Finite Group Databasebeta

← C7×#10335 #13187551

Subgroups — C7×#10335 #13187551

Order 1792.

574 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 574 conjugacy classes of subgroups — too many to draw. Only its 106 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×#3195C211yesmaximal, characteristic
2C7×#2503C211yesmaximal, characteristic
3C7×#2502C211yesmaximal, characteristic
4C7×#1821C211yesmaximal, characteristic
5C7×#1508C211yesmaximal, characteristic
6C7×#1508C211yesmaximal, characteristic
7C7×#1297C211yesmaximal, characteristic
8C2×C7×C22⋊Q8C2211yescharacteristic
9C2×C7×C22⋊Q8C2211yescharacteristic
10C22×C7×C4⋊C4C2211yescharacteristic
11C7×#483—21no—
12C7×#483—21no—
13C7×#470—22no—
14C7×#451—21no—
15C7×#451—21no—
16C7×#407C2211yescharacteristic
17C7×#407C2211yescharacteristic
18C7×#400—22no—
19C7×#398—22no—
20C7×#391—21no—
21C7×#391—21no—
22C7×#385—22no—
23C7×#384—21no—
24C7×#382—21no—
25C7×#381—22no—
26C4×C7×C22⋊C4C2211yescharacteristic
27C4×C7×C22⋊C4C2211yescharacteristic
28C7×#343—22no—
29C7×#328—22no—
30C7×#328—22no—
31#10335C711yesmaximal, characteristic, direct product factor, Sylow
32C22×C7×Q8D411yescharacteristic
33C22×C7×Q8D411yescharacteristic
34C23×C4×C7D412yes—
35C23×C4×C7C2311yescharacteristic
36C2×C7×Q16—22no—
37C2×C7×M16—21no—
38C2×C7×M16D412yes—
39C7×C22⋊Q8—21no—
40C7×C22⋊Q8—21no—
41C7×C22⋊Q8—21no—
42C7×C22⋊Q8—21no—
43C7×C22⋊Q8—21no—
44C7×C22⋊Q8—21no—
45C7×C22⋊Q8—21no—
46C7×C22⋊Q8—21no—
47C2×C7×C4⋊C4—21no—
48C2×C7×C4⋊C4—21no—
49C2×C7×C4⋊C4—21no—
50C2×C7×C4⋊C4—21no—
51C2×C7×C4⋊C4—21no—
52C2×C7×C4⋊C4—21no—
53C2×C7×C4⋊C4—21no—
54C2×C7×C4⋊C4—21no—
55C2×C7×C4⋊C4—24no—
56C2×C7×C4⋊C4D411yescharacteristic
57C2×C7×C4⋊C4D411yescharacteristic
58C2×C7×C22⋊C4—21no—
59C2×C7×C22⋊C4—21no—
60C2×C7×C22⋊C4—21no—
61C2×C7×C22⋊C4—21no—
62C2×C42×C7—21no—
63C2×C42×C7—21no—
64C7×#106—22no—
65C7×#103—22no—
66C7×#103—22no—
67C7×C22⋊C8—21no—
68C7×C22⋊C8—21no—
69C7×C22⋊C8—21no—
70C7×C22⋊C8—21no—
71C7×#95—24no—
72C7×#95—24no—
73C7×#95—24no—
74C7×#95D412yes—
75C7×#95D412yes—
76#3195C2×C711yescharacteristic
77#2503C2×C711yescharacteristic
78#2502C2×C711yescharacteristic
79#1821C2×C711yescharacteristic
80#1508C2×C711yescharacteristic
81#1508C2×C711yescharacteristic
82#1297C2×C711yescharacteristic
83C24×C7C2×D411yescharacteristic
84C2×C7×Q8—21no—
85C2×C7×Q8—21no—
86C2×C7×Q8—21no—
87C2×C7×Q8—21no—
88C2×C7×Q8—42no—
89C2×C7×Q8—42no—
90C22×C4×C7—22no—
91C22×C4×C7—22no—
92C22×C4×C7—21no—
93C22×C4×C7—21no—
94C22×C4×C7—21no—
95C22×C4×C7—22no—
96C22×C4×C7—22no—
97C22×C4×C7—21no—
98C22×C4×C7—22no—
99C22×C4×C7—42no—
100C22×C4×C7—44no—
101C22×C4×C7C2×D411yescharacteristic
102C22×C4×C7C2×D411yescharacteristic
103C22×C4×C7C2×D411yescharacteristic
104C22×C4×C7C2×D411yescharacteristic
105C22×C4×C7C2×D411yescharacteristic
106C22×C4×C7#4111yescharacteristic
107C7×Q16—82no—
108C7×M16—42no—
109C7×M16—42no—
110C2×C7×C8—22no—
111C2×C7×C8—41no—
112C7×C4⋊C4—21no—
113C7×C4⋊C4—21no—
114C7×C4⋊C4—21no—
115C7×C4⋊C4—21no—
116C7×C4⋊C4—41no—
117C7×C4⋊C4—41no—
118C7×C4⋊C4—41no—
119C7×C4⋊C4—41no—
120C7×C4⋊C4—41no—
121C7×C4⋊C4—41no—
122C7×C4⋊C4—41no—
123C7×C4⋊C4—41no—
124C7×C4⋊C4—84no—
125C7×C22⋊C4—24no—
126C7×C22⋊C4—42no—
127C7×C22⋊C4—41no—
128C7×C22⋊C4—41no—
129C7×C22⋊C4—41no—
130C7×C22⋊C4—41no—
131C42×C7—42no—
132C42×C7—42no—
133C2×C22⋊Q8C22×C711yescharacteristic
134C2×C22⋊Q8C22×C711yescharacteristic
135C22×C4⋊C4C22×C711yescharacteristic
136#483—21no—
137#483—21no—
138#470—22no—
139#451—21no—
140#451—21no—
141#407C22×C711yescharacteristic
142#407C22×C711yescharacteristic
143#400—22no—
144#398—22no—
145#391—21no—
146#391—21no—
147#385—22no—
148#384—21no—
149#382—21no—
150#381—22no—
151C4×C22⋊C4C22×C711yescharacteristic
152C4×C22⋊C4C22×C711yescharacteristic
153#343—22no—
154#328—22no—
155#328—22no—
156C23×C7—22no—
157C23×C7—22no—
158C23×C7—41no—
159C23×C7#12011yescharacteristic
160C23×C7#12011yescharacteristic
161C23×C7#12111yescharacteristic
162C7×Q8—42no—
163C7×Q8—42no—
164C7×Q8—81no—
165C7×Q8—81no—
166C2×C4×C7—22no—
167C2×C4×C7—22no—
168C2×C4×C7—22no—
169C2×C4×C7—24no—
170C2×C4×C7—41no—
171C2×C4×C7—41no—
172C2×C4×C7—41no—
173C2×C4×C7—42no—
174C2×C4×C7—42no—
175C2×C4×C7—42no—
176C2×C4×C7—42no—
177C2×C4×C7—44no—
178C2×C4×C7—42no—
179C2×C4×C7—42no—
180C2×C4×C7—41no—
181C2×C4×C7—41no—
182C2×C4×C7—41no—
183C2×C4×C7—41no—
184C2×C4×C7—42no—
185C2×C4×C7—42no—
186C2×C4×C7—44no—
187C2×C4×C7—42no—
188C2×C4×C7—42no—
189C2×C4×C7—82no—
190C2×C4×C7#12011yescharacteristic
191C2×C4×C7#12111yescharacteristic
192C2×C4×C7#12111yescharacteristic
193C2×C4×C7#12711yescharacteristic
194C7×C8—42no—
195C7×C8—81no—
196C22×Q8C7×D411yescharacteristic
197C22×Q8C7×D411yescharacteristic
198C23×C4C7×D412yes—
199C23×C4C23×C711yescharacteristic
200C2×Q16—22no—
201C2×M16—21no—
202C2×M16C7×D412yes—
203C22⋊Q8—21no—
204C22⋊Q8—21no—
205C22⋊Q8—21no—
206C22⋊Q8—21no—
207C22⋊Q8—21no—
208C22⋊Q8—21no—
209C22⋊Q8—21no—
210C22⋊Q8—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4C7×D411yescharacteristic
221C2×C4⋊C4C7×D411yescharacteristic
222C2×C22⋊C4—21no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—21no—
226C2×C42—21no—
227C2×C42—21no—
228#106—22no—
229#103—22no—
230#103—22no—
231C22⋊C8—21no—
232C22⋊C8—21no—
233C22⋊C8—21no—
234C22⋊C8—21no—
235#95—24no—
236#95—24no—
237#95—24no—
238#95C7×D412yes—
239#95C7×D412yes—
240C22×C7—21no—
241C22×C7—21no—
242C22×C7—21no—
243C22×C7—21no—
244C22×C7—21no—
245C22×C7—21no—
246C22×C7—21no—
247C22×C7—44no—
248C22×C7#39211yescharacteristic
249C22×C7#45512yes—
250C22×C7#45612yes—
251C4×C7—21no—
252C4×C7—21no—
253C4×C7—21no—
254C4×C7—44no—
255C4×C7—82no—
256C4×C7—81no—
257C4×C7—82no—
258C4×C7—81no—
259C4×C7—81no—
260C4×C7—81no—
261C4×C7#45812yes—
262C24C2×C7×D411yescharacteristic
263C2×Q8—21no—
264C2×Q8—21no—
265C2×Q8—21no—
266C2×Q8—21no—
267C2×Q8—42no—
268C2×Q8—42no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—21no—
277C22×C4—22no—
278C22×C4—42no—
279C22×C4—44no—
280C22×C4C2×C7×D411yescharacteristic
281C22×C4C2×C7×D411yescharacteristic
282C22×C4C2×C7×D411yescharacteristic
283C22×C4C2×C7×D411yescharacteristic
284C22×C4C2×C7×D411yescharacteristic
285C22×C4C7×#4111yescharacteristic
286Q16—82no—
287M16—42no—
288M16—42no—
289C2×C8—22no—
290C2×C8—41no—
291C4⋊C4—21no—
292C4⋊C4—21no—
293C4⋊C4—21no—
294C4⋊C4—21no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—84no—
304C22⋊C4—24no—
305C22⋊C4—42no—
306C22⋊C4—41no—
307C22⋊C4—41no—
308C22⋊C4—41no—
309C22⋊C4—41no—
310C42—42no—
311C42—42no—
312C2×C7—22no—
313C2×C7—22no—
314C2×C7—41no—
315C2×C7#200411yescharacteristic
316C2×C7#200511yescharacteristic
317C2×C7#202211yescharacteristic
318C23—22no—
319C23—22no—
320C23—41no—
321C23C7×#12011yescharacteristic
322C23C7×#12011yescharacteristic
323C23C7×#12111yescharacteristic
324Q8—42no—
325Q8—42no—
326Q8—81no—
327Q8—81no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—24no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—44no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—44no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—82no—
352C2×C4C7×#12011yescharacteristic
353C2×C4C7×#12111yescharacteristic
354C2×C4C7×#12111yescharacteristic
355C2×C4C7×#12711yescharacteristic
356C8—42no—
357C8—81no—
358C7#1033511yescharacteristic, direct product factor, Sylow
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—44no—
367C22C7×#39211yescharacteristic
368C22C7×#45512yes—
369C22C7×#45612yes—
370C4—21no—
371C4—21no—
372C4—21no—
373C4—44no—
374C4—82no—
375C4—82no—
376C4—81no—
377C4—81no—
378C4—81no—
379C4—81no—
380C4C7×#45812yes—
381C2—22no—
382C2—22no—
383C2—41no—
384C2C7×#200411yescharacteristic
385C2C7×#200511yescharacteristic
386C2C7×#202211yescharacteristic
subgroupG/H
center, socleC22×C7#392
commutator, Frattini, cosocleC23×C4C23×C7
prime radicalC24×C7C2×D4
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
31. #10335C7direct (N × H)
358. C7#10335direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#31952
C7×#25032
C7×#25022
C7×#18212
C7×#15082
C7×#12972
#103357