Finite Group Databasebeta

← #13249527

Subgroups — #13249527

Order 1792.

1010 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 1010 conjugacy classes of subgroups — too many to draw. Only its 151 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×#2848C211yesmaximal, characteristic
2C2×C4×C22⋊Dic7C211yesmaximal, characteristic
3#11729880C211yesmaximal, characteristic
4C2×#89333C212yesmaximal, complemented
5#11725430C212yesmaximal, complemented
6C24×C4×C7C2211yescharacteristic
7C2×C7×C22⋊Q8C412yescomplemented
8C2×C7×C22⋊Q8C2211yescharacteristic
9C22×C22⋊Dic7—21no—
10C22×C4×Dic7—21no—
11C7×#385—21no—
12C7×#385—22no—
13C7×#380C412yescomplemented
14C7×#380C2211yescharacteristic
15C7×#379—21no—
16C4×C22⋊Dic7—24no—
17C2×C7⋊#95C2212yes—
18C2×C22⋊(C7⋊C8)—22no—
19C2×C22⋊(C7⋊C8)C2212yes—
20#89333—24no—
21#9146—71nomaximal, Sylow
22C25×C7D411yescharacteristic
23C22×C7×Q8C2×C411yescharacteristic
24C23×C4×C7—22no—
25C23×C4×C7—22no—
26C23×C4×C7—44no—
27C23×C4×C7C2×C412yes—
28C23×C4×C7D411yescharacteristic
29C23×C4×C7D412yes—
30C23×C4×C7C2311yescharacteristic
31C23×Dic7—22no—
32C7×C22⋊Q8—22no—
33C7×C22⋊Q8—22no—
34C7×C22⋊Q8—44no—
35C2×C7×C4⋊C4—22no—
36C2×C7×C4⋊C4—21no—
37C2×C7×C4⋊C4C2×C411yescharacteristic
38C2×C7×C4⋊C4C2×C412yes—
39C2×C7×C22⋊C4—22no—
40C2×C7×C22⋊C4—22no—
41C2×C7×C22⋊C4—21no—
42C2×C7×C22⋊C4—21no—
43C2×C22⋊Dic7—24no—
44C2×C22⋊Dic7—28no—
45C2×C4×Dic7—24no—
46C2×C4×Dic7—44no—
47C22×C7⋊C8—22no—
48C7×#95—22no—
49C7×#95—21no—
50C7⋊#95—22no—
51C7⋊#95D414yes—
52C22⋊(C7⋊C8)—22no—
53C22⋊(C7⋊C8)—22no—
54C22⋊(C7⋊C8)—44no—
55#2848D711yescharacteristic
56C2×C4×C22⋊C4—71no—
57#1779—71no—
58C2×#328—72no—
59#1294—72no—
60C24×C7—24no—
61C24×C7—21no—
62C24×C7—21no—
63C24×C7—22no—
64C24×C7—22no—
65C24×C7—42no—
66C24×C7C22⋊C412yes—
67C24×C7C2×D411yescharacteristic
68C2×C7×Q8—22no—
69C2×C7×Q8—42no—
70C22×C4×C7—21no—
71C22×C4×C7—21no—
72C22×C4×C7—24no—
73C22×C4×C7—24no—
74C22×C4×C7—24no—
75C22×C4×C7—24no—
76C22×C4×C7—24no—
77C22×C4×C7—21no—
78C22×C4×C7—22no—
79C22×C4×C7—21no—
80C22×C4×C7—42no—
81C22×C4×C7—42no—
82C22×C4×C7—42no—
83C22×C4×C7—48no—
84C22×C4×C7—48no—
85C22×C4×C7C22⋊C412yes—
86C22×C4×C7C22×C411yescharacteristic
87C22×C4×C7C2×D411yescharacteristic
88C22×C4×C7C2×D412yes—
89C22×C4×C7#4112yes—
90C22×Dic7—28no—
91C22×Dic7—24no—
92C22×Dic7—44no—
93C22×Dic7—42no—
94C22×Dic7—42no—
95C7×C4⋊C4—22no—
96C7×C4⋊C4—42no—
97C7×C4⋊C4—44no—
98C7×C4⋊C4—42no—
99C7×C22⋊C4—44no—
100C7×C22⋊C4—44no—
101C7×C22⋊C4—41no—
102C7×C22⋊C4—41no—
103C7×C22⋊C4—42no—
104C22⋊Dic7—216no—
105C4×Dic7—48no—
106C2×C7⋊C8—44no—
107C2×C7⋊C8—42no—
108C24×C4C2×D711yescharacteristic
109C2×C22⋊Q8Dic712yescomplemented
110C2×C22⋊Q8C2×D711yescharacteristic
111C22×C22⋊C4—141no—
112C22×C42—141no—
113C2×C22⋊C8—72no—
114C2×C22⋊C8—142no—
115#385—21no—
116#385—22no—
117#380Dic712yescomplemented
118#380C2×D711yescharacteristic
119#379—21no—
120C4×C22⋊C4—144no—
121C2×#95—72no—
122#328—144no—
123C23×C7—22no—
124C23×C7—22no—
125C23×C7—24no—
126C23×C7—24no—
127C23×C7—22no—
128C23×C7—22no—
129C23×C7—22no—
130C23×C7—22no—
131C23×C7—22no—
132C23×C7—22no—
133C23×C7—48no—
134C23×C7—48no—
135C23×C7—42no—
136C23×C7—42no—
137C23×C7—42no—
138C23×C7—42no—
139C23×C7#10414yes—
140C23×C7C2×C22⋊C411yescharacteristic
141C23×C7#12011yescharacteristic
142C23×C7#12311yescharacteristic
143C23×C7#13512yes—
144C23×C7#13712yes—
145C7×Q8—82no—
146C2×C4×C7—24no—
147C2×C4×C7—24no—
148C2×C4×C7—22no—
149C2×C4×C7—22no—
150C2×C4×C7—22no—
151C2×C4×C7—22no—
152C2×C4×C7—42no—
153C2×C4×C7—42no—
154C2×C4×C7—48no—
155C2×C4×C7—48no—
156C2×C4×C7—41no—
157C2×C4×C7—41no—
158C2×C4×C7—42no—
159C2×C4×C7—42no—
160C2×C4×C7—42no—
161C2×C4×C7—42no—
162C2×C4×C7—42no—
163C2×C4×C7#9914yes—
164C2×C4×C7C4×D412yes—
165C2×C4×C7#12112yes—
166C2×Dic7—28no—
167C2×Dic7—44no—
168C2×Dic7—48no—
169C2×Dic7—48no—
170C2×Dic7—44no—
171C7⋊C8—82no—
172C25C22⋊D711yescharacteristic
173C22×Q8C2×Dic711yescharacteristic
174C23×C4—22no—
175C23×C4—22no—
176C23×C4—44no—
177C23×C4—142no—
178C23×C4C2×Dic712yes—
179C23×C4C22⋊D711yescharacteristic
180C23×C4C22⋊D712yes—
181C23×C4C22×D711yescharacteristic
182C22×C8—142no—
183C22⋊Q8—22no—
184C22⋊Q8—22no—
185C22⋊Q8—44no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4C2×Dic711yescharacteristic
189C2×C4⋊C4C2×Dic712yes—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—21no—
193C2×C22⋊C4—21no—
194C2×C22⋊C4—148no—
195C2×C22⋊C4—144no—
196C2×C42—144no—
197C2×C42—284no—
198C22⋊C8—142no—
199C22⋊C8—142no—
200C22⋊C8—284no—
201#95—22no—
202#95—21no—
203#95—74no—
204#95—142no—
205C22×C7—21no—
206C22×C7—21no—
207C22×C7—21no—
208C22×C7—21no—
209C22×C7—24no—
210C22×C7—24no—
211C22×C7—24no—
212C22×C7—24no—
213C22×C7—24no—
214C22×C7—48no—
215C22×C7—48no—
216C22×C7—42no—
217C22×C7—42no—
218C22×C7—42no—
219C22×C7—42no—
220C22×C7#38611yescharacteristic
221C22×C7C2×#9912yes—
222C22×C7#41611yescharacteristic
223C22×C7#41911yescharacteristic
224C22×C7C2×#10412yes—
225C22×C7#45212yes—
226C22×C7#48512yes—
227C4×C7—22no—
228C4×C7—24no—
229C4×C7—22no—
230C4×C7—81no—
231C4×C7—81no—
232C4×C7—82no—
233Dic7—48no—
234C24—24no—
235C24—21no—
236C24—21no—
237C24—22no—
238C24—22no—
239C24—42no—
240C24C22⋊Dic712yes—
241C24C2×C22⋊D711yescharacteristic
242C2×Q8—22no—
243C2×Q8—42no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—24no—
249C22×C4—24no—
250C22×C4—24no—
251C22×C4—22no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—48no—
258C22×C4—48no—
259C22×C4—148no—
260C22×C4—144no—
261C22×C4—284no—
262C22×C4—282no—
263C22×C4—282no—
264C22×C4C22⋊Dic712yes—
265C22×C4D4×D712yes—
266C22×C4#115912yes—
267C22×C4C22×Dic711yescharacteristic
268C22×C4C2×C22⋊D711yescharacteristic
269C2×C8—284no—
270C2×C8—282no—
271C4⋊C4—22no—
272C4⋊C4—42no—
273C4⋊C4—44no—
274C4⋊C4—42no—
275C22⋊C4—44no—
276C22⋊C4—44no—
277C22⋊C4—41no—
278C22⋊C4—41no—
279C22⋊C4—42no—
280C22⋊C4—1416no—
281C42—288no—
282C2×C7—22no—
283C2×C7—22no—
284C2×C7—44no—
285C2×C7#150112yes—
286C2×C7#150812yes—
287C2×C7#187311yescharacteristic
288C2×C7#189611yescharacteristic
289C2×C7#189811yescharacteristic
290C23—22no—
291C23—22no—
292C23—24no—
293C23—24no—
294C23—22no—
295C23—22no—
296C23—22no—
297C23—22no—
298C23—22no—
299C23—22no—
300C23—48no—
301C23—48no—
302C23—42no—
303C23—42no—
304C23—42no—
305C23—42no—
306C23#641114yes—
307C23#649811yescharacteristic
308C23#650011yescharacteristic
309C23#650512yes—
310C23#651312yes—
311C23C2×C22⋊Dic711yescharacteristic
312Q8—82no—
313C2×C4—24no—
314C2×C4—24no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—48no—
322C2×C4—48no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—148no—
331C2×C4—284no—
332C2×C4—288no—
333C2×C4—288no—
334C2×C4—284no—
335C2×C4#640814yes—
336C2×C4D4×Dic712yes—
337C2×C4#650212yes—
338C8—562no—
339C7#914611yescharacteristic, complemented, Sylow
340C22—21no—
341C22—21no—
342C22—24no—
343C22—24no—
344C22—21no—
345C22—21no—
346C22—24no—
347C22—24no—
348C22—24no—
349C22—48no—
350C22—48no—
351C22—42no—
352C22—42no—
353C22—42no—
354C22—42no—
355C22#8981712yes—
356C22#8981912yes—
357C22C2×#640812yes—
358C22#8999211yescharacteristic
359C22#8999711yescharacteristic
360C22#9000511yescharacteristic
361C22C2×#641112yes—
362C4—22no—
363C4—24no—
364C4—22no—
365C4—81no—
366C4—82no—
367C4—81no—
368C4—288no—
369C2—22no—
370C2—22no—
371C2—44no—
372C2#1172942411yescharacteristic
373C2#1172944212yes—
374C2#1172944711yescharacteristic
375C2#1172946112yes—
376C2#1172946511yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic7
commutatorC22×C4×C7C22×C4
FrattiniC23×C4C22×D7
FittingC7×#2848C2
socleC23×C7C2×C22⋊C4
prime radicalC25×C7D4
cosocleC23×C4×C7C23
hypercenter#2848D7
hypocenterC7#9146
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C24×C4 ◁ #2848

Complements

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

NH ≅ G/Ntype
4. C2×#89333C2semidirect
5. #11725430C2semidirect
7. C2×C7×C22⋊Q8C4semidirect
13. C7×#380C4semidirect
109. C2×C22⋊Q8Dic7semidirect
117. #380Dic7semidirect
339. C7#9146semidirect

Maximal subgroups

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

subgroupindex
C7×#28482
C2×C4×C22⋊Dic72
#117298802
C2×#893332
#117254302
#91467