Finite Group Databasebeta

← C7×#11906 #13189122

Subgroups — C7×#11906 #13189122

Order 1792.

440 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 114 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×#2564C211yesmaximal, characteristic
2C7×#1800C211yesmaximal, characteristic
3C7×#1459C211yesmaximal, characteristic
4C7×#1354C212yesmaximal, complemented
5C7×#1353C212yesmaximal, complemented
6C2×C7×C22⋊Q8C411yescharacteristic
7C2×C4×C7×D4C2211yescharacteristic
8C7×#423—21no—
9C2×C7×#100—21no—
10C2×C7×C22⋊C8—21no—
11C7×#400C2211yescharacteristic
12C7×#399C411yescharacteristic
13C7×#395C2211yescharacteristic
14C7×#393C411yescharacteristic
15C7×#387—21no—
16C7×#386—21no—
17C7×#385—21no—
18C7×#380—21no—
19C2×C7×#95C411yescharacteristic
20C7×#343C2212yes—
21C7×#324C2212yes—
22C7×#323—21no—
23#11906C711yesmaximal, characteristic, direct product factor, Sylow
24C22×C7×Q8C2×C411yescharacteristic
25C22×C7×D4D411yescharacteristic
26C23×C4×C7—21no—
27C23×C4×C7C2×C411yescharacteristic
28C23×C4×C7D411yescharacteristic
29C2×C7×M16—22no—
30C22×C7×C8—41no—
31C7×C22⋊Q8—41no—
32C7×C22⋊Q8—41no—
33C4×C7×D4—41no—
34C4×C7×D4—41no—
35C2×C7×C4⋊C4—21no—
36C2×C7×C4⋊C4—21no—
37C2×C7×C4⋊C4—21no—
38C2×C7×C4⋊C4C2×C411yescharacteristic
39C2×C7×C4⋊C4C2311yescharacteristic
40C2×C7×C22⋊C4—21no—
41C2×C7×C22⋊C4—21no—
42C2×C7×C22⋊C4—21no—
43C2×C7×C22⋊C4C2×C411yescharacteristic
44C2×C7×C22⋊C4D411yescharacteristic
45C2×C42×C7D411yescharacteristic
46C7×C4⋊C8—41no—
47C7×C4⋊C8—41no—
48C7×#100—42no—
49C7×C22⋊C8—22no—
50C7×C22⋊C8—41no—
51C7×#95—21no—
52C7×#95—21no—
53C7×#95—21no—
54C7×#95—21no—
55C7×#95—21no—
56C7×#95C2×C411yescharacteristic
57C7×#95C2×C411yescharacteristic
58#2564C2×C711yescharacteristic
59#1800C2×C711yescharacteristic
60#1459C2×C711yescharacteristic
61#1354C2×C712yescomplemented
62#1353C2×C712yescomplemented
63C24×C7C22⋊C411yescharacteristic
64C24×C7C22⋊C411yescharacteristic
65C2×C7×Q8—41no—
66C2×C7×Q8—81no—
67C2×C7×D4—41no—
68C2×C7×D4—41no—
69C2×C7×D4SD1612yes—
70C2×C7×D4Q1612yes—
71C22×C4×C7—21no—
72C22×C4×C7—21no—
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—42no—
80C22×C4×C7—42no—
81C22×C4×C7—81no—
82C22×C4×C7—81no—
83C22×C4×C7—81no—
84C22×C4×C7C22⋊C411yescharacteristic
85C22×C4×C7C22⋊C411yescharacteristic
86C22×C4×C7C22×C411yescharacteristic
87C22×C4×C7C2×D411yescharacteristic
88C22×C4×C7C2×D411yescharacteristic
89C7×M16—42no—
90C7×M16—42no—
91C2×C7×C8—42no—
92C2×C7×C8—42no—
93C2×C7×C8—82no—
94C7×C4⋊C4—41no—
95C7×C4⋊C4—41no—
96C7×C4⋊C4—42no—
97C7×C4⋊C4—41no—
98C7×C4⋊C4—41no—
99C7×C4⋊C4—81no—
100C7×C22⋊C4—41no—
101C7×C22⋊C4—41no—
102C7×C22⋊C4—42no—
103C7×C22⋊C4—42no—
104C7×C22⋊C4—81no—
105C42×C7—21no—
106C42×C7—21no—
107C2×C22⋊Q8C4×C711yescharacteristic
108C2×C4×D4C22×C711yescharacteristic
109#423—21no—
110C2×#100—21no—
111C2×C22⋊C8—21no—
112#400C22×C711yescharacteristic
113#399C4×C711yescharacteristic
114#395C22×C711yescharacteristic
115#393C4×C711yescharacteristic
116#387—21no—
117#386—21no—
118#385—21no—
119#380—21no—
120C2×#95C4×C711yescharacteristic
121#343C22×C712yes—
122#324C22×C712yes—
123#323—21no—
124C23×C7—21no—
125C23×C7—21no—
126C23×C7—21no—
127C23×C7—21no—
128C23×C7—41no—
129C23×C7—41no—
130C23×C7—41no—
131C23×C7—41no—
132C23×C7#10312yes—
133C23×C7#10312yes—
134C23×C7C2×C22⋊C411yescharacteristic
135C7×Q8—82no—
136C7×D4—42no—
137C7×D4—42no—
138C2×C4×C7—21no—
139C2×C4×C7—21no—
140C2×C4×C7—22no—
141C2×C4×C7—21no—
142C2×C4×C7—21no—
143C2×C4×C7—21no—
144C2×C4×C7—41no—
145C2×C4×C7—42no—
146C2×C4×C7—41no—
147C2×C4×C7—41no—
148C2×C4×C7—41no—
149C2×C4×C7—41no—
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—81no—
161C2×C4×C7—81no—
162C2×C4×C7—81no—
163C2×C4×C7—84no—
164C2×C4×C7#9911yescharacteristic
165C2×C4×C7#9911yescharacteristic
166C2×C4×C7C2×SD1611yescharacteristic
167C2×C4×C7C2×Q1611yescharacteristic
168C7×C8—82no—
169C7×C8—82no—
170C22×Q8C2×C4×C711yescharacteristic
171C22×D4C7×D411yescharacteristic
172C23×C4—21no—
173C23×C4C2×C4×C711yescharacteristic
174C23×C4C7×D411yescharacteristic
175C2×M16—22no—
176C22×C8—41no—
177C22⋊Q8—41no—
178C22⋊Q8—41no—
179C4×D4—41no—
180C4×D4—41no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4C2×C4×C711yescharacteristic
185C2×C4⋊C4C23×C711yescharacteristic
186C2×C22⋊C4—21no—
187C2×C22⋊C4—21no—
188C2×C22⋊C4—21no—
189C2×C22⋊C4C2×C4×C711yescharacteristic
190C2×C22⋊C4C7×D411yescharacteristic
191C2×C42C7×D411yescharacteristic
192C4⋊C8—41no—
193C4⋊C8—41no—
194#100—42no—
195C22⋊C8—22no—
196C22⋊C8—41no—
197#95—21no—
198#95—21no—
199#95—21no—
200#95—21no—
201#95—21no—
202#95C2×C4×C711yescharacteristic
203#95C2×C4×C711yescharacteristic
204C22×C7—21no—
205C22×C7—21no—
206C22×C7—22no—
207C22×C7—22no—
208C22×C7—42no—
209C22×C7—42no—
210C22×C7—42no—
211C22×C7—41no—
212C22×C7—41no—
213C22×C7—41no—
214C22×C7—41no—
215C22×C7—81no—
216C22×C7C2×#9911yescharacteristic
217C22×C7C2×#10311yescharacteristic
218C22×C7#42111yescharacteristic
219C4×C7—21no—
220C4×C7—21no—
221C4×C7—21no—
222C4×C7—21no—
223C4×C7—42no—
224C4×C7—41no—
225C4×C7—41no—
226C4×C7—41no—
227C4×C7—81no—
228C4×C7—81no—
229C4×C7—81no—
230C24C7×C22⋊C411yescharacteristic
231C24C7×C22⋊C411yescharacteristic
232C2×Q8—41no—
233C2×Q8—81no—
234C2×D4—41no—
235C2×D4—41no—
236C2×D4C7×SD1612yes—
237C2×D4C7×Q1612yes—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—81no—
249C22×C4—81no—
250C22×C4—81no—
251C22×C4C7×C22⋊C411yescharacteristic
252C22×C4C7×C22⋊C411yescharacteristic
253C22×C4C22×C4×C711yescharacteristic
254C22×C4C2×C7×D411yescharacteristic
255C22×C4C2×C7×D411yescharacteristic
256M16—42no—
257M16—42no—
258C2×C8—42no—
259C2×C8—42no—
260C2×C8—82no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—42no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—81no—
267C22⋊C4—41no—
268C22⋊C4—41no—
269C22⋊C4—42no—
270C22⋊C4—42no—
271C22⋊C4—81no—
272C42—21no—
273C42—21no—
274C2×C7—21no—
275C2×C7—21no—
276C2×C7—42no—
277C2×C7—42no—
278C2×C7#152011yescharacteristic
279C2×C7#212111yescharacteristic
280C2×C7#213211yescharacteristic
281C23—21no—
282C23—21no—
283C23—21no—
284C23—21no—
285C23—41no—
286C23—41no—
287C23—41no—
288C23—41no—
289C23C7×#10312yes—
290C23C7×#10312yes—
291C23C2×C7×C22⋊C411yescharacteristic
292Q8—82no—
293D4—42no—
294D4—42no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—22no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—41no—
302C2×C4—42no—
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×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—81no—
318C2×C4—81no—
319C2×C4—81no—
320C2×C4—84no—
321C2×C4C7×#9911yescharacteristic
322C2×C4C7×#9911yescharacteristic
323C2×C4C2×C7×SD1611yescharacteristic
324C2×C4C2×C7×Q1611yescharacteristic
325C8—82no—
326C8—82no—
327C7#1190611yescharacteristic, direct product factor, Sylow
328C22—21no—
329C22—21no—
330C22—22no—
331C22—22no—
332C22—42no—
333C22—42no—
334C22—42no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—81no—
340C22C2×C7×#9911yescharacteristic
341C22C2×C7×#10311yescharacteristic
342C22C7×#42111yescharacteristic
343C4—21no—
344C4—21no—
345C4—21no—
346C4—21no—
347C4—42no—
348C4—41no—
349C4—41no—
350C4—41no—
351C4—81no—
352C4—81no—
353C4—81no—
354C2—21no—
355C2—21no—
356C2—42no—
357C2—42no—
358C2C7×#152011yescharacteristic
359C2C7×#212111yescharacteristic
360C2C7×#213211yescharacteristic
subgroupG/H
center, socleC22×C7C2×#99
commutatorC22×C4C22×C4×C7
Frattini, cosocleC2×C4⋊C4C23×C7
prime radicalC22×C7×D4D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C7 ◁ C22×C4×C7 ◁ C2×C4×C7×D4 ◁ G

Complements

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

NH ≅ G/Ntype
4. C7×#1354C2semidirect
5. C7×#1353C2semidirect
23. #11906C7direct (N × H)
61. #1354C2×C7semidirect
62. #1353C2×C7semidirect
327. C7#11906direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#25642
C7×#18002
C7×#14592
C7×#13542
C7×#13532
#119067