Finite Group Databasebeta

← C7×#9297 #13186513

Subgroups — C7×#9297 #13186513

Order 1792.

454 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3209C211yesmaximal, characteristic
2C7×#2331C211yesmaximal, characteristic
3C7×#1995C211yesmaximal, characteristic
4C7×#1548C211yesmaximal, characteristic
5C7×#1518C211yesmaximal, characteristic
6C7×#1286C211yesmaximal, characteristic
7C7×#1274C211yesmaximal, characteristic
8C7×#533C2211yescharacteristic
9C2×C7×C4⋊Q8C2211yescharacteristic
10C7×#494C412yescomplemented
11C7×#487C412yescomplemented
12C7×#462—22no—
13C7×#451—22no—
14C7×#431C2211yescharacteristic
15C7×#431C2211yescharacteristic
16C7×#423C2211yescharacteristic
17C7×#391—22no—
18C7×#389—21no—
19C7×#387—22no—
20C7×#382C2211yescharacteristic
21C4×C7×C4⋊C4—21no—
22C7×#376C2211yescharacteristic
23C7×#339—21no—
24C7×#339—21no—
25C7×#333—22no—
26C7×#328—22no—
27#9297C711yesmaximal, characteristic, direct product factor, Sylow
28C22×C7×Q8D412yes—
29C2×C7×Q16—24no—
30C2×C7×M16—21no—
31C7×C4⋊Q8—21no—
32C7×C4⋊Q8—21no—
33C7×C4⋊Q8C2×C411yescharacteristic
34C7×C4⋊Q8C2×C411yescharacteristic
35C7×C4⋊Q8C2×C412yes—
36C7×C4⋊Q8D412yes—
37C7×#125—21no—
38C7×C22⋊Q8—21no—
39C7×C22⋊Q8—21no—
40C4×C7×Q8—21no—
41C4×C7×Q8—21no—
42C7×#117—21no—
43C2×C7×C4⋊C4—21no—
44C2×C7×C4⋊C4—21no—
45C2×C7×C4⋊C4—21no—
46C2×C7×C4⋊C4—22no—
47C2×C42×C7—22no—
48C2×C42×C7D411yescharacteristic
49C2×C42×C7D411yescharacteristic
50C2×C42×C7C2311yescharacteristic
51C7×C4⋊C8—21no—
52C7×C4⋊C8—21no—
53C7×C4⋊C8—21no—
54C7×C4⋊C8—21no—
55C7×#103—24no—
56C7×C22⋊C8—21no—
57C7×C22⋊C8—21no—
58C4×C7×C8—21no—
59C4×C7×C8C2×C412yes—
60C7×#95—22no—
61C7×#95—22no—
62C7×#95—21no—
63C7×#95—21no—
64C7×#95—21no—
65C7×#95D412yes—
66#3209C2×C711yescharacteristic
67#2331C2×C711yescharacteristic
68#1995C2×C711yescharacteristic
69#1548C2×C711yescharacteristic
70#1518C2×C711yescharacteristic
71#1286C2×C711yescharacteristic
72#1274C2×C711yescharacteristic
73C2×C7×Q8—22no—
74C2×C7×Q8—21no—
75C2×C7×Q8—21no—
76C2×C7×Q8—82no—
77C2×C7×Q8C22⋊C414yes—
78C22×C4×C7—21no—
79C22×C4×C7—22no—
80C22×C4×C7—21no—
81C22×C4×C7—22no—
82C22×C4×C7C2×D411yescharacteristic
83C22×C4×C7C2×D411yescharacteristic
84C22×C4×C7C2×D411yescharacteristic
85C7×Q16—84no—
86C7×M16—41no—
87C7×M16—41no—
88C2×C7×C8—22no—
89C2×C7×C8—41no—
90C2×C7×C8—41no—
91C7×C4⋊C4—22no—
92C7×C4⋊C4—22no—
93C7×C4⋊C4—22no—
94C7×C4⋊C4—22no—
95C7×C4⋊C4—21no—
96C7×C4⋊C4—21no—
97C7×C4⋊C4—21no—
98C7×C4⋊C4—21no—
99C7×C4⋊C4—41no—
100C7×C4⋊C4—41no—
101C7×C4⋊C4—41no—
102C7×C4⋊C4—41no—
103C7×C4⋊C4—82no—
104C7×C22⋊C4—41no—
105C42×C7—21no—
106C42×C7—21no—
107C42×C7—21no—
108C42×C7—41no—
109C42×C7—82no—
110C42×C7Q1612yescomplemented
111C42×C7C22×C411yescharacteristic
112C42×C7C2×D411yescharacteristic
113C42×C7#4111yescharacteristic
114C42×C7#4111yescharacteristic
115#533C22×C711yescharacteristic
116C2×C4⋊Q8C22×C711yescharacteristic
117#494C4×C712yescomplemented
118#487C4×C712yescomplemented
119#462—22no—
120#451—22no—
121#431C22×C711yescharacteristic
122#431C22×C711yescharacteristic
123#423C22×C711yescharacteristic
124#391—22no—
125#389—21no—
126#387—22no—
127#382C22×C711yescharacteristic
128C4×C4⋊C4—21no—
129#376C22×C711yescharacteristic
130#339—21no—
131#339—21no—
132#333—22no—
133#328—22no—
134C23×C7#12011yescharacteristic
135C7×Q8—44no—
136C7×Q8—42no—
137C7×Q8—82no—
138C7×Q8—81no—
139C2×C4×C7—24no—
140C2×C4×C7—22no—
141C2×C4×C7—41no—
142C2×C4×C7—41no—
143C2×C4×C7—41no—
144C2×C4×C7—42no—
145C2×C4×C7—42no—
146C2×C4×C7—41no—
147C2×C4×C7—42no—
148C2×C4×C7—41no—
149C2×C4×C7—41no—
150C2×C4×C7—42no—
151C2×C4×C7—41no—
152C2×C4×C7—41no—
153C2×C4×C7—41no—
154C2×C4×C7—42no—
155C2×C4×C7C2×C22⋊C411yescharacteristic
156C2×C4×C7C4×D411yescharacteristic
157C2×C4×C7C4×D411yescharacteristic
158C2×C4×C7#12111yescharacteristic
159C2×C4×C7#12111yescharacteristic
160C2×C4×C7#12311yescharacteristic
161C2×C4×C7C2×Q1611yescharacteristic
162C2×C4×C7#13511yescharacteristic
163C2×C4×C7#13711yescharacteristic
164C2×C4×C7#13711yescharacteristic
165C7×C8—42no—
166C7×C8—81no—
167C7×C8—81no—
168C22×Q8C7×D412yes—
169C2×Q16—24no—
170C2×M16—21no—
171C4⋊Q8—21no—
172C4⋊Q8—21no—
173C4⋊Q8C2×C4×C711yescharacteristic
174C4⋊Q8C2×C4×C711yescharacteristic
175C4⋊Q8C2×C4×C712yes—
176C4⋊Q8C7×D412yes—
177#125—21no—
178C22⋊Q8—21no—
179C22⋊Q8—21no—
180C4×Q8—21no—
181C4×Q8—21no—
182#117—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—22no—
187C2×C42—22no—
188C2×C42C7×D411yescharacteristic
189C2×C42C7×D411yescharacteristic
190C2×C42C23×C711yescharacteristic
191C4⋊C8—21no—
192C4⋊C8—21no—
193C4⋊C8—21no—
194C4⋊C8—21no—
195#103—24no—
196C22⋊C8—21no—
197C22⋊C8—21no—
198C4×C8—21no—
199C4×C8C2×C4×C712yes—
200#95—22no—
201#95—22no—
202#95—21no—
203#95—21no—
204#95—21no—
205#95C7×D412yes—
206C22×C7—21no—
207C22×C7—21no—
208C22×C7#38611yescharacteristic
209C22×C7#45111yescharacteristic
210C22×C7#45211yescharacteristic
211C4×C7—21no—
212C4×C7—21no—
213C4×C7—21no—
214C4×C7—21no—
215C4×C7—42no—
216C4×C7—44no—
217C4×C7—82no—
218C4×C7—81no—
219C4×C7—81no—
220C4×C7—81no—
221C4×C7C4×Q1611yescharacteristic
222C4×C7#44111yescharacteristic
223C4×C7#46211yescharacteristic
224C4×C7#46411yescharacteristic
225C2×Q8—22no—
226C2×Q8—21no—
227C2×Q8—21no—
228C2×Q8—82no—
229C2×Q8C7×C22⋊C414yes—
230C22×C4—21no—
231C22×C4—22no—
232C22×C4—21no—
233C22×C4—22no—
234C22×C4C2×C7×D411yescharacteristic
235C22×C4C2×C7×D411yescharacteristic
236C22×C4C2×C7×D411yescharacteristic
237Q16—84no—
238M16—41no—
239M16—41no—
240C2×C8—22no—
241C2×C8—41no—
242C2×C8—41no—
243C4⋊C4—22no—
244C4⋊C4—22no—
245C4⋊C4—22no—
246C4⋊C4—22no—
247C4⋊C4—21no—
248C4⋊C4—21no—
249C4⋊C4—21no—
250C4⋊C4—21no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—82no—
256C22⋊C4—41no—
257C42—21no—
258C42—21no—
259C42—21no—
260C42—41no—
261C42—82no—
262C42C7×Q1612yescomplemented
263C42C22×C4×C711yescharacteristic
264C42C2×C7×D411yescharacteristic
265C42C7×#4111yescharacteristic
266C42C7×#4111yescharacteristic
267C2×C7—21no—
268C2×C7—21no—
269C2×C7#187811yescharacteristic
270C2×C7#190511yescharacteristic
271C2×C7#191111yescharacteristic
272C23C7×#12011yescharacteristic
273Q8—44no—
274Q8—42no—
275Q8—82no—
276Q8—81no—
277C2×C4—24no—
278C2×C4—22no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—41no—
285C2×C4—42no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—42no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—42no—
293C2×C4C2×C7×C22⋊C411yescharacteristic
294C2×C4C4×C7×D411yescharacteristic
295C2×C4C4×C7×D411yescharacteristic
296C2×C4C7×#12111yescharacteristic
297C2×C4C7×#12111yescharacteristic
298C2×C4C7×#12311yescharacteristic
299C2×C4C2×C7×Q1611yescharacteristic
300C2×C4C7×#13511yescharacteristic
301C2×C4C7×#13711yescharacteristic
302C2×C4C7×#13711yescharacteristic
303C8—42no—
304C8—81no—
305C8—81no—
306C7#929711yescharacteristic, direct product factor, Sylow
307C22—21no—
308C22—21no—
309C22C7×#38611yescharacteristic
310C22C7×#45111yescharacteristic
311C22C7×#45211yescharacteristic
312C4—21no—
313C4—21no—
314C4—21no—
315C4—21no—
316C4—42no—
317C4—44no—
318C4—82no—
319C4—81no—
320C4—81no—
321C4—81no—
322C4C4×C7×Q1611yescharacteristic
323C4C7×#44111yescharacteristic
324C4C7×#46211yescharacteristic
325C4C7×#46411yescharacteristic
326C2—21no—
327C2—21no—
328C2C7×#187811yescharacteristic
329C2C7×#190511yescharacteristic
330C2C7×#191111yescharacteristic
subgroupG/H
center, socleC22×C7#386
commutatorC42C22×C4×C7
Frattini, cosocleC2×C42C23×C7
prime radicalC23×C7#120
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
10. C7×#494C4semidirect
11. C7×#487C4semidirect
27. #9297C7direct (N × H)
110. C42×C7Q16semidirect
117. #494C4×C7semidirect
118. #487C4×C7semidirect
262. C42C7×Q16semidirect
306. C7#9297direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#32092
C7×#23312
C7×#19952
C7×#15482
C7×#15182
C7×#12862
C7×#12742
#92977