Finite Group Databasebeta

← #13246774

Subgroups — #13246774

Order 1792.

456 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11741179C211yesmaximal, characteristic
2C7×#1810C211yesmaximal, characteristic, complemented
3#11729725C211yesmaximal, characteristic, complemented
4#11727733C212yesmaximal
5#11725491C212yesmaximal, complemented
6C23×C7×C8C2211yescharacteristic
7#90482C2211yescharacteristic
8C2×#6464C411yescharacteristic, complemented
9C2×C7×#108—21no—
10C7×#407C2212yes—
11#89984C2212yes—
12#89979C2211yescharacteristic
13#89883—21no—
14#89883—21no—
15#89882C411yescharacteristic, complemented
16C8×C22⋊D7—21no—
17C8×C22⋊D7—21no—
18C2×#6392—21no—
19#89874C411yescharacteristic, complemented
20#89873C411yescharacteristic, complemented
21C7×#341—21no—
22#89346—21no—
23#89265—22no—
24#89262—22no—
25#8627—71nomaximal, Sylow
26C23×C4×C7C2311yescharacteristic
27C2×#1157C2×C411yescharacteristic
28C2×C7×M16—22no—
29C22×C7×C8—42no—
30C22×C7×C8C2×C411yescharacteristic
31C22×C7×C8C2×C411yescharacteristic
32C22×C7×C8D411yescharacteristic
33C22×C7×C8D411yescharacteristic
34C22×C7×C8D411yescharacteristic
35C22×C7×C8D411yescharacteristic
36#6516—21no—
37C2×C22⋊Dic7—22no—
38#6493D411yescharacteristic
39#6493Q811yescharacteristic
40#6492—21no—
41C4×C22⋊D7—21no—
42C4×C22⋊D7—21no—
43#6489C2×C411yescharacteristic
44C2×C4⋊Dic7—22no—
45#6487D411yescharacteristic
46#6487Q811yescharacteristic
47C2×#1136C2×C411yescharacteristic
48C2×#1136C2×C411yescharacteristic
49#6464—42no—
50C2×#1131—21no—
51C2×C8×D7—21no—
52C7×#108—42no—
53C7×C22⋊C8—22no—
54C7×C22⋊C8—22no—
55C7⋊#95—22no—
56C22⋊(C7⋊C8)—21no—
57C22⋊(C7⋊C8)—21no—
58(C2×D7)⋊C8—21no—
59(C2×D7)⋊C8—21no—
60#6392—42no—
61#6389—21no—
62Dic7⋊C8—21no—
63Dic7⋊C8—21no—
64C8×Dic7—21no—
65#2929—71no—
66#1810D711yescharacteristic, complemented
67#1808—71no—
68#1513—72no—
69#1305—72no—
70C24×C7#4111yescharacteristic
71C22×C4×C7—41no—
72C22×C4×C7—41no—
73C22×C4×C7C22×C411yescharacteristic
74C22×C4×C7C2×D411yescharacteristic
75C22×C4×C7C2×D411yescharacteristic
76C22×C4×C7C2×D411yescharacteristic
77C22×C4×C7C2×Q811yescharacteristic
78C22×C4×C7#4111yescharacteristic
79C2×C22⋊D7—21no—
80C2×C22⋊D7—21no—
81C2×C22⋊D7—21no—
82C22×Dic7—42no—
83#1157—41no—
84#1157—41no—
85C2×D28C4⋊C411yescharacteristic
86C2×C4×D7—21no—
87C2×Dic14C4⋊C411yescharacteristic
88C7×M16—42no—
89C7×M16—42no—
90C2×C7×C8—22no—
91C2×C7×C8—22no—
92C2×C7×C8—22no—
93C2×C7×C8—41no—
94C2×C7×C8—41no—
95C2×C7×C8—41no—
96C2×C7×C8—42no—
97C2×C7×C8C22⋊C411yescharacteristic
98C2×C7×C8C22⋊C411yescharacteristic
99C2×C7×C8C22⋊C411yescharacteristic
100C2×C7×C8C22⋊C411yescharacteristic
101C22⋊Dic7—21no—
102C22⋊Dic7—21no—
103C22⋊Dic7—21no—
104C22⋊Dic7—82no—
105(C2×D7)⋊C4—21no—
106(C2×D7)⋊C4—21no—
107C4⋊Dic7—42no—
108C4⋊Dic7—42no—
109C4⋊Dic7C4⋊C411yescharacteristic
110C4⋊Dic7C4⋊C411yescharacteristic
111Dic7⋊C4—21no—
112Dic7⋊C4—21no—
113C4×Dic7—21no—
114#1136—41no—
115#1136—41no—
116C2×C7⋊C8—21no—
117C2×C7⋊C8—21no—
118#1131—42no—
119C8×D7—42no—
120C2×#131—71no—
121C23×C8C2×D711yescharacteristic
122#525—71no—
123#435—141no—
124#435—141no—
125C8×D4—141no—
126C8×D4—141no—
127C2×#108—21no—
128C2×#107—141no—
129#424—71no—
130#408—71no—
131#407—71no—
132#407C2×D712yes—
133#405—71no—
134#380—72no—
135#341—21no—
136#340—141no—
137#328—142no—
138#327—142no—
139C23×C7—21no—
140C23×C7—21no—
141C23×C7—41no—
142C23×C7—41no—
143C23×C7#11711yescharacteristic
144C23×C7#12111yescharacteristic
145C23×C7C22⋊Q811yescharacteristic
146C22×D7—21no—
147C2×C4×C7—21no—
148C2×C4×C7—21no—
149C2×C4×C7—21no—
150C2×C4×C7—21no—
151C2×C4×C7—21no—
152C2×C4×C7—21no—
153C2×C4×C7—41no—
154C2×C4×C7—41no—
155C2×C4×C7—41no—
156C2×C4×C7C2×C22⋊C411yescharacteristic
157C2×C4×C7C2×C4⋊C411yescharacteristic
158C2×C4×C7#12111yescharacteristic
159C2×C4×C7C22⋊Q811yescharacteristic
160C22⋊D7—42no—
161C22⋊D7—42no—
162C22⋊D7—41no—
163C22⋊D7—41no—
164C2×Dic7—21no—
165C2×Dic7—21no—
166C2×Dic7—21no—
167C2×Dic7—82no—
168C2×Dic7—82no—
169C2×Dic7—82no—
170D28—41no—
171C4×D7—41no—
172C4×D7—41no—
173Dic14—41no—
174C7×C8—21no—
175C7×C8—21no—
176C7×C8—21no—
177C7×C8—82no—
178C7×C8#9912yes—
179C7⋊C8—41no—
180C7⋊C8—41no—
181C2×#41—71no—
182C23×C4C22×D711yescharacteristic
183#131—282no—
184C2×M16—22no—
185C2×M16—71no—
186C2×M16—71no—
187C2×M16—141no—
188C22×C8—42no—
189C22×C8—141no—
190C22×C8C4×D711yescharacteristic
191C22×C8C4×D711yescharacteristic
192C22×C8D2811yescharacteristic
193C22×C8D2811yescharacteristic
194C22×C8C22⋊D711yescharacteristic
195C22×C8C22⋊D711yescharacteristic
196#123—141no—
197C22⋊Q8—71no—
198C22⋊Q8—71no—
199#121—71no—
200#121—71no—
201#120—141no—
202C4×D4—141no—
203C4×D4—141no—
204#117—71no—
205C2×C4⋊C4—142no—
206C2×C22⋊C4—142no—
207#108—42no—
208#107—282no—
209C4⋊C8—141no—
210C4⋊C8—141no—
211C22⋊C8—22no—
212C22⋊C8—22no—
213C22⋊C8—141no—
214C22⋊C8—141no—
215C22⋊C8—141no—
216C22⋊C8—141no—
217#97—141no—
218C4×C8—141no—
219#95—142no—
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—42no—
229C22×C7—42no—
230C22×C7#38011yescharacteristic
231C22×C7#47211yescharacteristic
232C22×C7#47311yescharacteristic
233C2×D7—41no—
234C2×D7—41no—
235C2×D7—41no—
236C4×C7—21no—
237C4×C7—21no—
238C4×C7—21no—
239C4×C7C2×#9911yescharacteristic
240C4×C7#41011yescharacteristic
241Dic7—41no—
242Dic7—41no—
243Dic7—41no—
244Dic7—162no—
245C24#116111yescharacteristic
246#41—281no—
247#41—281no—
248C2×Q8—71no—
249C2×D4—71no—
250C2×D4—141no—
251C2×D4—141no—
252C2×D4—141no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—141no—
256C22×C4—282no—
257C22×C4C2×C4×D711yescharacteristic
258C22×C4C2×D2811yescharacteristic
259C22×C4D4×D711yescharacteristic
260C22×C4#115911yescharacteristic
261C22×C4Q8×D711yescharacteristic
262C22×C4C2×C22⋊D711yescharacteristic
263M16—42no—
264M16—42no—
265M16—282no—
266M16—281no—
267M16—281no—
268C2×C8—22no—
269C2×C8—22no—
270C2×C8—22no—
271C2×C8—41no—
272C2×C8—41no—
273C2×C8—41no—
274C2×C8—42no—
275C2×C8—141no—
276C2×C8—141no—
277C2×C8—282no—
278C2×C8(C2×D7)⋊C411yescharacteristic
279C2×C8(C2×D7)⋊C411yescharacteristic
280C2×C8(C2×D7)⋊C411yescharacteristic
281C2×C8(C2×D7)⋊C411yescharacteristic
282C4⋊C4—71no—
283C4⋊C4—71no—
284C4⋊C4—141no—
285C4⋊C4—141no—
286C4⋊C4—282no—
287C4⋊C4—282no—
288C22⋊C4—141no—
289C22⋊C4—141no—
290C22⋊C4—141no—
291C22⋊C4—141no—
292C22⋊C4—141no—
293C22⋊C4—562no—
294C42—141no—
295C2×C7—21no—
296C2×C7—21no—
297C2×C7—41no—
298C2×C7—41no—
299C2×C7#179911yescharacteristic
300C2×C7#181411yescharacteristic
301C2×C7#181611yescharacteristic
302D7—81no—
303C23—21no—
304C23—21no—
305C23—41no—
306C23—41no—
307C23—141no—
308C23#645511yescharacteristic
309C23#645811yescharacteristic
310C23#650911yescharacteristic
311Q8—281no—
312D4—281no—
313D4—282no—
314D4—282no—
315D4—281no—
316D4—281no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—141no—
327C2×C4—141no—
328C2×C4—141no—
329C2×C4—281no—
330C2×C4—281no—
331C2×C4—562no—
332C2×C4—562no—
333C2×C4—562no—
334C2×C4D7×C4⋊C411yescharacteristic
335C2×C4#646011yescharacteristic
336C2×C4C2×(C2×D7)⋊C411yescharacteristic
337C2×C4#650111yescharacteristic
338C8—21no—
339C8—21no—
340C8—21no—
341C8—82no—
342C8—281no—
343C8—281no—
344C8#638012yes—
345C7#862711yescharacteristic, complemented, Sylow
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—42no—
355C22—42no—
356C22—281no—
357C22—281no—
358C22—281no—
359C22#8966811yescharacteristic
360C22#8966911yescharacteristic
361C22#8975911yescharacteristic
362C4—21no—
363C4—21no—
364C4—21no—
365C4—281no—
366C4—281no—
367C4—281no—
368C4—1122no—
369C4C2×#638011yescharacteristic
370C4#8977811yescharacteristic
371C2—21no—
372C2—21no—
373C2—41no—
374C2—41no—
375C2—561no—
376C2#1172728411yescharacteristic
377C2#1172847311yescharacteristic
378C2#1172847511yescharacteristic
subgroupG/H
centerC22#89759
commutatorC22×C4×C7C22×C4
FrattiniC23×C4C22×D7
FittingC7×#1810C2
socleC22×C7#380
prime radical#90482C22
cosocleC23×C4×C7C23
hypercenter#1810D7
hypocenterC7#8627
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. #11729725C2semidirect
5. #11725491C2semidirect
8. C2×#6464C4semidirect
15. #89882C4semidirect
19. #89874C4semidirect
20. #89873C4semidirect
66. #1810D7semidirect
345. C7#8627semidirect

Maximal subgroups

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

subgroupindex
#117411792
C7×#18102
#117297252
#117277332
#117254912
#86277