Finite Group Databasebeta

← #13054643

Subgroups — #13054643

Order 1344.

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

normal subgroups only

This group has 228 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
1C2×C7×#216C211yesmaximal, characteristic, complemented
2C3×#6509C211yesmaximal, characteristic
3#10620921C211yesmaximal, characteristic
4C2×D7×C3⋊C8C211yesmaximal, characteristic
5#10620674C211yesmaximal, characteristic, complemented
6#10620525C211yesmaximal, characteristic, complemented
7#10620507C211yesmaximal, characteristic, complemented
8#89960—31nomaximal, Hall
9C2×C3×C7×Q8C2211yescharacteristic
10C2×C7×Dic6C2211yescharacteristic
11C2×C3×C4×D7C2211yescharacteristic
12C4⋊Dic21C2211yescharacteristic
13C2×C3⋊(C7⋊C8)C2211yescharacteristic
14C7×#216—22no—
15C2×C7×C3⋊C8C2211yescharacteristic
16C3×(C2×D7)⋊C4—21no—
17C3×C4⋊Dic7C2211yescharacteristic
18C3×Dic7⋊C4—21no—
19#66577—21no—
20(C2×D7)⋊Dic3—21no—
21D7×C3⋊C8—22no—
22C2×C7×Q16—31no—
23#6509—31no—
24#6509D311yescharacteristic
25C2×C8×D7—31no—
26#6409—31no—
27#6409—31no—
28#6392—31no—
29#5051—71nomaximal, Hall
30C22×C3×D7D411yescharacteristic
31C3×C7×Q8—21no—
32C3×C7×Q8—21no—
33C2×C3×C4×C7—21no—
34C2×C3×C4×C7C2311yescharacteristic
35C2×Dic21—21no—
36C2×C7×Dic3—21no—
37C7×Dic6—21no—
38C7×Dic6—21no—
39C2×C3×Dic7—21no—
40C2×C3×Dic7D411yescharacteristic
41C3×C4×D7—21no—
42C3×C4×D7—21no—
43C3⋊(C7⋊C8)—21no—
44C7×C3⋊C8D412yes—
45C2×C7×Q8—31no—
46C2×C7×Q8C2×D311yescharacteristic
47C2×C4×D7C2×D311yescharacteristic
48C7×Q16—62no—
49C2×C7×C8—31no—
50(C2×D7)⋊C4—21no—
51(C2×D7)⋊C4—61no—
52C4⋊Dic7—31no—
53C4⋊Dic7C2×D311yescharacteristic
54Dic7⋊C4—21no—
55Dic7⋊C4—61no—
56C2×C7⋊C8—31no—
57C8×D7—62no—
58C3×C22⋊Q8—71no—
59C2×#216D711yescharacteristic, complemented
60#924—71no—
61C22×C3⋊C8—71no—
62#835—71no—
63#810—71no—
64#807—71no—
65C22×C3×C7C2×D411yescharacteristic
66C2×C3×D7—21no—
67C2×C3×D7—21no—
68C2×C3×D7Q1612yescomplemented
69C3×C4×C7—41no—
70C3×C4×C7C2×D411yescharacteristic
71C3×C4×C7#4111yescharacteristic
72Dic21—41no—
73C3×Dic7—21no—
74C3×Dic7—41no—
75C7×Dic3—41no—
76#467—211noSylow
77C22×D7#6711yescharacteristic
78C7×Q8—21no—
79C7×Q8—21no—
80C7×Q8—61no—
81C7×Q8—61no—
82C2×C4×C7—21no—
83C2×C4×C7—61no—
84C2×C4×C7C22×D311yescharacteristic
85C2×Dic7—21no—
86C2×Dic7—61no—
87C2×Dic7#6711yescharacteristic
88C4×D7—21no—
89C4×D7—21no—
90C7×C8—32no—
91C7⋊C8—61no—
92C2×C3×Q8C2×D711yescharacteristic
93C22×C3×C4—71no—
94C2×Dic6C2×D711yescharacteristic
95C3×C4⋊C4—71no—
96C3×C4⋊C4—141no—
97C3×C22⋊C4—141no—
98#217—141no—
99#216—22no—
100C4⋊Dic3—71no—
101Dic3⋊C4—141no—
102C2×C3⋊C8—71no—
103C2×C3⋊C8—142no—
104C2×C3⋊C8C2×D711yescharacteristic
105C2×C3×C7#12111yescharacteristic
106C2×C3×C7C2×Q1611yescharacteristic
107C2×C3×C7#13511yescharacteristic
108C3×D7—22no—
109C2×Q16—31no—
110C22×C8—211no—
111C22⋊Q8—71no—
112C22⋊Q8—211no—
113#107—211no—
114#103—211no—
115#103—211no—
116C22×C7C2×#6711yescharacteristic
117C2×D7—21no—
118C2×D7—21no—
119C2×D7#21612yescomplemented
120C4×C7—41no—
121C4×C7—121no—
122C4×C7D3×D411yescharacteristic
123C4×C7#23711yescharacteristic
124Dic7—21no—
125Dic7—41no—
126Dic7—121no—
127C23×C3—71no—
128C3×Q8—21no—
129C3×Q8—21no—
130C2×C3×C4—21no—
131C2×C3×C4—71no—
132C2×C3×C4—141no—
133C2×C3×C4—141no—
134C2×C3×C4—141no—
135C2×C3×C4C22×D711yescharacteristic
136C2×Dic3—21no—
137C2×Dic3—141no—
138Dic6—21no—
139Dic6—21no—
140C3⋊C8—141no—
141C3⋊C8C22⋊D712yes—
142C3×C7#46711yescharacteristic, complemented, Hall
143C2×Q8—31no—
144C2×Q8D3×D711yescharacteristic
145C22×C4—71no—
146Q16—62no—
147C2×C8—31no—
148C2×C8—211no—
149C2×C8—422no—
150C4⋊C4—71no—
151C4⋊C4—141no—
152C4⋊C4—211no—
153C4⋊C4—421no—
154C22⋊C4—141no—
155C22⋊C4—421no—
156C2×C7#93911yescharacteristic
157C2×C7C2×#21611yescharacteristic
158C2×C7#95011yescharacteristic
159D7—22no—
160C22×C3—72no—
161C22×C3—141no—
162C22×C3—141no—
163C22×C3D4×D711yescharacteristic
164C3×C4—41no—
165C3×C4—141no—
166C3×C4—281no—
167C3×C4#115911yescharacteristic
168C3×C4C2×C22⋊D711yescharacteristic
169Dic3—41no—
170Dic3—281no—
171C23—71no—
172Q8—21no—
173Q8—21no—
174Q8—61no—
175Q8—61no—
176C2×C4—21no—
177C2×C4—61no—
178C2×C4—71no—
179C2×C4—141no—
180C2×C4—141no—
181C2×C4—141no—
182C2×C4—421no—
183C2×C4C2×D3×D711yescharacteristic
184C8—32no—
185C8—421no—
186C7#505111yescharacteristic, complemented, Sylow
187C2×C3—142no—
188C2×C3D7×Q1611yescharacteristic
189C2×C3#648211yescharacteristic
190C2×C3#650111yescharacteristic
191C22—72no—
192C22—141no—
193C22—141no—
194C22D7×#6711yescharacteristic
195C4—41no—
196C4—121no—
197C4—141no—
198C4—281no—
199C4—841no—
200C4#6668411yescharacteristic
201C4D3×C22⋊D711yescharacteristic
202C3#8996011yescharacteristic, complemented, Sylow
203C2—142no—
204C2D7×#21611yescharacteristic
205C2#1062107811yescharacteristic
206C2#1062111111yescharacteristic
subgroupG/H
centerC22D7×#67
commutator, cosocleC2×C3×C4×C7C23
FrattiniC2×C4C2×D3×D7
FittingC2×C3×C7×Q8C22
socleC22×C3×C7C2×D4
prime radicalC22×C3×D7D4
hypercenterC2×Q8D3×D7
hypocenterC3×C7#467
LCSG ▷ C2×C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. C2×C7×#216C2semidirect
5. #10620674C2semidirect
6. #10620525C2semidirect
7. #10620507C2semidirect
59. C2×#216D7semidirect
68. C2×C3×D7Q16semidirect
119. C2×D7#216semidirect
142. C3×C7#467semidirect
186. C7#5051semidirect
202. C3#89960semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#2162
C3×#65092
#106209212
C2×D7×C3⋊C82
#106206742
#106205252
#106205072
#899603
#50517