Finite Group Databasebeta

← C7×#9303 #13186519

Subgroups — C7×#9303 #13186519

Order 1792.

488 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3077C211yesmaximal, characteristic
2C7×#2972C211yesmaximal, characteristic
3C7×#1804C211yesmaximal, characteristic
4C7×#1518C211yesmaximal, characteristic, complemented
5C7×#1481C211yesmaximal, characteristic, complemented
6C7×#1301C211yesmaximal, characteristic, complemented
7C7×#1286C211yesmaximal, characteristic, complemented
8C7×#533C2211yescharacteristic
9C2×C4×C7×D4C2211yescharacteristic
10C7×#478—22no—
11C7×#476C412yescomplemented
12C7×#474C412yescomplemented
13C7×C8×D4—22no—
14C7×C8×D4—22no—
15C7×#431C2211yescharacteristic
16C7×#431C2211yescharacteristic
17C7×#428—21no—
18C7×#423C2211yescharacteristic
19C2×C7×C4⋊C8C2211yescharacteristic
20C7×#418—21no—
21C2×C7×#102—21no—
22C2×C7×C22⋊C8—22no—
23C7×#382C2211yescharacteristic
24C7×#340—21no—
25C7×#328—22no—
26C7×Q8⋊C8—22no—
27#9303C711yesmaximal, characteristic, direct product factor, Sylow
28C22×C7×D4D411yescharacteristic
29C23×C4×C7—21no—
30C2×C7×M16—21no—
31C22×C7×C8—24no—
32C22×C7×C8—21no—
33C7×C4⋊Q8C2×C411yescharacteristic
34C7×C4⋊Q8C2×C411yescharacteristic
35C7×#125—21no—
36C7×C22⋊Q8—21no—
37C7×C22⋊Q8—21no—
38C4×C7×Q8—21no—
39C4×C7×Q8—21no—
40C4×C7×D4—21no—
41C4×C7×D4—21no—
42C4×C7×D4C2×C412yes—
43C4×C7×D4D412yes—
44C7×#117—21no—
45C2×C7×C4⋊C4—21no—
46C2×C7×C4⋊C4D411yescharacteristic
47C2×C7×C22⋊C4—21no—
48C2×C42×C7C2311yescharacteristic
49C7×#107—22no—
50C7×#106—22no—
51C7×C4⋊C8—21no—
52C7×C4⋊C8—21no—
53C7×C4⋊C8—21no—
54C7×C4⋊C8—21no—
55C7×C4⋊C8—21no—
56C7×C4⋊C8C2×C412yes—
57C7×#102—22no—
58C7×#102—22no—
59C7×#102—42no—
60C7×C22⋊C8—24no—
61C7×C22⋊C8—21no—
62C7×C22⋊C8D412yes—
63C4×C7×C8—21no—
64C4×C7×C8D412yes—
65C7×#95—21no—
66C7×#95—21no—
67#3077C2×C711yescharacteristic
68#2972C2×C711yescharacteristic
69#1804C2×C711yescharacteristic
70#1518C2×C711yescharacteristic, complemented
71#1481C2×C711yescharacteristic, complemented
72#1301C2×C711yescharacteristic, complemented
73#1286C2×C711yescharacteristic, complemented
74C24×C7—21no—
75C2×C7×Q8—21no—
76C2×C7×Q8—21no—
77C2×C7×D4—21no—
78C2×C7×D4—21no—
79C2×C7×D4—21no—
80C2×C7×D4C22⋊C412yes—
81C2×C7×D4SD1612yes—
82C2×C7×D4Q1612yes—
83C22×C4×C7—21no—
84C22×C4×C7—22no—
85C22×C4×C7—22no—
86C22×C4×C7—41no—
87C22×C4×C7—82no—
88C22×C4×C7C2×D411yescharacteristic
89C22×C4×C7C2×D411yescharacteristic
90C22×C4×C7#4111yescharacteristic
91C7×M16—41no—
92C7×M16—41no—
93C2×C7×C8—22no—
94C2×C7×C8—22no—
95C2×C7×C8—22no—
96C2×C7×C8—41no—
97C2×C7×C8—41no—
98C2×C7×C8—41no—
99C2×C7×C8—84no—
100C7×C4⋊C4—21no—
101C7×C4⋊C4—21no—
102C7×C4⋊C4—21no—
103C7×C4⋊C4—21no—
104C7×C4⋊C4—21no—
105C7×C4⋊C4—41no—
106C7×C4⋊C4—41no—
107C7×C4⋊C4—41no—
108C7×C4⋊C4—41no—
109C7×C4⋊C4C22⋊C412yes—
110C7×C22⋊C4—22no—
111C7×C22⋊C4—22no—
112C7×C22⋊C4—41no—
113C42×C7—41no—
114C42×C7C22×C411yescharacteristic
115C42×C7C2×D411yescharacteristic
116C42×C7C2×D411yescharacteristic
117C42×C7#4111yescharacteristic
118#533C22×C711yescharacteristic
119C2×C4×D4C22×C711yescharacteristic
120#478—22no—
121#476C4×C712yescomplemented
122#474C4×C712yescomplemented
123C8×D4—22no—
124C8×D4—22no—
125#431C22×C711yescharacteristic
126#431C22×C711yescharacteristic
127#428—21no—
128#423C22×C711yescharacteristic
129C2×C4⋊C8C22×C711yescharacteristic
130#418—21no—
131C2×#102—21no—
132C2×C22⋊C8—22no—
133#382C22×C711yescharacteristic
134#340—21no—
135#328—22no—
136Q8⋊C8—22no—
137C23×C7—22no—
138C23×C7—22no—
139C23×C7—22no—
140C23×C7—81no—
141C23×C7—81no—
142C23×C7#12111yescharacteristic
143C7×Q8—42no—
144C7×Q8—81no—
145C7×D4—22no—
146C7×D4—22no—
147C7×D4—22no—
148C7×D4#10314yes—
149C2×C4×C7—22no—
150C2×C4×C7—41no—
151C2×C4×C7—41no—
152C2×C4×C7—44no—
153C2×C4×C7—41no—
154C2×C4×C7—41no—
155C2×C4×C7—82no—
156C2×C4×C7—81no—
157C2×C4×C7—81no—
158C2×C4×C7—81no—
159C2×C4×C7—81no—
160C2×C4×C7C2×C22⋊C411yescharacteristic
161C2×C4×C7C4×D411yescharacteristic
162C2×C4×C7C4×D411yescharacteristic
163C2×C4×C7#12011yescharacteristic
164C2×C4×C7#12111yescharacteristic
165C2×C4×C7#12311yescharacteristic
166C2×C4×C7C2×SD1611yescharacteristic
167C2×C4×C7C2×Q1611yescharacteristic
168C2×C4×C7#13511yescharacteristic
169C2×C4×C7#13511yescharacteristic
170C7×C8—42no—
171C7×C8—42no—
172C7×C8—42no—
173C7×C8—81no—
174C22×D4C7×D411yescharacteristic
175C23×C4—21no—
176C2×M16—21no—
177C22×C8—24no—
178C22×C8—21no—
179C4⋊Q8C2×C4×C711yescharacteristic
180C4⋊Q8C2×C4×C711yescharacteristic
181#125—21no—
182C22⋊Q8—21no—
183C22⋊Q8—21no—
184C4×Q8—21no—
185C4×Q8—21no—
186C4×D4—21no—
187C4×D4—21no—
188C4×D4C2×C4×C712yes—
189C4×D4C7×D412yes—
190#117—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4C7×D411yescharacteristic
193C2×C22⋊C4—21no—
194C2×C42C23×C711yescharacteristic
195#107—22no—
196#106—22no—
197C4⋊C8—21no—
198C4⋊C8—21no—
199C4⋊C8—21no—
200C4⋊C8—21no—
201C4⋊C8—21no—
202C4⋊C8C2×C4×C712yes—
203#102—22no—
204#102—22no—
205#102—42no—
206C22⋊C8—24no—
207C22⋊C8—21no—
208C22⋊C8C7×D412yes—
209C4×C8—21no—
210C4×C8C7×D412yes—
211#95—21no—
212#95—21no—
213C22×C7—21no—
214C22×C7—21no—
215C22×C7—24no—
216C22×C7—42no—
217C22×C7—42no—
218C22×C7—82no—
219C22×C7—82no—
220C22×C7#38611yescharacteristic
221C22×C7#46511yescharacteristic
222C22×C7#46711yescharacteristic
223C4×C7—21no—
224C4×C7—21no—
225C4×C7—21no—
226C4×C7—21no—
227C4×C7—42no—
228C4×C7—81no—
229C4×C7—81no—
230C4×C7—161no—
231C4×C7C2×#10311yescharacteristic
232C4×C7#41611yescharacteristic
233C4×C7#46511yescharacteristic
234C4×C7#46711yescharacteristic
235C24—21no—
236C2×Q8—21no—
237C2×Q8—21no—
238C2×D4—21no—
239C2×D4—21no—
240C2×D4—21no—
241C2×D4C7×C22⋊C412yes—
242C2×D4C7×SD1612yes—
243C2×D4C7×Q1612yes—
244C22×C4—21no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—41no—
248C22×C4—82no—
249C22×C4C2×C7×D411yescharacteristic
250C22×C4C2×C7×D411yescharacteristic
251C22×C4C7×#4111yescharacteristic
252M16—41no—
253M16—41no—
254C2×C8—22no—
255C2×C8—22no—
256C2×C8—22no—
257C2×C8—41no—
258C2×C8—41no—
259C2×C8—41no—
260C2×C8—84no—
261C4⋊C4—21no—
262C4⋊C4—21no—
263C4⋊C4—21no—
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4C7×C22⋊C412yes—
271C22⋊C4—22no—
272C22⋊C4—22no—
273C22⋊C4—41no—
274C42—41no—
275C42C22×C4×C711yescharacteristic
276C42C2×C7×D411yescharacteristic
277C42C2×C7×D411yescharacteristic
278C42C7×#4111yescharacteristic
279C2×C7—21no—
280C2×C7—21no—
281C2×C7—44no—
282C2×C7#188111yescharacteristic
283C2×C7#189211yescharacteristic
284C2×C7#190411yescharacteristic
285C23—22no—
286C23—22no—
287C23—22no—
288C23—81no—
289C23—81no—
290C23C7×#12111yescharacteristic
291Q8—42no—
292Q8—81no—
293D4—22no—
294D4—22no—
295D4—22no—
296D4C7×#10314yes—
297C2×C4—22no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—44no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—82no—
304C2×C4—81no—
305C2×C4—81no—
306C2×C4—81no—
307C2×C4—81no—
308C2×C4C2×C7×C22⋊C411yescharacteristic
309C2×C4C4×C7×D411yescharacteristic
310C2×C4C4×C7×D411yescharacteristic
311C2×C4C7×#12011yescharacteristic
312C2×C4C7×#12111yescharacteristic
313C2×C4C7×#12311yescharacteristic
314C2×C4C2×C7×SD1611yescharacteristic
315C2×C4C2×C7×Q1611yescharacteristic
316C2×C4C7×#13511yescharacteristic
317C2×C4C7×#13511yescharacteristic
318C8—42no—
319C8—42no—
320C8—42no—
321C8—81no—
322C7#930311yescharacteristic, direct product factor, Sylow
323C22—21no—
324C22—21no—
325C22—24no—
326C22—42no—
327C22—42no—
328C22—82no—
329C22—82no—
330C22C7×#38611yescharacteristic
331C22C7×#46511yescharacteristic
332C22C7×#46711yescharacteristic
333C4—21no—
334C4—21no—
335C4—21no—
336C4—21no—
337C4—42no—
338C4—81no—
339C4—81no—
340C4—161no—
341C4C2×C7×#10311yescharacteristic
342C4C7×#41611yescharacteristic
343C4C7×#46511yescharacteristic
344C4C7×#46711yescharacteristic
345C2—21no—
346C2—21no—
347C2—44no—
348C2C7×#188111yescharacteristic
349C2C7×#189211yescharacteristic
350C2C7×#190411yescharacteristic
subgroupG/H
center, socleC22×C7#386
commutatorC42C22×C4×C7
Frattini, cosocleC2×C42C23×C7
prime radicalC22×C7×D4D4
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×#1518C2semidirect
5. C7×#1481C2semidirect
6. C7×#1301C2semidirect
7. C7×#1286C2semidirect
11. C7×#476C4semidirect
12. C7×#474C4semidirect
27. #9303C7direct (N × H)
70. #1518C2×C7semidirect
71. #1481C2×C7semidirect
72. #1301C2×C7semidirect
73. #1286C2×C7semidirect
121. #476C4×C7semidirect
122. #474C4×C7semidirect
322. C7#9303direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#30772
C7×#29722
C7×#18042
C7×#15182
C7×#14812
C7×#13012
C7×#12862
#93037