Finite Group Databasebeta

← #13264744

Subgroups — #13264744

Order 1792.

358 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 79 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#11738142C211yesmaximal, characteristic
2#11726847C211yesmaximal, characteristic, complemented
3#11726602C211yesmaximal, characteristic, complemented
4C7×#1347C211yesmaximal, characteristic, complemented
5#11725580C211yesmaximal, characteristic, complemented
6#11725332C211yesmaximal, characteristic, complemented
7#11725291C211yesmaximal, characteristic, complemented
8#90287—21no—
9C7×#466C2211yescharacteristic
10#89923C411yescharacteristic, complemented
11D8×Dic7C411yescharacteristic, complemented
12#89887C2211yescharacteristic
13C2×C8×Dic7C2211yescharacteristic
14#89810—21no—
15#89749—21no—
16#89646C411yescharacteristic, complemented
17#89644C411yescharacteristic, complemented
18#89532—21no—
19#89528—21no—
20C7×C22⋊C16C2211yescharacteristic
21C7×#340C2211yescharacteristic
22#89346C2211yescharacteristic
23C22⋊(C7⋊C16)C2211yescharacteristic
24#89324—21no—
25#89297—21no—
26Dic7⋊C16—21no—
27#11858—71nomaximal, Sylow
28C2×#1159—41no—
29C2×C7×D8C2×C411yescharacteristic
30C22×C7×C8C2311yescharacteristic
31C7×#121—21no—
32C2×C7×C4⋊C4—21no—
33#6503—21no—
34#6502—41no—
35D4×Dic7—21no—
36C2×#1141—21no—
37#6493—21no—
38C2×C4⋊Dic7—21no—
39C2×C4×Dic7D411yescharacteristic
40C22×C7⋊C8D411yescharacteristic
41C2×D56C2×C411yescharacteristic
42#6456—21no—
43#6451—21no—
44#6448—41no—
45(C22⋊D7)⋊C4—41no—
46C2×C7×C16—21no—
47C7×#107C2×C411yescharacteristic
48C7×#102—21no—
49#6406—21no—
50C7⋊#95—41no—
51C7⋊#95—41no—
52#6395—21no—
53#6392C2×C411yescharacteristic
54C8×Dic7C2×C411yescharacteristic
55C8×Dic7C2×C411yescharacteristic
56C8×Dic7D411yescharacteristic
57C8×Dic7D411yescharacteristic
58C2×C7⋊C16—21no—
59#6383—21no—
60#6381—21no—
61#3169—71no—
62#2163—71no—
63#1835—71no—
64#1347—71no—
65#1347—71no—
66#1347—71no—
67#1347D711yescharacteristic, complemented
68C2×C7×D4—21no—
69C2×C7×D4—41no—
70C22×C4×C7—41no—
71C22×C4×C7C2×D411yescharacteristic
72C2×C22⋊D7—41no—
73C2×C22⋊D7—41no—
74C2×C22⋊D7—41no—
75C22×Dic7—41no—
76C22×Dic7—41no—
77C22×Dic7D811yescharacteristic
78C22×Dic7D811yescharacteristic
79#1159—82no—
80#1159—82no—
81C2×D28—21no—
82C2×C4×D7—41no—
83C2×Dic14—41no—
84C7×D8—41no—
85C2×C7×C8—41no—
86C2×C7×C8C22×C411yescharacteristic
87C2×C7×C8C2×D411yescharacteristic
88C7×C4⋊C4—21no—
89C7×C4⋊C4—41no—
90C7×C4⋊C4—41no—
91C7×C22⋊C4—41no—
92C22⋊Dic7—41no—
93#1141—42no—
94(C2×D7)⋊C4—41no—
95(C2×D7)⋊C4—41no—
96C4⋊Dic7—21no—
97C4⋊Dic7—41no—
98C4⋊Dic7—41no—
99Dic7⋊C4—41no—
100C4×Dic7C22⋊C411yescharacteristic
101C4×Dic7C22⋊C411yescharacteristic
102C4×Dic7SD1611yescharacteristic
103C4×Dic7SD1611yescharacteristic
104C2×C7⋊C8—41no—
105C2×C7⋊C8C22⋊C411yescharacteristic
106C2×C7⋊C8C22⋊C411yescharacteristic
107D56—41no—
108C7×C16—41no—
109C7⋊C16—41no—
110#532—141no—
111#500—71no—
112#493—71no—
113#486—141no—
114#466—71no—
115#466—141no—
116#466C2×D711yescharacteristic
117C4×D8—71no—
118C4×D8—71no—
119C4×D8—141no—
120C2×C4×C8—71no—
121#382—141no—
122C4⋊C16—141no—
123C22⋊C16—71no—
124C22⋊C16C2×D711yescharacteristic
125#346—141no—
126#340—71no—
127#340—141no—
128#340C2×D711yescharacteristic
129C23×C7—41no—
130C23×C7C2×D811yescharacteristic
131C22×D7—41no—
132C7×D4—41no—
133C7×D4—41no—
134C7×D4—81no—
135C7×D4—81no—
136C2×C4×C7—21no—
137C2×C4×C7—21no—
138C2×C4×C7—41no—
139C2×C4×C7—81no—
140C2×C4×C7—81no—
141C2×C4×C7—81no—
142C2×C4×C7C2×C22⋊C411yescharacteristic
143C2×C4×C7C2×SD1611yescharacteristic
144C22⋊D7—82no—
145C22⋊D7—82no—
146C22⋊D7—81no—
147C22⋊D7—81no—
148C2×Dic7—21no—
149C2×Dic7—21no—
150C2×Dic7—41no—
151C2×Dic7—41no—
152C2×Dic7—82no—
153C2×Dic7—81no—
154C2×Dic7—81no—
155C2×Dic7—81no—
156C2×Dic7#10211yescharacteristic
157C2×Dic7#10211yescharacteristic
158C2×Dic7#10211yescharacteristic
159C2×Dic7#10211yescharacteristic
160D28—41no—
161D28—41no—
162C4×D7—82no—
163Dic14—82no—
164C7×C8—21no—
165C7×C8—21no—
166C7⋊C8—21no—
167C7⋊C8—21no—
168C2×#41—281no—
169C2×D8—71no—
170C2×D8—141no—
171C2×D8C4×D711yescharacteristic
172C22×C8—71no—
173C22×C8C22×D711yescharacteristic
174C4⋊Q8—141no—
175#127—141no—
176#124—281no—
177#121—21no—
178#121—141no—
179#121—281no—
180C4×D4—141no—
181C4×D4—141no—
182C4×D4—281no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—141no—
185C2×C42—71no—
186C2×C16—21no—
187C2×C16—141no—
188#107—71no—
189#107—141no—
190#107C4×D711yescharacteristic
191#102—21no—
192#102—141no—
193#102—141no—
194#102—141no—
195C4×C8—71no—
196C4×C8—71no—
197C4×C8—71no—
198C4×C8—71no—
199#95—281no—
200#95—281no—
201C22×C7—21no—
202C22×C7—21no—
203C22×C7—81no—
204C22×C7—81no—
205C22×C7—81no—
206C22×C7C2×#10211yescharacteristic
207C22×C7#50911yescharacteristic
208C22×C7#50911yescharacteristic
209C2×D7—81no—
210C2×D7—81no—
211C2×D7—81no—
212C4×C7—21no—
213C4×C7—81no—
214C4×C7—161no—
215C4×C7#41011yescharacteristic
216C4×C7#42111yescharacteristic
217Dic7—21no—
218Dic7—21no—
219Dic7—21no—
220Dic7—21no—
221Dic7—81no—
222Dic7—161no—
223#41—562no—
224#41—562no—
225C2×Q8—281no—
226C2×D4—21no—
227C2×D4—41no—
228C2×D4—141no—
229C2×D4—281no—
230C2×D4—281no—
231C2×D4—281no—
232C22×C4—41no—
233C22×C4—71no—
234C22×C4—71no—
235C22×C4—281no—
236C22×C4—281no—
237C22×C4—281no—
238C22×C4D4×D711yescharacteristic
239D8—41no—
240D8—281no—
241D8—282no—
242C2×C8—41no—
243C2×C8—71no—
244C2×C8—71no—
245C2×C8—281no—
246C2×C8C2×C4×D711yescharacteristic
247C2×C8D4×D711yescharacteristic
248C4⋊C4—21no—
249C4⋊C4—41no—
250C4⋊C4—41no—
251C4⋊C4—141no—
252C4⋊C4—281no—
253C4⋊C4—281no—
254C4⋊C4—281no—
255C22⋊C4—41no—
256C22⋊C4—281no—
257C22⋊C4—281no—
258C22⋊C4—281no—
259C42—71no—
260C42—71no—
261C42—71no—
262C42—71no—
263C16—41no—
264C16—281no—
265C2×C7—21no—
266C2×C7—21no—
267C2×C7—161no—
268C2×C7#151711yescharacteristic
269C2×C7#214011yescharacteristic
270C2×C7#214511yescharacteristic
271D7—161no—
272C23—41no—
273C23—281no—
274C23D7×D811yescharacteristic
275Q8—562no—
276D4—41no—
277D4—41no—
278D4—81no—
279D4—81no—
280D4—281no—
281D4—281no—
282D4—562no—
283D4—562no—
284D4—561no—
285D4—561no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—41no—
289C2×C4—71no—
290C2×C4—71no—
291C2×C4—71no—
292C2×C4—71no—
293C2×C4—81no—
294C2×C4—81no—
295C2×C4—81no—
296C2×C4—141no—
297C2×C4—141no—
298C2×C4—281no—
299C2×C4—281no—
300C2×C4—562no—
301C2×C4—562no—
302C2×C4—561no—
303C2×C4—561no—
304C2×C4—561no—
305C2×C4D7×C22⋊C411yescharacteristic
306C2×C4D7×SD1611yescharacteristic
307C8—21no—
308C8—21no—
309C8—141no—
310C8—141no—
311C7#1185811yescharacteristic, complemented, Sylow
312C22—21no—
313C22—21no—
314C22—81no—
315C22—81no—
316C22—81no—
317C22—561no—
318C22—561no—
319C22—561no—
320C22D7×#10211yescharacteristic
321C22#8968311yescharacteristic
322C22#8968611yescharacteristic
323C4—21no—
324C4—81no—
325C4—141no—
326C4—141no—
327C4—141no—
328C4—141no—
329C4—161no—
330C4—561no—
331C4—1121no—
332C4#8951211yescharacteristic
333C4#8959311yescharacteristic
334C2—21no—
335C2—21no—
336C2—161no—
337C2—1121no—
338C2#1172603611yescharacteristic
339C2#1172694411yescharacteristic
340C2#1172700511yescharacteristic
subgroupG/H
centerC22D7×#102
commutatorC2×C7×C8C22×C4
FrattiniC22×C8C22×D7
FittingC7×#1347C2
socleC22×C7C2×#102
prime radical#11738142C2
cosocleC22×C7×C8C23
hypercenter#1347D7
hypocenterC7#11858
LCSG ▷ C2×C7×C8 ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #1347

Complements

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

NH ≅ G/Ntype
2. #11726847C2semidirect
3. #11726602C2semidirect
4. C7×#1347C2semidirect
5. #11725580C2semidirect
6. #11725332C2semidirect
7. #11725291C2semidirect
10. #89923C4semidirect
11. D8×Dic7C4semidirect
16. #89646C4semidirect
17. #89644C4semidirect
67. #1347D7semidirect
311. C7#11858semidirect

Maximal subgroups

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

subgroupindex
#117381422
#117268472
#117266022
C7×#13472
#117255802
#117253322
#117252912
#118587