Finite Group Databasebeta

← #13056632

Subgroups — #13056632

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×#117C211yesmaximal, characteristic
2#10621407C211yesmaximal, characteristic
3#10620669C211yesmaximal, characteristic
4#10620649C211yesmaximal, characteristic, complemented
5#10620649C211yesmaximal, characteristic, complemented
6#10620647C211yesmaximal, characteristic, complemented
7#10620646C211yesmaximal, characteristic, complemented
8#89768—31nomaximal, Hall
9C22×C3×C4×C7C2211yescharacteristic
10C2×Dic42C2211yescharacteristic
11C3×C7×C4⋊C4C2211yescharacteristic
12C3×C7×C4⋊C4C2211yescharacteristic
13C3×C7×C22⋊C4—21no—
14C3×C42×C7—21no—
15#66636—21no—
16C4⋊Dic21C2211yescharacteristic
17#66629—21no—
18C2×C3⋊(C7⋊C8)C2211yescharacteristic
19C2×C3⋊(C7⋊C8)C2211yescharacteristic
20C7×#117D311yescharacteristic
21#6487—31no—
22C22⋊(C7⋊C8)—31no—
23#6384—31no—
24#6384—31no—
25#6382—31no—
26#6381—31no—
27#4996—71nomaximal, Hall
28C23×C3×C7D411yescharacteristic
29C2×C3×C4×C7—21no—
30C2×C3×C4×C7—21no—
31C2×C3×C4×C7—21no—
32C2×C3×C4×C7—21no—
33C2×C3×C4×C7D411yescharacteristic
34C2×C3×C4×C7C2311yescharacteristic
35C2×Dic21—21no—
36C2×Dic21—21no—
37Dic42—21no—
38Dic42—21no—
39C3⋊(C7⋊C8)—21no—
40C3⋊(C7⋊C8)—21no—
41C22×C4×C7C2×D311yescharacteristic
42C2×Dic14—31no—
43C7×C4⋊C4C2×D311yescharacteristic
44C7×C4⋊C4C2×D311yescharacteristic
45C7×C22⋊C4—21no—
46C42×C7—21no—
47C22⋊Dic7—61no—
48C4⋊Dic7—31no—
49Dic7⋊C4—61no—
50C2×C7⋊C8—31no—
51C2×C7⋊C8—31no—
52C3×#117D711yescharacteristic
53#924—71no—
54#830—71no—
55#810—71no—
56#810—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
72C2×C4×C7—21no—
73C2×C4×C7—21no—
74C2×C4×C7—21no—
75C2×C4×C7—21no—
76C2×C4×C7#6711yescharacteristic
77C2×C4×C7C22×D311yescharacteristic
78C2×Dic7—61no—
79C2×Dic7—61no—
80Dic14—61no—
81Dic14—61no—
82C7⋊C8—61no—
83C7⋊C8—61no—
84C22×C3×C4C2×D711yescharacteristic
85C2×Dic6—71no—
86C3×C4⋊C4C2×D711yescharacteristic
87C3×C4⋊C4C2×D711yescharacteristic
88C3×C22⋊C4—21no—
89C3×C42—21no—
90#217—141no—
91C4⋊Dic3—71no—
92Dic3⋊C4—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⋊Q8—211no—
100#117D2111yescharacteristic
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#23511yescharacteristic
114C4×C7#23511yescharacteristic
115Dic7—121no—
116Dic7—121no—
117C23×C3C22⋊D711yescharacteristic
118C2×C3×C4—21no—
119C2×C3×C4—21no—
120C2×C3×C4—21no—
121C2×C3×C4—21no—
122C2×C3×C4C22⋊D711yescharacteristic
123C2×C3×C4C22×D711yescharacteristic
124C2×Dic3—141no—
125C2×Dic3—141no—
126Dic6—141no—
127Dic6—141no—
128C3⋊C8—141no—
129C3⋊C8—141no—
130C3×C7#48511yescharacteristic, complemented, Hall
131C2×Q8—211no—
132C22×C4C2×D2111yescharacteristic
133C2×C8—211no—
134C2×C8—211no—
135C4⋊C4—211no—
136C4⋊C4—421no—
137C4⋊C4C2×D2111yescharacteristic
138C4⋊C4C2×D2111yescharacteristic
139C22⋊C4—21no—
140C22⋊C4—421no—
141C42—21no—
142C2×C7—41no—
143C2×C7#92911yescharacteristic
144C2×C7#95011yescharacteristic
145C2×C7#95111yescharacteristic
146C22×C3—21no—
147C22×C3—21no—
148C22×C3—21no—
149C22×C3C2×C22⋊D711yescharacteristic
150C3×C4—21no—
151C3×C4—41no—
152C3×C4—41no—
153C3×C4#115711yescharacteristic
154C3×C4#115711yescharacteristic
155Dic3—281no—
156Dic3—281no—
157C23#426811yescharacteristic
158Q8—421no—
159Q8—421no—
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#499611yescharacteristic, complemented, Sylow
171C2×C3—41no—
172C2×C3#649211yescharacteristic
173C2×C3#651311yescharacteristic
174C2×C3#651411yescharacteristic
175C22—21no—
176C22—21no—
177C22—21no—
178C22C2×#426811yescharacteristic
179C4—21no—
180C4—41no—
181C4—41no—
182C4—841no—
183C4—841no—
184C4#6672811yescharacteristic
185C4#6672811yescharacteristic
186C3#8976811yescharacteristic, complemented, Sylow
187C2—41no—
188C2#1062141211yescharacteristic
189C2#1062143311yescharacteristic
190C2#1062143411yescharacteristic
subgroupG/H
centerC22C2×#4268
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C22×D21
FittingC3×C7×#117C2
socleC22×C3×C7C2×D4
prime radicalC23×C3×C7D4
hypercenter#117D21
hypocenterC3×C7#485
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
4. #10620649C2semidirect
5. #10620649C2semidirect
6. #10620647C2semidirect
7. #10620646C2semidirect
130. C3×C7#485semidirect
170. C7#4996semidirect
186. C3#89768semidirect

Maximal subgroups

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

subgroupindex
C3×C7×#1172
#106214072
#106206692
#106206492
#106206472
#106206462
#897683
#49967