Finite Group Databasebeta

← C7×#12231 #13189447

Subgroups — C7×#12231 #13189447

Order 1792.

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

normal subgroups only

This group has 412 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×#2826C211yesmaximal, characteristic
2C7×#2062C211yesmaximal, characteristic, complemented
3C7×#1614C211yesmaximal, characteristic, complemented
4C7×#1596C211yesmaximal, characteristic
5C7×#1352C211yesmaximal, characteristic, complemented
6C7×#1343C211yesmaximal, characteristic
7C7×#1326C211yesmaximal, characteristic, complemented
8C7×#543C2211yescharacteristic
9C7×#484—21no—
10C7×#450—21no—
11C7×#400—21no—
12C7×#399C2211yescharacteristic
13C7×#397—21no—
14C7×#396—21no—
15C7×#394—21no—
16C7×#394—21no—
17C7×#392C2211yescharacteristic
18C7×#386—21no—
19C7×#385—21no—
20C7×#385C2211yescharacteristic
21C7×#342—21no—
22C7×#342C2211yescharacteristic
23C7×#328—21no—
24C7×#328—21no—
25C7×#323C2211yescharacteristic
26C7×#323C2211yescharacteristic
27#12231C711yesmaximal, characteristic, direct product factor, Sylow
28C22×C7×D4—21no—
29C22×C7×D4D411yescharacteristic
30C23×C4×C7D411yescharacteristic
31C2×C7×SD16—41no—
32C7×C22⋊Q8—21no—
33C7×C22⋊Q8—21no—
34C7×C22⋊Q8—41no—
35C7×C22⋊Q8—41no—
36C2×C7×C4⋊C4—21no—
37C2×C7×C4⋊C4—21no—
38C2×C7×C4⋊C4—21no—
39C2×C7×C4⋊C4—41no—
40C2×C7×C4⋊C4D411yescharacteristic
41C2×C7×C22⋊C4—21no—
42C2×C7×C22⋊C4—21no—
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⋊C4D411yescharacteristic
50C2×C7×C22⋊C4C2311yescharacteristic
51C7×#107—41no—
52C7×#103—41no—
53C7×#102—41no—
54C7×C22⋊C8—21no—
55C7×C22⋊C8—21no—
56C7×#95—21no—
57C7×#95—21no—
58C7×#95—21no—
59C7×#95—21no—
60C7×#95—41no—
61C7×#95D411yescharacteristic
62C7×#95D411yescharacteristic
63#2826C2×C711yescharacteristic
64#2062C2×C711yescharacteristic, complemented
65#1614C2×C711yescharacteristic, complemented
66#1596C2×C711yescharacteristic
67#1352C2×C711yescharacteristic, complemented
68#1343C2×C711yescharacteristic
69#1326C2×C711yescharacteristic, complemented
70C24×C7—21no—
71C24×C7C2×D411yescharacteristic
72C2×C7×Q8—21no—
73C2×C7×Q8—21no—
74C2×C7×D4—41no—
75C2×C7×D4—41no—
76C2×C7×D4—41no—
77C2×C7×D4—81no—
78C2×C7×D4—81no—
79C2×C7×D4—81no—
80C22×C4×C7—21no—
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—41no—
88C22×C4×C7—41no—
89C22×C4×C7—81no—
90C22×C4×C7C2×D411yescharacteristic
91C22×C4×C7C2×D411yescharacteristic
92C7×SD16—82no—
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—41no—
100C7×C4⋊C4—81no—
101C7×C4⋊C4—81no—
102C7×C4⋊C4—81no—
103C7×C4⋊C4—82no—
104C7×C22⋊C4—22no—
105C7×C22⋊C4—21no—
106C7×C22⋊C4—21no—
107C7×C22⋊C4—41no—
108C7×C22⋊C4—41no—
109C7×C22⋊C4—41no—
110C7×C22⋊C4—41no—
111C7×C22⋊C4—81no—
112C7×C22⋊C4—81no—
113C7×C22⋊C4—81no—
114C7×C22⋊C4—81no—
115C7×C22⋊C4—81no—
116C7×C22⋊C4—81no—
117#543C22×C711yescharacteristic
118#484—21no—
119#450—21no—
120#400—21no—
121#399C22×C711yescharacteristic
122#397—21no—
123#396—21no—
124#394—21no—
125#394—21no—
126#392C22×C711yescharacteristic
127#386—21no—
128#385—21no—
129#385C22×C711yescharacteristic
130#342—21no—
131#342C22×C711yescharacteristic
132#328—21no—
133#328—21no—
134#323C22×C711yescharacteristic
135#323C22×C711yescharacteristic
136C23×C7—21no—
137C23×C7—21no—
138C23×C7—41no—
139C23×C7—41no—
140C23×C7—41no—
141C23×C7—41no—
142C23×C7—41no—
143C23×C7—41no—
144C23×C7—41no—
145C23×C7—41no—
146C23×C7—41no—
147C23×C7#12011yescharacteristic
148C23×C7#13511yescharacteristic
149C23×C7#13711yescharacteristic
150C7×Q8—81no—
151C7×Q8—81no—
152C7×D4—42no—
153C7×D4—81no—
154C7×D4—82no—
155C7×D4—82no—
156C2×C4×C7—21no—
157C2×C4×C7—21no—
158C2×C4×C7—41no—
159C2×C4×C7—41no—
160C2×C4×C7—42no—
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—42no—
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—81no—
180C2×C4×C7—81no—
181C2×C4×C7—81no—
182C2×C4×C7—81no—
183C2×C4×C7—81no—
184C7×C8—81no—
185C7×C8—81no—
186C22×D4—21no—
187C22×D4C7×D411yescharacteristic
188C23×C4C7×D411yescharacteristic
189C2×SD16—41no—
190C22⋊Q8—21no—
191C22⋊Q8—21no—
192C22⋊Q8—41no—
193C22⋊Q8—41no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—41no—
198C2×C4⋊C4C7×D411yescharacteristic
199C2×C22⋊C4—21no—
200C2×C22⋊C4—21no—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—21no—
203C2×C22⋊C4—21no—
204C2×C22⋊C4—21no—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—21no—
207C2×C22⋊C4C7×D411yescharacteristic
208C2×C22⋊C4C23×C711yescharacteristic
209#107—41no—
210#103—41no—
211#102—41no—
212C22⋊C8—21no—
213C22⋊C8—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—81no—
231C22×C7—81no—
232C22×C7—81no—
233C22×C7—81no—
234C22×C7—81no—
235C22×C7—81no—
236C22×C7—81no—
237C22×C7#45211yescharacteristic
238C22×C7#45611yescharacteristic
239C22×C7B(4,𝔽2)11yescharacteristic
240C4×C7—41no—
241C4×C7—41no—
242C4×C7—81no—
243C4×C7—81no—
244C4×C7—81no—
245C4×C7—82no—
246C4×C7—81no—
247C4×C7—81no—
248C4×C7—81no—
249C4×C7—81no—
250C24—21no—
251C24C2×C7×D411yescharacteristic
252C2×Q8—21no—
253C2×Q8—21no—
254C2×D4—41no—
255C2×D4—41no—
256C2×D4—41no—
257C2×D4—81no—
258C2×D4—81no—
259C2×D4—81no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—81no—
270C22×C4C2×C7×D411yescharacteristic
271C22×C4C2×C7×D411yescharacteristic
272SD16—82no—
273C2×C8—41no—
274C2×C8—41no—
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—81no—
281C4⋊C4—81no—
282C4⋊C4—81no—
283C4⋊C4—82no—
284C22⋊C4—22no—
285C22⋊C4—21no—
286C22⋊C4—21no—
287C22⋊C4—41no—
288C22⋊C4—41no—
289C22⋊C4—41no—
290C22⋊C4—41no—
291C22⋊C4—81no—
292C22⋊C4—81no—
293C22⋊C4—81no—
294C22⋊C4—81no—
295C22⋊C4—81no—
296C22⋊C4—81no—
297C2×C7—21no—
298C2×C7—21no—
299C2×C7—41no—
300C2×C7—41no—
301C2×C7—161no—
302C2×C7#160311yescharacteristic
303C2×C7#220011yescharacteristic
304C2×C7#220011yescharacteristic
305C23—21no—
306C23—21no—
307C23—41no—
308C23—41no—
309C23—41no—
310C23—41no—
311C23—41no—
312C23—41no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23C7×#12011yescharacteristic
317C23C7×#13511yescharacteristic
318C23C7×#13711yescharacteristic
319Q8—81no—
320Q8—81no—
321D4—42no—
322D4—81no—
323D4—82no—
324D4—82no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—42no—
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—41no—
340C2×C4—42no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4—81no—
353C8—81no—
354C8—81no—
355C7#1223111yescharacteristic, direct product factor, Sylow
356C22—21no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—81no—
365C22—81no—
366C22—81no—
367C22—81no—
368C22—81no—
369C22—81no—
370C22—81no—
371C22—81no—
372C22C7×#45211yescharacteristic
373C22C7×#45611yescharacteristic
374C22C7×B(4,𝔽2)11yescharacteristic
375C4—41no—
376C4—41no—
377C4—81no—
378C4—81no—
379C4—81no—
380C4—82no—
381C4—81no—
382C4—81no—
383C4—81no—
384C4—81no—
385C2—21no—
386C2—21no—
387C2—41no—
388C2—41no—
389C2—161no—
390C2C7×#160311yescharacteristic
391C2C7×#220011yescharacteristic
392C2C7×#220011yescharacteristic
subgroupG/H
center, socleC22×C7B(4,𝔽2)
commutator, Frattini, cosocleC2×C22⋊C4C23×C7
prime radicalC7×#392C22
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
2. C7×#2062C2semidirect
3. C7×#1614C2semidirect
5. C7×#1352C2semidirect
7. C7×#1326C2semidirect
27. #12231C7direct (N × H)
64. #2062C2×C7semidirect
65. #1614C2×C7semidirect
67. #1352C2×C7semidirect
69. #1326C2×C7semidirect
355. C7#12231direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#28262
C7×#20622
C7×#16142
C7×#15962
C7×#13522
C7×#13432
C7×#13262
#122317