Finite Group Databasebeta

← #13264752

Subgroups — #13264752

Order 1792.

336 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 81 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#11738141C211yesmaximal, characteristic
2#11726847C211yesmaximal, characteristic, complemented
3#11726601C211yesmaximal, characteristic, complemented
4C7×#1347C211yesmaximal, characteristic
5#11725580C211yesmaximal, characteristic
6#11725334C211yesmaximal, characteristic, complemented
7#11725292C211yesmaximal, characteristic, complemented
8#90291—21no—
9C7×#466C2211yescharacteristic
10#89927C411yescharacteristic, complemented
11D8×Dic7C411yescharacteristic, complemented
12#89888C2211yescharacteristic
13C2×C8×Dic7C2211yescharacteristic
14#89813—21no—
15#89748—21no—
16#89650C411yescharacteristic, complemented
17#89645C411yescharacteristic, complemented
18#89540—21no—
19#89530—21no—
20C7×C22⋊C16C2211yescharacteristic
21C7×#340C2211yescharacteristic
22#89346C2211yescharacteristic
23C22⋊(C7⋊C16)C2211yescharacteristic
24#89324—21no—
25#89297—21no—
26Dic7⋊C16—21no—
27#11860—71nomaximal, Sylow
28C2×C7×D8C2×C411yescharacteristic
29C22×C7×C8C2311yescharacteristic
30C7×#121—21no—
31C2×C7×C4⋊C4—21no—
32#6499—21no—
33#6498—41no—
34D4×Dic7—21no—
35D4×Dic7—41no—
36C2×#1142—21no—
37C2×C4⋊Dic7—21no—
38#6487—21no—
39C2×Dic7⋊C4—41no—
40C2×Dic7⋊C4—41no—
41C2×C4×Dic7D411yescharacteristic
42C22×C7⋊C8D411yescharacteristic
43C2×Dic28C2×C411yescharacteristic
44#6453—21no—
45#6450—21no—
46#6442—41no—
47#6440—41no—
48C2×C7×C16—21no—
49C7×#107C2×C411yescharacteristic
50C7×#102—21no—
51#6406—21no—
52#6392C2×C411yescharacteristic
53#6390—21no—
54C8×Dic7C2×C411yescharacteristic
55C8×Dic7C2×C411yescharacteristic
56C8×Dic7D411yescharacteristic
57C8×Dic7D411yescharacteristic
58C2×C7⋊C16—21no—
59#6384—21no—
60#6382—21no—
61#3168—71no—
62#2163—71no—
63#1834—71no—
64#1348—71no—
65#1348—71no—
66#1347—71no—
67#1347D711yescharacteristic
68C2×C7×D4—21no—
69C2×C7×D4—41no—
70C22×C4×C7—41no—
71C22×C4×C7C2×D411yescharacteristic
72C22×Dic7—41no—
73C22×Dic7—41no—
74C22×Dic7D811yescharacteristic
75C22×Dic7D811yescharacteristic
76C2×Dic14—21no—
77C7×D8—41no—
78C2×C7×C8—41no—
79C2×C7×C8C22×C411yescharacteristic
80C2×C7×C8C2×D411yescharacteristic
81C7×C4⋊C4—21no—
82C7×C4⋊C4—41no—
83C7×C4⋊C4—41no—
84C7×C22⋊C4—41no—
85C22⋊Dic7—41no—
86C22⋊Dic7—41no—
87C22⋊Dic7—41no—
88C22⋊Dic7—41no—
89#1142—42no—
90C4⋊Dic7—21no—
91C4⋊Dic7—41no—
92C4⋊Dic7—41no—
93C4⋊Dic7—41no—
94Dic7⋊C4—41no—
95Dic7⋊C4—41no—
96Dic7⋊C4—41no—
97Dic7⋊C4—82no—
98Dic7⋊C4—82no—
99C4×Dic7—41no—
100C4×Dic7C22⋊C411yescharacteristic
101C4×Dic7C22⋊C411yescharacteristic
102C4×Dic7SD1611yescharacteristic
103C4×Dic7SD1611yescharacteristic
104C2×C7⋊C8—41no—
105C2×C7⋊C8C22⋊C411yescharacteristic
106C2×C7⋊C8C22⋊C411yescharacteristic
107Dic28—41no—
108C7×C16—41no—
109C7⋊C16—41no—
110#529—141no—
111#499—71no—
112#495—71no—
113#488—141no—
114#467—71no—
115#466C2×D711yescharacteristic
116#465—141no—
117C4×Q16—71no—
118C4×SD16—141no—
119C4×D8—71no—
120C2×C4×C8—71no—
121#384—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
131C7×D4—41no—
132C7×D4—41no—
133C7×D4—81no—
134C7×D4—81no—
135C2×C4×C7—21no—
136C2×C4×C7—21no—
137C2×C4×C7—41no—
138C2×C4×C7—81no—
139C2×C4×C7—81no—
140C2×C4×C7—81no—
141C2×C4×C7C2×C22⋊C411yescharacteristic
142C2×C4×C7C2×SD1611yescharacteristic
143C2×Dic7—21no—
144C2×Dic7—21no—
145C2×Dic7—22no—
146C2×Dic7—41no—
147C2×Dic7—41no—
148C2×Dic7—82no—
149C2×Dic7—81no—
150C2×Dic7—81no—
151C2×Dic7—81no—
152C2×Dic7#10211yescharacteristic
153C2×Dic7#10211yescharacteristic
154C2×Dic7#10211yescharacteristic
155C2×Dic7#10211yescharacteristic
156Dic14—41no—
157Dic14—41no—
158C7×C8—21no—
159C7×C8—21no—
160C7⋊C8—21no—
161C7⋊C8—21no—
162C2×Q16—71no—
163C2×SD16—141no—
164C2×D8C4×D711yescharacteristic
165C22×C8—71no—
166C22×C8C22×D711yescharacteristic
167#126—281no—
168#125—141no—
169#124—141no—
170#123—281no—
171C22⋊Q8—141no—
172#121—21no—
173C4×Q8—141no—
174C4×D4—141no—
175C4×D4—281no—
176#117—281no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—141no—
179C2×C4⋊C4—281no—
180C2×C4⋊C4—281no—
181C2×C42—71no—
182C2×C16—21no—
183C2×C16—141no—
184#107—71no—
185#107C4×D711yescharacteristic
186#106—141no—
187#103—141no—
188#103—141no—
189#102—21no—
190#102—141no—
191C4×C8—71no—
192C4×C8—71no—
193C4×C8—71no—
194C4×C8—71no—
195C22×C7—21no—
196C22×C7—21no—
197C22×C7—81no—
198C22×C7—81no—
199C22×C7—81no—
200C22×C7C2×#10211yescharacteristic
201C22×C7#50911yescharacteristic
202C22×C7#51011yescharacteristic
203C4×C7—21no—
204C4×C7—81no—
205C4×C7—161no—
206C4×C7#41011yescharacteristic
207C4×C7#42111yescharacteristic
208Dic7—21no—
209Dic7—21no—
210Dic7—21no—
211Dic7—81no—
212Dic7—81no—
213Dic7—161no—
214Dic7#36012yes—
215C2×Q8—141no—
216C2×D4—21no—
217C2×D4—41no—
218C22×C4—41no—
219C22×C4—71no—
220C22×C4—71no—
221C22×C4—281no—
222C22×C4—281no—
223C22×C4D4×D711yescharacteristic
224Q16—281no—
225SD16—282no—
226D8—41no—
227C2×C8—41no—
228C2×C8—71no—
229C2×C8—71no—
230C2×C8—281no—
231C2×C8C2×C4×D711yescharacteristic
232C2×C8D4×D711yescharacteristic
233C4⋊C4—21no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C4⋊C4—141no—
237C4⋊C4—281no—
238C4⋊C4—281no—
239C4⋊C4—281no—
240C4⋊C4—281no—
241C4⋊C4—281no—
242C4⋊C4—281no—
243C4⋊C4—562no—
244C4⋊C4—562no—
245C22⋊C4—41no—
246C22⋊C4—281no—
247C22⋊C4—281no—
248C22⋊C4—281no—
249C22⋊C4—281no—
250C42—71no—
251C42—71no—
252C42—71no—
253C42—71no—
254C42—281no—
255C16—41no—
256C16—281no—
257C2×C7—21no—
258C2×C7—21no—
259C2×C7—161no—
260C2×C7#151711yescharacteristic
261C2×C7#213911yescharacteristic
262C2×C7C2×#36011yescharacteristic
263C23—41no—
264C23D7×D811yescharacteristic
265Q8—281no—
266Q8—281no—
267D4—41no—
268D4—41no—
269D4—81no—
270D4—81no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—41no—
274C2×C4—71no—
275C2×C4—71no—
276C2×C4—71no—
277C2×C4—71no—
278C2×C4—81no—
279C2×C4—81no—
280C2×C4—81no—
281C2×C4—142no—
282C2×C4—141no—
283C2×C4—141no—
284C2×C4—281no—
285C2×C4—281no—
286C2×C4—562no—
287C2×C4—561no—
288C2×C4—561no—
289C2×C4—561no—
290C2×C4D7×C22⋊C411yescharacteristic
291C2×C4D7×SD1611yescharacteristic
292C8—21no—
293C8—21no—
294C8—141no—
295C8—141no—
296C7#1186011yescharacteristic, complemented, Sylow
297C22—21no—
298C22—21no—
299C22—81no—
300C22—81no—
301C22—81no—
302C22D7×#10211yescharacteristic
303C22#8968311yescharacteristic
304C22#8968711yescharacteristic
305C4—21no—
306C4—72no—
307C4—81no—
308C4—141no—
309C4—141no—
310C4—141no—
311C4—161no—
312C4—561no—
313C4—561no—
314C4—1121no—
315C4#8951211yescharacteristic
316C4#8959311yescharacteristic
317C2—21no—
318C2—21no—
319C2—161no—
320C2#1172603611yescharacteristic
321C2#1172694311yescharacteristic
322C2D7×#36011yescharacteristic
subgroupG/H
centerC22D7×#102
commutatorC2×C7×C8C22×C4
FrattiniC22×C8C22×D7
FittingC7×#1347C2
socleC22×C7C2×#102
prime radicalC7×#466C22
cosocleC22×C7×C8C23
hypercenter#1347D7
hypocenterC7#11860
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. #11726601C2semidirect
6. #11725334C2semidirect
7. #11725292C2semidirect
10. #89927C4semidirect
11. D8×Dic7C4semidirect
16. #89650C4semidirect
17. #89645C4semidirect
296. C7#11860semidirect

Maximal subgroups

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

subgroupindex
#117381412
#117268472
#117266012
C7×#13472
#117255802
#117253342
#117252922
#118607