Finite Group Databasebeta

← #13247214

Subgroups — #13247214

Order 1792.

630 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D7×#384C211yesmaximal, characteristic
2#11728229C211yesmaximal, characteristic
3#11728229C211yesmaximal, characteristic
4C7×#1296C211yesmaximal, characteristic, complemented
5#11725508C211yesmaximal, characteristic, complemented
6#11725508C211yesmaximal, characteristic, complemented
7#11725440C211yesmaximal, characteristic, complemented
8C2×D7×C4⋊C4—21no—
9C2×D7×C4⋊C4—21no—
10C2×D7×C4⋊C4C412yes—
11C2×C42×D7C2211yescharacteristic
12C2×C7×C4⋊C8C2211yescharacteristic
13C2×C7×C4⋊C8C2211yescharacteristic
14C7×#384C2211yescharacteristic
15C2×(C2×D7)⋊C8—21no—
16C2×(C2×D7)⋊C8—21no—
17#89748—21no—
18#89748—21no—
19#89748C412yes—
20#89707C2211yescharacteristic
21C2×C4⋊(C7⋊C8)C2211yescharacteristic
22C2×C4⋊(C7⋊C8)C2211yescharacteristic
23#89345—21no—
24#89345—21no—
25#8724—71nomaximal, Sylow
26C22×C4×D7—22no—
27C22×C4×D7C2×C411yescharacteristic
28C22×C4×D7D411yescharacteristic
29C22×C4×D7D411yescharacteristic
30C22×C7×C8—21no—
31C22×C7×C8—21no—
32C2×C7×C4⋊C4—21no—
33C2×C7×C4⋊C4—21no—
34C2×C7×C4⋊C4C2×C412yes—
35C2×C42×C7C2311yescharacteristic
36C2×C4⋊Dic7—21no—
37C2×C4⋊Dic7—21no—
38C2×C4⋊Dic7C2×C412yes—
39C2×Dic7⋊C4—22no—
40C2×Dic7⋊C4—22no—
41C2×Dic7⋊C4—22no—
42C2×C4×Dic7C2×C411yescharacteristic
43C2×C4×Dic7D411yescharacteristic
44C2×C4×Dic7D411yescharacteristic
45C22×C7⋊C8—21no—
46C22×C7⋊C8—21no—
47D7×C4⋊C4—42no—
48D7×C4⋊C4—42no—
49D7×C4⋊C4—42no—
50D7×C4⋊C4—42no—
51D7×C4⋊C4—44no—
52C42×D7—21no—
53C42×D7—21no—
54C42×D7—21no—
55C42×D7—21no—
56C7×C4⋊C8—21no—
57C7×C4⋊C8—21no—
58C7×C4⋊C8—21no—
59C7×C4⋊C8—21no—
60(C2×D7)⋊C8—42no—
61(C2×D7)⋊C8—42no—
62C4⋊(C7⋊C8)—21no—
63C4⋊(C7⋊C8)—21no—
64C4⋊(C7⋊C8)—21no—
65C4⋊(C7⋊C8)—21no—
66C2×#384—71no—
67#1797—71no—
68#1797—71no—
69#1296—71no—
70#1296—71no—
71#1296—71no—
72#1296D711yescharacteristic, complemented
73C23×D7C22⋊C411yescharacteristic
74C22×C4×C7—22no—
75C22×C4×C7C22×C411yescharacteristic
76C22×C4×C7C2×D411yescharacteristic
77C22×C4×C7C2×D411yescharacteristic
78C22×Dic7—22no—
79C22×Dic7C22⋊C411yescharacteristic
80C22×Dic7C22⋊C411yescharacteristic
81C22×Dic7C22⋊C411yescharacteristic
82C2×C4×D7—22no—
83C2×C4×D7—21no—
84C2×C4×D7—21no—
85C2×C4×D7—21no—
86C2×C4×D7—21no—
87C2×C4×D7—21no—
88C2×C4×D7—21no—
89C2×C4×D7—21no—
90C2×C4×D7—21no—
91C2×C4×D7—21no—
92C2×C4×D7—21no—
93C2×C4×D7—22no—
94C2×C4×D7—44no—
95C2×C4×D7—44no—
96C2×C4×D7—44no—
97C2×C4×D7D812yes—
98C2×C4×D7SD1612yes—
99C2×C4×D7SD1612yes—
100C2×C4×D7Q1612yes—
101C2×C7×C8—41no—
102C2×C7×C8—41no—
103C2×C7×C8—41no—
104C2×C7×C8—41no—
105C2×C7×C8—41no—
106C2×C7×C8—41no—
107C7×C4⋊C4—42no—
108C7×C4⋊C4—41no—
109C7×C4⋊C4—41no—
110C7×C4⋊C4—41no—
111C7×C4⋊C4—41no—
112C42×C7#4111yescharacteristic
113C42×C7#4111yescharacteristic
114C42×C7#4111yescharacteristic
115C42×C7#4111yescharacteristic
116C4⋊Dic7—41no—
117C4⋊Dic7—41no—
118C4⋊Dic7—41no—
119C4⋊Dic7—41no—
120C4⋊Dic7—42no—
121Dic7⋊C4—44no—
122Dic7⋊C4—44no—
123Dic7⋊C4—44no—
124C4×Dic7—21no—
125C4×Dic7—21no—
126C4×Dic7—21no—
127C4×Dic7—21no—
128C4×Dic7—21no—
129C4×Dic7—21no—
130C2×C7⋊C8—41no—
131C2×C7⋊C8—41no—
132C2×C7⋊C8—41no—
133C2×C7⋊C8—41no—
134C2×C7⋊C8—41no—
135C2×C7⋊C8—41no—
136C22×C4⋊C4—72no—
137C22×C4⋊C4—141no—
138C22×C4⋊C4—141no—
139C22×C42—71no—
140C2×C4⋊C8—71no—
141C2×C4⋊C8—71no—
142C2×C4⋊C8C2×D711yescharacteristic
143C2×C4⋊C8C2×D711yescharacteristic
144C2×C22⋊C8—141no—
145C2×C22⋊C8—141no—
146#384—72no—
147#384—71no—
148#384—141no—
149#384—141no—
150#384C2×D711yescharacteristic
151#336—141no—
152#336—141no—
153C23×C7C2×C22⋊C411yescharacteristic
154C22×D7—21no—
155C22×D7—21no—
156C22×D7—21no—
157C22×D7—21no—
158C22×D7#10112yes—
159C22×D7#10212yes—
160C22×D7#10312yes—
161C2×C4×C7—21no—
162C2×C4×C7—21no—
163C2×C4×C7—42no—
164C2×C4×C7—42no—
165C2×C4×C7—42no—
166C2×C4×C7#11711yescharacteristic
167C2×C4×C7#11711yescharacteristic
168C2×C4×C7#12311yescharacteristic
169C2×C4×C7#12311yescharacteristic
170C2×C4×C7#12311yescharacteristic
171C2×C4×C7#12311yescharacteristic
172C2×C4×C7C2×D811yescharacteristic
173C2×C4×C7C2×SD1611yescharacteristic
174C2×C4×C7C2×SD1611yescharacteristic
175C2×C4×C7C2×Q1611yescharacteristic
176C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
177C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
178C2×C4×C7#13711yescharacteristic
179C2×C4×C7#13711yescharacteristic
180C2×Dic7—21no—
181C2×Dic7—21no—
182C2×Dic7—21no—
183C2×Dic7—21no—
184C2×Dic7—21no—
185C2×Dic7—21no—
186C2×Dic7—42no—
187C2×Dic7—42no—
188C2×Dic7—42no—
189C2×Dic7#10012yes—
190C2×Dic7#10212yes—
191C2×Dic7#10312yes—
192C4×D7—24no—
193C4×D7—24no—
194C4×D7—22no—
195C4×D7—22no—
196C4×D7—22no—
197C4×D7—22no—
198C4×D7—22no—
199C4×D7—22no—
200C4×D7—22no—
201C4×D7—22no—
202C4×D7—88no—
203C7×C8—81no—
204C7×C8—81no—
205C7⋊C8—81no—
206C7⋊C8—81no—
207C23×C4—71no—
208C23×C4—71no—
209C23×C4—71no—
210C23×C4—142no—
211C22×C8—21no—
212C22×C8—21no—
213C22×C8—141no—
214C22×C8—141no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—72no—
218C2×C4⋊C4—142no—
219C2×C4⋊C4—142no—
220C2×C4⋊C4—142no—
221C2×C4⋊C4—141no—
222C2×C4⋊C4—141no—
223C2×C4⋊C4—282no—
224C2×C4⋊C4—282no—
225C2×C4⋊C4—282no—
226C2×C4⋊C4—282no—
227C2×C4⋊C4—284no—
228C2×C4⋊C4C4×D712yes—
229C2×C42—71no—
230C2×C42—71no—
231C2×C42—71no—
232C2×C42—141no—
233C2×C42—141no—
234C2×C42—141no—
235C2×C42—141no—
236C2×C42C22×D711yescharacteristic
237C4⋊C8—21no—
238C4⋊C8—21no—
239C4⋊C8—21no—
240C4⋊C8—21no—
241C4⋊C8—141no—
242C4⋊C8—141no—
243C4⋊C8—141no—
244C4⋊C8—141no—
245C22⋊C8—282no—
246C22⋊C8—282no—
247C22×C7#38111yescharacteristic
248C22×C7C2×#10011yescharacteristic
249C22×C7C2×#10111yescharacteristic
250C22×C7C2×#10211yescharacteristic
251C22×C7C2×#10311yescharacteristic
252C22×C7#41811yescharacteristic
253C22×C7#41911yescharacteristic
254C2×D7—22no—
255C2×D7—22no—
256C2×D7—22no—
257C2×D7—22no—
258C2×D7—22no—
259C2×D7—22no—
260C2×D7#33214yes—
261C4×C7—21no—
262C4×C7—21no—
263C4×C7—82no—
264C4×C7#48011yescharacteristic
265C4×C7#48011yescharacteristic
266C4×C7#48111yescharacteristic
267C4×C7#48111yescharacteristic
268C4×C7#48311yescharacteristic
269C4×C7#48311yescharacteristic
270C4×C7#48411yescharacteristic
271C4×C7#48411yescharacteristic
272Dic7—22no—
273Dic7—22no—
274Dic7—22no—
275Dic7—82no—
276C24—71no—
277C22×C4—22no—
278C22×C4—71no—
279C22×C4—71no—
280C22×C4—71no—
281C22×C4—72no—
282C22×C4—72no—
283C22×C4—72no—
284C22×C4—72no—
285C22×C4—142no—
286C22×C4—142no—
287C22×C4—141no—
288C22×C4—141no—
289C22×C4—141no—
290C22×C4—141no—
291C22×C4—141no—
292C22×C4—141no—
293C22×C4—141no—
294C22×C4—141no—
295C22×C4—141no—
296C22×C4—141no—
297C22×C4—142no—
298C22×C4—284no—
299C22×C4—284no—
300C22×C4—284no—
301C22×C4C2×C4×D711yescharacteristic
302C22×C4D4×D711yescharacteristic
303C22×C4D4×D711yescharacteristic
304C2×C8—41no—
305C2×C8—41no—
306C2×C8—41no—
307C2×C8—41no—
308C2×C8—41no—
309C2×C8—41no—
310C2×C8—281no—
311C2×C8—281no—
312C2×C8—281no—
313C2×C8—281no—
314C2×C8—281no—
315C2×C8—281no—
316C4⋊C4—42no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4—282no—
322C4⋊C4—284no—
323C4⋊C4—284no—
324C4⋊C4—284no—
325C4⋊C4—281no—
326C4⋊C4—281no—
327C4⋊C4—281no—
328C4⋊C4—281no—
329C42—141no—
330C42—141no—
331C42—141no—
332C42—141no—
333C42—141no—
334C42—141no—
335C42#115711yescharacteristic
336C42#115711yescharacteristic
337C42#115911yescharacteristic
338C42#116111yescharacteristic
339C2×C7C2×#33211yescharacteristic
340C2×C7#154511yescharacteristic
341C2×C7#154811yescharacteristic
342C2×C7#155611yescharacteristic
343C2×C7#182111yescharacteristic
344C2×C7#182311yescharacteristic
345C2×C7#182411yescharacteristic
346D7—24no—
347C23—72no—
348C23—72no—
349C23—72no—
350C23—141no—
351C23—141no—
352C23—141no—
353C23—141no—
354C23D7×C22⋊C411yescharacteristic
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—72no—
361C2×C4—72no—
362C2×C4—72no—
363C2×C4—144no—
364C2×C4—144no—
365C2×C4—141no—
366C2×C4—141no—
367C2×C4—141no—
368C2×C4—141no—
369C2×C4—141no—
370C2×C4—141no—
371C2×C4—142no—
372C2×C4—142no—
373C2×C4—142no—
374C2×C4—142no—
375C2×C4—142no—
376C2×C4—142no—
377C2×C4—142no—
378C2×C4—142no—
379C2×C4—282no—
380C2×C4—282no—
381C2×C4—282no—
382C2×C4—568no—
383C2×C4#643511yescharacteristic
384C2×C4#644611yescharacteristic
385C2×C4#644611yescharacteristic
386C2×C4#645511yescharacteristic
387C2×C4#645711yescharacteristic
388C2×C4#645711yescharacteristic
389C2×C4D7×D811yescharacteristic
390C2×C4#647411yescharacteristic
391C2×C4D7×SD1611yescharacteristic
392C2×C4D7×SD1611yescharacteristic
393C2×C4#647711yescharacteristic
394C2×C4#647811yescharacteristic
395C2×C4D7×Q1611yescharacteristic
396C2×C4#648111yescharacteristic
397C8—81no—
398C8—81no—
399C8—561no—
400C8—561no—
401C7#872411yescharacteristic, complemented, Sylow
402C22—74no—
403C22—142no—
404C22—142no—
405C22—142no—
406C22—142no—
407C22—142no—
408C22—142no—
409C22#8943611yescharacteristic
410C22D7×#10011yescharacteristic
411C22D7×#10111yescharacteristic
412C22D7×#10211yescharacteristic
413C22#8954211yescharacteristic
414C22D7×#10311yescharacteristic
415C22#8957411yescharacteristic
416C4—21no—
417C4—21no—
418C4—82no—
419C4—142no—
420C4—142no—
421C4—142no—
422C4—562no—
423C4#8954611yescharacteristic
424C4#8954911yescharacteristic
425C4#8957711yescharacteristic
426C4#8958111yescharacteristic
427C4#8963411yescharacteristic
428C4#8963511yescharacteristic
429C4#8965411yescharacteristic
430C4#8965611yescharacteristic
431C2—144no—
432C2#1172613711yescharacteristic
433C2D7×#33211yescharacteristic
434C2#1172618211yescharacteristic
435C2#1172623711yescharacteristic
436C2#1172662511yescharacteristic
437C2#1172662611yescharacteristic
438C2#1172674411yescharacteristic
subgroupG/H
centerC23D7×C22⋊C4
commutatorC22×C4×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#1296C2
socleC23×C7C2×C22⋊C4
prime radicalC23×D7C22⋊C4
cosocleC2×C42×C7C23
hypercenter#1296D7
hypocenterC7#8724
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1296

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. C7×#1296C2semidirect
5. #11725508C2semidirect
6. #11725508C2semidirect
7. #11725440C2semidirect
72. #1296D7semidirect
401. C7#8724semidirect

Maximal subgroups

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

subgroupindex
D7×#3842
#117282292
C7×#12962
#117255082
#117254402
#87247