Finite Group Databasebeta

← #13274213

Subgroups — #13274213

Order 1792.

316 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 67 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#11740854C211yesmaximal, characteristic
2C7×#2135C211yesmaximal, characteristic, complemented
3#11729626C211yesmaximal, characteristic, complemented
4#11725358C212yesmaximal, complemented
5#11725357C212yesmaximal, complemented
6C7×#536C2211yescharacteristic
7#90226C411yescharacteristic
8C7×#413—21no—
9#90010—21no—
10#89999—21no—
11C2×#6410—21no—
12#89857C2211yescharacteristic
13#89855—21no—
14#89854C2211yescharacteristic
15#89830C411yescharacteristic
16#89829—21no—
17#89828C411yescharacteristic
18#89772C411yescharacteristic
19C7×#356—21no—
20C7×#356C2212yes—
21#89343C2212yes—
22#89337—21no—
23#13611—71nomaximal, Sylow
24C2×C7×#41D411yescharacteristic
25C22×C7×Q8D411yescharacteristic
26C2×C7×M16—41no—
27C7×C4⋊Q8D411yescharacteristic
28C7×C4⋊Q8C2311yescharacteristic
29C7×#124C2×C411yescharacteristic
30C7×#124D411yescharacteristic
31C7×#123—41no—
32C7×C22⋊Q8—41no—
33C7×#117C2×C411yescharacteristic
34C2×#1143—21no—
35C2×#1143—21no—
36C2×#1143—21no—
37C2×#1141—21no—
38#6493—21no—
39C2×#1136—41no—
40#6457—41no—
41#6447—41no—
42#6437C2×C411yescharacteristic
43C4×D28—21no—
44#6433C2×C411yescharacteristic
45C7×#101—22no—
46C7×#100—41no—
47#6410—22no—
48#6410—41no—
49#6409—21no—
50#6409—21no—
51#6409—21no—
52#6406—21no—
53C22⋊(C7⋊C8)—21no—
54C4⋊(C7⋊C8)—21no—
55#6377C2×C411yescharacteristic
56C4×C7⋊C8C2×C411yescharacteristic
57#3221—71no—
58#2135D711yescharacteristic, complemented
59#2133—71no—
60#1413—72no—
61#1410—72no—
62C7×#41—81no—
63C2×C7×Q8—21no—
64C2×C7×Q8—81no—
65C2×C7×Q8C22⋊C411yescharacteristic
66C2×C7×Q8C2×D411yescharacteristic
67C2×C7×Q8C2×D411yescharacteristic
68C2×C7×D4—21no—
69C2×C7×D4C22⋊C411yescharacteristic
70C22×C4×C7—21no—
71C22×C4×C7—21no—
72C22×C4×C7C22⋊C411yescharacteristic
73C2×C22⋊D7—41no—
74C2×D28—21no—
75C2×D28—41no—
76C2×C4×D7—41no—
77C7×M16—42no—
78C7×M16—81no—
79C2×C7×C8—81no—
80C7×C4⋊C4—21no—
81C7×C4⋊C4—41no—
82C7×C4⋊C4—41no—
83C7×C22⋊C4—21no—
84C7×C22⋊C4—41no—
85C7×C22⋊C4—41no—
86C42×C7C22⋊C411yescharacteristic
87C42×C7C22×C411yescharacteristic
88#1143—81no—
89#1143—81no—
90#1143—81no—
91#1141—81no—
92(C2×D7)⋊C4—41no—
93(C2×D7)⋊C4—41no—
94C4⋊Dic7—21no—
95Dic7⋊C4—41no—
96#1136—42no—
97#1136—81no—
98C2×C7⋊C8—21no—
99C2×C7⋊C8—21no—
100C2×C7⋊C8—81no—
101#540—71no—
102#536C2×D711yescharacteristic
103#496—71no—
104#492—71no—
105#491—71no—
106#488—71no—
107#463—141no—
108#461—141no—
109#449—141no—
110#448—141no—
111#433—71no—
112#413—21no—
113C2×#101—141no—
114#356—21no—
115#356—72no—
116#356C2×D712yes—
117#355—141no—
118C23×C7—21no—
119C23×C7#9911yescharacteristic
120C22×D7—41no—
121C7×Q8—41no—
122C7×Q8—41no—
123C7×Q8—41no—
124C7×Q8—81no—
125C7×D4—41no—
126C7×D4—81no—
127C2×C4×C7—21no—
128C2×C4×C7—21no—
129C2×C4×C7—21no—
130C2×C4×C7—21no—
131C2×C4×C7—21no—
132C2×C4×C7—41no—
133C2×C4×C7—41no—
134C2×C4×C7—41no—
135C2×C4×C7—41no—
136C2×C4×C7—41no—
137C2×C4×C7#9911yescharacteristic
138C2×C4×C7C2×C22⋊C411yescharacteristic
139C22⋊D7—161no—
140C2×Dic7—41no—
141D28—41no—
142D28—41no—
143D28—161no—
144C4×D7—161no—
145C7×C8—82no—
146C7⋊C8—41no—
147C7⋊C8—41no—
148C7⋊C8—82no—
149C2×#41C22⋊D711yescharacteristic
150C22×Q8D2811yescharacteristic
151C2×SD16—141no—
152C2×SD16—141no—
153C2×SD16—141no—
154C2×D8—141no—
155C2×M16—41no—
156C2×M16—281no—
157C4⋊Q8C22⋊D711yescharacteristic
158C4⋊Q8C22×D711yescharacteristic
159#127—71no—
160#125—71no—
161#124C4×D711yescharacteristic
162#124D2811yescharacteristic
163#123—41no—
164#123—281no—
165C22⋊Q8—41no—
166#121—141no—
167#121—281no—
168C4×D4—141no—
169#117C4×D711yescharacteristic
170C4⋊C8—141no—
171#103—141no—
172#103—141no—
173#103—141no—
174#102—141no—
175#101—22no—
176#101—142no—
177#101—281no—
178#100—41no—
179C22⋊C8—141no—
180#97—71no—
181C4×C8—71no—
182C22×C7—21no—
183C22×C7—41no—
184C22×C7—41no—
185C22×C7—81no—
186C22×C7C2×#9911yescharacteristic
187C2×D7—81no—
188C2×D7—81no—
189C2×D7—81no—
190C4×C7—21no—
191C4×C7—21no—
192C4×C7—41no—
193C4×C7—41no—
194C4×C7—41no—
195C4×C7—41no—
196C4×C7—41no—
197Dic7—81no—
198#41—81no—
199C2×Q8—21no—
200C2×Q8—81no—
201C2×Q8(C2×D7)⋊C411yescharacteristic
202C2×Q8C2×D2811yescharacteristic
203C2×Q8C2×C22⋊D711yescharacteristic
204C2×D4—21no—
205C2×D4—141no—
206C2×D4—281no—
207C2×D4—281no—
208C2×D4(C2×D7)⋊C411yescharacteristic
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—281no—
212C22×C4(C2×D7)⋊C411yescharacteristic
213SD16—561no—
214SD16—561no—
215SD16—561no—
216D8—561no—
217M16—42no—
218M16—81no—
219M16—282no—
220M16—561no—
221C2×C8—81no—
222C2×C8—141no—
223C2×C8—141no—
224C2×C8—561no—
225C4⋊C4—21no—
226C4⋊C4—41no—
227C4⋊C4—41no—
228C4⋊C4—141no—
229C4⋊C4—281no—
230C22⋊C4—21no—
231C22⋊C4—41no—
232C22⋊C4—41no—
233C22⋊C4—281no—
234C22⋊C4—281no—
235C42(C2×D7)⋊C411yescharacteristic
236C42C2×C4×D711yescharacteristic
237C2×C7—21no—
238C2×C7—41no—
239C2×C7—81no—
240C2×C7#212711yescharacteristic
241D7—161no—
242C23—21no—
243C23—281no—
244C23#638011yescharacteristic
245Q8—41no—
246Q8—41no—
247Q8—41no—
248Q8—81no—
249D4—41no—
250D4—81no—
251D4—281no—
252D4—281no—
253D4—1121no—
254D4—1121no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—281no—
266C2×C4—1121no—
267C2×C4#638011yescharacteristic
268C2×C4C2×(C2×D7)⋊C411yescharacteristic
269C8—82no—
270C8—281no—
271C8—281no—
272C8—562no—
273C7#1361111yescharacteristic, complemented, Sylow
274C22—21no—
275C22—41no—
276C22—41no—
277C22—81no—
278C22—561no—
279C22—561no—
280C22—561no—
281C22C2×#638011yescharacteristic
282C4—21no—
283C4—21no—
284C4—41no—
285C4—41no—
286C4—41no—
287C4—41no—
288C4—41no—
289C4—561no—
290C2—21no—
291C2—41no—
292C2—81no—
293C2—1121no—
294C2#1172777311yescharacteristic
subgroupG/H
centerC2#11727773
commutatorC42×C7C22×C4
FrattiniC4⋊Q8C22×D7
FittingC7×#2135C2
socleC2×C7#2127
prime radical#11740854C2
cosocleC7×C4⋊Q8C23
hypercenter#2135D7
hypocenterC7#13611
LCSG ▷ C42×C7 ▷ C2×C4×C7 ▷ C22×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2 ◁ C22 ◁ C22×C4 ◁ #536 ◁ #2135

Complements

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

NH ≅ G/Ntype
2. C7×#2135C2semidirect
3. #11729626C2semidirect
4. #11725358C2semidirect
5. #11725357C2semidirect
58. #2135D7semidirect
273. C7#13611semidirect

Maximal subgroups

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

subgroupindex
#117408542
C7×#21352
#117296262
#117253582
#117253572
#136117