Finite Group Databasebeta

← #13264206

Subgroups — #13264206

Order 1792.

350 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 89 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#11726846C211yesmaximal, characteristic
3#11726539C211yesmaximal, 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—
15C4⋊C4×Dic7—21no—
16C2×#6381—21no—
17#89646C411yescharacteristic, complemented
18#89645C411yescharacteristic, complemented
19#89564—21no—
20#89562—21no—
21C7×C22⋊C16C2211yescharacteristic
22C7×#340C2211yescharacteristic
23#89346C2211yescharacteristic
24C22⋊(C7⋊C16)C2211yescharacteristic
25Dic7⋊C16—21no—
26C16×Dic7—21no—
27#11743—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×Dic7—41no—
38C2×C4×Dic7D411yescharacteristic
39C22×C7⋊C8D411yescharacteristic
40C2×Dic28C2×C411yescharacteristic
41#6452—41no—
42#6451—21no—
43#6450—21no—
44#6450—41no—
45#6441—41no—
46#6440—41no—
47C2×C7×C16—21no—
48C7×#107C2×C411yescharacteristic
49C7×#103—21no—
50#6409—21no—
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—24no—
61#6381—21no—
62#3170—71no—
63#2162—71no—
64C4×#107—71no—
65#1348—71no—
66#1348—71no—
67#1348—71no—
68#1348D711yescharacteristic
69C2×C7×Q8—21no—
70C22×C4×C7—41no—
71C22×C4×C7C2×D411yescharacteristic
72C22×Dic7—41no—
73C22×Dic7D811yescharacteristic
74C22×Dic7D811yescharacteristic
75C2×Dic14—21no—
76C2×Dic14—41no—
77C7×Q16—41no—
78C2×C7×C8—41no—
79C2×C7×C8C22×C411yescharacteristic
80C2×C7×C8C2×D411yescharacteristic
81C7×C4⋊C4—22no—
82C7×C4⋊C4—21no—
83C7×C4⋊C4—41no—
84C7×C4⋊C4—41no—
85C7×C22⋊C4—41no—
86C22⋊Dic7—41no—
87#1144—42no—
88C4⋊Dic7—22no—
89C4⋊Dic7—21no—
90C4⋊Dic7—41no—
91C4⋊Dic7—41no—
92Dic7⋊C4—41no—
93Dic7⋊C4—41no—
94Dic7⋊C4—41no—
95Dic7⋊C4—41no—
96Dic7⋊C4—41no—
97C4×Dic7—41no—
98C4×Dic7—41no—
99C4×Dic7—82no—
100C4×Dic7C22⋊C411yescharacteristic
101C4×Dic7C22⋊C411yescharacteristic
102C4×Dic7SD1611yescharacteristic
103C4×Dic7SD1611yescharacteristic
104C2×C7⋊C8—22no—
105C2×C7⋊C8C22⋊C411yescharacteristic
106C2×C7⋊C8C22⋊C411yescharacteristic
107Dic28—41no—
108C7×C16—41no—
109C7⋊C16—41no—
110#533—141no—
111#500—71no—
112#494—71no—
113#487—141no—
114#467—71no—
115#467—141no—
116#467C2×D711yescharacteristic
117C4×Q16—71no—
118C4×Q16—71no—
119C4×Q16—141no—
120C2×#107—141no—
121C2×C4×C8—71no—
122C4×C4⋊C4—141no—
123C4⋊C16—141no—
124C22⋊C16—71no—
125C22⋊C16C2×D711yescharacteristic
126C4×C16—141no—
127#340—71no—
128#340C2×D711yescharacteristic
129C23×C7C2×D811yescharacteristic
130C7×Q8—41no—
131C7×Q8—41no—
132C2×C4×C7—21no—
133C2×C4×C7—21no—
134C2×C4×C7—42no—
135C2×C4×C7—41no—
136C2×C4×C7—41no—
137C2×C4×C7—81no—
138C2×C4×C7—81no—
139C2×C4×C7C2×C22⋊C411yescharacteristic
140C2×C4×C7C2×SD1611yescharacteristic
141C2×Dic7—41no—
142C2×Dic7—42no—
143C2×Dic7—41no—
144C2×Dic7—41no—
145C2×Dic7—81no—
146C2×Dic7—81no—
147C2×Dic7#10211yescharacteristic
148C2×Dic7#10211yescharacteristic
149C2×Dic7#10211yescharacteristic
150C2×Dic7#10211yescharacteristic
151C2×Dic7SD3212yes—
152C2×Dic7Q3212yes—
153Dic14—41no—
154Dic14—41no—
155Dic14—82no—
156C7×C8—21no—
157C7×C8—21no—
158C7⋊C8—21no—
159C7⋊C8#10412yes—
160C2×Q16—71no—
161C2×Q16—141no—
162C2×Q16C4×D711yescharacteristic
163C22×C8—71no—
164C22×C8C22×D711yescharacteristic
165C4⋊Q8—141no—
166C4⋊Q8—141no—
167#125—281no—
168C22⋊Q8—21no—
169C22⋊Q8—141no—
170C22⋊Q8—281no—
171C4×Q8—141no—
172C4×Q8—141no—
173C4×Q8—281no—
174#117—281no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—141no—
177C2×C42—71no—
178C2×C42—281no—
179C2×C16—21no—
180C2×C16—141no—
181#107—71no—
182#107—141no—
183#107—144no—
184#107C4×D711yescharacteristic
185#103—21no—
186#103—141no—
187#103—141no—
188#103—141no—
189C4×C8—71no—
190C4×C8—71no—
191C4×C8—71no—
192C4×C8—71no—
193#95—281no—
194C22×C7—21no—
195C22×C7—21no—
196C22×C7C2×#10211yescharacteristic
197C22×C7C2×SD3211yescharacteristic
198C22×C7C2×Q3211yescharacteristic
199C4×C7—21no—
200C4×C7—82no—
201C4×C7—81no—
202C4×C7—81no—
203C4×C7#41011yescharacteristic
204C4×C7C2×#10411yescharacteristic
205Dic7—21no—
206Dic7—21no—
207Dic7—82no—
208Dic7—81no—
209Dic7—81no—
210Dic7#35812yes—
211Dic7#35812yes—
212C2×Q8—21no—
213C2×Q8—141no—
214C2×Q8—281no—
215C22×C4—41no—
216C22×C4—71no—
217C22×C4—71no—
218C22×C4—281no—
219C22×C4D4×D711yescharacteristic
220Q16—41no—
221Q16—281no—
222Q16—282no—
223C2×C8—41no—
224C2×C8—71no—
225C2×C8—71no—
226C2×C8—142no—
227C2×C8C2×C4×D711yescharacteristic
228C2×C8D4×D711yescharacteristic
229C4⋊C4—22no—
230C4⋊C4—21no—
231C4⋊C4—41no—
232C4⋊C4—41no—
233C4⋊C4—142no—
234C4⋊C4—141no—
235C4⋊C4—281no—
236C4⋊C4—281no—
237C4⋊C4—281no—
238C4⋊C4—281no—
239C4⋊C4—281no—
240C4⋊C4—281no—
241C4⋊C4—281no—
242C22⋊C4—41no—
243C22⋊C4—281no—
244C42—71no—
245C42—71no—
246C42—71no—
247C42—71no—
248C42—281no—
249C42—281no—
250C42—562no—
251C16—41no—
252C16—281no—
253C2×C7—21no—
254C2×C7—21no—
255C2×C7#150211yescharacteristic
256C2×C7C2×#35811yescharacteristic
257C2×C7#214511yescharacteristic
258C23D7×D811yescharacteristic
259Q8—41no—
260Q8—41no—
261Q8—281no—
262Q8—281no—
263Q8—562no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—42no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—72no—
270C2×C4—72no—
271C2×C4—71no—
272C2×C4—71no—
273C2×C4—71no—
274C2×C4—71no—
275C2×C4—81no—
276C2×C4—81no—
277C2×C4—281no—
278C2×C4—282no—
279C2×C4—281no—
280C2×C4—281no—
281C2×C4—561no—
282C2×C4—561no—
283C2×C4D7×C22⋊C411yescharacteristic
284C2×C4D7×SD1611yescharacteristic
285C8—21no—
286C8—21no—
287C8—72no—
288C8—141no—
289C7#1174311yescharacteristic, complemented, Sylow
290C22—21no—
291C22—21no—
292C22D7×#10211yescharacteristic
293C22D7×SD3211yescharacteristic
294C22D7×Q3211yescharacteristic
295C4—21no—
296C4—72no—
297C4—72no—
298C4—82no—
299C4—81no—
300C4—81no—
301C4—141no—
302C4—141no—
303C4—562no—
304C4—561no—
305C4—561no—
306C4#8951211yescharacteristic
307C4D7×#10411yescharacteristic
308C2—21no—
309C2—21no—
310C2#1172603111yescharacteristic
311C2D7×#35811yescharacteristic
312C2#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#11743
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
17. #89646C4semidirect
18. #89645C4semidirect
289. C7#11743semidirect

Maximal subgroups

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

subgroupindex
#117381862
#117268462
#117265392
C7×#13482
#117255822
#117253342
#117252922
#117437