Finite Group Databasebeta

← C7×#12179 #13189395

Subgroups — C7×#12179 #13189395

Order 1792.

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

normal subgroups only

This group has 412 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×#2807C211yesmaximal, characteristic, complemented
2C7×#2044C211yesmaximal, characteristic
3C7×#1613C211yesmaximal, characteristic
4C7×#1613C211yesmaximal, characteristic
5C7×#1352C211yesmaximal, characteristic, complemented
6C7×#1352C211yesmaximal, characteristic, complemented
7C7×#1326C211yesmaximal, characteristic, complemented
8C7×#534C2211yescharacteristic
9C7×#481—21no—
10C7×#481—21no—
11C2×C7×#99—21no—
12C7×#400—21no—
13C7×#399C2211yescharacteristic
14C7×#399C2211yescharacteristic
15C7×#398—21no—
16C7×#389—21no—
17C7×#389—21no—
18C7×#388—21no—
19C7×#386—21no—
20C7×#385—21no—
21C7×#385C2211yescharacteristic
22C7×#342C2211yescharacteristic
23C7×#327—21no—
24C7×#327—21no—
25C7×#323C2211yescharacteristic
26C7×#323C2211yescharacteristic
27#12179C711yesmaximal, characteristic, direct product factor, Sylow
28C22×C7×D4—21no—
29C23×C4×C7D411yescharacteristic
30C7×#123—41no—
31C7×#121—21no—
32C7×#121—21no—
33C7×#120—41no—
34C2×C7×C4⋊C4—21no—
35C2×C7×C4⋊C4—21no—
36C2×C7×C4⋊C4—21no—
37C2×C7×C4⋊C4—21no—
38C2×C7×C4⋊C4—21no—
39C2×C7×C4⋊C4—21no—
40C2×C7×C4⋊C4D411yescharacteristic
41C2×C7×C4⋊C4D411yescharacteristic
42C2×C7×C22⋊C4—21no—
43C2×C7×C22⋊C4—21no—
44C2×C7×C22⋊C4—21no—
45C2×C7×C22⋊C4—41no—
46C2×C7×C22⋊C4D411yescharacteristic
47C2×C7×C22⋊C4C2311yescharacteristic
48C2×C42×C7—21no—
49C7×#106—41no—
50C7×#106—41no—
51C7×#102—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—41no—
62C7×#95D411yescharacteristic
63C7×#95D411yescharacteristic
64#2807C2×C711yescharacteristic, complemented
65#2044C2×C711yescharacteristic
66#1613C2×C711yescharacteristic
67#1613C2×C711yescharacteristic
68#1352C2×C711yescharacteristic, complemented
69#1352C2×C711yescharacteristic, complemented
70#1326C2×C711yescharacteristic, complemented
71C24×C7—21no—
72C24×C7C2×D411yescharacteristic
73C2×C7×D4—21no—
74C2×C7×D4—21no—
75C2×C7×D4—41no—
76C2×C7×D4—41no—
77C2×C7×D4—41no—
78C2×C7×D4—41no—
79C2×C7×D4—41no—
80C2×C7×D4—41no—
81C22×C4×C7—21no—
82C22×C4×C7—21no—
83C22×C4×C7—21no—
84C22×C4×C7—21no—
85C22×C4×C7—21no—
86C22×C4×C7—21no—
87C22×C4×C7—21no—
88C22×C4×C7—41no—
89C22×C4×C7—41no—
90C22×C4×C7—81no—
91C22×C4×C7C2×D411yescharacteristic
92C22×C4×C7C2×D411yescharacteristic
93C2×C7×C8—41no—
94C2×C7×C8—41no—
95C7×C4⋊C4—21no—
96C7×C4⋊C4—21no—
97C7×C4⋊C4—41no—
98C7×C4⋊C4—41no—
99C7×C4⋊C4—42no—
100C7×C4⋊C4—42no—
101C7×C4⋊C4—81no—
102C7×C4⋊C4—81no—
103C7×C4⋊C4—81no—
104C7×C4⋊C4—81no—
105C7×C22⋊C4—21no—
106C7×C22⋊C4—41no—
107C7×C22⋊C4—42no—
108C7×C22⋊C4—41no—
109C7×C22⋊C4—41no—
110C7×C22⋊C4—41no—
111C7×C22⋊C4—41no—
112C7×C22⋊C4—81no—
113C7×C22⋊C4—82no—
114C7×C22⋊C4SD1612yes—
115C42×C7—81no—
116#534C22×C711yescharacteristic
117#481—21no—
118#481—21no—
119C2×#99—21no—
120#400—21no—
121#399C22×C711yescharacteristic
122#399C22×C711yescharacteristic
123#398—21no—
124#389—21no—
125#389—21no—
126#388—21no—
127#386—21no—
128#385—21no—
129#385C22×C711yescharacteristic
130#342C22×C711yescharacteristic
131#327—21no—
132#327—21no—
133#323C22×C711yescharacteristic
134#323C22×C711yescharacteristic
135C23×C7—21no—
136C23×C7—21no—
137C23×C7—41no—
138C23×C7—41no—
139C23×C7—41no—
140C23×C7—41no—
141C23×C7—41no—
142C23×C7—81no—
143C23×C7—81no—
144C23×C7#12011yescharacteristic
145C23×C7C2×SD1611yescharacteristic
146C23×C7#13711yescharacteristic
147C7×D4—81no—
148C7×D4—81no—
149C7×D4—81no—
150C7×D4—81no—
151C7×D4—81no—
152C7×D4—81no—
153C2×C4×C7—22no—
154C2×C4×C7—22no—
155C2×C4×C7—21no—
156C2×C4×C7—21no—
157C2×C4×C7—41no—
158C2×C4×C7—41no—
159C2×C4×C7—41no—
160C2×C4×C7—41no—
161C2×C4×C7—41no—
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—41no—
169C2×C4×C7—41no—
170C2×C4×C7—41no—
171C2×C4×C7—41no—
172C2×C4×C7—41no—
173C2×C4×C7—42no—
174C2×C4×C7—41no—
175C2×C4×C7—41no—
176C2×C4×C7—41no—
177C2×C4×C7—41no—
178C2×C4×C7—41no—
179C2×C4×C7—81no—
180C2×C4×C7—81no—
181C2×C4×C7—81no—
182C2×C4×C7—81no—
183C7×C8—81no—
184C7×C8—81no—
185C22×D4—21no—
186C23×C4C7×D411yescharacteristic
187#123—41no—
188#121—21no—
189#121—21no—
190#120—41no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4C7×D411yescharacteristic
198C2×C4⋊C4C7×D411yescharacteristic
199C2×C22⋊C4—21no—
200C2×C22⋊C4—21no—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—41no—
203C2×C22⋊C4C7×D411yescharacteristic
204C2×C22⋊C4C23×C711yescharacteristic
205C2×C42—21no—
206#106—41no—
207#106—41no—
208#102—41no—
209#102—41no—
210#99—42no—
211C22⋊C8—21no—
212C22⋊C8—21no—
213#95—21no—
214#95—21no—
215#95—21no—
216#95—21no—
217#95—21no—
218#95—41no—
219#95C7×D411yescharacteristic
220#95C7×D411yescharacteristic
221C22×C7—21no—
222C22×C7—21no—
223C22×C7—21no—
224C22×C7—21no—
225C22×C7—41no—
226C22×C7—41no—
227C22×C7—41no—
228C22×C7—41no—
229C22×C7—81no—
230C22×C7—82no—
231C22×C7—81no—
232C22×C7—81no—
233C22×C7—81no—
234C22×C7—161no—
235C22×C7#44811yescharacteristic
236C22×C7#45411yescharacteristic
237C22×C7B(4,𝔽2)11yescharacteristic
238C4×C7—42no—
239C4×C7—42no—
240C4×C7—41no—
241C4×C7—41no—
242C4×C7—81no—
243C4×C7—81no—
244C4×C7—81no—
245C4×C7—81no—
246C4×C7—81no—
247C4×C7—161no—
248C24—21no—
249C24C2×C7×D411yescharacteristic
250C2×D4—21no—
251C2×D4—21no—
252C2×D4—41no—
253C2×D4—41no—
254C2×D4—41no—
255C2×D4—41no—
256C2×D4—41no—
257C2×D4—41no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—81no—
268C22×C4C2×C7×D411yescharacteristic
269C22×C4C2×C7×D411yescharacteristic
270C2×C8—41no—
271C2×C8—41no—
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—42no—
277C4⋊C4—42no—
278C4⋊C4—81no—
279C4⋊C4—81no—
280C4⋊C4—81no—
281C4⋊C4—81no—
282C22⋊C4—21no—
283C22⋊C4—41no—
284C22⋊C4—42no—
285C22⋊C4—41no—
286C22⋊C4—41no—
287C22⋊C4—41no—
288C22⋊C4—41no—
289C22⋊C4—81no—
290C22⋊C4—82no—
291C22⋊C4C7×SD1612yes—
292C42—81no—
293C2×C7—21no—
294C2×C7—21no—
295C2×C7—41no—
296C2×C7—41no—
297C2×C7—161no—
298C2×C7#159911yescharacteristic
299C2×C7#219811yescharacteristic
300C2×C7#220011yescharacteristic
301C23—21no—
302C23—21no—
303C23—41no—
304C23—41no—
305C23—41no—
306C23—41no—
307C23—41no—
308C23—81no—
309C23—81no—
310C23C7×#12011yescharacteristic
311C23C2×C7×SD1611yescharacteristic
312C23C7×#13711yescharacteristic
313D4—81no—
314D4—81no—
315D4—81no—
316D4—81no—
317D4—81no—
318D4—81no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—42no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—81no—
346C2×C4—81no—
347C2×C4—81no—
348C2×C4—81no—
349C8—81no—
350C8—81no—
351C7#1217911yescharacteristic, direct product factor, Sylow
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—81no—
361C22—82no—
362C22—81no—
363C22—81no—
364C22—81no—
365C22—161no—
366C22C7×#44811yescharacteristic
367C22C7×#45411yescharacteristic
368C22C7×B(4,𝔽2)11yescharacteristic
369C4—42no—
370C4—42no—
371C4—41no—
372C4—41no—
373C4—81no—
374C4—81no—
375C4—81no—
376C4—81no—
377C4—81no—
378C4—161no—
379C2—21no—
380C2—21no—
381C2—41no—
382C2—41no—
383C2—161no—
384C2C7×#159911yescharacteristic
385C2C7×#219811yescharacteristic
386C2C7×#220011yescharacteristic
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×#2807C2semidirect
5. C7×#1352C2semidirect
6. C7×#1352C2semidirect
7. C7×#1326C2semidirect
27. #12179C7direct (N × H)
64. #2807C2×C7semidirect
68. #1352C2×C7semidirect
69. #1352C2×C7semidirect
70. #1326C2×C7semidirect
351. C7#12179direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#28072
C7×#20442
C7×#16132
C7×#13522
C7×#13262
#121797