Finite Group Databasebeta

← C7×#10324 #13187540

Subgroups — C7×#10324 #13187540

Order 1792.

910 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 910 conjugacy classes of subgroups — too many to draw. Only its 122 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×#3189C211yesmaximal, characteristic, complemented
2C7×#2848C212yesmaximal
3C7×#1779C211yesmaximal, characteristic
4C7×#1520C212yesmaximal, complemented
5C7×#1289C211yesmaximal, characteristic, complemented
6C24×C4×C7C2211yescharacteristic
7C2×C7×C22⋊Q8—22no—
8C2×C7×C22⋊Q8C2212yes—
9C7×#484—22no—
10C7×#467—22no—
11C7×#451—22no—
12C2×C7×C22⋊C8—22no—
13C2×C7×C22⋊C8C2211yescharacteristic
14C2×C7×C22⋊C8C2211yescharacteristic
15C7×#385—22no—
16C7×#385—24no—
17C7×#380—22no—
18C7×#380C2212yes—
19C7×#379—22no—
20C7×#328—24no—
21#10324C711yesmaximal, characteristic, direct product factor, Sylow
22C25×C7D411yescharacteristic
23C22×C7×Q8D412yes—
24C23×C4×C7—22no—
25C23×C4×C7—21no—
26C23×C4×C7—22no—
27C23×C4×C7—21no—
28C23×C4×C7—44no—
29C23×C4×C7D411yescharacteristic
30C23×C4×C7C2311yescharacteristic
31C2×C7×Q16—22no—
32C22×C7×C8—21no—
33C22×C7×C8D412yes—
34C7×C22⋊Q8—22no—
35C7×C22⋊Q8—22no—
36C7×C22⋊Q8—22no—
37C7×C22⋊Q8—22no—
38C7×C22⋊Q8—48no—
39C2×C7×C4⋊C4—24no—
40C2×C7×C4⋊C4—22no—
41C2×C7×C4⋊C4—22no—
42C2×C7×C4⋊C4D412yes—
43C2×C7×C22⋊C4—24no—
44C2×C7×C22⋊C4—24no—
45C2×C7×C22⋊C4—22no—
46C2×C7×C22⋊C4—22no—
47C7×#107—22no—
48C7×#103—24no—
49C7×C22⋊C8—22no—
50C7×C22⋊C8—22no—
51C7×C22⋊C8—44no—
52C7×#95—24no—
53C7×#95D414yes—
54#3189C2×C711yescharacteristic, complemented
55#2848C2×C712yes—
56#1779C2×C711yescharacteristic
57#1520C2×C712yescomplemented
58#1289C2×C711yescharacteristic, complemented
59C24×C7—22no—
60C24×C7—21no—
61C24×C7—21no—
62C24×C7—22no—
63C24×C7—42no—
64C24×C7—42no—
65C24×C7Q1612yescomplemented
66C24×C7C2×D411yescharacteristic
67C2×C7×Q8—22no—
68C2×C7×Q8—22no—
69C2×C7×Q8—44no—
70C22×C4×C7—24no—
71C22×C4×C7—28no—
72C22×C4×C7—22no—
73C22×C4×C7—21no—
74C22×C4×C7—22no—
75C22×C4×C7—24no—
76C22×C4×C7—22no—
77C22×C4×C7—42no—
78C22×C4×C7—42no—
79C22×C4×C7—44no—
80C22×C4×C7—42no—
81C22×C4×C7—44no—
82C22×C4×C7—42no—
83C22×C4×C7—44no—
84C22×C4×C7—42no—
85C22×C4×C7—41no—
86C22×C4×C7—42no—
87C22×C4×C7—41no—
88C22×C4×C7C2×D412yes—
89C22×C4×C7C2×D411yescharacteristic
90C22×C4×C7C2×D412yes—
91C22×C4×C7#4111yescharacteristic
92C7×Q16—82no—
93C2×C7×C8—22no—
94C2×C7×C8—42no—
95C2×C7×C8—42no—
96C2×C7×C8—41no—
97C7×C4⋊C4—22no—
98C7×C4⋊C4—22no—
99C7×C4⋊C4—44no—
100C7×C4⋊C4—48no—
101C7×C4⋊C4—42no—
102C7×C4⋊C4—42no—
103C7×C22⋊C4—48no—
104C7×C22⋊C4—48no—
105C7×C22⋊C4—42no—
106C7×C22⋊C4—42no—
107C7×C22⋊C4—42no—
108C7×C22⋊C4—42no—
109C24×C4C22×C711yescharacteristic
110C2×C22⋊Q8—22no—
111C2×C22⋊Q8C22×C712yes—
112#484—22no—
113#467—22no—
114#451—22no—
115C2×C22⋊C8—22no—
116C2×C22⋊C8C22×C711yescharacteristic
117C2×C22⋊C8C22×C711yescharacteristic
118#385—22no—
119#385—24no—
120#380—22no—
121#380C22×C712yes—
122#379—22no—
123#328—24no—
124C23×C7—24no—
125C23×C7—24no—
126C23×C7—22no—
127C23×C7—22no—
128C23×C7—24no—
129C23×C7—22no—
130C23×C7—41no—
131C23×C7—44no—
132C23×C7—41no—
133C23×C7—42no—
134C23×C7—41no—
135C23×C7—42no—
136C23×C7—41no—
137C23×C7—44no—
138C23×C7—48no—
139C23×C7—41no—
140C23×C7—42no—
141C23×C7—41no—
142C23×C7#12012yes—
143C23×C7#12111yescharacteristic
144C23×C7C2×Q1611yescharacteristic
145C23×C7#13511yescharacteristic
146C23×C7#13712yes—
147C7×Q8—44no—
148C7×Q8—82no—
149C2×C4×C7—24no—
150C2×C4×C7—22no—
151C2×C4×C7—24no—
152C2×C4×C7—22no—
153C2×C4×C7—42no—
154C2×C4×C7—41no—
155C2×C4×C7—44no—
156C2×C4×C7—48no—
157C2×C4×C7—42no—
158C2×C4×C7—41no—
159C2×C4×C7—41no—
160C2×C4×C7—44no—
161C2×C4×C7—44no—
162C2×C4×C7—44no—
163C2×C4×C7—44no—
164C2×C4×C7—44no—
165C2×C4×C7—42no—
166C2×C4×C7—42no—
167C2×C4×C7—44no—
168C2×C4×C7#12011yescharacteristic
169C2×C4×C7#12112yes—
170C2×C4×C7#12711yescharacteristic
171C7×C8—42no—
172C7×C8—81no—
173C25C7×D411yescharacteristic
174C22×Q8C7×D412yes—
175C23×C4—22no—
176C23×C4—21no—
177C23×C4—22no—
178C23×C4—21no—
179C23×C4—44no—
180C23×C4C7×D411yescharacteristic
181C23×C4C23×C711yescharacteristic
182C2×Q16—22no—
183C22×C8—21no—
184C22×C8C7×D412yes—
185C22⋊Q8—22no—
186C22⋊Q8—22no—
187C22⋊Q8—22no—
188C22⋊Q8—22no—
189C22⋊Q8—48no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4C7×D412yes—
194C2×C22⋊C4—24no—
195C2×C22⋊C4—24no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198#107—22no—
199#103—24no—
200C22⋊C8—22no—
201C22⋊C8—22no—
202C22⋊C8—44no—
203#95—24no—
204#95C7×D414yes—
205C22×C7—22no—
206C22×C7—21no—
207C22×C7—22no—
208C22×C7—21no—
209C22×C7—28no—
210C22×C7—24no—
211C22×C7—41no—
212C22×C7—41no—
213C22×C7—42no—
214C22×C7—44no—
215C22×C7—44no—
216C22×C7—44no—
217C22×C7—42no—
218C22×C7—42no—
219C22×C7—42no—
220C22×C7—42no—
221C22×C7—42no—
222C22×C7—42no—
223C22×C7#39211yescharacteristic
224C22×C7#45112yes—
225C22×C7#45212yes—
226C22×C7#46711yescharacteristic
227C22×C7#47011yescharacteristic
228C4×C7—21no—
229C4×C7—22no—
230C4×C7—42no—
231C4×C7—84no—
232C4×C7—82no—
233C4×C7—82no—
234C4×C7B(4,𝔽2)12yes—
235C24—22no—
236C24—21no—
237C24—21no—
238C24—22no—
239C24—42no—
240C24—42no—
241C24C7×Q1612yescomplemented
242C24C2×C7×D411yescharacteristic
243C2×Q8—22no—
244C2×Q8—22no—
245C2×Q8—44no—
246C22×C4—24no—
247C22×C4—28no—
248C22×C4—22no—
249C22×C4—21no—
250C22×C4—22no—
251C22×C4—24no—
252C22×C4—22no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—44no—
256C22×C4—42no—
257C22×C4—44no—
258C22×C4—42no—
259C22×C4—44no—
260C22×C4—42no—
261C22×C4—41no—
262C22×C4—42no—
263C22×C4—41no—
264C22×C4C2×C7×D412yes—
265C22×C4C2×C7×D411yescharacteristic
266C22×C4C2×C7×D412yes—
267C22×C4C7×#4111yescharacteristic
268Q16—82no—
269C2×C8—22no—
270C2×C8—42no—
271C2×C8—42no—
272C2×C8—41no—
273C4⋊C4—22no—
274C4⋊C4—22no—
275C4⋊C4—44no—
276C4⋊C4—48no—
277C4⋊C4—42no—
278C4⋊C4—42no—
279C22⋊C4—48no—
280C22⋊C4—48no—
281C22⋊C4—42no—
282C22⋊C4—42no—
283C22⋊C4—42no—
284C22⋊C4—42no—
285C2×C7—22no—
286C2×C7—22no—
287C2×C7—21no—
288C2×C7—21no—
289C2×C7—44no—
290C2×C7#200111yescharacteristic
291C2×C7#201011yescharacteristic
292C2×C7#202111yescharacteristic
293C23—24no—
294C23—24no—
295C23—22no—
296C23—22no—
297C23—22no—
298C23—24no—
299C23—41no—
300C23—44no—
301C23—41no—
302C23—42no—
303C23—41no—
304C23—42no—
305C23—41no—
306C23—44no—
307C23—48no—
308C23—41no—
309C23—41no—
310C23—42no—
311C23C7×#12012yes—
312C23C7×#12111yescharacteristic
313C23C2×C7×Q1611yescharacteristic
314C23C7×#13511yescharacteristic
315C23C7×#13712yes—
316Q8—44no—
317Q8—82no—
318C2×C4—24no—
319C2×C4—22no—
320C2×C4—24no—
321C2×C4—22no—
322C2×C4—42no—
323C2×C4—41no—
324C2×C4—44no—
325C2×C4—48no—
326C2×C4—42no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—44no—
330C2×C4—44no—
331C2×C4—44no—
332C2×C4—44no—
333C2×C4—44no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—44no—
337C2×C4C7×#12011yescharacteristic
338C2×C4C7×#12112yes—
339C2×C4C7×#12711yescharacteristic
340C8—42no—
341C8—81no—
342C7#1032411yescharacteristic, direct product factor, Sylow
343C22—22no—
344C22—21no—
345C22—22no—
346C22—21no—
347C22—24no—
348C22—28no—
349C22—41no—
350C22—41no—
351C22—42no—
352C22—44no—
353C22—44no—
354C22—42no—
355C22—44no—
356C22—42no—
357C22—42no—
358C22—42no—
359C22—42no—
360C22—42no—
361C22C7×#39211yescharacteristic
362C22C7×#45112yes—
363C22C7×#45212yes—
364C22C7×#46711yescharacteristic
365C22C7×#47011yescharacteristic
366C4—21no—
367C4—22no—
368C4—42no—
369C4—84no—
370C4—82no—
371C4—82no—
372C4C7×B(4,𝔽2)12yes—
373C2—22no—
374C2—21no—
375C2—22no—
376C2—21no—
377C2—44no—
378C2C7×#200111yescharacteristic
379C2C7×#201011yescharacteristic
380C2C7×#202111yescharacteristic
subgroupG/H
center, socleC22×C7#392
commutator, Frattini, cosocleC23×C4C23×C7
prime radicalC25×C7D4
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×#3189C2semidirect
4. C7×#1520C2semidirect
5. C7×#1289C2semidirect
21. #10324C7direct (N × H)
54. #3189C2×C7semidirect
57. #1520C2×C7semidirect
58. #1289C2×C7semidirect
65. C24×C7Q16semidirect
241. C24C7×Q16semidirect
342. C7#10324direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#31892
C7×#28482
C7×#17792
C7×#15202
C7×#12892
#103247