Finite Group Databasebeta

← C7×#12193 #13189409

Subgroups — C7×#12193 #13189409

Order 1792.

442 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 70 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×#2764C211yesmaximal, characteristic, complemented
2C7×#2022C211yesmaximal, characteristic
3C7×#1607C211yesmaximal, characteristic
4C7×#1601C211yesmaximal, characteristic
5C7×#1351C211yesmaximal, characteristic, complemented
6C7×#1344C211yesmaximal, characteristic, complemented
7C7×#1317C211yesmaximal, characteristic, complemented
8C7×#534C2211yescharacteristic
9C2×C7×#123—21no—
10C7×#480—21no—
11C7×#448—21no—
12C2×C7×#99—21no—
13C7×#400—21no—
14C7×#398—21no—
15C7×#397C2211yescharacteristic
16C7×#393C2211yescharacteristic
17C7×#391—21no—
18C7×#388—21no—
19C7×#385—21no—
20C7×#383—21no—
21C7×#381C2211yescharacteristic
22C7×#342C2211yescharacteristic
23C7×#327—21no—
24C7×#327—21no—
25C7×#323C2211yescharacteristic
26C7×#323C2211yescharacteristic
27#12193C711yesmaximal, characteristic, direct product factor, Sylow
28C22×C7×Q8D411yescharacteristic
29C22×C7×D4—21no—
30C23×C4×C7D411yescharacteristic
31C2×C7×SD16—41no—
32C7×#123—42no—
33C7×#123—41no—
34C7×#123—42no—
35C7×#121—21no—
36C7×#121—21no—
37C7×#120—41no—
38C2×C7×C4⋊C4—21no—
39C2×C7×C4⋊C4—21no—
40C2×C7×C4⋊C4—21no—
41C2×C7×C4⋊C4—41no—
42C2×C7×C4⋊C4D411yescharacteristic
43C2×C7×C22⋊C4—21no—
44C2×C7×C22⋊C4—21no—
45C2×C7×C22⋊C4—21no—
46C2×C7×C22⋊C4—41no—
47C2×C7×C22⋊C4D411yescharacteristic
48C2×C7×C22⋊C4C2311yescharacteristic
49C2×C42×C7—21no—
50C7×#107—41no—
51C7×#103—41no—
52C7×#102—41no—
53C7×#99—42no—
54C7×C22⋊C8—21no—
55C7×C22⋊C8—21no—
56C7×#95—21no—
57C7×#95—21no—
58C7×#95—21no—
59C7×#95—21no—
60C7×#95—21no—
61C7×#95—21no—
62C7×#95—21no—
63C7×#95—21no—
64C7×#95D411yescharacteristic
65C7×#95D411yescharacteristic
66#2764C2×C711yescharacteristic, complemented
67#2022C2×C711yescharacteristic
68#1607C2×C711yescharacteristic
69#1601C2×C711yescharacteristic
70#1351C2×C711yescharacteristic, complemented
71#1344C2×C711yescharacteristic, complemented
72#1317C2×C711yescharacteristic, complemented
73C24×C7—21no—
74C24×C7C2×D411yescharacteristic
75C2×C7×Q8—42no—
76C2×C7×Q8—41no—
77C2×C7×D4—21no—
78C2×C7×D4—21no—
79C2×C7×D4—41no—
80C2×C7×D4—41no—
81C2×C7×D4—41no—
82C2×C7×D4—41no—
83C2×C7×D4—41no—
84C2×C7×D4—41no—
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—21no—
92C22×C4×C7—21no—
93C22×C4×C7—21no—
94C22×C4×C7—41no—
95C22×C4×C7—41no—
96C22×C4×C7—41no—
97C22×C4×C7C2×D411yescharacteristic
98C22×C4×C7C2×D411yescharacteristic
99C7×SD16—82no—
100C2×C7×C8—41no—
101C2×C7×C8—41no—
102C7×C4⋊C4—21no—
103C7×C4⋊C4—21no—
104C7×C4⋊C4—41no—
105C7×C4⋊C4—44no—
106C7×C4⋊C4—81no—
107C7×C4⋊C4—81no—
108C7×C4⋊C4—81no—
109C7×C22⋊C4—22no—
110C7×C22⋊C4—41no—
111C7×C22⋊C4—42no—
112C7×C22⋊C4—41no—
113C7×C22⋊C4—41no—
114C7×C22⋊C4—44no—
115C7×C22⋊C4—41no—
116C7×C22⋊C4—41no—
117C7×C22⋊C4—41no—
118C7×C22⋊C4—41no—
119C42×C7—42no—
120#534C22×C711yescharacteristic
121C2×#123—21no—
122#480—21no—
123#448—21no—
124C2×#99—21no—
125#400—21no—
126#398—21no—
127#397C22×C711yescharacteristic
128#393C22×C711yescharacteristic
129#391—21no—
130#388—21no—
131#385—21no—
132#383—21no—
133#381C22×C711yescharacteristic
134#342C22×C711yescharacteristic
135#327—21no—
136#327—21no—
137#323C22×C711yescharacteristic
138#323C22×C711yescharacteristic
139C23×C7—21no—
140C23×C7—21no—
141C23×C7—41no—
142C23×C7—41no—
143C23×C7—41no—
144C23×C7—41no—
145C23×C7—41no—
146C23×C7—81no—
147C23×C7—81no—
148C23×C7#12011yescharacteristic
149C23×C7#13511yescharacteristic
150C23×C7AGL(1,ℤ/8)11yescharacteristic
151C7×Q8—42no—
152C7×Q8—81no—
153C7×D4—81no—
154C7×D4—81no—
155C7×D4—81no—
156C7×D4—81no—
157C7×D4—81no—
158C7×D4—81no—
159C2×C4×C7—22no—
160C2×C4×C7—21no—
161C2×C4×C7—21no—
162C2×C4×C7—41no—
163C2×C4×C7—41no—
164C2×C4×C7—41no—
165C2×C4×C7—41no—
166C2×C4×C7—41no—
167C2×C4×C7—41no—
168C2×C4×C7—42no—
169C2×C4×C7—42no—
170C2×C4×C7—41no—
171C2×C4×C7—41no—
172C2×C4×C7—41no—
173C2×C4×C7—41no—
174C2×C4×C7—41no—
175C2×C4×C7—41no—
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—81no—
187C2×C4×C7—81no—
188C2×C4×C7—81no—
189C7×C8—81no—
190C7×C8—81no—
191C22×Q8C7×D411yescharacteristic
192C22×D4—21no—
193C23×C4C7×D411yescharacteristic
194C2×SD16—41no—
195#123—42no—
196#123—41no—
197#123—42no—
198#121—21no—
199#121—21no—
200#120—41no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—41no—
205C2×C4⋊C4C7×D411yescharacteristic
206C2×C22⋊C4—21no—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—21no—
209C2×C22⋊C4—41no—
210C2×C22⋊C4C7×D411yescharacteristic
211C2×C22⋊C4C23×C711yescharacteristic
212C2×C42—21no—
213#107—41no—
214#103—41no—
215#102—41no—
216#99—42no—
217C22⋊C8—21no—
218C22⋊C8—21no—
219#95—21no—
220#95—21no—
221#95—21no—
222#95—21no—
223#95—21no—
224#95—21no—
225#95—21no—
226#95—21no—
227#95C7×D411yescharacteristic
228#95C7×D411yescharacteristic
229C22×C7—21no—
230C22×C7—21no—
231C22×C7—21no—
232C22×C7—21no—
233C22×C7—41no—
234C22×C7—41no—
235C22×C7—41no—
236C22×C7—41no—
237C22×C7—81no—
238C22×C7—82no—
239C22×C7—81no—
240C22×C7—81no—
241C22×C7—81no—
242C22×C7—161no—
243C22×C7#44911yescharacteristic
244C22×C7#45311yescharacteristic
245C22×C7B(4,𝔽2)11yescharacteristic
246C4×C7—41no—
247C4×C7—42no—
248C4×C7—41no—
249C4×C7—81no—
250C4×C7—81no—
251C4×C7—81no—
252C4×C7—82no—
253C4×C7—81no—
254C4×C7—81no—
255C4×C7—81no—
256C24—21no—
257C24C2×C7×D411yescharacteristic
258C2×Q8—42no—
259C2×Q8—41no—
260C2×D4—21no—
261C2×D4—21no—
262C2×D4—41no—
263C2×D4—41no—
264C2×D4—41no—
265C2×D4—41no—
266C2×D4—41no—
267C2×D4—41no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4C2×C7×D411yescharacteristic
281C22×C4C2×C7×D411yescharacteristic
282SD16—82no—
283C2×C8—41no—
284C2×C8—41no—
285C4⋊C4—21no—
286C4⋊C4—21no—
287C4⋊C4—41no—
288C4⋊C4—44no—
289C4⋊C4—81no—
290C4⋊C4—81no—
291C4⋊C4—81no—
292C22⋊C4—22no—
293C22⋊C4—41no—
294C22⋊C4—42no—
295C22⋊C4—41no—
296C22⋊C4—41no—
297C22⋊C4—44no—
298C22⋊C4—41no—
299C22⋊C4—41no—
300C22⋊C4—41no—
301C22⋊C4—41no—
302C42—42no—
303C2×C7—21no—
304C2×C7—21no—
305C2×C7—41no—
306C2×C7—41no—
307C2×C7—161no—
308C2×C7#160011yescharacteristic
309C2×C7#219711yescharacteristic
310C2×C7#220111yescharacteristic
311C23—21no—
312C23—21no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23—41no—
317C23—41no—
318C23—81no—
319C23—81no—
320C23C7×#12011yescharacteristic
321C23C7×#13511yescharacteristic
322C23C7×AGL(1,ℤ/8)11yescharacteristic
323Q8—42no—
324Q8—81no—
325D4—81no—
326D4—81no—
327D4—81no—
328D4—81no—
329D4—81no—
330D4—81no—
331C2×C4—22no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—81no—
361C8—81no—
362C8—81no—
363C7#1219311yescharacteristic, direct product factor, Sylow
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—41no—
369C22—41no—
370C22—41no—
371C22—41no—
372C22—81no—
373C22—82no—
374C22—81no—
375C22—81no—
376C22—81no—
377C22—161no—
378C22C7×#44911yescharacteristic
379C22C7×#45311yescharacteristic
380C22C7×B(4,𝔽2)11yescharacteristic
381C4—41no—
382C4—42no—
383C4—41no—
384C4—81no—
385C4—81no—
386C4—81no—
387C4—82no—
388C4—81no—
389C4—81no—
390C4—81no—
391C2—21no—
392C2—21no—
393C2—41no—
394C2—41no—
395C2—161no—
396C2C7×#160011yescharacteristic
397C2C7×#219711yescharacteristic
398C2C7×#220111yescharacteristic
subgroupG/H
center, socleC22×C7B(4,𝔽2)
commutator, Frattini, cosocleC2×C22⋊C4C23×C7
prime radicalC7×#534C22
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
1. C7×#2764C2semidirect
5. C7×#1351C2semidirect
6. C7×#1344C2semidirect
7. C7×#1317C2semidirect
27. #12193C7direct (N × H)
66. #2764C2×C7semidirect
70. #1351C2×C7semidirect
71. #1344C2×C7semidirect
72. #1317C2×C7semidirect
363. C7#12193direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#27642
C7×#20222
C7×#16072
C7×#16012
C7×#13512
C7×#13442
C7×#13172
#121937