Finite Group Databasebeta

← #13263391

Subgroups — #13263391

Order 1792.

750 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 750 conjugacy classes of subgroups — too many to draw. Only its 95 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
1C7×#2408C211yesmaximal, characteristic
2#11729357C211yesmaximal, characteristic, complemented
3#11729354C211yesmaximal, characteristic, complemented
4#11725428C214yesmaximal, complemented
5C23×C7×D4C411yescharacteristic
6C2×C7×#121—21no—
7C2×C7×#120C412yes—
8C2×C4×C7×D4C2211yescharacteristic
9C7×#394—21no—
10C7×#392—21no—
11C7×#390—21no—
12C7×#380C411yescharacteristic
13C7×#379C2212yes—
14C2×C22⋊(C7⋊C8)—21no—
15#89698—21no—
16#89323—21no—
17#89320—21no—
18#89320C2212yes—
19#89320C2212yes—
20#11557—71nomaximal, Sylow
21C25×C7C2×C412yes—
22C22×C7×D4—22no—
23C22×C7×D4—21no—
24C22×C7×D4—21no—
25C22×C7×D4—41no—
26C22×C7×D4—41no—
27C22×C7×D4—44no—
28C22×C7×D4C2×C411yescharacteristic
29C23×C4×C7C2×C411yescharacteristic
30C23×C4×C7D411yescharacteristic
31C7×#121—44no—
32C7×#120—44no—
33C4×C7×D4—41no—
34C4×C7×D4—41no—
35C2×C7×C4⋊C4—21no—
36C2×C7×C4⋊C4D411yescharacteristic
37C2×C7×C22⋊C4—22no—
38C2×C7×C22⋊C4—22no—
39C2×C7×C22⋊C4—22no—
40C2×C7×C22⋊C4C2×C412yes—
41C2×C7×C22⋊C4D411yescharacteristic
42C2×C7×C22⋊C4C2311yescharacteristic
43C2×C42×C7D411yescharacteristic
44C22×C7⋊C8—41no—
45C7×#95—21no—
46C22⋊(C7⋊C8)—22no—
47C22⋊(C7⋊C8)—22no—
48C22⋊(C7⋊C8)—41no—
49C4⋊(C7⋊C8)—41no—
50#6377—41no—
51#2408D711yescharacteristic
52#1466—71no—
53#1459—71no—
54#1316—74no—
55C24×C7—22no—
56C24×C7—24no—
57C24×C7—42no—
58C24×C7—44no—
59C24×C7—42no—
60C24×C7—82no—
61C24×C7C22⋊C411yescharacteristic
62C24×C7C22×C411yescharacteristic
63C2×C7×D4—24no—
64C2×C7×D4—22no—
65C2×C7×D4—22no—
66C2×C7×D4—41no—
67C2×C7×D4—42no—
68C2×C7×D4—42no—
69C2×C7×D4—44no—
70C2×C7×D4—48no—
71C2×C7×D4—44no—
72C2×C7×D4—44no—
73C2×C7×D4—44no—
74C2×C7×D4—44no—
75C2×C7×D4—41no—
76C2×C7×D4—41no—
77C2×C7×D4—82no—
78C22×C4×C7—21no—
79C22×C4×C7—21no—
80C22×C4×C7—21no—
81C22×C4×C7—21no—
82C22×C4×C7—22no—
83C22×C4×C7—42no—
84C22×C4×C7—42no—
85C22×C4×C7C22⋊C411yescharacteristic
86C22×C4×C7C22⋊C411yescharacteristic
87C22×C4×C7C22⋊C411yescharacteristic
88C22×C4×C7C2×D411yescharacteristic
89C22×C4×C7C2×D411yescharacteristic
90C7×C4⋊C4—41no—
91C7×C4⋊C4—42no—
92C7×C22⋊C4—41no—
93C7×C22⋊C4—41no—
94C7×C22⋊C4—44no—
95C7×C22⋊C4—44no—
96C7×C22⋊C4—42no—
97C7×C22⋊C4—82no—
98C42×C7—21no—
99C42×C7—21no—
100C2×C7⋊C8—42no—
101C2×C7⋊C8—42no—
102C2×C7⋊C8—82no—
103C23×D4Dic711yescharacteristic
104C2×#121—21no—
105C2×#120Dic712yes—
106C2×C4×D4C2×D711yescharacteristic
107#432—141no—
108C2×C22⋊C8—141no—
109#394—21no—
110#392—21no—
111#390—21no—
112#380Dic711yescharacteristic
113#379C2×D712yes—
114#324—141no—
115#323—72no—
116#323—72no—
117#323—141no—
118C23×C7—21no—
119C23×C7—21no—
120C23×C7—21no—
121C23×C7—21no—
122C23×C7—24no—
123C23×C7—41no—
124C23×C7—41no—
125C23×C7—41no—
126C23×C7—41no—
127C23×C7—42no—
128C23×C7—42no—
129C23×C7—42no—
130C23×C7—44no—
131C23×C7—44no—
132C23×C7—42no—
133C23×C7—42no—
134C23×C7—42no—
135C23×C7—84no—
136C23×C7—88no—
137C23×C7—88no—
138C23×C7—84no—
139C23×C7#10012yes—
140C23×C7C2×C22⋊C411yescharacteristic
141C23×C7#13112yes—
142C7×D4—44no—
143C7×D4—44no—
144C7×D4—48no—
145C7×D4—44no—
146C7×D4—88no—
147C2×C4×C7—22no—
148C2×C4×C7—21no—
149C2×C4×C7—21no—
150C2×C4×C7—21no—
151C2×C4×C7—21no—
152C2×C4×C7—21no—
153C2×C4×C7—21no—
154C2×C4×C7—42no—
155C2×C4×C7—42no—
156C2×C4×C7—42no—
157C2×C4×C7—41no—
158C2×C4×C7—41no—
159C2×C4×C7—41no—
160C2×C4×C7—42no—
161C2×C4×C7—42no—
162C2×C4×C7—42no—
163C2×C4×C7—81no—
164C2×C4×C7#9911yescharacteristic
165C2×C4×C7#9911yescharacteristic
166C7⋊C8—82no—
167C7⋊C8—82no—
168C25C2×Dic712yes—
169C22×D4—22no—
170C22×D4—21no—
171C22×D4—21no—
172C22×D4—41no—
173C22×D4—41no—
174C22×D4—44no—
175C22×D4C2×Dic711yescharacteristic
176C23×C4C2×Dic711yescharacteristic
177C23×C4C22⋊D711yescharacteristic
178C22×C8—281no—
179#121—44no—
180#120—44no—
181C4×D4—41no—
182C4×D4—41no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4C22⋊D711yescharacteristic
185C2×C22⋊C4—22no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4C2×Dic712yes—
189C2×C22⋊C4C22⋊D711yescharacteristic
190C2×C22⋊C4C22×D711yescharacteristic
191C2×C42C22⋊D711yescharacteristic
192C4⋊C8—281no—
193C22⋊C8—142no—
194C22⋊C8—142no—
195C22⋊C8—281no—
196#97—281no—
197#95—21no—
198C22×C7—21no—
199C22×C7—21no—
200C22×C7—22no—
201C22×C7—22no—
202C22×C7—42no—
203C22×C7—42no—
204C22×C7—42no—
205C22×C7—44no—
206C22×C7—44no—
207C22×C7—41no—
208C22×C7—41no—
209C22×C7—41no—
210C22×C7—41no—
211C22×C7—44no—
212C22×C7—81no—
213C22×C7—82no—
214C22×C7—82no—
215C22×C7—88no—
216C22×C7—88no—
217C22×C7—82no—
218C22×C7—82no—
219C22×C7#40811yescharacteristic
220C22×C7C2×#9911yescharacteristic
221C22×C7C2×#10011yescharacteristic
222C4×C7—21no—
223C4×C7—41no—
224C4×C7—41no—
225C4×C7—41no—
226C4×C7—41no—
227C4×C7—82no—
228C4×C7AΣL(1,𝔽16)12yes—
229C24—22no—
230C24—24no—
231C24—42no—
232C24—44no—
233C24—42no—
234C24—82no—
235C24C22⋊Dic711yescharacteristic
236C24C22×Dic711yescharacteristic
237C2×D4—24no—
238C2×D4—22no—
239C2×D4—22no—
240C2×D4—41no—
241C2×D4—42no—
242C2×D4—42no—
243C2×D4—44no—
244C2×D4—48no—
245C2×D4—44no—
246C2×D4—44no—
247C2×D4—44no—
248C2×D4—44no—
249C2×D4—41no—
250C2×D4—41no—
251C2×D4—82no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—22no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4C22⋊Dic711yescharacteristic
260C22×C4C22⋊Dic711yescharacteristic
261C22×C4C22⋊Dic711yescharacteristic
262C22×C4C2×C22⋊D711yescharacteristic
263C22×C4C2×C22⋊D711yescharacteristic
264C2×C8—282no—
265C2×C8—282no—
266C2×C8—562no—
267C4⋊C4—41no—
268C4⋊C4—42no—
269C22⋊C4—41no—
270C22⋊C4—41no—
271C22⋊C4—44no—
272C22⋊C4—44no—
273C22⋊C4—42no—
274C22⋊C4—82no—
275C42—21no—
276C42—21no—
277C2×C7—21no—
278C2×C7—21no—
279C2×C7—42no—
280C2×C7—42no—
281C2×C7—84no—
282C2×C7#147111yescharacteristic
283C2×C7C2×AΣL(1,𝔽16)11yescharacteristic
284C2×C7#212211yescharacteristic
285C23—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23—24no—
290C23—41no—
291C23—41no—
292C23—41no—
293C23—41no—
294C23—42no—
295C23—42no—
296C23—42no—
297C23—44no—
298C23—44no—
299C23—42no—
300C23—42no—
301C23—42no—
302C23—84no—
303C23—88no—
304C23—88no—
305C23—84no—
306C23#640712yes—
307C23#651112yes—
308C23C2×C22⋊Dic711yescharacteristic
309D4—44no—
310D4—44no—
311D4—48no—
312D4—44no—
313D4—88no—
314C2×C4—22no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—81no—
331C2×C4#640811yescharacteristic
332C2×C4#640811yescharacteristic
333C8—562no—
334C8—562no—
335C7#1155711yescharacteristic, complemented, Sylow
336C22—21no—
337C22—21no—
338C22—22no—
339C22—22no—
340C22—42no—
341C22—42no—
342C22—42no—
343C22—44no—
344C22—44no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—44no—
350C22—81no—
351C22—82no—
352C22—82no—
353C22—88no—
354C22—88no—
355C22—82no—
356C22—82no—
357C22C2×#640711yescharacteristic
358C22C2×#640811yescharacteristic
359C22#9000611yescharacteristic
360C4—21no—
361C4—41no—
362C4—41no—
363C4—41no—
364C4—41no—
365C4—82no—
366C4#8933112yes—
367C2—21no—
368C2—21no—
369C2—42no—
370C2—42no—
371C2—84no—
372C2C2×#8933111yescharacteristic
373C2#1172943011yescharacteristic
374C2#1172961311yescharacteristic
subgroupG/H
centerC22C2×#6408
commutatorC24×C7C22×C4
FrattiniC2×C22⋊C4C22×D7
Fitting, prime radicalC7×#2408C2
socleC22×C7C2×#99
cosocleC2×C7×C22⋊C4C23
hypercenter#2408D7
hypocenterC7#11557
LCSG ▷ C24×C7 ▷ C22×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2408

Complements

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

NH ≅ G/Ntype
2. #11729357C2semidirect
3. #11729354C2semidirect
4. #11725428C2semidirect
335. C7#11557semidirect

Maximal subgroups

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

subgroupindex
C7×#24082
#117293572
#117293542
#117254282
#115577