Finite Group Databasebeta

← #13054657

Subgroups — #13054657

Order 1344.

296 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 76 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×C3×Q8×D7C211yesmaximal, characteristic
2C2×#66635C211yesmaximal, characteristic, complemented
3#10620922C211yesmaximal, characteristic
4C2×#66570C211yesmaximal, characteristic, complemented
5C7×#835C211yesmaximal, characteristic, complemented
6#10620505C211yesmaximal, characteristic, complemented
7(C2×D7)⋊(C3⋊C8)C211yesmaximal, characteristic
8#89580—31nomaximal, Hall
9C2×C3×C7×Q8C2211yescharacteristic
10C2×Dic42C2211yescharacteristic
11C3×Q8×D7—22no—
12C3×Q8×D7—22no—
13C2×C3×C4×D7—21no—
14C2×C3×C4×D7C2211yescharacteristic
15C2×C3×Dic14—21no—
16C2×C3×Dic14C2211yescharacteristic
17#66635—22no—
18C2×C3⋊(C7⋊C8)C2211yescharacteristic
19C7×C4⋊Dic3C2211yescharacteristic
20C2×C7×C3⋊C8C2211yescharacteristic
21Dic3⋊Dic7—21no—
22(C2×D7)⋊Dic3—21no—
23#66570—22no—
24C2×Q8×D7D311yescharacteristic
25C2×#1144—31no—
26C2×Dic28—31no—
27#6460—31no—
28C7×#103—31no—
29(C2×D7)⋊C8—31no—
30#6384—31no—
31#5228—71nomaximal, Hall
32C22×C3×D7D411yescharacteristic
33C3×C7×Q8—21no—
34C3×C7×Q8D412yes—
35C2×C3×C4×C7—21no—
36C2×C3×C4×C7C2311yescharacteristic
37C2×Dic21—21no—
38Dic42—21no—
39Dic42—21no—
40C2×C7×Dic3—21no—
41C2×C3×Dic7—21no—
42C2×C3×Dic7D411yescharacteristic
43C3×C4×D7—24no—
44C3×C4×D7—21no—
45C3×C4×D7—21no—
46C3×Dic14—24no—
47C3×Dic14—21no—
48C3×Dic14D412yes—
49C3⋊(C7⋊C8)—21no—
50C7×C3⋊C8—21no—
51C2×C7×Q8C2×D311yescharacteristic
52Q8×D7—22no—
53Q8×D7—22no—
54C2×C4×D7—21no—
55C2×C4×D7C2×D311yescharacteristic
56C2×Dic14—21no—
57C2×Dic14—31no—
58C2×Dic14C2×D311yescharacteristic
59C2×C7×C8—31no—
60C7×C4⋊C4—31no—
61#1144—62no—
62(C2×D7)⋊C4—61no—
63C4⋊Dic7—61no—
64C2×C7⋊C8—31no—
65Dic28—62no—
66C22×C3×Q8—71no—
67C2×#216—71no—
68C2×#216—71no—
69#924—71no—
70#835—71no—
71#835D711yescharacteristic, complemented
72#830—71no—
73C22×C3×C7C2×D411yescharacteristic
74C2×C3×D7—21no—
75C2×C3×D7—21no—
76C2×C3×D7Q1612yescomplemented
77C3×C4×C7—22no—
78C3×C4×C7C2×D411yescharacteristic
79C3×C4×C7C2×D411yescharacteristic
80Dic21—41no—
81C3×Dic7—21no—
82C3×Dic7—22no—
83C7×Dic3—41no—
84#451—211noSylow
85C22×D7#6711yescharacteristic
86C7×Q8—21no—
87C7×Q8#6712yes—
88C2×C4×C7—21no—
89C2×C4×C7—61no—
90C2×C4×C7C22×D311yescharacteristic
91C2×Dic7—21no—
92C2×Dic7—61no—
93C2×Dic7#6711yescharacteristic
94C4×D7—24no—
95C4×D7—21no—
96C4×D7—21no—
97Dic14—24no—
98Dic14—21no—
99Dic14—61no—
100Dic14—61no—
101Dic14#6712yes—
102C7×C8—61no—
103C7⋊C8—61no—
104C2×C3×Q8—71no—
105C2×C3×Q8—141no—
106C2×C3×Q8—142no—
107C2×C3×Q8—142no—
108C2×C3×Q8C2×D711yescharacteristic
109C22×C3×C4—71no—
110C22×C3×C4—141no—
111C2×Dic6—71no—
112#217—141no—
113#216—142no—
114#216—142no—
115C4⋊Dic3C2×D711yescharacteristic
116Dic3⋊C4—141no—
117C2×C3⋊C8—71no—
118C2×C3⋊C8C2×D711yescharacteristic
119C2×C3×C7#12011yescharacteristic
120C2×C3×C7C2×Q1611yescharacteristic
121C2×C3×C7#13711yescharacteristic
122C3×D7—22no—
123C22×Q8—71no—
124C2×Q16—211no—
125C2×Q16—211no—
126C22⋊Q8—211no—
127#103—31no—
128#103—211no—
129C22⋊C8—211no—
130C22×C7C2×#6711yescharacteristic
131C2×D7—21no—
132C2×D7—21no—
133C2×D7#21612yescomplemented
134C4×C7—22no—
135C4×C7—121no—
136C4×C7C2×#6711yescharacteristic
137C4×C7C2×#6711yescharacteristic
138Dic7—21no—
139Dic7—22no—
140Dic7—121no—
141C23×C3—71no—
142C3×Q8—21no—
143C3×Q8—72no—
144C3×Q8—144no—
145C3×Q8—141no—
146C3×Q8D2812yes—
147C2×C3×C4—21no—
148C2×C3×C4—71no—
149C2×C3×C4—144no—
150C2×C3×C4—141no—
151C2×C3×C4—141no—
152C2×C3×C4—141no—
153C2×C3×C4C22×D711yescharacteristic
154C2×Dic3—21no—
155C2×Dic3—141no—
156Dic6—141no—
157Dic6—141no—
158C3⋊C8—21no—
159C3⋊C8—141no—
160C3×C7#45111yescharacteristic, complemented, Hall
161C2×Q8—71no—
162C2×Q8—141no—
163C2×Q8—142no—
164C2×Q8—142no—
165C2×Q8—211no—
166C2×Q8D3×D711yescharacteristic
167C22×C4—71no—
168C22×C4—141no—
169Q16—422no—
170Q16—422no—
171C2×C8—31no—
172C2×C8—211no—
173C4⋊C4—31no—
174C4⋊C4—421no—
175C22⋊C4—421no—
176C2×C7#94211yescharacteristic
177C2×C7C2×#21611yescharacteristic
178C2×C7#95311yescharacteristic
179D7—22no—
180C22×C3—72no—
181C22×C3—141no—
182C22×C3—141no—
183C22×C3D4×D711yescharacteristic
184C3×C4—22no—
185C3×C4—141no—
186C3×C4—142no—
187C3×C4C2×D2811yescharacteristic
188C3×C4D4×D711yescharacteristic
189Dic3—41no—
190Dic3—281no—
191C23—71no—
192Q8—21no—
193Q8—72no—
194Q8—144no—
195Q8—141no—
196Q8—421no—
197Q8—421no—
198Q8#424612yes—
199C2×C4—21no—
200C2×C4—61no—
201C2×C4—71no—
202C2×C4—144no—
203C2×C4—141no—
204C2×C4—141no—
205C2×C4—141no—
206C2×C4—421no—
207C2×C4C2×D3×D711yescharacteristic
208C8—61no—
209C8—421no—
210C7#522811yescharacteristic, complemented, Sylow
211C2×C3—142no—
212C2×C3#644511yescharacteristic
213C2×C3#647811yescharacteristic
214C2×C3D7×Q1611yescharacteristic
215C22—72no—
216C22—141no—
217C22—141no—
218C22D7×#6711yescharacteristic
219C4—22no—
220C4—121no—
221C4—141no—
222C4—142no—
223C4—841no—
224C4C2×#424611yescharacteristic
225C4D7×#6711yescharacteristic
226C3#8958011yescharacteristic, complemented, Sylow
227C2—142no—
228C2D7×#21611yescharacteristic
229C2#1062107511yescharacteristic
230C2#1062112611yescharacteristic
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#451
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
2. C2×#66635C2semidirect
4. C2×#66570C2semidirect
5. C7×#835C2semidirect
6. #10620505C2semidirect
71. #835D7semidirect
76. C2×C3×D7Q16semidirect
133. C2×D7#216semidirect
160. C3×C7#451semidirect
210. C7#5228semidirect
226. C3#89580semidirect

Maximal subgroups

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

subgroupindex
C2×C3×Q8×D72
C2×#666352
#106209222
C2×#665702
C7×#8352
#106205052
(C2×D7)⋊(C3⋊C8)2
#895803
#52287