Finite Group Databasebeta

← C7×#10523 #13187739

Subgroups — C7×#10523 #13187739

Order 1792.

382 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 114 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×#3249C211yesmaximal, characteristic
2C7×#1994C212yesmaximal
3C7×#1831C211yesmaximal, characteristic
4C7×#1550C212yesmaximal
5C7×#1292C211yesmaximal, characteristic
6C7×#533C2212yes—
7C7×#477—22no—
8C7×#467—22no—
9C7×#462—22no—
10C7×#423C2212yes—
11C2×C7×C4⋊C8C2211yescharacteristic
12C2×C7×C4⋊C8C2211yescharacteristic
13C43×C7C2211yescharacteristic
14C7×#339—22no—
15C7×#339—22no—
16C7×#336—22no—
17C7×#333—22no—
18C7×#332—22no—
19#10523C711yesmaximal, characteristic, direct product factor, Sylow
20C2×C7×Q16—22no—
21C2×C7×M16—22no—
22C22×C7×C8—21no—
23C22×C7×C8D412yes—
24C7×C4⋊Q8D412yes—
25C7×C4⋊Q8D412yes—
26C7×#125—22no—
27C7×C22⋊Q8—22no—
28C7×C22⋊Q8—22no—
29C4×C7×Q8—22no—
30C4×C7×Q8—22no—
31C7×#117—22no—
32C2×C42×C7—21no—
33C2×C42×C7—21no—
34C2×C42×C7D411yescharacteristic
35C2×C42×C7D411yescharacteristic
36C2×C42×C7C2311yescharacteristic
37C7×#107—22no—
38C7×C4⋊C8—22no—
39C7×C4⋊C8—22no—
40C7×C4⋊C8—22no—
41C7×C4⋊C8—22no—
42C7×#103—24no—
43#3249C2×C711yescharacteristic
44#1994C2×C712yes—
45#1831C2×C711yescharacteristic
46#1550C2×C712yes—
47#1292C2×C711yescharacteristic
48C2×C7×Q8—22no—
49C2×C7×Q8—22no—
50C22×C4×C7—21no—
51C22×C4×C7—21no—
52C22×C4×C7C2×D411yescharacteristic
53C22×C4×C7C2×D411yescharacteristic
54C22×C4×C7#4111yescharacteristic
55C7×Q16—82no—
56C7×M16—42no—
57C7×M16—42no—
58C2×C7×C8—22no—
59C2×C7×C8—42no—
60C2×C7×C8—42no—
61C2×C7×C8—41no—
62C2×C7×C8—42no—
63C7×C4⋊C4—22no—
64C7×C4⋊C4—22no—
65C7×C4⋊C4—22no—
66C7×C4⋊C4—22no—
67C7×C4⋊C4—42no—
68C7×C4⋊C4—42no—
69C7×C22⋊C4—42no—
70C42×C7—21no—
71C42×C7—22no—
72C42×C7—41no—
73C42×C7—41no—
74C42×C7—42no—
75C42×C7—42no—
76C42×C7Q1612yescomplemented
77C42×C7C2×D412yes—
78C42×C7#4112yes—
79#533C22×C712yes—
80#477—22no—
81#467—22no—
82#462—22no—
83#423C22×C712yes—
84C2×C4⋊C8C22×C711yescharacteristic
85C2×C4⋊C8C22×C711yescharacteristic
86C43C22×C711yescharacteristic
87#339—22no—
88#339—22no—
89#336—22no—
90#333—22no—
91#332—22no—
92C23×C7#12111yescharacteristic
93C7×Q8—42no—
94C7×Q8—42no—
95C7×Q8—82no—
96C2×C4×C7—21no—
97C2×C4×C7—21no—
98C2×C4×C7—22no—
99C2×C4×C7—22no—
100C2×C4×C7—41no—
101C2×C4×C7—41no—
102C2×C4×C7—41no—
103C2×C4×C7—41no—
104C2×C4×C7—41no—
105C2×C4×C7—42no—
106C2×C4×C7—42no—
107C2×C4×C7#12011yescharacteristic
108C2×C4×C7#12112yes—
109C2×C4×C7#12312yes—
110C2×C4×C7#12411yescharacteristic
111C2×C4×C7C2×Q1611yescharacteristic
112C2×C4×C7#13511yescharacteristic
113C2×C4×C7#13712yes—
114C7×C8—42no—
115C7×C8—81no—
116C7×C8—82no—
117C2×Q16—22no—
118C2×M16—22no—
119C22×C8—21no—
120C22×C8C7×D412yes—
121C4⋊Q8C7×D412yes—
122C4⋊Q8C7×D412yes—
123#125—22no—
124C22⋊Q8—22no—
125C22⋊Q8—22no—
126C4×Q8—22no—
127C4×Q8—22no—
128#117—22no—
129C2×C42—21no—
130C2×C42—21no—
131C2×C42C7×D411yescharacteristic
132C2×C42C7×D411yescharacteristic
133C2×C42C23×C711yescharacteristic
134#107—22no—
135C4⋊C8—22no—
136C4⋊C8—22no—
137C4⋊C8—22no—
138C4⋊C8—22no—
139#103—24no—
140C22×C7—21no—
141C22×C7—21no—
142C22×C7#39411yescharacteristic
143C22×C7#46711yescharacteristic
144C22×C7#47011yescharacteristic
145C4×C7—22no—
146C4×C7—21no—
147C4×C7—42no—
148C4×C7—42no—
149C4×C7—82no—
150C4×C7—82no—
151C4×C7#45411yescharacteristic
152C4×C7#45511yescharacteristic
153C4×C7#46212yes—
154C4×C7#46412yes—
155C2×Q8—22no—
156C2×Q8—22no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4C2×C7×D411yescharacteristic
160C22×C4C2×C7×D411yescharacteristic
161C22×C4C7×#4111yescharacteristic
162Q16—82no—
163M16—42no—
164M16—42no—
165C2×C8—22no—
166C2×C8—42no—
167C2×C8—42no—
168C2×C8—41no—
169C2×C8—42no—
170C4⋊C4—22no—
171C4⋊C4—22no—
172C4⋊C4—22no—
173C4⋊C4—22no—
174C4⋊C4—42no—
175C4⋊C4—42no—
176C22⋊C4—42no—
177C42—21no—
178C42—22no—
179C42—41no—
180C42—41no—
181C42—42no—
182C42—42no—
183C42C7×Q1612yescomplemented
184C42C2×C7×D412yes—
185C42C7×#4112yes—
186C2×C7—21no—
187C2×C7—21no—
188C2×C7#204011yescharacteristic
189C2×C7#205011yescharacteristic
190C2×C7#205611yescharacteristic
191C23C7×#12111yescharacteristic
192Q8—42no—
193Q8—42no—
194Q8—82no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—41no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4C7×#12011yescharacteristic
207C2×C4C7×#12112yes—
208C2×C4C7×#12312yes—
209C2×C4C7×#12411yescharacteristic
210C2×C4C2×C7×Q1611yescharacteristic
211C2×C4C7×#13511yescharacteristic
212C2×C4C7×#13712yes—
213C8—42no—
214C8—81no—
215C8—82no—
216C7#1052311yescharacteristic, direct product factor, Sylow
217C22—21no—
218C22—21no—
219C22C7×#39411yescharacteristic
220C22C7×#46711yescharacteristic
221C22C7×#47011yescharacteristic
222C4—21no—
223C4—22no—
224C4—42no—
225C4—42no—
226C4—82no—
227C4—82no—
228C4C7×#45411yescharacteristic
229C4C7×#45511yescharacteristic
230C4C7×#46212yes—
231C4C7×#46412yes—
232C2—21no—
233C2—21no—
234C2C7×#204011yescharacteristic
235C2C7×#205011yescharacteristic
236C2C7×#205611yescharacteristic
subgroupG/H
center, socleC22×C7#394
commutator, Frattini, cosocleC2×C42C23×C7
prime radicalC23×C7#121
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C7 ◁ C2×C42×C7 ◁ G

Complements

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

NH ≅ G/Ntype
19. #10523C7direct (N × H)
76. C42×C7Q16semidirect
183. C42C7×Q16semidirect
216. C7#10523direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#32492
C7×#19942
C7×#18312
C7×#15502
C7×#12922
#105237