Finite Group Databasebeta

← #13265255

Subgroups — #13265255

Order 1792.

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

normal subgroups only

This group has 312 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#11739254C211yesmaximal, characteristic
2D7×C4⋊C16C211yesmaximal, characteristic, complemented
3#11726284C211yesmaximal, characteristic, complemented
4C7×#1360C211yesmaximal, characteristic, complemented
5#11725602C211yesmaximal, characteristic, complemented
6#11725369C211yesmaximal, characteristic, complemented
7#11725274C211yesmaximal, characteristic, complemented
8#90371—21no—
9C7×#495C2211yescharacteristic
10#89960C411yescharacteristic
11Q16×Dic7C411yescharacteristic
12#89943—21no—
13SD16×Dic7—21no—
14#89929C411yescharacteristic
15D8×Dic7C411yescharacteristic
16#89828—21no—
17C2×D7×C16—21no—
18#89607—21no—
19C4×C8×D7C2211yescharacteristic
20#89455C2211yescharacteristic
21C7×C4⋊C16C2211yescharacteristic
22C7×#334C2211yescharacteristic
23Dic7⋊C16—21no—
24#89274—21no—
25C4⋊(C7⋊C16)C2211yescharacteristic
26#89252C2211yescharacteristic
27#11978—71nomaximal, Sylow
28C2×C7×Q16C2×C411yescharacteristic
29C2×C7×SD16—21no—
30C2×C7×D8C2×C411yescharacteristic
31C7×#124—21no—
32#6509—21no—
33Q8×Dic7—21no—
34#6501—21no—
35D4×Dic7—21no—
36C2×C8×D7C2×C411yescharacteristic
37C2×C8×D7D411yescharacteristic
38#6446—41no—
39(C22⋊D7)⋊C4—41no—
40#6442—41no—
41#6440—41no—
42C42×D7D411yescharacteristic
43#6433—21no—
44C2×C7×C16—21no—
45C7×C4⋊C8—21no—
46C4×C7×C8C2311yescharacteristic
47#6409—21no—
48#6409—21no—
49#6406—21no—
50#6406—21no—
51(C2×D7)⋊C8—41no—
52#6392C2×C411yescharacteristic
53#6392C2×C411yescharacteristic
54#6391—21no—
55#6389—41no—
56C8×Dic7C2×C411yescharacteristic
57C8×Dic7D411yescharacteristic
58C2×C7⋊C16—21no—
59C4⋊(C7⋊C8)—21no—
60C4×C7⋊C8D411yescharacteristic
61D7×C16—42no—
62#3168—71no—
63C2×C4⋊C16—71no—
64#1568—71no—
65#1363—71no—
66#1363—71no—
67#1360—71no—
68#1360D711yescharacteristic, complemented
69C2×C7×Q8—21no—
70C2×C7×D4—21no—
71C2×C22⋊D7—41no—
72C22×Dic7—41no—
73C2×C4×D7C22⋊C411yescharacteristic
74C2×C4×D7D811yescharacteristic
75C2×C4×D7SD1611yescharacteristic
76C7×Q16—41no—
77C7×SD16—41no—
78C7×SD16—41no—
79C7×D8—41no—
80C2×C7×C8—41no—
81C2×C7×C8C22×C411yescharacteristic
82C2×C7×C8C2×D411yescharacteristic
83C7×C22⋊C4—41no—
84C7×C22⋊C4—41no—
85C42×C7C2×D411yescharacteristic
86C22⋊Dic7—41no—
87C22⋊Dic7—41no—
88(C2×D7)⋊C4—41no—
89C4⋊Dic7—21no—
90C4⋊Dic7—21no—
91C4⋊Dic7—41no—
92Dic7⋊C4—41no—
93Dic7⋊C4—41no—
94Dic7⋊C4—41no—
95C4×Dic7—41no—
96C4×Dic7C22⋊C411yescharacteristic
97C4×Dic7D811yescharacteristic
98C4×Dic7SD1611yescharacteristic
99C2×C7⋊C8—41no—
100C2×C7⋊C8C22⋊C411yescharacteristic
101C2×C7⋊C8C22⋊C411yescharacteristic
102C8×D7—21no—
103C8×D7—21no—
104C8×D7—41no—
105C7×C16—41no—
106C7⋊C16—41no—
107#529—141no—
108C22×C16—141no—
109#499—71no—
110#495C2×D711yescharacteristic
111#488—141no—
112#467—71no—
113#466—71no—
114#465—141no—
115C4×Q16—71no—
116C4×SD16—141no—
117C4×D8—71no—
118#432—141no—
119C2×C4×C8—71no—
120C4⋊C16—71no—
121C4⋊C16—141no—
122C4⋊C16C2×D711yescharacteristic
123#335—141no—
124#334—71no—
125#334C2×D711yescharacteristic
126C23×C7—41no—
127C22×D7#10211yescharacteristic
128C7×Q8—41no—
129C7×Q8—41no—
130C7×D4—41no—
131C7×D4—41no—
132C2×C4×C7—41no—
133C2×C4×C7C2×C22⋊C411yescharacteristic
134C2×C4×C7C2×D811yescharacteristic
135C2×C4×C7C2×SD1611yescharacteristic
136C22⋊D7—82no—
137C2×Dic7—41no—
138C2×Dic7—41no—
139C2×Dic7—82no—
140C2×Dic7#10211yescharacteristic
141C2×Dic7#10211yescharacteristic
142C2×Dic7#10211yescharacteristic
143C4×D7—22no—
144C4×D7—21no—
145C4×D7—21no—
146C4×D7—41no—
147C7×C8—21no—
148C7×C8—81no—
149C7×C8AGL(1,ℤ/8)11yescharacteristic
150C7×C8#13711yescharacteristic
151C7⋊C8—21no—
152C7⋊C8—21no—
153C7⋊C8—81no—
154C2×Q16C4×D711yescharacteristic
155C2×SD16—21no—
156C2×D8C4×D711yescharacteristic
157C22×C8—71no—
158C22×C8—71no—
159#126—281no—
160#125—141no—
161#124—21no—
162#123—281no—
163C22⋊Q8—141no—
164#121—141no—
165C4×Q8—141no—
166C4×D4—141no—
167C4×D4—281no—
168#117—281no—
169C2×C42—71no—
170C2×C16—21no—
171C2×C16—141no—
172C2×C16—282no—
173#107—71no—
174#107—71no—
175#106—141no—
176C4⋊C8—21no—
177C4⋊C8—141no—
178#103—141no—
179#103—141no—
180#102—141no—
181#102—141no—
182C22⋊C8—281no—
183#97—281no—
184C4×C8—71no—
185C4×C8—71no—
186C4×C8—71no—
187C4×C8C22×D711yescharacteristic
188C22×C7—81no—
189C22×C7—81no—
190C22×C7—81no—
191C22×C7C2×#10211yescharacteristic
192C2×D7—21no—
193C2×D7—21no—
194C2×D7#35912yes—
195C4×C7—21no—
196C4×C7—21no—
197C4×C7—81no—
198C4×C7#41311yescharacteristic
199C4×C7#41711yescharacteristic
200Dic7—21no—
201Dic7—21no—
202Dic7—21no—
203Dic7—81no—
204Dic7—81no—
205C2×Q8—21no—
206C2×D4—21no—
207C2×D4—281no—
208C22×C4—71no—
209C22×C4—71no—
210C22×C4—71no—
211C22×C4—281no—
212Q16—41no—
213SD16—41no—
214SD16—41no—
215D8—41no—
216C2×C8—41no—
217C2×C8—71no—
218C2×C8—71no—
219C2×C8—141no—
220C2×C8—141no—
221C2×C8—281no—
222C2×C8—281no—
223C2×C8C2×C4×D711yescharacteristic
224C2×C8D4×D711yescharacteristic
225C4⋊C4—141no—
226C4⋊C4—141no—
227C4⋊C4—281no—
228C4⋊C4—281no—
229C4⋊C4—281no—
230C4⋊C4—281no—
231C22⋊C4—41no—
232C22⋊C4—41no—
233C22⋊C4—281no—
234C22⋊C4—281no—
235C22⋊C4—281no—
236C42—71no—
237C42—71no—
238C42—71no—
239C42—281no—
240C42D4×D711yescharacteristic
241C16—41no—
242C16—281no—
243C2×C7—161no—
244C2×C7#154711yescharacteristic
245C2×C7C2×#35911yescharacteristic
246C2×C7#214511yescharacteristic
247D7—22no—
248C23—41no—
249C23—71no—
250Q8—41no—
251Q8—41no—
252D4—41no—
253D4—41no—
254D4—562no—
255C2×C4—41no—
256C2×C4—71no—
257C2×C4—71no—
258C2×C4—71no—
259C2×C4—142no—
260C2×C4—141no—
261C2×C4—141no—
262C2×C4—281no—
263C2×C4—281no—
264C2×C4—281no—
265C2×C4—562no—
266C2×C4D7×C22⋊C411yescharacteristic
267C2×C4D7×D811yescharacteristic
268C2×C4D7×SD1611yescharacteristic
269C8—21no—
270C8—81no—
271C8—141no—
272C8—141no—
273C8—561no—
274C8#647411yescharacteristic
275C8#647811yescharacteristic
276C7#1197811yescharacteristic, complemented, Sylow
277C22—72no—
278C22—81no—
279C22—81no—
280C22—81no—
281C22—141no—
282C22—141no—
283C22D7×#10211yescharacteristic
284C4—21no—
285C4—21no—
286C4—81no—
287C4—141no—
288C4—141no—
289C4—141no—
290C4—561no—
291C4—561no—
292C4#8951711yescharacteristic
293C4#8954311yescharacteristic
294C2—142no—
295C2—161no—
296C2#1172613811yescharacteristic
297C2D7×#35911yescharacteristic
298C2#1172700511yescharacteristic
subgroupG/H
centerC22D7×#102
commutatorC2×C7×C8C22×C4
FrattiniC4×C8C22×D7
FittingC7×#1360C2
socleC22×C7C2×#102
prime radical#89929C4
cosocleC4×C7×C8C23
hypercenter#1360D7
hypocenterC7#11978
LCSG ▷ C2×C7×C8 ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #1360

Complements

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

NH ≅ G/Ntype
2. D7×C4⋊C16C2semidirect
3. #11726284C2semidirect
4. C7×#1360C2semidirect
5. #11725602C2semidirect
6. #11725369C2semidirect
7. #11725274C2semidirect
68. #1360D7semidirect
276. C7#11978semidirect

Maximal subgroups

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

subgroupindex
#117392542
D7×C4⋊C162
#117262842
C7×#13602
#117256022
#117253692
#117252742
#119787