Finite Group Databasebeta

← C7×#10002 #13187218

Subgroups — C7×#10002 #13187218

Order 1792.

548 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3104C211yesmaximal, characteristic, complemented
2C7×#3077C211yesmaximal, characteristic
3C7×#2580C211yesmaximal, characteristic
4C7×#1518C211yesmaximal, characteristic, complemented
5C7×#1487C211yesmaximal, characteristic, complemented
6C7×#1478C211yesmaximal, characteristic
7C7×#1286C211yesmaximal, characteristic, complemented
8C7×#533C2211yescharacteristic
9C2×C4×C7×Q8C2211yescharacteristic
10C2×C4×C7×D4C2211yescharacteristic
11C7×#484—21no—
12C7×#478—22no—
13C7×#477—21no—
14C7×#476—21no—
15C7×#474—21no—
16C7×#462—21no—
17C7×#451—21no—
18C4×C7×Q16—22no—
19C7×#431C2211yescharacteristic
20C7×#431C2211yescharacteristic
21C7×#428—21no—
22C7×#423C2211yescharacteristic
23C7×#418—21no—
24C7×#399—21no—
25C7×#393—21no—
26C7×#390—21no—
27C7×#389—21no—
28C7×#385—22no—
29C7×#382C2211yescharacteristic
30C7×#328—22no—
31C7×Q8⋊C8—22no—
32C7×D4⋊C8—22no—
33#10002C711yesmaximal, characteristic, direct product factor, Sylow
34C22×C7×Q8D411yescharacteristic
35C22×C7×D4D411yescharacteristic
36C23×C4×C7—21no—
37C2×C7×Q16—22no—
38C2×C7×M16—21no—
39C7×C4⋊Q8D411yescharacteristic
40C7×C4⋊Q8D411yescharacteristic
41C7×#125—21no—
42C7×C22⋊Q8—21no—
43C7×C22⋊Q8—21no—
44C4×C7×Q8—21no—
45C4×C7×Q8—21no—
46C4×C7×Q8—21no—
47C4×C7×Q8—21no—
48C4×C7×Q8—21no—
49C4×C7×Q8—21no—
50C4×C7×D4—21no—
51C4×C7×D4—21no—
52C4×C7×D4—21no—
53C4×C7×D4—21no—
54C7×#117—21no—
55C2×C7×C4⋊C4—21no—
56C2×C7×C4⋊C4—21no—
57C2×C7×C4⋊C4—22no—
58C2×C7×C4⋊C4D411yescharacteristic
59C2×C7×C4⋊C4D411yescharacteristic
60C2×C7×C22⋊C4—22no—
61C2×C7×C22⋊C4—21no—
62C2×C7×C22⋊C4—22no—
63C2×C42×C7—21no—
64C2×C42×C7C2311yescharacteristic
65C7×#107—22no—
66C7×#107—22no—
67C7×#106—22no—
68C7×C4⋊C8—21no—
69C7×C4⋊C8—21no—
70C7×C4⋊C8—21no—
71C7×C4⋊C8D412yes—
72C7×#103—22no—
73C7×#103—22no—
74C7×#102—22no—
75C7×#102—22no—
76C7×C22⋊C8—21no—
77C7×C22⋊C8—21no—
78C4×C7×C8—21no—
79C4×C7×C8D412yes—
80C7×#95—22no—
81C7×#95—21no—
82C7×#95D412yes—
83#3104C2×C711yescharacteristic, complemented
84#3077C2×C711yescharacteristic
85#2580C2×C711yescharacteristic
86#1518C2×C711yescharacteristic, complemented
87#1487C2×C711yescharacteristic, complemented
88#1478C2×C711yescharacteristic
89#1286C2×C711yescharacteristic, complemented
90C24×C7—21no—
91C2×C7×Q8—21no—
92C2×C7×Q8—21no—
93C2×C7×Q8—21no—
94C2×C7×Q8—21no—
95C2×C7×Q8—42no—
96C2×C7×D4—21no—
97C2×C7×D4—21no—
98C2×C7×D4—21no—
99C2×C7×D4—21no—
100C2×C7×D4—21no—
101C2×C7×D4Q1612yescomplemented
102C22×C4×C7—21no—
103C22×C4×C7—22no—
104C22×C4×C7—21no—
105C22×C4×C7—21no—
106C22×C4×C7—41no—
107C22×C4×C7—41no—
108C22×C4×C7—82no—
109C22×C4×C7C2×D411yescharacteristic
110C22×C4×C7C2×D411yescharacteristic
111C22×C4×C7C2×D411yescharacteristic
112C7×Q16—44no—
113C7×M16—41no—
114C7×M16—41no—
115C2×C7×C8—22no—
116C2×C7×C8—22no—
117C2×C7×C8—41no—
118C7×C4⋊C4—21no—
119C7×C4⋊C4—21no—
120C7×C4⋊C4—21no—
121C7×C4⋊C4—21no—
122C7×C4⋊C4—21no—
123C7×C4⋊C4—21no—
124C7×C4⋊C4—21no—
125C7×C4⋊C4—21no—
126C7×C4⋊C4—41no—
127C7×C4⋊C4—41no—
128C7×C4⋊C4—41no—
129C7×C4⋊C4—41no—
130C7×C4⋊C4—41no—
131C7×C4⋊C4—41no—
132C7×C4⋊C4—82no—
133C7×C22⋊C4—44no—
134C7×C22⋊C4—41no—
135C7×C22⋊C4—41no—
136C7×C22⋊C4—41no—
137C7×C22⋊C4—82no—
138C42×C7—41no—
139C42×C7—41no—
140C42×C7—41no—
141C42×C7C2×D411yescharacteristic
142C42×C7C2×D411yescharacteristic
143C42×C7C2×D411yescharacteristic
144C42×C7#4111yescharacteristic
145#533C22×C711yescharacteristic
146C2×C4×Q8C22×C711yescharacteristic
147C2×C4×D4C22×C711yescharacteristic
148#484—21no—
149#478—22no—
150#477—21no—
151#476—21no—
152#474—21no—
153#462—21no—
154#451—21no—
155C4×Q16—22no—
156#431C22×C711yescharacteristic
157#431C22×C711yescharacteristic
158#428—21no—
159#423C22×C711yescharacteristic
160#418—21no—
161#399—21no—
162#393—21no—
163#390—21no—
164#389—21no—
165#385—22no—
166#382C22×C711yescharacteristic
167#328—22no—
168Q8⋊C8—22no—
169D4⋊C8—22no—
170C23×C7—22no—
171C23×C7—42no—
172C23×C7—42no—
173C23×C7—41no—
174C23×C7—41no—
175C23×C7#12011yescharacteristic
176C7×Q8—22no—
177C7×Q8—22no—
178C7×Q8—41no—
179C7×Q8—41no—
180C7×Q8—42no—
181C7×Q8—81no—
182C7×D4—22no—
183C7×D4—22no—
184C7×D4—41no—
185C7×D4—41no—
186C2×C4×C7—22no—
187C2×C4×C7—41no—
188C2×C4×C7—41no—
189C2×C4×C7—41no—
190C2×C4×C7—41no—
191C2×C4×C7—42no—
192C2×C4×C7—41no—
193C2×C4×C7—42no—
194C2×C4×C7—41no—
195C2×C4×C7—42no—
196C2×C4×C7—41no—
197C2×C4×C7—41no—
198C2×C4×C7—41no—
199C2×C4×C7—41no—
200C2×C4×C7—42no—
201C2×C4×C7—82no—
202C2×C4×C7—82no—
203C2×C4×C7#12011yescharacteristic
204C2×C4×C7#12011yescharacteristic
205C2×C4×C7#12111yescharacteristic
206C2×C4×C7#12111yescharacteristic
207C2×C4×C7#12111yescharacteristic
208C2×C4×C7#12711yescharacteristic
209C2×C4×C7C2×Q1611yescharacteristic
210C2×C4×C7#13511yescharacteristic
211C2×C4×C7#13711yescharacteristic
212C2×C4×C7#13711yescharacteristic
213C7×C8—42no—
214C7×C8—42no—
215C7×C8—81no—
216C22×Q8C7×D411yescharacteristic
217C22×D4C7×D411yescharacteristic
218C23×C4—21no—
219C2×Q16—22no—
220C2×M16—21no—
221C4⋊Q8C7×D411yescharacteristic
222C4⋊Q8C7×D411yescharacteristic
223#125—21no—
224C22⋊Q8—21no—
225C22⋊Q8—21no—
226C4×Q8—21no—
227C4×Q8—21no—
228C4×Q8—21no—
229C4×Q8—21no—
230C4×Q8—21no—
231C4×Q8—21no—
232C4×D4—21no—
233C4×D4—21no—
234C4×D4—21no—
235C4×D4—21no—
236#117—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4C7×D411yescharacteristic
241C2×C4⋊C4C7×D411yescharacteristic
242C2×C22⋊C4—22no—
243C2×C22⋊C4—21no—
244C2×C22⋊C4—22no—
245C2×C42—21no—
246C2×C42C23×C711yescharacteristic
247#107—22no—
248#107—22no—
249#106—22no—
250C4⋊C8—21no—
251C4⋊C8—21no—
252C4⋊C8—21no—
253C4⋊C8C7×D412yes—
254#103—22no—
255#103—22no—
256#102—22no—
257#102—22no—
258C22⋊C8—21no—
259C22⋊C8—21no—
260C4×C8—21no—
261C4×C8C7×D412yes—
262#95—22no—
263#95—21no—
264#95C7×D412yes—
265C22×C7—21no—
266C22×C7—21no—
267C22×C7—44no—
268C22×C7—42no—
269C22×C7—82no—
270C22×C7—81no—
271C22×C7—81no—
272C22×C7#39211yescharacteristic
273C22×C7#45111yescharacteristic
274C22×C7#45211yescharacteristic
275C4×C7—21no—
276C4×C7—21no—
277C4×C7—21no—
278C4×C7—21no—
279C4×C7—42no—
280C4×C7—82no—
281C4×C7—81no—
282C4×C7—81no—
283C4×C7—81no—
284C4×C7—81no—
285C4×C7#45111yescharacteristic
286C4×C7#45211yescharacteristic
287C4×C7#46711yescharacteristic
288C4×C7#47011yescharacteristic
289C24—21no—
290C2×Q8—21no—
291C2×Q8—21no—
292C2×Q8—21no—
293C2×Q8—21no—
294C2×Q8—42no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4—21no—
298C2×D4—21no—
299C2×D4—21no—
300C2×D4C7×Q1612yescomplemented
301C22×C4—21no—
302C22×C4—22no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—82no—
308C22×C4C2×C7×D411yescharacteristic
309C22×C4C2×C7×D411yescharacteristic
310C22×C4C2×C7×D411yescharacteristic
311Q16—44no—
312M16—41no—
313M16—41no—
314C2×C8—22no—
315C2×C8—22no—
316C2×C8—41no—
317C4⋊C4—21no—
318C4⋊C4—21no—
319C4⋊C4—21no—
320C4⋊C4—21no—
321C4⋊C4—21no—
322C4⋊C4—21no—
323C4⋊C4—21no—
324C4⋊C4—21no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—82no—
332C22⋊C4—44no—
333C22⋊C4—41no—
334C22⋊C4—41no—
335C22⋊C4—41no—
336C22⋊C4—82no—
337C42—41no—
338C42—41no—
339C42—41no—
340C42C2×C7×D411yescharacteristic
341C42C2×C7×D411yescharacteristic
342C42C2×C7×D411yescharacteristic
343C42C7×#4111yescharacteristic
344C2×C7—21no—
345C2×C7—21no—
346C2×C7—82no—
347C2×C7#200111yescharacteristic
348C2×C7#200811yescharacteristic
349C2×C7#201011yescharacteristic
350C23—22no—
351C23—42no—
352C23—42no—
353C23—41no—
354C23—41no—
355C23C7×#12011yescharacteristic
356Q8—22no—
357Q8—22no—
358Q8—41no—
359Q8—41no—
360Q8—42no—
361Q8—81no—
362D4—22no—
363D4—22no—
364D4—41no—
365D4—41no—
366C2×C4—22no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—42no—
372C2×C4—41no—
373C2×C4—42no—
374C2×C4—41no—
375C2×C4—42no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—42no—
381C2×C4—82no—
382C2×C4—82no—
383C2×C4C7×#12011yescharacteristic
384C2×C4C7×#12011yescharacteristic
385C2×C4C7×#12111yescharacteristic
386C2×C4C7×#12111yescharacteristic
387C2×C4C7×#12111yescharacteristic
388C2×C4C7×#12711yescharacteristic
389C2×C4C2×C7×Q1611yescharacteristic
390C2×C4C7×#13511yescharacteristic
391C2×C4C7×#13711yescharacteristic
392C2×C4C7×#13711yescharacteristic
393C8—42no—
394C8—42no—
395C8—81no—
396C7#1000211yescharacteristic, direct product factor, Sylow
397C22—21no—
398C22—21no—
399C22—44no—
400C22—42no—
401C22—82no—
402C22—81no—
403C22—81no—
404C22C7×#39211yescharacteristic
405C22C7×#45111yescharacteristic
406C22C7×#45211yescharacteristic
407C4—21no—
408C4—21no—
409C4—21no—
410C4—21no—
411C4—42no—
412C4—82no—
413C4—81no—
414C4—81no—
415C4—81no—
416C4—81no—
417C4C7×#45111yescharacteristic
418C4C7×#45211yescharacteristic
419C4C7×#46711yescharacteristic
420C4C7×#47011yescharacteristic
421C2—21no—
422C2—21no—
423C2—82no—
424C2C7×#200111yescharacteristic
425C2C7×#200811yescharacteristic
426C2C7×#201011yescharacteristic
subgroupG/H
center, socleC22×C7#392
commutator, Frattini, cosocleC2×C42C23×C7
prime radicalC22×C7×D4D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C7 ◁ C2×C42×C7 ◁ G

Complements

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

NH ≅ G/Ntype
1. C7×#3104C2semidirect
4. C7×#1518C2semidirect
5. C7×#1487C2semidirect
7. C7×#1286C2semidirect
33. #10002C7direct (N × H)
83. #3104C2×C7semidirect
86. #1518C2×C7semidirect
87. #1487C2×C7semidirect
89. #1286C2×C7semidirect
101. C2×C7×D4Q16semidirect
300. C2×D4C7×Q16semidirect
396. C7#10002direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#31042
C7×#30772
C7×#25802
C7×#15182
C7×#14872
C7×#14782
C7×#12862
#100027