Finite Group Databasebeta

← #13272110

Subgroups — #13272110

Order 1792.

552 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 111 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×#2547C211yesmaximal, characteristic
2C2×#89336C214yesmaximal
3#11729373C211yesmaximal, characteristic
4C4×#6408C211yesmaximal, characteristic
5C2×C7×C4⋊Q8C411yescharacteristic, complemented
6C2×C7×#124C2212yes—
7C2×C4×C7×D4C2211yescharacteristic
8C7×#397—21no—
9C7×#397—21no—
10C7×#391C412yescomplemented
11C7×#380—21no—
12C7×#380—21no—
13C4×C7×C4⋊C4C411yescharacteristic, complemented
14C2×#6408C2212yes—
15C2×#6408C2212yes—
16#89984—21no—
17C4×C22⋊Dic7—21no—
18#89336—28no—
19#89321—21no—
20#89321—21no—
21#13224—71nomaximal, Sylow
22C22×C7×Q8C2×C412yes—
23C22×C7×D4C2311yescharacteristic
24C23×C4×C7D411yescharacteristic
25C23×C4×C7D411yescharacteristic
26C7×C4⋊Q8—41no—
27C7×C4⋊Q8—41no—
28C7×#124—24no—
29C7×#124—24no—
30C4×C7×D4—41no—
31C4×C7×D4—41no—
32C2×C7×C4⋊C4—21no—
33C2×C7×C4⋊C4—21no—
34C2×C7×C4⋊C4—21no—
35C2×C7×C4⋊C4C2×C411yescharacteristic
36C2×C7×C22⋊C4—22no—
37C2×C7×C22⋊C4—22no—
38C2×C7×C22⋊C4D411yescharacteristic
39C2×C7×C22⋊C4D411yescharacteristic
40C2×C42×C7C2×C412yes—
41C2×C42×C7C2×C411yescharacteristic
42C2×C22⋊Dic7—22no—
43C2×C22⋊Dic7—22no—
44C2×C4⋊Dic7—41no—
45C2×C4×Dic7—41no—
46C7×#95—21no—
47C7×#95—22no—
48C7×#95—22no—
49#6408—22no—
50#6408—24no—
51#6408D414yes—
52C7⋊#95—41no—
53C7⋊#95—41no—
54#2547D711yescharacteristic
55C2×#354—74no—
56#1799—71no—
57C4×#99—71no—
58C24×C7C2×D411yescharacteristic
59C24×C7C2×D411yescharacteristic
60C2×C7×Q8—24no—
61C2×C7×Q8—42no—
62C2×C7×Q8—42no—
63C2×C7×D4—41no—
64C2×C7×D4—41no—
65C2×C7×D4C2×D412yes—
66C2×C7×D4#4112yes—
67C22×C4×C7—21no—
68C22×C4×C7—21no—
69C22×C4×C7—21no—
70C22×C4×C7—21no—
71C22×C4×C7—22no—
72C22×C4×C7—22no—
73C22×C4×C7—42no—
74C22×C4×C7—42no—
75C22×C4×C7C22⋊C411yescharacteristic
76C22×C4×C7C22⋊C411yescharacteristic
77C22×C4×C7C22⋊C411yescharacteristic
78C22×C4×C7C22⋊C411yescharacteristic
79C22×C4×C7C22×C411yescharacteristic
80C22×Dic7—42no—
81C22×Dic7—42no—
82C7×C4⋊C4—41no—
83C7×C4⋊C4—42no—
84C7×C4⋊C4—42no—
85C7×C4⋊C4—41no—
86C7×C4⋊C4—41no—
87C7×C22⋊C4—41no—
88C7×C22⋊C4—41no—
89C7×C22⋊C4—44no—
90C7×C22⋊C4—42no—
91C7×C22⋊C4—42no—
92C42×C7—21no—
93C42×C7—21no—
94C42×C7—24no—
95C22⋊Dic7—24no—
96C22⋊Dic7—42no—
97C22⋊Dic7—42no—
98C22⋊Dic7—42no—
99C4⋊Dic7—82no—
100C4×Dic7—82no—
101C2×C4⋊Q8Dic711yescharacteristic, complemented
102C2×#124C2×D712yes—
103C2×C4×D4C2×D711yescharacteristic
104C2×#99—72no—
105C2×#99—72no—
106#397—21no—
107#397—21no—
108#391Dic712yescomplemented
109#380—21no—
110#380—21no—
111#380—141no—
112C4×C4⋊C4Dic711yescharacteristic, complemented
113C4×C22⋊C4—141no—
114#354—148no—
115#342—141no—
116#342—141no—
117C23×C7—21no—
118C23×C7—21no—
119C23×C7—21no—
120C23×C7—21no—
121C23×C7—41no—
122C23×C7—41no—
123C23×C7—41no—
124C23×C7—41no—
125C23×C7C2×C22⋊C411yescharacteristic
126C23×C7#12112yes—
127C23×C7#12411yescharacteristic
128C23×C7#12711yescharacteristic
129C7×Q8—84no—
130C7×D4—42no—
131C7×D4—42no—
132C2×C4×C7—21no—
133C2×C4×C7—21no—
134C2×C4×C7—22no—
135C2×C4×C7—21no—
136C2×C4×C7—21no—
137C2×C4×C7—21no—
138C2×C4×C7—42no—
139C2×C4×C7—42no—
140C2×C4×C7—41no—
141C2×C4×C7—41no—
142C2×C4×C7—41no—
143C2×C4×C7—42no—
144C2×C4×C7—42no—
145C2×C4×C7—42no—
146C2×C4×C7—42no—
147C2×C4×C7—42no—
148C2×C4×C7—42no—
149C2×C4×C7—81no—
150C2×C4×C7—81no—
151C2×C4×C7#9911yescharacteristic
152C2×C4×C7#9911yescharacteristic
153C2×C4×C7C4×D412yes—
154C2×Dic7—44no—
155C2×Dic7—82no—
156C2×Dic7—82no—
157C2×Dic7—82no—
158C2×Dic7—82no—
159C2×Dic7—82no—
160C22×Q8C2×Dic712yes—
161C22×D4C22×D711yescharacteristic
162C23×C4C22⋊D711yescharacteristic
163C23×C4C22⋊D711yescharacteristic
164C4⋊Q8—41no—
165C4⋊Q8—41no—
166#124—24no—
167#124—24no—
168C4×D4—41no—
169C4×D4—41no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—281no—
174C2×C4⋊C4C2×Dic711yescharacteristic
175C2×C22⋊C4—22no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—142no—
178C2×C22⋊C4—142no—
179C2×C22⋊C4C22⋊D711yescharacteristic
180C2×C22⋊C4C22⋊D711yescharacteristic
181C2×C42—281no—
182C2×C42C2×Dic712yes—
183C2×C42C2×Dic711yescharacteristic
184#99—74no—
185#99—142no—
186#99—144no—
187#95—21no—
188#95—22no—
189#95—22no—
190#95—281no—
191#95—281no—
192C22×C7—21no—
193C22×C7—21no—
194C22×C7—22no—
195C22×C7—21no—
196C22×C7—21no—
197C22×C7—42no—
198C22×C7—41no—
199C22×C7—41no—
200C22×C7—41no—
201C22×C7—41no—
202C22×C7—41no—
203C22×C7—41no—
204C22×C7—81no—
205C22×C7—81no—
206C22×C7#39011yescharacteristic
207C22×C7C2×#9911yescharacteristic
208C22×C7#41011yescharacteristic
209C4×C7—21no—
210C4×C7—22no—
211C4×C7—41no—
212C4×C7—41no—
213C4×C7—41no—
214C4×C7—82no—
215C4×C7—82no—
216C4×C7#35412yescomplemented
217Dic7—84no—
218Dic7—162no—
219C24C2×C22⋊D711yescharacteristic
220C24C2×C22⋊D711yescharacteristic
221C2×Q8—24no—
222C2×Q8—42no—
223C2×Q8—42no—
224C2×D4—41no—
225C2×D4—41no—
226C2×D4D4×D712yes—
227C2×D4#115912yes—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—282no—
237C22×C4—282no—
238C22×C4C22⋊Dic711yescharacteristic
239C22×C4C22⋊Dic711yescharacteristic
240C22×C4C22⋊Dic711yescharacteristic
241C22×C4C22⋊Dic711yescharacteristic
242C22×C4C22×Dic711yescharacteristic
243C4⋊C4—41no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—562no—
249C22⋊C4—41no—
250C22⋊C4—41no—
251C22⋊C4—42no—
252C22⋊C4—44no—
253C22⋊C4—42no—
254C22⋊C4—144no—
255C22⋊C4—282no—
256C22⋊C4—282no—
257C22⋊C4—282no—
258C42—21no—
259C42—21no—
260C42—24no—
261C42—562no—
262C2×C7—21no—
263C2×C7—21no—
264C2×C7—42no—
265C2×C7—41no—
266C2×C7—41no—
267C2×C7#196411yescharacteristic
268C2×C7C2×#35411yescharacteristic
269C2×C7#212911yescharacteristic
270C23—21no—
271C23—21no—
272C23—21no—
273C23—21no—
274C23—41no—
275C23—41no—
276C23—41no—
277C23—41no—
278C23#649911yescharacteristic
279C23#650112yes—
280C23#650311yescharacteristic
281C23C2×C22⋊Dic711yescharacteristic
282Q8—84no—
283D4—42no—
284D4—42no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—22no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—81no—
303C2×C4—81no—
304C2×C4—284no—
305C2×C4—562no—
306C2×C4—562no—
307C2×C4—562no—
308C2×C4—562no—
309C2×C4—562no—
310C2×C4#640811yescharacteristic
311C2×C4#640811yescharacteristic
312C2×C4D4×Dic712yes—
313C7#1322411yescharacteristic, complemented, Sylow
314C22—21no—
315C22—21no—
316C22—22no—
317C22—21no—
318C22—21no—
319C22—41no—
320C22—42no—
321C22—41no—
322C22—41no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—81no—
327C22—81no—
328C22C2×#640811yescharacteristic
329C22#8999311yescharacteristic
330C22#9001211yescharacteristic
331C4—22no—
332C4—21no—
333C4—41no—
334C4—41no—
335C4—41no—
336C4—82no—
337C4—82no—
338C4—564no—
339C4—1122no—
340C4#8933612yescomplemented
341C2—21no—
342C2—21no—
343C2—42no—
344C2—41no—
345C2—41no—
346C2#1172944011yescharacteristic
347C2C2×#8933611yescharacteristic
348C2#1172962511yescharacteristic
subgroupG/H
centerC22C2×#6408
commutatorC22×C4×C7C22×C4
FrattiniC22×D4C22×D7
FittingC7×#2547C2
socleC22×C7C2×#99
prime radical, cosocleC22×C7×D4C23
hypercenter#2547D7
hypocenterC7#13224
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2547

Complements

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

NH ≅ G/Ntype
5. C2×C7×C4⋊Q8C4semidirect
10. C7×#391C4semidirect
13. C4×C7×C4⋊C4C4semidirect
101. C2×C4⋊Q8Dic7semidirect
108. #391Dic7semidirect
112. C4×C4⋊C4Dic7semidirect
216. C4×C7#354semidirect
313. C7#13224semidirect
340. C4#89336semidirect

Maximal subgroups

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

subgroupindex
C7×#25472
C2×#893362
#117293732
C4×#64082
#132247