Finite Group Databasebeta

← C7×#8734 #13185950

Subgroups — C7×#8734 #13185950

Order 1792.

622 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 622 conjugacy classes of subgroups — too many to draw. Only its 166 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×#2273C211yesmaximal, characteristic
2C2×C7×#323C212yesmaximal
3C7×#1282C214yesmaximal, complemented
4C22×C7×C22⋊C4C411yescharacteristic
5C22×C7×C22⋊C4C411yescharacteristic
6C22×C7×C22⋊C4C2211yescharacteristic
7C2×C7×C22⋊C8C2214yes—
8C4×C7×C22⋊C4—22no—
9C4×C7×C22⋊C4—22no—
10C4×C7×C22⋊C4C412yes—
11C4×C7×C22⋊C4C2212yes—
12C7×#323—22no—
13C7×#323—22no—
14#8734C711yesmaximal, characteristic, direct product factor, Sylow
15C25×C7C2×C411yescharacteristic
16C23×C4×C7—21no—
17C23×C4×C7—21no—
18C23×C4×C7C2×C411yescharacteristic
19C23×C4×C7C2311yescharacteristic
20C22×C7×C8—24no—
21C2×C7×C22⋊C4—21no—
22C2×C7×C22⋊C4—21no—
23C2×C7×C22⋊C4—22no—
24C2×C7×C22⋊C4—24no—
25C2×C7×C22⋊C4—24no—
26C2×C7×C22⋊C4C814yescomplemented
27C2×C7×C22⋊C4C2×C412yes—
28C2×C7×C22⋊C4C2×C412yes—
29C2×C7×C22⋊C4D412yes—
30C2×C7×C22⋊C4D412yes—
31C2×C42×C7—22no—
32C2×C42×C7—22no—
33C7×C22⋊C8—24no—
34C7×C22⋊C8—24no—
35C7×#95—22no—
36C7×#95—22no—
37#2273C2×C711yescharacteristic
38C2×#323C2×C712yes—
39#1282C2×C714yescomplemented
40C24×C7—22no—
41C24×C7—22no—
42C24×C7—81no—
43C24×C7—81no—
44C24×C7C22⋊C411yescharacteristic
45C24×C7C22⋊C411yescharacteristic
46C24×C7C2×C812yes—
47C24×C7M1612yes—
48C24×C7C22×C411yescharacteristic
49C22×C4×C7—22no—
50C22×C4×C7—22no—
51C22×C4×C7—24no—
52C22×C4×C7—24no—
53C22×C4×C7—22no—
54C22×C4×C7—81no—
55C22×C4×C7—81no—
56C22×C4×C7—81no—
57C22×C4×C7—81no—
58C22×C4×C7—81no—
59C22×C4×C7—81no—
60C22×C4×C7C22⋊C412yes—
61C22×C4×C7C2×D412yes—
62C22×C4×C7#4114yes—
63C2×C7×C8—44no—
64C2×C7×C8—84no—
65C7×C22⋊C4—24no—
66C7×C22⋊C4—24no—
67C7×C22⋊C4—22no—
68C7×C22⋊C4—22no—
69C7×C22⋊C4—24no—
70C7×C22⋊C4—24no—
71C7×C22⋊C4C2×C814yes—
72C7×C22⋊C4M1614yes—
73C42×C7—82no—
74C42×C7—82no—
75C22×C22⋊C4C4×C711yescharacteristic
76C22×C22⋊C4C4×C711yescharacteristic
77C22×C22⋊C4C22×C711yescharacteristic
78C2×C22⋊C8C22×C714yes—
79C4×C22⋊C4—22no—
80C4×C22⋊C4—22no—
81C4×C22⋊C4C4×C712yes—
82C4×C22⋊C4C22×C712yes—
83#323—22no—
84#323—22no—
85C23×C7—21no—
86C23×C7—21no—
87C23×C7—21no—
88C23×C7—21no—
89C23×C7—24no—
90C23×C7—24no—
91C23×C7—41no—
92C23×C7—41no—
93C23×C7—41no—
94C23×C7—41no—
95C23×C7—41no—
96C23×C7—41no—
97C23×C7—82no—
98C23×C7—82no—
99C23×C7—82no—
100C23×C7—82no—
101C23×C7—84no—
102C23×C7C22⋊C812yes—
103C23×C7C22⋊C812yes—
104C23×C7C2×C22⋊C411yescharacteristic
105C23×C7#11712yes—
106C23×C7C22×C811yescharacteristic
107C23×C7C2×M1611yescharacteristic
108C23×C7C2×M1612yes—
109C2×C4×C7—22no—
110C2×C4×C7—24no—
111C2×C4×C7—24no—
112C2×C4×C7—42no—
113C2×C4×C7—44no—
114C2×C4×C7—42no—
115C2×C4×C7—42no—
116C2×C4×C7—42no—
117C2×C4×C7—42no—
118C2×C4×C7—42no—
119C2×C4×C7—82no—
120C2×C4×C7—84no—
121C2×C4×C7—84no—
122C2×C4×C7#12314yes—
123C7×C8—84no—
124C25C2×C4×C711yescharacteristic
125C23×C4—21no—
126C23×C4—21no—
127C23×C4C2×C4×C711yescharacteristic
128C23×C4C23×C711yescharacteristic
129C22×C8—24no—
130C2×C22⋊C4—21no—
131C2×C22⋊C4—21no—
132C2×C22⋊C4—22no—
133C2×C22⋊C4—24no—
134C2×C22⋊C4—24no—
135C2×C22⋊C4C7×C814yescomplemented
136C2×C22⋊C4C2×C4×C712yes—
137C2×C22⋊C4C2×C4×C712yes—
138C2×C22⋊C4C7×D412yes—
139C2×C22⋊C4C7×D412yes—
140C2×C42—22no—
141C2×C42—22no—
142C22⋊C8—24no—
143C22⋊C8—24no—
144#95—22no—
145#95—22no—
146C22×C7—21no—
147C22×C7—22no—
148C22×C7—21no—
149C22×C7—22no—
150C22×C7—22no—
151C22×C7—22no—
152C22×C7—41no—
153C22×C7—41no—
154C22×C7—42no—
155C22×C7—42no—
156C22×C7—42no—
157C22×C7—42no—
158C22×C7—42no—
159C22×C7—41no—
160C22×C7—41no—
161C22×C7—41no—
162C22×C7—41no—
163C22×C7—84no—
164C22×C7—84no—
165C22×C7#38111yescharacteristic
166C22×C7C2×C22⋊C811yescharacteristic
167C22×C7#40711yescharacteristic
168C22×C7#43112yes—
169C22×C7#43212yes—
170C4×C7—24no—
171C4×C7—44no—
172C4×C7—44no—
173C4×C7—42no—
174C24—22no—
175C24—22no—
176C24—81no—
177C24—81no—
178C24C7×C22⋊C411yescharacteristic
179C24C7×C22⋊C411yescharacteristic
180C24C2×C7×C812yes—
181C24C7×M1612yes—
182C24C22×C4×C711yescharacteristic
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—24no—
186C22×C4—24no—
187C22×C4—22no—
188C22×C4—81no—
189C22×C4—81no—
190C22×C4—81no—
191C22×C4—81no—
192C22×C4—81no—
193C22×C4—81no—
194C22×C4C7×C22⋊C412yes—
195C22×C4C2×C7×D412yes—
196C22×C4C7×#4114yes—
197C2×C8—44no—
198C2×C8—84no—
199C22⋊C4—24no—
200C22⋊C4—24no—
201C22⋊C4—22no—
202C22⋊C4—22no—
203C22⋊C4—24no—
204C22⋊C4—24no—
205C22⋊C4C2×C7×C814yes—
206C22⋊C4C7×M1614yes—
207C42—82no—
208C42—82no—
209C2×C7—21no—
210C2×C7—21no—
211C2×C7—22no—
212C2×C7—22no—
213C2×C7—42no—
214C2×C7—42no—
215C2×C7#181711yescharacteristic
216C2×C7#182811yescharacteristic
217C2×C7#182911yescharacteristic
218C23—21no—
219C23—21no—
220C23—21no—
221C23—21no—
222C23—24no—
223C23—24no—
224C23—41no—
225C23—41no—
226C23—41no—
227C23—41no—
228C23—41no—
229C23—41no—
230C23—82no—
231C23—82no—
232C23—82no—
233C23—82no—
234C23—84no—
235C23C7×C22⋊C812yes—
236C23C7×C22⋊C812yes—
237C23C2×C7×C22⋊C411yescharacteristic
238C23C7×#11712yes—
239C23C22×C7×C811yescharacteristic
240C23C2×C7×M1611yescharacteristic
241C23C2×C7×M1612yes—
242C2×C4—22no—
243C2×C4—24no—
244C2×C4—24no—
245C2×C4—42no—
246C2×C4—44no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—82no—
253C2×C4—84no—
254C2×C4—84no—
255C2×C4C7×#12314yes—
256C8—84no—
257C7#873411yescharacteristic, direct product factor, Sylow
258C22—21no—
259C22—22no—
260C22—21no—
261C22—22no—
262C22—22no—
263C22—22no—
264C22—41no—
265C22—41no—
266C22—42no—
267C22—42no—
268C22—42no—
269C22—42no—
270C22—42no—
271C22—41no—
272C22—41no—
273C22—41no—
274C22—41no—
275C22—84no—
276C22—84no—
277C22C7×#38111yescharacteristic
278C22C2×C7×C22⋊C811yescharacteristic
279C22C7×#40711yescharacteristic
280C22C7×#43112yes—
281C22C7×#43212yes—
282C4—24no—
283C4—44no—
284C4—44no—
285C4—42no—
286C2—21no—
287C2—21no—
288C2—22no—
289C2—22no—
290C2—42no—
291C2—42no—
292C2C7×#181711yescharacteristic
293C2C7×#182811yescharacteristic
294C2C7×#182911yescharacteristic
subgroupG/H
center, socleC22×C7#381
commutatorC23C22×C7×C8
Frattini, cosocleC23×C4C23×C7
prime radicalC25×C7C2×C4
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C7 ◁ C23×C4×C7 ◁ G

Complements

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

NH ≅ G/Ntype
3. C7×#1282C2semidirect
14. #8734C7direct (N × H)
26. C2×C7×C22⋊C4C8semidirect
39. #1282C2×C7semidirect
135. C2×C22⋊C4C7×C8semidirect
257. C7#8734direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#22732
C2×C7×#3232
C7×#12822
#87347