊Dic21C2211yescharacteristic
16C4⋊Dic21C2211yescharacteristic
17C4×Dic21—21no—
18C2×C3⋊(C7⋊C8)C2211yescharacteristic
19C2×C3⋊(C7⋊C8)C2211yescharacteristic
20C7×C22⋊Q8D311yescharacteristic
21#6489—31no—
22#6409—31no—
23#6409—31no—
24C22⋊(C7⋊C8)—31no—
25#6382—31no—
26#6381—31no—
27#5045—71nomaximal, Hall
28C23×C3×C7D411yescharacteristic
29C3×C7×Q8—21no—
30C3×C7×Q8—21no—
31C2×C3×C4×C7—21no—
32C2×C3×C4×C7—21no—
33C2×C3×C4×C7—21no—
34C2×C3×C4×C7—21no—
35C2×C3×C4×C7D411yescharacteristic
36C2×C3×C4×C7C2311yescharacteristic
37C2×Dic21—21no—
38C2×Dic21—21no—
39C3⋊(C7⋊C8)—21no—
40C3⋊(C7⋊C8)—21no—
41C2×C7×Q8C2×D311yescharacteristic
42C22×C4×C7C2×D311yescharacteristic
43C7×C4⋊C4—21no—
44C7×C4⋊C4C2×D311yescharacteristic
45C7×C22⋊C4—21no—
46C22⋊Dic7—61no—
47C4⋊Dic7—31no—
48C4⋊Dic7—31no—
49C4×Dic7—61no—
50C2×C7⋊C8—31no—
51C2×C7⋊C8—31no—
52C3×C22⋊Q8D711yescharacteristic
53#926—71no—
54#835—71no—
55#835—71no—
56#830—71no—
57#808—71no—
58#807—71no—
59C22×C3×C7—21no—
60C22×C3×C7—21no—
61C22×C3×C7—21no—
62C22×C3×C7C2×D411yescharacteristic
63C3×C4×C7—21no—
64C3×C4×C7—41no—
65C3×C4×C7—41no—
66C3×C4×C7#4111yescharacteristic
67C3×C4×C7#4111yescharacteristic
68Dic21—41no—
69Dic21—41no—
70#485—211noSylow
71C23×C7#6711yescharacteristic
72C7×Q8—21no—
73C7×Q8—21no—
74C2×C4×C7—21no—
75C2×C4×C7—21no—
76C2×C4×C7—21no—
77C2×C4×C7—21no—
78C2×C4×C7#6711yescharacteristic
79C2×C4×C7C22×D311yescharacteristic
80C2×Dic7—61no—
81C2×Dic7—61no—
82C7⋊C8—61no—
83C7⋊C8—61no—
84C2×C3×Q8C2×D711yescharacteristic
85C22×C3×C4C2×D711yescharacteristic
86C3×C4⋊C4—21no—
87C3×C4⋊C4C2×D711yescharacteristic
88C3×C22⋊C4—21no—
89#217—141no—
90C4⋊Dic3—71no—
91C4⋊Dic3—71no—
92C4×Dic3—141no—
93C2×C3⋊C8—71no—
94C2×C3⋊C8—71no—
95C2×C3×C7—41no—
96C2×C3×C7#12311yescharacteristic
97C2×C3×C7#13511yescharacteristic
98C2×C3×C7#13711yescharacteristic
99C22⋊Q8D2111yescharacteristic
100#117—211no—
101#107—211no—
102#106—211no—
103#103—211no—
104#103—211no—
105C22⋊C8—211no—
106C22×C7—21no—
107C22×C7—21no—
108C22×C7—21no—
109C22×C7C2×#6711yescharacteristic
110C4×C7—21no—
111C4×C7—41no—
112C4×C7—41no—
113C4×C7#23711yescharacteristic
114C4×C7#23711yescharacteristic
115Dic7—121no—
116Dic7—121no—
117C23×C3C22⋊D711yescharacteristic
118C3×Q8—21no—
119C3×Q8—21no—
120C2×C3×C4—21no—
121C2×C3×C4—21no—
122C2×C3×C4—21no—
123C2×C3×C4—21no—
124C2×C3×C4C22⋊D711yescharacteristic
125C2×C3×C4C22×D711yescharacteristic
126C2×Dic3—141no—
127C2×Dic3—141no—
128C3⋊C8—141no—
129C3⋊C8—141no—
130C3×C7#48511yescharacteristic, complemented, Hall
131C2×Q8C2×D2111yescharacteristic
132C22×C4C2×D2111yescharacteristic
133C2×C8—211no—
134C2×C8—211no—
135C4⋊C4—21no—
136C4⋊C4—211no—
137C4⋊C4—211no—
138C4⋊C4C2×D2111yescharacteristic
139C22⋊C4—21no—
140C22⋊C4—421no—
141C42—421no—
142C2×C7—41no—
143C2×C7#93511yescharacteristic
144C2×C7#94211yescharacteristic
145C2×C7#95011yescharacteristic
146C22×C3—21no—
147C22×C3—21no—
148C22×C3—21no—
149C22×C3C2×C22⋊D711yescharacteristic
150C3×C4—21no—
151C3×C4—41no—
152C3×C4—41no—
153C3×C4#115911yescharacteristic
154C3×C4#115911yescharacteristic
155Dic3—281no—
156Dic3—281no—
157C23#426811yescharacteristic
158Q8—21no—
159Q8—21no—
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—421no—
165C2×C4—421no—
166C2×C4#426811yescharacteristic
167C2×C4C22×D2111yescharacteristic
168C8—421no—
169C8—421no—
170C7#504511yescharacteristic, complemented, Sylow
171C2×C3—41no—
172C2×C3#649811yescharacteristic
173C2×C3#650511yescharacteristic
174C2×C3#651311yescharacteristic
175C22—21no—
176C22—21no—
177C22—21no—
178C22C2×#426811yescharacteristic
179C4—21no—
180C4—41no—
181C4—41no—
182C4—841no—
183C4—841no—
184C4#6673011yescharacteristic
185C4#6673011yescharacteristic
186C3#8981711yescharacteristic, complemented, Sylow
187C2—41no—
188C2#1062141811yescharacteristic
189C2#1062142511yescharacteristic
190C2#1062143311yescharacteristic
subgroupG/H
centerC22C2×#4268
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C22×D21
FittingC3×C7×C22⋊Q8C2
socleC22×C3×C7C2×D4
prime radicalC23×C3×C7D4
hypercenterC22⋊Q8D21
hypocenterC3×C7#485
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. #10620674C2semidirect
4. #10620674C2semidirect
6. #10620647C2semidirect
7. #10620646C2semidirect
130. C3×C7#485semidirect
170. C7#5045semidirect
186. C3#89817semidirect

Maximal subgroups

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

subgroupindex
C3×C7×C22⋊Q82
#106214092
#106206742
#106206692
#106206472
#106206462
#898173
#50457