Finite Group Databasebeta

← #13264748

Subgroups — #13264748

Order 1792.

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

normal subgroups only

This group has 314 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#11738186C211yesmaximal, characteristic
2#11726847C211yesmaximal, characteristic
3#11726602C211yesmaximal, characteristic
4C7×#1348C211yesmaximal, characteristic
5#11725582C211yesmaximal, characteristic
6#11725334C211yesmaximal, characteristic
7#11725292C211yesmaximal, characteristic
8#90313—21no—
9C7×#467C2211yescharacteristic
10Q16×Dic7C411yescharacteristic, complemented
11#89953C411yescharacteristic, complemented
12#89888C2211yescharacteristic
13C2×C8×Dic7C2211yescharacteristic
14#89823—21no—
15#89749—21no—
16#89646C411yescharacteristic, complemented
17#89645C411yescharacteristic, complemented
18#89564—21no—
19#89562—21no—
20C7×C22⋊C16C2211yescharacteristic
21C7×#340C2211yescharacteristic
22#89346C2211yescharacteristic
23C22⋊(C7⋊C16)C2211yescharacteristic
24#89324—21no—
25#89297—21no—
26Dic7⋊C16—21no—
27#11859—71nomaximal, Sylow
28C2×C7×Q16C2×C411yescharacteristic
29C22×C7×C8C2311yescharacteristic
30C7×C22⋊Q8—21no—
31C2×C7×C4⋊C4—21no—
32Q8×Dic7—21no—
33#6507—21no—
34C2×#1144—21no—
35C2×C4⋊Dic7—21no—
36#6487—21no—
37C2×C4×Dic7D411yescharacteristic
38C22×C7⋊C8D411yescharacteristic
39C2×Dic28C2×C411yescharacteristic
40#6452—41no—
41#6451—21no—
42#6450—21no—
43#6450—41no—
44#6441—41no—
45#6440—41no—
46C2×C7×C16—21no—
47C7×#107C2×C411yescharacteristic
48C7×#103—21no—
49#6409—21no—
50C7⋊#95—41no—
51C7⋊#95—41no—
52#6392C2×C411yescharacteristic
53#6390—21no—
54C8×Dic7C2×C411yescharacteristic
55C8×Dic7C2×C411yescharacteristic
56C8×Dic7D411yescharacteristic
57C8×Dic7D411yescharacteristic
58C2×C7⋊C16—21no—
59#6384—21no—
60#6381—21no—
61#3170—71no—
62#2163—71no—
63#1835—71no—
64#1348—71no—
65#1348—71no—
66#1348—71no—
67#1348D711yescharacteristic
68C2×C7×Q8—21no—
69C22×C4×C7—41no—
70C22×C4×C7C2×D411yescharacteristic
71C22×Dic7—41no—
72C22×Dic7D811yescharacteristic
73C22×Dic7D811yescharacteristic
74C2×Dic14—21no—
75C2×Dic14—41no—
76C7×Q16—41no—
77C2×C7×C8—41no—
78C2×C7×C8C22×C411yescharacteristic
79C2×C7×C8C2×D411yescharacteristic
80C7×C4⋊C4—21no—
81C7×C4⋊C4—41no—
82C7×C4⋊C4—41no—
83C7×C4⋊C4—41no—
84C7×C22⋊C4—41no—
85C22⋊Dic7—41no—
86#1144—42no—
87C4⋊Dic7—21no—
88C4⋊Dic7—41no—
89C4⋊Dic7—41no—
90C4⋊Dic7—41no—
91Dic7⋊C4—41no—
92Dic7⋊C4—41no—
93Dic7⋊C4—41no—
94Dic7⋊C4—41no—
95Dic7⋊C4—41no—
96C4×Dic7—41no—
97C4×Dic7—41no—
98C4×Dic7C22⋊C411yescharacteristic
99C4×Dic7C22⋊C411yescharacteristic
100C4×Dic7SD1611yescharacteristic
101C4×Dic7SD1611yescharacteristic
102C2×C7⋊C8—41no—
103C2×C7⋊C8C22⋊C411yescharacteristic
104C2×C7⋊C8C22⋊C411yescharacteristic
105Dic28—41no—
106C7×C16—41no—
107C7⋊C16—41no—
108#533—141no—
109#500—71no—
110#494—71no—
111#487—141no—
112#467—71no—
113#467—141no—
114#467C2×D711yescharacteristic
115C4×Q16—71no—
116C4×Q16—71no—
117C4×Q16—141no—
118C2×C4×C8—71no—
119#382—141no—
120C4⋊C16—141no—
121C22⋊C16—71no—
122C22⋊C16C2×D711yescharacteristic
123#346—141no—
124#340—71no—
125#340—141no—
126#340C2×D711yescharacteristic
127C23×C7C2×D811yescharacteristic
128C7×Q8—41no—
129C7×Q8—41no—
130C2×C4×C7—21no—
131C2×C4×C7—21no—
132C2×C4×C7—41no—
133C2×C4×C7—41no—
134C2×C4×C7—81no—
135C2×C4×C7—81no—
136C2×C4×C7—81no—
137C2×C4×C7C2×C22⋊C411yescharacteristic
138C2×C4×C7C2×SD1611yescharacteristic
139C2×Dic7—21no—
140C2×Dic7—21no—
141C2×Dic7—41no—
142C2×Dic7—41no—
143C2×Dic7—41no—
144C2×Dic7—81no—
145C2×Dic7—81no—
146C2×Dic7—81no—
147C2×Dic7#10211yescharacteristic
148C2×Dic7#10211yescharacteristic
149C2×Dic7#10211yescharacteristic
150C2×Dic7#10211yescharacteristic
151Dic14—41no—
152Dic14—41no—
153Dic14—82no—
154C7×C8—21no—
155C7×C8—21no—
156C7⋊C8—21no—
157C7⋊C8—21no—
158C2×Q16—71no—
159C2×Q16—141no—
160C2×Q16C4×D711yescharacteristic
161C22×C8—71no—
162C22×C8C22×D711yescharacteristic
163C4⋊Q8—141no—
164C4⋊Q8—141no—
165#125—281no—
166C22⋊Q8—21no—
167C22⋊Q8—141no—
168C22⋊Q8—281no—
169C4×Q8—141no—
170C4×Q8—141no—
171C4×Q8—281no—
172#117—281no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—141no—
175C2×C42—71no—
176C2×C16—21no—
177C2×C16—141no—
178#107—71no—
179#107—141no—
180#107C4×D711yescharacteristic
181#103—21no—
182#103—141no—
183#103—141no—
184#103—141no—
185C4×C8—71no—
186C4×C8—71no—
187C4×C8—71no—
188C4×C8—71no—
189#95—281no—
190#95—281no—
191C22×C7—21no—
192C22×C7—21no—
193C22×C7C2×#10211yescharacteristic
194C22×C7#51011yescharacteristic
195C22×C7#51011yescharacteristic
196C4×C7—21no—
197C4×C7—81no—
198C4×C7—81no—
199C4×C7—161no—
200C4×C7#41011yescharacteristic
201C4×C7#42111yescharacteristic
202Dic7—21no—
203Dic7—21no—
204Dic7—21no—
205Dic7—21no—
206Dic7—81no—
207Dic7—81no—
208Dic7—161no—
209C2×Q8—21no—
210C2×Q8—141no—
211C2×Q8—281no—
212C22×C4—41no—
213C22×C4—71no—
214C22×C4—71no—
215C22×C4—281no—
216C22×C4D4×D711yescharacteristic
217Q16—41no—
218Q16—281no—
219Q16—282no—
220C2×C8—41no—
221C2×C8—71no—
222C2×C8—71no—
223C2×C8—281no—
224C2×C8C2×C4×D711yescharacteristic
225C2×C8D4×D711yescharacteristic
226C4⋊C4—21no—
227C4⋊C4—41no—
228C4⋊C4—41no—
229C4⋊C4—41no—
230C4⋊C4—141no—
231C4⋊C4—281no—
232C4⋊C4—281no—
233C4⋊C4—281no—
234C4⋊C4—281no—
235C4⋊C4—281no—
236C4⋊C4—281no—
237C4⋊C4—281no—
238C4⋊C4—281no—
239C22⋊C4—41no—
240C22⋊C4—281no—
241C42—71no—
242C42—71no—
243C42—71no—
244C42—71no—
245C42—281no—
246C42—281no—
247C16—41no—
248C16—281no—
249C2×C7—21no—
250C2×C7—21no—
251C2×C7#151711yescharacteristic
252C2×C7#214111yescharacteristic
253C2×C7#214511yescharacteristic
254C23D7×D811yescharacteristic
255Q8—41no—
256Q8—41no—
257Q8—281no—
258Q8—281no—
259Q8—562no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—71no—
265C2×C4—71no—
266C2×C4—71no—
267C2×C4—71no—
268C2×C4—81no—
269C2×C4—81no—
270C2×C4—81no—
271C2×C4—141no—
272C2×C4—141no—
273C2×C4—281no—
274C2×C4—281no—
275C2×C4—281no—
276C2×C4—561no—
277C2×C4—561no—
278C2×C4—561no—
279C2×C4D7×C22⋊C411yescharacteristic
280C2×C4D7×SD1611yescharacteristic
281C8—21no—
282C8—21no—
283C8—141no—
284C8—141no—
285C7#1185911yescharacteristic, complemented, Sylow
286C22—21no—
287C22—21no—
288C22D7×#10211yescharacteristic
289C22#8968711yescharacteristic
290C22#8969011yescharacteristic
291C4—21no—
292C4—81no—
293C4—81no—
294C4—141no—
295C4—141no—
296C4—141no—
297C4—141no—
298C4—161no—
299C4—561no—
300C4—561no—
301C4—1121no—
302C4#8951211yescharacteristic
303C4#8959311yescharacteristic
304C2—21no—
305C2—21no—
306C2#1172603611yescharacteristic
307C2#1172697611yescharacteristic
308C2#1172700511yescharacteristic
subgroupG/H
centerC22D7×#102
commutatorC2×C7×C8C22×C4
FrattiniC22×C8C22×D7
FittingC7×#1348C2
socleC22×C7C2×#102
prime radicalC23×C7C2×D8
cosocleC22×C7×C8C23
hypercenter#1348D7
hypocenterC7#11859
LCSG ▷ C2×C7×C8 ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #1348

Complements

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

NH ≅ G/Ntype
10. Q16×Dic7C4semidirect
11. #89953C4semidirect
16. #89646C4semidirect
17. #89645C4semidirect
285. C7#11859semidirect

Maximal subgroups

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

subgroupindex
#117381862
#117268472
#117266022
C7×#13482
#117255822
#117253342
#117252922
#118597