Finite Group Databasebeta

← C7×#9487 #13186703

Subgroups — C7×#9487 #13186703

Order 1792.

518 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 518 conjugacy classes of subgroups — too many to draw. Only its 134 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×#2975C211yesmaximal, characteristic
2C7×#2580C211yesmaximal, characteristic
3C7×#1800C211yesmaximal, characteristic
4C7×#1487C212yesmaximal, complemented
5C7×#1286C212yesmaximal, complemented
6C2×C4×C7×Q8C2211yescharacteristic
7C2×C4×C7×D4C2211yescharacteristic
8C7×#436—24no—
9C7×#431C2212yes—
10C7×#423C2212yes—
11C2×C7×#100—21no—
12C7×#407—22no—
13C7×#399C412yes—
14C7×#393C412yes—
15C7×#390—21no—
16C7×#389—21no—
17C7×#385—22no—
18C7×#382C2211yescharacteristic
19C7×#343—21no—
20C7×Q8⋊C8—24no—
21#9487C711yesmaximal, characteristic, direct product factor, Sylow
22C22×C7×Q8C2×C411yescharacteristic
23C22×C7×D4D411yescharacteristic
24C23×C4×C7D412yes—
25C2×C7×M16—24no—
26C2×C7×M16—22no—
27C4×C7×Q8—22no—
28C4×C7×Q8—22no—
29C4×C7×D4—22no—
30C4×C7×D4—22no—
31C2×C7×C4⋊C4—21no—
32C2×C7×C4⋊C4—21no—
33C2×C7×C4⋊C4—22no—
34C2×C7×C4⋊C4C2×C411yescharacteristic
35C2×C7×C4⋊C4D411yescharacteristic
36C2×C7×C22⋊C4—22no—
37C2×C7×C22⋊C4—22no—
38C2×C7×C22⋊C4C2×C412yes—
39C2×C42×C7—21no—
40C2×C42×C7C2311yescharacteristic
41C7×C4⋊C8—22no—
42C7×C4⋊C8—22no—
43C7×C4⋊C8—22no—
44C7×#100—42no—
45C7×C22⋊C8—24no—
46C7×C22⋊C8—22no—
47C4×C7×C8D414yes—
48C7×#95—22no—
49C7×#95—21no—
50C7×#95C2×C412yes—
51#2975C2×C711yescharacteristic
52#2580C2×C711yescharacteristic
53#1800C2×C711yescharacteristic
54#1487C2×C712yescomplemented
55#1286C2×C712yescomplemented
56C24×C7C22⋊C412yes—
57C2×C7×Q8—22no—
58C2×C7×Q8—42no—
59C2×C7×D4—22no—
60C2×C7×D4—21no—
61C2×C7×D4—22no—
62C2×C7×D4—21no—
63C22×C4×C7—21no—
64C22×C4×C7—22no—
65C22×C4×C7—22no—
66C22×C4×C7—22no—
67C22×C4×C7—21no—
68C22×C4×C7—82no—
69C22×C4×C7C22⋊C412yes—
70C22×C4×C7C22×C411yescharacteristic
71C22×C4×C7C2×D411yescharacteristic
72C22×C4×C7C2×D411yescharacteristic
73C7×M16—42no—
74C7×M16—42no—
75C7×M16—84no—
76C2×C7×C8—24no—
77C2×C7×C8—42no—
78C7×C4⋊C4—22no—
79C7×C4⋊C4—22no—
80C7×C4⋊C4—41no—
81C7×C4⋊C4—41no—
82C7×C4⋊C4—41no—
83C7×C4⋊C4—41no—
84C7×C4⋊C4—82no—
85C7×C22⋊C4—24no—
86C7×C22⋊C4—44no—
87C7×C22⋊C4—82no—
88C42×C7—42no—
89C42×C7C2×D412yes—
90C42×C7#4112yes—
91C2×C4×Q8C22×C711yescharacteristic
92C2×C4×D4C22×C711yescharacteristic
93#436—24no—
94#431C22×C712yes—
95#423C22×C712yes—
96C2×#100—21no—
97#407—22no—
98#399C4×C712yes—
99#393C4×C712yes—
100#390—21no—
101#389—21no—
102#385—22no—
103#382C22×C711yescharacteristic
104#343—21no—
105Q8⋊C8—24no—
106C23×C7—22no—
107C23×C7—22no—
108C23×C7—22no—
109C23×C7—42no—
110C23×C7—42no—
111C23×C7C2×C22⋊C411yescharacteristic
112C7×Q8—24no—
113C7×Q8—42no—
114C7×D4—42no—
115C7×D4—42no—
116C2×C4×C7—22no—
117C2×C4×C7—22no—
118C2×C4×C7—41no—
119C2×C4×C7—41no—
120C2×C4×C7—44no—
121C2×C4×C7—41no—
122C2×C4×C7—41no—
123C2×C4×C7—42no—
124C2×C4×C7—42no—
125C2×C4×C7—41no—
126C2×C4×C7—42no—
127C2×C4×C7—41no—
128C2×C4×C7—41no—
129C2×C4×C7—42no—
130C2×C4×C7—82no—
131C2×C4×C7C4×D412yes—
132C2×C4×C7#12011yescharacteristic
133C2×C4×C7#12112yes—
134C2×C4×C7#12311yescharacteristic
135C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
136C2×C4×C7#13712yes—
137C2×C4×C7#13711yescharacteristic
138C7×C8—44no—
139C7×C8—82no—
140C22×Q8C2×C4×C711yescharacteristic
141C22×D4C7×D411yescharacteristic
142C23×C4C7×D412yes—
143C2×M16—24no—
144C2×M16—22no—
145C4×Q8—22no—
146C4×Q8—22no—
147C4×D4—22no—
148C4×D4—22no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4C2×C4×C711yescharacteristic
153C2×C4⋊C4C7×D411yescharacteristic
154C2×C22⋊C4—22no—
155C2×C22⋊C4—22no—
156C2×C22⋊C4C2×C4×C712yes—
157C2×C42—21no—
158C2×C42C23×C711yescharacteristic
159C4⋊C8—22no—
160C4⋊C8—22no—
161C4⋊C8—22no—
162#100—42no—
163C22⋊C8—24no—
164C22⋊C8—22no—
165C4×C8C7×D414yes—
166#95—22no—
167#95—21no—
168#95C2×C4×C712yes—
169C22×C7—21no—
170C22×C7—21no—
171C22×C7—44no—
172C22×C7—42no—
173C22×C7—42no—
174C22×C7—42no—
175C22×C7—82no—
176C22×C7#38611yescharacteristic
177C22×C7#41711yescharacteristic
178C22×C7#41911yescharacteristic
179C4×C7—22no—
180C4×C7—21no—
181C4×C7—21no—
182C4×C7—42no—
183C4×C7—42no—
184C4×C7—82no—
185C4×C7—81no—
186C4×C7#46812yes—
187C4×C7#47012yes—
188C24C7×C22⋊C412yes—
189C2×Q8—22no—
190C2×Q8—42no—
191C2×D4—22no—
192C2×D4—21no—
193C2×D4—22no—
194C2×D4—21no—
195C22×C4—21no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—21no—
200C22×C4—82no—
201C22×C4C7×C22⋊C412yes—
202C22×C4C22×C4×C711yescharacteristic
203C22×C4C2×C7×D411yescharacteristic
204C22×C4C2×C7×D411yescharacteristic
205M16—42no—
206M16—42no—
207M16—84no—
208C2×C8—24no—
209C2×C8—42no—
210C4⋊C4—22no—
211C4⋊C4—22no—
212C4⋊C4—41no—
213C4⋊C4—41no—
214C4⋊C4—41no—
215C4⋊C4—41no—
216C4⋊C4—82no—
217C22⋊C4—24no—
218C22⋊C4—44no—
219C22⋊C4—82no—
220C42—42no—
221C42C2×C7×D412yes—
222C42C7×#4112yes—
223C2×C7—21no—
224C2×C7—21no—
225C2×C7—82no—
226C2×C7#189111yescharacteristic
227C2×C7#189411yescharacteristic
228C2×C7#191111yescharacteristic
229C23—22no—
230C23—22no—
231C23—22no—
232C23—42no—
233C23—42no—
234C23C2×C7×C22⋊C411yescharacteristic
235Q8—24no—
236Q8—42no—
237D4—42no—
238D4—42no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—44no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—41no—
249C2×C4—42no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—42no—
253C2×C4—82no—
254C2×C4C4×C7×D412yes—
255C2×C4C7×#12011yescharacteristic
256C2×C4C7×#12112yes—
257C2×C4C7×#12311yescharacteristic
258C2×C4C7×AGL(1,ℤ/8)11yescharacteristic
259C2×C4C7×#13712yes—
260C2×C4C7×#13711yescharacteristic
261C8—44no—
262C8—82no—
263C7#948711yescharacteristic, direct product factor, Sylow
264C22—21no—
265C22—21no—
266C22—44no—
267C22—42no—
268C22—42no—
269C22—42no—
270C22—82no—
271C22C7×#38611yescharacteristic
272C22C7×#41711yescharacteristic
273C22C7×#41911yescharacteristic
274C4—22no—
275C4—21no—
276C4—21no—
277C4—42no—
278C4—42no—
279C4—82no—
280C4—81no—
281C4C7×#46812yes—
282C4C7×#47012yes—
283C2—21no—
284C2—21no—
285C2—82no—
286C2C7×#189111yescharacteristic
287C2C7×#189411yescharacteristic
288C2C7×#191111yescharacteristic
subgroupG/H
center, socleC22×C7#386
commutatorC22×C4C22×C4×C7
Frattini, cosocleC2×C42C23×C7
prime radicalC22×C7×D4D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C7 ◁ C2×C42×C7 ◁ G

Complements

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

NH ≅ G/Ntype
4. C7×#1487C2semidirect
5. C7×#1286C2semidirect
21. #9487C7direct (N × H)
54. #1487C2×C7semidirect
55. #1286C2×C7semidirect
263. C7#9487direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#29752
C7×#25802
C7×#18002
C7×#14872
C7×#12862
#94877