Finite Group Databasebeta

← C7×#9433 #13186649

Subgroups — C7×#9433 #13186649

Order 1792.

440 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3141C211yesmaximal, characteristic
2C7×#2979C211yesmaximal, characteristic
3C7×#1560C211yesmaximal, characteristic
4C7×#1548C211yesmaximal, characteristic, complemented
5C7×#1546C211yesmaximal, characteristic, complemented
6C7×#1299C211yesmaximal, characteristic, complemented
7C7×#1299C211yesmaximal, characteristic, complemented
8C7×#532C2211yescharacteristic
9C2×C7×C4⋊Q8C2211yescharacteristic
10C7×#489C414yes—
11C7×#487—21no—
12C7×#486—21no—
13C7×#436—21no—
14C7×#436—21no—
15C7×C8×D4—21no—
16C7×C8×D4—21no—
17C7×#431C2211yescharacteristic
18C7×#431C2211yescharacteristic
19C7×#423C2211yescharacteristic
20C7×#423C2211yescharacteristic
21C7×#417—22no—
22C7×#408—22no—
23C4×C7×M16C2211yescharacteristic
24C7×#333—21no—
25C7×#333—21no—
26C7×#333—21no—
27C7×#332—21no—
28C7×#332—21no—
29C7×#331—21no—
30#9433C711yesmaximal, characteristic, direct product factor, Sylow
31C2×C7×#41D412yes—
32C22×C7×Q8—21no—
33C2×C7×M16—21no—
34C2×C7×M16—22no—
35C2×C7×M16—21no—
36C2×C7×M16—21no—
37C22×C7×C8—22no—
38C7×C4⋊Q8—21no—
39C7×C4⋊Q8—21no—
40C7×C4⋊Q8—21no—
41C7×C4⋊Q8C2×C412yes—
42C7×C4⋊Q8D411yescharacteristic
43C7×#127D411yescharacteristic
44C7×#124C2×C412yes—
45C7×#121—21no—
46C7×#121—21no—
47C4×C7×D4—21no—
48C4×C7×D4—21no—
49C2×C7×C4⋊C4—21no—
50C2×C7×C4⋊C4—21no—
51C2×C42×C7C2311yescharacteristic
52C7×C4⋊C8—21no—
53C7×C4⋊C8—21no—
54C7×C4⋊C8—21no—
55C7×C4⋊C8—21no—
56C7×C4⋊C8D411yescharacteristic
57C7×C4⋊C8D411yescharacteristic
58C7×C4⋊C8D411yescharacteristic
59C7×C4⋊C8D411yescharacteristic
60C7×#103—24no—
61C7×#102—24no—
62C7×C22⋊C8—22no—
63C7×C22⋊C8—22no—
64C7×C22⋊C8—21no—
65C7×C22⋊C8—21no—
66C7×#97C2×C412yes—
67C4×C7×C8—21no—
68C4×C7×C8—21no—
69C4×C7×C8—21no—
70#3141C2×C711yescharacteristic
71#2979C2×C711yescharacteristic
72#1560C2×C711yescharacteristic
73#1548C2×C711yescharacteristic, complemented
74#1546C2×C711yescharacteristic, complemented
75#1299C2×C711yescharacteristic, complemented
76#1299C2×C711yescharacteristic, complemented
77C7×#41—42no—
78C7×#41—42no—
79C2×C7×Q8—22no—
80C2×C7×Q8—41no—
81C2×C7×Q8—81no—
82C2×C7×Q8—81no—
83C2×C7×Q8C22⋊C412yes—
84C2×C7×D4—22no—
85C2×C7×D4—21no—
86C2×C7×D4—21no—
87C2×C7×D4C22⋊C412yes—
88C22×C4×C7—22no—
89C22×C4×C7—41no—
90C22×C4×C7C2×D411yescharacteristic
91C22×C4×C7#4111yescharacteristic
92C22×C4×C7#4111yescharacteristic
93C7×M16—41no—
94C7×M16—41no—
95C7×M16—82no—
96C7×M16—81no—
97C7×M16—81no—
98C2×C7×C8—22no—
99C2×C7×C8—21no—
100C2×C7×C8—21no—
101C2×C7×C8—21no—
102C2×C7×C8—21no—
103C2×C7×C8—41no—
104C2×C7×C8—41no—
105C2×C7×C8—82no—
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×C22⋊C4—22no—
113C7×C22⋊C4—22no—
114C42×C7C22×C411yescharacteristic
115C42×C7C2×D411yescharacteristic
116C42×C7C2×D411yescharacteristic
117C42×C7C2×D411yescharacteristic
118#532C22×C711yescharacteristic
119C2×C4⋊Q8C22×C711yescharacteristic
120#489C4×C714yes—
121#487—21no—
122#486—21no—
123#436—21no—
124#436—21no—
125C8×D4—21no—
126C8×D4—21no—
127#431C22×C711yescharacteristic
128#431C22×C711yescharacteristic
129#423C22×C711yescharacteristic
130#423C22×C711yescharacteristic
131#417—22no—
132#408—22no—
133C4×M16C22×C711yescharacteristic
134#333—21no—
135#333—21no—
136#333—21no—
137#332—21no—
138#332—21no—
139#331—21no—
140C23×C7—22no—
141C23×C7#12311yescharacteristic
142C7×Q8—42no—
143C7×Q8—82no—
144C7×Q8—81no—
145C7×Q8—81no—
146C7×D4—42no—
147C7×D4—41no—
148C7×D4—41no—
149C7×D4—41no—
150C7×D4—41no—
151C7×D4—82no—
152C2×C4×C7—21no—
153C2×C4×C7—21no—
154C2×C4×C7—21no—
155C2×C4×C7—21no—
156C2×C4×C7—21no—
157C2×C4×C7—21no—
158C2×C4×C7—22no—
159C2×C4×C7—42no—
160C2×C4×C7—42no—
161C2×C4×C7—42no—
162C2×C4×C7—82no—
163C2×C4×C7C2×C22⋊C411yescharacteristic
164C2×C4×C7C4×D411yescharacteristic
165C2×C4×C7C4×D411yescharacteristic
166C2×C4×C7#12011yescharacteristic
167C2×C4×C7#12111yescharacteristic
168C2×C4×C7#12111yescharacteristic
169C7×C8—42no—
170C7×C8—41no—
171C7×C8—41no—
172C7×C8—41no—
173C7×C8—41no—
174C7×C8—81no—
175C7×C8—81no—
176C2×#41C7×D412yes—
177C22×Q8—21no—
178C2×M16—21no—
179C2×M16—22no—
180C2×M16—21no—
181C2×M16—21no—
182C22×C8—22no—
183C4⋊Q8—21no—
184C4⋊Q8—21no—
185C4⋊Q8—21no—
186C4⋊Q8C2×C4×C712yes—
187C4⋊Q8C7×D411yescharacteristic
188#127C7×D411yescharacteristic
189#124C2×C4×C712yes—
190#121—21no—
191#121—21no—
192C4×D4—21no—
193C4×D4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C42C23×C711yescharacteristic
197C4⋊C8—21no—
198C4⋊C8—21no—
199C4⋊C8—21no—
200C4⋊C8—21no—
201C4⋊C8C7×D411yescharacteristic
202C4⋊C8C7×D411yescharacteristic
203C4⋊C8C7×D411yescharacteristic
204C4⋊C8C7×D411yescharacteristic
205#103—24no—
206#102—24no—
207C22⋊C8—22no—
208C22⋊C8—22no—
209C22⋊C8—21no—
210C22⋊C8—21no—
211#97C2×C4×C712yes—
212C4×C8—21no—
213C4×C8—21no—
214C4×C8—21no—
215C22×C7—21no—
216C22×C7—21no—
217C22×C7—21no—
218C22×C7—42no—
219C22×C7—42no—
220C22×C7—42no—
221C22×C7#38611yescharacteristic
222C4×C7—21no—
223C4×C7—21no—
224C4×C7—21no—
225C4×C7—42no—
226C4×C7—82no—
227C4×C7#45511yescharacteristic
228C4×C7#45511yescharacteristic
229C4×C7#47111yescharacteristic
230C4×C7#47111yescharacteristic
231C4×C7#47211yescharacteristic
232C4×C7#47311yescharacteristic
233#41—42no—
234#41—42no—
235C2×Q8—22no—
236C2×Q8—41no—
237C2×Q8—81no—
238C2×Q8—81no—
239C2×Q8C7×C22⋊C412yes—
240C2×D4—22no—
241C2×D4—21no—
242C2×D4—21no—
243C2×D4C7×C22⋊C412yes—
244C22×C4—22no—
245C22×C4—41no—
246C22×C4C2×C7×D411yescharacteristic
247C22×C4C7×#4111yescharacteristic
248C22×C4C7×#4111yescharacteristic
249M16—41no—
250M16—41no—
251M16—82no—
252M16—81no—
253M16—81no—
254C2×C8—22no—
255C2×C8—21no—
256C2×C8—21no—
257C2×C8—21no—
258C2×C8—21no—
259C2×C8—41no—
260C2×C8—41no—
261C2×C8—82no—
262C4⋊C4—21no—
263C4⋊C4—21no—
264C4⋊C4—22no—
265C4⋊C4—22no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C22⋊C4—22no—
269C22⋊C4—22no—
270C42C22×C4×C711yescharacteristic
271C42C2×C7×D411yescharacteristic
272C42C2×C7×D411yescharacteristic
273C42C2×C7×D411yescharacteristic
274C2×C7—41no—
275C2×C7—82no—
276C2×C7#188511yescharacteristic
277C2×C7#190711yescharacteristic
278C2×C7#190811yescharacteristic
279C23—22no—
280C23C7×#12311yescharacteristic
281Q8—42no—
282Q8—82no—
283Q8—81no—
284Q8—81no—
285D4—42no—
286D4—41no—
287D4—41no—
288D4—41no—
289D4—41no—
290D4—82no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—22no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—82no—
302C2×C4C2×C7×C22⋊C411yescharacteristic
303C2×C4C4×C7×D411yescharacteristic
304C2×C4C4×C7×D411yescharacteristic
305C2×C4C7×#12011yescharacteristic
306C2×C4C7×#12111yescharacteristic
307C2×C4C7×#12111yescharacteristic
308C8—42no—
309C8—41no—
310C8—41no—
311C8—41no—
312C8—41no—
313C8—81no—
314C8—81no—
315C7#943311yescharacteristic, direct product factor, Sylow
316C22—21no—
317C22—21no—
318C22—21no—
319C22—42no—
320C22—42no—
321C22—42no—
322C22C7×#38611yescharacteristic
323C4—21no—
324C4—21no—
325C4—21no—
326C4—42no—
327C4—82no—
328C4C7×#45511yescharacteristic
329C4C7×#45511yescharacteristic
330C4C7×#47111yescharacteristic
331C4C7×#47111yescharacteristic
332C4C7×#47211yescharacteristic
333C4C7×#47311yescharacteristic
334C2—41no—
335C2—82no—
336C2C7×#188511yescharacteristic
337C2C7×#190711yescharacteristic
338C2C7×#190811yescharacteristic
subgroupG/H
center, socleC22×C7#386
commutatorC42C22×C4×C7
Frattini, cosocleC2×C42C23×C7
prime radicalC7×#532C22
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
4. C7×#1548C2semidirect
5. C7×#1546C2semidirect
6. C7×#1299C2semidirect
7. C7×#1299C2semidirect
30. #9433C7direct (N × H)
73. #1548C2×C7semidirect
74. #1546C2×C7semidirect
75. #1299C2×C7semidirect
76. #1299C2×C7semidirect
315. C7#9433direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#31412
C7×#29792
C7×#15602
C7×#15482
C7×#15462
C7×#12992
#94337