Finite Group Databasebeta

← C7×#10024 #13187240

Subgroups — C7×#10024 #13187240

Order 1792.

512 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 98 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#3140C211yesmaximal, characteristic
2C7×#3113C211yesmaximal, characteristic, complemented
3C7×#2632C211yesmaximal, characteristic
4C7×#1533C211yesmaximal, characteristic
5C7×#1515C211yesmaximal, characteristic, complemented
6C7×#1491C211yesmaximal, characteristic, complemented
7C7×#1288C211yesmaximal, characteristic, complemented
8C7×#533C2211yescharacteristic
9C2×C7×#124C2211yescharacteristic
10C2×C4×C7×Q8C2211yescharacteristic
11C7×#489—21no—
12C7×#489—21no—
13C7×#488—22no—
14C7×#483—21no—
15C7×#475—21no—
16C7×#462—21no—
17C7×#451—21no—
18C7×#441—22no—
19C7×#432C2211yescharacteristic
20C7×#432C2211yescharacteristic
21C7×#419—21no—
22C7×#418—21no—
23C4×C7×M16C2211yescharacteristic
24C7×#399—21no—
25C7×#395—21no—
26C7×#394—21no—
27C7×#393—21no—
28C7×#388—22no—
29C7×#383C2211yescharacteristic
30C7×#329—22no—
31C7×#328—22no—
32C7×Q8⋊C8—22no—
33#10024C711yesmaximal, characteristic, direct product factor, Sylow
34C22×C7×Q8D411yescharacteristic
35C22×C7×Q8D411yescharacteristic
36C22×C7×D4D411yescharacteristic
37C2×C7×Q16—22no—
38C2×C7×M16—21no—
39C7×C4⋊Q8D411yescharacteristic
40C7×C4⋊Q8D411yescharacteristic
41C7×#125—21no—
42C7×#124—21no—
43C7×#124—21no—
44C7×#124—21no—
45C7×#124—21no—
46C7×C22⋊Q8—21no—
47C7×C22⋊Q8—21no—
48C4×C7×Q8—21no—
49C4×C7×Q8—21no—
50C4×C7×Q8—21no—
51C4×C7×Q8—21no—
52C4×C7×Q8—21no—
53C4×C7×Q8—21no—
54C7×#117—21no—
55C2×C7×C4⋊C4—21no—
56C2×C7×C4⋊C4—22no—
57C2×C7×C4⋊C4D411yescharacteristic
58C2×C7×C22⋊C4—22no—
59C2×C7×C22⋊C4—21no—
60C2×C7×C22⋊C4—21no—
61C2×C7×C22⋊C4—22no—
62C2×C42×C7—21no—
63C2×C42×C7C2311yescharacteristic
64C7×#106—22no—
65C7×C4⋊C8—21no—
66C7×C4⋊C8—21no—
67C7×#103—22no—
68C7×#103—22no—
69C7×#103—22no—
70C7×#103—22no—
71C7×#102—22no—
72C7×#102—22no—
73C7×C22⋊C8—21no—
74C7×C22⋊C8—21no—
75C7×#97—21no—
76C7×#97—21no—
77C7×#97D412yes—
78C4×C7×C8D412yes—
79C7×#95—22no—
80C7×#95—21no—
81C7×#95—21no—
82C7×#95D412yes—
83#3140C2×C711yescharacteristic
84#3113C2×C711yescharacteristic, complemented
85#2632C2×C711yescharacteristic
86#1533C2×C711yescharacteristic
87#1515C2×C711yescharacteristic, complemented
88#1491C2×C711yescharacteristic, complemented
89#1288C2×C711yescharacteristic, complemented
90C24×C7—21no—
91C2×C7×Q8—21no—
92C2×C7×Q8—21no—
93C2×C7×Q8—21no—
94C2×C7×Q8—21no—
95C2×C7×Q8—21no—
96C2×C7×Q8—21no—
97C2×C7×Q8—42no—
98C2×C7×Q8—42no—
99C2×C7×D4—21no—
100C2×C7×D4—21no—
101C2×C7×D4—42no—
102C22×C4×C7—21no—
103C22×C4×C7—22no—
104C22×C4×C7—21no—
105C22×C4×C7—21no—
106C22×C4×C7C2×D411yescharacteristic
107C22×C4×C7C2×D411yescharacteristic
108C22×C4×C7C2×D411yescharacteristic
109C7×Q16—44no—
110C7×M16—42no—
111C2×C7×C8—22no—
112C2×C7×C8—22no—
113C2×C7×C8—41no—
114C7×C4⋊C4—21no—
115C7×C4⋊C4—21no—
116C7×C4⋊C4—21no—
117C7×C4⋊C4—21no—
118C7×C4⋊C4—21no—
119C7×C4⋊C4—21no—
120C7×C4⋊C4—41no—
121C7×C4⋊C4—41no—
122C7×C4⋊C4—41no—
123C7×C4⋊C4—41no—
124C7×C4⋊C4—82no—
125C7×C22⋊C4—44no—
126C7×C22⋊C4—41no—
127C7×C22⋊C4—41no—
128C7×C22⋊C4—41no—
129C7×C22⋊C4—41no—
130C7×C22⋊C4—41no—
131C7×C22⋊C4—82no—
132C42×C7—41no—
133C42×C7—41no—
134C42×C7—41no—
135C42×C7C2×D411yescharacteristic
136C42×C7C2×D411yescharacteristic
137C42×C7C2×D411yescharacteristic
138C42×C7#4111yescharacteristic
139#533C22×C711yescharacteristic
140C2×#124C22×C711yescharacteristic
141C2×C4×Q8C22×C711yescharacteristic
142#489—21no—
143#489—21no—
144#488—22no—
145#483—21no—
146#475—21no—
147#462—21no—
148#451—21no—
149#441—22no—
150#432C22×C711yescharacteristic
151#432C22×C711yescharacteristic
152#419—21no—
153#418—21no—
154C4×M16C22×C711yescharacteristic
155#399—21no—
156#395—21no—
157#394—21no—
158#393—21no—
159#388—22no—
160#383C22×C711yescharacteristic
161#329—22no—
162#328—22no—
163Q8⋊C8—22no—
164C23×C7—42no—
165C23×C7—42no—
166C23×C7—41no—
167C23×C7—42no—
168C23×C7#12011yescharacteristic
169C7×Q8—22no—
170C7×Q8—22no—
171C7×Q8—41no—
172C7×Q8—41no—
173C7×Q8—42no—
174C7×Q8—42no—
175C7×Q8—81no—
176C7×Q8—81no—
177C7×D4—42no—
178C7×D4—81no—
179C2×C4×C7—22no—
180C2×C4×C7—22no—
181C2×C4×C7—41no—
182C2×C4×C7—41no—
183C2×C4×C7—41no—
184C2×C4×C7—41no—
185C2×C4×C7—42no—
186C2×C4×C7—42no—
187C2×C4×C7—41no—
188C2×C4×C7—41no—
189C2×C4×C7—41no—
190C2×C4×C7—41no—
191C2×C4×C7—42no—
192C2×C4×C7—42no—
193C2×C4×C7—42no—
194C2×C4×C7#12011yescharacteristic
195C2×C4×C7#12011yescharacteristic
196C2×C4×C7#12111yescharacteristic
197C2×C4×C7#12111yescharacteristic
198C2×C4×C7#12111yescharacteristic
199C2×C4×C7#12711yescharacteristic
200C7×C8—42no—
201C7×C8—42no—
202C7×C8—81no—
203C22×Q8C7×D411yescharacteristic
204C22×Q8C7×D411yescharacteristic
205C22×D4C7×D411yescharacteristic
206C2×Q16—22no—
207C2×M16—21no—
208C4⋊Q8C7×D411yescharacteristic
209C4⋊Q8C7×D411yescharacteristic
210#125—21no—
211#124—21no—
212#124—21no—
213#124—21no—
214#124—21no—
215C22⋊Q8—21no—
216C22⋊Q8—21no—
217C4×Q8—21no—
218C4×Q8—21no—
219C4×Q8—21no—
220C4×Q8—21no—
221C4×Q8—21no—
222C4×Q8—21no—
223#117—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4C7×D411yescharacteristic
227C2×C22⋊C4—22no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—22no—
231C2×C42—21no—
232C2×C42C23×C711yescharacteristic
233#106—22no—
234C4⋊C8—21no—
235C4⋊C8—21no—
236#103—22no—
237#103—22no—
238#103—22no—
239#103—22no—
240#102—22no—
241#102—22no—
242C22⋊C8—21no—
243C22⋊C8—21no—
244#97—21no—
245#97—21no—
246#97C7×D412yes—
247C4×C8C7×D412yes—
248#95—22no—
249#95—21no—
250#95—21no—
251#95C7×D412yes—
252C22×C7—21no—
253C22×C7—21no—
254C22×C7—84no—
255C22×C7—81no—
256C22×C7—81no—
257C22×C7—81no—
258C22×C7#39211yescharacteristic
259C22×C7#45512yes—
260C4×C7—21no—
261C4×C7—21no—
262C4×C7—21no—
263C4×C7—21no—
264C4×C7—21no—
265C4×C7—21no—
266C4×C7—42no—
267C4×C7—82no—
268C4×C7—81no—
269C4×C7—81no—
270C4×C7—81no—
271C4×C7—81no—
272C24—21no—
273C2×Q8—21no—
274C2×Q8—21no—
275C2×Q8—21no—
276C2×Q8—21no—
277C2×Q8—21no—
278C2×Q8—21no—
279C2×Q8—42no—
280C2×Q8—42no—
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—42no—
284C22×C4—21no—
285C22×C4—22no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4C2×C7×D411yescharacteristic
289C22×C4C2×C7×D411yescharacteristic
290C22×C4C2×C7×D411yescharacteristic
291Q16—44no—
292M16—42no—
293C2×C8—22no—
294C2×C8—22no—
295C2×C8—41no—
296C4⋊C4—21no—
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4—21no—
300C4⋊C4—21no—
301C4⋊C4—21no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—82no—
307C22⋊C4—44no—
308C22⋊C4—41no—
309C22⋊C4—41no—
310C22⋊C4—41no—
311C22⋊C4—41no—
312C22⋊C4—41no—
313C22⋊C4—82no—
314C42—41no—
315C42—41no—
316C42—41no—
317C42C2×C7×D411yescharacteristic
318C42C2×C7×D411yescharacteristic
319C42C2×C7×D411yescharacteristic
320C42C7×#4111yescharacteristic
321C2×C7—22no—
322C2×C7—161no—
323C2×C7#200411yescharacteristic
324C2×C7#200911yescharacteristic
325C2×C7#201011yescharacteristic
326C23—42no—
327C23—42no—
328C23—41no—
329C23—42no—
330C23C7×#12011yescharacteristic
331Q8—22no—
332Q8—22no—
333Q8—41no—
334Q8—41no—
335Q8—42no—
336Q8—42no—
337Q8—81no—
338Q8—81no—
339D4—42no—
340D4—81no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4C7×#12011yescharacteristic
357C2×C4C7×#12011yescharacteristic
358C2×C4C7×#12111yescharacteristic
359C2×C4C7×#12111yescharacteristic
360C2×C4C7×#12111yescharacteristic
361C2×C4C7×#12711yescharacteristic
362C8—42no—
363C8—42no—
364C8—81no—
365C7#1002411yescharacteristic, direct product factor, Sylow
366C22—21no—
367C22—21no—
368C22—84no—
369C22—81no—
370C22—81no—
371C22—81no—
372C22C7×#39211yescharacteristic
373C22C7×#45512yes—
374C4—21no—
375C4—21no—
376C4—21no—
377C4—21no—
378C4—21no—
379C4—21no—
380C4—42no—
381C4—82no—
382C4—81no—
383C4—81no—
384C4—81no—
385C4—81no—
386C2—22no—
387C2—161no—
388C2C7×#200411yescharacteristic
389C2C7×#200911yescharacteristic
390C2C7×#201011yescharacteristic
subgroupG/H
center, socleC22×C7#392
commutator, Frattini, cosocleC2×C42C23×C7
prime radicalC22×C7×D4D4
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×#3113C2semidirect
5. C7×#1515C2semidirect
6. C7×#1491C2semidirect
7. C7×#1288C2semidirect
33. #10024C7direct (N × H)
84. #3113C2×C7semidirect
87. #1515C2×C7semidirect
88. #1491C2×C7semidirect
89. #1288C2×C7semidirect
365. C7#10024direct (N × H)

Maximal subgroups

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

subgroupindex
C7×#31402
C7×#31132
C7×#26322
C7×#15332
C7×#15152
C7×#14912
C7×#12882
#100247