Finite Group Databasebeta

← #13254509

Subgroups — #13254509

Order 1792.

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

normal subgroups only

This group has 536 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×#484C211yesmaximal, characteristic, complemented
2#11740183C211yesmaximal, characteristic
3C2×#89739C211yesmaximal, characteristic
4#11729445C211yesmaximal, characteristic, complemented
5#11729316C211yesmaximal, characteristic, complemented
6#11727675C211yesmaximal, characteristic
7#11725486C211yesmaximal, characteristic, complemented
8C2×C7×C22⋊Q8C2211yescharacteristic
9C22×C7×C4⋊C4C2211yescharacteristic
10C2×#6493C2211yescharacteristic
11C7×#484—22no—
12C7×#484—22no—
13C2×C7×#107—21no—
14C2×C7×#103—21no—
15C2×C7×C22⋊C8C2211yescharacteristic
16#89999—21no—
17#89979C2211yescharacteristic
18#89979C2211yescharacteristic
19C2×C22⋊(C7⋊C8)C2211yescharacteristic
20#89821—22no—
21#89818—21no—
22#89816—21no—
23#89739—21no—
24#89739—22no—
25#89739—22no—
26C2×#6383—21no—
27C2×#6381—21no—
28#89325—22no—
29#89323—22no—
30#89263—22no—
31#9994—71nomaximal, Sylow
32C22×C7×Q8D411yescharacteristic
33C23×C4×C7—21no—
34C23×C4×C7C2311yescharacteristic
35C22×C22⋊D7—21no—
36C22×D28D411yescharacteristic
37C22×C7×C8—21no—
38C7×C22⋊Q8—21no—
39C7×C22⋊Q8—21no—
40C7×C22⋊Q8—21no—
41C7×C22⋊Q8—21no—
42C2×C7×C4⋊C4—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×#1143—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×#107—24no—
62C7×#103—24no—
63C7×C22⋊C8—21no—
64C7×C22⋊C8D412yes—
65#6409—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#6381—24no—
75#6381—22no—
76#3193—71no—
77C2×#484D711yescharacteristic, complemented
78C2×#480—71no—
79#1821—71no—
80#1472—71no—
81#1471—71no—
82#1298—71no—
83C24×C7#4111yescharacteristic
84C23×D7—21no—
85C2×C7×Q8—21no—
86C2×C7×Q8—21no—
87C2×C7×Q8—42no—
88C22×C4×C7—21no—
89C22×C4×C7—21no—
90C22×C4×C7—22no—
91C22×C4×C7—42no—
92C22×C4×C7—41no—
93C22×C4×C7—41no—
94C22×C4×C7—82no—
95C22×C4×C7C2×D411yescharacteristic
96C22×C4×C7C2×D411yescharacteristic
97C22×C4×C7C2×D411yescharacteristic
98C22×C4×C7C2×D411yescharacteristic
99C22×C4×C7C2×D411yescharacteristic
100C22×C4×C7C2×D411yescharacteristic
101C2×C22⋊D7—41no—
102C2×C22⋊D7—41no—
103C2×C22⋊D7—82no—
104C22×Dic7—21no—
105C2×D28—21no—
106C2×D28—21no—
107C2×D28—42no—
108C2×C7×C8—22no—
109C2×C7×C8—42no—
110C7×C4⋊C4—22no—
111C7×C4⋊C4—21no—
112C7×C4⋊C4—22no—
113C7×C4⋊C4—21no—
114C7×C4⋊C4—41no—
115C7×C4⋊C4—41no—
116C7×C4⋊C4—41no—
117C7×C4⋊C4—41no—
118C7×C22⋊C4—41no—
119C7×C22⋊C4—41no—
120#1143—82no—
121(C2×D7)⋊C4—41no—
122(C2×D7)⋊C4—41no—
123C4⋊Dic7—21no—
124C4⋊Dic7—21no—
125#1136—42no—
126#1136—42no—
127C2×C7⋊C8—22no—
128C2×C7⋊C8—22no—
129C2×C7⋊C8—42no—
130C2×C7⋊C8—41no—
131C2×C22⋊Q8C2×D711yescharacteristic
132C2×#121—71no—
133C22×C4⋊C4C2×D711yescharacteristic
134#484—22no—
135#484—22no—
136#484—141no—
137#480—142no—
138#480—142no—
139#480—141no—
140#468—142no—
141#450—141no—
142#448—141no—
143C2×#107—21no—
144C2×#107—141no—
145C2×#103—21no—
146C2×#102—141no—
147#407—71no—
148#407—71no—
149C2×C22⋊C8—71no—
150C2×C22⋊C8C2×D711yescharacteristic
151#343—142no—
152#324—142no—
153#323—142no—
154C23×C7—22no—
155C23×C7—22no—
156C23×C7—41no—
157C23×C7#12111yescharacteristic
158C23×C7#12111yescharacteristic
159C23×C7#12111yescharacteristic
160C22×D7—41no—
161C22×D7—42no—
162C22×D7—81no—
163C22×D7—81no—
164C7×Q8—42no—
165C7×Q8—42no—
166C2×C4×C7—22no—
167C2×C4×C7—22no—
168C2×C4×C7—22no—
169C2×C4×C7—41no—
170C2×C4×C7—41no—
171C2×C4×C7—41no—
172C2×C4×C7—41no—
173C2×C4×C7—41no—
174C2×C4×C7—42no—
175C2×C4×C7—42no—
176C2×C4×C7—82no—
177C2×C4×C7—82no—
178C2×C4×C7—82no—
179C2×C4×C7—81no—
180C2×C4×C7#12011yescharacteristic
181C2×C4×C7#12011yescharacteristic
182C2×C4×C7#12011yescharacteristic
183C2×C4×C7#12711yescharacteristic
184C22⋊D7—82no—
185C22⋊D7—82no—
186C2×Dic7—41no—
187C2×Dic7—81no—
188D28—42no—
189D28—42no—
190C7×C8—42no—
191C7⋊C8—42no—
192C7⋊C8—42no—
193C7⋊C8—81no—
194C22×Q8C22⋊D711yescharacteristic
195C22×D4—71no—
196C22×D4—141no—
197C23×C4—21no—
198C23×C4C22×D711yescharacteristic
199C2×SD16—142no—
200C2×M16—72no—
201C2×M16—141no—
202C22×C8—21no—
203C22×C8—141no—
204C22⋊Q8—21no—
205C22⋊Q8—21no—
206C22⋊Q8—21no—
207C22⋊Q8—21no—
208#121—141no—
209#121—141no—
210#121—141no—
211#121—141no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—71no—
219C2×C4⋊C4C22⋊D712yes—
220C2×C4⋊C4C22⋊D711yescharacteristic
221C2×C22⋊C4—21no—
222C2×C22⋊C4—141no—
223#107—24no—
224#107—144no—
225#107—142no—
226#103—24no—
227#103—142no—
228#102—144no—
229#102—142no—
230C22⋊C8—21no—
231C22⋊C8—72no—
232C22⋊C8—141no—
233C22⋊C8—141no—
234C22⋊C8—141no—
235C22⋊C8—141no—
236C22⋊C8—141no—
237C22⋊C8C22⋊D712yes—
238C22×C7—21no—
239C22×C7—21no—
240C22×C7—21no—
241C22×C7—21no—
242C22×C7—21no—
243C22×C7—21no—
244C22×C7—21no—
245C22×C7—44no—
246C22×C7#39211yescharacteristic
247C22×C7#47212yes—
248C22×C7#47312yes—
249C2×D7—81no—
250C2×D7—81no—
251C2×D7—81no—
252C2×D7—162no—
253C4×C7—21no—
254C4×C7—21no—
255C4×C7—21no—
256C4×C7—81no—
257C4×C7—82no—
258C4×C7—81no—
259C4×C7#45812yes—
260Dic7—81no—
261C24—141no—
262C24#115911yescharacteristic
263C2×Q8—21no—
264C2×Q8—21no—
265C2×Q8—42no—
266C2×D4—141no—
267C2×D4—141no—
268C2×D4—282no—
269C2×D4—281no—
270C2×D4—281no—
271C2×D4—562no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—22no—
275C22×C4—42no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—82no—
279C22×C4—141no—
280C22×C4D4×D711yescharacteristic
281C22×C4D4×D711yescharacteristic
282C22×C4D4×D711yescharacteristic
283C22×C4C2×C22⋊D711yescharacteristic
284C22×C4C2×C22⋊D711yescharacteristic
285C22×C4C2×C22⋊D711yescharacteristic
286SD16—562no—
287M16—282no—
288M16—282no—
289C2×C8—22no—
290C2×C8—42no—
291C2×C8—142no—
292C2×C8—142no—
293C2×C8—282no—
294C2×C8—281no—
295C4⋊C4—22no—
296C4⋊C4—21no—
297C4⋊C4—22no—
298C4⋊C4—21no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—141no—
304C4⋊C4—141no—
305C22⋊C4—41no—
306C22⋊C4—41no—
307C22⋊C4—281no—
308C22⋊C4—281no—
309C2×C7—22no—
310C2×C7—22no—
311C2×C7—41no—
312C2×C7#201811yescharacteristic
313C2×C7#202011yescharacteristic
314C2×C7#202211yescharacteristic
315D7—161no—
316C23—22no—
317C23—22no—
318C23—41no—
319C23—281no—
320C23—282no—
321C23—561no—
322C23—561no—
323C23#650111yescharacteristic
324C23#650211yescharacteristic
325C23#650211yescharacteristic
326Q8—42no—
327Q8—42no—
328D4—282no—
329D4—282no—
330D4—562no—
331D4—562no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—42no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—42no—
342C2×C4—82no—
343C2×C4—82no—
344C2×C4—82no—
345C2×C4—81no—
346C2×C4—281no—
347C2×C4—561no—
348C2×C4#650011yescharacteristic
349C2×C4#650011yescharacteristic
350C2×C4#650311yescharacteristic
351C2×C4#651611yescharacteristic
352C8—42no—
353C8—282no—
354C8—282no—
355C8—561no—
356C7#999411yescharacteristic, complemented, Sylow
357C22—21no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—44no—
365C22—561no—
366C22—561no—
367C22—561no—
368C22—1122no—
369C22#8996412yes—
370C22#8997112yes—
371C22#8999511yescharacteristic
372C4—21no—
373C4—21no—
374C4—21no—
375C4—81no—
376C4—81no—
377C4—82no—
378C4—561no—
379C4#8982512yes—
380C2—22no—
381C2—22no—
382C2—41no—
383C2—1121no—
384C2#1172946011yescharacteristic
385C2#1172982811yescharacteristic
386C2#1172984911yescharacteristic
subgroupG/H
centerC22#89995
commutator, cosocleC23×C4×C7C23
FrattiniC23×C4C22×D7
FittingC2×C7×#484C2
socleC22×C7#392
prime radicalC2×#6493C22
hypercenterC2×#484D7
hypocenterC7#9994
LCSG ▷ C23×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#484

Complements

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

NH ≅ G/Ntype
1. C2×C7×#484C2semidirect
4. #11729445C2semidirect
5. #11729316C2semidirect
7. #11725486C2semidirect
77. C2×#484D7semidirect
356. C7#9994semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#4842
#117401832
C2×#897392
#117294452
#117293162
#117276752
#117254862
#99947