Finite Group Databasebeta

← C7×#10169 #13187385

Subgroups — C7×#10169 #13187385

Order 1792.

556 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3208C211yesmaximal, characteristic
2C7×#3139C211yesmaximal, characteristic, complemented
3C2×C7×#486C211yesmaximal, characteristic
4C7×#1965C211yesmaximal, characteristic
5C7×#1546C211yesmaximal, characteristic, complemented
6C7×#1493C211yesmaximal, characteristic, complemented
7C7×#1278C211yesmaximal, characteristic, complemented
8C7×#533C2211yescharacteristic
9C2×C7×C4⋊Q8C2211yescharacteristic
10C2×C7×#127C2211yescharacteristic
11C7×#492—21no—
12C7×#491—22no—
13C7×#487—21no—
14C7×#487—21no—
15C7×#486—21no—
16C7×#486—22no—
17C7×#486—22no—
18C7×#460—22no—
19C7×#450—22no—
20C7×#431C2211yescharacteristic
21C7×#423C2211yescharacteristic
22C7×#419—22no—
23C2×C7×#102—22no—
24C2×C7×#100—22no—
25C4×C7×M16C2211yescharacteristic
26C2×C4×C7×C8C2211yescharacteristic
27C7×#332—22no—
28C7×Q8⋊C8—22no—
29#10169C711yesmaximal, characteristic, direct product factor, Sylow
30C22×C7×Q8—21no—
31C22×C7×D4—21no—
32C22×C7×D4—21no—
33C22×C7×D4D412yes—
34C2×C7×SD16—24no—
35C2×C7×M16—21no—
36C2×C7×M16D412yes—
37C22×C7×C8—21no—
38C7×C4⋊Q8—21no—
39C7×C4⋊Q8—21no—
40C7×C4⋊Q8—21no—
41C7×C4⋊Q8D411yescharacteristic
42C7×C4⋊Q8D412yes—
43C7×C4⋊Q8D411yescharacteristic
44C7×#127—21no—
45C7×#127—21no—
46C7×#127—21no—
47C7×#127—21no—
48C7×#125—21no—
49C7×C22⋊Q8—21no—
50C7×C22⋊Q8—21no—
51C4×C7×Q8—21no—
52C4×C7×Q8—21no—
53C7×#117—21no—
54C2×C7×C4⋊C4—21no—
55C2×C7×C4⋊C4—21no—
56C2×C42×C7C2311yescharacteristic
57C7×C4⋊C8—21no—
58C7×C4⋊C8—21no—
59C7×C4⋊C8D412yes—
60C7×#103—24no—
61C7×#103—24no—
62C7×#102—28no—
63C7×#102—24no—
64C7×#100—44no—
65C7×C22⋊C8—21no—
66C7×#97—21no—
67C4×C7×C8—21no—
68C4×C7×C8—21no—
69C4×C7×C8—21no—
70C4×C7×C8D412yes—
71#3208C2×C711yescharacteristic
72#3139C2×C711yescharacteristic, complemented
73C2×#486C2×C711yescharacteristic
74#1965C2×C711yescharacteristic
75#1546C2×C711yescharacteristic, complemented
76#1493C2×C711yescharacteristic, complemented
77#1278C2×C711yescharacteristic, complemented
78C24×C7—22no—
79C2×C7×Q8—22no—
80C2×C7×Q8—21no—
81C2×C7×Q8—21no—
82C2×C7×Q8—41no—
83C2×C7×Q8—42no—
84C2×C7×Q8—42no—
85C2×C7×D4—22no—
86C2×C7×D4—22no—
87C2×C7×D4—44no—
88C2×C7×D4—41no—
89C2×C7×D4—41no—
90C2×C7×D4—41no—
91C2×C7×D4—41no—
92C2×C7×D4—42no—
93C2×C7×D4—42no—
94C2×C7×D4—82no—
95C22×C4×C7—41no—
96C22×C4×C7C2×D411yescharacteristic
97C22×C4×C7C2×D411yescharacteristic
98C22×C4×C7#4111yescharacteristic
99C7×SD16—84no—
100C7×M16—22no—
101C7×M16—22no—
102C7×M16—42no—
103C2×C7×C8—22no—
104C2×C7×C8—22no—
105C2×C7×C8—22no—
106C2×C7×C8—42no—
107C2×C7×C8—41no—
108C7×C4⋊C4—22no—
109C7×C4⋊C4—21no—
110C7×C4⋊C4—21no—
111C7×C4⋊C4—21no—
112C7×C4⋊C4—22no—
113C7×C4⋊C4—21no—
114C7×C4⋊C4—41no—
115C7×C4⋊C4—41no—
116C7×C4⋊C4—41no—
117C7×C4⋊C4—41no—
118C7×C22⋊C4—41no—
119C42×C7—41no—
120C42×C7C2×D411yescharacteristic
121C42×C7C2×D411yescharacteristic
122C42×C7C2×D411yescharacteristic
123C42×C7C2×D411yescharacteristic
124#533C22×C711yescharacteristic
125C2×C4⋊Q8C22×C711yescharacteristic
126C2×#127C22×C711yescharacteristic
127#492—21no—
128#491—22no—
129#487—21no—
130#487—21no—
131#486—22no—
132#486—21no—
133#486—22no—
134#460—22no—
135#450—22no—
136#431C22×C711yescharacteristic
137#423C22×C711yescharacteristic
138#419—22no—
139C2×#102—22no—
140C2×#100—22no—
141C4×M16C22×C711yescharacteristic
142C2×C4×C8C22×C711yescharacteristic
143#332—22no—
144Q8⋊C8—22no—
145C23×C7—42no—
146C23×C7—44no—
147C23×C7—82no—
148C23×C7—82no—
149C23×C7#12111yescharacteristic
150C7×Q8—44no—
151C7×Q8—42no—
152C7×Q8—42no—
153C7×Q8—81no—
154C7×Q8—81no—
155C7×D4—44no—
156C7×D4—44no—
157C7×D4—82no—
158C7×D4—82no—
159C7×D4—82no—
160C7×D4—81no—
161C7×D4—81no—
162C2×C4×C7—22no—
163C2×C4×C7—22no—
164C2×C4×C7—41no—
165C2×C4×C7—41no—
166C2×C4×C7—42no—
167C2×C4×C7—82no—
168C2×C4×C7#12011yescharacteristic
169C2×C4×C7#12011yescharacteristic
170C2×C4×C7#12011yescharacteristic
171C2×C4×C7#12111yescharacteristic
172C2×C4×C7#12111yescharacteristic
173C2×C4×C7#12711yescharacteristic
174C2×C4×C7#13512yes—
175C2×C4×C7AGL(1,ℤ/8)12yes—
176C7×C8—42no—
177C7×C8—42no—
178C7×C8—42no—
179C7×C8—81no—
180C22×Q8—21no—
181C22×D4—21no—
182C22×D4—21no—
183C22×D4C7×D412yes—
184C2×SD16—24no—
185C2×M16—21no—
186C2×M16C7×D412yes—
187C22×C8—21no—
188C4⋊Q8—21no—
189C4⋊Q8—21no—
190C4⋊Q8—21no—
191C4⋊Q8C7×D412yes—
192C4⋊Q8C7×D411yescharacteristic
193C4⋊Q8C7×D411yescharacteristic
194#127—21no—
195#127—21no—
196#127—21no—
197#127—21no—
198#125—21no—
199C22⋊Q8—21no—
200C22⋊Q8—21no—
201C4×Q8—21no—
202C4×Q8—21no—
203#117—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C42C23×C711yescharacteristic
207C4⋊C8—21no—
208C4⋊C8—21no—
209C4⋊C8C7×D412yes—
210#103—24no—
211#103—24no—
212#102—28no—
213#102—24no—
214#100—44no—
215C22⋊C8—21no—
216#97—21no—
217C4×C8—21no—
218C4×C8—21no—
219C4×C8—21no—
220C4×C8C7×D412yes—
221C22×C7—21no—
222C22×C7—21no—
223C22×C7—82no—
224C22×C7—82no—
225C22×C7—82no—
226C22×C7—164no—
227C22×C7#39211yescharacteristic
228C22×C7#46312yes—
229C4×C7—21no—
230C4×C7—21no—
231C4×C7—81no—
232C4×C7—82no—
233C4×C7—81no—
234C4×C7#44912yes—
235C4×C7#45512yes—
236C4×C7#47112yes—
237C4×C7#49511yescharacteristic
238C4×C7#49611yescharacteristic
239C24—22no—
240C2×Q8—22no—
241C2×Q8—21no—
242C2×Q8—21no—
243C2×Q8—41no—
244C2×Q8—42no—
245C2×Q8—42no—
246C2×D4—22no—
247C2×D4—22no—
248C2×D4—44no—
249C2×D4—41no—
250C2×D4—41no—
251C2×D4—41no—
252C2×D4—41no—
253C2×D4—42no—
254C2×D4—42no—
255C2×D4—82no—
256C22×C4—41no—
257C22×C4C2×C7×D411yescharacteristic
258C22×C4C2×C7×D411yescharacteristic
259C22×C4C7×#4111yescharacteristic
260SD16—84no—
261M16—22no—
262M16—22no—
263M16—42no—
264C2×C8—22no—
265C2×C8—22no—
266C2×C8—22no—
267C2×C8—42no—
268C2×C8—41no—
269C4⋊C4—22no—
270C4⋊C4—21no—
271C4⋊C4—21no—
272C4⋊C4—21no—
273C4⋊C4—22no—
274C4⋊C4—21no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C22⋊C4—41no—
280C42—41no—
281C42C2×C7×D411yescharacteristic
282C42C2×C7×D411yescharacteristic
283C42C2×C7×D411yescharacteristic
284C42C2×C7×D411yescharacteristic
285C2×C7—22no—
286C2×C7—162no—
287C2×C7#200411yescharacteristic
288C2×C7#201211yescharacteristic
289C2×C7#201911yescharacteristic
290C23—42no—
291C23—44no—
292C23—82no—
293C23—82no—
294C23C7×#12111yescharacteristic
295Q8—42no—
296Q8—44no—
297Q8—42no—
298Q8—81no—
299Q8—81no—
300D4—44no—
301D4—44no—
302D4—82no—
303D4—82no—
304D4—81no—
305D4—81no—
306D4—82no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—42no—
312C2×C4—82no—
313C2×C4C7×#12011yescharacteristic
314C2×C4C7×#12011yescharacteristic
315C2×C4C7×#12011yescharacteristic
316C2×C4C7×#12111yescharacteristic
317C2×C4C7×#12111yescharacteristic
318C2×C4C7×#12711yescharacteristic
319C2×C4C7×#13512yes—
320C2×C4C7×AGL(1,ℤ/8)12yes—
321C8—42no—
322C8—42no—
323C8—42no—
324C8—81no—
325C7#1016911yescharacteristic, direct product factor, Sylow
326C22—21no—
327C22—21no—
328C22—82no—
329C22—82no—
330C22—82no—
331C22—164no—
332C22C7×#39211yescharacteristic
333C22C7×#46312yes—
334C4—21no—
335C4—21no—
336C4—81no—
337C4—82no—
338C4—81no—
339C4C7×#44912yes—
340C4C7×#45512yes—
341C4C7×#47112yes—
342C4C7×#49511yescharacteristic
343C4C7×#49611yescharacteristic
344C2—22no—
345C2—162no—
346C2C7×#200411yescharacteristic
347C2C7×#201211yescharacteristic
348C2C7×#201911yescharacteristic
subgroupG/H
center, socleC22×C7#392
commutator, Frattini, cosocleC2×C42C23×C7
prime radicalC2×C7×#127C22
LCSG ▷ C2×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×#3139C2semidirect
5. C7×#1546C2semidirect
6. C7×#1493C2semidirect
7. C7×#1278C2semidirect
29. #10169C7direct (N × H)
72. #3139C2×C7semidirect
75. #1546C2×C7semidirect
76. #1493C2×C7semidirect
77. #1278C2×C7semidirect
325. C7#10169direct (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×#31392
C2×C7×#4862
C7×#19652
C7×#15462
C7×#14932
C7×#12782
#101697