Finite Group Databasebeta

← C7×#10349 #13187565

Subgroups — C7×#10349 #13187565

Order 1792.

504 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 154 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×#2551C211yesmaximal, characteristic
2C7×#1987C212yesmaximal
3C7×#1956C211yesmaximal, characteristic
4C7×#1296C211yesmaximal, characteristic
5C7×#1279C212yesmaximal
6C2×C7×C4⋊Q8—22no—
7C2×C7×C4⋊Q8C2211yescharacteristic
8C2×C7×C4⋊C8C2212yes—
9C2×C7×#103—22no—
10C2×C4×C7×C8C2211yescharacteristic
11C7×#395—22no—
12C7×#389—22no—
13C7×#384C2211yescharacteristic
14C4×C7×C4⋊C4C2212yes—
15C7×#340—22no—
16C7×#340—22no—
17#10349C711yesmaximal, characteristic, direct product factor, Sylow
18C22×C7×Q8—22no—
19C22×C7×Q8—21no—
20C22×C7×C8—21no—
21C22×C7×C8—22no—
22C7×C4⋊Q8—22no—
23C7×C4⋊Q8—21no—
24C7×C4⋊Q8—44no—
25C7×C4⋊Q8—44no—
26C7×C4⋊Q8Q812yescomplemented
27C2×C7×C4⋊C4—22no—
28C2×C7×C4⋊C4—24no—
29C2×C7×C4⋊C4—22no—
30C2×C7×C4⋊C4D412yes—
31C2×C7×C4⋊C4D412yes—
32C2×C7×C4⋊C4D412yes—
33C2×C42×C7—22no—
34C2×C42×C7C2311yescharacteristic
35C7×C4⋊C8—22no—
36C7×C4⋊C8—22no—
37C7×#103—44no—
38C4×C7×C8—22no—
39C7×#95—22no—
40C7×#95—22no—
41#2551C2×C711yescharacteristic
42#1987C2×C712yes—
43#1956C2×C711yescharacteristic
44#1296C2×C711yescharacteristic
45#1279C2×C712yes—
46C2×C7×Q8—22no—
47C2×C7×Q8—48no—
48C2×C7×Q8—42no—
49C2×C7×Q8—42no—
50C2×C7×Q8—41no—
51C2×C7×Q8—82no—
52C22×C4×C7—22no—
53C22×C4×C7—22no—
54C22×C4×C7—22no—
55C22×C4×C7—41no—
56C22×C4×C7C2×D411yescharacteristic
57C22×C4×C7C2×D412yes—
58C2×C7×C8—22no—
59C2×C7×C8—41no—
60C2×C7×C8—41no—
61C2×C7×C8—42no—
62C2×C7×C8—42no—
63C2×C7×C8—42no—
64C7×C4⋊C4—22no—
65C7×C4⋊C4—22no—
66C7×C4⋊C4—24no—
67C7×C4⋊C4—42no—
68C7×C4⋊C4—48no—
69C7×C4⋊C4—42no—
70C7×C4⋊C4—42no—
71C7×C4⋊C4—42no—
72C7×C4⋊C4SD1614yes—
73C7×C4⋊C4Q1614yes—
74C42×C7—42no—
75C42×C7—42no—
76C42×C7C2×Q811yescharacteristic
77C42×C7#4112yes—
78C42×C7#4111yescharacteristic
79C2×C4⋊Q8—22no—
80C2×C4⋊Q8C22×C711yescharacteristic
81C2×C4⋊C8C22×C712yes—
82C2×#103—22no—
83C2×C4×C8C22×C711yescharacteristic
84#395—22no—
85#389—22no—
86#384C22×C711yescharacteristic
87C4×C4⋊C4C22×C712yes—
88#340—22no—
89#340—22no—
90C23×C7#12011yescharacteristic
91C7×Q8—44no—
92C7×Q8—82no—
93C7×Q8—88no—
94C2×C4×C7—21no—
95C2×C4×C7—21no—
96C2×C4×C7—24no—
97C2×C4×C7—24no—
98C2×C4×C7—42no—
99C2×C4×C7—42no—
100C2×C4×C7—42no—
101C2×C4×C7—42no—
102C2×C4×C7—42no—
103C2×C4×C7—44no—
104C2×C4×C7—42no—
105C2×C4×C7—81no—
106C2×C4×C7—81no—
107C2×C4×C7C22⋊Q811yescharacteristic
108C2×C4×C7C22⋊Q812yes—
109C2×C4×C7#12411yescharacteristic
110C2×C4×C7#12412yes—
111C2×C4×C7C2×SD1612yes—
112C2×C4×C7C2×Q1612yes—
113C2×C4×C7AGL(1,ℤ/8)12yes—
114C2×C4×C7#13712yes—
115C7×C8—42no—
116C7×C8—82no—
117C22×Q8—22no—
118C22×Q8—21no—
119C22×C8—21no—
120C22×C8—22no—
121C4⋊Q8—22no—
122C4⋊Q8—21no—
123C4⋊Q8—44no—
124C4⋊Q8—44no—
125C4⋊Q8C7×Q812yescomplemented
126C2×C4⋊C4—22no—
127C2×C4⋊C4—24no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4C7×D412yes—
130C2×C4⋊C4C7×D412yes—
131C2×C4⋊C4C7×D412yes—
132C2×C42—22no—
133C2×C42C23×C711yescharacteristic
134C4⋊C8—22no—
135C4⋊C8—22no—
136#103—44no—
137C4×C8—22no—
138#95—22no—
139#95—22no—
140C22×C7#39311yescharacteristic
141C22×C7#45012yes—
142C22×C7#45112yes—
143C22×C7#45311yescharacteristic
144C22×C7#45411yescharacteristic
145C4×C7—21no—
146C4×C7—21no—
147C4×C7—44no—
148C4×C7—44no—
149C4×C7—82no—
150C4×C7—82no—
151C4×C7#47512yes—
152C4×C7#47712yes—
153C4×C7#48712yes—
154C4×C7#48912yes—
155C2×Q8—22no—
156C2×Q8—48no—
157C2×Q8—42no—
158C2×Q8—42no—
159C2×Q8—41no—
160C2×Q8—82no—
161C22×C4—22no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—41no—
165C22×C4C2×C7×D411yescharacteristic
166C22×C4C2×C7×D412yes—
167C2×C8—22no—
168C2×C8—41no—
169C2×C8—41no—
170C2×C8—42no—
171C2×C8—42no—
172C2×C8—42no—
173C4⋊C4—22no—
174C4⋊C4—22no—
175C4⋊C4—24no—
176C4⋊C4—42no—
177C4⋊C4—48no—
178C4⋊C4—42no—
179C4⋊C4—42no—
180C4⋊C4—42no—
181C4⋊C4C7×SD1614yes—
182C4⋊C4C7×Q1614yes—
183C42—42no—
184C42—42no—
185C42C2×C7×Q811yescharacteristic
186C42C7×#4112yes—
187C42C7×#4111yescharacteristic
188C2×C7#162911yescharacteristic
189C2×C7#163311yescharacteristic
190C2×C7#163712yes—
191C2×C7#202412yes—
192C2×C7#202711yescharacteristic
193C23C7×#12011yescharacteristic
194Q8—44no—
195Q8—82no—
196Q8—88no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—24no—
200C2×C4—24no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—44no—
207C2×C4—42no—
208C2×C4—81no—
209C2×C4—81no—
210C2×C4C7×C22⋊Q811yescharacteristic
211C2×C4C7×C22⋊Q812yes—
212C2×C4C7×#12411yescharacteristic
213C2×C4C7×#12412yes—
214C2×C4C2×C7×SD1612yes—
215C2×C4C2×C7×Q1612yes—
216C2×C4C7×AGL(1,ℤ/8)12yes—
217C2×C4C7×#13712yes—
218C8—42no—
219C8—82no—
220C7#1034911yescharacteristic, direct product factor, Sylow
221C22C7×#39311yescharacteristic
222C22C7×#45012yes—
223C22C7×#45112yes—
224C22C7×#45311yescharacteristic
225C22C7×#45411yescharacteristic
226C4—21no—
227C4—21no—
228C4—44no—
229C4—44no—
230C4—82no—
231C4—82no—
232C4C7×#47512yes—
233C4C7×#47712yes—
234C4C7×#48712yes—
235C4C7×#48912yes—
236C2C7×#162911yescharacteristic
237C2C7×#163311yescharacteristic
238C2C7×#163712yes—
239C2C7×#202412yes—
240C2C7×#202711yescharacteristic
subgroupG/H
center, socle, prime radicalC23×C7#120
commutator, Frattini, cosocleC2×C42C23×C7
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23×C7 ◁ C2×C42×C7 ◁ G

Complements

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

NH ≅ G/Ntype
17. #10349C7direct (N × H)
26. C7×C4⋊Q8Q8semidirect
125. C4⋊Q8C7×Q8semidirect
220. C7#10349direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#25512
C7×#19872
C7×#19562
C7×#12962
C7×#12792
#103497