Finite Group Databasebeta

← #13242271

Subgroups — #13242271

Order 1792.

464 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 103 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
1#11733870C211yesmaximal, characteristic
2#11726357C211yesmaximal, characteristic, complemented
3#11726356C211yesmaximal, characteristic, complemented
4C7×#1270C211yesmaximal, characteristic
5#11725445C211yesmaximal, characteristic
6#11725284C211yesmaximal, characteristic, complemented
7#11725283C211yesmaximal, characteristic, complemented
8C2×#6442—21no—
9C7×#390C2211yescharacteristic
10#89993C411yescharacteristic
11#89993C411yescharacteristic
12C2×#6389—21no—
13C2×Dic7⋊C8—21no—
14C2×Dic7⋊C8—21no—
15C2×C8×Dic7—21no—
16#89757C411yescharacteristic
17#89745C411yescharacteristic
18#89723—21no—
19#89717—21no—
20#89716—21no—
21C42×Dic7C2211yescharacteristic
22#89700C2211yescharacteristic
23C7×#336C2211yescharacteristic
24C7×#336C2211yescharacteristic
25#89319C2211yescharacteristic
26#89319C2211yescharacteristic
27#7686—71nomaximal, Sylow
28C22×C7×D4C2×C411yescharacteristic
29C22×C7×C8—21no—
30C22×C7×C8—21no—
31C2×C7×C4⋊C4C2×C411yescharacteristic
32C2×C7×C22⋊C4—21no—
33C2×C7×C22⋊C4—21no—
34C2×C42×C7C2311yescharacteristic
35C2×C22⋊Dic7—21no—
36C2×C22⋊Dic7—21no—
37C2×C22⋊Dic7—21no—
38C2×C22⋊Dic7—21no—
39C2×C4⋊Dic7—21no—
40C2×C4⋊Dic7C2×C411yescharacteristic
41C2×C4⋊Dic7C2×C411yescharacteristic
42C2×Dic7⋊C4—21no—
43C2×Dic7⋊C4—21no—
44C2×C4×Dic7C2×C411yescharacteristic
45C2×C4×Dic7C2×C411yescharacteristic
46C2×C4×Dic7D411yescharacteristic
47C2×C4×Dic7D411yescharacteristic
48C2×C4×Dic7D411yescharacteristic
49C2×C4×Dic7D411yescharacteristic
50C22×C7⋊C8—21no—
51C22×C7⋊C8—21no—
52#6442—42no—
53#6442—42no—
54C7⋊#95—21no—
55C7⋊#95—21no—
56C7⋊#95—21no—
57#6389—42no—
58Dic7⋊C8—24no—
59Dic7⋊C8—24no—
60C8×Dic7—42no—
61#2860—71no—
62#1767—71no—
63C4×C4⋊C8—71no—
64#1271—71no—
65#1271—71no—
66#1270—71no—
67#1270D711yescharacteristic
68C24×C7—21no—
69C2×C7×D4—21no—
70C2×C7×D4—21no—
71C2×C7×D4—41no—
72C2×C7×D4—41no—
73C22×C4×C7—21no—
74C22×C4×C7C22×C411yescharacteristic
75C22×C4×C7C2×D411yescharacteristic
76C22×C4×C7C2×D411yescharacteristic
77C22×Dic7—21no—
78C22×Dic7—21no—
79C22×Dic7C22⋊C411yescharacteristic
80C22×Dic7C22⋊C411yescharacteristic
81C22×Dic7C22⋊C411yescharacteristic
82C22×Dic7C22⋊C411yescharacteristic
83C2×C7×C8—22no—
84C2×C7×C8—22no—
85C2×C7×C8—41no—
86C2×C7×C8—41no—
87C2×C7×C8—41no—
88C2×C7×C8—41no—
89C7×C4⋊C4—41no—
90C7×C22⋊C4—41no—
91C7×C22⋊C4—41no—
92C7×C22⋊C4—41no—
93C7×C22⋊C4—41no—
94C42×C7—41no—
95C22⋊Dic7—42no—
96C22⋊Dic7—42no—
97C22⋊Dic7—42no—
98C22⋊Dic7—42no—
99C4⋊Dic7—21no—
100C4⋊Dic7—21no—
101C4⋊Dic7—42no—
102C4⋊Dic7—41no—
103Dic7⋊C4—41no—
104Dic7⋊C4—41no—
105Dic7⋊C4—42no—
106C4×Dic7—21no—
107C4×Dic7—21no—
108C4×Dic7—21no—
109C4×Dic7—21no—
110C4×Dic7—22no—
111C4×Dic7—22no—
112C4×Dic7—41no—
113C4×Dic7D812yes—
114C4×Dic7SD1612yes—
115C2×C7⋊C8—22no—
116C2×C7⋊C8—22no—
117C2×C7⋊C8—41no—
118C2×C7⋊C8—41no—
119C2×C7⋊C8—41no—
120C2×C7⋊C8—41no—
121C2×#126—141no—
122C2×C4⋊C8—141no—
123C2×C4⋊C8—141no—
124C2×#97—141no—
125C2×C4×C8—141no—
126#390—71no—
127#390—71no—
128#390C2×D711yescharacteristic
129#389—71no—
130#389—71no—
131#389—71no—
132#388—141no—
133#388—141no—
134#388—141no—
135C43—71no—
136#336—71no—
137#336—71no—
138#336C2×D711yescharacteristic
139#336C2×D711yescharacteristic
140C23×C7—41no—
141C23×C7—41no—
142C23×C7—41no—
143C23×C7—41no—
144C23×C7—41no—
145C23×C7—41no—
146C23×C7—41no—
147C23×C7C2×C22⋊C411yescharacteristic
148C7×D4—81no—
149C7×D4—81no—
150C2×C4×C7—21no—
151C2×C4×C7—21no—
152C2×C4×C7—21no—
153C2×C4×C7—21no—
154C2×C4×C7—21no—
155C2×C4×C7—21no—
156C2×C4×C7—41no—
157C2×C4×C7—41no—
158C2×C4×C7—41no—
159C2×C4×C7#13111yescharacteristic
160C2×C4×C7#13111yescharacteristic
161C2×C4×C7C2×D811yescharacteristic
162C2×C4×C7C2×SD1611yescharacteristic
163C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
164C2×C4×C7#13711yescharacteristic
165C2×Dic7—21no—
166C2×Dic7—21no—
167C2×Dic7—21no—
168C2×Dic7—21no—
169C2×Dic7—21no—
170C2×Dic7—21no—
171C2×Dic7—21no—
172C2×Dic7—21no—
173C2×Dic7—41no—
174C2×Dic7—41no—
175C2×Dic7—41no—
176C2×Dic7—41no—
177C2×Dic7—41no—
178C2×Dic7—41no—
179C2×Dic7#10212yes—
180C2×Dic7#10212yes—
181C2×Dic7#10412yes—
182C2×Dic7#10412yes—
183C7×C8—42no—
184C7×C8—42no—
185C7⋊C8—42no—
186C7⋊C8—42no—
187C22×D4C4×D711yescharacteristic
188C22×C8—21no—
189C22×C8—21no—
190C22×C8—141no—
191C22×C8—141no—
192#126—282no—
193#126—282no—
194C2×C4⋊C4—71no—
195C2×C4⋊C4—71no—
196C2×C4⋊C4—141no—
197C2×C4⋊C4—141no—
198C2×C4⋊C4—141no—
199C2×C4⋊C4C4×D711yescharacteristic
200C2×C22⋊C4—21no—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—141no—
203C2×C22⋊C4—141no—
204C2×C22⋊C4—141no—
205C2×C22⋊C4—141no—
206C2×C42—71no—
207C2×C42—71no—
208C2×C42—71no—
209C2×C42—71no—
210C2×C42—71no—
211C2×C42—71no—
212C2×C42C22×D711yescharacteristic
213C4⋊C8—144no—
214C4⋊C8—144no—
215#97—282no—
216C4×C8—282no—
217#95—141no—
218#95—141no—
219#95—141no—
220C22×C7—81no—
221C22×C7—81no—
222C22×C7—81no—
223C22×C7—81no—
224C22×C7—81no—
225C22×C7—81no—
226C22×C7—81no—
227C22×C7#40811yescharacteristic
228C22×C7#41011yescharacteristic
229C22×C7#41311yescharacteristic
230C22×C7C2×#10211yescharacteristic
231C22×C7#41711yescharacteristic
232C22×C7C2×#10411yescharacteristic
233C22×C7C2×#10411yescharacteristic
234C4×C7—21no—
235C4×C7—21no—
236C4×C7—21no—
237C4×C7—21no—
238C4×C7—21no—
239C4×C7—21no—
240C4×C7—81no—
241Dic7—22no—
242Dic7—22no—
243Dic7—22no—
244Dic7—22no—
245Dic7—81no—
246Dic7—81no—
247C24—21no—
248C2×D4—21no—
249C2×D4—21no—
250C2×D4—41no—
251C2×D4—41no—
252C22×C4—21no—
253C22×C4—71no—
254C22×C4—71no—
255C22×C4—71no—
256C22×C4—71no—
257C22×C4—141no—
258C22×C4—141no—
259C22×C4C2×C4×D711yescharacteristic
260C22×C4D4×D711yescharacteristic
261C22×C4D4×D711yescharacteristic
262C2×C8—22no—
263C2×C8—22no—
264C2×C8—41no—
265C2×C8—41no—
266C2×C8—41no—
267C2×C8—41no—
268C2×C8—142no—
269C2×C8—142no—
270C2×C8—281no—
271C2×C8—281no—
272C2×C8—281no—
273C2×C8—281no—
274C4⋊C4—41no—
275C4⋊C4—141no—
276C4⋊C4—141no—
277C4⋊C4—281no—
278C4⋊C4—282no—
279C4⋊C4—282no—
280C4⋊C4—281no—
281C4⋊C4—281no—
282C22⋊C4—41no—
283C22⋊C4—41no—
284C22⋊C4—41no—
285C22⋊C4—41no—
286C22⋊C4—282no—
287C22⋊C4—282no—
288C22⋊C4—282no—
289C22⋊C4—282no—
290C42—41no—
291C42—72no—
292C42—72no—
293C42—142no—
294C42—142no—
295C42—141no—
296C42—141no—
297C42—141no—
298C42—141no—
299C42—281no—
300C2×C7—161no—
301C2×C7#147311yescharacteristic
302C2×C7#148311yescharacteristic
303C2×C7#149211yescharacteristic
304C2×C7#150211yescharacteristic
305C2×C7#150911yescharacteristic
306C2×C7#152411yescharacteristic
307C2×C7#154711yescharacteristic
308C23—41no—
309C23—41no—
310C23—41no—
311C23—41no—
312C23—41no—
313C23—41no—
314C23—41no—
315C23D7×C22⋊C411yescharacteristic
316D4—81no—
317D4—81no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—72no—
328C2×C4—72no—
329C2×C4—72no—
330C2×C4—72no—
331C2×C4—141no—
332C2×C4—141no—
333C2×C4—141no—
334C2×C4—141no—
335C2×C4—141no—
336C2×C4—141no—
337C2×C4—141no—
338C2×C4—141no—
339C2×C4—281no—
340C2×C4—281no—
341C2×C4—281no—
342C2×C4—281no—
343C2×C4—281no—
344C2×C4—281no—
345C2×C4#646411yescharacteristic
346C2×C4#647011yescharacteristic
347C2×C4D7×D811yescharacteristic
348C2×C4#647411yescharacteristic
349C2×C4D7×SD1611yescharacteristic
350C2×C4#647811yescharacteristic
351C8—42no—
352C8—42no—
353C8—282no—
354C8—282no—
355C7#768611yescharacteristic, complemented, Sylow
356C22—81no—
357C22—81no—
358C22—81no—
359C22—81no—
360C22—81no—
361C22—81no—
362C22—81no—
363C22#8949911yescharacteristic
364C22#8951211yescharacteristic
365C22#8951711yescharacteristic
366C22D7×#10211yescharacteristic
367C22#8954311yescharacteristic
368C22D7×#10411yescharacteristic
369C22D7×#10411yescharacteristic
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—81no—
377C4—142no—
378C4—142no—
379C4—142no—
380C4—142no—
381C4—561no—
382C4—561no—
383C2—161no—
384C2#1172585911yescharacteristic
385C2#1172592411yescharacteristic
386C2#1172593011yescharacteristic
387C2#1172603111yescharacteristic
388C2#1172603311yescharacteristic
389C2#1172609511yescharacteristic
390C2#1172613811yescharacteristic
subgroupG/H
centerC23D7×C22⋊C4
commutatorC22×C4×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#1270C2
socleC23×C7C2×C22⋊C4
prime radicalC22×C7×D4C2×C4
cosocleC2×C42×C7C23
hypercenter#1270D7
hypocenterC7#7686
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1270

Complements

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

NH ≅ G/Ntype
2. #11726357C2semidirect
3. #11726356C2semidirect
6. #11725284C2semidirect
7. #11725283C2semidirect
355. C7#7686semidirect

Maximal subgroups

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

subgroupindex
#117338702
#117263572
#117263562
C7×#12702
#117254452
#117252842
#117252832
#76867