Finite Group Databasebeta

← C7×#9449 #13186665

Subgroups — C7×#9449 #13186665

Order 1792.

490 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 122 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×#2973C211yesmaximal, characteristic
2C7×#2566C211yesmaximal, characteristic
3C7×#1534C212yesmaximal, complemented
4C7×#1466C211yesmaximal, characteristic
5C7×#1285C212yesmaximal, complemented
6C2×C7×#125C2211yescharacteristic
7C2×C4×C7×D4C2211yescharacteristic
8C7×#435—24no—
9C7×#432C2212yes—
10C7×#432C2212yes—
11C7×#407—21no—
12C2×C7×C22⋊C8—21no—
13C7×#396—21no—
14C7×#396—21no—
15C7×#390C411yescharacteristic
16C7×#390C411yescharacteristic
17C7×#385—21no—
18C7×#385—21no—
19C7×#384C411yescharacteristic
20C7×#382C411yescharacteristic
21C4×C7×C4⋊C4C2211yescharacteristic
22C7×#330—24no—
23C7×#323—21no—
24C7×#323—21no—
25#9449C711yesmaximal, characteristic, direct product factor, Sylow
26C22×C7×D4C2×C411yescharacteristic
27C23×C4×C7D411yescharacteristic
28C23×C4×C7D411yescharacteristic
29C2×C7×M16—22no—
30C22×C7×C8—22no—
31C7×#125—22no—
32C7×#125—22no—
33C4×C7×D4—22no—
34C4×C7×D4—22no—
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⋊C4C2×C411yescharacteristic
41C2×C7×C4⋊C4C2×C411yescharacteristic
42C2×C7×C4⋊C4C2×C411yescharacteristic
43C2×C7×C22⋊C4—21no—
44C2×C7×C22⋊C4—21no—
45C2×C7×C22⋊C4—21no—
46C2×C7×C22⋊C4—21no—
47C2×C7×C22⋊C4D411yescharacteristic
48C2×C7×C22⋊C4D411yescharacteristic
49C2×C42×C7C2×C411yescharacteristic
50C2×C42×C7C2×C411yescharacteristic
51C2×C42×C7C2311yescharacteristic
52C7×C4⋊C8—22no—
53C7×C4⋊C8—22no—
54C7×C22⋊C8—22no—
55C7×C22⋊C8—22no—
56C7×C22⋊C8—22no—
57C7×C22⋊C8—22no—
58C7×#97—22no—
59C7×#97D414yes—
60C7×#95—21no—
61C7×#95—21no—
62C7×#95—21no—
63#2973C2×C711yescharacteristic
64#2566C2×C711yescharacteristic
65#1534C2×C712yescomplemented
66#1466C2×C711yescharacteristic
67#1285C2×C712yescomplemented
68C24×C7C22⋊C411yescharacteristic
69C24×C7C22⋊C411yescharacteristic
70C2×C7×D4—22no—
71C2×C7×D4—22no—
72C2×C7×D4—41no—
73C22×C4×C7—21no—
74C22×C4×C7—21no—
75C22×C4×C7—21no—
76C22×C4×C7—21no—
77C22×C4×C7—21no—
78C22×C4×C7—21no—
79C22×C4×C7—21no—
80C22×C4×C7—21no—
81C22×C4×C7—81no—
82C22×C4×C7—81no—
83C22×C4×C7C22⋊C411yescharacteristic
84C22×C4×C7C22⋊C411yescharacteristic
85C22×C4×C7C22×C411yescharacteristic
86C22×C4×C7C2×D411yescharacteristic
87C22×C4×C7C2×D411yescharacteristic
88C7×M16—82no—
89C2×C7×C8—24no—
90C2×C7×C8—42no—
91C2×C7×C8—82no—
92C7×C4⋊C4—22no—
93C7×C4⋊C4—22no—
94C7×C4⋊C4—22no—
95C7×C4⋊C4—42no—
96C7×C4⋊C4—41no—
97C7×C4⋊C4—42no—
98C7×C4⋊C4—41no—
99C7×C4⋊C4—81no—
100C7×C4⋊C4—81no—
101C7×C22⋊C4—22no—
102C7×C22⋊C4—22no—
103C7×C22⋊C4—42no—
104C7×C22⋊C4—42no—
105C7×C22⋊C4—81no—
106C7×C22⋊C4—81no—
107C42×C7—41no—
108C42×C7—41no—
109C42×C7C2×D412yes—
110C42×C7#4112yes—
111C2×#125C22×C711yescharacteristic
112C2×C4×D4C22×C711yescharacteristic
113#435—24no—
114#432C22×C712yes—
115#432C22×C712yes—
116#407—21no—
117C2×C22⋊C8—21no—
118#396—21no—
119#396—21no—
120#390C4×C711yescharacteristic
121#390C4×C711yescharacteristic
122#385—21no—
123#385—21no—
124#384C4×C711yescharacteristic
125#382C4×C711yescharacteristic
126C4×C4⋊C4C22×C711yescharacteristic
127#330—24no—
128#323—21no—
129#323—21no—
130C23×C7—21no—
131C23×C7—21no—
132C23×C7—21no—
133C23×C7—21no—
134C23×C7—21no—
135C23×C7—41no—
136C23×C7—41no—
137C23×C7—41no—
138C23×C7—41no—
139C23×C7#10112yes—
140C23×C7C2×C22⋊C411yescharacteristic
141C7×D4—44no—
142C2×C4×C7—21no—
143C2×C4×C7—21no—
144C2×C4×C7—22no—
145C2×C4×C7—21no—
146C2×C4×C7—21no—
147C2×C4×C7—41no—
148C2×C4×C7—41no—
149C2×C4×C7—42no—
150C2×C4×C7—42no—
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—81no—
165C2×C4×C7—81no—
166C2×C4×C7C4×D412yes—
167C2×C4×C7#12011yescharacteristic
168C2×C4×C7#12112yes—
169C2×C4×C7#12311yescharacteristic
170C7×C8—44no—
171C7×C8—82no—
172C22×D4C2×C4×C711yescharacteristic
173C23×C4C7×D411yescharacteristic
174C23×C4C7×D411yescharacteristic
175C2×M16—22no—
176C22×C8—22no—
177#125—22no—
178#125—22no—
179C4×D4—22no—
180C4×D4—22no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4C2×C4×C711yescharacteristic
187C2×C4⋊C4C2×C4×C711yescharacteristic
188C2×C4⋊C4C2×C4×C711yescharacteristic
189C2×C22⋊C4—21no—
190C2×C22⋊C4—21no—
191C2×C22⋊C4—21no—
192C2×C22⋊C4—21no—
193C2×C22⋊C4C7×D411yescharacteristic
194C2×C22⋊C4C7×D411yescharacteristic
195C2×C42C2×C4×C711yescharacteristic
196C2×C42C2×C4×C711yescharacteristic
197C2×C42C23×C711yescharacteristic
198C4⋊C8—22no—
199C4⋊C8—22no—
200C22⋊C8—22no—
201C22⋊C8—22no—
202C22⋊C8—22no—
203C22⋊C8—22no—
204#97—22no—
205#97C7×D414yes—
206#95—21no—
207#95—21no—
208#95—21no—
209C22×C7—21no—
210C22×C7—21no—
211C22×C7—22no—
212C22×C7—42no—
213C22×C7—42no—
214C22×C7—41no—
215C22×C7—41no—
216C22×C7—41no—
217C22×C7—41no—
218C22×C7—41no—
219C22×C7—81no—
220C22×C7—81no—
221C22×C7#38611yescharacteristic
222C22×C7C2×#10111yescharacteristic
223C22×C7#41311yescharacteristic
224C4×C7—22no—
225C4×C7—22no—
226C4×C7—21no—
227C4×C7—21no—
228C4×C7—42no—
229C4×C7—41no—
230C4×C7—41no—
231C4×C7—81no—
232C4×C7—81no—
233C4×C7—81no—
234C24C7×C22⋊C411yescharacteristic
235C24C7×C22⋊C411yescharacteristic
236C2×D4—22no—
237C2×D4—22no—
238C2×D4—41no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—81no—
248C22×C4—81no—
249C22×C4C7×C22⋊C411yescharacteristic
250C22×C4C7×C22⋊C411yescharacteristic
251C22×C4C22×C4×C711yescharacteristic
252C22×C4C2×C7×D411yescharacteristic
253C22×C4C2×C7×D411yescharacteristic
254M16—82no—
255C2×C8—24no—
256C2×C8—42no—
257C2×C8—82no—
258C4⋊C4—22no—
259C4⋊C4—22no—
260C4⋊C4—22no—
261C4⋊C4—42no—
262C4⋊C4—41no—
263C4⋊C4—42no—
264C4⋊C4—41no—
265C4⋊C4—81no—
266C4⋊C4—81no—
267C22⋊C4—22no—
268C22⋊C4—22no—
269C22⋊C4—42no—
270C22⋊C4—42no—
271C22⋊C4—81no—
272C22⋊C4—81no—
273C42—41no—
274C42—41no—
275C42C2×C7×D412yes—
276C42C7×#4112yes—
277C2×C7—21no—
278C2×C7—21no—
279C2×C7—42no—
280C2×C7—81no—
281C2×C7#188411yescharacteristic
282C2×C7#190111yescharacteristic
283C2×C7#191111yescharacteristic
284C23—21no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23—41no—
290C23—41no—
291C23—41no—
292C23—41no—
293C23C7×#10112yes—
294C23C2×C7×C22⋊C411yescharacteristic
295D4—44no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—22no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—42no—
304C2×C4—42no—
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×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—81no—
319C2×C4—81no—
320C2×C4C4×C7×D412yes—
321C2×C4C7×#12011yescharacteristic
322C2×C4C7×#12112yes—
323C2×C4C7×#12311yescharacteristic
324C8—44no—
325C8—82no—
326C7#944911yescharacteristic, direct product factor, Sylow
327C22—21no—
328C22—21no—
329C22—22no—
330C22—42no—
331C22—42no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—81no—
338C22—81no—
339C22C7×#38611yescharacteristic
340C22C2×C7×#10111yescharacteristic
341C22C7×#41311yescharacteristic
342C4—22no—
343C4—22no—
344C4—21no—
345C4—21no—
346C4—42no—
347C4—41no—
348C4—41no—
349C4—81no—
350C4—81no—
351C4—81no—
352C2—21no—
353C2—21no—
354C2—42no—
355C2—81no—
356C2C7×#188411yescharacteristic
357C2C7×#190111yescharacteristic
358C2C7×#191111yescharacteristic
subgroupG/H
center, socleC22×C7#386
commutatorC22×C4C22×C4×C7
Frattini, cosocleC2×C42C23×C7
prime radicalC22×C7×D4C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C7 ◁ C2×C42×C7 ◁ G

Complements

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

NH ≅ G/Ntype
3. C7×#1534C2semidirect
5. C7×#1285C2semidirect
25. #9449C7direct (N × H)
65. #1534C2×C7semidirect
67. #1285C2×C7semidirect
326. C7#9449direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#29732
C7×#25662
C7×#15342
C7×#14662
C7×#12852
#94497