Finite Group Databasebeta

← C7×#12214 #13189430

Subgroups — C7×#12214 #13189430

Order 1792.

578 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 578 conjugacy classes of subgroups — too many to draw. Only its 74 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×#2437C211yesmaximal, characteristic
2C7×#2029C211yesmaximal, characteristic
3C7×#1614C211yesmaximal, characteristic, complemented
4C7×#1602C211yesmaximal, characteristic, complemented
5C7×#1352C211yesmaximal, characteristic, complemented
6C7×#1351C211yesmaximal, characteristic, complemented
7C7×#1316C211yesmaximal, characteristic
8C7×#543C2211yescharacteristic
9C2×C7×C22⋊Q8—21no—
10C22×C7×C22⋊C4—21no—
11C7×#484—21no—
12C7×#451—21no—
13C7×#399C2211yescharacteristic
14C7×#396—21no—
15C7×#393C2211yescharacteristic
16C7×#385—21no—
17C7×#385—21no—
18C7×#380—21no—
19C7×#379—21no—
20C7×#379C2211yescharacteristic
21C7×#342—21no—
22C7×#342C2211yescharacteristic
23C7×#328—21no—
24C7×#328—21no—
25C7×#323C2211yescharacteristic
26C7×#323C2211yescharacteristic
27#12214C711yesmaximal, characteristic, direct product factor, Sylow
28C25×C7D411yescharacteristic
29C22×C7×Q8D411yescharacteristic
30C23×C4×C7—21no—
31C23×C4×C7—21no—
32C2×C7×Q16—41no—
33C7×C22⋊Q8—21no—
34C7×C22⋊Q8—21no—
35C7×C22⋊Q8—44no—
36C7×C22⋊Q8—41no—
37C7×C22⋊Q8—41no—
38C2×C7×C4⋊C4—21no—
39C2×C7×C4⋊C4—21no—
40C2×C7×C4⋊C4—21no—
41C2×C7×C4⋊C4—21no—
42C2×C7×C4⋊C4D411yescharacteristic
43C2×C7×C22⋊C4—21no—
44C2×C7×C22⋊C4—21no—
45C2×C7×C22⋊C4—21no—
46C2×C7×C22⋊C4—21no—
47C2×C7×C22⋊C4—21no—
48C2×C7×C22⋊C4—21no—
49C2×C7×C22⋊C4—21no—
50C2×C7×C22⋊C4—21no—
51C2×C7×C22⋊C4—44no—
52C2×C7×C22⋊C4—41no—
53C2×C7×C22⋊C4—41no—
54C2×C7×C22⋊C4D411yescharacteristic
55C2×C7×C22⋊C4C2311yescharacteristic
56C7×#107—41no—
57C7×#103—41no—
58C7×#103—41no—
59C7×C22⋊C8—21no—
60C7×C22⋊C8—21no—
61C7×#95—21no—
62C7×#95—21no—
63C7×#95D411yescharacteristic
64C7×#95D411yescharacteristic
65#2437C2×C711yescharacteristic
66#2029C2×C711yescharacteristic
67#1614C2×C711yescharacteristic, complemented
68#1602C2×C711yescharacteristic, complemented
69#1352C2×C711yescharacteristic, complemented
70#1351C2×C711yescharacteristic, complemented
71#1316C2×C711yescharacteristic
72C24×C7—21no—
73C24×C7—21no—
74C24×C7—42no—
75C24×C7—41no—
76C24×C7—42no—
77C24×C7—41no—
78C24×C7Q1612yescomplemented
79C24×C7C2×D411yescharacteristic
80C2×C7×Q8—21no—
81C2×C7×Q8—21no—
82C2×C7×Q8—42no—
83C2×C7×Q8—41no—
84C22×C4×C7—21no—
85C22×C4×C7—21no—
86C22×C4×C7—21no—
87C22×C4×C7—21no—
88C22×C4×C7—21no—
89C22×C4×C7—21no—
90C22×C4×C7—21no—
91C22×C4×C7—42no—
92C22×C4×C7—42no—
93C22×C4×C7—42no—
94C22×C4×C7—42no—
95C22×C4×C7—41no—
96C22×C4×C7—81no—
97C22×C4×C7—81no—
98C22×C4×C7C2×D411yescharacteristic
99C22×C4×C7C2×D411yescharacteristic
100C7×Q16—82no—
101C2×C7×C8—41no—
102C2×C7×C8—41no—
103C7×C4⋊C4—21no—
104C7×C4⋊C4—21no—
105C7×C4⋊C4—41no—
106C7×C4⋊C4—42no—
107C7×C4⋊C4—42no—
108C7×C4⋊C4—42no—
109C7×C4⋊C4—41no—
110C7×C4⋊C4—41no—
111C7×C4⋊C4—81no—
112C7×C22⋊C4—24no—
113C7×C22⋊C4—41no—
114C7×C22⋊C4—42no—
115C7×C22⋊C4—42no—
116C7×C22⋊C4—42no—
117C7×C22⋊C4—42no—
118C7×C22⋊C4—42no—
119C7×C22⋊C4—41no—
120C7×C22⋊C4—41no—
121C7×C22⋊C4—41no—
122C7×C22⋊C4—41no—
123C7×C22⋊C4—41no—
124C7×C22⋊C4—82no—
125C7×C22⋊C4—82no—
126#543C22×C711yescharacteristic
127C2×C22⋊Q8—21no—
128C22×C22⋊C4—21no—
129#484—21no—
130#451—21no—
131#399C22×C711yescharacteristic
132#396—21no—
133#393C22×C711yescharacteristic
134#385—21no—
135#385—21no—
136#380—21no—
137#379—21no—
138#379C22×C711yescharacteristic
139#342—21no—
140#342C22×C711yescharacteristic
141#328—21no—
142#328—21no—
143#323C22×C711yescharacteristic
144#323C22×C711yescharacteristic
145C23×C7—22no—
146C23×C7—22no—
147C23×C7—22no—
148C23×C7—21no—
149C23×C7—21no—
150C23×C7—41no—
151C23×C7—41no—
152C23×C7—41no—
153C23×C7—44no—
154C23×C7—42no—
155C23×C7—41no—
156C23×C7—44no—
157C23×C7—41no—
158C23×C7—41no—
159C23×C7—42no—
160C23×C7—42no—
161C23×C7—81no—
162C23×C7—84no—
163C23×C7—81no—
164C23×C7—81no—
165C23×C7#12011yescharacteristic
166C23×C7C2×Q1611yescharacteristic
167C23×C7#13711yescharacteristic
168C7×Q8—42no—
169C7×Q8—81no—
170C7×Q8—81no—
171C7×Q8—81no—
172C2×C4×C7—22no—
173C2×C4×C7—22no—
174C2×C4×C7—21no—
175C2×C4×C7—21no—
176C2×C4×C7—41no—
177C2×C4×C7—41no—
178C2×C4×C7—41no—
179C2×C4×C7—41no—
180C2×C4×C7—41no—
181C2×C4×C7—41no—
182C2×C4×C7—41no—
183C2×C4×C7—41no—
184C2×C4×C7—41no—
185C2×C4×C7—41no—
186C2×C4×C7—41no—
187C2×C4×C7—41no—
188C2×C4×C7—41no—
189C2×C4×C7—41no—
190C2×C4×C7—41no—
191C2×C4×C7—41no—
192C2×C4×C7—41no—
193C2×C4×C7—84no—
194C2×C4×C7—84no—
195C2×C4×C7—81no—
196C2×C4×C7—81no—
197C2×C4×C7—81no—
198C2×C4×C7—81no—
199C2×C4×C7—81no—
200C7×C8—81no—
201C7×C8—81no—
202C25C7×D411yescharacteristic
203C22×Q8C7×D411yescharacteristic
204C23×C4—21no—
205C23×C4—21no—
206C2×Q16—41no—
207C22⋊Q8—21no—
208C22⋊Q8—21no—
209C22⋊Q8—44no—
210C22⋊Q8—41no—
211C22⋊Q8—41no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4C7×D411yescharacteristic
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—44no—
226C2×C22⋊C4—41no—
227C2×C22⋊C4—41no—
228C2×C22⋊C4C7×D411yescharacteristic
229C2×C22⋊C4C23×C711yescharacteristic
230#107—41no—
231#103—41no—
232#103—41no—
233C22⋊C8—21no—
234C22⋊C8—21no—
235#95—21no—
236#95—21no—
237#95C7×D411yescharacteristic
238#95C7×D411yescharacteristic
239C22×C7—21no—
240C22×C7—21no—
241C22×C7—24no—
242C22×C7—21no—
243C22×C7—21no—
244C22×C7—41no—
245C22×C7—41no—
246C22×C7—42no—
247C22×C7—42no—
248C22×C7—42no—
249C22×C7—44no—
250C22×C7—42no—
251C22×C7—41no—
252C22×C7—42no—
253C22×C7—41no—
254C22×C7—42no—
255C22×C7—81no—
256C22×C7—82no—
257C22×C7—82no—
258C22×C7—82no—
259C22×C7#45111yescharacteristic
260C22×C7#45511yescharacteristic
261C22×C7B(4,𝔽2)11yescharacteristic
262C4×C7—42no—
263C4×C7—41no—
264C4×C7—42no—
265C4×C7—41no—
266C4×C7—81no—
267C4×C7—81no—
268C4×C7—81no—
269C4×C7—81no—
270C4×C7—81no—
271C4×C7—161no—
272C24—21no—
273C24—21no—
274C24—42no—
275C24—41no—
276C24—42no—
277C24—41no—
278C24C7×Q1612yescomplemented
279C24C2×C7×D411yescharacteristic
280C2×Q8—21no—
281C2×Q8—21no—
282C2×Q8—42no—
283C2×Q8—41no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—41no—
296C22×C4—81no—
297C22×C4—81no—
298C22×C4C2×C7×D411yescharacteristic
299C22×C4C2×C7×D411yescharacteristic
300Q16—82no—
301C2×C8—41no—
302C2×C8—41no—
303C4⋊C4—21no—
304C4⋊C4—21no—
305C4⋊C4—41no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—81no—
312C22⋊C4—24no—
313C22⋊C4—41no—
314C22⋊C4—42no—
315C22⋊C4—42no—
316C22⋊C4—42no—
317C22⋊C4—42no—
318C22⋊C4—41no—
319C22⋊C4—42no—
320C22⋊C4—41no—
321C22⋊C4—41no—
322C22⋊C4—41no—
323C22⋊C4—41no—
324C22⋊C4—82no—
325C22⋊C4—82no—
326C2×C7—21no—
327C2×C7—21no—
328C2×C7—41no—
329C2×C7—44no—
330C2×C7—41no—
331C2×C7#160211yescharacteristic
332C2×C7#219611yescharacteristic
333C2×C7#219711yescharacteristic
334C23—22no—
335C23—22no—
336C23—22no—
337C23—21no—
338C23—21no—
339C23—41no—
340C23—41no—
341C23—41no—
342C23—44no—
343C23—42no—
344C23—41no—
345C23—44no—
346C23—41no—
347C23—41no—
348C23—42no—
349C23—42no—
350C23—81no—
351C23—84no—
352C23—81no—
353C23—81no—
354C23C7×#12011yescharacteristic
355C23C2×C7×Q1611yescharacteristic
356C23C7×#13711yescharacteristic
357Q8—42no—
358Q8—81no—
359Q8—81no—
360Q8—81no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—84no—
383C2×C4—84no—
384C2×C4—81no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—81no—
389C8—81no—
390C8—81no—
391C7#1221411yescharacteristic, direct product factor, Sylow
392C22—21no—
393C22—21no—
394C22—24no—
395C22—21no—
396C22—21no—
397C22—41no—
398C22—41no—
399C22—42no—
400C22—42no—
401C22—42no—
402C22—44no—
403C22—42no—
404C22—41no—
405C22—42no—
406C22—41no—
407C22—42no—
408C22—81no—
409C22—82no—
410C22—82no—
411C22—82no—
412C22C7×#45111yescharacteristic
413C22C7×#45511yescharacteristic
414C22C7×B(4,𝔽2)11yescharacteristic
415C4—42no—
416C4—41no—
417C4—42no—
418C4—41no—
419C4—81no—
420C4—81no—
421C4—81no—
422C4—81no—
423C4—81no—
424C4—161no—
425C2—21no—
426C2—21no—
427C2—41no—
428C2—44no—
429C2—41no—
430C2C7×#160211yescharacteristic
431C2C7×#219611yescharacteristic
432C2C7×#219711yescharacteristic
subgroupG/H
center, socleC22×C7B(4,𝔽2)
commutator, Frattini, cosocleC2×C22⋊C4C23×C7
prime radicalC25×C7D4
LCSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C7 ◁ C23×C7 ◁ C2×C7×C22⋊C4 ◁ G

Complements

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

NH ≅ G/Ntype
3. C7×#1614C2semidirect
4. C7×#1602C2semidirect
5. C7×#1352C2semidirect
6. C7×#1351C2semidirect
27. #12214C7direct (N × H)
67. #1614C2×C7semidirect
68. #1602C2×C7semidirect
69. #1352C2×C7semidirect
70. #1351C2×C7semidirect
78. C24×C7Q16semidirect
278. C24C7×Q16semidirect
391. C7#12214direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#24372
C7×#20292
C7×#16142
C7×#16022
C7×#13522
C7×#13512
C7×#13162
#122147