Finite Group Databasebeta

← C7×#10522 #13187738

Subgroups — C7×#10522 #13187738

Order 1792.

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

normal subgroups only

This group has 402 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×#1842C211yesmaximal, characteristic
4C7×#1550C212yesmaximal
5C7×#1292C211yesmaximal, characteristic
6C7×#533C2212yes—
7C7×#477—22no—
8C7×#467—22no—
9C7×#462—22no—
10C7×#423C2212yes—
11C2×C7×C4⋊C8—22no—
12C2×C7×C4⋊C8C2211yescharacteristic
13C2×C7×C4⋊C8C2211yescharacteristic
14C43×C7C2211yescharacteristic
15C7×#339—22no—
16C7×#339—22no—
17C7×#333—22no—
18C7×#332—22no—
19#10522C711yesmaximal, 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×C4⋊C8—44no—
43C7×#103—24no—
44#3249C2×C711yescharacteristic
45#1994C2×C712yes—
46#1842C2×C711yescharacteristic
47#1550C2×C712yes—
48#1292C2×C711yescharacteristic
49C2×C7×Q8—22no—
50C2×C7×Q8—22no—
51C22×C4×C7—21no—
52C22×C4×C7—21no—
53C22×C4×C7C2×D411yescharacteristic
54C22×C4×C7C2×D411yescharacteristic
55C22×C4×C7#4111yescharacteristic
56C7×Q16—82no—
57C7×M16—42no—
58C7×M16—42no—
59C2×C7×C8—22no—
60C2×C7×C8—42no—
61C2×C7×C8—42no—
62C2×C7×C8—41no—
63C2×C7×C8—42no—
64C7×C4⋊C4—22no—
65C7×C4⋊C4—22no—
66C7×C4⋊C4—22no—
67C7×C4⋊C4—22no—
68C7×C4⋊C4—42no—
69C7×C4⋊C4—42no—
70C7×C22⋊C4—42no—
71C42×C7—22no—
72C42×C7—22no—
73C42×C7—21no—
74C42×C7—22no—
75C42×C7—42no—
76C42×C7—42no—
77C42×C7Q1612yescomplemented
78C42×C7C2×D412yes—
79C42×C7#4112yes—
80#533C22×C712yes—
81#477—22no—
82#467—22no—
83#462—22no—
84#423C22×C712yes—
85C2×C4⋊C8—22no—
86C2×C4⋊C8C22×C711yescharacteristic
87C2×C4⋊C8C22×C711yescharacteristic
88C43C22×C711yescharacteristic
89#339—22no—
90#339—22no—
91#333—22no—
92#332—22no—
93C23×C7#12111yescharacteristic
94C7×Q8—42no—
95C7×Q8—42no—
96C7×Q8—82no—
97C2×C4×C7—22no—
98C2×C4×C7—22no—
99C2×C4×C7—22no—
100C2×C4×C7—21no—
101C2×C4×C7—21no—
102C2×C4×C7—22no—
103C2×C4×C7—41no—
104C2×C4×C7—41no—
105C2×C4×C7—41no—
106C2×C4×C7—42no—
107C2×C4×C7—42no—
108C2×C4×C7#12011yescharacteristic
109C2×C4×C7#12112yes—
110C2×C4×C7#12312yes—
111C2×C4×C7#12411yescharacteristic
112C2×C4×C7C2×Q1611yescharacteristic
113C2×C4×C7#13511yescharacteristic
114C2×C4×C7#13712yes—
115C7×C8—42no—
116C7×C8—81no—
117C7×C8—82no—
118C2×Q16—22no—
119C2×M16—22no—
120C22×C8—21no—
121C22×C8C7×D412yes—
122C4⋊Q8C7×D412yes—
123C4⋊Q8C7×D412yes—
124#125—22no—
125C22⋊Q8—22no—
126C22⋊Q8—22no—
127C4×Q8—22no—
128C4×Q8—22no—
129#117—22no—
130C2×C42—21no—
131C2×C42—21no—
132C2×C42C7×D411yescharacteristic
133C2×C42C7×D411yescharacteristic
134C2×C42C23×C711yescharacteristic
135#107—22no—
136C4⋊C8—22no—
137C4⋊C8—22no—
138C4⋊C8—22no—
139C4⋊C8—22no—
140C4⋊C8—44no—
141#103—24no—
142C22×C7—21no—
143C22×C7—21no—
144C22×C7#39411yescharacteristic
145C22×C7#46711yescharacteristic
146C22×C7#47011yescharacteristic
147C4×C7—24no—
148C4×C7—22no—
149C4×C7—21no—
150C4×C7—42no—
151C4×C7—82no—
152C4×C7—82no—
153C4×C7#45311yescharacteristic
154C4×C7#45611yescharacteristic
155C4×C7#46212yes—
156C4×C7#46412yes—
157C2×Q8—22no—
158C2×Q8—22no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4C2×C7×D411yescharacteristic
162C22×C4C2×C7×D411yescharacteristic
163C22×C4C7×#4111yescharacteristic
164Q16—82no—
165M16—42no—
166M16—42no—
167C2×C8—22no—
168C2×C8—42no—
169C2×C8—42no—
170C2×C8—41no—
171C2×C8—42no—
172C4⋊C4—22no—
173C4⋊C4—22no—
174C4⋊C4—22no—
175C4⋊C4—22no—
176C4⋊C4—42no—
177C4⋊C4—42no—
178C22⋊C4—42no—
179C42—22no—
180C42—22no—
181C42—21no—
182C42—22no—
183C42—42no—
184C42—42no—
185C42C7×Q1612yescomplemented
186C42C2×C7×D412yes—
187C42C7×#4112yes—
188C2×C7—21no—
189C2×C7—21no—
190C2×C7#203911yescharacteristic
191C2×C7#205011yescharacteristic
192C2×C7#205611yescharacteristic
193C23C7×#12111yescharacteristic
194Q8—42no—
195Q8—42no—
196Q8—82no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—22no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4C7×#12011yescharacteristic
209C2×C4C7×#12112yes—
210C2×C4C7×#12312yes—
211C2×C4C7×#12411yescharacteristic
212C2×C4C2×C7×Q1611yescharacteristic
213C2×C4C7×#13511yescharacteristic
214C2×C4C7×#13712yes—
215C8—42no—
216C8—81no—
217C8—82no—
218C7#1052211yescharacteristic, direct product factor, Sylow
219C22—21no—
220C22—21no—
221C22C7×#39411yescharacteristic
222C22C7×#46711yescharacteristic
223C22C7×#47011yescharacteristic
224C4—24no—
225C4—21no—
226C4—22no—
227C4—42no—
228C4—82no—
229C4—82no—
230C4C7×#45311yescharacteristic
231C4C7×#45611yescharacteristic
232C4C7×#46212yes—
233C4C7×#46412yes—
234C2—21no—
235C2—21no—
236C2C7×#203911yescharacteristic
237C2C7×#205011yescharacteristic
238C2C7×#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. #10522C7direct (N × H)
77. C42×C7Q16semidirect
185. C42C7×Q16semidirect
218. C7#10522direct (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×#18422
C7×#15502
C7×#12922
#105227