Finite Group Databasebeta

← #13257641

Subgroups — #13257641

Order 1792.

474 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89856C211yesmaximal, characteristic
2C7×#1990C211yesmaximal, characteristic, complemented
3C2×#89341C211yesmaximal, characteristic, complemented
4#11727541C211yesmaximal, characteristic
5C2×#89252C211yesmaximal, characteristic, complemented
6#11727338C211yesmaximal, characteristic
7#11725440C211yesmaximal, characteristic, complemented
8C2×C7×C4⋊Q8C2211yescharacteristic
9C2×C4×D28C2211yescharacteristic
10C2×C4×C7×C8C2211yescharacteristic
11C7×#384C2211yescharacteristic
12C2×(C2×D7)⋊C8—21no—
13C2×(C2×D7)⋊C8—21no—
14#89856—22no—
15#89856—22no—
16C2×#6383—21no—
17C2×#6383—21no—
18C2×#6382—21no—
19C2×C4⋊(C7⋊C8)C2211yescharacteristic
20C2×C4⋊(C7⋊C8)C2211yescharacteristic
21C2×C4⋊(C7⋊C8)C2211yescharacteristic
22C7×#340—21no—
23C7×#340—21no—
24#89341—22no—
25#89324—21no—
26#89252—22no—
27#10480—71nomaximal, Sylow
28C22×C7×Q8—21no—
29C22×D28D411yescharacteristic
30C22×C4×D7—21no—
31C22×C7×C8D411yescharacteristic
32C22×C7×C8D411yescharacteristic
33C7×C4⋊Q8—21no—
34C7×C4⋊Q8—21no—
35C7×C4⋊Q8—21no—
36C7×C4⋊Q8D412yes—
37C2×C7×C4⋊C4—21no—
38C2×C7×C4⋊C4—21no—
39C2×C7×C4⋊C4—21no—
40C2×C7×C4⋊C4—21no—
41C2×C7×C4⋊C4—21no—
42C2×C42×C7C2311yescharacteristic
43C2×(C2×D7)⋊C4—21no—
44C2×C4⋊Dic7D411yescharacteristic
45C22×C7⋊C8—21no—
46C22×C7⋊C8—21no—
47C22×C7⋊C8—21no—
48C4×D28—21no—
49C4×D28—21no—
50C4×D28—21no—
51C4×D28—21no—
52C4×C7×C8—21no—
53C4×C7×C8—21no—
54(C2×D7)⋊C8—42no—
55(C2×D7)⋊C8—42no—
56#6383—24no—
57#6383—42no—
58#6382—24no—
59C4⋊(C7⋊C8)—21no—
60C4⋊(C7⋊C8)—21no—
61C4⋊(C7⋊C8)—21no—
62C4⋊(C7⋊C8)—21no—
63C4⋊(C7⋊C8)—21no—
64C4⋊(C7⋊C8)D412yes—
65C2×#476—71no—
66#1990D711yescharacteristic, complemented
67#1954—71no—
68#1802—71no—
69C2×#334—71no—
70C2×#333—71no—
71#1296—71no—
72C23×D7—21no—
73C2×C7×Q8—22no—
74C2×C7×Q8—41no—
75C2×C7×Q8—81no—
76C2×C7×Q8—81no—
77C22×C4×C7—41no—
78C22×C4×C7—41no—
79C22×C4×C7C2×D411yescharacteristic
80C22×C4×C7C2×D411yescharacteristic
81C22×C4×C7#4111yescharacteristic
82C22×Dic7—21no—
83C2×D28—21no—
84C2×D28—21no—
85C2×D28—21no—
86C2×D28—21no—
87C2×D28—21no—
88C2×D28SD1612yes—
89C2×C4×D7—41no—
90C2×C4×D7—41no—
91C2×C4×D7—41no—
92C2×C4×D7—41no—
93C2×C4×D7—41no—
94C2×C4×D7—41no—
95C2×C7×C8—21no—
96C2×C7×C8—21no—
97C2×C7×C8—21no—
98C2×C7×C8—21no—
99C2×C7×C8—21no—
100C2×C7×C8SD1612yes—
101C7×C4⋊C4—22no—
102C7×C4⋊C4—22no—
103C7×C4⋊C4—41no—
104C7×C4⋊C4—41no—
105C7×C4⋊C4—41no—
106C7×C4⋊C4—41no—
107C7×C4⋊C4—41no—
108C7×C4⋊C4—41no—
109C7×C4⋊C4—81no—
110C42×C7C2×D411yescharacteristic
111C42×C7C2×D411yescharacteristic
112C42×C7#4111yescharacteristic
113C42×C7#4111yescharacteristic
114(C2×D7)⋊C4—41no—
115(C2×D7)⋊C4—41no—
116C4⋊Dic7—21no—
117C4⋊Dic7—21no—
118C2×C7⋊C8—22no—
119C2×C7⋊C8—41no—
120C2×C7⋊C8—41no—
121C2×C7⋊C8—41no—
122C2×C7⋊C8—41no—
123C2×C7⋊C8—41no—
124C2×C7⋊C8—41no—
125C2×C7⋊C8—41no—
126C2×C7⋊C8—41no—
127C2×C4⋊Q8C2×D711yescharacteristic
128C2×C4×D4—71no—
129#476—142no—
130#476—142no—
131C2×#106—141no—
132C2×C4⋊C8—71no—
133C2×C4⋊C8—71no—
134C2×C4⋊C8—71no—
135C2×#102—141no—
136C2×#102—141no—
137C2×C22⋊C8—141no—
138C2×C22⋊C8—141no—
139C2×C4×C8C2×D711yescharacteristic
140#384C2×D711yescharacteristic
141#340—21no—
142#340—21no—
143#340—141no—
144#334—142no—
145#333—142no—
146C23×C7#12111yescharacteristic
147C22×D7—22no—
148C22×D7—41no—
149C22×D7—41no—
150C22×D7—41no—
151C22×D7—41no—
152C22×D7—41no—
153C22×D7—41no—
154C7×Q8—82no—
155C7×Q8—82no—
156C2×C4×C7—21no—
157C2×C4×C7—21no—
158C2×C4×C7—42no—
159C2×C4×C7—81no—
160C2×C4×C7—81no—
161C2×C4×C7—81no—
162C2×C4×C7—81no—
163C2×C4×C7—81no—
164C2×C4×C7#12011yescharacteristic
165C2×C4×C7#12111yescharacteristic
166C2×C4×C7#12111yescharacteristic
167C2×C4×C7#12311yescharacteristic
168C2×C4×C7#12311yescharacteristic
169C2×C4×C7#12411yescharacteristic
170C2×C4×C7C2×SD1611yescharacteristic
171C2×C4×C7C2×SD1611yescharacteristic
172C2×C4×C7#13511yescharacteristic
173C2×C4×C7#13511yescharacteristic
174C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
175C2×C4×C7#13711yescharacteristic
176C2×C4×C7#13711yescharacteristic
177C2×C4×C7#13711yescharacteristic
178C2×Dic7—41no—
179C2×Dic7—41no—
180C2×Dic7—41no—
181D28—22no—
182D28—22no—
183D28—41no—
184D28—41no—
185C4×D7—81no—
186C4×D7—81no—
187C4×D7—81no—
188C4×D7—81no—
189C7×C8—22no—
190C7×C8—41no—
191C7⋊C8—42no—
192C7⋊C8—81no—
193C7⋊C8—81no—
194C22×Q8—21no—
195C22×D4—71no—
196C23×C4—141no—
197C22×C8—141no—
198C22×C8—141no—
199C22×C8—141no—
200C22×C8C22⋊D711yescharacteristic
201C22×C8C22⋊D711yescharacteristic
202C4⋊Q8—21no—
203C4⋊Q8—21no—
204C4⋊Q8—21no—
205C4⋊Q8D2812yes—
206C4×D4—141no—
207C4×D4—141no—
208C4×D4—141no—
209C4×D4—141no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—71no—
216C2×C22⋊C4—141no—
217C2×C42C22×D711yescharacteristic
218#106—144no—
219C4⋊C8—72no—
220C4⋊C8—141no—
221C4⋊C8—141no—
222C4⋊C8—141no—
223C4⋊C8—141no—
224C4⋊C8—141no—
225#102—144no—
226#102—282no—
227C22⋊C8—282no—
228C22⋊C8—282no—
229C4×C8—21no—
230C4×C8—21no—
231C22×C7#39411yescharacteristic
232C22×C7#46111yescharacteristic
233C22×C7#46311yescharacteristic
234C22×C7#46511yescharacteristic
235C22×C7#47011yescharacteristic
236C22×C7#47111yescharacteristic
237C22×C7#47211yescharacteristic
238C2×D7—42no—
239C2×D7—42no—
240C2×D7—42no—
241C2×D7—81no—
242C2×D7—81no—
243C2×D7—81no—
244C2×D7—81no—
245C4×C7—21no—
246C4×C7—21no—
247C4×C7—82no—
248C4×C7—161no—
249C4×C7#44811yescharacteristic
250C4×C7#45211yescharacteristic
251C4×C7#46011yescharacteristic
252C4×C7#46411yescharacteristic
253C4×C7#46511yescharacteristic
254C4×C7#46811yescharacteristic
255C4×C7#48311yescharacteristic
256C4×C7#48511yescharacteristic
257Dic7—81no—
258C24—141no—
259C2×Q8—22no—
260C2×Q8—41no—
261C2×Q8—81no—
262C2×Q8—81no—
263C2×D4—72no—
264C2×D4—141no—
265C2×D4—141no—
266C2×D4—141no—
267C2×D4—141no—
268C2×D4—141no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—141no—
272C22×C4—281no—
273C22×C4—281no—
274C22×C4—281no—
275C22×C4—281no—
276C22×C4—281no—
277C22×C4—281no—
278C22×C4D4×D711yescharacteristic
279C22×C4#115911yescharacteristic
280C22×C4C2×C22⋊D711yescharacteristic
281C2×C8—21no—
282C2×C8—21no—
283C2×C8—21no—
284C2×C8—21no—
285C2×C8—21no—
286C2×C8—142no—
287C2×C8—281no—
288C2×C8—281no—
289C2×C8—281no—
290C2×C8—281no—
291C2×C8—281no—
292C2×C8—281no—
293C2×C8—281no—
294C2×C8—281no—
295C2×C8#114312yes—
296C4⋊C4—22no—
297C4⋊C4—22no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—81no—
305C4⋊C4—141no—
306C4⋊C4—141no—
307C22⋊C4—281no—
308C22⋊C4—281no—
309C42C2×D2811yescharacteristic
310C42#115711yescharacteristic
311C42D4×D711yescharacteristic
312C42#115911yescharacteristic
313C2×C7#167111yescharacteristic
314C2×C7#167711yescharacteristic
315C2×C7#169511yescharacteristic
316C2×C7#170311yescharacteristic
317C2×C7#203511yescharacteristic
318C2×C7#205111yescharacteristic
319C2×C7#205511yescharacteristic
320D7—82no—
321C23—142no—
322C23—281no—
323C23—281no—
324C23—281no—
325C23—281no—
326C23—281no—
327C23—281no—
328C23#650111yescharacteristic
329Q8—82no—
330Q8—82no—
331D4—142no—
332D4—142no—
333D4—281no—
334D4—281no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—42no—
338C2×C4—81no—
339C2×C4—81no—
340C2×C4—81no—
341C2×C4—81no—
342C2×C4—81no—
343C2×C4—281no—
344C2×C4—281no—
345C2×C4—281no—
346C2×C4—561no—
347C2×C4—561no—
348C2×C4—561no—
349C2×C4—561no—
350C2×C4#644511yescharacteristic
351C2×C4#644611yescharacteristic
352C2×C4#644811yescharacteristic
353C2×C4#644911yescharacteristic
354C2×C4D7×SD1611yescharacteristic
355C2×C4#647911yescharacteristic
356C2×C4#648111yescharacteristic
357C2×C4#648111yescharacteristic
358C2×C4#649311yescharacteristic
359C2×C4#650211yescharacteristic
360C2×C4C2×#114311yescharacteristic
361C2×C4#650511yescharacteristic
362C2×C4#651211yescharacteristic
363C2×C4#651311yescharacteristic
364C8—22no—
365C8—41no—
366C8—282no—
367C8—561no—
368C8—561no—
369C7#1048011yescharacteristic, complemented, Sylow
370C22—282no—
371C22—282no—
372C22—282no—
373C22—561no—
374C22—561no—
375C22—561no—
376C22—561no—
377C22#8973311yescharacteristic
378C22#8982911yescharacteristic
379C22#8985511yescharacteristic
380C22#8993211yescharacteristic
381C22#8994311yescharacteristic
382C22#8994911yescharacteristic
383C22#8996111yescharacteristic
384C4—21no—
385C4—21no—
386C4—82no—
387C4—161no—
388C4—561no—
389C4#8957711yescharacteristic
390C4#8957811yescharacteristic
391C4#8958211yescharacteristic
392C4#8958711yescharacteristic
393C4#8979911yescharacteristic
394C4#8980011yescharacteristic
395C4#8982011yescharacteristic
396C4#8982111yescharacteristic
397C2—562no—
398C2#1172743411yescharacteristic
399C2#1172795011yescharacteristic
400C2#1172799711yescharacteristic
401C2#1172895411yescharacteristic
402C2#1172898911yescharacteristic
403C2#1172909811yescharacteristic
404C2#1172912811yescharacteristic
subgroupG/H
centerC23#6501
commutator, cosocleC2×C42×C7C23
FrattiniC2×C42C22×D7
FittingC7×#1990C2
socleC23×C7#121
prime radicalC22×D28D4
hypercenter#1990D7
hypocenterC7#10480
LCSG ▷ C2×C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1990

Complements

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

NH ≅ G/Ntype
2. C7×#1990C2semidirect
3. C2×#89341C2semidirect
5. C2×#89252C2semidirect
7. #11725440C2semidirect
66. #1990D7semidirect
369. C7#10480semidirect

Maximal subgroups

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

subgroupindex
C2×#898562
C7×#19902
C2×#893412
#117275412
C2×#892522
#117273382
#117254402
#104807