Finite Group Databasebeta

← #13249541

Subgroups — #13249541

Order 1792.

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

normal subgroups only

This group has 756 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×#376C211yesmaximal, characteristic, complemented
2#11739525C211yesmaximal, characteristic
3#11729285C211yesmaximal, characteristic, complemented
4C2×#89246C212yesmaximal
5#11725418C212yesmaximal, complemented
6C22×C42×C7—21no—
7C22×C42×C7C2211yescharacteristic
8C2×#6493C2211yescharacteristic
9C2×#6492—21no—
10C7×#376—24no—
11C2×C7×#95—21no—
12C2×C7×#95C2212yes—
13#89985—21no—
14#89984C2211yescharacteristic
15C2×C22⋊(C7⋊C8)C2212yes—
16#89712—22no—
17#89711C412yescomplemented
18#89700C412yescomplemented
19#89319—22no—
20#89246—24no—
21#9149—71nomaximal, Sylow
22C23×C4×C7—22no—
23C23×C4×C7D411yescharacteristic
24C23×C4×C7D411yescharacteristic
25C23×C4×C7C2311yescharacteristic
26C22×C22⋊D7—21no—
27C22×D28C2×C411yescharacteristic
28C2×C42×C7—28no—
29C2×C42×C7—24no—
30C2×C42×C7—22no—
31C2×C42×C7—22no—
32C2×C42×C7C2×C412yes—
33C2×C42×C7D412yes—
34C2×C22⋊Dic7—21no—
35#6493—22no—
36#6493—22no—
37#6492—44no—
38C2×(C2×D7)⋊C4—22no—
39C2×(C2×D7)⋊C4—22no—
40C2×(C2×D7)⋊C4—21no—
41C2×C4⋊Dic7C2×C411yescharacteristic
42C2×C4⋊Dic7C2×C412yes—
43C2×Dic7⋊C4—22no—
44C22×C7⋊C8—22no—
45C7×#95—24no—
46C7×#95—22no—
47C7×#95D414yes—
48C7⋊#95—22no—
49C7⋊#95—21no—
50C22⋊(C7⋊C8)—22no—
51C22⋊(C7⋊C8)—22no—
52#2854—71no—
53C2×#376D711yescharacteristic, complemented
54#1817—71no—
55C2×#327—72no—
56#1294—72no—
57C24×C7C2×D411yescharacteristic
58C23×D7—21no—
59C22×C4×C7—21no—
60C22×C4×C7—21no—
61C22×C4×C7—24no—
62C22×C4×C7—28no—
63C22×C4×C7—21no—
64C22×C4×C7—24no—
65C22×C4×C7—21no—
66C22×C4×C7—22no—
67C22×C4×C7—22no—
68C22×C4×C7—42no—
69C22×C4×C7—44no—
70C22×C4×C7—42no—
71C22×C4×C7—42no—
72C22×C4×C7C22⋊C412yes—
73C22×C4×C7C22⋊C412yes—
74C22×C4×C7C22×C411yescharacteristic
75C22×C4×C7C2×D411yescharacteristic
76C22×C4×C7C2×D412yes—
77C22×C4×C7#4112yes—
78C2×C22⋊D7—41no—
79C2×C22⋊D7—41no—
80C2×C22⋊D7—42no—
81C2×C22⋊D7—42no—
82C22×Dic7—21no—
83C22×Dic7—22no—
84C2×D28—22no—
85C2×D28—42no—
86C42×C7—216no—
87C42×C7—24no—
88C42×C7—24no—
89C22⋊Dic7—42no—
90(C2×D7)⋊C4—44no—
91(C2×D7)⋊C4—44no—
92(C2×D7)⋊C4—42no—
93C4⋊Dic7—22no—
94C4⋊Dic7—42no—
95Dic7⋊C4—44no—
96C2×C7⋊C8—44no—
97C2×C7⋊C8—42no—
98C2×#123—141no—
99C2×#121—71no—
100C22×C42—21no—
101C22×C42C2×D711yescharacteristic
102C2×C22⋊C8—72no—
103#390—72no—
104#389—72no—
105#388—142no—
106#386—141no—
107#380—71no—
108#376—24no—
109C2×#95—21no—
110C2×#95C2×D712yes—
111#336—142no—
112#327—144no—
113C23×C7—22no—
114C23×C7—22no—
115C23×C7C2×C22⋊C411yescharacteristic
116C23×C7#12011yescharacteristic
117C23×C7#12311yescharacteristic
118C23×C7#13512yes—
119C23×C7AGL(1,ℤ/8)12yes—
120C22×D7—41no—
121C22×D7—42no—
122C22×D7—41no—
123C22×D7—41no—
124C22×D7—42no—
125C2×C4×C7—24no—
126C2×C4×C7—24no—
127C2×C4×C7—22no—
128C2×C4×C7—22no—
129C2×C4×C7—22no—
130C2×C4×C7—22no—
131C2×C4×C7—28no—
132C2×C4×C7—22no—
133C2×C4×C7—22no—
134C2×C4×C7—22no—
135C2×C4×C7—42no—
136C2×C4×C7—42no—
137C2×C4×C7—44no—
138C2×C4×C7—48no—
139C2×C4×C7—48no—
140C2×C4×C7—42no—
141C2×C4×C7—44no—
142C2×C4×C7—42no—
143C2×C4×C7#10414yes—
144C2×C4×C7C4×D412yes—
145C2×C4×C7#12112yes—
146C22⋊D7—82no—
147C22⋊D7—82no—
148C2×Dic7—41no—
149C2×Dic7—42no—
150C2×Dic7—41no—
151C2×Dic7—41no—
152C2×Dic7—42no—
153C2×Dic7—42no—
154D28—82no—
155C7⋊C8—82no—
156C22×D4—71no—
157C22×D4—141no—
158C23×C4—22no—
159C23×C4C22⋊D711yescharacteristic
160C23×C4C22⋊D711yescharacteristic
161C23×C4C22×D711yescharacteristic
162C22×C8—142no—
163#123—284no—
164#121—142no—
165#121—142no—
166C2×C4⋊C4—71no—
167C2×C4⋊C4—72no—
168C2×C4⋊C4—142no—
169C2×C22⋊C4—141no—
170C2×C22⋊C4—142no—
171C2×C22⋊C4—142no—
172C2×C22⋊C4—141no—
173C2×C42—24no—
174C2×C42—28no—
175C2×C42—22no—
176C2×C42—22no—
177C2×C42C4×D712yes—
178C2×C42D2812yes—
179C22⋊C8—142no—
180C22⋊C8—142no—
181#95—24no—
182#95—22no—
183#95—142no—
184#95—141no—
185#95C22⋊D714yes—
186C22×C7—21no—
187C22×C7—21no—
188C22×C7—21no—
189C22×C7—21no—
190C22×C7—24no—
191C22×C7—24no—
192C22×C7#38611yescharacteristic
193C22×C7#41012yes—
194C22×C7#41611yescharacteristic
195C22×C7#41811yescharacteristic
196C22×C7C2×#10412yes—
197C22×C7#44912yes—
198C22×C7#48212yes—
199C2×D7—81no—
200C2×D7—81no—
201C2×D7—81no—
202C2×D7—82no—
203C2×D7—82no—
204C4×C7—24no—
205C4×C7—22no—
206C4×C7—22no—
207C4×C7—48no—
208C4×C7—42no—
209Dic7—81no—
210Dic7—82no—
211C24—141no—
212C24C2×C22⋊D711yescharacteristic
213C2×D4—142no—
214C2×D4—282no—
215C2×D4—281no—
216C2×D4—282no—
217C2×D4—282no—
218C2×D4—281no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—24no—
222C22×C4—28no—
223C22×C4—21no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—21no—
227C22×C4—24no—
228C22×C4—42no—
229C22×C4—44no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—142no—
233C22×C4—141no—
234C22×C4(C2×D7)⋊C412yes—
235C22×C4(C2×D7)⋊C412yes—
236C22×C4C2×C4×D711yescharacteristic
237C22×C4C2×D2811yescharacteristic
238C22×C4#115712yes—
239C22×C4C2×C22⋊D712yes—
240C2×C8—284no—
241C2×C8—282no—
242C4⋊C4—142no—
243C4⋊C4—282no—
244C4⋊C4—284no—
245C22⋊C4—284no—
246C22⋊C4—282no—
247C22⋊C4—282no—
248C22⋊C4—284no—
249C42—216no—
250C42—24no—
251C42—24no—
252C2×C7—22no—
253C2×C7—22no—
254C2×C7#150012yes—
255C2×C7#150512yes—
256C2×C7#187511yescharacteristic
257C2×C7#189611yescharacteristic
258C2×C7#189911yescharacteristic
259D7—161no—
260C23—22no—
261C23—22no—
262C23—282no—
263C23—281no—
264C23—281no—
265C23—281no—
266C23—282no—
267C23C2×(C2×D7)⋊C411yescharacteristic
268C23#649211yescharacteristic
269C23#651212yes—
270C23#651312yes—
271C23#651611yescharacteristic
272D4—562no—
273D4—562no—
274D4—562no—
275C2×C4—24no—
276C2×C4—24no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—28no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—48no—
288C2×C4—48no—
289C2×C4—44no—
290C2×C4—42no—
291C2×C4—44no—
292C2×C4—42no—
293C2×C4—282no—
294C2×C4—281no—
295C2×C4—282no—
296C2×C4—281no—
297C2×C4—281no—
298C2×C4—282no—
299C2×C4#637914yes—
300C2×C4C4×C22⋊D712yes—
301C2×C4#649312yes—
302C8—562no—
303C7#914911yescharacteristic, complemented, Sylow
304C22—24no—
305C22—24no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—562no—
311C22—562no—
312C22—561no—
313C22—561no—
314C22—561no—
315C22C2×#637912yes—
316C22#8977311yescharacteristic
317C22#8977411yescharacteristic
318C22#8977512yes—
319C22#8977812yes—
320C22#8998511yescharacteristic
321C22#9001012yes—
322C4—24no—
323C4—22no—
324C4—22no—
325C4—48no—
326C4—42no—
327C4—562no—
328C4—561no—
329C2—22no—
330C2—22no—
331C2—1121no—
332C2#1172718512yes—
333C2#1172718612yes—
334C2#1172928611yescharacteristic
335C2#1172934011yescharacteristic
336C2#1172934911yescharacteristic
subgroupG/H
centerC23C2×(C2×D7)⋊C4
commutatorC22×C4×C7C22×C4
FrattiniC23×C4C22×D7
FittingC2×C7×#376C2
socleC23×C7C2×C22⋊C4
prime radicalC2×#6493C22
cosocleC23×C4×C7C23
hypercenterC2×#376D7
hypocenterC7#9149
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#376

Complements

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

NH ≅ G/Ntype
1. C2×C7×#376C2semidirect
3. #11729285C2semidirect
5. #11725418C2semidirect
17. #89711C4semidirect
18. #89700C4semidirect
53. C2×#376D7semidirect
303. C7#9149semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#3762
#117395252
#117292852
C2×#892462
#117254182
#91497