Finite Group Databasebeta

← #13246772

Subgroups — #13246772

Order 1792.

456 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 97 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#11736071C211yesmaximal, characteristic
2#11727736C211yesmaximal, characteristic, complemented
3#11727733C211yesmaximal, characteristic
4#11727266C211yesmaximal, characteristic, complemented
5#11727262C211yesmaximal, characteristic
6C7×#1305C211yesmaximal, characteristic, complemented
7#11725490C211yesmaximal, characteristic, complemented
8#90482C2211yescharacteristic
9C23×C7⋊C8C2211yescharacteristic
10C2×#6470C411yescharacteristic, complemented
11C7×#407C2211yescharacteristic
12C7×#407C2211yescharacteristic
13C7×#380C2211yescharacteristic
14#89984C2211yescharacteristic
15#89979C2211yescharacteristic
16#89901C411yescharacteristic, complemented
17C2×#6396—21no—
18#89894C411yescharacteristic, complemented
19#89891C411yescharacteristic, complemented
20C2×#6381—21no—
21#89502—21no—
22#89502—21no—
23(C22⋊D7)⋊C8—21no—
24(C22⋊D7)⋊C8—21no—
25#89353—21no—
26#89324—21no—
27#89265—21no—
28#89262—21no—
29#89247—21no—
30#89246—21no—
31#8627—71nomaximal, Sylow
32C23×C4×C7C2311yescharacteristic
33C2×#1157C2×C411yescharacteristic
34C2×C7×M16—21no—
35C2×C7×M16C2×C411yescharacteristic
36C2×C7×M16C2×C411yescharacteristic
37C2×C7×C4⋊C4—21no—
38C2×C7×C22⋊C4—21no—
39#6516—21no—
40C2×C22⋊Dic7—21no—
41#6493D411yescharacteristic
42#6493Q811yescharacteristic
43#6492—21no—
44C4×C22⋊D7—21no—
45C4×C22⋊D7—21no—
46#6489C2×C411yescharacteristic
47C2×C4⋊Dic7—21no—
48#6487D411yescharacteristic
49#6487Q811yescharacteristic
50C2×#1136—21no—
51C22×C7⋊C8—42no—
52C22×C7⋊C8C2×C411yescharacteristic
53C22×C7⋊C8C2×C411yescharacteristic
54C22×C7⋊C8D411yescharacteristic
55C22×C7⋊C8D411yescharacteristic
56C22×C7⋊C8D411yescharacteristic
57C22×C7⋊C8D411yescharacteristic
58#6470—42no—
59C2×#1131—21no—
60C2×C8×D7—21no—
61C7×C22⋊C8—21no—
62C7×C22⋊C8—21no—
63C7×C22⋊C8—21no—
64C7×C22⋊C8—21no—
65C7×#95—21no—
66C7⋊#95—21no—
67C22⋊(C7⋊C8)—21no—
68C22⋊(C7⋊C8)—21no—
69#6396—42no—
70(C2×D7)⋊C8—21no—
71(C2×D7)⋊C8—21no—
72#6389—21no—
73Dic7⋊C8—21no—
74Dic7⋊C8—21no—
75C8×Dic7—21no—
76#6381—42no—
77#2929—71no—
78#1810—71no—
79#1808—71no—
80#1513—71no—
81#1513—71no—
82#1305—71no—
83#1305D711yescharacteristic, complemented
84C24×C7#4111yescharacteristic
85C22×C4×C7—41no—
86C22×C4×C7—41no—
87C22×C4×C7—41no—
88C22×C4×C7C22×C411yescharacteristic
89C22×C4×C7C2×D411yescharacteristic
90C22×C4×C7C2×D411yescharacteristic
91C22×C4×C7C2×D411yescharacteristic
92C22×C4×C7C2×Q811yescharacteristic
93C22×C4×C7#4111yescharacteristic
94C2×C22⋊D7—21no—
95C2×C22⋊D7—21no—
96C2×C22⋊D7—21no—
97C22×Dic7—41no—
98#1157—41no—
99#1157—41no—
100C2×D28C4⋊C411yescharacteristic
101C2×C4×D7—21no—
102C2×Dic14C4⋊C411yescharacteristic
103C7×M16—41no—
104C7×M16—41no—
105C7×M16—41no—
106C7×M16—41no—
107C2×C7×C8—21no—
108C2×C7×C8—21no—
109C2×C7×C8—41no—
110C7×C4⋊C4—41no—
111C7×C4⋊C4—41no—
112C7×C22⋊C4—81no—
113C22⋊Dic7—21no—
114C22⋊Dic7—21no—
115C22⋊Dic7—21no—
116C22⋊Dic7—81no—
117(C2×D7)⋊C4—21no—
118(C2×D7)⋊C4—21no—
119C4⋊Dic7—41no—
120C4⋊Dic7—41no—
121C4⋊Dic7C4⋊C411yescharacteristic
122C4⋊Dic7C4⋊C411yescharacteristic
123Dic7⋊C4—21no—
124Dic7⋊C4—21no—
125C4×Dic7—21no—
126#1136—41no—
127#1136—41no—
128C2×C7⋊C8—22no—
129C2×C7⋊C8—22no—
130C2×C7⋊C8—22no—
131C2×C7⋊C8—41no—
132C2×C7⋊C8—41no—
133C2×C7⋊C8—41no—
134C2×C7⋊C8—41no—
135C2×C7⋊C8C22⋊C411yescharacteristic
136C2×C7⋊C8C22⋊C411yescharacteristic
137C2×C7⋊C8C22⋊C411yescharacteristic
138C2×C7⋊C8C22⋊C411yescharacteristic
139#1131—42no—
140C8×D7—42no—
141C2×#131—71no—
142C23×C8—71no—
143#525—71no—
144#435—141no—
145#435—141no—
146C8×D4—141no—
147C8×D4—141no—
148C2×#108—141no—
149C2×#107—141no—
150#424—71no—
151#408—71no—
152#407—71no—
153#407C2×D711yescharacteristic
154#407C2×D711yescharacteristic
155#405—71no—
156#380—71no—
157#380C2×D711yescharacteristic
158#341—141no—
159#340—141no—
160#328—141no—
161#328—141no—
162#327—141no—
163#327—141no—
164C23×C7—21no—
165C23×C7—21no—
166C23×C7—41no—
167C23×C7—41no—
168C23×C7#11711yescharacteristic
169C23×C7#12111yescharacteristic
170C23×C7C22⋊Q811yescharacteristic
171C22×D7—21no—
172C2×C4×C7—21no—
173C2×C4×C7—21no—
174C2×C4×C7—21no—
175C2×C4×C7—21no—
176C2×C4×C7—21no—
177C2×C4×C7—21no—
178C2×C4×C7—41no—
179C2×C4×C7—41no—
180C2×C4×C7—41no—
181C2×C4×C7—81no—
182C2×C4×C7—81no—
183C2×C4×C7—81no—
184C2×C4×C7C2×C22⋊C411yescharacteristic
185C2×C4×C7C2×C4⋊C411yescharacteristic
186C2×C4×C7#12111yescharacteristic
187C2×C4×C7C22⋊Q811yescharacteristic
188C22⋊D7—42no—
189C22⋊D7—42no—
190C22⋊D7—41no—
191C22⋊D7—41no—
192C2×Dic7—21no—
193C2×Dic7—21no—
194C2×Dic7—21no—
195C2×Dic7—81no—
196C2×Dic7—81no—
197C2×Dic7—81no—
198D28—41no—
199C4×D7—41no—
200C4×D7—41no—
201Dic14—41no—
202C7×C8—41no—
203C7×C8—41no—
204C7×C8—81no—
205C7⋊C8—21no—
206C7⋊C8—21no—
207C7⋊C8—21no—
208C7⋊C8—81no—
209C7⋊C8#9912yes—
210C2×#41—71no—
211C23×C4C22×D711yescharacteristic
212#131—282no—
213C2×M16—21no—
214C2×M16—141no—
215C2×M16—141no—
216C2×M16C4×D711yescharacteristic
217C2×M16C4×D711yescharacteristic
218C22×C8—71no—
219C22×C8—71no—
220C22×C8—71no—
221C22×C8—71no—
222C22×C8—71no—
223C22×C8—71no—
224C22×C8—141no—
225C22×C8—282no—
226#123—141no—
227C22⋊Q8—71no—
228C22⋊Q8—71no—
229#121—71no—
230#121—71no—
231#120—141no—
232C4×D4—141no—
233C4×D4—141no—
234#117—71no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—141no—
237C2×C22⋊C4—21no—
238C2×C22⋊C4—141no—
239#108—282no—
240#107—282no—
241C4⋊C8—141no—
242C4⋊C8—141no—
243C22⋊C8—21no—
244C22⋊C8—21no—
245C22⋊C8—21no—
246C22⋊C8—21no—
247C22⋊C8—141no—
248C22⋊C8—141no—
249C22⋊C8—141no—
250C22⋊C8—141no—
251#97—141no—
252C4×C8—141no—
253#95—21no—
254#95—141no—
255C22×C7—21no—
256C22×C7—21no—
257C22×C7—21no—
258C22×C7—21no—
259C22×C7—21no—
260C22×C7—21no—
261C22×C7—21no—
262C22×C7—21no—
263C22×C7—42no—
264C22×C7—42no—
265C22×C7#38011yescharacteristic
266C22×C7#47211yescharacteristic
267C22×C7#47311yescharacteristic
268C2×D7—41no—
269C2×D7—41no—
270C2×D7—41no—
271C4×C7—21no—
272C4×C7—21no—
273C4×C7—21no—
274C4×C7—161no—
275C4×C7C2×#9911yescharacteristic
276C4×C7#41011yescharacteristic
277Dic7—41no—
278Dic7—41no—
279Dic7—41no—
280Dic7—161no—
281C24#115711yescharacteristic
282#41—281no—
283#41—281no—
284C2×Q8—71no—
285C2×D4—71no—
286C2×D4—141no—
287C2×D4—141no—
288C2×D4—141no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—141no—
293C22×C4—281no—
294C22×C4C2×C4×D711yescharacteristic
295C22×C4#115711yescharacteristic
296C22×C4D4×D711yescharacteristic
297C22×C4D4×D711yescharacteristic
298C22×C4D4×D711yescharacteristic
299C22×C4Q8×D711yescharacteristic
300M16—41no—
301M16—41no—
302M16—41no—
303M16—41no—
304M16—282no—
305M16—281no—
306M16—281no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—41no—
310C2×C8—71no—
311C2×C8—71no—
312C2×C8—71no—
313C2×C8—71no—
314C2×C8—142no—
315C2×C8—142no—
316C2×C8—142no—
317C2×C8—282no—
318C2×C8—281no—
319C2×C8—281no—
320C2×C8—281no—
321C2×C8—281no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—71no—
325C4⋊C4—71no—
326C4⋊C4—141no—
327C4⋊C4—141no—
328C4⋊C4—281no—
329C4⋊C4—281no—
330C22⋊C4—81no—
331C22⋊C4—141no—
332C22⋊C4—141no—
333C22⋊C4—141no—
334C22⋊C4—141no—
335C22⋊C4—141no—
336C22⋊C4—561no—
337C42—141no—
338C2×C7—21no—
339C2×C7—21no—
340C2×C7—41no—
341C2×C7—41no—
342C2×C7#179911yescharacteristic
343C2×C7#181411yescharacteristic
344C2×C7#181611yescharacteristic
345D7—81no—
346C23—21no—
347C23—21no—
348C23—41no—
349C23—41no—
350C23—141no—
351C23#643511yescharacteristic
352C23#644711yescharacteristic
353C23#645911yescharacteristic
354Q8—281no—
355D4—281no—
356D4—282no—
357D4—282no—
358D4—281no—
359D4—281no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—141no—
373C2×C4—141no—
374C2×C4—141no—
375C2×C4—281no—
376C2×C4—281no—
377C2×C4—561no—
378C2×C4—561no—
379C2×C4—561no—
380C2×C4D7×C22⋊C411yescharacteristic
381C2×C4#644711yescharacteristic
382C2×C4D7×C4⋊C411yescharacteristic
383C2×C4#645911yescharacteristic
384C8—41no—
385C8—41no—
386C8—72no—
387C8—81no—
388C8—141no—
389C8—141no—
390C8—141no—
391C8—561no—
392C7#862711yescharacteristic, complemented, Sylow
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—42no—
402C22—42no—
403C22—281no—
404C22—281no—
405C22—281no—
406C22#8943911yescharacteristic
407C22#8952111yescharacteristic
408C22#8952711yescharacteristic
409C4—21no—
410C4—21no—
411C4—21no—
412C4—161no—
413C4—281no—
414C4—281no—
415C4—281no—
416C4—1121no—
417C4#8951211yescharacteristic
418C4D7×#9911yescharacteristic
419C2—21no—
420C2—21no—
421C2—41no—
422C2—41no—
423C2—561no—
424C2#1172671311yescharacteristic
425C2#1172675511yescharacteristic
426C2#1172678611yescharacteristic
subgroupG/H
centerC22#89439
commutatorC22×C4×C7C22×C4
FrattiniC23×C4C22×D7
FittingC7×#1305C2
socleC22×C7#380
prime radical#90482C22
cosocleC23×C4×C7C23
hypercenter#1305D7
hypocenterC7#8627
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1305

Complements

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

NH ≅ G/Ntype
2. #11727736C2semidirect
4. #11727266C2semidirect
6. C7×#1305C2semidirect
7. #11725490C2semidirect
10. C2×#6470C4semidirect
16. #89901C4semidirect
18. #89894C4semidirect
19. #89891C4semidirect
83. #1305D7semidirect
392. C7#8627semidirect

Maximal subgroups

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

subgroupindex
#117360712
#117277362
#117277332
#117272662
#117272622
C7×#13052
#117254902
#86277