Finite Group Databasebeta

← #13264006

Subgroups — #13264006

Order 1792.

552 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 107 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#11739731C211yesmaximal, characteristic
2C7×#1742C211yesmaximal, characteristic, complemented
3#11729325C211yesmaximal, characteristic, complemented
4#11725299C214yesmaximal, complemented
5C22×C7×M16—21no—
6C2×C7×#117C2211yescharacteristic
7C2×#6493—21no—
8C2×#6437C411yescharacteristic
9#89763—21no—
10#89763C2212yes—
11#89732—21no—
12#89732C412yes—
13#89711—21no—
14#89708C411yescharacteristic
15#89697—21no—
16C7×#343—21no—
17C7×#343C2212yes—
18#89323C2212yes—
19#89320—21no—
20#11699—71nomaximal, Sylow
21C23×C4×C7D411yescharacteristic
22C22×C22⋊D7—22no—
23C22×D28—21no—
24C22×D28—21no—
25C22×D28C2×C412yes—
26C2×C7×M16—24no—
27C2×C7×M16—22no—
28C2×C7×M16—41no—
29C2×C7×M16—42no—
30C22×C7×C8—41no—
31C7×#117—41no—
32C7×#117—41no—
33C2×C7×C4⋊C4D411yescharacteristic
34C2×C7×C4⋊C4C2311yescharacteristic
35C2×C7×C22⋊C4C2×C411yescharacteristic
36C2×C7×C22⋊C4D411yescharacteristic
37C2×C42×C7C2×C411yescharacteristic
38C2×C42×C7D411yescharacteristic
39#6493—42no—
40#6493—42no—
41C2×(C2×D7)⋊C4—22no—
42C2×(C2×D7)⋊C4—22no—
43C2×(C2×D7)⋊C4—22no—
44C2×C4⋊Dic7—21no—
45#6437—41no—
46#6437—41no—
47C7⋊#95—21no—
48C7⋊#95C2×C412yes—
49C22⋊(C7⋊C8)—22no—
50C4⋊(C7⋊C8)—41no—
51C4×C7⋊C8—41no—
52#2619—71no—
53#1742D711yescharacteristic, complemented
54#1458—71no—
55#1345—74no—
56C24×C7C22⋊C411yescharacteristic
57C23×D7—22no—
58C22×C4×C7—21no—
59C22×C4×C7—21no—
60C22×C4×C7—22no—
61C22×C4×C7C22⋊C411yescharacteristic
62C22×C4×C7C22⋊C411yescharacteristic
63C22×C4×C7C22⋊C411yescharacteristic
64C22×C4×C7D812yes—
65C22×C4×C7SD1612yes—
66C22×C4×C7C22×C411yescharacteristic
67C22×C4×C7C2×D411yescharacteristic
68C22×C4×C7C2×D411yescharacteristic
69C2×C22⋊D7—44no—
70C2×C22⋊D7—82no—
71C2×C22⋊D7—82no—
72C22×Dic7—22no—
73C2×D28—42no—
74C2×D28—42no—
75C2×D28—41no—
76C2×D28—41no—
77C2×D28—42no—
78C2×D28—42no—
79C2×D28—82no—
80C2×D28—82no—
81C7×M16—24no—
82C7×M16—44no—
83C7×M16—42no—
84C2×C7×C8—42no—
85C2×C7×C8—44no—
86C7×C4⋊C4—41no—
87C7×C4⋊C4—41no—
88C7×C22⋊C4—41no—
89C7×C22⋊C4—41no—
90C42×C7—21no—
91C42×C7—21no—
92C42×C7—21no—
93C42×C7—21no—
94(C2×D7)⋊C4—42no—
95(C2×D7)⋊C4—42no—
96(C2×D7)⋊C4—82no—
97(C2×D7)⋊C4—82no—
98C4⋊Dic7—42no—
99C2×C7⋊C8—42no—
100C22×M16—21no—
101C2×#127—71no—
102C2×#121—141no—
103C2×#117C2×D711yescharacteristic
104#431—141no—
105#394—72no—
106#394—141no—
107#392—72no—
108#392—141no—
109#390—141no—
110#382—71no—
111#343—21no—
112#343C2×D712yes—
113#324—72no—
114#323—141no—
115C23×C7—21no—
116C23×C7—21no—
117C23×C7—41no—
118C23×C7—41no—
119C23×C7#10212yes—
120C23×C7C2×C22⋊C411yescharacteristic
121C22×D7—42no—
122C22×D7—42no—
123C22×D7—42no—
124C22×D7—42no—
125C22×D7—42no—
126C22×D7—42no—
127C22×D7—42no—
128C2×C4×C7—21no—
129C2×C4×C7—21no—
130C2×C4×C7—22no—
131C2×C4×C7—22no—
132C2×C4×C7—22no—
133C2×C4×C7—22no—
134C2×C4×C7—21no—
135C2×C4×C7—21no—
136C2×C4×C7—21no—
137C2×C4×C7—21no—
138C2×C4×C7—41no—
139C2×C4×C7—41no—
140C2×C4×C7—41no—
141C2×C4×C7—41no—
142C2×C4×C7#9911yescharacteristic
143C2×C4×C7#9911yescharacteristic
144C2×C4×C7#10212yes—
145C2×C4×C7#10412yes—
146C2×C4×C7C2×D811yescharacteristic
147C2×C4×C7C2×SD1611yescharacteristic
148C22⋊D7—88no—
149C2×Dic7—42no—
150C2×Dic7—42no—
151C2×Dic7—42no—
152D28—84no—
153D28—84no—
154D28—82no—
155D28—82no—
156C7×C8—44no—
157C7⋊C8—82no—
158C22×D4—72no—
159C22×D4—142no—
160C22×D4—141no—
161C22×D4—141no—
162C23×C4C22⋊D711yescharacteristic
163C2×M16—24no—
164C2×M16—22no—
165C2×M16—41no—
166C2×M16—42no—
167C22×C8—41no—
168#127—281no—
169#127—281no—
170#121—282no—
171#121—282no—
172#117—41no—
173#117—41no—
174C2×C4⋊C4—141no—
175C2×C4⋊C4C22⋊D711yescharacteristic
176C2×C4⋊C4C22×D711yescharacteristic
177C2×C22⋊C4—142no—
178C2×C22⋊C4—142no—
179C2×C22⋊C4—142no—
180C2×C22⋊C4C4×D711yescharacteristic
181C2×C22⋊C4D2811yescharacteristic
182C2×C42C4×D711yescharacteristic
183C2×C42D2811yescharacteristic
184C4⋊C8—281no—
185C22⋊C8—142no—
186C4×C8—281no—
187#95—72no—
188#95—141no—
189C22×C7—21no—
190C22×C7—21no—
191C22×C7—22no—
192C22×C7—42no—
193C22×C7—41no—
194C22×C7—41no—
195C22×C7—81no—
196C22×C7C2×#9911yescharacteristic
197C22×C7C2×#10211yescharacteristic
198C22×C7C2×#10411yescharacteristic
199C2×D7—82no—
200C2×D7—82no—
201C2×D7—82no—
202C2×D7—82no—
203C2×D7—82no—
204C2×D7—82no—
205C2×D7—82no—
206C4×C7—21no—
207C4×C7—21no—
208C4×C7—41no—
209C4×C7—41no—
210C4×C7—41no—
211C4×C7—41no—
212C4×C7#32712yes—
213C4×C7#32712yes—
214Dic7—82no—
215C24—142no—
216C24(C2×D7)⋊C411yescharacteristic
217C2×D4—282no—
218C2×D4—282no—
219C2×D4—284no—
220C2×D4—282no—
221C2×D4—281no—
222C2×D4—281no—
223C2×D4—282no—
224C2×D4—562no—
225C2×D4—562no—
226C2×D4—562no—
227C2×D4—562no—
228C22×C4—22no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—142no—
232C22×C4(C2×D7)⋊C411yescharacteristic
233C22×C4(C2×D7)⋊C411yescharacteristic
234C22×C4(C2×D7)⋊C411yescharacteristic
235C22×C4#114112yes—
236C22×C4#114312yes—
237C22×C4C2×C4×D711yescharacteristic
238C22×C4C2×D2811yescharacteristic
239C22×C4C2×C22⋊D711yescharacteristic
240M16—24no—
241M16—44no—
242M16—42no—
243C2×C8—42no—
244C2×C8—44no—
245C2×C8—282no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—282no—
249C22⋊C4—41no—
250C22⋊C4—41no—
251C22⋊C4—282no—
252C22⋊C4—282no—
253C22⋊C4—562no—
254C22⋊C4—562no—
255C42—21no—
256C42—21no—
257C42—21no—
258C42—21no—
259C2×C7—21no—
260C2×C7—21no—
261C2×C7—42no—
262C2×C7C2×#32711yescharacteristic
263C2×C7#212211yescharacteristic
264C2×C7#213411yescharacteristic
265D7—162no—
266C23—21no—
267C23—21no—
268C23—41no—
269C23—41no—
270C23—282no—
271C23—282no—
272C23—282no—
273C23—282no—
274C23—282no—
275C23—282no—
276C23—282no—
277C23#638312yes—
278C23C2×(C2×D7)⋊C411yescharacteristic
279D4—564no—
280D4—564no—
281D4—568no—
282D4—562no—
283D4—562no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—282no—
299C2×C4—282no—
300C2×C4—282no—
301C2×C4#637912yes—
302C2×C4#638011yescharacteristic
303C2×C4#638011yescharacteristic
304C2×C4#638312yes—
305C2×C4C2×#114111yescharacteristic
306C2×C4C2×#114311yescharacteristic
307C8—44no—
308C8—562no—
309C7#1169911yescharacteristic, complemented, Sylow
310C22—21no—
311C22—21no—
312C22—22no—
313C22—42no—
314C22—41no—
315C22—41no—
316C22—81no—
317C22—562no—
318C22—562no—
319C22—562no—
320C22—562no—
321C22—562no—
322C22—562no—
323C22—562no—
324C22C2×#637911yescharacteristic
325C22C2×#638011yescharacteristic
326C22C2×#638311yescharacteristic
327C4—21no—
328C4—21no—
329C4—41no—
330C4—41no—
331C4—41no—
332C4—41no—
333C4—562no—
334C4#8924612yes—
335C4#8924612yes—
336C2—21no—
337C2—21no—
338C2—42no—
339C2—1122no—
340C2C2×#8924611yescharacteristic
341C2#1172790011yescharacteristic
342C2#1172790311yescharacteristic
subgroupG/H
centerC22C2×#6380
commutatorC22×C4×C7C22×C4
FrattiniC2×C4⋊C4C22×D7
FittingC7×#1742C2
socleC22×C7C2×#99
prime radical#11739731C2
cosocleC2×C7×C4⋊C4C23
hypercenter#1742D7
hypocenterC7#11699
LCSG ▷ C22×C4×C7 ▷ C23×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1742

Complements

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

NH ≅ G/Ntype
2. C7×#1742C2semidirect
3. #11729325C2semidirect
4. #11725299C2semidirect
53. #1742D7semidirect
309. C7#11699semidirect

Maximal subgroups

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

subgroupindex
#117397312
C7×#17422
#117293252
#117252992
#116997