Finite Group Databasebeta

← C7×#12272 #13189488

Subgroups — C7×#12272 #13189488

Order 1792.

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

normal subgroups only

This group has 372 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×#2593C211yesmaximal, characteristic
2C7×#2027C211yesmaximal, characteristic
3C7×#1617C211yesmaximal, characteristic, complemented
4C7×#1611C211yesmaximal, characteristic, complemented
5C7×#1353C211yesmaximal, characteristic, complemented
6C7×#1346C211yesmaximal, characteristic, complemented
7C7×#1318C211yesmaximal, characteristic
8C7×#544C2211yescharacteristic
9C7×#484—21no—
10C7×#483—21no—
11C7×#418—21no—
12C7×#400C2211yescharacteristic
13C7×#398C2211yescharacteristic
14C7×#394—21no—
15C7×#390C2211yescharacteristic
16C7×#388—21no—
17C7×#388—21no—
18C7×#388—21no—
19C7×#388—21no—
20C4×C7×C4⋊C4—21no—
21C7×#343C2211yescharacteristic
22C7×#339—21no—
23C7×#328—21no—
24C7×#328—21no—
25C7×#324C2211yescharacteristic
26C7×#324C2211yescharacteristic
27#12272C711yesmaximal, characteristic, direct product factor, Sylow
28C22×C7×D4D411yescharacteristic
29C2×C7×M16—21no—
30C7×C4⋊Q8—41no—
31C7×#125—41no—
32C7×C22⋊Q8—21no—
33C7×C22⋊Q8—21no—
34C7×#117—21no—
35C2×C7×C4⋊C4—21no—
36C2×C7×C4⋊C4—21no—
37C2×C7×C4⋊C4D411yescharacteristic
38C2×C7×C4⋊C4D411yescharacteristic
39C2×C7×C4⋊C4C2311yescharacteristic
40C2×C7×C22⋊C4—21no—
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×C42×C7—21no—
47C2×C42×C7—21no—
48C2×C42×C7D411yescharacteristic
49C7×#107—41no—
50C7×#106—41no—
51C7×#103—41no—
52C7×#103—41no—
53C7×#102—41no—
54C7×#102—41no—
55C7×C22⋊C8—21no—
56C7×C22⋊C8—21no—
57C7×#95—21no—
58C7×#95—21no—
59C7×#95—21no—
60C7×#95—21no—
61C7×#95D411yescharacteristic
62C7×#95D411yescharacteristic
63#2593C2×C711yescharacteristic
64#2027C2×C711yescharacteristic
65#1617C2×C711yescharacteristic, complemented
66#1611C2×C711yescharacteristic, complemented
67#1353C2×C711yescharacteristic, complemented
68#1346C2×C711yescharacteristic, complemented
69#1318C2×C711yescharacteristic
70C24×C7—21no—
71C2×C7×Q8—21no—
72C2×C7×Q8—21no—
73C2×C7×D4—21no—
74C2×C7×D4—21no—
75C2×C7×D4—41no—
76C2×C7×D4—41no—
77C22×C4×C7—21no—
78C22×C4×C7—21no—
79C22×C4×C7—21no—
80C22×C4×C7—21no—
81C22×C4×C7—21no—
82C22×C4×C7C2×D411yescharacteristic
83C22×C4×C7C2×D411yescharacteristic
84C22×C4×C7C2×D411yescharacteristic
85C7×M16—41no—
86C7×M16—41no—
87C2×C7×C8—41no—
88C2×C7×C8—41no—
89C2×C7×C8—41no—
90C7×C4⋊C4—21no—
91C7×C4⋊C4—21no—
92C7×C4⋊C4—21no—
93C7×C4⋊C4—21no—
94C7×C4⋊C4—41no—
95C7×C4⋊C4—41no—
96C7×C4⋊C4—41no—
97C7×C4⋊C4—42no—
98C7×C4⋊C4—41no—
99C7×C4⋊C4—41no—
100C7×C4⋊C4—41no—
101C7×C4⋊C4—41no—
102C7×C22⋊C4—42no—
103C7×C22⋊C4—42no—
104C7×C22⋊C4—42no—
105C7×C22⋊C4—42no—
106C7×C22⋊C4—41no—
107C7×C22⋊C4—81no—
108C7×C22⋊C4—81no—
109C42×C7—41no—
110C42×C7—42no—
111C42×C7—41no—
112C42×C7—81no—
113#544C22×C711yescharacteristic
114#484—21no—
115#483—21no—
116#418—21no—
117#400C22×C711yescharacteristic
118#398C22×C711yescharacteristic
119#394—21no—
120#390C22×C711yescharacteristic
121#388—21no—
122#388—21no—
123#388—21no—
124#388—21no—
125C4×C4⋊C4—21no—
126#343C22×C711yescharacteristic
127#339—21no—
128#328—21no—
129#328—21no—
130#324C22×C711yescharacteristic
131#324C22×C711yescharacteristic
132C23×C7—41no—
133C23×C7—41no—
134C23×C7—41no—
135C23×C7—41no—
136C23×C7—41no—
137C23×C7—41no—
138C23×C7—41no—
139C23×C7#12011yescharacteristic
140C7×Q8—81no—
141C7×Q8—81no—
142C7×D4—42no—
143C7×D4—81no—
144C2×C4×C7—21no—
145C2×C4×C7—21no—
146C2×C4×C7—22no—
147C2×C4×C7—22no—
148C2×C4×C7—21no—
149C2×C4×C7—21no—
150C2×C4×C7—41no—
151C2×C4×C7—41no—
152C2×C4×C7—41no—
153C2×C4×C7—41no—
154C2×C4×C7—41no—
155C2×C4×C7—41no—
156C2×C4×C7—41no—
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#13511yescharacteristic
168C2×C4×C7#13711yescharacteristic
169C7×C8—81no—
170C7×C8—81no—
171C7×C8—81no—
172C22×D4C7×D411yescharacteristic
173C2×M16—21no—
174C4⋊Q8—41no—
175#125—41no—
176C22⋊Q8—21no—
177C22⋊Q8—21no—
178#117—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4C7×D411yescharacteristic
182C2×C4⋊C4C7×D411yescharacteristic
183C2×C4⋊C4C23×C711yescharacteristic
184C2×C22⋊C4—21no—
185C2×C22⋊C4—21no—
186C2×C22⋊C4—21no—
187C2×C22⋊C4—21no—
188C2×C22⋊C4—21no—
189C2×C22⋊C4—21no—
190C2×C42—21no—
191C2×C42—21no—
192C2×C42C7×D411yescharacteristic
193#107—41no—
194#106—41no—
195#103—41no—
196#103—41no—
197#102—41no—
198#102—41no—
199C22⋊C8—21no—
200C22⋊C8—21no—
201#95—21no—
202#95—21no—
203#95—21no—
204#95—21no—
205#95C7×D411yescharacteristic
206#95C7×D411yescharacteristic
207C22×C7—21no—
208C22×C7—21no—
209C22×C7—81no—
210C22×C7—81no—
211C22×C7—81no—
212C22×C7—81no—
213C22×C7—81no—
214C22×C7—81no—
215C22×C7—81no—
216C22×C7#45211yescharacteristic
217C22×C7#45511yescharacteristic
218C22×C7B(4,𝔽2)11yescharacteristic
219C4×C7—21no—
220C4×C7—21no—
221C4×C7—42no—
222C4×C7—42no—
223C4×C7—41no—
224C4×C7—41no—
225C4×C7—81no—
226C4×C7—81no—
227C4×C7—81no—
228C4×C7—81no—
229C4×C7—81no—
230C24—21no—
231C2×Q8—21no—
232C2×Q8—21no—
233C2×D4—21no—
234C2×D4—21no—
235C2×D4—41no—
236C2×D4—41no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4C2×C7×D411yescharacteristic
243C22×C4C2×C7×D411yescharacteristic
244C22×C4C2×C7×D411yescharacteristic
245M16—41no—
246M16—41no—
247C2×C8—41no—
248C2×C8—41no—
249C2×C8—41no—
250C4⋊C4—21no—
251C4⋊C4—21no—
252C4⋊C4—21no—
253C4⋊C4—21no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—42no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C4⋊C4—41no—
261C4⋊C4—41no—
262C22⋊C4—42no—
263C22⋊C4—42no—
264C22⋊C4—42no—
265C22⋊C4—42no—
266C22⋊C4—41no—
267C22⋊C4—81no—
268C22⋊C4—81no—
269C42—41no—
270C42—42no—
271C42—41no—
272C42—81no—
273C2×C7—21no—
274C2×C7—21no—
275C2×C7—161no—
276C2×C7#160611yescharacteristic
277C2×C7#219811yescharacteristic
278C2×C7#220411yescharacteristic
279C23—41no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23C7×#12011yescharacteristic
287Q8—81no—
288Q8—81no—
289D4—42no—
290D4—81no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4C7×#13511yescharacteristic
315C2×C4C7×#13711yescharacteristic
316C8—81no—
317C8—81no—
318C8—81no—
319C7#1227211yescharacteristic, direct product factor, Sylow
320C22—21no—
321C22—21no—
322C22—81no—
323C22—81no—
324C22—81no—
325C22—81no—
326C22—81no—
327C22—81no—
328C22—81no—
329C22C7×#45211yescharacteristic
330C22C7×#45511yescharacteristic
331C22C7×B(4,𝔽2)11yescharacteristic
332C4—21no—
333C4—21no—
334C4—42no—
335C4—42no—
336C4—41no—
337C4—41no—
338C4—81no—
339C4—81no—
340C4—81no—
341C4—81no—
342C4—81no—
343C2—21no—
344C2—21no—
345C2—161no—
346C2C7×#160611yescharacteristic
347C2C7×#219811yescharacteristic
348C2C7×#220411yescharacteristic
subgroupG/H
center, socleC22×C7B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23×C7
prime radicalC22×C7×D4D4
LCSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C7 ◁ C23×C7 ◁ C2×C7×C4⋊C4 ◁ G

Complements

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

NH ≅ G/Ntype
3. C7×#1617C2semidirect
4. C7×#1611C2semidirect
5. C7×#1353C2semidirect
6. C7×#1346C2semidirect
27. #12272C7direct (N × H)
65. #1617C2×C7semidirect
66. #1611C2×C7semidirect
67. #1353C2×C7semidirect
68. #1346C2×C7semidirect
319. C7#12272direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#25932
C7×#20272
C7×#16172
C7×#16112
C7×#13532
C7×#13462
C7×#13182
#122727