Finite Group Databasebeta

← #13250089

Subgroups — #13250089

Order 1792.

588 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2293C211yesmaximal, characteristic, complemented
2#11739753C211yesmaximal, characteristic
3#11729352C211yesmaximal, characteristic, complemented
4#11727223C211yesmaximal, characteristic, complemented
5#11727149C211yesmaximal, characteristic, complemented
6#11725426C211yesmaximal, characteristic, complemented
7#11725417C211yesmaximal, characteristic, complemented
8C2×C4×C7×D4C2211yescharacteristic
9C2×C7×#117—21no—
10#90168C2211yescharacteristic
11C7×#390—21no—
12C7×#389—21no—
13C7×#388—22no—
14C4×C7×C4⋊C4C2211yescharacteristic
15C4×C7×C22⋊C4—21no—
16C43×C7C2211yescharacteristic
17#89989—21no—
18#89787C412yescomplemented
19C4×#1141—22no—
20#89779C412yescomplemented
21#89739—21no—
22C2×#6379—21no—
23C2×#6379—21no—
24#89697C2211yescharacteristic
25#89697C2211yescharacteristic
26#89696C2211yescharacteristic
27D4⋊(C7⋊C8)—22no—
28#89246—22no—
29#9237—71nomaximal, Sylow
30C22×C7×D4D411yescharacteristic
31C23×C4×C7—21no—
32C2×#1157—21no—
33C4×C7×D4—21no—
34C4×C7×D4—21no—
35C4×C7×D4C2×C412yes—
36C4×C7×D4D412yes—
37C7×#117—42no—
38C7×#117—42no—
39C2×C7×C4⋊C4—22no—
40C2×C7×C4⋊C4—21no—
41C2×C7×C4⋊C4D411yescharacteristic
42C2×C7×C22⋊C4—21no—
43C2×C7×C22⋊C4—22no—
44C2×C7×C22⋊C4—22no—
45C2×C42×C7—21no—
46C2×C42×C7—21no—
47C2×C42×C7—21no—
48C2×C42×C7D411yescharacteristic
49C2×C42×C7D411yescharacteristic
50C2×C42×C7C2311yescharacteristic
51C2×#1141—22no—
52#6493—21no—
53#6493—21no—
54C2×#1136—21no—
55#6438—21no—
56#6437C2×C411yescharacteristic
57C4×D28—21no—
58C4×D28—21no—
59#6432C2×C411yescharacteristic
60C7×#95—22no—
61C7×#95D412yes—
62#6406—22no—
63C22⋊(C7⋊C8)—21no—
64C22⋊(C7⋊C8)—21no—
65#6383—22no—
66#6381—22no—
67#6379—42no—
68#6379—42no—
69C4⋊(C7⋊C8)—21no—
70C4⋊(C7⋊C8)—21no—
71C4⋊(C7⋊C8)—21no—
72C4⋊(C7⋊C8)C2×C412yes—
73C4×C7⋊C8—21no—
74C4×C7⋊C8—21no—
75#3100—71no—
76#2293D711yescharacteristic, complemented
77#1962—71no—
78#1502—71no—
79#1486—71no—
80#1303—71no—
81#1276—71no—
82C24×C7—21no—
83C2×C7×D4—21no—
84C2×C7×D4—42no—
85C2×C7×D4C22⋊C412yes—
86C22×C4×C7—21no—
87C22×C4×C7—21no—
88C22×C4×C7—21no—
89C22×C4×C7—22no—
90C22×C4×C7—22no—
91C22×C4×C7—22no—
92C22×C4×C7—41no—
93C22×C4×C7—41no—
94C22×C4×C7—41no—
95C22×C4×C7—41no—
96C22×C4×C7C2×D411yescharacteristic
97C22×C4×C7C2×D411yescharacteristic
98C22×C4×C7C2×D411yescharacteristic
99C2×C22⋊D7—41no—
100#1157—41no—
101#1157—41no—
102#1157—41no—
103#1157—41no—
104C2×D28—21no—
105C2×D28—21no—
106C2×D28—21no—
107C2×C4×D7—41no—
108C2×Dic14—21no—
109C7×C4⋊C4—21no—
110C7×C4⋊C4—42no—
111C7×C4⋊C4—44no—
112C7×C4⋊C4C22⋊C412yes—
113C7×C22⋊C4—22no—
114C7×C22⋊C4—22no—
115C7×C22⋊C4—44no—
116C7×C22⋊C4—44no—
117C42×C7—21no—
118C42×C7—21no—
119C42×C7—21no—
120C42×C7—42no—
121C42×C7—42no—
122C42×C7—41no—
123C42×C7—41no—
124C42×C7D812yescomplemented
125C42×C7C22×C411yescharacteristic
126C42×C7C2×D411yescharacteristic
127C42×C7#4111yescharacteristic
128C42×C7#4111yescharacteristic
129#1141—44no—
130(C2×D7)⋊C4—41no—
131(C2×D7)⋊C4—41no—
132C4⋊Dic7—21no—
133C4⋊Dic7—21no—
134#1136—41no—
135#1136—41no—
136C2×C7⋊C8—22no—
137C2×C7⋊C8—41no—
138C2×C7⋊C8—41no—
139#532—71no—
140C2×C4×D4C2×D711yescharacteristic
141C2×#117—21no—
142#480—141no—
143#474—72no—
144#459—72no—
145#447—141no—
146C4×D8—142no—
147#431—71no—
148#431—71no—
149#423—71no—
150C2×#104—141no—
151C2×#104—141no—
152#390—21no—
153#389—21no—
154#388—22no—
155C4×C4⋊C4C2×D711yescharacteristic
156C4×C22⋊C4—21no—
157C43C2×D711yescharacteristic
158#327—142no—
159D4⋊C8—142no—
160C23×C7—22no—
161C23×C7—42no—
162C23×C7—42no—
163C23×C7—41no—
164C23×C7—41no—
165C23×C7#12011yescharacteristic
166C22×D7—41no—
167C7×D4—22no—
168C7×D4—22no—
169C7×D4—42no—
170C2×C4×C7—22no—
171C2×C4×C7—22no—
172C2×C4×C7—22no—
173C2×C4×C7—22no—
174C2×C4×C7—41no—
175C2×C4×C7—41no—
176C2×C4×C7—41no—
177C2×C4×C7—41no—
178C2×C4×C7—41no—
179C2×C4×C7—41no—
180C2×C4×C7—41no—
181C2×C4×C7—42no—
182C2×C4×C7—42no—
183C2×C4×C7—42no—
184C2×C4×C7—42no—
185C2×C4×C7—42no—
186C2×C4×C7—42no—
187C2×C4×C7—42no—
188C2×C4×C7C2×C22⋊C411yescharacteristic
189C2×C4×C7C4×D411yescharacteristic
190C2×C4×C7C4×D411yescharacteristic
191C2×C4×C7#12111yescharacteristic
192C2×C4×C7#12111yescharacteristic
193C2×C4×C7#12311yescharacteristic
194C2×C4×C7C2×D811yescharacteristic
195C2×C4×C7#13511yescharacteristic
196C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
197C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
198C22⋊D7—81no—
199C22⋊D7—81no—
200C2×Dic7—41no—
201D28—42no—
202D28—41no—
203D28—41no—
204D28—81no—
205C4×D7—81no—
206C4×D7—81no—
207Dic14—41no—
208Dic14—41no—
209C7⋊C8—42no—
210C7⋊C8—81no—
211C7⋊C8—81no—
212C2×#41—141no—
213C22×D4C22⋊D711yescharacteristic
214C23×C4—21no—
215C2×D8—142no—
216C2×M16—141no—
217C4⋊Q8—71no—
218#127—71no—
219#124—141no—
220#121—141no—
221#121—141no—
222C4×D4—21no—
223C4×D4—21no—
224C4×D4—141no—
225C4×D4—141no—
226C4×D4C4×D712yes—
227C4×D4D2812yes—
228#117—42no—
229#117—42no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4C22⋊D711yescharacteristic
233C2×C22⋊C4—21no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—22no—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42C22⋊D711yescharacteristic
240C2×C42C22⋊D711yescharacteristic
241C2×C42C22×D711yescharacteristic
242#107—142no—
243C4⋊C8—72no—
244C4⋊C8—141no—
245C4⋊C8—141no—
246C4⋊C8—141no—
247#104—282no—
248#104—282no—
249#102—142no—
250#102—142no—
251C22⋊C8—141no—
252C22⋊C8—141no—
253C4×C8—141no—
254C4×C8—141no—
255#95—22no—
256#95C22⋊D712yes—
257C22×C7—21no—
258C22×C7—21no—
259C22×C7—42no—
260C22×C7—42no—
261C22×C7—42no—
262C22×C7—82no—
263C22×C7—82no—
264C22×C7#38611yescharacteristic
265C22×C7#44711yescharacteristic
266C22×C7#44911yescharacteristic
267C2×D7—81no—
268C2×D7—81no—
269C2×D7—81no—
270C4×C7—21no—
271C4×C7—21no—
272C4×C7—21no—
273C4×C7—42no—
274C4×C7—42no—
275C4×C7—42no—
276C4×C7—82no—
277C4×C7#42112yes—
278C4×C7C4×D811yescharacteristic
279C4×C7#44211yescharacteristic
280C4×C7#45911yescharacteristic
281C4×C7#46311yescharacteristic
282Dic7—81no—
283C24—21no—
284#41—281no—
285#41—281no—
286#41—281no—
287#41—281no—
288C2×Q8—141no—
289C2×D4—21no—
290C2×D4—42no—
291C2×D4—141no—
292C2×D4—141no—
293C2×D4—141no—
294C2×D4—281no—
295C2×D4(C2×D7)⋊C412yes—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—281no—
307C22×C4C2×C22⋊D711yescharacteristic
308C22×C4C2×C22⋊D711yescharacteristic
309C22×C4C2×C22⋊D711yescharacteristic
310D8—284no—
311M16—281no—
312M16—281no—
313C2×C8—142no—
314C2×C8—281no—
315C2×C8—281no—
316C4⋊C4—21no—
317C4⋊C4—44no—
318C4⋊C4—42no—
319C4⋊C4—141no—
320C4⋊C4—141no—
321C4⋊C4(C2×D7)⋊C412yes—
322C22⋊C4—22no—
323C22⋊C4—22no—
324C22⋊C4—44no—
325C22⋊C4—44no—
326C22⋊C4—281no—
327C22⋊C4—281no—
328C42—21no—
329C42—21no—
330C42—21no—
331C42—42no—
332C42—42no—
333C42—41no—
334C42—41no—
335C42#114112yescomplemented
336C42C2×C4×D711yescharacteristic
337C42C2×D2811yescharacteristic
338C42#115711yescharacteristic
339C42#115711yescharacteristic
340C2×C7—21no—
341C2×C7—21no—
342C2×C7—82no—
343C2×C7#187911yescharacteristic
344C2×C7#189711yescharacteristic
345C2×C7#190611yescharacteristic
346D7—161no—
347C23—22no—
348C23—42no—
349C23—42no—
350C23—41no—
351C23—41no—
352C23—281no—
353C23#651611yescharacteristic
354Q8—281no—
355Q8—281no—
356D4—22no—
357D4—22no—
358D4—42no—
359D4—282no—
360D4—281no—
361D4—281no—
362D4—561no—
363D4—561no—
364D4—561no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—281no—
384C2×C4—561no—
385C2×C4—561no—
386C2×C4C2×(C2×D7)⋊C411yescharacteristic
387C2×C4C4×C22⋊D711yescharacteristic
388C2×C4C4×C22⋊D711yescharacteristic
389C2×C4#649211yescharacteristic
390C2×C4#649311yescharacteristic
391C2×C4#649311yescharacteristic
392C2×C4C2×#114111yescharacteristic
393C2×C4#649511yescharacteristic
394C2×C4#651211yescharacteristic
395C2×C4#651311yescharacteristic
396C8—282no—
397C8—561no—
398C8—561no—
399C7#923711yescharacteristic, complemented, Sylow
400C22—21no—
401C22—21no—
402C22—42no—
403C22—42no—
404C22—42no—
405C22—82no—
406C22—82no—
407C22—561no—
408C22—561no—
409C22—561no—
410C22#8998511yescharacteristic
411C22#8998911yescharacteristic
412C22#9001011yescharacteristic
413C4—21no—
414C4—21no—
415C4—21no—
416C4—42no—
417C4—42no—
418C4—42no—
419C4—82no—
420C4—561no—
421C4#8977712yes—
422C4C4×#114111yescharacteristic
423C4#8978611yescharacteristic
424C4#8978711yescharacteristic
425C4#8978811yescharacteristic
426C2—21no—
427C2—21no—
428C2—82no—
429C2—1121no—
430C2#1172934511yescharacteristic
431C2#1172936711yescharacteristic
432C2#1172939911yescharacteristic
subgroupG/H
centerC2×C4#6493
commutatorC42×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#2293C2
socleC22×C7#386
prime radical#11739753C2
cosocleC2×C42×C7C23
hypercenter#2293D7
hypocenterC7#9237
LCSG ▷ C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2293

Complements

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

NH ≅ G/Ntype
1. C7×#2293C2semidirect
3. #11729352C2semidirect
4. #11727223C2semidirect
5. #11727149C2semidirect
6. #11725426C2semidirect
7. #11725417C2semidirect
18. #89787C4semidirect
20. #89779C4semidirect
76. #2293D7semidirect
124. C42×C7D8semidirect
335. C42#1141semidirect
399. C7#9237semidirect

Maximal subgroups

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

subgroupindex
C7×#22932
#117397532
#117293522
#117272232
#117271492
#117254262
#117254172
#92377