Finite Group Databasebeta

← #13269278

Subgroups — #13269278

Order 1792.

976 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 976 conjugacy classes of subgroups — too many to draw. Only its 139 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×#536C211yesmaximal, characteristic
2#11729893C211yesmaximal, characteristic
3#11729891C211yesmaximal, characteristic
4C2×#89334C214yesmaximal, complemented
5C22×C7×#41C2211yescharacteristic
6C23×C7×Q8C411yescharacteristic, complemented
7C7×#536—28no—
8C2×C7×C4⋊Q8—21no—
9C2×C7×#124—21no—
10C2×C7×#123C412yescomplemented
11C2×C7×#123C2212yes—
12C2×C7×C22⋊Q8—21no—
13C2×C7×C22⋊Q8—21no—
14C2×C7×#117C411yescharacteristic, complemented
15C2×#6410—21no—
16C2×#6408—21no—
17C2×#6408C2212yes—
18C2×#6407C2212yes—
19#89983—21no—
20#89979—21no—
21#89334—28no—
22#12692—71nomaximal, Sylow
23C2×C7×#41—24no—
24C2×C7×#41—42no—
25C2×C7×#41—42no—
26C22×C7×Q8—21no—
27C22×C7×Q8—21no—
28C22×C7×Q8—24no—
29C22×C7×Q8—42no—
30C22×C7×Q8—42no—
31C22×C7×Q8D411yescharacteristic
32C22×C7×D4C2×C411yescharacteristic
33C22×C7×D4D411yescharacteristic
34C22×C7×D4C2311yescharacteristic
35C23×C4×C7C2×C411yescharacteristic
36C23×C4×C7C2×C412yes—
37C23×C4×C7D411yescharacteristic
38C23×C4×C7D411yescharacteristic
39C7×C4⋊Q8—28no—
40C7×#124—28no—
41C7×#123—24no—
42C7×#123—24no—
43C7×#123—24no—
44C7×#123D418yes—
45C7×C22⋊Q8—28no—
46C7×C22⋊Q8—28no—
47C7×#117—24no—
48C2×C7×C4⋊C4—21no—
49C2×C7×C4⋊C4—22no—
50C2×C7×C4⋊C4—22no—
51C2×C7×C22⋊C4—22no—
52C2×C7×C22⋊C4—22no—
53C2×C7×C22⋊C4C2×C412yes—
54C2×C42×C7—21no—
55C2×C22⋊Dic7—22no—
56C2×#1136—22no—
57#6410—42no—
58#6408—24no—
59#6408—42no—
60#6407—24no—
61C7⋊#95—42no—
62C22⋊(C7⋊C8)—42no—
63C2×#536D711yescharacteristic
64C2×#352—74no—
65#1785—71no—
66#1782—71no—
67C24×C7C22⋊C411yescharacteristic
68C24×C7C22×C411yescharacteristic
69C24×C7C2×D411yescharacteristic
70C7×#41—416no—
71C2×C7×Q8—24no—
72C2×C7×Q8—24no—
73C2×C7×Q8—22no—
74C2×C7×Q8—22no—
75C2×C7×Q8—41no—
76C2×C7×Q8—416no—
77C2×C7×Q8—42no—
78C2×C7×Q8—42no—
79C2×C7×Q8—42no—
80C2×C7×Q8—42no—
81C2×C7×D4—22no—
82C2×C7×D4—22no—
83C2×C7×D4—22no—
84C2×C7×D4—41no—
85C2×C7×D4—41no—
86C2×C7×D4—41no—
87C2×C7×D4—41no—
88C2×C7×D4—41no—
89C2×C7×D4C2×D414yes—
90C22×C4×C7—22no—
91C22×C4×C7—22no—
92C22×C4×C7—22no—
93C22×C4×C7—22no—
94C22×C4×C7—22no—
95C22×C4×C7—22no—
96C22×C4×C7—22no—
97C22×C4×C7—22no—
98C22×C4×C7—22no—
99C22×C4×C7—22no—
100C22×C4×C7—41no—
101C22×C4×C7—41no—
102C22×C4×C7—42no—
103C22×C4×C7—42no—
104C22×C4×C7C22⋊C411yescharacteristic
105C22×C4×C7C22⋊C411yescharacteristic
106C22×C4×C7C22⋊C411yescharacteristic
107C22×C4×C7C22⋊C414yes—
108C22×C4×C7C2×D411yescharacteristic
109C22×Dic7—42no—
110C7×C4⋊C4—24no—
111C7×C4⋊C4—28no—
112C7×C4⋊C4—28no—
113C7×C22⋊C4—28no—
114C7×C22⋊C4—22no—
115C7×C22⋊C4—28no—
116C7×C22⋊C4C22⋊C414yes—
117C42×C7—24no—
118C22⋊Dic7—42no—
119C22⋊Dic7—42no—
120#1136—42no—
121#1136—42no—
122C2×C7⋊C8—42no—
123C22×#41C2×D711yescharacteristic
124C23×Q8Dic711yescharacteristic, complemented
125#536—28no—
126C2×C4⋊Q8—21no—
127C2×#124—21no—
128C2×#123Dic712yescomplemented
129C2×#123C2×D712yes—
130C2×C22⋊Q8—21no—
131C2×C22⋊Q8—21no—
132C2×#117Dic711yescharacteristic, complemented
133C2×#101—141no—
134C2×#100—72no—
135C2×#99—72no—
136C2×#99—141no—
137#407—141no—
138#381—141no—
139#352—148no—
140C23×C7—21no—
141C23×C7—21no—
142C23×C7—21no—
143C23×C7—21no—
144C23×C7—21no—
145C23×C7—21no—
146C23×C7—21no—
147C23×C7—42no—
148C23×C7—42no—
149C23×C7—42no—
150C23×C7C2×C22⋊C411yescharacteristic
151C23×C7C2×C22⋊C412yes—
152C23×C7#12012yes—
153C7×Q8—44no—
154C7×Q8—48no—
155C7×Q8—48no—
156C7×D4—44no—
157C7×D4—44no—
158C7×D4—44no—
159C2×C4×C7—28no—
160C2×C4×C7—21no—
161C2×C4×C7—21no—
162C2×C4×C7—21no—
163C2×C4×C7—21no—
164C2×C4×C7—21no—
165C2×C4×C7—21no—
166C2×C4×C7—24no—
167C2×C4×C7—24no—
168C2×C4×C7—21no—
169C2×C4×C7—21no—
170C2×C4×C7—21no—
171C2×C4×C7—44no—
172C2×C4×C7—44no—
173C2×C4×C7—44no—
174C2×C4×C7—44no—
175C2×C4×C7—42no—
176C2×C4×C7—42no—
177C2×C4×C7—44no—
178C2×C4×C7#9912yes—
179C2×C4×C7#9912yes—
180C2×C4×C7#12012yes—
181C2×Dic7—82no—
182C2×Dic7—84no—
183C7⋊C8—82no—
184C2×#41—24no—
185C2×#41—42no—
186C2×#41—42no—
187C22×Q8—21no—
188C22×Q8—24no—
189C22×Q8—21no—
190C22×Q8—42no—
191C22×Q8—42no—
192C22×Q8C22⋊D711yescharacteristic
193C22×D4C2×Dic711yescharacteristic
194C22×D4C22⋊D711yescharacteristic
195C22×D4C22×D711yescharacteristic
196C23×C4C2×Dic711yescharacteristic
197C23×C4C2×Dic712yes—
198C23×C4C22⋊D711yescharacteristic
199C23×C4C22⋊D711yescharacteristic
200C2×M16—142no—
201C4⋊Q8—28no—
202#124—28no—
203#123—24no—
204#123—24no—
205#123—24no—
206#123C22⋊D718yes—
207C22⋊Q8—28no—
208C22⋊Q8—28no—
209#117—24no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—142no—
216C2×C22⋊C4C2×Dic712yes—
217C2×C42—21no—
218#101—282no—
219#100—144no—
220#99—144no—
221#99—282no—
222C22⋊C8—282no—
223#95—282no—
224C22×C7—22no—
225C22×C7—22no—
226C22×C7—22no—
227C22×C7—42no—
228C22×C7—42no—
229C22×C7—42no—
230C22×C7—42no—
231C22×C7—42no—
232C22×C7—42no—
233C22×C7—42no—
234C22×C7—41no—
235C22×C7—81no—
236C22×C7—81no—
237C22×C7#37911yescharacteristic
238C22×C7C2×#9911yescharacteristic
239C22×C7C2×#9911yescharacteristic
240C4×C7—22no—
241C4×C7—22no—
242C4×C7—22no—
243C4×C7—44no—
244C4×C7—41no—
245C4×C7—44no—
246Dic7—162no—
247C24C22⋊Dic711yescharacteristic
248C24C22×Dic711yescharacteristic
249C24C2×C22⋊D711yescharacteristic
250#41—416no—
251C2×Q8—24no—
252C2×Q8—24no—
253C2×Q8—22no—
254C2×Q8—22no—
255C2×Q8—41no—
256C2×Q8—416no—
257C2×Q8—42no—
258C2×Q8—42no—
259C2×Q8—42no—
260C2×Q8—42no—
261C2×D4—22no—
262C2×D4—22no—
263C2×D4—22no—
264C2×D4—41no—
265C2×D4—41no—
266C2×D4—41no—
267C2×D4—41no—
268C2×D4—41no—
269C2×D4C2×C22⋊D714yes—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—282no—
285C22×C4C22⋊Dic711yescharacteristic
286C22×C4C22⋊Dic711yescharacteristic
287C22×C4C22⋊Dic714yes—
288C22×C4C22⋊Dic711yescharacteristic
289C22×C4C2×C22⋊D711yescharacteristic
290M16—282no—
291M16—282no—
292C2×C8—282no—
293C4⋊C4—24no—
294C4⋊C4—28no—
295C4⋊C4—28no—
296C22⋊C4—28no—
297C22⋊C4—22no—
298C22⋊C4—28no—
299C22⋊C4—282no—
300C22⋊C4—282no—
301C22⋊C4C22⋊Dic714yes—
302C42—24no—
303C2×C7—21no—
304C2×C7—21no—
305C2×C7—42no—
306C2×C7—42no—
307C2×C7—81no—
308C2×C7#178111yescharacteristic
309C2×C7#212312yes—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—42no—
318C23—42no—
319C23—42no—
320C23C2×C22⋊Dic711yescharacteristic
321C23C2×C22⋊Dic712yes—
322C23#651612yes—
323Q8—44no—
324Q8—48no—
325Q8—48no—
326D4—44no—
327D4—44no—
328D4—44no—
329C2×C4—28no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—24no—
339C2×C4—24no—
340C2×C4—21no—
341C2×C4—44no—
342C2×C4—44no—
343C2×C4—44no—
344C2×C4—44no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—44no—
348C2×C4—562no—
349C2×C4—564no—
350C2×C4#640812yes—
351C2×C4#640812yes—
352C2×C4#651612yes—
353C8—562no—
354C7#1269211yescharacteristic, complemented, Sylow
355C22—22no—
356C22—22no—
357C22—22no—
358C22—42no—
359C22—42no—
360C22—42no—
361C22—42no—
362C22—42no—
363C22—42no—
364C22—42no—
365C22—41no—
366C22—81no—
367C22—81no—
368C22C2×#640811yescharacteristic
369C22C2×#640811yescharacteristic
370C22#9001911yescharacteristic
371C4—22no—
372C4—22no—
373C4—22no—
374C4—44no—
375C4—41no—
376C4—44no—
377C4—1122no—
378C2—21no—
379C2—21no—
380C2—42no—
381C2—42no—
382C2—81no—
383C2#1172962412yes—
384C2#1172988411yescharacteristic
subgroupG/H
centerC22C2×#6408
commutatorC24×C7C22×C4
FrattiniC22×D4C22×D7
FittingC2×C7×#536C2
socleC22×C7C2×#99
prime radicalC22×C7×#41C22
cosocleC22×C7×D4C23
hypercenterC2×#536D7
hypocenterC7#12692
LCSG ▷ C24×C7 ▷ C22×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ C2×#536

Complements

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

NH ≅ G/Ntype
4. C2×#89334C2semidirect
6. C23×C7×Q8C4semidirect
10. C2×C7×#123C4semidirect
14. C2×C7×#117C4semidirect
124. C23×Q8Dic7semidirect
128. C2×#123Dic7semidirect
132. C2×#117Dic7semidirect
354. C7#12692semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#5362
#117298932
#117298912
C2×#893342
#126927