Finite Group Databasebeta

← #11734864

Subgroups — #11734864

Order 896.

648 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#6460C211yesmaximal, characteristic, complemented
2C2×D7×C4⋊C4C211yesmaximal, characteristic, complemented
3C2×(C22⋊D7)⋊C4C211yesmaximal, characteristic
4C2×#6441C211yesmaximal, characteristic, complemented
5C7×#396C211yesmaximal, characteristic, complemented
6#89761C211yesmaximal, characteristic, complemented
7#89757C211yesmaximal, characteristic, complemented
8#89751C211yesmaximal, characteristic, complemented
9#89750C211yesmaximal, characteristic, complemented
10#89728C211yesmaximal, characteristic
11#89728C211yesmaximal, characteristic
12#89722C211yesmaximal, characteristic, complemented
13#89721C211yesmaximal, characteristic, complemented
14#89440C211yesmaximal, characteristic, complemented
15#89429C211yesmaximal, characteristic, complemented
16C22×C22⋊D7C2211yescharacteristic
17C23×Dic7C2211yescharacteristic
18C22×C4×D7C2211yescharacteristic
19C22×C4×D7C2211yescharacteristic
20C22×C4×D7C2211yescharacteristic
21C22×Dic14C2211yescharacteristic
22C2×C7×C4⋊C4C2211yescharacteristic
23C2×C7×C4⋊C4C2211yescharacteristic
24C2×C7×C4⋊C4C2211yescharacteristic
25C2×C7×C22⋊C4C2211yescharacteristic
26C2×C7×C22⋊C4C2211yescharacteristic
27C2×C7×C22⋊C4C2211yescharacteristic
28C2×C22⋊Dic7C2211yescharacteristic
29C2×C22⋊Dic7C2211yescharacteristic
30C2×(C2×D7)⋊C4C2211yescharacteristic
31C2×(C2×D7)⋊C4C2211yescharacteristic
32C2×(C2×D7)⋊C4C2211yescharacteristic
33C2×C4⋊Dic7C2211yescharacteristic
34C2×C4⋊Dic7C2211yescharacteristic
35C2×C4⋊Dic7C2211yescharacteristic
36C2×C4⋊Dic7C2211yescharacteristic
37C2×Dic7⋊C4C2211yescharacteristic
38C2×Dic7⋊C4C2211yescharacteristic
39C2×Dic7⋊C4C2211yescharacteristic
40C2×Dic7⋊C4C2211yescharacteristic
41C2×Dic7⋊C4C2211yescharacteristic
42C2×C4×Dic7C2211yescharacteristic
43#6460—24no—
44D7×C4⋊C4—24no—
45(C22⋊D7)⋊C4—24no—
46#6441—24no—
47C7×#95C2211yescharacteristic
48C7⋊#95C2211yescharacteristic
49C7⋊#95C2211yescharacteristic
50C7⋊#95C2211yescharacteristic
51C7⋊#95C2211yescharacteristic
52C7⋊#95C2211yescharacteristic
53C7⋊#95C2211yescharacteristic
54C7⋊#95C2211yescharacteristic
55#2449—71nomaximal, Sylow
56C24×C7C2311yescharacteristic
57C23×D7C2311yescharacteristic
58C22×C4×C7C2311yescharacteristic
59C22×C4×C7C2311yescharacteristic
60C22×C4×C7C2311yescharacteristic
61C22×C4×C7C2311yescharacteristic
62C22×C4×C7C2311yescharacteristic
63C22×C4×C7C2311yescharacteristic
64C2×C22⋊D7—22no—
65C2×C22⋊D7—22no—
66C2×C22⋊D7Q814yescomplemented
67C22×Dic7—22no—
68C22×Dic7—22no—
69C22×Dic7—41no—
70C22×Dic7C2311yescharacteristic
71C22×Dic7C2311yescharacteristic
72C22×Dic7C2311yescharacteristic
73C22×Dic7C2311yescharacteristic
74C22×Dic7C2311yescharacteristic
75C22×Dic7C2311yescharacteristic
76C22×Dic7C2311yescharacteristic
77C2×C4×D7—22no—
78C2×C4×D7—22no—
79C2×C4×D7—22no—
80C2×C4×D7—22no—
81C2×C4×D7—22no—
82C2×C4×D7—22no—
83C2×C4×D7—22no—
84C2×C4×D7—41no—
85C2×C4×D7—41no—
86C2×Dic14—22no—
87C2×Dic14—22no—
88C2×Dic14—41no—
89C7×C4⋊C4—22no—
90C7×C4⋊C4—22no—
91C7×C4⋊C4—22no—
92C7×C22⋊C4—22no—
93C7×C22⋊C4—22no—
94C7×C22⋊C4—22no—
95C22⋊Dic7—22no—
96C22⋊Dic7—41no—
97(C2×D7)⋊C4—22no—
98(C2×D7)⋊C4—22no—
99(C2×D7)⋊C4D414yes—
100C4⋊Dic7—22no—
101C4⋊Dic7—22no—
102C4⋊Dic7—22no—
103C4⋊Dic7—22no—
104Dic7⋊C4—22no—
105Dic7⋊C4—22no—
106Dic7⋊C4—22no—
107Dic7⋊C4—41no—
108Dic7⋊C4D414yescomplemented
109C4×Dic7—22no—
110C2×C22⋊Q8—71no—
111C2×C22⋊Q8—71no—
112C2×C4×D4—71no—
113C22×C4⋊C4—71no—
114#399—71no—
115#398—71no—
116#396D711yescharacteristic, complemented
117#395—71no—
118#389—71no—
119#387—71no—
120#386—71no—
121#386—71no—
122#385—71no—
123#385—71no—
124#385—71no—
125C23×C7—21no—
126C23×C7—21no—
127C23×C7—21no—
128C23×C7—21no—
129C23×C7—21no—
130C23×C7—21no—
131C23×C7C2×Q812yes—
132C23×C7C2411yescharacteristic
133C22×D7—21no—
134C22×D7—21no—
135C22×D7—21no—
136C22×D7—21no—
137C22×D7C2×D412yes—
138C22×D7C2×Q812yes—
139C22×D7#4112yes—
140C2×C4×C7—21no—
141C2×C4×C7—21no—
142C2×C4×C7—21no—
143C2×C4×C7—21no—
144C2×C4×C7—21no—
145C2×C4×C7—21no—
146C2×C4×C7—21no—
147C2×C4×C7—21no—
148C2×C4×C7—21no—
149C2×C4×C7—21no—
150C2×C4×C7—21no—
151C2×C4×C7—21no—
152C2×C4×C7—21no—
153C2×C4×C7—21no—
154C2×C4×C7—21no—
155C2×C4×C7C2×D412yes—
156C2×C4×C7C2×D412yes—
157C2×C4×C7#4112yes—
158C22⋊D7—28no—
159C2×Dic7—21no—
160C2×Dic7—21no—
161C2×Dic7—21no—
162C2×Dic7—21no—
163C2×Dic7—21no—
164C2×Dic7—21no—
165C2×Dic7—21no—
166C2×Dic7—21no—
167C2×Dic7—21no—
168C2×Dic7—21no—
169C2×Dic7—21no—
170C2×Dic7—21no—
171C2×Dic7—21no—
172C2×Dic7—21no—
173C2×Dic7—21no—
174C2×Dic7—21no—
175C2×Dic7—21no—
176C2×Dic7—44no—
177C2×Dic7C2×D412yes—
178C2×Dic7C2×D412yes—
179C2×Dic7C2×D412yes—
180C2×Dic7C2×Q812yes—
181C4×D7—28no—
182C4×D7—44no—
183C4×D7—44no—
184Dic14—44no—
185C22×Q8—71no—
186C22×D4—71no—
187C23×C4—71no—
188C23×C4—71no—
189C23×C4—71no—
190C23×C4—71no—
191C22⋊Q8—144no—
192C22⋊Q8—144no—
193C4×D4—144no—
194C2×C4⋊C4—71no—
195C2×C4⋊C4—71no—
196C2×C4⋊C4—71no—
197C2×C4⋊C4—71no—
198C2×C4⋊C4—71no—
199C2×C4⋊C4—71no—
200C2×C4⋊C4—71no—
201C2×C4⋊C4—71no—
202C2×C4⋊C4—71no—
203C2×C4⋊C4—144no—
204C2×C4⋊C4C2×D711yescharacteristic
205C2×C4⋊C4C2×D711yescharacteristic
206C2×C4⋊C4C2×D711yescharacteristic
207C2×C22⋊C4—71no—
208C2×C22⋊C4—71no—
209C2×C22⋊C4—71no—
210C2×C22⋊C4—71no—
211C2×C22⋊C4—71no—
212C2×C22⋊C4C2×D711yescharacteristic
213C2×C22⋊C4C2×D711yescharacteristic
214C2×C22⋊C4C2×D711yescharacteristic
215C2×C42—71no—
216#95—71no—
217#95—71no—
218#95—71no—
219#95—71no—
220#95—71no—
221#95—71no—
222#95—71no—
223#95C2×D711yescharacteristic
224C22×C7—22no—
225C22×C7—22no—
226C22×C7—22no—
227C22×C7—41no—
228C22×C7—41no—
229C22×C7—41no—
230C22×C7—41no—
231C22×C7C22×D411yescharacteristic
232C22×C7C22×D411yescharacteristic
233C22×C7C22×Q811yescharacteristic
234C22×C7C2×#4111yescharacteristic
235C22×C7C2×#4111yescharacteristic
236C22×C7H(5,𝔽2)11yescharacteristic
237C22×C72−1+411yescharacteristic
238C2×D7—22no—
239C2×D7—22no—
240C2×D7—22no—
241C2×D7—22no—
242C2×D7—22no—
243C2×D7—22no—
244C2×D7C22⋊Q814yescomplemented
245C4×C7—22no—
246C4×C7—22no—
247C4×C7—22no—
248C4×C7—41no—
249C4×C7—41no—
250C4×C7—41no—
251Dic7—22no—
252Dic7—22no—
253Dic7—22no—
254Dic7—22no—
255Dic7—41no—
256Dic7—41no—
257Dic7—41no—
258C24—71no—
259C24C22×D711yescharacteristic
260C2×Q8—142no—
261C2×Q8—142no—
262C2×Q8—281no—
263C2×D4—74no—
264C2×D4—142no—
265C2×D4—142no—
266C22×C4—71no—
267C22×C4—71no—
268C22×C4—71no—
269C22×C4—71no—
270C22×C4—71no—
271C22×C4—71no—
272C22×C4—71no—
273C22×C4—142no—
274C22×C4—142no—
275C22×C4—142no—
276C22×C4—142no—
277C22×C4—142no—
278C22×C4—142no—
279C22×C4—142no—
280C22×C4—142no—
281C22×C4—142no—
282C22×C4—281no—
283C22×C4—281no—
284C22×C4—281no—
285C22×C4C22×D711yescharacteristic
286C22×C4C22×D711yescharacteristic
287C22×C4C22×D711yescharacteristic
288C22×C4C22×D711yescharacteristic
289C22×C4C22×D711yescharacteristic
290C22×C4C22×D711yescharacteristic
291C4⋊C4—22no—
292C4⋊C4—22no—
293C4⋊C4—22no—
294C4⋊C4—74no—
295C4⋊C4—142no—
296C4⋊C4—142no—
297C4⋊C4—142no—
298C4⋊C4—142no—
299C4⋊C4—142no—
300C4⋊C4—142no—
301C4⋊C4—142no—
302C4⋊C4—281no—
303C22⋊C4—22no—
304C22⋊C4—22no—
305C22⋊C4—22no—
306C22⋊C4—74no—
307C22⋊C4—142no—
308C22⋊C4—142no—
309C22⋊C4—142no—
310C22⋊C4—281no—
311C42—142no—
312C2×C7—42no—
313C2×C7C2×C22⋊Q811yescharacteristic
314C2×C7#52511yescharacteristic
315C2×C7#53911yescharacteristic
316C2×C7D4211yescharacteristic
317C2×C7#54711yescharacteristic
318C2×C7D4×Q811yescharacteristic
319C2×C7#55411yescharacteristic
320D7—24no—
321C23—21no—
322C23—21no—
323C23—21no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23—72no—
328C23—72no—
329C23—72no—
330C23—141no—
331C23—141no—
332C23—141no—
333C23—141no—
334C23Q8×D712yes—
335C23C23×D711yescharacteristic
336Q8—284no—
337D4—148no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—72no—
354C2×C4—72no—
355C2×C4—72no—
356C2×C4—72no—
357C2×C4—148no—
358C2×C4—141no—
359C2×C4—141no—
360C2×C4—141no—
361C2×C4—141no—
362C2×C4—141no—
363C2×C4—141no—
364C2×C4—141no—
365C2×C4—141no—
366C2×C4—141no—
367C2×C4—141no—
368C2×C4—141no—
369C2×C4—141no—
370C2×C4—141no—
371C2×C4—141no—
372C2×C4—141no—
373C2×C4—141no—
374C2×C4—141no—
375C2×C4—284no—
376C2×C4—284no—
377C2×C4—284no—
378C2×C4D4×D712yes—
379C2×C4D4×D712yes—
380C2×C4#115912yes—
381C7#244911yescharacteristic, complemented, Sylow
382C22—22no—
383C22—22no—
384C22—22no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—74no—
390C22—142no—
391C22—142no—
392C22—142no—
393C22—142no—
394C22—142no—
395C22—142no—
396C22C2×D4×D711yescharacteristic
397C22C2×D4×D711yescharacteristic
398C22C2×#115911yescharacteristic
399C22C2×Q8×D711yescharacteristic
400C22D7×#4111yescharacteristic
401C22#655311yescharacteristic
402C22#655411yescharacteristic
403C4—22no—
404C4—22no—
405C4—22no—
406C4—41no—
407C4—41no—
408C4—41no—
409C4—142no—
410C4—142no—
411C4—142no—
412C4—142no—
413C4—281no—
414C4—281no—
415C4—281no—
416C2—42no—
417C2—144no—
418C2#9029711yescharacteristic
419C2#9030011yescharacteristic
420C2#9030211yescharacteristic
421C2D7×C22⋊Q811yescharacteristic
422C2#9032311yescharacteristic
423C2#9032611yescharacteristic
424C2#9039811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#396C2
prime radicalC22×C22⋊D7C22
hypercenter#396D7
hypocenterC7#2449
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #396

Complements

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

NH ≅ G/Ntype
1. C2×#6460C2semidirect
2. C2×D7×C4⋊C4C2semidirect
4. C2×#6441C2semidirect
5. C7×#396C2semidirect
6. #89761C2semidirect
7. #89757C2semidirect
8. #89751C2semidirect
9. #89750C2semidirect
12. #89722C2semidirect
13. #89721C2semidirect
14. #89440C2semidirect
15. #89429C2semidirect
66. C2×C22⋊D7Q8semidirect
108. Dic7⋊C4D4semidirect
116. #396D7semidirect
244. C2×D7C22⋊Q8semidirect
381. C7#2449semidirect

Maximal subgroups

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

subgroupindex
C2×#64602
C2×D7×C4⋊C42
C2×(C22⋊D7)⋊C42
C2×#64412
C7×#3962
#897612
#897572
#897512
#897502
#897282
#897222
#897212
#894402
#894292
#24497