Finite Group Databasebeta

← #13246712

Subgroups — #13246712

Order 1792.

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

normal subgroups only

This group has 460 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
1C2×#89883C211yesmaximal, characteristic
2C7×#1810C211yesmaximal, characteristic, complemented
3#11729731C211yesmaximal, characteristic, complemented
4#11727722C212yesmaximal
5#11725493C212yesmaximal, complemented
6C23×C7×C8C2211yescharacteristic
7C2×C4×C22⋊D7C2211yescharacteristic
8C22×#1131C411yescharacteristic
9C2×C7×#108—21no—
10C7×#407C2212yes—
11#89979C2212yes—
12C2×C22⋊(C7⋊C8)C2211yescharacteristic
13#89883—21no—
14#89883—21no—
15#89883—21no—
16#89883—21no—
17C2×(C2×D7)⋊C8C411yescharacteristic
18C2×#6393—21no—
19C2×#6389C411yescharacteristic
20C2×Dic7⋊C8C411yescharacteristic
21C7×#341—21no—
22#89347—21no—
23#89264—22no—
24#89263—22no—
25#8613—71nomaximal, Sylow
26C23×C4×C7C2311yescharacteristic
27C22×C22⋊D7Q811yescharacteristic
28C22×C4×D7C2×C411yescharacteristic
29C2×C7×M16—22no—
30C22×C7×C8—42no—
31C22×C7×C8C2×C411yescharacteristic
32C22×C7×C8C2×C411yescharacteristic
33C22×C7×C8D411yescharacteristic
34C22×C7×C8D411yescharacteristic
35C22×C7×C8D411yescharacteristic
36C22×C7×C8D411yescharacteristic
37C2×C22⋊Dic7Q811yescharacteristic
38C4×C22⋊D7—21no—
39C4×C22⋊D7—21no—
40C4×C22⋊D7—21no—
41C4×C22⋊D7—21no—
42C2×(C2×D7)⋊C4D411yescharacteristic
43C2×Dic7⋊C4D411yescharacteristic
44C2×C4×Dic7C2×C411yescharacteristic
45C2×#1136—22no—
46C22×C7⋊C8C2×C411yescharacteristic
47C22×C7⋊C8C2×C411yescharacteristic
48C2×#1131—21no—
49C2×#1131—21no—
50C2×#1131—42no—
51C7×#108—42no—
52C7×C22⋊C8—22no—
53C7×C22⋊C8—22no—
54C22⋊(C7⋊C8)—21no—
55C22⋊(C7⋊C8)—21no—
56C22⋊(C7⋊C8)—22no—
57C22⋊(C7⋊C8)—22no—
58(C2×D7)⋊C8—21no—
59(C2×D7)⋊C8—21no—
60#6393—42no—
61#6389—21no—
62#6389—21no—
63Dic7⋊C8—21no—
64Dic7⋊C8—21no—
65C2×#435—71no—
66#1810—71no—
67#1810D711yescharacteristic, complemented
68#1465—72no—
69#1293—72no—
70C24×C7C2×Q811yescharacteristic
71C23×D7C4⋊C411yescharacteristic
72C22×C4×C7—41no—
73C22×C4×C7—41no—
74C22×C4×C7C22×C411yescharacteristic
75C22×C4×C7C2×D411yescharacteristic
76C22×C4×C7C2×D411yescharacteristic
77C22×C4×C7C2×D411yescharacteristic
78C22×C4×C7#4111yescharacteristic
79C22×C4×C7#4111yescharacteristic
80C2×C22⋊D7—21no—
81C2×C22⋊D7—21no—
82C2×C22⋊D7—42no—
83C22×Dic7C4⋊C411yescharacteristic
84C22×Dic7C4⋊C411yescharacteristic
85C22×Dic7C4⋊C411yescharacteristic
86C2×C4×D7—21no—
87C2×C4×D7—21no—
88C2×C4×D7—41no—
89C2×C4×D7—41no—
90C7×M16—42no—
91C7×M16—42no—
92C2×C7×C8—22no—
93C2×C7×C8—22no—
94C2×C7×C8—22no—
95C2×C7×C8—41no—
96C2×C7×C8—41no—
97C2×C7×C8—41no—
98C2×C7×C8—42no—
99C2×C7×C8C22⋊C411yescharacteristic
100C2×C7×C8C22⋊C411yescharacteristic
101C2×C7×C8C22⋊C411yescharacteristic
102C2×C7×C8C22⋊C411yescharacteristic
103C22⋊Dic7—21no—
104C22⋊Dic7—21no—
105(C2×D7)⋊C4—21no—
106(C2×D7)⋊C4—21no—
107Dic7⋊C4—21no—
108Dic7⋊C4—21no—
109C4×Dic7—21no—
110C4×Dic7—21no—
111#1136—42no—
112#1136—42no—
113C2×C7⋊C8—21no—
114C2×C7⋊C8—21no—
115C2×C7⋊C8—41no—
116C2×C7⋊C8—41no—
117C2×C7⋊C8—42no—
118#1131—42no—
119#1131—42no—
120C22×M16—71no—
121C23×C8C2×D711yescharacteristic
122C2×C4×D4—71no—
123#435—141no—
124#435—141no—
125#435—141no—
126#435—141no—
127C2×#108—21no—
128C2×#108—141no—
129C2×C4⋊C8—71no—
130#407—72no—
131#407C2×D712yes—
132C2×C22⋊C8—71no—
133C2×C22⋊C8—71no—
134C2×#97—71no—
135#341—21no—
136#341—141no—
137#324—142no—
138#323—142no—
139C23×C7—22no—
140C23×C7—41no—
141C23×C7—41no—
142C23×C7C2×C4⋊C411yescharacteristic
143C23×C7C22⋊Q811yescharacteristic
144C23×C7C22⋊Q811yescharacteristic
145C22×D7—21no—
146C22×D7—22no—
147C22×D7—41no—
148C22×D7—41no—
149C2×C4×C7—21no—
150C2×C4×C7—21no—
151C2×C4×C7—21no—
152C2×C4×C7—21no—
153C2×C4×C7—21no—
154C2×C4×C7—21no—
155C2×C4×C7—41no—
156C2×C4×C7—41no—
157C2×C4×C7—41no—
158C2×C4×C7C2×C22⋊C411yescharacteristic
159C2×C4×C7#11711yescharacteristic
160C2×C4×C7#12111yescharacteristic
161C2×C4×C7#12111yescharacteristic
162C22⋊D7—42no—
163C22⋊D7—42no—
164C2×Dic7—21no—
165C2×Dic7—21no—
166C2×Dic7—21no—
167C2×Dic7—22no—
168C2×Dic7—41no—
169C2×Dic7—41no—
170C4×D7—42no—
171C4×D7—41no—
172C4×D7—41no—
173C7×C8—21no—
174C7×C8—21no—
175C7×C8—21no—
176C7×C8—82no—
177C7×C8#9912yes—
178C7⋊C8—41no—
179C7⋊C8—41no—
180C7⋊C8—82no—
181C22×D4—71no—
182C23×C4—71no—
183C23×C4C22×D711yescharacteristic
184C2×M16—22no—
185C2×M16—141no—
186C2×M16—141no—
187C2×M16—142no—
188C2×M16—282no—
189C22×C8—42no—
190C22×C8—71no—
191C22×C8—71no—
192C22×C8C4×D711yescharacteristic
193C22×C8C4×D711yescharacteristic
194C22×C8D2811yescharacteristic
195C22×C8D2811yescharacteristic
196C22×C8C22⋊D711yescharacteristic
197C22×C8C22⋊D711yescharacteristic
198C4×D4—141no—
199C4×D4—141no—
200C4×D4—141no—
201C4×D4—141no—
202C2×C4⋊C4—71no—
203C2×C22⋊C4—71no—
204C2×C22⋊C4—71no—
205C2×C42—71no—
206#108—42no—
207#108—282no—
208C4⋊C8—141no—
209C4⋊C8—141no—
210C22⋊C8—22no—
211C22⋊C8—22no—
212C22⋊C8—141no—
213C22⋊C8—141no—
214C22⋊C8—141no—
215C22⋊C8—141no—
216C22⋊C8—142no—
217C22⋊C8—142no—
218#97—141no—
219#97—141no—
220C22×C7—21no—
221C22×C7—21no—
222C22×C7—21no—
223C22×C7—21no—
224C22×C7—21no—
225C22×C7—21no—
226C22×C7—21no—
227C22×C7—21no—
228C22×C7—44no—
229C22×C7#38011yescharacteristic
230C22×C7#43012yes—
231C2×D7—41no—
232C2×D7—41no—
233C2×D7—41no—
234C2×D7—82no—
235C4×C7—21no—
236C4×C7—21no—
237C4×C7—21no—
238C4×C7C2×#9911yescharacteristic
239C4×C7#41011yescharacteristic
240Dic7—41no—
241Dic7—41no—
242Dic7—41no—
243C24—71no—
244C24Q8×D711yescharacteristic
245C2×D4—141no—
246C2×D4—141no—
247C2×D4—282no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—71no—
251C22×C4—71no—
252C22×C4—71no—
253C22×C4—141no—
254C22×C4—141no—
255C22×C4—281no—
256C22×C4—281no—
257C22×C4C2×C4×D711yescharacteristic
258C22×C4C2×D2811yescharacteristic
259C22×C4D4×D711yescharacteristic
260C22×C4#115911yescharacteristic
261C22×C4#116111yescharacteristic
262C22×C4C2×C22⋊D711yescharacteristic
263M16—42no—
264M16—42no—
265M16—282no—
266M16—282no—
267M16—282no—
268M16—282no—
269C2×C8—22no—
270C2×C8—22no—
271C2×C8—22no—
272C2×C8—41no—
273C2×C8—41no—
274C2×C8—41no—
275C2×C8—42no—
276C2×C8—141no—
277C2×C8—141no—
278C2×C8—281no—
279C2×C8—281no—
280C2×C8—282no—
281C2×C8(C2×D7)⋊C411yescharacteristic
282C2×C8(C2×D7)⋊C411yescharacteristic
283C2×C8(C2×D7)⋊C411yescharacteristic
284C2×C8(C2×D7)⋊C411yescharacteristic
285C4⋊C4—141no—
286C4⋊C4—141no—
287C22⋊C4—141no—
288C22⋊C4—141no—
289C22⋊C4—141no—
290C22⋊C4—141no—
291C42—141no—
292C42—141no—
293C2×C7—22no—
294C2×C7—41no—
295C2×C7—41no—
296C2×C7#179911yescharacteristic
297C2×C7#181111yescharacteristic
298C2×C7#181311yescharacteristic
299D7—81no—
300C23—22no—
301C23—41no—
302C23—41no—
303C23—141no—
304C23—142no—
305C23—281no—
306C23—281no—
307C23D7×C4⋊C411yescharacteristic
308C23#646011yescharacteristic
309C23#650911yescharacteristic
310D4—282no—
311D4—282no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—142no—
322C2×C4—141no—
323C2×C4—141no—
324C2×C4—141no—
325C2×C4—282no—
326C2×C4—281no—
327C2×C4—281no—
328C2×C4—281no—
329C2×C4—281no—
330C2×C4#645511yescharacteristic
331C2×C4#645811yescharacteristic
332C2×C4C2×(C2×D7)⋊C411yescharacteristic
333C2×C4#650111yescharacteristic
334C8—21no—
335C8—21no—
336C8—21no—
337C8—82no—
338C8—281no—
339C8—281no—
340C8—562no—
341C8#638012yes—
342C7#861311yescharacteristic, complemented, Sylow
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—44no—
352C22—281no—
353C22—281no—
354C22—281no—
355C22—562no—
356C22#8966512yes—
357C22#8975911yescharacteristic
358C4—21no—
359C4—21no—
360C4—21no—
361C4—281no—
362C4—281no—
363C4—281no—
364C4C2×#638011yescharacteristic
365C4#8977811yescharacteristic
366C2—22no—
367C2—41no—
368C2—41no—
369C2—561no—
370C2#1172728411yescharacteristic
371C2#1172846611yescharacteristic
372C2#1172847211yescharacteristic
subgroupG/H
centerC22#89759
commutatorC22×C4×C7C22×C4
FrattiniC23×C4C22×D7
FittingC7×#1810C2
socleC22×C7#380
prime radicalC22×C22⋊D7Q8
cosocleC23×C4×C7C23
hypercenter#1810D7
hypocenterC7#8613
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1810

Complements

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

NH ≅ G/Ntype
2. C7×#1810C2semidirect
3. #11729731C2semidirect
5. #11725493C2semidirect
67. #1810D7semidirect
342. C7#8613semidirect

Maximal subgroups

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

subgroupindex
C2×#898832
C7×#18102
#117297312
#117277222
#117254932
#86137