Finite Group Databasebeta

← #13254498

Subgroups — #13254498

Order 1792.

562 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 562 conjugacy classes of subgroups — too many to draw. Only its 89 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
1C2×C7×#481C211yesmaximal, characteristic, complemented
2#11740072C211yesmaximal, characteristic
3C2×#89739C211yesmaximal, characteristic, complemented
4#11729430C211yesmaximal, characteristic, complemented
5#11729316C211yesmaximal, characteristic, complemented
6#11727675C211yesmaximal, characteristic, complemented
7#11725486C211yesmaximal, characteristic, complemented
8C2×C7×#121C2211yescharacteristic
9C22×C7×C4⋊C4C2211yescharacteristic
10C2×#6493C2211yescharacteristic
11C7×#481—22no—
12C7×#481—22no—
13C2×C7×#106—21no—
14C2×C7×#102—21no—
15C2×C7×C22⋊C8C2211yescharacteristic
16#89989—21no—
17#89979C2211yescharacteristic
18#89979C2211yescharacteristic
19C2×C22⋊(C7⋊C8)C2211yescharacteristic
20#89811—22no—
21#89808—21no—
22#89806—21no—
23#89740—21no—
24#89739—22no—
25#89739—22no—
26C2×#6383—21no—
27C2×#6381—21no—
28#89325—22no—
29#89320—22no—
30#89263—22no—
31#9992—71nomaximal, Sylow
32C22×C7×D4—21no—
33C22×C7×D4D411yescharacteristic
34C23×C4×C7—21no—
35C23×C4×C7C2311yescharacteristic
36C22×C22⋊D7—21no—
37C22×D28D411yescharacteristic
38C22×C7×C8—21no—
39C7×#121—21no—
40C7×#121—21no—
41C7×#121—21no—
42C7×#121—21no—
43C2×C7×C4⋊C4—21no—
44C2×C7×C4⋊C4—21no—
45C2×C7×C4⋊C4—21no—
46C2×C7×C4⋊C4—21no—
47C2×C7×C4⋊C4—21no—
48C2×C7×C4⋊C4D412yes—
49C2×C7×C4⋊C4D411yescharacteristic
50C2×C7×C22⋊C4—21no—
51C2×#1141—22no—
52#6493—21no—
53#6493—21no—
54#6493—21no—
55#6493—21no—
56C2×(C2×D7)⋊C4—21no—
57C2×C4⋊Dic7D411yescharacteristic
58C2×#1136—21no—
59C2×#1136D412yes—
60C22×C7⋊C8—21no—
61C7×#106—24no—
62C7×#102—24no—
63C7×C22⋊C8—21no—
64C7×C22⋊C8D412yes—
65#6406—22no—
66C22⋊(C7⋊C8)—21no—
67C22⋊(C7⋊C8)—21no—
68C22⋊(C7⋊C8)—21no—
69C22⋊(C7⋊C8)—21no—
70C22⋊(C7⋊C8)—21no—
71C22⋊(C7⋊C8)D412yes—
72#6383—24no—
73#6383—22no—
74#6382—22no—
75#6381—24no—
76#3192—71no—
77C2×#481D711yescharacteristic, complemented
78C2×#480—71no—
79#1821—71no—
80#1471—71no—
81#1471—71no—
82#1298—71no—
83C24×C7—21no—
84C24×C7#4111yescharacteristic
85C23×D7—21no—
86C2×C7×D4—21no—
87C2×C7×D4—21no—
88C2×C7×D4—42no—
89C2×C7×D4—41no—
90C2×C7×D4—41no—
91C2×C7×D4—82no—
92C22×C4×C7—21no—
93C22×C4×C7—22no—
94C22×C4×C7—42no—
95C22×C4×C7—41no—
96C22×C4×C7—41no—
97C22×C4×C7—82no—
98C22×C4×C7C2×D411yescharacteristic
99C22×C4×C7C2×D411yescharacteristic
100C22×C4×C7C2×D411yescharacteristic
101C22×C4×C7C2×D411yescharacteristic
102C22×C4×C7C2×D411yescharacteristic
103C22×C4×C7C2×D411yescharacteristic
104C2×C22⋊D7—41no—
105C2×C22⋊D7—41no—
106C2×C22⋊D7—82no—
107C22×Dic7—21no—
108C2×D28—21no—
109C2×D28—21no—
110C2×D28—42no—
111C2×C7×C8—22no—
112C2×C7×C8—42no—
113C7×C4⋊C4—22no—
114C7×C4⋊C4—21no—
115C7×C4⋊C4—22no—
116C7×C4⋊C4—21no—
117C7×C4⋊C4—41no—
118C7×C4⋊C4—41no—
119C7×C22⋊C4—41no—
120C7×C22⋊C4—41no—
121#1141—82no—
122(C2×D7)⋊C4—41no—
123(C2×D7)⋊C4—41no—
124C4⋊Dic7—21no—
125C4⋊Dic7—21no—
126#1136—42no—
127#1136—42no—
128C2×C7⋊C8—22no—
129C2×C7⋊C8—22no—
130C2×C7⋊C8—42no—
131C2×C7⋊C8—41no—
132C2×#121—71no—
133C2×#121C2×D711yescharacteristic
134C22×C4⋊C4C2×D711yescharacteristic
135#481—22no—
136#481—22no—
137#481—141no—
138#481—141no—
139#480—142no—
140#480—142no—
141#469—142no—
142#447—141no—
143#447—141no—
144C2×#107—141no—
145C2×#106—21no—
146C2×#102—21no—
147C2×#102—141no—
148#407—71no—
149#407—71no—
150C2×C22⋊C8—71no—
151C2×C22⋊C8C2×D711yescharacteristic
152#343—142no—
153#323—142no—
154#323—142no—
155C23×C7—22no—
156C23×C7—22no—
157C23×C7—41no—
158C23×C7—41no—
159C23×C7—42no—
160C23×C7—81no—
161C23×C7—81no—
162C23×C7#12111yescharacteristic
163C23×C7#12111yescharacteristic
164C23×C7#12111yescharacteristic
165C22×D7—41no—
166C22×D7—42no—
167C22×D7—81no—
168C22×D7—81no—
169C7×D4—42no—
170C7×D4—42no—
171C7×D4—82no—
172C7×D4—82no—
173C2×C4×C7—22no—
174C2×C4×C7—22no—
175C2×C4×C7—22no—
176C2×C4×C7—41no—
177C2×C4×C7—41no—
178C2×C4×C7—41no—
179C2×C4×C7—41no—
180C2×C4×C7—42no—
181C2×C4×C7—82no—
182C2×C4×C7—82no—
183C2×C4×C7—82no—
184C2×C4×C7—81no—
185C2×C4×C7#12011yescharacteristic
186C2×C4×C7#12011yescharacteristic
187C2×C4×C7#12011yescharacteristic
188C2×C4×C7#12711yescharacteristic
189C22⋊D7—82no—
190C22⋊D7—82no—
191C2×Dic7—41no—
192C2×Dic7—81no—
193D28—42no—
194D28—42no—
195C7×C8—42no—
196C7⋊C8—42no—
197C7⋊C8—42no—
198C7⋊C8—81no—
199C22×D4—21no—
200C22×D4—71no—
201C22×D4—141no—
202C22×D4C22⋊D711yescharacteristic
203C23×C4—21no—
204C23×C4C22×D711yescharacteristic
205C2×D8—142no—
206C2×M16—72no—
207C2×M16—141no—
208C22×C8—21no—
209C22×C8—141no—
210#121—21no—
211#121—21no—
212#121—21no—
213#121—21no—
214#121—141no—
215#121—141no—
216#121—141no—
217#121—141no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—71no—
224C2×C4⋊C4C22⋊D712yes—
225C2×C4⋊C4C22⋊D711yescharacteristic
226C2×C22⋊C4—21no—
227C2×C22⋊C4—141no—
228#107—144no—
229#106—24no—
230#106—142no—
231#102—24no—
232#102—144no—
233#102—142no—
234#102—142no—
235C22⋊C8—21no—
236C22⋊C8—72no—
237C22⋊C8—141no—
238C22⋊C8—141no—
239C22⋊C8—141no—
240C22⋊C8—141no—
241C22⋊C8—141no—
242C22⋊C8C22⋊D712yes—
243C22×C7—21no—
244C22×C7—21no—
245C22×C7—21no—
246C22×C7—21no—
247C22×C7—21no—
248C22×C7—21no—
249C22×C7—21no—
250C22×C7—44no—
251C22×C7—81no—
252C22×C7—81no—
253C22×C7—81no—
254C22×C7—162no—
255C22×C7#39211yescharacteristic
256C22×C7#47112yes—
257C22×C7#47212yes—
258C2×D7—81no—
259C2×D7—81no—
260C2×D7—81no—
261C2×D7—162no—
262C4×C7—21no—
263C4×C7—21no—
264C4×C7—21no—
265C4×C7—82no—
266C4×C7—81no—
267C4×C7#45812yes—
268Dic7—81no—
269C24—21no—
270C24—141no—
271C24#115911yescharacteristic
272C2×D4—21no—
273C2×D4—21no—
274C2×D4—42no—
275C2×D4—41no—
276C2×D4—41no—
277C2×D4—82no—
278C2×D4—141no—
279C2×D4—141no—
280C2×D4—282no—
281C2×D4—281no—
282C2×D4—281no—
283C2×D4—562no—
284C22×C4—21no—
285C22×C4—22no—
286C22×C4—42no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—82no—
290C22×C4—141no—
291C22×C4D4×D711yescharacteristic
292C22×C4D4×D711yescharacteristic
293C22×C4D4×D711yescharacteristic
294C22×C4C2×C22⋊D711yescharacteristic
295C22×C4C2×C22⋊D711yescharacteristic
296C22×C4C2×C22⋊D711yescharacteristic
297D8—562no—
298M16—282no—
299M16—282no—
300C2×C8—22no—
301C2×C8—42no—
302C2×C8—142no—
303C2×C8—142no—
304C2×C8—282no—
305C2×C8—281no—
306C4⋊C4—21no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—21no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—141no—
313C4⋊C4—141no—
314C22⋊C4—41no—
315C22⋊C4—41no—
316C22⋊C4—281no—
317C22⋊C4—281no—
318C2×C7—22no—
319C2×C7—22no—
320C2×C7—41no—
321C2×C7—161no—
322C2×C7#201811yescharacteristic
323C2×C7#201911yescharacteristic
324C2×C7#202211yescharacteristic
325D7—161no—
326C23—22no—
327C23—22no—
328C23—41no—
329C23—41no—
330C23—42no—
331C23—81no—
332C23—81no—
333C23—281no—
334C23—282no—
335C23—561no—
336C23—561no—
337C23#650111yescharacteristic
338C23#650211yescharacteristic
339C23#650211yescharacteristic
340D4—42no—
341D4—42no—
342D4—82no—
343D4—82no—
344D4—282no—
345D4—282no—
346D4—562no—
347D4—562no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—42no—
356C2×C4—82no—
357C2×C4—82no—
358C2×C4—82no—
359C2×C4—81no—
360C2×C4—281no—
361C2×C4—561no—
362C2×C4#650011yescharacteristic
363C2×C4#650011yescharacteristic
364C2×C4#650311yescharacteristic
365C2×C4#651611yescharacteristic
366C8—42no—
367C8—282no—
368C8—282no—
369C8—561no—
370C7#999211yescharacteristic, complemented, Sylow
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—44no—
379C22—81no—
380C22—81no—
381C22—81no—
382C22—162no—
383C22—561no—
384C22—561no—
385C22—561no—
386C22—1122no—
387C22#8994912yes—
388C22#8997112yes—
389C22#8999511yescharacteristic
390C4—21no—
391C4—21no—
392C4—21no—
393C4—81no—
394C4—82no—
395C4—561no—
396C4#8982512yes—
397C2—22no—
398C2—22no—
399C2—41no—
400C2—161no—
401C2—1121no—
402C2#1172946011yescharacteristic
403C2#1172981311yescharacteristic
404C2#1172984911yescharacteristic
subgroupG/H
centerC22#89995
commutator, cosocleC23×C4×C7C23
FrattiniC23×C4C22×D7
FittingC2×C7×#481C2
socleC22×C7#392
prime radical#11740072C2
hypercenterC2×#481D7
hypocenterC7#9992
LCSG ▷ C23×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#481

Complements

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

NH ≅ G/Ntype
1. C2×C7×#481C2semidirect
3. C2×#89739C2semidirect
4. #11729430C2semidirect
5. #11729316C2semidirect
6. #11727675C2semidirect
7. #11725486C2semidirect
77. C2×#481D7semidirect
370. C7#9992semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#4812
#117400722
C2×#897392
#117294302
#117293162
#117276752
#117254862
#99927