Finite Group Databasebeta

← C7×#7396 #13184612

Subgroups — C7×#7396 #13184612

Order 1792.

296 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 42 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
1C7×#2010C211yesmaximal, characteristic
2C7×#1320C211yesmaximal, characteristic, complemented
3C7×#1297C211yesmaximal, characteristic, complemented
4C7×#575C411yescharacteristic, complemented
5C7×#525—21no—
6C7×#421—21no—
7C7×#407C2211yescharacteristic
8C4×C7×C22⋊C4—21no—
9C7×#349—21no—
10C7×H(3,ℤ/4)C411yescharacteristic, complemented
11#7396C711yesmaximal, characteristic, direct product factor, Sylow
12C2×C7×#41—21no—
13C23×C4×C7D411yescharacteristic
14C7×#137—41no—
15C7×AGL(1,ℤ/8)—41no—
16C7×#135—41no—
17C2×C7×M16C2×C411yescharacteristic
18C2×C7×M16D411yescharacteristic
19C7×#123—41no—
20C7×C22⋊Q8—41no—
21C7×#121—41no—
22C7×#120—41no—
23C4×C7×D4—42no—
24C7×#117—21no—
25C2×C7×C22⋊C4—41no—
26C2×C42×C7—41no—
27C7×M32—41no—
28C7×#104—42no—
29C7×C22⋊C8—41no—
30C7×#95—22no—
31#2010C2×C711yescharacteristic
32#1320C2×C711yescharacteristic, complemented
33#1297C2×C711yescharacteristic, complemented
34C24×C7D811yescharacteristic
35C7×#41—41no—
36C7×#41—41no—
37C7×#41—41no—
38C7×#41—41no—
39C2×C7×Q8—21no—
40C2×C7×D4—21no—
41C2×C7×D4—42no—
42C2×C7×D4—41no—
43C22×C4×C7—21no—
44C22×C4×C7—21no—
45C22×C4×C7—41no—
46C22×C4×C7—41no—
47C22×C4×C7—41no—
48C22×C4×C7—42no—
49C22×C4×C7C22⋊C411yescharacteristic
50C22×C4×C7SD1611yescharacteristic
51C7×Q16—41no—
52C7×SD16—42no—
53C7×D8—41no—
54C7×M16—41no—
55C7×M16—41no—
56C2×C7×C8—21no—
57C2×C7×C8—21no—
58C7×C4⋊C4—41no—
59C7×C4⋊C4—42no—
60C7×C22⋊C4—44no—
61C7×C22⋊C4—42no—
62C7×C22⋊C4—42no—
63C7×C22⋊C4—41no—
64C42×C7—22no—
65C42×C7—81no—
66C42×C7—81no—
67C7×C16—81no—
68#575C4×C711yescharacteristic, complemented
69#525—21no—
70#421—21no—
71#407C22×C711yescharacteristic
72C4×C22⋊C4—21no—
73#349—21no—
74H(3,ℤ/4)C4×C711yescharacteristic, complemented
75C23×C7—21no—
76C23×C7—41no—
77C23×C7—42no—
78C23×C7—41no—
79C23×C7#10211yescharacteristic
80C7×Q8—41no—
81C7×Q8—41no—
82C7×D4—41no—
83C7×D4—41no—
84C7×D4—81no—
85C7×D4—84no—
86C7×D4—81no—
87C2×C4×C7—21no—
88C2×C4×C7—21no—
89C2×C4×C7—21no—
90C2×C4×C7—41no—
91C2×C4×C7—41no—
92C2×C4×C7—41no—
93C2×C4×C7—42no—
94C2×C4×C7—41no—
95C2×C4×C7—44no—
96C2×C4×C7—81no—
97C2×C4×C7—81no—
98C2×C4×C7—82no—
99C2×C4×C7—82no—
100C2×C4×C7—81no—
101C2×C4×C7#9911yescharacteristic
102C2×C4×C7#10411yescharacteristic
103C7×C8—41no—
104C7×C8—41no—
105C2×#41—21no—
106C23×C4C7×D411yescharacteristic
107#137—41no—
108AGL(1,ℤ/8)—41no—
109#135—41no—
110C2×M16C2×C4×C711yescharacteristic
111C2×M16C7×D411yescharacteristic
112#123—41no—
113C22⋊Q8—41no—
114#121—41no—
115#120—41no—
116C4×D4—42no—
117#117—21no—
118C2×C22⋊C4—41no—
119C2×C42—41no—
120M32—41no—
121#104—42no—
122C22⋊C8—41no—
123#95—22no—
124C22×C7—22no—
125C22×C7—21no—
126C22×C7—41no—
127C22×C7—41no—
128C22×C7—42no—
129C22×C7—41no—
130C22×C7—81no—
131C22×C7—82no—
132C22×C7—81no—
133C22×C7#32711yescharacteristic
134C4×C7—21no—
135C4×C7—41no—
136C4×C7—84no—
137C4×C7—82no—
138C4×C7—81no—
139C4×C7AΣL(1,𝔽16)11yescharacteristic
140C4×C7#35311yescharacteristic
141C24C7×D811yescharacteristic
142#41—41no—
143#41—41no—
144#41—41no—
145#41—41no—
146C2×Q8—21no—
147C2×D4—21no—
148C2×D4—42no—
149C2×D4—41no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—41no—
153C22×C4—41no—
154C22×C4—41no—
155C22×C4—42no—
156C22×C4C7×C22⋊C411yescharacteristic
157C22×C4C7×SD1611yescharacteristic
158Q16—41no—
159SD16—42no—
160D8—41no—
161M16—41no—
162M16—41no—
163C2×C8—21no—
164C2×C8—21no—
165C4⋊C4—41no—
166C4⋊C4—42no—
167C22⋊C4—44no—
168C22⋊C4—42no—
169C22⋊C4—42no—
170C22⋊C4—41no—
171C42—22no—
172C42—81no—
173C42—81no—
174C16—81no—
175C2×C7—21no—
176C2×C7—42no—
177C2×C7—41no—
178C2×C7—161no—
179C2×C7#134311yescharacteristic
180C23—21no—
181C23—41no—
182C23—42no—
183C23—41no—
184C23C7×#10211yescharacteristic
185Q8—41no—
186Q8—41no—
187D4—41no—
188D4—41no—
189D4—81no—
190D4—84no—
191D4—81no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—41no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—42no—
199C2×C4—41no—
200C2×C4—44no—
201C2×C4—81no—
202C2×C4—81no—
203C2×C4—82no—
204C2×C4—82no—
205C2×C4—81no—
206C2×C4C7×#9911yescharacteristic
207C2×C4C7×#10411yescharacteristic
208C8—41no—
209C8—41no—
210C7#739611yescharacteristic, direct product factor, Sylow
211C22—22no—
212C22—21no—
213C22—41no—
214C22—41no—
215C22—42no—
216C22—41no—
217C22—81no—
218C22—82no—
219C22—81no—
220C22C7×#32711yescharacteristic
221C4—21no—
222C4—41no—
223C4—84no—
224C4—82no—
225C4—81no—
226C4C7×AΣL(1,𝔽16)11yescharacteristic
227C4C7×#35311yescharacteristic
228C2—21no—
229C2—41no—
230C2—42no—
231C2—161no—
232C2C7×#134311yescharacteristic
subgroupG/H
centerC4×C7#353
commutatorC2×M16C2×C4×C7
Frattini, cosocle#407C22×C7
socleC2×C7#1343
prime radicalC7×#2010C2
LCSG ▷ C2×M16 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C4×C7 ◁ C2×C4×C7 ◁ C22×C4×C7 ◁ C7×#407 ◁ G

Complements

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

NH ≅ G/Ntype
2. C7×#1320C2semidirect
3. C7×#1297C2semidirect
4. C7×#575C4semidirect
10. C7×H(3,ℤ/4)C4semidirect
11. #7396C7direct (N × H)
32. #1320C2×C7semidirect
33. #1297C2×C7semidirect
68. #575C4×C7semidirect
74. H(3,ℤ/4)C4×C7semidirect
210. C7#7396direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#20102
C7×#13202
C7×#12972
#73967