Finite Group Databasebeta

← C7×#10386 #13187602

Subgroups — C7×#10386 #13187602

Order 1792.

602 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 602 conjugacy classes of subgroups — too many to draw. Only its 90 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×#2607C211yesmaximal, characteristic, complemented
2C7×#1789C212yesmaximal
3C7×#1782C211yesmaximal, characteristic
4C7×#1306C211yesmaximal, characteristic, complemented
5C7×#1284C212yesmaximal, complemented
6C22×C7×#41C2211yescharacteristic
7C2×C7×#104—22no—
8C2×C7×#104—22no—
9C2×C7×#99—21no—
10C2×C7×#99—21no—
11C7×#407C2212yes—
12C7×#399—21no—
13C7×#393—21no—
14C7×#391—22no—
15C7×#387—22no—
16C7×#381C2211yescharacteristic
17C7×#381C2211yescharacteristic
18C4×C7×C22⋊C4C2212yes—
19#10386C711yesmaximal, characteristic, direct product factor, Sylow
20C2×C7×#41—22no—
21C2×C7×#41—22no—
22C2×C7×#41—42no—
23C2×C7×#41—42no—
24C22×C7×Q8—21no—
25C22×C7×Q8Q811yescharacteristic, complemented
26C22×C7×D4—21no—
27C22×C7×D4Q811yescharacteristic, complemented
28C23×C4×C7—21no—
29C23×C4×C7C2311yescharacteristic
30C2×C7×M16—22no—
31C2×C7×C4⋊C4—22no—
32C2×C7×C4⋊C4—21no—
33C2×C7×C22⋊C4—22no—
34C2×C7×C22⋊C4—21no—
35C2×C7×C22⋊C4D411yescharacteristic
36C2×C7×C22⋊C4D411yescharacteristic
37C2×C42×C7D412yes—
38C2×C42×C7D412yes—
39C7×#104—44no—
40C7×#104—44no—
41C7×#99—42no—
42C7×#99—42no—
43C7×C22⋊C8—22no—
44C7×C22⋊C8—22no—
45C7×#95—21no—
46C7×#95—22no—
47C7×#95—21no—
48C7×#95—22no—
49C7×#95—22no—
50C7×#95—22no—
51#2607C2×C711yescharacteristic, complemented
52#1789C2×C712yes—
53#1782C2×C711yescharacteristic
54#1306C2×C711yescharacteristic, complemented
55#1284C2×C712yescomplemented
56C24×C7—21no—
57C24×C7C2×D411yescharacteristic
58C7×#41—24no—
59C7×#41—24no—
60C7×#41—42no—
61C7×#41—42no—
62C7×#41—44no—
63C7×#41—44no—
64C2×C7×Q8—22no—
65C2×C7×Q8—42no—
66C2×C7×Q8—44no—
67C2×C7×Q8—81no—
68C2×C7×D4—22no—
69C2×C7×D4—22no—
70C2×C7×D4—22no—
71C2×C7×D4—42no—
72C2×C7×D4—41no—
73C2×C7×D4—41no—
74C2×C7×D4—41no—
75C2×C7×D4—41no—
76C22×C4×C7—22no—
77C22×C4×C7—21no—
78C22×C4×C7—21no—
79C22×C4×C7—22no—
80C22×C4×C7—21no—
81C22×C4×C7—41no—
82C22×C4×C7—41no—
83C22×C4×C7—42no—
84C22×C4×C7—42no—
85C22×C4×C7—42no—
86C22×C4×C7—41no—
87C22×C4×C7C2×D412yes—
88C22×C4×C7C2×Q811yescharacteristic
89C22×C4×C7#4112yes—
90C22×C4×C7#4111yescharacteristic
91C7×M16—42no—
92C7×M16—42no—
93C2×C7×C8—42no—
94C7×C4⋊C4—44no—
95C7×C4⋊C4—81no—
96C7×C22⋊C4—44no—
97C7×C22⋊C4—41no—
98C7×C22⋊C4—41no—
99C7×C22⋊C4—41no—
100C7×C22⋊C4—41no—
101C42×C7—42no—
102C42×C7—42no—
103C22×#41C22×C711yescharacteristic
104C2×#104—22no—
105C2×#104—22no—
106C2×#99—21no—
107C2×#99—21no—
108#407C22×C712yes—
109#399—21no—
110#393—21no—
111#391—22no—
112#387—22no—
113#381C22×C711yescharacteristic
114#381C22×C711yescharacteristic
115C4×C22⋊C4C22×C712yes—
116C23×C7—21no—
117C23×C7—21no—
118C23×C7—22no—
119C23×C7—42no—
120C23×C7—41no—
121C23×C7—41no—
122C23×C7—82no—
123C23×C7#12011yescharacteristic
124C23×C7C22⋊Q811yescharacteristic
125C23×C7#12411yescharacteristic
126C7×Q8—24no—
127C7×Q8—42no—
128C7×Q8—84no—
129C7×D4—24no—
130C7×D4—42no—
131C7×D4—42no—
132C7×D4—42no—
133C7×D4—42no—
134C7×D4—42no—
135C2×C4×C7—22no—
136C2×C4×C7—22no—
137C2×C4×C7—24no—
138C2×C4×C7—22no—
139C2×C4×C7—41no—
140C2×C4×C7—41no—
141C2×C4×C7—42no—
142C2×C4×C7—44no—
143C2×C4×C7—44no—
144C2×C4×C7—41no—
145C2×C4×C7—42no—
146C2×C4×C7—41no—
147C2×C4×C7—42no—
148C2×C4×C7—42no—
149C2×C4×C7—42no—
150C2×C4×C7—42no—
151C2×C4×C7—42no—
152C2×C4×C7—42no—
153C2×C4×C7—41no—
154C2×C4×C7—41no—
155C2×C4×C7—81no—
156C2×C4×C7—82no—
157C2×C4×C7C22⋊Q812yes—
158C2×C4×C7#12412yes—
159C7×C8—82no—
160C2×#41—22no—
161C2×#41—22no—
162C2×#41—42no—
163C2×#41—42no—
164C22×Q8—21no—
165C22×Q8C7×Q811yescharacteristic, complemented
166C22×D4—21no—
167C22×D4C7×Q811yescharacteristic, complemented
168C23×C4—21no—
169C23×C4C23×C711yescharacteristic
170C2×M16—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—21no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—21no—
175C2×C22⋊C4C7×D411yescharacteristic
176C2×C22⋊C4C7×D411yescharacteristic
177C2×C42C7×D412yes—
178C2×C42C7×D412yes—
179#104—44no—
180#104—44no—
181#99—42no—
182#99—42no—
183C22⋊C8—22no—
184C22⋊C8—22no—
185#95—21no—
186#95—22no—
187#95—21no—
188#95—22no—
189#95—22no—
190#95—22no—
191C22×C7—22no—
192C22×C7—21no—
193C22×C7—21no—
194C22×C7—22no—
195C22×C7—22no—
196C22×C7—42no—
197C22×C7—81no—
198C22×C7—81no—
199C22×C7—82no—
200C22×C7—82no—
201C22×C7—82no—
202C22×C7#39311yescharacteristic
203C22×C7B(4,𝔽2)11yescharacteristic
204C22×C7B(4,𝔽2)11yescharacteristic
205C4×C7—22no—
206C4×C7—22no—
207C4×C7—44no—
208C4×C7—42no—
209C4×C7—82no—
210C4×C7—81no—
211C4×C7—81no—
212C4×C7—161no—
213C24—21no—
214C24C2×C7×D411yescharacteristic
215#41—24no—
216#41—24no—
217#41—42no—
218#41—42no—
219#41—44no—
220#41—44no—
221C2×Q8—22no—
222C2×Q8—42no—
223C2×Q8—44no—
224C2×Q8—81no—
225C2×D4—22no—
226C2×D4—22no—
227C2×D4—22no—
228C2×D4—42no—
229C2×D4—41no—
230C2×D4—41no—
231C2×D4—41no—
232C2×D4—41no—
233C22×C4—22no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—22no—
237C22×C4—21no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—41no—
244C22×C4C2×C7×D412yes—
245C22×C4C2×C7×Q811yescharacteristic
246C22×C4C7×#4112yes—
247C22×C4C7×#4111yescharacteristic
248M16—42no—
249M16—42no—
250C2×C8—42no—
251C4⋊C4—44no—
252C4⋊C4—81no—
253C22⋊C4—44no—
254C22⋊C4—41no—
255C22⋊C4—41no—
256C22⋊C4—41no—
257C22⋊C4—41no—
258C42—42no—
259C42—42no—
260C2×C7—21no—
261C2×C7—21no—
262C2×C7—41no—
263C2×C7—41no—
264C2×C7—82no—
265C2×C7#202911yescharacteristic
266C2×C7#203112yes—
267C23—21no—
268C23—21no—
269C23—22no—
270C23—42no—
271C23—41no—
272C23—41no—
273C23—82no—
274C23C7×#12011yescharacteristic
275C23C7×C22⋊Q811yescharacteristic
276C23C7×#12411yescharacteristic
277Q8—24no—
278Q8—42no—
279Q8—84no—
280D4—24no—
281D4—42no—
282D4—42no—
283D4—42no—
284D4—42no—
285D4—42no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—24no—
289C2×C4—22no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—42no—
293C2×C4—44no—
294C2×C4—44no—
295C2×C4—41no—
296C2×C4—42no—
297C2×C4—41no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—81no—
307C2×C4—82no—
308C2×C4C7×C22⋊Q812yes—
309C2×C4C7×#12412yes—
310C8—82no—
311C7#1038611yescharacteristic, direct product factor, Sylow
312C22—22no—
313C22—21no—
314C22—21no—
315C22—22no—
316C22—22no—
317C22—42no—
318C22—81no—
319C22—81no—
320C22—82no—
321C22—82no—
322C22—82no—
323C22C7×#39311yescharacteristic
324C22C7×B(4,𝔽2)11yescharacteristic
325C22C7×B(4,𝔽2)11yescharacteristic
326C4—22no—
327C4—22no—
328C4—44no—
329C4—42no—
330C4—81no—
331C4—82no—
332C4—81no—
333C4—161no—
334C2—21no—
335C2—21no—
336C2—41no—
337C2—41no—
338C2—82no—
339C2C7×#202911yescharacteristic
340C2C7×#203112yes—
subgroupG/H
center, socleC22×C7#393
commutator, Frattini, cosocleC23×C4C23×C7
prime radicalC22×C7×#41C22
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C7 ◁ C23×C4×C7 ◁ G

Complements

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

NH ≅ G/Ntype
1. C7×#2607C2semidirect
4. C7×#1306C2semidirect
5. C7×#1284C2semidirect
19. #10386C7direct (N × H)
25. C22×C7×Q8Q8semidirect
27. C22×C7×D4Q8semidirect
51. #2607C2×C7semidirect
54. #1306C2×C7semidirect
55. #1284C2×C7semidirect
165. C22×Q8C7×Q8semidirect
167. C22×D4C7×Q8semidirect
311. C7#10386direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#26072
C7×#17892
C7×#17822
C7×#13062
C7×#12842
#103867