Finite Group Databasebeta

← D7×A4⋊Q8 #13062031

Subgroups — D7×A4⋊Q8 #13062031

Order 1344.

330 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 47 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
1C4×A4×D7C211yesmaximal, characteristic
2#10621496C211yesmaximal, characteristic, complemented
3C7×A4⋊Q8C211yesmaximal, characteristic, complemented
4D7×SL(2,ℤ/4)C212yesmaximal
5#10621477C212yesmaximal, complemented
6D7×C22⋊Q8—31nomaximal, Hall
7C2×A4×D7C2211yescharacteristic
8C4×C7×A4C2211yescharacteristic
9D7×Dic6—41nomaximal
10A4×Dic7C2211yescharacteristic
11C7⋊SL(2,ℤ/4)C2212yes—
12C7×SL(2,ℤ/4)C2212yes—
13C2×Q8×D7—31no—
14C22×C4×D7D311yescharacteristic
15C7×C22⋊Q8—31no—
16#6509—31no—
17#6487—31no—
18#6460—31no—
19#6459—32no—
20D7×C4⋊C4—31no—
21D7×C4⋊C4—32no—
22D7×C22⋊C4—32no—
23#6441—32no—
24C2×A4⋊Q8—71nomaximal, Hall
25C2×C7×A4C2311yescharacteristic
26A4×D7Q812yescomplemented
27Dic42—41no—
28C7×Dic6—41no—
29C3×C4×D7—41no—
30#4248—42no—
31Dic3×D7—42no—
32C23×D7C2×D311yescharacteristic
33C2×C7×Q8—31no—
34C22×C4×C7C2×D311yescharacteristic
35C22×Dic7C2×D311yescharacteristic
36Q8×D7—61no—
37Q8×D7—61no—
38Q8×D7—61no—
39Q8×D7—61no—
40C2×C4×D7—31no—
41C2×C4×D7—31no—
42C2×C4×D7—31no—
43C2×C4×D7—31no—
44C2×C4×D7—32no—
45C2×C4×D7—32no—
46C2×Dic14—32no—
47C2×Dic14—31no—
48C7×C4⋊C4—32no—
49C7×C4⋊C4—31no—
50C7×C22⋊C4—32no—
51C22⋊Dic7—32no—
52(C2×D7)⋊C4—32no—
53(C2×D7)⋊C4—32no—
54C4⋊Dic7—32no—
55C4⋊Dic7—31no—
56Dic7⋊C4—32no—
57Dic7⋊C4—32no—
58Dic7⋊C4—32no—
59C2×C4×A4—71no—
60C2×SL(2,ℤ/4)—72no—
61A4⋊Q8—71no—
62A4⋊Q8—72no—
63A4⋊Q8D711yescharacteristic, direct product factor
64C2×C3×D7—41no—
65C7×A4C2×Q811yescharacteristic, complemented
66C3×C4×C7—41no—
67Dic21—42no—
68C3×Dic7—41no—
69C7×Dic3—42no—
70C2×C22⋊Q8—211noSylow
71C23×C7C22×D311yescharacteristic
72C22×D7—31no—
73C22×D7—31no—
74C22×D7—32no—
75C22×D7Dic612yescomplemented
76C7×Q8—61no—
77C7×Q8—61no—
78C2×C4×C7—31no—
79C2×C4×C7—31no—
80C2×C4×C7—32no—
81C2×C4×C7—32no—
82C2×Dic7—31no—
83C2×Dic7—31no—
84C2×Dic7—32no—
85C2×Dic7—32no—
86C4×D7—31no—
87C4×D7—31no—
88C4×D7—31no—
89C4×D7—61no—
90C4×D7—64no—
91C4×D7—62no—
92C4×D7—62no—
93C4×D7S411yescharacteristic
94Dic14—62no—
95Dic14—61no—
96Dic14—61no—
97Dic14—62no—
98C22×A4—71no—
99C2×Dic6—281no—
100C4×A4—71no—
101C4×A4C2×D711yescharacteristic
102SL(2,ℤ/4)—72no—
103SL(2,ℤ/4)C2×D712yes—
104C2×C3×C7—41no—
105C3×D7—42no—
106C22×Q8—211no—
107C23×C4—71no—
108C22⋊Q8—31no—
109C22⋊Q8—212no—
110C22⋊Q8—212no—
111C22⋊Q8—211no—
112C22⋊Q8—211no—
113C22⋊Q8—211no—
114C2×C4⋊C4—211no—
115C2×C4⋊C4—212no—
116C2×C22⋊C4—212no—
117C22×C7—31no—
118C22×C7—31no—
119C22×C7C2×Dic611yescharacteristic, complemented
120C2×D7—31no—
121C2×D7—32no—
122C2×D7—32no—
123C2×D7—32no—
124C2×D7—61no—
125C2×D7C2×S411yescharacteristic
126C4×C7—31no—
127C4×C7—62no—
128C4×C7—62no—
129C4×C7C2×S411yescharacteristic
130Dic7—31no—
131Dic7—62no—
132Dic7—62no—
133Dic7C2×S411yescharacteristic
134C2×A4—72no—
135C2×A4C22×D711yescharacteristic
136C2×C3×C4—281no—
137C2×Dic3—282no—
138Dic6—41no—
139Dic6—282no—
140Dic6—281no—
141C3×C7—41noHall
142C24—71no—
143C2×Q8—31no—
144C2×Q8—211no—
145C2×Q8—212no—
146C2×Q8—421no—
147C2×Q8—421no—
148C2×Q8—421no—
149C2×Q8—421no—
150C22×C4—71no—
151C22×C4—211no—
152C22×C4—211no—
153C22×C4—211no—
154C22×C4—211no—
155C22×C4—212no—
156C22×C4—212no—
157C22×C4D3×D711yescharacteristic
158C4⋊C4—31no—
159C4⋊C4—32no—
160C4⋊C4—212no—
161C4⋊C4—212no—
162C4⋊C4—212no—
163C4⋊C4—211no—
164C4⋊C4—212no—
165C22⋊C4—32no—
166C22⋊C4—212no—
167C22⋊C4—212no—
168C22⋊C4—212no—
169C2×C7—31no—
170C2×C7—31no—
171C2×C7C22×S411yescharacteristic
172D7—32no—
173D7A4⋊Q812yesdirect product factor
174C22×C3—281no—
175A4Q8×D711yescharacteristic, complemented
176C3×C4—41no—
177C3×C4—281no—
178Dic3—42no—
179Dic3—282no—
180C23—72no—
181C23—211no—
182C23—211no—
183C23—212no—
184C23C2×D3×D711yescharacteristic
185Q8—61no—
186Q8—61no—
187Q8—421no—
188Q8—422no—
189Q8—422no—
190Q8—421no—
191C2×C4—31no—
192C2×C4—31no—
193C2×C4—32no—
194C2×C4—32no—
195C2×C4—71no—
196C2×C4—211no—
197C2×C4—211no—
198C2×C4—211no—
199C2×C4—211no—
200C2×C4—211no—
201C2×C4—212no—
202C2×C4—212no—
203C2×C4—421no—
204C2×C4—422no—
205C2×C4—424no—
206C2×C4—422no—
207C7C2×A4⋊Q811yescharacteristic, complemented, Sylow
208C2×C3—41no—
209C2×C3—282no—
210C22—31no—
211C22—31no—
212C22—71no—
213C22—212no—
214C22—212no—
215C22—212no—
216C22—211no—
217C22—421no—
218C22D7×Dic611yescharacteristic, complemented
219C4—31no—
220C4—62no—
221C4—62no—
222C4—71no—
223C4—211no—
224C4—422no—
225C4—422no—
226C4D7×S411yescharacteristic
227C3—41noSylow
228C2—31no—
229C2—31no—
230C2—72no—
231C2—212no—
232C2C2×D7×S411yescharacteristic
subgroupG/H
center, FrattiniC2C2×D7×S4
commutator, cosocleC2×C7×A4C23
FittingC22×C4×C7C2×D3
socleC23×C7C22×D3
prime radicalC2×A4×D7C22
hypercenterC4D7×S4
hypocenterC7×A4C2×Q8
LCSG ▷ C2×C7×A4 ▷ C7×A4
UCSC1 ◁ C2 ◁ C4

Complements

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

NH ≅ G/Ntype
2. #10621496C2semidirect
3. C7×A4⋊Q8C2semidirect
5. #10621477C2semidirect
26. A4×D7Q8semidirect
63. A4⋊Q8D7direct (N × H)
65. C7×A4C2×Q8semidirect
75. C22×D7Dic6semidirect
119. C22×C7C2×Dic6semidirect
173. D7A4⋊Q8direct (N × H)
175. A4Q8×D7semidirect
207. C7C2×A4⋊Q8semidirect
218. C22D7×Dic6semidirect

Maximal subgroups

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

subgroupindex
C4×A4×D72
#106214962
C7×A4⋊Q82
D7×SL(2,ℤ/4)2
#106214772
D7×C22⋊Q83
D7×Dic64
C2×A4⋊Q87