Finite Group Databasebeta

← C7×#19118 #13196334

Subgroups — C7×#19118 #13196334

Order 1792.

606 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3160C211yesmaximal, characteristic
2C7×#3104C212yesmaximal
3C2×C7×C8×Q8C211yesmaximal, characteristic
4C7×#1676C212yesmaximal, complemented
5C7×#1674C212yesmaximal, complemented
6C7×#1672C214yesmaximal, complemented
7C7×#1564C211yesmaximal, characteristic
8C7×#1542C212yesmaximal
9C7×#533C2212yes—
10C2×C4×C7×Q8C2211yescharacteristic
11C7×#500C2211yescharacteristic
12C7×#500C2211yescharacteristic
13C7×#499—21no—
14C7×#499—21no—
15C7×#498C2212yes—
16C7×#484—22no—
17C7×#477C2214yes—
18C7×#462C2214yes—
19C7×#451—22no—
20C7×C8×Q8—21no—
21C7×C8×Q8—21no—
22C7×C8×Q8C2212yes—
23C7×C8×Q8C2212yes—
24C4×C7×Q16—24no—
25C7×#431C2212yes—
26C7×#427—21no—
27C7×#427—21no—
28C2×C7×C4⋊C8—21no—
29C2×C7×C4⋊C8C2211yescharacteristic
30C2×C4×C7×C8—21no—
31C2×C4×C7×C8C2211yescharacteristic
32C7×#335C2212yes—
33C7×#334C2212yes—
34C7×#333C2214yes—
35C7×#332C2214yes—
36#19118C711yesmaximal, characteristic, direct product factor, Sylow
37C22×C7×Q8D411yescharacteristic
38C2×C7×Q16—24no—
39C22×C7×C8—21no—
40C22×C7×C8D411yescharacteristic
41C22×C7×C8D411yescharacteristic
42C7×C4⋊Q8C2312yes—
43C7×C4⋊Q8C2312yes—
44C7×#125—22no—
45C7×C22⋊Q8—22no—
46C7×C22⋊Q8—22no—
47C4×C7×Q8—21no—
48C4×C7×Q8—21no—
49C4×C7×Q8—22no—
50C4×C7×Q8—22no—
51C4×C7×Q8D412yes—
52C4×C7×Q8C2312yes—
53C7×#117—22no—
54C2×C7×C4⋊C4—21no—
55C2×C7×C4⋊C4D411yescharacteristic
56C2×C42×C7—21no—
57C2×C42×C7C2311yescharacteristic
58C7×#107—21no—
59C7×#107—21no—
60C7×#107—21no—
61C7×#107—21no—
62C7×#107—24no—
63C7×#106—22no—
64C7×#106—22no—
65C7×C4⋊C8—22no—
66C7×C4⋊C8—22no—
67C7×C4⋊C8—21no—
68C7×C4⋊C8C2314yes—
69C7×C4⋊C8C2312yes—
70C7×#103—24no—
71C7×#103—24no—
72C7×C22⋊C8—22no—
73C4×C7×C8—22no—
74C4×C7×C8—22no—
75C4×C7×C8—22no—
76C4×C7×C8D411yescharacteristic
77C4×C7×C8D411yescharacteristic
78C4×C7×C8C2311yescharacteristic
79C4×C7×C8C2311yescharacteristic
80#3160C2×C711yescharacteristic
81#3104C2×C712yes—
82C2×C8×Q8C2×C711yescharacteristic
83#1676C2×C712yescomplemented
84#1674C2×C712yescomplemented
85#1672C2×C714yescomplemented
86#1564C2×C711yescharacteristic
87#1542C2×C712yes—
88C2×C7×Q8—24no—
89C2×C7×Q8—21no—
90C2×C7×Q8—22no—
91C2×C7×Q8—22no—
92C2×C7×Q8C2×D412yes—
93C22×C4×C7—21no—
94C22×C4×C7—21no—
95C22×C4×C7C2×D411yescharacteristic
96C22×C4×C7C2×D411yescharacteristic
97C22×C4×C7#4111yescharacteristic
98C7×Q16—48no—
99C2×C7×C8—22no—
100C2×C7×C8—22no—
101C2×C7×C8—24no—
102C2×C7×C8—42no—
103C2×C7×C8Q1614yescomplemented
104C2×C7×C8C2×D411yescharacteristic
105C2×C7×C8C2×D411yescharacteristic
106C2×C7×C8C2×D411yescharacteristic
107C2×C7×C8C2×D411yescharacteristic
108C7×C4⋊C4—22no—
109C7×C4⋊C4—21no—
110C7×C4⋊C4—22no—
111C7×C4⋊C4—22no—
112C7×C4⋊C4—22no—
113C7×C4⋊C4—22no—
114C7×C4⋊C4—22no—
115C7×C4⋊C4—42no—
116C7×C4⋊C4—42no—
117C7×C4⋊C4C2×D412yes—
118C7×C22⋊C4—42no—
119C42×C7—22no—
120C42×C7—22no—
121C42×C7—42no—
122C42×C7C2×D411yescharacteristic
123C42×C7C2×D411yescharacteristic
124C42×C7#4111yescharacteristic
125C42×C7C2411yescharacteristic
126#533C22×C712yes—
127C2×C4×Q8C22×C711yescharacteristic
128#500C22×C711yescharacteristic
129#500C22×C711yescharacteristic
130#499—21no—
131#499—21no—
132#498C22×C712yes—
133#484—22no—
134#477C22×C714yes—
135#462C22×C714yes—
136#451—22no—
137C8×Q8—21no—
138C8×Q8—21no—
139C8×Q8C22×C712yes—
140C8×Q8C22×C712yes—
141C4×Q16—24no—
142#431C22×C712yes—
143#427—21no—
144#427—21no—
145C2×C4⋊C8—21no—
146C2×C4⋊C8C22×C711yescharacteristic
147C2×C4×C8—21no—
148C2×C4×C8C22×C711yescharacteristic
149#335C22×C712yes—
150#334C22×C712yes—
151#333C22×C714yes—
152#332C22×C714yes—
153C23×C7#12111yescharacteristic
154C7×Q8—24no—
155C7×Q8—22no—
156C7×Q8—44no—
157C7×Q8—82no—
158C7×Q8#13514yes—
159C2×C4×C7—24no—
160C2×C4×C7—22no—
161C2×C4×C7—22no—
162C2×C4×C7—22no—
163C2×C4×C7—41no—
164C2×C4×C7—42no—
165C2×C4×C7—42no—
166C2×C4×C7—42no—
167C2×C4×C7#12111yescharacteristic
168C2×C4×C7#12111yescharacteristic
169C2×C4×C7#12111yescharacteristic
170C2×C4×C7C2×Q1612yes—
171C2×C4×C7#13712yes—
172C2×C4×C7C22×D411yescharacteristic
173C2×C4×C7C22×D411yescharacteristic
174C2×C4×C7C2×#4111yescharacteristic
175C7×C8—21no—
176C7×C8—21no—
177C7×C8—44no—
178C7×C8—42no—
179C7×C8C2×Q1612yes—
180C7×C8C2×Q1612yes—
181C22×Q8C7×D411yescharacteristic
182C2×Q16—24no—
183C22×C8—21no—
184C22×C8C7×D411yescharacteristic
185C22×C8C7×D411yescharacteristic
186C4⋊Q8C23×C712yes—
187C4⋊Q8C23×C712yes—
188#125—22no—
189C22⋊Q8—22no—
190C22⋊Q8—22no—
191C4×Q8—21no—
192C4×Q8—21no—
193C4×Q8—22no—
194C4×Q8—22no—
195C4×Q8C7×D412yes—
196C4×Q8C23×C712yes—
197#117—22no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4C7×D411yescharacteristic
200C2×C42—21no—
201C2×C42C23×C711yescharacteristic
202#107—21no—
203#107—21no—
204#107—21no—
205#107—21no—
206#107—24no—
207#106—22no—
208#106—22no—
209C4⋊C8—22no—
210C4⋊C8—22no—
211C4⋊C8—21no—
212C4⋊C8C23×C712yes—
213C4⋊C8C23×C714yes—
214#103—24no—
215#103—24no—
216C22⋊C8—22no—
217C4×C8—22no—
218C4×C8—22no—
219C4×C8—22no—
220C4×C8C7×D411yescharacteristic
221C4×C8C7×D411yescharacteristic
222C4×C8C23×C711yescharacteristic
223C4×C8C23×C711yescharacteristic
224C22×C7—21no—
225C22×C7—21no—
226C22×C7#46212yes—
227C22×C7C2×#12111yescharacteristic
228C4×C7—24no—
229C4×C7—21no—
230C4×C7—21no—
231C4×C7—21no—
232C4×C7—42no—
233C4×C7—82no—
234C4×C7—82no—
235C4×C7#46212yes—
236C4×C7C22×Q1611yescharacteristic
237C4×C7C2×#13512yes—
238C4×C7C2×#13711yescharacteristic
239C2×Q8—24no—
240C2×Q8—21no—
241C2×Q8—22no—
242C2×Q8—22no—
243C2×Q8C2×C7×D412yes—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4C2×C7×D411yescharacteristic
247C22×C4C2×C7×D411yescharacteristic
248C22×C4C7×#4111yescharacteristic
249Q16—48no—
250C2×C8—22no—
251C2×C8—22no—
252C2×C8—24no—
253C2×C8—42no—
254C2×C8C7×Q1614yescomplemented
255C2×C8C2×C7×D411yescharacteristic
256C2×C8C2×C7×D411yescharacteristic
257C2×C8C2×C7×D411yescharacteristic
258C2×C8C2×C7×D411yescharacteristic
259C4⋊C4—22no—
260C4⋊C4—21no—
261C4⋊C4—22no—
262C4⋊C4—22no—
263C4⋊C4—22no—
264C4⋊C4—22no—
265C4⋊C4—22no—
266C4⋊C4—42no—
267C4⋊C4—42no—
268C4⋊C4C2×C7×D412yes—
269C22⋊C4—42no—
270C42—22no—
271C42—22no—
272C42—42no—
273C42C2×C7×D411yescharacteristic
274C42C2×C7×D411yescharacteristic
275C42C7×#4111yescharacteristic
276C42C24×C711yescharacteristic
277C2×C7—22no—
278C2×C7C2×#46211yescharacteristic
279C2×C7#305011yescharacteristic
280C2×C7#306711yescharacteristic
281C23C7×#12111yescharacteristic
282Q8—24no—
283Q8—22no—
284Q8—44no—
285Q8—82no—
286Q8C7×#13514yes—
287C2×C4—24no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—41no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4C7×#12111yescharacteristic
296C2×C4C7×#12111yescharacteristic
297C2×C4C7×#12111yescharacteristic
298C2×C4C2×C7×Q1612yes—
299C2×C4C7×#13712yes—
300C2×C4C22×C7×D411yescharacteristic
301C2×C4C22×C7×D411yescharacteristic
302C2×C4C2×C7×#4111yescharacteristic
303C8—21no—
304C8—21no—
305C8—44no—
306C8—42no—
307C8C2×C7×Q1612yes—
308C8C2×C7×Q1612yes—
309C7#1911811yescharacteristic, direct product factor, Sylow
310C22—21no—
311C22—21no—
312C22C7×#46212yes—
313C22C2×C7×#12111yescharacteristic
314C4—24no—
315C4—21no—
316C4—21no—
317C4—21no—
318C4—42no—
319C4—82no—
320C4—82no—
321C4C7×#46212yes—
322C4C22×C7×Q1611yescharacteristic
323C4C2×C7×#13512yes—
324C4C2×C7×#13711yescharacteristic
325C2—22no—
326C2C2×C7×#46211yescharacteristic
327C2C7×#305011yescharacteristic
328C2C7×#306711yescharacteristic
subgroupG/H
centerC2×C4×C7#121
commutator, Frattini, cosocleC42C24×C7
socleC22×C7C2×#121
prime radicalC23×C7#121
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4×C7 ◁ C2×C42×C7 ◁ G

Complements

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

NH ≅ G/Ntype
4. C7×#1676C2semidirect
5. C7×#1674C2semidirect
6. C7×#1672C2semidirect
36. #19118C7direct (N × H)
83. #1676C2×C7semidirect
84. #1674C2×C7semidirect
85. #1672C2×C7semidirect
103. C2×C7×C8Q16semidirect
254. C2×C8C7×Q16semidirect
309. C7#19118direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#31602
C7×#31042
C2×C7×C8×Q82
C7×#16762
C7×#16742
C7×#16722
C7×#15642
C7×#15422
#191187