Finite Group Databasebeta

← #13264231

Subgroups — #13264231

Order 1792.

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

normal subgroups only

This group has 330 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#11738141C211yesmaximal, characteristic
2#11726846C211yesmaximal, characteristic, complemented
3#11726670C211yesmaximal, characteristic, complemented
4C7×#1349C211yesmaximal, characteristic
5#11725581C211yesmaximal, characteristic
6#11725335C211yesmaximal, characteristic, complemented
7#11725294C211yesmaximal, characteristic, complemented
8#90291—21no—
9C7×#466C2211yescharacteristic
10#89927C411yescharacteristic
11D8×Dic7C411yescharacteristic
12C2×#6396—21no—
13M16×Dic7—21no—
14#89888C2211yescharacteristic
15C2×C8×Dic7C2211yescharacteristic
16#89813—21no—
17#89650C411yescharacteristic
18#89645C411yescharacteristic
19#89540—21no—
20#89530—21no—
21C7×C22⋊C16C2211yescharacteristic
22C7×#341C2211yescharacteristic
23#89347C2211yescharacteristic
24C22⋊(C7⋊C16)C2211yescharacteristic
25Dic7⋊C16—21no—
26C16×Dic7—21no—
27#11749—71nomaximal, Sylow
28C2×C7×D8C2×C411yescharacteristic
29C2×C7×M16—21no—
30C22×C7×C8C2311yescharacteristic
31C7×#121—21no—
32#6499—21no—
33#6498—41no—
34D4×Dic7—21no—
35D4×Dic7—41no—
36C2×#1142—21no—
37#6487—21no—
38C2×C4×Dic7D411yescharacteristic
39C2×#1136—21no—
40C22×C7⋊C8D411yescharacteristic
41C2×Dic28C2×C411yescharacteristic
42#6453—21no—
43#6450—21no—
44#6442—41no—
45#6440—41no—
46C2×C7×C16—21no—
47C7×#107C2×C411yescharacteristic
48C7×#102—21no—
49#6406—21no—
50#6396—24no—
51#6392C2×C411yescharacteristic
52#6390—21no—
53#6389—41no—
54C8×Dic7—41no—
55C8×Dic7C2×C411yescharacteristic
56C8×Dic7C2×C411yescharacteristic
57C8×Dic7D411yescharacteristic
58C8×Dic7D411yescharacteristic
59C2×C7⋊C16—21no—
60#6384—21no—
61#6382—21no—
62#3168—71no—
63#2162—71no—
64C4×#108—71no—
65#1350—71no—
66#1350—71no—
67#1349—71no—
68#1349D711yescharacteristic
69C2×C7×D4—21no—
70C2×C7×D4—41no—
71C22×C4×C7C2×D411yescharacteristic
72C22×Dic7—41no—
73C22×Dic7SD1611yescharacteristic
74C22×Dic7SD1611yescharacteristic
75C2×Dic14—21no—
76C7×D8—41no—
77C7×M16—22no—
78C7×M16—41no—
79C2×C7×C8—41no—
80C2×C7×C8—41no—
81C2×C7×C8C22×C411yescharacteristic
82C2×C7×C8C2×D411yescharacteristic
83C7×C4⋊C4—21no—
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—
92Dic7⋊C4—41no—
93Dic7⋊C4—41no—
94Dic7⋊C4—41no—
95C4×Dic7—41no—
96C4×Dic7C22⋊C411yescharacteristic
97C4×Dic7C22⋊C411yescharacteristic
98C4×Dic7D811yescharacteristic
99C4×Dic7D811yescharacteristic
100#1136—22no—
101#1136—41no—
102C2×C7⋊C8—22no—
103C2×C7⋊C8—41no—
104C2×C7⋊C8C22⋊C411yescharacteristic
105C2×C7⋊C8C22⋊C411yescharacteristic
106Dic28—41no—
107C7×C16—41no—
108C7⋊C16—41no—
109#529—141no—
110#499—71no—
111#495—71no—
112#488—141no—
113#467—71no—
114#466C2×D711yescharacteristic
115#465—141no—
116C4×Q16—71no—
117C4×SD16—141no—
118C4×D8—71no—
119C2×#108—141no—
120C4×M16—141no—
121C2×C4×C8—71no—
122C4⋊C16—141no—
123C22⋊C16—71no—
124C22⋊C16C2×D711yescharacteristic
125C4×C16—141no—
126#341—71no—
127#341C2×D711yescharacteristic
128C23×C7—41no—
129C23×C7C2×SD1611yescharacteristic
130C7×D4—41no—
131C7×D4—41no—
132C7×D4—82no—
133C2×C4×C7—21no—
134C2×C4×C7—21no—
135C2×C4×C7—41no—
136C2×C4×C7C2×C22⋊C411yescharacteristic
137C2×C4×C7C2×D811yescharacteristic
138C2×Dic7—22no—
139C2×Dic7—41no—
140C2×Dic7—41no—
141C2×Dic7—41no—
142C2×Dic7—82no—
143C2×Dic7#10211yescharacteristic
144C2×Dic7#10211yescharacteristic
145C2×Dic7#10211yescharacteristic
146C2×Dic7#10211yescharacteristic
147Dic14—41no—
148Dic14—41no—
149C7×C8—21no—
150C7×C8—21no—
151C7×C8—42no—
152C7⋊C8—21no—
153C7⋊C8—42no—
154C7⋊C8#10412yes—
155C2×Q16—71no—
156C2×SD16—141no—
157C2×D8C4×D711yescharacteristic
158C2×M16—21no—
159C2×M16—141no—
160C22×C8—71no—
161C22×C8C22×D711yescharacteristic
162#126—281no—
163#125—141no—
164#124—141no—
165#123—281no—
166C22⋊Q8—141no—
167#121—21no—
168C4×Q8—141no—
169C4×D4—141no—
170C4×D4—281no—
171#117—281no—
172C2×C42—71no—
173C2×C16—21no—
174C2×C16—141no—
175#108—144no—
176#107—71no—
177#107C4×D711yescharacteristic
178#106—141no—
179#103—141no—
180#103—141no—
181#102—21no—
182#102—141no—
183#97—281no—
184C4×C8—71no—
185C4×C8—71no—
186C4×C8—71no—
187C4×C8—71no—
188C4×C8—281no—
189C22×C7—21no—
190C22×C7—21no—
191C22×C7—21no—
192C22×C7—81no—
193C22×C7—81no—
194C22×C7—81no—
195C22×C7C2×#10211yescharacteristic
196C4×C7—21no—
197C4×C7—81no—
198C4×C7#41011yescharacteristic
199C4×C7C2×#10411yescharacteristic
200Dic7—21no—
201Dic7—21no—
202Dic7—21no—
203Dic7—81no—
204Dic7—81no—
205Dic7#35912yes—
206C2×Q8—141no—
207C2×D4—21no—
208C2×D4—41no—
209C22×C4—71no—
210C22×C4—71no—
211C22×C4—281no—
212C22×C4D4×D711yescharacteristic
213Q16—281no—
214SD16—282no—
215D8—41no—
216M16—22no—
217M16—41no—
218M16—142no—
219M16—281no—
220C2×C8—41no—
221C2×C8—41no—
222C2×C8—71no—
223C2×C8—71no—
224C2×C8—142no—
225C2×C8—281no—
226C2×C8C2×C4×D711yescharacteristic
227C2×C8D4×D711yescharacteristic
228C4⋊C4—21no—
229C4⋊C4—141no—
230C4⋊C4—281no—
231C4⋊C4—281no—
232C4⋊C4—281no—
233C4⋊C4—281no—
234C22⋊C4—41no—
235C22⋊C4—281no—
236C22⋊C4—281no—
237C22⋊C4—281no—
238C22⋊C4—281no—
239C42—71no—
240C42—71no—
241C42—71no—
242C42—71no—
243C42—281no—
244C16—41no—
245C16—281no—
246C2×C7—41no—
247C2×C7—161no—
248C2×C7#150211yescharacteristic
249C2×C7C2×#35911yescharacteristic
250C2×C7#214811yescharacteristic
251C23—41no—
252C23D7×SD1611yescharacteristic
253Q8—281no—
254Q8—281no—
255D4—41no—
256D4—41no—
257D4—82no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—41no—
261C2×C4—71no—
262C2×C4—71no—
263C2×C4—71no—
264C2×C4—71no—
265C2×C4—142no—
266C2×C4—281no—
267C2×C4—281no—
268C2×C4—281no—
269C2×C4—562no—
270C2×C4D7×C22⋊C411yescharacteristic
271C2×C4D7×D811yescharacteristic
272C8—21no—
273C8—21no—
274C8—42no—
275C8—72no—
276C8—141no—
277C8—282no—
278C7#1174911yescharacteristic, complemented, Sylow
279C22—21no—
280C22—21no—
281C22—21no—
282C22—81no—
283C22—81no—
284C22—81no—
285C22D7×#10211yescharacteristic
286C4—21no—
287C4—72no—
288C4—81no—
289C4—141no—
290C4—141no—
291C4—141no—
292C4—561no—
293C4—561no—
294C4#8951211yescharacteristic
295C4D7×#10411yescharacteristic
296C2—41no—
297C2—161no—
298C2#1172603111yescharacteristic
299C2D7×#35911yescharacteristic
300C2#1172701011yescharacteristic
subgroupG/H
centerC22D7×#102
commutatorC2×C7×C8C22×C4
FrattiniC22×C8C22×D7
FittingC7×#1349C2
socleC22×C7C2×#102
prime radicalC7×#466C22
cosocleC22×C7×C8C23
hypercenter#1349D7
hypocenterC7#11749
LCSG ▷ C2×C7×C8 ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #1349

Complements

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

NH ≅ G/Ntype
2. #11726846C2semidirect
3. #11726670C2semidirect
6. #11725335C2semidirect
7. #11725294C2semidirect
278. C7#11749semidirect

Maximal subgroups

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

subgroupindex
#117381412
#117268462
#117266702
C7×#13492
#117255812
#117253352
#117252942
#117497