Finite Group Databasebeta

← #13264842

Subgroups — #13264842

Order 1792.

764 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 764 conjugacy classes of subgroups — too many to draw. Only its 107 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#11729371C211yesmaximal, characteristic, complemented
3#11729354C211yesmaximal, characteristic, complemented
4#11725453C214yesmaximal, complemented
5C23×C7×D4C411yescharacteristic
6C2×C7×#121—21no—
7C2×C7×#120—21no—
8C2×C4×C7×D4C2211yescharacteristic
9C7×#394C2212yes—
10C7×#392C412yes—
11C7×#390—21no—
12C7×#380C411yescharacteristic
13C7×#379—21no—
14C2×#6407—21no—
15C2×C22⋊(C7⋊C8)—21no—
16#89696—21no—
17#89325C2212yescomplemented
18#89323C2212yescomplemented
19#89320—21no—
20#11880—71nomaximal, Sylow
21C25×C7—21no—
22C22×C7×D4—22no—
23C22×C7×D4—21no—
24C22×C7×D4—41no—
25C22×C7×D4—41no—
26C22×C7×D4—44no—
27C22×C7×D4C2×C412yes—
28C22×C7×D4D411yescharacteristic
29C23×C4×C7C2×C411yescharacteristic
30C23×C4×C7D411yescharacteristic
31C7×#121—44no—
32C7×#120—42no—
33C7×#120—42no—
34C4×C7×D4—41no—
35C4×C7×D4—41no—
36C2×C7×C4⋊C4—21no—
37C2×C7×C4⋊C4C2311yescharacteristic
38C2×C7×C22⋊C4—21no—
39C2×C7×C22⋊C4—22no—
40C2×C7×C22⋊C4—22no—
41C2×C7×C22⋊C4—22no—
42C2×C7×C22⋊C4C2×C411yescharacteristic
43C2×C7×C22⋊C4D411yescharacteristic
44C2×C42×C7D411yescharacteristic
45C2×#1136—22no—
46C22×C7⋊C8—41no—
47C7×#95C2×C412yes—
48#6407—42no—
49C22⋊(C7⋊C8)—22no—
50C22⋊(C7⋊C8)—41no—
51C4⋊(C7⋊C8)—41no—
52C4⋊(C7⋊C8)—41no—
53#2408D711yescharacteristic
54#1800—71no—
55#1459—71no—
56#1345—74no—
57C24×C7—22no—
58C24×C7—42no—
59C24×C7—44no—
60C24×C7—42no—
61C24×C7—42no—
62C24×C7—82no—
63C24×C7C22⋊C411yescharacteristic
64C24×C7C22⋊C411yescharacteristic
65C2×C7×D4—24no—
66C2×C7×D4—41no—
67C2×C7×D4—41no—
68C2×C7×D4—42no—
69C2×C7×D4—42no—
70C2×C7×D4—48no—
71C2×C7×D4—44no—
72C2×C7×D4—44no—
73C2×C7×D4—44no—
74C2×C7×D4—44no—
75C2×C7×D4—44no—
76C2×C7×D4—41no—
77C2×C7×D4—41no—
78C2×C7×D4—82no—
79C2×C7×D4D812yes—
80C2×C7×D4SD1612yes—
81C22×C4×C7—21no—
82C22×C4×C7—21no—
83C22×C4×C7—21no—
84C22×C4×C7—21no—
85C22×C4×C7—22no—
86C22×C4×C7—42no—
87C22×C4×C7—42no—
88C22×C4×C7C22⋊C411yescharacteristic
89C22×C4×C7C22⋊C411yescharacteristic
90C22×C4×C7C22×C411yescharacteristic
91C22×C4×C7C2×D411yescharacteristic
92C22×C4×C7C2×D411yescharacteristic
93C7×C4⋊C4—41no—
94C7×C4⋊C4—42no—
95C7×C22⋊C4—41no—
96C7×C22⋊C4—41no—
97C7×C22⋊C4—44no—
98C7×C22⋊C4—42no—
99C7×C22⋊C4—42no—
100C7×C22⋊C4—42no—
101C7×C22⋊C4—82no—
102C42×C7—21no—
103C42×C7—21no—
104#1136—42no—
105#1136—42no—
106C2×C7⋊C8—42no—
107C2×C7⋊C8—42no—
108C2×C7⋊C8—82no—
109C23×D4Dic711yescharacteristic
110C2×#121—21no—
111C2×#120—21no—
112C2×C4×D4C2×D711yescharacteristic
113#423—141no—
114C2×#100—141no—
115C2×C22⋊C8—141no—
116#394C2×D712yes—
117#392Dic712yes—
118#390—21no—
119#380Dic711yescharacteristic
120#379—21no—
121#343—72no—
122#324—72no—
123#323—141no—
124C23×C7—21no—
125C23×C7—21no—
126C23×C7—21no—
127C23×C7—21no—
128C23×C7—24no—
129C23×C7—41no—
130C23×C7—41no—
131C23×C7—41no—
132C23×C7—41no—
133C23×C7—42no—
134C23×C7—42no—
135C23×C7—42no—
136C23×C7—48no—
137C23×C7—42no—
138C23×C7—42no—
139C23×C7—42no—
140C23×C7—84no—
141C23×C7—88no—
142C23×C7—88no—
143C23×C7—84no—
144C23×C7#10212yes—
145C23×C7#10212yes—
146C23×C7C2×C22⋊C411yescharacteristic
147C7×D4—42no—
148C7×D4—42no—
149C7×D4—44no—
150C7×D4—44no—
151C7×D4—44no—
152C7×D4—44no—
153C7×D4—88no—
154C2×C4×C7—21no—
155C2×C4×C7—21no—
156C2×C4×C7—22no—
157C2×C4×C7—21no—
158C2×C4×C7—21no—
159C2×C4×C7—21no—
160C2×C4×C7—42no—
161C2×C4×C7—42no—
162C2×C4×C7—42no—
163C2×C4×C7—41no—
164C2×C4×C7—41no—
165C2×C4×C7—41no—
166C2×C4×C7—42no—
167C2×C4×C7—42no—
168C2×C4×C7—42no—
169C2×C4×C7—81no—
170C2×C4×C7#9911yescharacteristic
171C2×C4×C7#9911yescharacteristic
172C2×C4×C7C2×D811yescharacteristic
173C2×C4×C7C2×SD1611yescharacteristic
174C7⋊C8—82no—
175C7⋊C8—82no—
176C25—21no—
177C22×D4—22no—
178C22×D4—21no—
179C22×D4—41no—
180C22×D4—41no—
181C22×D4—44no—
182C22×D4C2×Dic712yes—
183C22×D4C22⋊D711yescharacteristic
184C23×C4C2×Dic711yescharacteristic
185C23×C4C22⋊D711yescharacteristic
186C2×M16—142no—
187C22×C8—281no—
188#121—44no—
189#120—42no—
190#120—42no—
191C4×D4—41no—
192C4×D4—41no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4C22×D711yescharacteristic
195C2×C22⋊C4—21no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4C2×Dic711yescharacteristic
200C2×C22⋊C4C22⋊D711yescharacteristic
201C2×C42C22⋊D711yescharacteristic
202C4⋊C8—281no—
203C4⋊C8—281no—
204#100—282no—
205C22⋊C8—142no—
206C22⋊C8—281no—
207#95C2×Dic712yes—
208C22×C7—21no—
209C22×C7—21no—
210C22×C7—22no—
211C22×C7—22no—
212C22×C7—42no—
213C22×C7—42no—
214C22×C7—42no—
215C22×C7—44no—
216C22×C7—44no—
217C22×C7—41no—
218C22×C7—41no—
219C22×C7—41no—
220C22×C7—41no—
221C22×C7—44no—
222C22×C7—81no—
223C22×C7—82no—
224C22×C7—82no—
225C22×C7—816no—
226C22×C7—82no—
227C22×C7—82no—
228C22×C7C2×#9911yescharacteristic
229C22×C7C2×#10211yescharacteristic
230C22×C7#42111yescharacteristic
231C4×C7—21no—
232C4×C7—21no—
233C4×C7—21no—
234C4×C7—41no—
235C4×C7—41no—
236C4×C7—41no—
237C4×C7—82no—
238C4×C7AΣL(1,𝔽16)12yes—
239C24—22no—
240C24—42no—
241C24—44no—
242C24—42no—
243C24—42no—
244C24—82no—
245C24C22⋊Dic711yescharacteristic
246C24C22⋊Dic711yescharacteristic
247C2×D4—24no—
248C2×D4—41no—
249C2×D4—41no—
250C2×D4—42no—
251C2×D4—42no—
252C2×D4—48no—
253C2×D4—44no—
254C2×D4—44no—
255C2×D4—44no—
256C2×D4—44no—
257C2×D4—44no—
258C2×D4—41no—
259C2×D4—41no—
260C2×D4—82no—
261C2×D4#114112yes—
262C2×D4#114212yes—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—22no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4C22⋊Dic711yescharacteristic
271C22×C4C22⋊Dic711yescharacteristic
272C22×C4C22×Dic711yescharacteristic
273C22×C4C2×C22⋊D711yescharacteristic
274C22×C4C2×C22⋊D711yescharacteristic
275M16—282no—
276M16—282no—
277C2×C8—282no—
278C2×C8—282no—
279C2×C8—562no—
280C4⋊C4—41no—
281C4⋊C4—42no—
282C22⋊C4—41no—
283C22⋊C4—41no—
284C22⋊C4—44no—
285C22⋊C4—42no—
286C22⋊C4—42no—
287C22⋊C4—42no—
288C22⋊C4—82no—
289C42—21no—
290C42—21no—
291C2×C7—21no—
292C2×C7—21no—
293C2×C7—42no—
294C2×C7—42no—
295C2×C7—84no—
296C2×C7#151911yescharacteristic
297C2×C7C2×AΣL(1,𝔽16)11yescharacteristic
298C2×C7#213411yescharacteristic
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—24no—
304C23—41no—
305C23—41no—
306C23—41no—
307C23—41no—
308C23—42no—
309C23—42no—
310C23—42no—
311C23—48no—
312C23—42no—
313C23—42no—
314C23—42no—
315C23—84no—
316C23—88no—
317C23—88no—
318C23—84no—
319C23#640612yes—
320C23#640612yes—
321C23C2×C22⋊Dic711yescharacteristic
322D4—42no—
323D4—42no—
324D4—44no—
325D4—44no—
326D4—44no—
327D4—44no—
328D4—88no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—22no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—81no—
345C2×C4#640811yescharacteristic
346C2×C4#640811yescharacteristic
347C2×C4C2×#114111yescharacteristic
348C2×C4C2×#114211yescharacteristic
349C8—562no—
350C8—562no—
351C7#1188011yescharacteristic, complemented, Sylow
352C22—21no—
353C22—21no—
354C22—22no—
355C22—22no—
356C22—42no—
357C22—42no—
358C22—42no—
359C22—44no—
360C22—44no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—44no—
366C22—81no—
367C22—82no—
368C22—82no—
369C22—816no—
370C22—82no—
371C22—82no—
372C22C2×#640611yescharacteristic
373C22C2×#640811yescharacteristic
374C22#9000811yescharacteristic
375C4—21no—
376C4—21no—
377C4—21no—
378C4—41no—
379C4—41no—
380C4—41no—
381C4—82no—
382C4#8933112yes—
383C2—21no—
384C2—21no—
385C2—42no—
386C2—42no—
387C2—84no—
388C2C2×#8933111yescharacteristic
389C2#1172942911yescharacteristic
390C2#1172961611yescharacteristic
subgroupG/H
centerC22C2×#6408
commutatorC22×C4×C7C22×C4
FrattiniC2×C4⋊C4C22×D7
Fitting, prime radicalC7×#2408C2
socleC22×C7C2×#99
cosocleC2×C7×C4⋊C4C23
hypercenter#2408D7
hypocenterC7#11880
LCSG ▷ C22×C4×C7 ▷ C23×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. #11729371C2semidirect
3. #11729354C2semidirect
4. #11725453C2semidirect
17. #89325C22semidirect
18. #89323C22semidirect
351. C7#11880semidirect

Maximal subgroups

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

subgroupindex
C7×#24082
#117293712
#117293542
#117254532
#118807