Finite Group Databasebeta

← #13245144

Subgroups — #13245144

Order 1792.

1238 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 1238 conjugacy classes of subgroups — too many to draw. Only its 171 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
1C22×C7×#123C211yesmaximal, characteristic
2#11729880C212yesmaximal, complemented
3C2×#89323C212yesmaximal, complemented
4C2×#89320C212yesmaximal, complemented
5C23×C7×D4C411yescharacteristic
6C24×C4×C7C2211yescharacteristic
7C2×C7×#123—24no—
8C2×C7×#123—24no—
9C2×C7×#123—24no—
10C22×C7×C4⋊C4C411yescharacteristic
11C22×C7×C4⋊C4C2211yescharacteristic
12C22×C7×C22⋊C4C411yescharacteristic
13C22×C7×C22⋊C4C411yescharacteristic
14C22×C7×C22⋊C4C2211yescharacteristic
15C2×C22⋊(C7⋊C8)—24no—
16C2×C22⋊(C7⋊C8)C2214yes—
17#89323—24no—
18#89320—24no—
19#8304—71nomaximal, Sylow
20C25×C7C2×C411yescharacteristic
21C25×C7C2×C411yescharacteristic
22C22×C7×D4—22no—
23C22×C7×D4—22no—
24C22×C7×D4—22no—
25C22×C7×D4—44no—
26C23×C4×C7—22no—
27C23×C4×C7—22no—
28C23×C4×C7—44no—
29C23×C4×C7C2×C411yescharacteristic
30C23×C4×C7C2×C411yescharacteristic
31C23×C4×C7C2×C411yescharacteristic
32C23×C4×C7C2×C411yescharacteristic
33C23×C4×C7D414yes—
34C23×C4×C7C2311yescharacteristic
35C7×#123—232no—
36C2×C7×C4⋊C4—22no—
37C2×C7×C4⋊C4—22no—
38C2×C7×C4⋊C4—22no—
39C2×C7×C4⋊C4—22no—
40C2×C7×C4⋊C4—22no—
41C2×C7×C4⋊C4D414yes—
42C2×C7×C22⋊C4—22no—
43C2×C7×C22⋊C4—22no—
44C2×C7×C22⋊C4—22no—
45C2×C7×C22⋊C4—22no—
46C2×C7×C22⋊C4—22no—
47C2×C7×C22⋊C4—22no—
48C2×C7×C22⋊C4—22no—
49C2×C7×C22⋊C4—22no—
50C2×C7×C22⋊C4D414yes—
51C22×C7⋊C8—24no—
52C22⋊(C7⋊C8)—24no—
53C22⋊(C7⋊C8)—24no—
54C22⋊(C7⋊C8)—48no—
55C22×#123D711yescharacteristic
56#1779—72no—
57C2×#324—72no—
58C2×#323—72no—
59C24×C7—24no—
60C24×C7—21no—
61C24×C7—21no—
62C24×C7—21no—
63C24×C7—21no—
64C24×C7—24no—
65C24×C7—24no—
66C24×C7—42no—
67C24×C7—42no—
68C24×C7—42no—
69C24×C7C22⋊C412yes—
70C24×C7C22⋊C412yes—
71C24×C7C22×C411yescharacteristic
72C2×C7×D4—28no—
73C2×C7×D4—48no—
74C2×C7×D4—44no—
75C2×C7×D4—48no—
76C2×C7×D4—44no—
77C22×C4×C7—28no—
78C22×C4×C7—21no—
79C22×C4×C7—21no—
80C22×C4×C7—21no—
81C22×C4×C7—21no—
82C22×C4×C7—21no—
83C22×C4×C7—21no—
84C22×C4×C7—21no—
85C22×C4×C7—21no—
86C22×C4×C7—24no—
87C22×C4×C7—28no—
88C22×C4×C7—42no—
89C22×C4×C7—44no—
90C22×C4×C7—42no—
91C22×C4×C7—48no—
92C22×C4×C7—48no—
93C22×C4×C7—42no—
94C22×C4×C7—42no—
95C22×C4×C7C22⋊C412yes—
96C22×C4×C7C22⋊C412yes—
97C22×C4×C7C22⋊C412yes—
98C22×C4×C7C22⋊C412yes—
99C22×C4×C7C2×D412yes—
100C22×C4×C7C2×D412yes—
101C22×C4×C7C2×D412yes—
102C7×C4⋊C4—28no—
103C7×C4⋊C4—28no—
104C7×C22⋊C4—28no—
105C7×C22⋊C4—28no—
106C7×C22⋊C4—28no—
107C2×C7⋊C8—48no—
108C2×C7⋊C8—44no—
109C23×D4Dic711yescharacteristic
110C24×C4C2×D711yescharacteristic
111C2×#123—24no—
112C2×#123—24no—
113C2×#123—24no—
114C22×C4⋊C4Dic711yescharacteristic
115C22×C4⋊C4C2×D711yescharacteristic
116C22×C22⋊C4Dic711yescharacteristic
117C22×C22⋊C4Dic711yescharacteristic
118C22×C22⋊C4C2×D711yescharacteristic
119C2×C22⋊C8—74no—
120C2×C22⋊C8—144no—
121#324—144no—
122#323—144no—
123C23×C7—22no—
124C23×C7—22no—
125C23×C7—24no—
126C23×C7—24no—
127C23×C7—22no—
128C23×C7—24no—
129C23×C7—22no—
130C23×C7—24no—
131C23×C7—22no—
132C23×C7—24no—
133C23×C7—416no—
134C23×C7—42no—
135C23×C7—42no—
136C23×C7—42no—
137C23×C7—42no—
138C23×C7—42no—
139C23×C7—42no—
140C23×C7—42no—
141C23×C7—42no—
142C23×C7—42no—
143C23×C7—42no—
144C23×C7—42no—
145C23×C7—42no—
146C23×C7—88no—
147C23×C7#9914yes—
148C23×C7C2×C22⋊C411yescharacteristic
149C23×C7C2×C22⋊C411yescharacteristic
150C23×C7C2×C22⋊C411yescharacteristic
151C23×C7#13114yes—
152C7×D4—416no—
153C2×C4×C7—24no—
154C2×C4×C7—22no—
155C2×C4×C7—22no—
156C2×C4×C7—22no—
157C2×C4×C7—22no—
158C2×C4×C7—24no—
159C2×C4×C7—24no—
160C2×C4×C7—42no—
161C2×C4×C7—42no—
162C2×C4×C7—416no—
163C2×C4×C7—42no—
164C2×C4×C7—42no—
165C2×C4×C7—42no—
166C2×C4×C7—42no—
167C2×C4×C7—42no—
168C2×C4×C7—42no—
169C2×C4×C7—42no—
170C2×C4×C7—42no—
171C2×C4×C7—42no—
172C2×C4×C7—42no—
173C2×C4×C7#12014yes—
174C7⋊C8—84no—
175C25C2×Dic711yescharacteristic
176C25C2×Dic711yescharacteristic
177C22×D4—22no—
178C22×D4—22no—
179C22×D4—22no—
180C22×D4—44no—
181C23×C4—22no—
182C23×C4—22no—
183C23×C4—44no—
184C23×C4C2×Dic711yescharacteristic
185C23×C4C2×Dic711yescharacteristic
186C23×C4C2×Dic711yescharacteristic
187C23×C4C2×Dic711yescharacteristic
188C23×C4C22⋊D714yes—
189C23×C4C22×D711yescharacteristic
190C22×C8—144no—
191#123—232no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4C22⋊D714yes—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4C22⋊D714yes—
207C22⋊C8—144no—
208C22⋊C8—144no—
209C22⋊C8—288no—
210C22×C7—28no—
211C22×C7—21no—
212C22×C7—21no—
213C22×C7—21no—
214C22×C7—21no—
215C22×C7—28no—
216C22×C7—24no—
217C22×C7—48no—
218C22×C7—48no—
219C22×C7—44no—
220C22×C7—42no—
221C22×C7—42no—
222C22×C7—42no—
223C22×C7—42no—
224C22×C7—42no—
225C22×C7—84no—
226C22×C7—84no—
227C22×C7—84no—
228C22×C7#37911yescharacteristic
229C22×C7#40812yes—
230C22×C7#40812yes—
231C22×C7#40812yes—
232C22×C7C2×#9912yes—
233C22×C7#41312yes—
234C4×C7—24no—
235C4×C7—42no—
236C4×C7—42no—
237C4×C7—42no—
238C4×C7—42no—
239C24—24no—
240C24—21no—
241C24—21no—
242C24—24no—
243C24—21no—
244C24—21no—
245C24—24no—
246C24—42no—
247C24—42no—
248C24—42no—
249C24C22⋊Dic712yes—
250C24C22⋊Dic712yes—
251C24C22×Dic711yescharacteristic
252C2×D4—28no—
253C2×D4—48no—
254C2×D4—44no—
255C2×D4—48no—
256C2×D4—44no—
257C22×C4—28no—
258C22×C4—24no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—28no—
268C22×C4—42no—
269C22×C4—44no—
270C22×C4—42no—
271C22×C4—48no—
272C22×C4—48no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4C22⋊Dic712yes—
276C22×C4C22⋊Dic712yes—
277C22×C4C22⋊Dic712yes—
278C22×C4C22⋊Dic712yes—
279C22×C4C2×C22⋊D712yes—
280C22×C4C2×C22⋊D712yes—
281C22×C4C2×C22⋊D712yes—
282C2×C8—288no—
283C2×C8—284no—
284C4⋊C4—28no—
285C4⋊C4—28no—
286C22⋊C4—28no—
287C22⋊C4—28no—
288C22⋊C4—28no—
289C2×C7—22no—
290C2×C7—22no—
291C2×C7—44no—
292C2×C7—82no—
293C2×C7#146814yes—
294C2×C7#178011yescharacteristic
295C2×C7#178111yescharacteristic
296C2×C7#178511yescharacteristic
297C23—22no—
298C23—22no—
299C23—24no—
300C23—24no—
301C23—22no—
302C23—24no—
303C23—24no—
304C23—22no—
305C23—22no—
306C23—24no—
307C23—416no—
308C23—42no—
309C23—42no—
310C23—42no—
311C23—42no—
312C23—42no—
313C23—42no—
314C23—42no—
315C23—42no—
316C23—42no—
317C23—42no—
318C23—42no—
319C23—42no—
320C23—88no—
321C23#640814yes—
322C23#651114yes—
323C23C2×C22⋊Dic711yescharacteristic
324C23C2×C22⋊Dic711yescharacteristic
325C23C2×C22⋊Dic711yescharacteristic
326D4—416no—
327C2×C4—24no—
328C2×C4—24no—
329C2×C4—24no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—416no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4#651614yes—
348C8—564no—
349C7#830411yescharacteristic, complemented, Sylow
350C22—28no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—28no—
356C22—24no—
357C22—48no—
358C22—48no—
359C22—44no—
360C22—42no—
361C22—42no—
362C22—42no—
363C22—42no—
364C22—42no—
365C22—84no—
366C22—84no—
367C22—84no—
368C22C2×#640812yes—
369C22#9000612yes—
370C22#9000612yes—
371C22#9000612yes—
372C22#9000912yes—
373C22#9001911yescharacteristic
374C4—24no—
375C4—42no—
376C4—42no—
377C4—42no—
378C4—42no—
379C2—22no—
380C2—22no—
381C2—44no—
382C2—82no—
383C2#1172945614yes—
384C2#1172988411yescharacteristic
385C2#1172989011yescharacteristic
386C2#1172989111yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic7
commutatorC24×C7C22×C4
FrattiniC23×C4C22×D7
FittingC22×C7×#123C2
socleC23×C7C2×C22⋊C4
prime radicalC23×C7×D4C4
cosocleC23×C4×C7C23
hypercenterC22×#123D7
hypocenterC7#8304
LCSG ▷ C24×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C24×C4 ◁ C22×#123

Complements

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

NH ≅ G/Ntype
2. #11729880C2semidirect
3. C2×#89323C2semidirect
4. C2×#89320C2semidirect
349. C7#8304semidirect

Maximal subgroups

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

subgroupindex
C22×C7×#1232
#117298802
C2×#893232
C2×#893202
#83047