Finite Group Databasebeta

← #13264366

Subgroups — #13264366

Order 1792.

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

normal subgroups only

This group has 652 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×#2541C211yesmaximal, characteristic
2C4×#6408C211yesmaximal, characteristic, complemented
3#11729356C211yesmaximal, characteristic, complemented
4#11725451C214yesmaximal, complemented
5C2×C7×#127—21no—
6C2×C7×#127C411yescharacteristic, complemented
7C2×C7×#121—21no—
8C2×C7×#121—21no—
9C2×C4×C7×D4C2211yescharacteristic
10C7×#392C412yescomplemented
11C7×#392C2212yes—
12C7×#390—21no—
13C4×C7×C4⋊C4C411yescharacteristic, complemented
14C2×#6408—21no—
15C4×C22⋊Dic7—21no—
16#89979—21no—
17#89323—21no—
18#89321C2212yes—
19#89320C2212yes—
20#11776—71nomaximal, Sylow
21C22×C7×D4—22no—
22C22×C7×D4—22no—
23C22×C7×D4—21no—
24C22×C7×D4—21no—
25C22×C7×D4C2×C412yes—
26C22×C7×D4D411yescharacteristic
27C23×C4×C7D411yescharacteristic
28C23×C4×C7D411yescharacteristic
29C7×#127—41no—
30C7×#127—41no—
31C7×#127—44no—
32C7×#121—44no—
33C7×#121—44no—
34C4×C7×D4—41no—
35C4×C7×D4—41no—
36C2×C7×C4⋊C4—21no—
37C2×C7×C4⋊C4D411yescharacteristic
38C2×C7×C22⋊C4—22no—
39C2×C7×C22⋊C4—22no—
40C2×C7×C22⋊C4C2×C411yescharacteristic
41C2×C7×C22⋊C4C2311yescharacteristic
42C2×C42×C7—21no—
43C2×C42×C7C2×C411yescharacteristic
44C2×C22⋊Dic7—22no—
45C2×C4×Dic7—41no—
46C2×#1136—41no—
47C7×#95C2×C412yes—
48#6408—24no—
49C7⋊#95—41no—
50C22⋊(C7⋊C8)—22no—
51C22⋊(C7⋊C8)—41no—
52#2541D711yescharacteristic
53C4×#99—71no—
54#1465—71no—
55#1343—74no—
56C24×C7—22no—
57C24×C7C22⋊C411yescharacteristic
58C24×C7C2×D411yescharacteristic
59C2×C7×D4—22no—
60C2×C7×D4—22no—
61C2×C7×D4—41no—
62C2×C7×D4—42no—
63C2×C7×D4—42no—
64C2×C7×D4—44no—
65C2×C7×D4—44no—
66C2×C7×D4—42no—
67C2×C7×D4—42no—
68C2×C7×D4—42no—
69C2×C7×D4—44no—
70C2×C7×D4—44no—
71C2×C7×D4—41no—
72C2×C7×D4—41no—
73C2×C7×D4—42no—
74C2×C7×D4—82no—
75C2×C7×D4—82no—
76C2×C7×D4—82no—
77C22×C4×C7—22no—
78C22×C4×C7—21no—
79C22×C4×C7—21no—
80C22×C4×C7—21no—
81C22×C4×C7—21no—
82C22×C4×C7—42no—
83C22×C4×C7—42no—
84C22×C4×C7C22⋊C411yescharacteristic
85C22×C4×C7C22⋊C411yescharacteristic
86C22×C4×C7C22⋊C411yescharacteristic
87C22×C4×C7C22×C411yescharacteristic
88C22×C4×C7C2×D411yescharacteristic
89C22×Dic7—42no—
90C7×C4⋊C4—24no—
91C7×C4⋊C4—41no—
92C7×C22⋊C4—41no—
93C7×C22⋊C4—41no—
94C7×C22⋊C4—44no—
95C7×C22⋊C4—44no—
96C42×C7—21no—
97C42×C7—21no—
98C42×C7—42no—
99C22⋊Dic7—24no—
100C22⋊Dic7—42no—
101C4×Dic7—82no—
102#1136—82no—
103C2×C7⋊C8—42no—
104C2×#127—21no—
105C2×#127Dic711yescharacteristic, complemented
106C2×#121—21no—
107C2×#121—21no—
108C2×C4×D4C2×D711yescharacteristic
109C2×#99—141no—
110#407—141no—
111#392Dic712yescomplemented
112#392C2×D712yes—
113#390—21no—
114C4×C4⋊C4Dic711yescharacteristic, complemented
115C4×C22⋊C4—141no—
116#342—72no—
117#324—141no—
118#323—72no—
119C23×C7—21no—
120C23×C7—21no—
121C23×C7—21no—
122C23×C7—21no—
123C23×C7—21no—
124C23×C7—41no—
125C23×C7—41no—
126C23×C7—41no—
127C23×C7—41no—
128C23×C7—42no—
129C23×C7—42no—
130C23×C7—42no—
131C23×C7—42no—
132C23×C7—42no—
133C23×C7—42no—
134C23×C7—42no—
135C23×C7C2×C22⋊C411yescharacteristic
136C23×C7AGL(1,ℤ/8)12yes—
137C7×D4—44no—
138C7×D4—82no—
139C7×D4—84no—
140C7×D4—88no—
141C7×D4—88no—
142C7×D4—84no—
143C7×D4—82no—
144C2×C4×C7—21no—
145C2×C4×C7—21no—
146C2×C4×C7—22no—
147C2×C4×C7—24no—
148C2×C4×C7—21no—
149C2×C4×C7—21no—
150C2×C4×C7—21no—
151C2×C4×C7—42no—
152C2×C4×C7—42no—
153C2×C4×C7—42no—
154C2×C4×C7—42no—
155C2×C4×C7—41no—
156C2×C4×C7—41no—
157C2×C4×C7—41no—
158C2×C4×C7—81no—
159C2×C4×C7—81no—
160C2×C4×C7#9911yescharacteristic
161C2×C4×C7#9911yescharacteristic
162C2×C4×C7#10412yes—
163C2×Dic7—44no—
164C2×Dic7—82no—
165C2×Dic7—82no—
166C7⋊C8—82no—
167C22×D4—22no—
168C22×D4—22no—
169C22×D4—21no—
170C22×D4—21no—
171C22×D4C2×Dic712yes—
172C22×D4C22⋊D711yescharacteristic
173C23×C4C22⋊D711yescharacteristic
174C23×C4C22⋊D711yescharacteristic
175C2×M16—281no—
176#127—41no—
177#127—41no—
178#127—44no—
179#121—44no—
180#121—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—142no—
188C2×C22⋊C4C2×Dic711yescharacteristic
189C2×C22⋊C4C22×D711yescharacteristic
190C2×C42—21no—
191C2×C42—281no—
192C2×C42C2×Dic711yescharacteristic
193#99—144no—
194C22⋊C8—142no—
195C22⋊C8—281no—
196#95—281no—
197#95C2×Dic712yes—
198C22×C7—21no—
199C22×C7—21no—
200C22×C7—22no—
201C22×C7—42no—
202C22×C7—42no—
203C22×C7—41no—
204C22×C7—41no—
205C22×C7—41no—
206C22×C7—41no—
207C22×C7—41no—
208C22×C7—81no—
209C22×C7—81no—
210C22×C7—82no—
211C22×C7—82no—
212C22×C7—82no—
213C22×C7—82no—
214C22×C7—82no—
215C22×C7—82no—
216C22×C7—82no—
217C22×C7C2×#9911yescharacteristic
218C22×C7#41811yescharacteristic
219C22×C7C2×#10411yescharacteristic
220C4×C7—21no—
221C4×C7—22no—
222C4×C7—44no—
223C4×C7—41no—
224C4×C7—41no—
225C4×C7—41no—
226C4×C7#35312yes—
227Dic7—84no—
228C24—22no—
229C24C22⋊Dic711yescharacteristic
230C24C2×C22⋊D711yescharacteristic
231C2×D4—22no—
232C2×D4—22no—
233C2×D4—41no—
234C2×D4—42no—
235C2×D4—42no—
236C2×D4—42no—
237C2×D4—44no—
238C2×D4—44no—
239C2×D4—42no—
240C2×D4—42no—
241C2×D4—44no—
242C2×D4—44no—
243C2×D4—41no—
244C2×D4—41no—
245C2×D4—42no—
246C2×D4—82no—
247C2×D4—82no—
248C2×D4—82no—
249C22×C4—22no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—282no—
257C22×C4C22⋊Dic711yescharacteristic
258C22×C4C22⋊Dic711yescharacteristic
259C22×C4C22⋊Dic711yescharacteristic
260C22×C4C22×Dic711yescharacteristic
261C22×C4C2×C22⋊D711yescharacteristic
262M16—562no—
263C2×C8—282no—
264C4⋊C4—24no—
265C4⋊C4—41no—
266C22⋊C4—41no—
267C22⋊C4—41no—
268C22⋊C4—44no—
269C22⋊C4—44no—
270C22⋊C4—144no—
271C22⋊C4—282no—
272C42—21no—
273C42—21no—
274C42—42no—
275C42—562no—
276C2×C7—21no—
277C2×C7—21no—
278C2×C7—42no—
279C2×C7—81no—
280C2×C7—162no—
281C2×C7#150711yescharacteristic
282C2×C7#212211yescharacteristic
283C2×C7C2×#35311yescharacteristic
284C23—21no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23—41no—
290C23—41no—
291C23—41no—
292C23—41no—
293C23—42no—
294C23—42no—
295C23—42no—
296C23—42no—
297C23—42no—
298C23—42no—
299C23—42no—
300C23#649512yes—
301C23C2×C22⋊Dic711yescharacteristic
302D4—44no—
303D4—84no—
304D4—82no—
305D4—88no—
306D4—88no—
307D4—84no—
308D4—82no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—22no—
312C2×C4—24no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—81no—
324C2×C4—81no—
325C2×C4—284no—
326C2×C4—562no—
327C2×C4—562no—
328C2×C4#640811yescharacteristic
329C2×C4#640811yescharacteristic
330C2×C4#641112yes—
331C8—562no—
332C7#1177611yescharacteristic, complemented, Sylow
333C22—21no—
334C22—21no—
335C22—22no—
336C22—42no—
337C22—42no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—81no—
344C22—81no—
345C22—82no—
346C22—82no—
347C22—82no—
348C22—82no—
349C22—82no—
350C22—82no—
351C22—82no—
352C22#8998711yescharacteristic
353C22C2×#640811yescharacteristic
354C22C2×#641111yescharacteristic
355C4—22no—
356C4—21no—
357C4—44no—
358C4—41no—
359C4—41no—
360C4—41no—
361C4—564no—
362C4#8934012yes—
363C2—21no—
364C2—21no—
365C2—42no—
366C2—81no—
367C2—162no—
368C2#1172942711yescharacteristic
369C2C2×#8934011yescharacteristic
370C2#1172961311yescharacteristic
subgroupG/H
centerC22C2×#6408
commutatorC22×C4×C7C22×C4
FrattiniC2×C22⋊C4C22×D7
Fitting, prime radicalC7×#2541C2
socleC22×C7C2×#99
cosocleC2×C7×C22⋊C4C23
hypercenter#2541D7
hypocenterC7#11776
LCSG ▷ C22×C4×C7 ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2541

Complements

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

NH ≅ G/Ntype
2. C4×#6408C2semidirect
3. #11729356C2semidirect
4. #11725451C2semidirect
6. C2×C7×#127C4semidirect
10. C7×#392C4semidirect
13. C4×C7×C4⋊C4C4semidirect
105. C2×#127Dic7semidirect
111. #392Dic7semidirect
114. C4×C4⋊C4Dic7semidirect
332. C7#11776semidirect

Maximal subgroups

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

subgroupindex
C7×#25412
C4×#64082
#117293562
#117254512
#117767