Finite Group Databasebeta

← C7×#10332 #13187548

Subgroups — C7×#10332 #13187548

Order 1792.

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

normal subgroups only

This group has 600 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×#3193C211yesmaximal, characteristic
2C7×#2504C211yesmaximal, characteristic
3C7×#2503C211yesmaximal, characteristic, complemented
4C7×#1821C211yesmaximal, characteristic, complemented
5C7×#1508C211yesmaximal, characteristic, complemented
6C7×#1507C211yesmaximal, characteristic
7C7×#1297C211yesmaximal, characteristic, complemented
8C2×C7×C22⋊Q8C2211yescharacteristic
9C2×C7×#121C2211yescharacteristic
10C22×C7×C4⋊C4C2211yescharacteristic
11C7×#484—21no—
12C7×#480—21no—
13C7×#468—22no—
14C7×#450—21no—
15C7×#448—21no—
16C7×#407C2211yescharacteristic
17C7×#407C2211yescharacteristic
18C7×#400—22no—
19C7×#397—22no—
20C7×#391—21no—
21C7×#390—21no—
22C7×#386—22no—
23C7×#382—21no—
24C7×#382—21no—
25C7×#381—22no—
26C4×C7×C22⋊C4C2211yescharacteristic
27C4×C7×C22⋊C4C2211yescharacteristic
28C7×#343—22no—
29C7×#328—22no—
30C7×#327—22no—
31#10332C711yesmaximal, characteristic, direct product factor, Sylow
32C22×C7×Q8D411yescharacteristic
33C22×C7×D4—21no—
34C22×C7×D4D411yescharacteristic
35C23×C4×C7D412yes—
36C23×C4×C7C2311yescharacteristic
37C2×C7×SD16—22no—
38C2×C7×M16—21no—
39C2×C7×M16D412yes—
40C7×C22⋊Q8—21no—
41C7×C22⋊Q8—21no—
42C7×C22⋊Q8—21no—
43C7×C22⋊Q8—21no—
44C7×#121—21no—
45C7×#121—21no—
46C7×#121—21no—
47C7×#121—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⋊C4D411yescharacteristic
56C2×C7×C4⋊C4D411yescharacteristic
57C2×C7×C22⋊C4—21no—
58C2×C7×C22⋊C4—21no—
59C2×C7×C22⋊C4—21no—
60C2×C7×C22⋊C4—21no—
61C2×C7×C22⋊C4—24no—
62C2×C42×C7—21no—
63C2×C42×C7—21no—
64C7×#107—22no—
65C7×#103—22no—
66C7×#102—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#3193C2×C711yescharacteristic
77#2504C2×C711yescharacteristic
78#2503C2×C711yescharacteristic, complemented
79#1821C2×C711yescharacteristic, complemented
80#1508C2×C711yescharacteristic, complemented
81#1507C2×C711yescharacteristic
82#1297C2×C711yescharacteristic, complemented
83C24×C7—21no—
84C24×C7C2×D411yescharacteristic
85C2×C7×Q8—21no—
86C2×C7×Q8—21no—
87C2×C7×Q8—42no—
88C2×C7×D4—21no—
89C2×C7×D4—21no—
90C2×C7×D4—42no—
91C2×C7×D4—44no—
92C2×C7×D4—41no—
93C2×C7×D4—41no—
94C22×C4×C7—22no—
95C22×C4×C7—21no—
96C22×C4×C7—21no—
97C22×C4×C7—21no—
98C22×C4×C7—22no—
99C22×C4×C7—22no—
100C22×C4×C7—22no—
101C22×C4×C7—22no—
102C22×C4×C7—42no—
103C22×C4×C7—44no—
104C22×C4×C7C2×D411yescharacteristic
105C22×C4×C7C2×D411yescharacteristic
106C22×C4×C7C2×D411yescharacteristic
107C22×C4×C7C2×D411yescharacteristic
108C22×C4×C7C2×D411yescharacteristic
109C22×C4×C7#4111yescharacteristic
110C7×SD16—82no—
111C7×M16—42no—
112C7×M16—42no—
113C2×C7×C8—22no—
114C2×C7×C8—41no—
115C7×C4⋊C4—21no—
116C7×C4⋊C4—21no—
117C7×C4⋊C4—21no—
118C7×C4⋊C4—21no—
119C7×C4⋊C4—41no—
120C7×C4⋊C4—41no—
121C7×C4⋊C4—41no—
122C7×C4⋊C4—41no—
123C7×C4⋊C4—41no—
124C7×C4⋊C4—41no—
125C7×C22⋊C4—42no—
126C7×C22⋊C4—42no—
127C7×C22⋊C4—41no—
128C7×C22⋊C4—41no—
129C7×C22⋊C4—41no—
130C7×C22⋊C4—41no—
131C7×C22⋊C4—84no—
132C42×C7—42no—
133C42×C7—42no—
134C2×C22⋊Q8C22×C711yescharacteristic
135C2×#121C22×C711yescharacteristic
136C22×C4⋊C4C22×C711yescharacteristic
137#484—21no—
138#480—21no—
139#468—22no—
140#450—21no—
141#448—21no—
142#407C22×C711yescharacteristic
143#407C22×C711yescharacteristic
144#400—22no—
145#397—22no—
146#391—21no—
147#390—21no—
148#386—22no—
149#382—21no—
150#382—21no—
151#381—22no—
152C4×C22⋊C4C22×C711yescharacteristic
153C4×C22⋊C4C22×C711yescharacteristic
154#343—22no—
155#328—22no—
156#327—22no—
157C23×C7—22no—
158C23×C7—22no—
159C23×C7—41no—
160C23×C7—42no—
161C23×C7—42no—
162C23×C7—41no—
163C23×C7—42no—
164C23×C7#12011yescharacteristic
165C23×C7#12011yescharacteristic
166C23×C7#12111yescharacteristic
167C7×Q8—42no—
168C7×Q8—81no—
169C7×D4—42no—
170C7×D4—81no—
171C7×D4—82no—
172C7×D4—82no—
173C2×C4×C7—22no—
174C2×C4×C7—22no—
175C2×C4×C7—22no—
176C2×C4×C7—41no—
177C2×C4×C7—41no—
178C2×C4×C7—41no—
179C2×C4×C7—42no—
180C2×C4×C7—42no—
181C2×C4×C7—42no—
182C2×C4×C7—42no—
183C2×C4×C7—44no—
184C2×C4×C7—42no—
185C2×C4×C7—42no—
186C2×C4×C7—41no—
187C2×C4×C7—41no—
188C2×C4×C7—41no—
189C2×C4×C7—42no—
190C2×C4×C7—42no—
191C2×C4×C7—42no—
192C2×C4×C7—42no—
193C2×C4×C7—44no—
194C2×C4×C7—82no—
195C2×C4×C7#12011yescharacteristic
196C2×C4×C7#12111yescharacteristic
197C2×C4×C7#12111yescharacteristic
198C2×C4×C7#12711yescharacteristic
199C7×C8—42no—
200C7×C8—81no—
201C22×Q8C7×D411yescharacteristic
202C22×D4—21no—
203C22×D4C7×D411yescharacteristic
204C23×C4C7×D412yes—
205C23×C4C23×C711yescharacteristic
206C2×SD16—22no—
207C2×M16—21no—
208C2×M16C7×D412yes—
209C22⋊Q8—21no—
210C22⋊Q8—21no—
211C22⋊Q8—21no—
212C22⋊Q8—21no—
213#121—21no—
214#121—21no—
215#121—21no—
216#121—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4C7×D411yescharacteristic
225C2×C4⋊C4C7×D411yescharacteristic
226C2×C22⋊C4—21no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—24no—
231C2×C42—21no—
232C2×C42—21no—
233#107—22no—
234#103—22no—
235#102—22no—
236C22⋊C8—21no—
237C22⋊C8—21no—
238C22⋊C8—21no—
239C22⋊C8—21no—
240#95—24no—
241#95—24no—
242#95—24no—
243#95C7×D412yes—
244#95C7×D412yes—
245C22×C7—21no—
246C22×C7—21no—
247C22×C7—21no—
248C22×C7—21no—
249C22×C7—21no—
250C22×C7—21no—
251C22×C7—21no—
252C22×C7—44no—
253C22×C7—84no—
254C22×C7—81no—
255C22×C7—81no—
256C22×C7—81no—
257C22×C7#39211yescharacteristic
258C22×C7#45412yes—
259C22×C7#45512yes—
260C4×C7—21no—
261C4×C7—21no—
262C4×C7—21no—
263C4×C7—82no—
264C4×C7—82no—
265C4×C7—81no—
266C4×C7—81no—
267C4×C7—82no—
268C4×C7—81no—
269C4×C7#45812yes—
270C24—21no—
271C24C2×C7×D411yescharacteristic
272C2×Q8—21no—
273C2×Q8—21no—
274C2×Q8—42no—
275C2×D4—21no—
276C2×D4—21no—
277C2×D4—42no—
278C2×D4—44no—
279C2×D4—41no—
280C2×D4—41no—
281C22×C4—22no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—42no—
290C22×C4—44no—
291C22×C4C2×C7×D411yescharacteristic
292C22×C4C2×C7×D411yescharacteristic
293C22×C4C2×C7×D411yescharacteristic
294C22×C4C2×C7×D411yescharacteristic
295C22×C4C2×C7×D411yescharacteristic
296C22×C4C7×#4111yescharacteristic
297SD16—82no—
298M16—42no—
299M16—42no—
300C2×C8—22no—
301C2×C8—41no—
302C4⋊C4—21no—
303C4⋊C4—21no—
304C4⋊C4—21no—
305C4⋊C4—21no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C22⋊C4—42no—
313C22⋊C4—42no—
314C22⋊C4—41no—
315C22⋊C4—41no—
316C22⋊C4—41no—
317C22⋊C4—41no—
318C22⋊C4—84no—
319C42—42no—
320C42—42no—
321C2×C7—22no—
322C2×C7—22no—
323C2×C7—41no—
324C2×C7—161no—
325C2×C7#200311yescharacteristic
326C2×C7#200411yescharacteristic
327C2×C7#202211yescharacteristic
328C23—22no—
329C23—22no—
330C23—41no—
331C23—42no—
332C23—42no—
333C23—41no—
334C23—42no—
335C23C7×#12011yescharacteristic
336C23C7×#12011yescharacteristic
337C23C7×#12111yescharacteristic
338Q8—42no—
339Q8—81no—
340D4—42no—
341D4—81no—
342D4—82no—
343D4—82no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—44no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—44no—
365C2×C4—82no—
366C2×C4C7×#12011yescharacteristic
367C2×C4C7×#12111yescharacteristic
368C2×C4C7×#12111yescharacteristic
369C2×C4C7×#12711yescharacteristic
370C8—42no—
371C8—81no—
372C7#1033211yescharacteristic, direct product factor, Sylow
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—44no—
381C22—84no—
382C22—81no—
383C22—81no—
384C22—81no—
385C22C7×#39211yescharacteristic
386C22C7×#45412yes—
387C22C7×#45512yes—
388C4—21no—
389C4—21no—
390C4—21no—
391C4—82no—
392C4—82no—
393C4—81no—
394C4—81no—
395C4—82no—
396C4—81no—
397C4C7×#45812yes—
398C2—22no—
399C2—22no—
400C2—41no—
401C2—161no—
402C2C7×#200311yescharacteristic
403C2C7×#200411yescharacteristic
404C2C7×#202211yescharacteristic
subgroupG/H
center, socleC22×C7#392
commutator, Frattini, cosocleC23×C4C23×C7
prime radicalC2×C7×#121C22
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
3. C7×#2503C2semidirect
4. C7×#1821C2semidirect
5. C7×#1508C2semidirect
7. C7×#1297C2semidirect
31. #10332C7direct (N × H)
78. #2503C2×C7semidirect
79. #1821C2×C7semidirect
80. #1508C2×C7semidirect
82. #1297C2×C7semidirect
372. C7#10332direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#31932
C7×#25042
C7×#25032
C7×#18212
C7×#15082
C7×#15072
C7×#12972
#103327