Finite Group Databasebeta

← C7×#9294 #13186510

Subgroups — C7×#9294 #13186510

Order 1792.

490 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3208C211yesmaximal, characteristic
2C7×#2961C211yesmaximal, characteristic, complemented
3C7×#1961C211yesmaximal, characteristic
4C7×#1545C211yesmaximal, characteristic
5C7×#1518C211yesmaximal, characteristic, complemented
6C7×#1286C211yesmaximal, characteristic, complemented
7C7×#1275C211yesmaximal, characteristic, complemented
8C7×#533C2211yescharacteristic
9C2×C7×#127C2211yescharacteristic
10C7×#492C412yescomplemented
11C7×#486C412yescomplemented
12C7×#460—22no—
13C7×#450—22no—
14C7×#446—22no—
15C7×#446—22no—
16C7×#431C2211yescharacteristic
17C7×#431C2211yescharacteristic
18C7×#424—21no—
19C7×#423C2211yescharacteristic
20C2×C7×#102—21no—
21C2×C7×#102—21no—
22C7×#405—21no—
23C2×C7×#97C2211yescharacteristic
24C7×#382C2211yescharacteristic
25C7×#331—22no—
26C7×#328—22no—
27#9294C711yesmaximal, characteristic, direct product factor, Sylow
28C22×C7×D4—21no—
29C22×C7×D4—21no—
30C22×C7×D4—21no—
31C2×C7×SD16—24no—
32C2×C7×M16—21no—
33C22×C7×C8D411yescharacteristic
34C22×C7×C8D411yescharacteristic
35C7×C4⋊Q8C2×C411yescharacteristic
36C7×C4⋊Q8C2×C411yescharacteristic
37C7×#127—21no—
38C7×#127—21no—
39C7×#127—21no—
40C7×#127C2×C412yes—
41C7×#125—21no—
42C7×C22⋊Q8—21no—
43C7×C22⋊Q8D412yes—
44C4×C7×Q8—21no—
45C4×C7×Q8D412yes—
46C7×#117—21no—
47C2×C7×C4⋊C4—21no—
48C2×C42×C7C2311yescharacteristic
49C7×C4⋊C8—24no—
50C7×C4⋊C8—21no—
51C7×C4⋊C8—21no—
52C7×C4⋊C8—21no—
53C7×C4⋊C8D412yes—
54C7×#102—24no—
55C7×#102—42no—
56C7×#102—42no—
57C7×C22⋊C8—21no—
58C7×C22⋊C8—21no—
59C7×#97—22no—
60C7×#97—21no—
61C7×#97—21no—
62C4×C7×C8—22no—
63C4×C7×C8—21no—
64C4×C7×C8C2×C412yes—
65C7×#95—21no—
66C7×#95—21no—
67#3208C2×C711yescharacteristic
68#2961C2×C711yescharacteristic, complemented
69#1961C2×C711yescharacteristic
70#1545C2×C711yescharacteristic
71#1518C2×C711yescharacteristic, complemented
72#1286C2×C711yescharacteristic, complemented
73#1275C2×C711yescharacteristic, complemented
74C24×C7—41no—
75C2×C7×Q8—21no—
76C2×C7×Q8C22⋊C412yes—
77C2×C7×D4—22no—
78C2×C7×D4—22no—
79C2×C7×D4—22no—
80C2×C7×D4—41no—
81C2×C7×D4—41no—
82C2×C7×D4—41no—
83C2×C7×D4—41no—
84C2×C7×D4—41no—
85C2×C7×D4—41no—
86C2×C7×D4—41no—
87C2×C7×D4—82no—
88C2×C7×D4—82no—
89C22×C4×C7—41no—
90C22×C4×C7C2×D411yescharacteristic
91C22×C4×C7C2×D411yescharacteristic
92C22×C4×C7#4111yescharacteristic
93C7×SD16—84no—
94C7×M16—41no—
95C7×M16—41no—
96C2×C7×C8—21no—
97C2×C7×C8—21no—
98C2×C7×C8—22no—
99C2×C7×C8—22no—
100C2×C7×C8—21no—
101C2×C7×C8—21no—
102C2×C7×C8—21no—
103C2×C7×C8—41no—
104C2×C7×C8SD1612yescomplemented
105C7×C4⋊C4—21no—
106C7×C4⋊C4—21no—
107C7×C4⋊C4—21no—
108C7×C4⋊C4—22no—
109C7×C4⋊C4—22no—
110C7×C4⋊C4—41no—
111C7×C4⋊C4—41no—
112C7×C4⋊C4C22⋊C412yes—
113C7×C22⋊C4—22no—
114C42×C7—22no—
115C42×C7C22×C411yescharacteristic
116C42×C7C2×D411yescharacteristic
117C42×C7C2×D411yescharacteristic
118C42×C7#4111yescharacteristic
119#533C22×C711yescharacteristic
120C2×#127C22×C711yescharacteristic
121#492C4×C712yescomplemented
122#486C4×C712yescomplemented
123#460—22no—
124#450—22no—
125#446—22no—
126#446—22no—
127#431C22×C711yescharacteristic
128#431C22×C711yescharacteristic
129#424—21no—
130#423C22×C711yescharacteristic
131C2×#102—21no—
132C2×#102—21no—
133#405—21no—
134C2×#97C22×C711yescharacteristic
135#382C22×C711yescharacteristic
136#331—22no—
137#328—22no—
138C23×C7—42no—
139C23×C7—82no—
140C23×C7—82no—
141C23×C7—81no—
142C23×C7—81no—
143C23×C7#12111yescharacteristic
144C7×Q8—42no—
145C7×Q8—81no—
146C7×D4—44no—
147C7×D4—42no—
148C7×D4—42no—
149C7×D4—82no—
150C7×D4—82no—
151C7×D4—82no—
152C7×D4—81no—
153C7×D4—81no—
154C2×C4×C7—22no—
155C2×C4×C7—22no—
156C2×C4×C7—41no—
157C2×C4×C7—41no—
158C2×C4×C7—81no—
159C2×C4×C7—81no—
160C2×C4×C7—81no—
161C2×C4×C7C2×C22⋊C411yescharacteristic
162C2×C4×C7C4×D411yescharacteristic
163C2×C4×C7C4×D411yescharacteristic
164C2×C4×C7#12011yescharacteristic
165C2×C4×C7#12111yescharacteristic
166C2×C4×C7#12311yescharacteristic
167C2×C4×C7C2×SD1611yescharacteristic
168C2×C4×C7#13511yescharacteristic
169C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
170C2×C4×C7#13711yescharacteristic
171C7×C8—22no—
172C7×C8—42no—
173C7×C8—41no—
174C7×C8—42no—
175C7×C8—81no—
176C22×D4—21no—
177C22×D4—21no—
178C22×D4—21no—
179C2×SD16—24no—
180C2×M16—21no—
181C22×C8C7×D411yescharacteristic
182C22×C8C7×D411yescharacteristic
183C4⋊Q8C2×C4×C711yescharacteristic
184C4⋊Q8C2×C4×C711yescharacteristic
185#127—21no—
186#127—21no—
187#127—21no—
188#127C2×C4×C712yes—
189#125—21no—
190C22⋊Q8—21no—
191C22⋊Q8C7×D412yes—
192C4×Q8—21no—
193C4×Q8C7×D412yes—
194#117—21no—
195C2×C4⋊C4—21no—
196C2×C42C23×C711yescharacteristic
197C4⋊C8—24no—
198C4⋊C8—21no—
199C4⋊C8—21no—
200C4⋊C8—21no—
201C4⋊C8C7×D412yes—
202#102—24no—
203#102—42no—
204#102—42no—
205C22⋊C8—21no—
206C22⋊C8—21no—
207#97—22no—
208#97—21no—
209#97—21no—
210C4×C8—22no—
211C4×C8—21no—
212C4×C8C2×C4×C712yes—
213#95—21no—
214#95—21no—
215C22×C7—21no—
216C22×C7—21no—
217C22×C7—82no—
218C22×C7—82no—
219C22×C7—82no—
220C22×C7—162no—
221C22×C7—162no—
222C22×C7#38611yescharacteristic
223C22×C7#46011yescharacteristic
224C22×C7#46411yescharacteristic
225C4×C7—21no—
226C4×C7—21no—
227C4×C7—21no—
228C4×C7—21no—
229C4×C7—42no—
230C4×C7—42no—
231C4×C7—161no—
232C4×C7C4×SD1611yescharacteristic
233C4×C7#44011yescharacteristic
234C4×C7#44811yescharacteristic
235C4×C7#44911yescharacteristic
236C24—41no—
237C2×Q8—21no—
238C2×Q8C7×C22⋊C412yes—
239C2×D4—22no—
240C2×D4—22no—
241C2×D4—22no—
242C2×D4—41no—
243C2×D4—41no—
244C2×D4—41no—
245C2×D4—41no—
246C2×D4—41no—
247C2×D4—41no—
248C2×D4—41no—
249C2×D4—82no—
250C2×D4—82no—
251C22×C4—41no—
252C22×C4C2×C7×D411yescharacteristic
253C22×C4C2×C7×D411yescharacteristic
254C22×C4C7×#4111yescharacteristic
255SD16—84no—
256M16—41no—
257M16—41no—
258C2×C8—21no—
259C2×C8—21no—
260C2×C8—22no—
261C2×C8—22no—
262C2×C8—21no—
263C2×C8—21no—
264C2×C8—21no—
265C2×C8—41no—
266C2×C8C7×SD1612yescomplemented
267C4⋊C4—21no—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—22no—
271C4⋊C4—22no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4C7×C22⋊C412yes—
275C22⋊C4—22no—
276C42—22no—
277C42C22×C4×C711yescharacteristic
278C42C2×C7×D411yescharacteristic
279C42C2×C7×D411yescharacteristic
280C42C7×#4111yescharacteristic
281C2×C7—21no—
282C2×C7—21no—
283C2×C7—162no—
284C2×C7#188011yescharacteristic
285C2×C7#190311yescharacteristic
286C2×C7#191011yescharacteristic
287C23—42no—
288C23—82no—
289C23—82no—
290C23—81no—
291C23—81no—
292C23C7×#12111yescharacteristic
293Q8—42no—
294Q8—81no—
295D4—44no—
296D4—42no—
297D4—42no—
298D4—82no—
299D4—82no—
300D4—82no—
301D4—81no—
302D4—81no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—81no—
308C2×C4—81no—
309C2×C4—81no—
310C2×C4C2×C7×C22⋊C411yescharacteristic
311C2×C4C4×C7×D411yescharacteristic
312C2×C4C4×C7×D411yescharacteristic
313C2×C4C7×#12011yescharacteristic
314C2×C4C7×#12111yescharacteristic
315C2×C4C7×#12311yescharacteristic
316C2×C4C2×C7×SD1611yescharacteristic
317C2×C4C7×#13511yescharacteristic
318C2×C4C7×AGL(1,ℤ/8)11yescharacteristic
319C2×C4C7×#13711yescharacteristic
320C8—22no—
321C8—42no—
322C8—41no—
323C8—42no—
324C8—81no—
325C7#929411yescharacteristic, direct product factor, Sylow
326C22—21no—
327C22—21no—
328C22—82no—
329C22—82no—
330C22—82no—
331C22—162no—
332C22—162no—
333C22C7×#38611yescharacteristic
334C22C7×#46011yescharacteristic
335C22C7×#46411yescharacteristic
336C4—21no—
337C4—21no—
338C4—21no—
339C4—21no—
340C4—42no—
341C4—42no—
342C4—161no—
343C4C4×C7×SD1611yescharacteristic
344C4C7×#44011yescharacteristic
345C4C7×#44811yescharacteristic
346C4C7×#44911yescharacteristic
347C2—21no—
348C2—21no—
349C2—162no—
350C2C7×#188011yescharacteristic
351C2C7×#190311yescharacteristic
352C2C7×#191011yescharacteristic
subgroupG/H
center, socleC22×C7#386
commutatorC42C22×C4×C7
Frattini, cosocleC2×C42C23×C7
prime radicalC2×C7×#127C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C7 ◁ C2×C42×C7 ◁ G

Complements

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

NH ≅ G/Ntype
2. C7×#2961C2semidirect
5. C7×#1518C2semidirect
6. C7×#1286C2semidirect
7. C7×#1275C2semidirect
10. C7×#492C4semidirect
11. C7×#486C4semidirect
27. #9294C7direct (N × H)
68. #2961C2×C7semidirect
71. #1518C2×C7semidirect
72. #1286C2×C7semidirect
73. #1275C2×C7semidirect
104. C2×C7×C8SD16semidirect
121. #492C4×C7semidirect
122. #486C4×C7semidirect
266. C2×C8C7×SD16semidirect
325. C7#9294direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#32082
C7×#29612
C7×#19612
C7×#15452
C7×#15182
C7×#12862
C7×#12752
#92947